TPTP Problem File: SLH0635^1.p
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%------------------------------------------------------------------------------
% File : SLH0000^1 : TPTP v8.2.0. Released v8.2.0.
% Domain : Archive of Formal Proofs
% Problem :
% Version : Especial.
% English :
% Refs : [Des23] Desharnais (2023), Email to Geoff Sutcliffe
% Source : [Des23]
% Names : Number_Theoretic_Transform/0006_Preliminary_Lemmas/prob_00251_009124__14065174_1 [Des23]
% Status : Theorem
% Rating : ? v8.2.0
% Syntax : Number of formulae : 1340 ( 613 unt; 64 typ; 0 def)
% Number of atoms : 3305 (1507 equ; 0 cnn)
% Maximal formula atoms : 10 ( 2 avg)
% Number of connectives : 9789 ( 356 ~; 105 |; 192 &;8020 @)
% ( 0 <=>;1116 =>; 0 <=; 0 <~>)
% Maximal formula depth : 17 ( 6 avg)
% Number of types : 7 ( 6 usr)
% Number of type conns : 188 ( 188 >; 0 *; 0 +; 0 <<)
% Number of symbols : 61 ( 58 usr; 14 con; 0-4 aty)
% Number of variables : 3133 ( 71 ^;2906 !; 156 ?;3133 :)
% SPC : TH0_THM_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2023-01-18 16:37:55.202
%------------------------------------------------------------------------------
% Could-be-implicit typings (6)
thf(ty_n_t__Finite____Field__Omod____ring_Itf__a_J,type,
finite_mod_ring_a: $tType ).
thf(ty_n_t__Set__Oset_It__Nat__Onat_J,type,
set_nat: $tType ).
thf(ty_n_t__itself_Itf__a_J,type,
itself_a: $tType ).
thf(ty_n_t__Real__Oreal,type,
real: $tType ).
thf(ty_n_t__Nat__Onat,type,
nat: $tType ).
thf(ty_n_t__Int__Oint,type,
int: $tType ).
% Explicit typings (58)
thf(sy_c_Cong_Ounique__euclidean__semiring__class_Ocong_001t__Int__Oint,type,
unique651150874487253600ng_int: int > int > int > $o ).
thf(sy_c_Cong_Ounique__euclidean__semiring__class_Ocong_001t__Nat__Onat,type,
unique653641344996303876ng_nat: nat > nat > nat > $o ).
thf(sy_c_Factorial__Ring_Onormalization__semidom__class_Oprime_001t__Int__Oint,type,
factor1798656936486255268me_int: int > $o ).
thf(sy_c_Factorial__Ring_Onormalization__semidom__class_Oprime_001t__Nat__Onat,type,
factor1801147406995305544me_nat: nat > $o ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Int__Oint,type,
minus_minus_int: int > int > int ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Nat__Onat,type,
minus_minus_nat: nat > nat > nat ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Real__Oreal,type,
minus_minus_real: real > real > real ).
thf(sy_c_Groups_Oone__class_Oone_001t__Int__Oint,type,
one_one_int: int ).
thf(sy_c_Groups_Oone__class_Oone_001t__Nat__Onat,type,
one_one_nat: nat ).
thf(sy_c_Groups_Oone__class_Oone_001t__Real__Oreal,type,
one_one_real: real ).
thf(sy_c_Groups_Oplus__class_Oplus_001t__Int__Oint,type,
plus_plus_int: int > int > int ).
thf(sy_c_Groups_Oplus__class_Oplus_001t__Nat__Onat,type,
plus_plus_nat: nat > nat > nat ).
thf(sy_c_Groups_Oplus__class_Oplus_001t__Real__Oreal,type,
plus_plus_real: real > real > real ).
thf(sy_c_Groups_Otimes__class_Otimes_001t__Finite____Field__Omod____ring_Itf__a_J,type,
times_5121417576591743744ring_a: finite_mod_ring_a > finite_mod_ring_a > finite_mod_ring_a ).
thf(sy_c_Groups_Otimes__class_Otimes_001t__Int__Oint,type,
times_times_int: int > int > int ).
thf(sy_c_Groups_Otimes__class_Otimes_001t__Nat__Onat,type,
times_times_nat: nat > nat > nat ).
thf(sy_c_Groups_Otimes__class_Otimes_001t__Real__Oreal,type,
times_times_real: real > real > real ).
thf(sy_c_Groups_Ozero__class_Ozero_001t__Int__Oint,type,
zero_zero_int: int ).
thf(sy_c_Groups_Ozero__class_Ozero_001t__Nat__Onat,type,
zero_zero_nat: nat ).
thf(sy_c_Groups_Ozero__class_Ozero_001t__Real__Oreal,type,
zero_zero_real: real ).
thf(sy_c_If_001t__Int__Oint,type,
if_int: $o > int > int > int ).
thf(sy_c_If_001t__Nat__Onat,type,
if_nat: $o > nat > nat > nat ).
thf(sy_c_If_001t__Real__Oreal,type,
if_real: $o > real > real > real ).
thf(sy_c_Nat_OSuc,type,
suc: nat > nat ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Int__Oint,type,
ord_less_int: int > int > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Nat__Onat,type,
ord_less_nat: nat > nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Real__Oreal,type,
ord_less_real: real > real > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Int__Oint,type,
ord_less_eq_int: int > int > $o ).
thf(sy_c_Pocklington_Oord_001t__Int__Oint,type,
ord_int: int > int > nat ).
thf(sy_c_Pocklington_Oord_001t__Nat__Onat,type,
ord_nat: nat > nat > nat ).
thf(sy_c_Power_Opower__class_Opower_001t__Finite____Field__Omod____ring_Itf__a_J,type,
power_6826135765519566523ring_a: finite_mod_ring_a > nat > finite_mod_ring_a ).
thf(sy_c_Power_Opower__class_Opower_001t__Int__Oint,type,
power_power_int: int > nat > int ).
thf(sy_c_Power_Opower__class_Opower_001t__Nat__Onat,type,
power_power_nat: nat > nat > nat ).
thf(sy_c_Power_Opower__class_Opower_001t__Real__Oreal,type,
power_power_real: real > nat > real ).
thf(sy_c_Preliminary__Lemmas_Opreliminary_001tf__a,type,
prelim7757304714281691100nary_a: itself_a > nat > nat > nat > $o ).
thf(sy_c_Prime__Powers_Oaprimedivisor_001t__Int__Oint,type,
prime_1887421117182150092or_int: int > int ).
thf(sy_c_Prime__Powers_Oaprimedivisor_001t__Nat__Onat,type,
prime_1889911587691200368or_nat: nat > nat ).
thf(sy_c_Prime__Powers_Omangoldt_001t__Real__Oreal,type,
prime_mangoldt_real: nat > real ).
thf(sy_c_Prime__Powers_Oprimepow_001t__Int__Oint,type,
prime_primepow_int: int > $o ).
thf(sy_c_Prime__Powers_Oprimepow_001t__Nat__Onat,type,
prime_primepow_nat: nat > $o ).
thf(sy_c_Pure_Otype_001tf__a,type,
type_a: itself_a ).
thf(sy_c_Residue__Primitive__Roots_OCarmichael,type,
residu178308219970301372ichael: nat > nat ).
thf(sy_c_Residue__Primitive__Roots_Ocyclic__moduli,type,
residu3389958895863328978moduli: set_nat ).
thf(sy_c_Residue__Primitive__Roots_Oresidue__primroot,type,
residu2993863765933214154imroot: nat > nat > $o ).
thf(sy_c_Rings_Odivide__class_Odivide_001t__Int__Oint,type,
divide_divide_int: int > int > int ).
thf(sy_c_Rings_Odivide__class_Odivide_001t__Nat__Onat,type,
divide_divide_nat: nat > nat > nat ).
thf(sy_c_Rings_Odivide__class_Odivide_001t__Real__Oreal,type,
divide_divide_real: real > real > real ).
thf(sy_c_Rings_Odvd__class_Odvd_001t__Int__Oint,type,
dvd_dvd_int: int > int > $o ).
thf(sy_c_Rings_Odvd__class_Odvd_001t__Nat__Onat,type,
dvd_dvd_nat: nat > nat > $o ).
thf(sy_c_Rings_Odvd__class_Odvd_001t__Real__Oreal,type,
dvd_dvd_real: real > real > $o ).
thf(sy_c_Set_OCollect_001t__Nat__Onat,type,
collect_nat: ( nat > $o ) > set_nat ).
thf(sy_c_Totient_Ototatives,type,
totatives: nat > set_nat ).
thf(sy_c_Totient_Ototient,type,
totient: nat > nat ).
thf(sy_c_member_001t__Nat__Onat,type,
member_nat: nat > set_nat > $o ).
thf(sy_v_g____,type,
g: nat ).
thf(sy_v_k,type,
k: nat ).
thf(sy_v_n,type,
n: nat ).
thf(sy_v_p,type,
p: nat ).
% Relevant facts (1268)
thf(fact_0_g__Def,axiom,
( ( residu2993863765933214154imroot @ p @ g )
& ( g != one_one_nat ) ) ).
% g_Def
thf(fact_1__092_060open_062_092_060And_062thesis_O_A_I_092_060And_062g_O_Aresidue__primroot_Ap_Ag_A_092_060and_062_Ag_A_092_060noteq_062_A1_A_092_060Longrightarrow_062_Athesis_J_A_092_060Longrightarrow_062_Athesis_092_060close_062,axiom,
~ ! [G: nat] :
~ ( ( residu2993863765933214154imroot @ p @ G )
& ( G != one_one_nat ) ) ).
% \<open>\<And>thesis. (\<And>g. residue_primroot p g \<and> g \<noteq> 1 \<Longrightarrow> thesis) \<Longrightarrow> thesis\<close>
thf(fact_2_cong__1,axiom,
! [B: nat,C: nat] : ( unique653641344996303876ng_nat @ B @ C @ one_one_nat ) ).
% cong_1
thf(fact_3_cong__1,axiom,
! [B: int,C: int] : ( unique651150874487253600ng_int @ B @ C @ one_one_int ) ).
% cong_1
thf(fact_4_test,axiom,
factor1801147406995305544me_nat @ p ).
% test
thf(fact_5_cong__sym,axiom,
! [B: nat,C: nat,A: nat] :
( ( unique653641344996303876ng_nat @ B @ C @ A )
=> ( unique653641344996303876ng_nat @ C @ B @ A ) ) ).
% cong_sym
thf(fact_6_cong__sym,axiom,
! [B: int,C: int,A: int] :
( ( unique651150874487253600ng_int @ B @ C @ A )
=> ( unique651150874487253600ng_int @ C @ B @ A ) ) ).
% cong_sym
thf(fact_7_cong__refl,axiom,
! [B: nat,A: nat] : ( unique653641344996303876ng_nat @ B @ B @ A ) ).
% cong_refl
thf(fact_8_cong__refl,axiom,
! [B: int,A: int] : ( unique651150874487253600ng_int @ B @ B @ A ) ).
% cong_refl
thf(fact_9_cong__trans,axiom,
! [B: nat,C: nat,A: nat,D: nat] :
( ( unique653641344996303876ng_nat @ B @ C @ A )
=> ( ( unique653641344996303876ng_nat @ C @ D @ A )
=> ( unique653641344996303876ng_nat @ B @ D @ A ) ) ) ).
% cong_trans
thf(fact_10_cong__trans,axiom,
! [B: int,C: int,A: int,D: int] :
( ( unique651150874487253600ng_int @ B @ C @ A )
=> ( ( unique651150874487253600ng_int @ C @ D @ A )
=> ( unique651150874487253600ng_int @ B @ D @ A ) ) ) ).
% cong_trans
thf(fact_11_cong__sym__eq,axiom,
( unique653641344996303876ng_nat
= ( ^ [B2: nat,C2: nat] : ( unique653641344996303876ng_nat @ C2 @ B2 ) ) ) ).
% cong_sym_eq
thf(fact_12_cong__sym__eq,axiom,
( unique651150874487253600ng_int
= ( ^ [B2: int,C2: int] : ( unique651150874487253600ng_int @ C2 @ B2 ) ) ) ).
% cong_sym_eq
thf(fact_13_one__natural_Orsp,axiom,
one_one_nat = one_one_nat ).
% one_natural.rsp
thf(fact_14_one__reorient,axiom,
! [X: nat] :
( ( one_one_nat = X )
= ( X = one_one_nat ) ) ).
% one_reorient
thf(fact_15_one__reorient,axiom,
! [X: int] :
( ( one_one_int = X )
= ( X = one_one_int ) ) ).
% one_reorient
thf(fact_16_one__reorient,axiom,
! [X: real] :
( ( one_one_real = X )
= ( X = one_one_real ) ) ).
% one_reorient
thf(fact_17_ord__1,axiom,
! [N: nat] :
( ( ord_nat @ one_one_nat @ N )
= one_one_nat ) ).
% ord_1
thf(fact_18_ord__1,axiom,
! [N: int] :
( ( ord_int @ one_one_int @ N )
= one_one_nat ) ).
% ord_1
thf(fact_19_residue__primroot__cong,axiom,
! [X: nat,X2: nat,N: nat] :
( ( unique653641344996303876ng_nat @ X @ X2 @ N )
=> ( ( residu2993863765933214154imroot @ N @ X )
= ( residu2993863765933214154imroot @ N @ X2 ) ) ) ).
% residue_primroot_cong
thf(fact_20_primroot__ord,axiom,
! [G2: nat] :
( ( residu2993863765933214154imroot @ p @ G2 )
=> ( ( ord_nat @ p @ G2 )
= ( minus_minus_nat @ p @ one_one_nat ) ) ) ).
% primroot_ord
thf(fact_21_ord__1__right,axiom,
! [N: nat] :
( ( ord_nat @ N @ one_one_nat )
= one_one_nat ) ).
% ord_1_right
thf(fact_22_diff__eq__diff__eq,axiom,
! [A: int,B: int,C: int,D: int] :
( ( ( minus_minus_int @ A @ B )
= ( minus_minus_int @ C @ D ) )
=> ( ( A = B )
= ( C = D ) ) ) ).
% diff_eq_diff_eq
thf(fact_23_cancel__ab__semigroup__add__class_Odiff__right__commute,axiom,
! [A: nat,C: nat,B: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ A @ C ) @ B )
= ( minus_minus_nat @ ( minus_minus_nat @ A @ B ) @ C ) ) ).
% cancel_ab_semigroup_add_class.diff_right_commute
thf(fact_24_cancel__ab__semigroup__add__class_Odiff__right__commute,axiom,
! [A: int,C: int,B: int] :
( ( minus_minus_int @ ( minus_minus_int @ A @ C ) @ B )
= ( minus_minus_int @ ( minus_minus_int @ A @ B ) @ C ) ) ).
% cancel_ab_semigroup_add_class.diff_right_commute
thf(fact_25_cong__diff,axiom,
! [B: int,C: int,A: int,D: int,E: int] :
( ( unique651150874487253600ng_int @ B @ C @ A )
=> ( ( unique651150874487253600ng_int @ D @ E @ A )
=> ( unique651150874487253600ng_int @ ( minus_minus_int @ B @ D ) @ ( minus_minus_int @ C @ E ) @ A ) ) ) ).
% cong_diff
thf(fact_26_ord__cong,axiom,
! [K1: nat,K2: nat,N: nat] :
( ( unique653641344996303876ng_nat @ K1 @ K2 @ N )
=> ( ( ord_nat @ N @ K1 )
= ( ord_nat @ N @ K2 ) ) ) ).
% ord_cong
thf(fact_27_ord__cong,axiom,
! [K1: int,K2: int,N: int] :
( ( unique651150874487253600ng_int @ K1 @ K2 @ N )
=> ( ( ord_int @ N @ K1 )
= ( ord_int @ N @ K2 ) ) ) ).
% ord_cong
thf(fact_28_not__prime__1,axiom,
~ ( factor1801147406995305544me_nat @ one_one_nat ) ).
% not_prime_1
thf(fact_29_not__prime__1,axiom,
~ ( factor1798656936486255268me_int @ one_one_int ) ).
% not_prime_1
thf(fact_30_residue__prime__has__primroot,axiom,
! [P: nat] :
( ( factor1801147406995305544me_nat @ P )
=> ? [X3: nat] :
( ( member_nat @ X3 @ ( totatives @ P ) )
& ( ( ord_nat @ P @ X3 )
= ( minus_minus_nat @ P @ one_one_nat ) ) ) ) ).
% residue_prime_has_primroot
thf(fact_31_residue__primroot__iff__in__cyclic__moduli,axiom,
! [M: nat] :
( ( ? [X4: nat] : ( residu2993863765933214154imroot @ M @ X4 ) )
= ( member_nat @ M @ residu3389958895863328978moduli ) ) ).
% residue_primroot_iff_in_cyclic_moduli
thf(fact_32_Carmichael__prime,axiom,
! [P: nat] :
( ( factor1801147406995305544me_nat @ P )
=> ( ( residu178308219970301372ichael @ P )
= ( minus_minus_nat @ P @ one_one_nat ) ) ) ).
% Carmichael_prime
thf(fact_33_prime__primitive__root__exists,axiom,
! [N: nat] :
( ( ord_less_nat @ one_one_nat @ N )
=> ( ( factor1801147406995305544me_nat @ N )
=> ? [X_1: nat] : ( residu2993863765933214154imroot @ N @ X_1 ) ) ) ).
% prime_primitive_root_exists
thf(fact_34_ord,axiom,
! [A: nat,N: nat] : ( unique653641344996303876ng_nat @ ( power_power_nat @ A @ ( ord_nat @ N @ A ) ) @ one_one_nat @ N ) ).
% ord
thf(fact_35_diff__commute,axiom,
! [I: nat,J: nat,K: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ I @ J ) @ K )
= ( minus_minus_nat @ ( minus_minus_nat @ I @ K ) @ J ) ) ).
% diff_commute
thf(fact_36_cong__to__1__nat,axiom,
! [A: nat,N: nat] :
( ( unique653641344996303876ng_nat @ A @ one_one_nat @ N )
=> ( dvd_dvd_nat @ N @ ( minus_minus_nat @ A @ one_one_nat ) ) ) ).
% cong_to_1_nat
thf(fact_37_diff__left__imp__eq,axiom,
! [A: int,B: int,C: int] :
( ( ( minus_minus_int @ A @ B )
= ( minus_minus_int @ A @ C ) )
=> ( B = C ) ) ).
% diff_left_imp_eq
thf(fact_38_not__residue__primroot__0__right,axiom,
! [N: nat] :
( ( residu2993863765933214154imroot @ N @ zero_zero_nat )
= ( N = one_one_nat ) ) ).
% not_residue_primroot_0_right
thf(fact_39_not__gr__zero,axiom,
! [N: nat] :
( ( ~ ( ord_less_nat @ zero_zero_nat @ N ) )
= ( N = zero_zero_nat ) ) ).
% not_gr_zero
thf(fact_40_cancel__comm__monoid__add__class_Odiff__cancel,axiom,
! [A: real] :
( ( minus_minus_real @ A @ A )
= zero_zero_real ) ).
% cancel_comm_monoid_add_class.diff_cancel
thf(fact_41_cancel__comm__monoid__add__class_Odiff__cancel,axiom,
! [A: nat] :
( ( minus_minus_nat @ A @ A )
= zero_zero_nat ) ).
% cancel_comm_monoid_add_class.diff_cancel
thf(fact_42_cancel__comm__monoid__add__class_Odiff__cancel,axiom,
! [A: int] :
( ( minus_minus_int @ A @ A )
= zero_zero_int ) ).
% cancel_comm_monoid_add_class.diff_cancel
thf(fact_43_diff__zero,axiom,
! [A: real] :
( ( minus_minus_real @ A @ zero_zero_real )
= A ) ).
% diff_zero
thf(fact_44_diff__zero,axiom,
! [A: nat] :
( ( minus_minus_nat @ A @ zero_zero_nat )
= A ) ).
% diff_zero
thf(fact_45_diff__zero,axiom,
! [A: int] :
( ( minus_minus_int @ A @ zero_zero_int )
= A ) ).
% diff_zero
thf(fact_46_zero__diff,axiom,
! [A: nat] :
( ( minus_minus_nat @ zero_zero_nat @ A )
= zero_zero_nat ) ).
% zero_diff
thf(fact_47_diff__0__right,axiom,
! [A: real] :
( ( minus_minus_real @ A @ zero_zero_real )
= A ) ).
% diff_0_right
thf(fact_48_diff__0__right,axiom,
! [A: int] :
( ( minus_minus_int @ A @ zero_zero_int )
= A ) ).
% diff_0_right
thf(fact_49_diff__self,axiom,
! [A: real] :
( ( minus_minus_real @ A @ A )
= zero_zero_real ) ).
% diff_self
thf(fact_50_diff__self,axiom,
! [A: int] :
( ( minus_minus_int @ A @ A )
= zero_zero_int ) ).
% diff_self
thf(fact_51_less__nat__zero__code,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% less_nat_zero_code
thf(fact_52_neq0__conv,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
= ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% neq0_conv
thf(fact_53_bot__nat__0_Onot__eq__extremum,axiom,
! [A: nat] :
( ( A != zero_zero_nat )
= ( ord_less_nat @ zero_zero_nat @ A ) ) ).
% bot_nat_0.not_eq_extremum
thf(fact_54_diff__self__eq__0,axiom,
! [M: nat] :
( ( minus_minus_nat @ M @ M )
= zero_zero_nat ) ).
% diff_self_eq_0
thf(fact_55_diff__0__eq__0,axiom,
! [N: nat] :
( ( minus_minus_nat @ zero_zero_nat @ N )
= zero_zero_nat ) ).
% diff_0_eq_0
thf(fact_56_cong__0,axiom,
! [B: nat,C: nat] :
( ( unique653641344996303876ng_nat @ B @ C @ zero_zero_nat )
= ( B = C ) ) ).
% cong_0
thf(fact_57_cong__0,axiom,
! [B: int,C: int] :
( ( unique651150874487253600ng_int @ B @ C @ zero_zero_int )
= ( B = C ) ) ).
% cong_0
thf(fact_58_nat__dvd__1__iff__1,axiom,
! [M: nat] :
( ( dvd_dvd_nat @ M @ one_one_nat )
= ( M = one_one_nat ) ) ).
% nat_dvd_1_iff_1
thf(fact_59_mem__Collect__eq,axiom,
! [A: nat,P2: nat > $o] :
( ( member_nat @ A @ ( collect_nat @ P2 ) )
= ( P2 @ A ) ) ).
% mem_Collect_eq
thf(fact_60_Collect__mem__eq,axiom,
! [A2: set_nat] :
( ( collect_nat
@ ^ [X5: nat] : ( member_nat @ X5 @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_61_Carmichael__1,axiom,
( ( residu178308219970301372ichael @ one_one_nat )
= one_one_nat ) ).
% Carmichael_1
thf(fact_62_k__bound,axiom,
ord_less_nat @ zero_zero_nat @ k ).
% k_bound
thf(fact_63_diff__gt__0__iff__gt,axiom,
! [A: int,B: int] :
( ( ord_less_int @ zero_zero_int @ ( minus_minus_int @ A @ B ) )
= ( ord_less_int @ B @ A ) ) ).
% diff_gt_0_iff_gt
thf(fact_64_diff__gt__0__iff__gt,axiom,
! [A: real,B: real] :
( ( ord_less_real @ zero_zero_real @ ( minus_minus_real @ A @ B ) )
= ( ord_less_real @ B @ A ) ) ).
% diff_gt_0_iff_gt
thf(fact_65_less__one,axiom,
! [N: nat] :
( ( ord_less_nat @ N @ one_one_nat )
= ( N = zero_zero_nat ) ) ).
% less_one
thf(fact_66_zero__less__diff,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( minus_minus_nat @ N @ M ) )
= ( ord_less_nat @ M @ N ) ) ).
% zero_less_diff
thf(fact_67_ord__0__right__nat,axiom,
! [N: nat] :
( ( ( N = one_one_nat )
=> ( ( ord_nat @ N @ zero_zero_nat )
= one_one_nat ) )
& ( ( N != one_one_nat )
=> ( ( ord_nat @ N @ zero_zero_nat )
= zero_zero_nat ) ) ) ).
% ord_0_right_nat
thf(fact_68_ord__0__nat,axiom,
! [N: nat] :
( ( ( N = one_one_nat )
=> ( ( ord_nat @ zero_zero_nat @ N )
= one_one_nat ) )
& ( ( N != one_one_nat )
=> ( ( ord_nat @ zero_zero_nat @ N )
= zero_zero_nat ) ) ) ).
% ord_0_nat
thf(fact_69_Carmichael__pos,axiom,
! [N: nat] : ( ord_less_nat @ zero_zero_nat @ ( residu178308219970301372ichael @ N ) ) ).
% Carmichael_pos
thf(fact_70_Carmichael__0,axiom,
( ( residu178308219970301372ichael @ zero_zero_nat )
= one_one_nat ) ).
% Carmichael_0
thf(fact_71_prime__power__eq__one__iff,axiom,
! [P: nat,N: nat] :
( ( factor1801147406995305544me_nat @ P )
=> ( ( ( power_power_nat @ P @ N )
= one_one_nat )
= ( N = zero_zero_nat ) ) ) ).
% prime_power_eq_one_iff
thf(fact_72_prime__power__eq__one__iff,axiom,
! [P: int,N: nat] :
( ( factor1798656936486255268me_int @ P )
=> ( ( ( power_power_int @ P @ N )
= one_one_int )
= ( N = zero_zero_nat ) ) ) ).
% prime_power_eq_one_iff
thf(fact_73_one__eq__prime__power__iff,axiom,
! [P: nat,N: nat] :
( ( factor1801147406995305544me_nat @ P )
=> ( ( one_one_nat
= ( power_power_nat @ P @ N ) )
= ( N = zero_zero_nat ) ) ) ).
% one_eq_prime_power_iff
thf(fact_74_one__eq__prime__power__iff,axiom,
! [P: int,N: nat] :
( ( factor1798656936486255268me_int @ P )
=> ( ( one_one_int
= ( power_power_int @ P @ N ) )
= ( N = zero_zero_nat ) ) ) ).
% one_eq_prime_power_iff
thf(fact_75_prime__power__inj_H_I2_J,axiom,
! [P: nat,Q: nat,M: nat,N: nat] :
( ( factor1801147406995305544me_nat @ P )
=> ( ( factor1801147406995305544me_nat @ Q )
=> ( ( ( power_power_nat @ P @ M )
= ( power_power_nat @ Q @ N ) )
=> ( ( ord_less_nat @ zero_zero_nat @ M )
=> ( ( ord_less_nat @ zero_zero_nat @ N )
=> ( M = N ) ) ) ) ) ) ).
% prime_power_inj'(2)
thf(fact_76_prime__power__inj_H_I2_J,axiom,
! [P: int,Q: int,M: nat,N: nat] :
( ( factor1798656936486255268me_int @ P )
=> ( ( factor1798656936486255268me_int @ Q )
=> ( ( ( power_power_int @ P @ M )
= ( power_power_int @ Q @ N ) )
=> ( ( ord_less_nat @ zero_zero_nat @ M )
=> ( ( ord_less_nat @ zero_zero_nat @ N )
=> ( M = N ) ) ) ) ) ) ).
% prime_power_inj'(2)
thf(fact_77_prime__power__inj_H_I1_J,axiom,
! [P: nat,Q: nat,M: nat,N: nat] :
( ( factor1801147406995305544me_nat @ P )
=> ( ( factor1801147406995305544me_nat @ Q )
=> ( ( ( power_power_nat @ P @ M )
= ( power_power_nat @ Q @ N ) )
=> ( ( ord_less_nat @ zero_zero_nat @ M )
=> ( ( ord_less_nat @ zero_zero_nat @ N )
=> ( P = Q ) ) ) ) ) ) ).
% prime_power_inj'(1)
thf(fact_78_prime__power__inj_H_I1_J,axiom,
! [P: int,Q: int,M: nat,N: nat] :
( ( factor1798656936486255268me_int @ P )
=> ( ( factor1798656936486255268me_int @ Q )
=> ( ( ( power_power_int @ P @ M )
= ( power_power_int @ Q @ N ) )
=> ( ( ord_less_nat @ zero_zero_nat @ M )
=> ( ( ord_less_nat @ zero_zero_nat @ N )
=> ( P = Q ) ) ) ) ) ) ).
% prime_power_inj'(1)
thf(fact_79_zero__natural_Orsp,axiom,
zero_zero_nat = zero_zero_nat ).
% zero_natural.rsp
thf(fact_80_gr__zeroI,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
=> ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% gr_zeroI
thf(fact_81_prime__power__inj_H_H,axiom,
! [P: nat,Q: nat,M: nat,N: nat] :
( ( factor1801147406995305544me_nat @ P )
=> ( ( factor1801147406995305544me_nat @ Q )
=> ( ( ( power_power_nat @ P @ M )
= ( power_power_nat @ Q @ N ) )
= ( ( ( M = zero_zero_nat )
& ( N = zero_zero_nat ) )
| ( ( P = Q )
& ( M = N ) ) ) ) ) ) ).
% prime_power_inj''
thf(fact_82_prime__power__inj_H_H,axiom,
! [P: int,Q: int,M: nat,N: nat] :
( ( factor1798656936486255268me_int @ P )
=> ( ( factor1798656936486255268me_int @ Q )
=> ( ( ( power_power_int @ P @ M )
= ( power_power_int @ Q @ N ) )
= ( ( ( M = zero_zero_nat )
& ( N = zero_zero_nat ) )
| ( ( P = Q )
& ( M = N ) ) ) ) ) ) ).
% prime_power_inj''
thf(fact_83_not__less__zero,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% not_less_zero
thf(fact_84_prime__dvd__power,axiom,
! [P: nat,X: nat,N: nat] :
( ( factor1801147406995305544me_nat @ P )
=> ( ( dvd_dvd_nat @ P @ ( power_power_nat @ X @ N ) )
=> ( dvd_dvd_nat @ P @ X ) ) ) ).
% prime_dvd_power
thf(fact_85_prime__dvd__power,axiom,
! [P: int,X: int,N: nat] :
( ( factor1798656936486255268me_int @ P )
=> ( ( dvd_dvd_int @ P @ ( power_power_int @ X @ N ) )
=> ( dvd_dvd_int @ P @ X ) ) ) ).
% prime_dvd_power
thf(fact_86_prime__dvd__power__iff,axiom,
! [P: nat,N: nat,X: nat] :
( ( factor1801147406995305544me_nat @ P )
=> ( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( dvd_dvd_nat @ P @ ( power_power_nat @ X @ N ) )
= ( dvd_dvd_nat @ P @ X ) ) ) ) ).
% prime_dvd_power_iff
thf(fact_87_prime__dvd__power__iff,axiom,
! [P: int,N: nat,X: int] :
( ( factor1798656936486255268me_int @ P )
=> ( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( dvd_dvd_int @ P @ ( power_power_int @ X @ N ) )
= ( dvd_dvd_int @ P @ X ) ) ) ) ).
% prime_dvd_power_iff
thf(fact_88_gr__implies__not__zero,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( N != zero_zero_nat ) ) ).
% gr_implies_not_zero
thf(fact_89_zero__less__iff__neq__zero,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
= ( N != zero_zero_nat ) ) ).
% zero_less_iff_neq_zero
thf(fact_90_diff__less,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_nat @ zero_zero_nat @ M )
=> ( ord_less_nat @ ( minus_minus_nat @ M @ N ) @ M ) ) ) ).
% diff_less
thf(fact_91_dvd__minus__self,axiom,
! [M: nat,N: nat] :
( ( dvd_dvd_nat @ M @ ( minus_minus_nat @ N @ M ) )
= ( ( ord_less_nat @ N @ M )
| ( dvd_dvd_nat @ M @ N ) ) ) ).
% dvd_minus_self
thf(fact_92_zero__reorient,axiom,
! [X: nat] :
( ( zero_zero_nat = X )
= ( X = zero_zero_nat ) ) ).
% zero_reorient
thf(fact_93_zero__reorient,axiom,
! [X: int] :
( ( zero_zero_int = X )
= ( X = zero_zero_int ) ) ).
% zero_reorient
thf(fact_94_zero__reorient,axiom,
! [X: real] :
( ( zero_zero_real = X )
= ( X = zero_zero_real ) ) ).
% zero_reorient
thf(fact_95_nat__exists__least__iff,axiom,
( ( ^ [P3: nat > $o] :
? [X6: nat] : ( P3 @ X6 ) )
= ( ^ [P4: nat > $o] :
? [N2: nat] :
( ( P4 @ N2 )
& ! [M2: nat] :
( ( ord_less_nat @ M2 @ N2 )
=> ~ ( P4 @ M2 ) ) ) ) ) ).
% nat_exists_least_iff
thf(fact_96_Carmichael__nonzero,axiom,
! [N: nat] :
( ( residu178308219970301372ichael @ N )
!= zero_zero_nat ) ).
% Carmichael_nonzero
thf(fact_97_Carmichael__root__exists,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ~ ! [G: nat] :
( ( member_nat @ G @ ( totatives @ N ) )
=> ( ( ord_nat @ N @ G )
!= ( residu178308219970301372ichael @ N ) ) ) ) ).
% Carmichael_root_exists
thf(fact_98_linorder__neqE__nat,axiom,
! [X: nat,Y: nat] :
( ( X != Y )
=> ( ~ ( ord_less_nat @ X @ Y )
=> ( ord_less_nat @ Y @ X ) ) ) ).
% linorder_neqE_nat
thf(fact_99_infinite__descent0,axiom,
! [P2: nat > $o,N: nat] :
( ( P2 @ zero_zero_nat )
=> ( ! [N3: nat] :
( ( ord_less_nat @ zero_zero_nat @ N3 )
=> ( ~ ( P2 @ N3 )
=> ? [M3: nat] :
( ( ord_less_nat @ M3 @ N3 )
& ~ ( P2 @ M3 ) ) ) )
=> ( P2 @ N ) ) ) ).
% infinite_descent0
thf(fact_100_nat__dvd__not__less,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ M )
=> ( ( ord_less_nat @ M @ N )
=> ~ ( dvd_dvd_nat @ N @ M ) ) ) ).
% nat_dvd_not_less
thf(fact_101_infinite__descent,axiom,
! [P2: nat > $o,N: nat] :
( ! [N3: nat] :
( ~ ( P2 @ N3 )
=> ? [M3: nat] :
( ( ord_less_nat @ M3 @ N3 )
& ~ ( P2 @ M3 ) ) )
=> ( P2 @ N ) ) ).
% infinite_descent
thf(fact_102_nat__less__induct,axiom,
! [P2: nat > $o,N: nat] :
( ! [N3: nat] :
( ! [M3: nat] :
( ( ord_less_nat @ M3 @ N3 )
=> ( P2 @ M3 ) )
=> ( P2 @ N3 ) )
=> ( P2 @ N ) ) ).
% nat_less_induct
thf(fact_103_less__irrefl__nat,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ N ) ).
% less_irrefl_nat
thf(fact_104_gr__implies__not0,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( N != zero_zero_nat ) ) ).
% gr_implies_not0
thf(fact_105_less__not__refl3,axiom,
! [S: nat,T: nat] :
( ( ord_less_nat @ S @ T )
=> ( S != T ) ) ).
% less_not_refl3
thf(fact_106_less__not__refl2,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ N @ M )
=> ( M != N ) ) ).
% less_not_refl2
thf(fact_107_less__not__refl,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ N ) ).
% less_not_refl
thf(fact_108_nat__neq__iff,axiom,
! [M: nat,N: nat] :
( ( M != N )
= ( ( ord_less_nat @ M @ N )
| ( ord_less_nat @ N @ M ) ) ) ).
% nat_neq_iff
thf(fact_109_dvd__antisym,axiom,
! [M: nat,N: nat] :
( ( dvd_dvd_nat @ M @ N )
=> ( ( dvd_dvd_nat @ N @ M )
=> ( M = N ) ) ) ).
% dvd_antisym
thf(fact_110_less__zeroE,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% less_zeroE
thf(fact_111_not__less0,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% not_less0
thf(fact_112_not__gr0,axiom,
! [N: nat] :
( ( ~ ( ord_less_nat @ zero_zero_nat @ N ) )
= ( N = zero_zero_nat ) ) ).
% not_gr0
thf(fact_113_gr0I,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
=> ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% gr0I
thf(fact_114_bot__nat__0_Oextremum__strict,axiom,
! [A: nat] :
~ ( ord_less_nat @ A @ zero_zero_nat ) ).
% bot_nat_0.extremum_strict
thf(fact_115_less__iff__diff__less__0,axiom,
( ord_less_int
= ( ^ [A3: int,B2: int] : ( ord_less_int @ ( minus_minus_int @ A3 @ B2 ) @ zero_zero_int ) ) ) ).
% less_iff_diff_less_0
thf(fact_116_less__iff__diff__less__0,axiom,
( ord_less_real
= ( ^ [A3: real,B2: real] : ( ord_less_real @ ( minus_minus_real @ A3 @ B2 ) @ zero_zero_real ) ) ) ).
% less_iff_diff_less_0
thf(fact_117_cong__0__iff,axiom,
! [B: nat,A: nat] :
( ( unique653641344996303876ng_nat @ B @ zero_zero_nat @ A )
= ( dvd_dvd_nat @ A @ B ) ) ).
% cong_0_iff
thf(fact_118_cong__0__iff,axiom,
! [B: int,A: int] :
( ( unique651150874487253600ng_int @ B @ zero_zero_int @ A )
= ( dvd_dvd_int @ A @ B ) ) ).
% cong_0_iff
thf(fact_119_dvd__diff__nat,axiom,
! [K: nat,M: nat,N: nat] :
( ( dvd_dvd_nat @ K @ M )
=> ( ( dvd_dvd_nat @ K @ N )
=> ( dvd_dvd_nat @ K @ ( minus_minus_nat @ M @ N ) ) ) ) ).
% dvd_diff_nat
thf(fact_120_less__imp__diff__less,axiom,
! [J: nat,K: nat,N: nat] :
( ( ord_less_nat @ J @ K )
=> ( ord_less_nat @ ( minus_minus_nat @ J @ N ) @ K ) ) ).
% less_imp_diff_less
thf(fact_121_diff__less__mono2,axiom,
! [M: nat,N: nat,L: nat] :
( ( ord_less_nat @ M @ N )
=> ( ( ord_less_nat @ M @ L )
=> ( ord_less_nat @ ( minus_minus_nat @ L @ N ) @ ( minus_minus_nat @ L @ M ) ) ) ) ).
% diff_less_mono2
thf(fact_122_prime__power__inj,axiom,
! [A: nat,M: nat,N: nat] :
( ( factor1801147406995305544me_nat @ A )
=> ( ( ( power_power_nat @ A @ M )
= ( power_power_nat @ A @ N ) )
=> ( M = N ) ) ) ).
% prime_power_inj
thf(fact_123_prime__power__inj,axiom,
! [A: int,M: nat,N: nat] :
( ( factor1798656936486255268me_int @ A )
=> ( ( ( power_power_int @ A @ M )
= ( power_power_int @ A @ N ) )
=> ( M = N ) ) ) ).
% prime_power_inj
thf(fact_124_primes__dvd__imp__eq,axiom,
! [P: nat,Q: nat] :
( ( factor1801147406995305544me_nat @ P )
=> ( ( factor1801147406995305544me_nat @ Q )
=> ( ( dvd_dvd_nat @ P @ Q )
=> ( P = Q ) ) ) ) ).
% primes_dvd_imp_eq
thf(fact_125_primes__dvd__imp__eq,axiom,
! [P: int,Q: int] :
( ( factor1798656936486255268me_int @ P )
=> ( ( factor1798656936486255268me_int @ Q )
=> ( ( dvd_dvd_int @ P @ Q )
=> ( P = Q ) ) ) ) ).
% primes_dvd_imp_eq
thf(fact_126_diffs0__imp__equal,axiom,
! [M: nat,N: nat] :
( ( ( minus_minus_nat @ M @ N )
= zero_zero_nat )
=> ( ( ( minus_minus_nat @ N @ M )
= zero_zero_nat )
=> ( M = N ) ) ) ).
% diffs0_imp_equal
thf(fact_127_minus__nat_Odiff__0,axiom,
! [M: nat] :
( ( minus_minus_nat @ M @ zero_zero_nat )
= M ) ).
% minus_nat.diff_0
thf(fact_128_not__prime__0,axiom,
~ ( factor1801147406995305544me_nat @ zero_zero_nat ) ).
% not_prime_0
thf(fact_129_not__prime__0,axiom,
~ ( factor1798656936486255268me_int @ zero_zero_int ) ).
% not_prime_0
thf(fact_130_prime__divisor__exists,axiom,
! [A: nat] :
( ( A != zero_zero_nat )
=> ( ~ ( dvd_dvd_nat @ A @ one_one_nat )
=> ? [B3: nat] :
( ( dvd_dvd_nat @ B3 @ A )
& ( factor1801147406995305544me_nat @ B3 ) ) ) ) ).
% prime_divisor_exists
thf(fact_131_prime__divisor__exists,axiom,
! [A: int] :
( ( A != zero_zero_int )
=> ( ~ ( dvd_dvd_int @ A @ one_one_int )
=> ? [B3: int] :
( ( dvd_dvd_int @ B3 @ A )
& ( factor1798656936486255268me_int @ B3 ) ) ) ) ).
% prime_divisor_exists
thf(fact_132_prime__divisorE,axiom,
! [A: nat] :
( ( A != zero_zero_nat )
=> ( ~ ( dvd_dvd_nat @ A @ one_one_nat )
=> ~ ! [P5: nat] :
( ( factor1801147406995305544me_nat @ P5 )
=> ~ ( dvd_dvd_nat @ P5 @ A ) ) ) ) ).
% prime_divisorE
thf(fact_133_prime__divisorE,axiom,
! [A: int] :
( ( A != zero_zero_int )
=> ( ~ ( dvd_dvd_int @ A @ one_one_int )
=> ~ ! [P5: int] :
( ( factor1798656936486255268me_int @ P5 )
=> ~ ( dvd_dvd_int @ P5 @ A ) ) ) ) ).
% prime_divisorE
thf(fact_134_ord__minimal,axiom,
! [M: nat,N: nat,A: nat] :
( ( ord_less_nat @ zero_zero_nat @ M )
=> ( ( ord_less_nat @ M @ ( ord_nat @ N @ A ) )
=> ~ ( unique653641344996303876ng_nat @ ( power_power_nat @ A @ M ) @ one_one_nat @ N ) ) ) ).
% ord_minimal
thf(fact_135_ord__works,axiom,
! [A: nat,N: nat] :
( ( unique653641344996303876ng_nat @ ( power_power_nat @ A @ ( ord_nat @ N @ A ) ) @ one_one_nat @ N )
& ! [M3: nat] :
( ( ( ord_less_nat @ zero_zero_nat @ M3 )
& ( ord_less_nat @ M3 @ ( ord_nat @ N @ A ) ) )
=> ~ ( unique653641344996303876ng_nat @ ( power_power_nat @ A @ M3 ) @ one_one_nat @ N ) ) ) ).
% ord_works
thf(fact_136_diff__strict__right__mono,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_int @ A @ B )
=> ( ord_less_int @ ( minus_minus_int @ A @ C ) @ ( minus_minus_int @ B @ C ) ) ) ).
% diff_strict_right_mono
thf(fact_137_diff__strict__right__mono,axiom,
! [A: real,B: real,C: real] :
( ( ord_less_real @ A @ B )
=> ( ord_less_real @ ( minus_minus_real @ A @ C ) @ ( minus_minus_real @ B @ C ) ) ) ).
% diff_strict_right_mono
thf(fact_138_diff__strict__left__mono,axiom,
! [B: int,A: int,C: int] :
( ( ord_less_int @ B @ A )
=> ( ord_less_int @ ( minus_minus_int @ C @ A ) @ ( minus_minus_int @ C @ B ) ) ) ).
% diff_strict_left_mono
thf(fact_139_diff__strict__left__mono,axiom,
! [B: real,A: real,C: real] :
( ( ord_less_real @ B @ A )
=> ( ord_less_real @ ( minus_minus_real @ C @ A ) @ ( minus_minus_real @ C @ B ) ) ) ).
% diff_strict_left_mono
thf(fact_140_diff__eq__diff__less,axiom,
! [A: int,B: int,C: int,D: int] :
( ( ( minus_minus_int @ A @ B )
= ( minus_minus_int @ C @ D ) )
=> ( ( ord_less_int @ A @ B )
= ( ord_less_int @ C @ D ) ) ) ).
% diff_eq_diff_less
thf(fact_141_diff__eq__diff__less,axiom,
! [A: real,B: real,C: real,D: real] :
( ( ( minus_minus_real @ A @ B )
= ( minus_minus_real @ C @ D ) )
=> ( ( ord_less_real @ A @ B )
= ( ord_less_real @ C @ D ) ) ) ).
% diff_eq_diff_less
thf(fact_142_diff__strict__mono,axiom,
! [A: int,B: int,D: int,C: int] :
( ( ord_less_int @ A @ B )
=> ( ( ord_less_int @ D @ C )
=> ( ord_less_int @ ( minus_minus_int @ A @ C ) @ ( minus_minus_int @ B @ D ) ) ) ) ).
% diff_strict_mono
thf(fact_143_diff__strict__mono,axiom,
! [A: real,B: real,D: real,C: real] :
( ( ord_less_real @ A @ B )
=> ( ( ord_less_real @ D @ C )
=> ( ord_less_real @ ( minus_minus_real @ A @ C ) @ ( minus_minus_real @ B @ D ) ) ) ) ).
% diff_strict_mono
thf(fact_144_cong__pow,axiom,
! [B: nat,C: nat,A: nat,N: nat] :
( ( unique653641344996303876ng_nat @ B @ C @ A )
=> ( unique653641344996303876ng_nat @ ( power_power_nat @ B @ N ) @ ( power_power_nat @ C @ N ) @ A ) ) ).
% cong_pow
thf(fact_145_cong__pow,axiom,
! [B: int,C: int,A: int,N: nat] :
( ( unique651150874487253600ng_int @ B @ C @ A )
=> ( unique651150874487253600ng_int @ ( power_power_int @ B @ N ) @ ( power_power_int @ C @ N ) @ A ) ) ).
% cong_pow
thf(fact_146_cong__dvd__modulus,axiom,
! [X: int,Y: int,M: int,N: int] :
( ( unique651150874487253600ng_int @ X @ Y @ M )
=> ( ( dvd_dvd_int @ N @ M )
=> ( unique651150874487253600ng_int @ X @ Y @ N ) ) ) ).
% cong_dvd_modulus
thf(fact_147_cong__dvd__iff,axiom,
! [B: nat,C: nat,A: nat] :
( ( unique653641344996303876ng_nat @ B @ C @ A )
=> ( ( dvd_dvd_nat @ A @ B )
= ( dvd_dvd_nat @ A @ C ) ) ) ).
% cong_dvd_iff
thf(fact_148_cong__dvd__iff,axiom,
! [B: int,C: int,A: int] :
( ( unique651150874487253600ng_int @ B @ C @ A )
=> ( ( dvd_dvd_int @ A @ B )
= ( dvd_dvd_int @ A @ C ) ) ) ).
% cong_dvd_iff
thf(fact_149_eq__iff__diff__eq__0,axiom,
( ( ^ [Y2: real,Z: real] : ( Y2 = Z ) )
= ( ^ [A3: real,B2: real] :
( ( minus_minus_real @ A3 @ B2 )
= zero_zero_real ) ) ) ).
% eq_iff_diff_eq_0
thf(fact_150_eq__iff__diff__eq__0,axiom,
( ( ^ [Y2: int,Z: int] : ( Y2 = Z ) )
= ( ^ [A3: int,B2: int] :
( ( minus_minus_int @ A3 @ B2 )
= zero_zero_int ) ) ) ).
% eq_iff_diff_eq_0
thf(fact_151_cong__dvd__modulus__nat,axiom,
! [X: nat,Y: nat,M: nat,N: nat] :
( ( unique653641344996303876ng_nat @ X @ Y @ M )
=> ( ( dvd_dvd_nat @ N @ M )
=> ( unique653641344996303876ng_nat @ X @ Y @ N ) ) ) ).
% cong_dvd_modulus_nat
thf(fact_152_cong__less__modulus__unique__nat,axiom,
! [X: nat,Y: nat,M: nat] :
( ( unique653641344996303876ng_nat @ X @ Y @ M )
=> ( ( ord_less_nat @ X @ M )
=> ( ( ord_less_nat @ Y @ M )
=> ( X = Y ) ) ) ) ).
% cong_less_modulus_unique_nat
thf(fact_153_not__prime__unit,axiom,
! [X: nat] :
( ( dvd_dvd_nat @ X @ one_one_nat )
=> ~ ( factor1801147406995305544me_nat @ X ) ) ).
% not_prime_unit
thf(fact_154_not__prime__unit,axiom,
! [X: int] :
( ( dvd_dvd_int @ X @ one_one_int )
=> ~ ( factor1798656936486255268me_int @ X ) ) ).
% not_prime_unit
thf(fact_155_prime__power__iff,axiom,
! [P: nat,N: nat] :
( ( factor1801147406995305544me_nat @ ( power_power_nat @ P @ N ) )
= ( ( factor1801147406995305544me_nat @ P )
& ( N = one_one_nat ) ) ) ).
% prime_power_iff
thf(fact_156_prime__power__iff,axiom,
! [P: int,N: nat] :
( ( factor1798656936486255268me_int @ ( power_power_int @ P @ N ) )
= ( ( factor1798656936486255268me_int @ P )
& ( N = one_one_nat ) ) ) ).
% prime_power_iff
thf(fact_157_not__residue__primroot__0,axiom,
! [X: nat] :
~ ( residu2993863765933214154imroot @ zero_zero_nat @ X ) ).
% not_residue_primroot_0
thf(fact_158_ord__divides,axiom,
! [A: nat,D: nat,N: nat] :
( ( unique653641344996303876ng_nat @ ( power_power_nat @ A @ D ) @ one_one_nat @ N )
= ( dvd_dvd_nat @ ( ord_nat @ N @ A ) @ D ) ) ).
% ord_divides
thf(fact_159_cong__iff__dvd__diff,axiom,
( unique651150874487253600ng_int
= ( ^ [A3: int,B2: int,M2: int] : ( dvd_dvd_int @ M2 @ ( minus_minus_int @ A3 @ B2 ) ) ) ) ).
% cong_iff_dvd_diff
thf(fact_160_prime__prime__factor,axiom,
( factor1801147406995305544me_nat
= ( ^ [N2: nat] :
( ( N2 != one_one_nat )
& ! [P6: nat] :
( ( ( factor1801147406995305544me_nat @ P6 )
& ( dvd_dvd_nat @ P6 @ N2 ) )
=> ( P6 = N2 ) ) ) ) ) ).
% prime_prime_factor
thf(fact_161_cong__diff__iff__cong__0,axiom,
! [B: int,C: int,A: int] :
( ( unique651150874487253600ng_int @ ( minus_minus_int @ B @ C ) @ zero_zero_int @ A )
= ( unique651150874487253600ng_int @ B @ C @ A ) ) ).
% cong_diff_iff_cong_0
thf(fact_162_cong__0__1__nat,axiom,
! [N: nat] :
( ( unique653641344996303876ng_nat @ zero_zero_nat @ one_one_nat @ N )
= ( N = one_one_nat ) ) ).
% cong_0_1_nat
thf(fact_163_power__strict__decreasing__iff,axiom,
! [B: nat,M: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ B )
=> ( ( ord_less_nat @ B @ one_one_nat )
=> ( ( ord_less_nat @ ( power_power_nat @ B @ M ) @ ( power_power_nat @ B @ N ) )
= ( ord_less_nat @ N @ M ) ) ) ) ).
% power_strict_decreasing_iff
thf(fact_164_power__strict__decreasing__iff,axiom,
! [B: int,M: nat,N: nat] :
( ( ord_less_int @ zero_zero_int @ B )
=> ( ( ord_less_int @ B @ one_one_int )
=> ( ( ord_less_int @ ( power_power_int @ B @ M ) @ ( power_power_int @ B @ N ) )
= ( ord_less_nat @ N @ M ) ) ) ) ).
% power_strict_decreasing_iff
thf(fact_165_power__strict__decreasing__iff,axiom,
! [B: real,M: nat,N: nat] :
( ( ord_less_real @ zero_zero_real @ B )
=> ( ( ord_less_real @ B @ one_one_real )
=> ( ( ord_less_real @ ( power_power_real @ B @ M ) @ ( power_power_real @ B @ N ) )
= ( ord_less_nat @ N @ M ) ) ) ) ).
% power_strict_decreasing_iff
thf(fact_166_pow__divides__pow__iff,axiom,
! [N: nat,A: nat,B: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( dvd_dvd_nat @ ( power_power_nat @ A @ N ) @ ( power_power_nat @ B @ N ) )
= ( dvd_dvd_nat @ A @ B ) ) ) ).
% pow_divides_pow_iff
thf(fact_167_pow__divides__pow__iff,axiom,
! [N: nat,A: int,B: int] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( dvd_dvd_int @ ( power_power_int @ A @ N ) @ ( power_power_int @ B @ N ) )
= ( dvd_dvd_int @ A @ B ) ) ) ).
% pow_divides_pow_iff
thf(fact_168_power__eq__0__iff,axiom,
! [A: nat,N: nat] :
( ( ( power_power_nat @ A @ N )
= zero_zero_nat )
= ( ( A = zero_zero_nat )
& ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% power_eq_0_iff
thf(fact_169_power__eq__0__iff,axiom,
! [A: int,N: nat] :
( ( ( power_power_int @ A @ N )
= zero_zero_int )
= ( ( A = zero_zero_int )
& ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% power_eq_0_iff
thf(fact_170_power__eq__0__iff,axiom,
! [A: real,N: nat] :
( ( ( power_power_real @ A @ N )
= zero_zero_real )
= ( ( A = zero_zero_real )
& ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% power_eq_0_iff
thf(fact_171_power__strict__increasing__iff,axiom,
! [B: nat,X: nat,Y: nat] :
( ( ord_less_nat @ one_one_nat @ B )
=> ( ( ord_less_nat @ ( power_power_nat @ B @ X ) @ ( power_power_nat @ B @ Y ) )
= ( ord_less_nat @ X @ Y ) ) ) ).
% power_strict_increasing_iff
thf(fact_172_power__strict__increasing__iff,axiom,
! [B: int,X: nat,Y: nat] :
( ( ord_less_int @ one_one_int @ B )
=> ( ( ord_less_int @ ( power_power_int @ B @ X ) @ ( power_power_int @ B @ Y ) )
= ( ord_less_nat @ X @ Y ) ) ) ).
% power_strict_increasing_iff
thf(fact_173_power__strict__increasing__iff,axiom,
! [B: real,X: nat,Y: nat] :
( ( ord_less_real @ one_one_real @ B )
=> ( ( ord_less_real @ ( power_power_real @ B @ X ) @ ( power_power_real @ B @ Y ) )
= ( ord_less_nat @ X @ Y ) ) ) ).
% power_strict_increasing_iff
thf(fact_174_nat__zero__less__power__iff,axiom,
! [X: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( power_power_nat @ X @ N ) )
= ( ( ord_less_nat @ zero_zero_nat @ X )
| ( N = zero_zero_nat ) ) ) ).
% nat_zero_less_power_iff
thf(fact_175_power__inject__exp,axiom,
! [A: nat,M: nat,N: nat] :
( ( ord_less_nat @ one_one_nat @ A )
=> ( ( ( power_power_nat @ A @ M )
= ( power_power_nat @ A @ N ) )
= ( M = N ) ) ) ).
% power_inject_exp
thf(fact_176_power__inject__exp,axiom,
! [A: int,M: nat,N: nat] :
( ( ord_less_int @ one_one_int @ A )
=> ( ( ( power_power_int @ A @ M )
= ( power_power_int @ A @ N ) )
= ( M = N ) ) ) ).
% power_inject_exp
thf(fact_177_power__inject__exp,axiom,
! [A: real,M: nat,N: nat] :
( ( ord_less_real @ one_one_real @ A )
=> ( ( ( power_power_real @ A @ M )
= ( power_power_real @ A @ N ) )
= ( M = N ) ) ) ).
% power_inject_exp
thf(fact_178_diff__numeral__special_I9_J,axiom,
( ( minus_minus_real @ one_one_real @ one_one_real )
= zero_zero_real ) ).
% diff_numeral_special(9)
thf(fact_179_diff__numeral__special_I9_J,axiom,
( ( minus_minus_int @ one_one_int @ one_one_int )
= zero_zero_int ) ).
% diff_numeral_special(9)
thf(fact_180_fermat__theorem,axiom,
! [P: nat,A: nat] :
( ( factor1801147406995305544me_nat @ P )
=> ( ~ ( dvd_dvd_nat @ P @ A )
=> ( unique653641344996303876ng_nat @ ( power_power_nat @ A @ ( minus_minus_nat @ P @ one_one_nat ) ) @ one_one_nat @ P ) ) ) ).
% fermat_theorem
thf(fact_181_prime__dvd__power__nat__iff,axiom,
! [P: nat,N: nat,X: nat] :
( ( factor1801147406995305544me_nat @ P )
=> ( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( dvd_dvd_nat @ P @ ( power_power_nat @ X @ N ) )
= ( dvd_dvd_nat @ P @ X ) ) ) ) ).
% prime_dvd_power_nat_iff
thf(fact_182_power__one__right,axiom,
! [A: nat] :
( ( power_power_nat @ A @ one_one_nat )
= A ) ).
% power_one_right
thf(fact_183_power__one__right,axiom,
! [A: int] :
( ( power_power_int @ A @ one_one_nat )
= A ) ).
% power_one_right
thf(fact_184_power__one__right,axiom,
! [A: real] :
( ( power_power_real @ A @ one_one_nat )
= A ) ).
% power_one_right
thf(fact_185_prime__power__not__one,axiom,
! [P: nat,K: nat] :
( ( factor1801147406995305544me_nat @ P )
=> ( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ( power_power_nat @ P @ K )
!= one_one_nat ) ) ) ).
% prime_power_not_one
thf(fact_186_prime__power__not__one,axiom,
! [P: int,K: nat] :
( ( factor1798656936486255268me_int @ P )
=> ( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ( power_power_int @ P @ K )
!= one_one_int ) ) ) ).
% prime_power_not_one
thf(fact_187_power__one,axiom,
! [N: nat] :
( ( power_power_nat @ one_one_nat @ N )
= one_one_nat ) ).
% power_one
thf(fact_188_power__one,axiom,
! [N: nat] :
( ( power_power_int @ one_one_int @ N )
= one_one_int ) ).
% power_one
thf(fact_189_power__one,axiom,
! [N: nat] :
( ( power_power_real @ one_one_real @ N )
= one_one_real ) ).
% power_one
thf(fact_190_prime__dvd__power__int__iff,axiom,
! [P: int,N: nat,X: int] :
( ( factor1798656936486255268me_int @ P )
=> ( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( dvd_dvd_int @ P @ ( power_power_int @ X @ N ) )
= ( dvd_dvd_int @ P @ X ) ) ) ) ).
% prime_dvd_power_int_iff
thf(fact_191_gcd__nat_Oasym,axiom,
! [A: nat,B: nat] :
( ( ( dvd_dvd_nat @ A @ B )
& ( A != B ) )
=> ~ ( ( dvd_dvd_nat @ B @ A )
& ( B != A ) ) ) ).
% gcd_nat.asym
thf(fact_192_gcd__nat_Orefl,axiom,
! [A: nat] : ( dvd_dvd_nat @ A @ A ) ).
% gcd_nat.refl
thf(fact_193_gcd__nat_Otrans,axiom,
! [A: nat,B: nat,C: nat] :
( ( dvd_dvd_nat @ A @ B )
=> ( ( dvd_dvd_nat @ B @ C )
=> ( dvd_dvd_nat @ A @ C ) ) ) ).
% gcd_nat.trans
thf(fact_194_gcd__nat_Oeq__iff,axiom,
( ( ^ [Y2: nat,Z: nat] : ( Y2 = Z ) )
= ( ^ [A3: nat,B2: nat] :
( ( dvd_dvd_nat @ A3 @ B2 )
& ( dvd_dvd_nat @ B2 @ A3 ) ) ) ) ).
% gcd_nat.eq_iff
thf(fact_195_gcd__nat_Oirrefl,axiom,
! [A: nat] :
~ ( ( dvd_dvd_nat @ A @ A )
& ( A != A ) ) ).
% gcd_nat.irrefl
thf(fact_196_gcd__nat_Oantisym,axiom,
! [A: nat,B: nat] :
( ( dvd_dvd_nat @ A @ B )
=> ( ( dvd_dvd_nat @ B @ A )
=> ( A = B ) ) ) ).
% gcd_nat.antisym
thf(fact_197_gcd__nat_Ostrict__trans,axiom,
! [A: nat,B: nat,C: nat] :
( ( ( dvd_dvd_nat @ A @ B )
& ( A != B ) )
=> ( ( ( dvd_dvd_nat @ B @ C )
& ( B != C ) )
=> ( ( dvd_dvd_nat @ A @ C )
& ( A != C ) ) ) ) ).
% gcd_nat.strict_trans
thf(fact_198_gcd__nat_Ostrict__trans1,axiom,
! [A: nat,B: nat,C: nat] :
( ( dvd_dvd_nat @ A @ B )
=> ( ( ( dvd_dvd_nat @ B @ C )
& ( B != C ) )
=> ( ( dvd_dvd_nat @ A @ C )
& ( A != C ) ) ) ) ).
% gcd_nat.strict_trans1
thf(fact_199_gcd__nat_Ostrict__trans2,axiom,
! [A: nat,B: nat,C: nat] :
( ( ( dvd_dvd_nat @ A @ B )
& ( A != B ) )
=> ( ( dvd_dvd_nat @ B @ C )
=> ( ( dvd_dvd_nat @ A @ C )
& ( A != C ) ) ) ) ).
% gcd_nat.strict_trans2
thf(fact_200_gcd__nat_Ostrict__iff__not,axiom,
! [A: nat,B: nat] :
( ( ( dvd_dvd_nat @ A @ B )
& ( A != B ) )
= ( ( dvd_dvd_nat @ A @ B )
& ~ ( dvd_dvd_nat @ B @ A ) ) ) ).
% gcd_nat.strict_iff_not
thf(fact_201_gcd__nat_Oorder__iff__strict,axiom,
( dvd_dvd_nat
= ( ^ [A3: nat,B2: nat] :
( ( ( dvd_dvd_nat @ A3 @ B2 )
& ( A3 != B2 ) )
| ( A3 = B2 ) ) ) ) ).
% gcd_nat.order_iff_strict
thf(fact_202_gcd__nat_Ostrict__iff__order,axiom,
! [A: nat,B: nat] :
( ( ( dvd_dvd_nat @ A @ B )
& ( A != B ) )
= ( ( dvd_dvd_nat @ A @ B )
& ( A != B ) ) ) ).
% gcd_nat.strict_iff_order
thf(fact_203_gcd__nat_Ostrict__implies__order,axiom,
! [A: nat,B: nat] :
( ( ( dvd_dvd_nat @ A @ B )
& ( A != B ) )
=> ( dvd_dvd_nat @ A @ B ) ) ).
% gcd_nat.strict_implies_order
thf(fact_204_gcd__nat_Ostrict__implies__not__eq,axiom,
! [A: nat,B: nat] :
( ( ( dvd_dvd_nat @ A @ B )
& ( A != B ) )
=> ( A != B ) ) ).
% gcd_nat.strict_implies_not_eq
thf(fact_205_gcd__nat_Onot__eq__order__implies__strict,axiom,
! [A: nat,B: nat] :
( ( A != B )
=> ( ( dvd_dvd_nat @ A @ B )
=> ( ( dvd_dvd_nat @ A @ B )
& ( A != B ) ) ) ) ).
% gcd_nat.not_eq_order_implies_strict
thf(fact_206_less__numeral__extra_I3_J,axiom,
~ ( ord_less_nat @ zero_zero_nat @ zero_zero_nat ) ).
% less_numeral_extra(3)
thf(fact_207_less__numeral__extra_I3_J,axiom,
~ ( ord_less_int @ zero_zero_int @ zero_zero_int ) ).
% less_numeral_extra(3)
thf(fact_208_less__numeral__extra_I3_J,axiom,
~ ( ord_less_real @ zero_zero_real @ zero_zero_real ) ).
% less_numeral_extra(3)
thf(fact_209_less__numeral__extra_I4_J,axiom,
~ ( ord_less_nat @ one_one_nat @ one_one_nat ) ).
% less_numeral_extra(4)
thf(fact_210_less__numeral__extra_I4_J,axiom,
~ ( ord_less_int @ one_one_int @ one_one_int ) ).
% less_numeral_extra(4)
thf(fact_211_less__numeral__extra_I4_J,axiom,
~ ( ord_less_real @ one_one_real @ one_one_real ) ).
% less_numeral_extra(4)
thf(fact_212_power__not__zero,axiom,
! [A: nat,N: nat] :
( ( A != zero_zero_nat )
=> ( ( power_power_nat @ A @ N )
!= zero_zero_nat ) ) ).
% power_not_zero
thf(fact_213_power__not__zero,axiom,
! [A: int,N: nat] :
( ( A != zero_zero_int )
=> ( ( power_power_int @ A @ N )
!= zero_zero_int ) ) ).
% power_not_zero
thf(fact_214_power__not__zero,axiom,
! [A: real,N: nat] :
( ( A != zero_zero_real )
=> ( ( power_power_real @ A @ N )
!= zero_zero_real ) ) ).
% power_not_zero
thf(fact_215_dvd__power__same,axiom,
! [X: nat,Y: nat,N: nat] :
( ( dvd_dvd_nat @ X @ Y )
=> ( dvd_dvd_nat @ ( power_power_nat @ X @ N ) @ ( power_power_nat @ Y @ N ) ) ) ).
% dvd_power_same
thf(fact_216_dvd__power__same,axiom,
! [X: int,Y: int,N: nat] :
( ( dvd_dvd_int @ X @ Y )
=> ( dvd_dvd_int @ ( power_power_int @ X @ N ) @ ( power_power_int @ Y @ N ) ) ) ).
% dvd_power_same
thf(fact_217_dvd__power__same,axiom,
! [X: real,Y: real,N: nat] :
( ( dvd_dvd_real @ X @ Y )
=> ( dvd_dvd_real @ ( power_power_real @ X @ N ) @ ( power_power_real @ Y @ N ) ) ) ).
% dvd_power_same
thf(fact_218_gcd__nat_Oextremum,axiom,
! [A: nat] : ( dvd_dvd_nat @ A @ zero_zero_nat ) ).
% gcd_nat.extremum
thf(fact_219_gcd__nat_Oextremum__strict,axiom,
! [A: nat] :
~ ( ( dvd_dvd_nat @ zero_zero_nat @ A )
& ( zero_zero_nat != A ) ) ).
% gcd_nat.extremum_strict
thf(fact_220_gcd__nat_Oextremum__unique,axiom,
! [A: nat] :
( ( dvd_dvd_nat @ zero_zero_nat @ A )
= ( A = zero_zero_nat ) ) ).
% gcd_nat.extremum_unique
thf(fact_221_gcd__nat_Onot__eq__extremum,axiom,
! [A: nat] :
( ( A != zero_zero_nat )
= ( ( dvd_dvd_nat @ A @ zero_zero_nat )
& ( A != zero_zero_nat ) ) ) ).
% gcd_nat.not_eq_extremum
thf(fact_222_gcd__nat_Oextremum__uniqueI,axiom,
! [A: nat] :
( ( dvd_dvd_nat @ zero_zero_nat @ A )
=> ( A = zero_zero_nat ) ) ).
% gcd_nat.extremum_uniqueI
thf(fact_223_bigger__prime,axiom,
! [N: nat] :
? [P5: nat] :
( ( factor1801147406995305544me_nat @ P5 )
& ( ord_less_nat @ N @ P5 ) ) ).
% bigger_prime
thf(fact_224_less__numeral__extra_I1_J,axiom,
ord_less_nat @ zero_zero_nat @ one_one_nat ).
% less_numeral_extra(1)
thf(fact_225_less__numeral__extra_I1_J,axiom,
ord_less_int @ zero_zero_int @ one_one_int ).
% less_numeral_extra(1)
thf(fact_226_less__numeral__extra_I1_J,axiom,
ord_less_real @ zero_zero_real @ one_one_real ).
% less_numeral_extra(1)
thf(fact_227_zero__less__power,axiom,
! [A: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ A )
=> ( ord_less_nat @ zero_zero_nat @ ( power_power_nat @ A @ N ) ) ) ).
% zero_less_power
thf(fact_228_zero__less__power,axiom,
! [A: int,N: nat] :
( ( ord_less_int @ zero_zero_int @ A )
=> ( ord_less_int @ zero_zero_int @ ( power_power_int @ A @ N ) ) ) ).
% zero_less_power
thf(fact_229_zero__less__power,axiom,
! [A: real,N: nat] :
( ( ord_less_real @ zero_zero_real @ A )
=> ( ord_less_real @ zero_zero_real @ ( power_power_real @ A @ N ) ) ) ).
% zero_less_power
thf(fact_230_power__0,axiom,
! [A: nat] :
( ( power_power_nat @ A @ zero_zero_nat )
= one_one_nat ) ).
% power_0
thf(fact_231_power__0,axiom,
! [A: int] :
( ( power_power_int @ A @ zero_zero_nat )
= one_one_int ) ).
% power_0
thf(fact_232_power__0,axiom,
! [A: real] :
( ( power_power_real @ A @ zero_zero_nat )
= one_one_real ) ).
% power_0
thf(fact_233_dvd__pos__nat,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( dvd_dvd_nat @ M @ N )
=> ( ord_less_nat @ zero_zero_nat @ M ) ) ) ).
% dvd_pos_nat
thf(fact_234_nat__power__less__imp__less,axiom,
! [I: nat,M: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ I )
=> ( ( ord_less_nat @ ( power_power_nat @ I @ M ) @ ( power_power_nat @ I @ N ) )
=> ( ord_less_nat @ M @ N ) ) ) ).
% nat_power_less_imp_less
thf(fact_235_prime__gt__0__nat,axiom,
! [P: nat] :
( ( factor1801147406995305544me_nat @ P )
=> ( ord_less_nat @ zero_zero_nat @ P ) ) ).
% prime_gt_0_nat
thf(fact_236_prime__gt__1__nat,axiom,
! [P: nat] :
( ( factor1801147406995305544me_nat @ P )
=> ( ord_less_nat @ one_one_nat @ P ) ) ).
% prime_gt_1_nat
thf(fact_237_prime__power__exp__nat,axiom,
! [P: nat,N: nat,X: nat,K: nat] :
( ( factor1801147406995305544me_nat @ P )
=> ( ( N != zero_zero_nat )
=> ( ( ( power_power_nat @ X @ N )
= ( power_power_nat @ P @ K ) )
=> ? [I2: nat] :
( X
= ( power_power_nat @ P @ I2 ) ) ) ) ) ).
% prime_power_exp_nat
thf(fact_238_prime__factor__nat,axiom,
! [N: nat] :
( ( N != one_one_nat )
=> ? [P5: nat] :
( ( factor1801147406995305544me_nat @ P5 )
& ( dvd_dvd_nat @ P5 @ N ) ) ) ).
% prime_factor_nat
thf(fact_239_prime__dvd__power__nat,axiom,
! [P: nat,X: nat,N: nat] :
( ( factor1801147406995305544me_nat @ P )
=> ( ( dvd_dvd_nat @ P @ ( power_power_nat @ X @ N ) )
=> ( dvd_dvd_nat @ P @ X ) ) ) ).
% prime_dvd_power_nat
thf(fact_240_power__0__left,axiom,
! [N: nat] :
( ( ( N = zero_zero_nat )
=> ( ( power_power_nat @ zero_zero_nat @ N )
= one_one_nat ) )
& ( ( N != zero_zero_nat )
=> ( ( power_power_nat @ zero_zero_nat @ N )
= zero_zero_nat ) ) ) ).
% power_0_left
thf(fact_241_power__0__left,axiom,
! [N: nat] :
( ( ( N = zero_zero_nat )
=> ( ( power_power_int @ zero_zero_int @ N )
= one_one_int ) )
& ( ( N != zero_zero_nat )
=> ( ( power_power_int @ zero_zero_int @ N )
= zero_zero_int ) ) ) ).
% power_0_left
thf(fact_242_power__0__left,axiom,
! [N: nat] :
( ( ( N = zero_zero_nat )
=> ( ( power_power_real @ zero_zero_real @ N )
= one_one_real ) )
& ( ( N != zero_zero_nat )
=> ( ( power_power_real @ zero_zero_real @ N )
= zero_zero_real ) ) ) ).
% power_0_left
thf(fact_243_power__strict__increasing,axiom,
! [N: nat,N4: nat,A: nat] :
( ( ord_less_nat @ N @ N4 )
=> ( ( ord_less_nat @ one_one_nat @ A )
=> ( ord_less_nat @ ( power_power_nat @ A @ N ) @ ( power_power_nat @ A @ N4 ) ) ) ) ).
% power_strict_increasing
thf(fact_244_power__strict__increasing,axiom,
! [N: nat,N4: nat,A: int] :
( ( ord_less_nat @ N @ N4 )
=> ( ( ord_less_int @ one_one_int @ A )
=> ( ord_less_int @ ( power_power_int @ A @ N ) @ ( power_power_int @ A @ N4 ) ) ) ) ).
% power_strict_increasing
thf(fact_245_power__strict__increasing,axiom,
! [N: nat,N4: nat,A: real] :
( ( ord_less_nat @ N @ N4 )
=> ( ( ord_less_real @ one_one_real @ A )
=> ( ord_less_real @ ( power_power_real @ A @ N ) @ ( power_power_real @ A @ N4 ) ) ) ) ).
% power_strict_increasing
thf(fact_246_power__less__imp__less__exp,axiom,
! [A: nat,M: nat,N: nat] :
( ( ord_less_nat @ one_one_nat @ A )
=> ( ( ord_less_nat @ ( power_power_nat @ A @ M ) @ ( power_power_nat @ A @ N ) )
=> ( ord_less_nat @ M @ N ) ) ) ).
% power_less_imp_less_exp
thf(fact_247_power__less__imp__less__exp,axiom,
! [A: int,M: nat,N: nat] :
( ( ord_less_int @ one_one_int @ A )
=> ( ( ord_less_int @ ( power_power_int @ A @ M ) @ ( power_power_int @ A @ N ) )
=> ( ord_less_nat @ M @ N ) ) ) ).
% power_less_imp_less_exp
thf(fact_248_power__less__imp__less__exp,axiom,
! [A: real,M: nat,N: nat] :
( ( ord_less_real @ one_one_real @ A )
=> ( ( ord_less_real @ ( power_power_real @ A @ M ) @ ( power_power_real @ A @ N ) )
=> ( ord_less_nat @ M @ N ) ) ) ).
% power_less_imp_less_exp
thf(fact_249_zero__power,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( power_power_nat @ zero_zero_nat @ N )
= zero_zero_nat ) ) ).
% zero_power
thf(fact_250_zero__power,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( power_power_int @ zero_zero_int @ N )
= zero_zero_int ) ) ).
% zero_power
thf(fact_251_zero__power,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( power_power_real @ zero_zero_real @ N )
= zero_zero_real ) ) ).
% zero_power
thf(fact_252_is__unit__power__iff,axiom,
! [A: nat,N: nat] :
( ( dvd_dvd_nat @ ( power_power_nat @ A @ N ) @ one_one_nat )
= ( ( dvd_dvd_nat @ A @ one_one_nat )
| ( N = zero_zero_nat ) ) ) ).
% is_unit_power_iff
thf(fact_253_is__unit__power__iff,axiom,
! [A: int,N: nat] :
( ( dvd_dvd_int @ ( power_power_int @ A @ N ) @ one_one_int )
= ( ( dvd_dvd_int @ A @ one_one_int )
| ( N = zero_zero_nat ) ) ) ).
% is_unit_power_iff
thf(fact_254_prime__nat__iff,axiom,
( factor1801147406995305544me_nat
= ( ^ [N2: nat] :
( ( ord_less_nat @ one_one_nat @ N2 )
& ! [M2: nat] :
( ( dvd_dvd_nat @ M2 @ N2 )
=> ( ( M2 = one_one_nat )
| ( M2 = N2 ) ) ) ) ) ) ).
% prime_nat_iff
thf(fact_255_prime__nat__not__dvd,axiom,
! [P: nat,N: nat] :
( ( factor1801147406995305544me_nat @ P )
=> ( ( ord_less_nat @ N @ P )
=> ( ( N != one_one_nat )
=> ~ ( dvd_dvd_nat @ N @ P ) ) ) ) ).
% prime_nat_not_dvd
thf(fact_256_power__strict__decreasing,axiom,
! [N: nat,N4: nat,A: nat] :
( ( ord_less_nat @ N @ N4 )
=> ( ( ord_less_nat @ zero_zero_nat @ A )
=> ( ( ord_less_nat @ A @ one_one_nat )
=> ( ord_less_nat @ ( power_power_nat @ A @ N4 ) @ ( power_power_nat @ A @ N ) ) ) ) ) ).
% power_strict_decreasing
thf(fact_257_power__strict__decreasing,axiom,
! [N: nat,N4: nat,A: int] :
( ( ord_less_nat @ N @ N4 )
=> ( ( ord_less_int @ zero_zero_int @ A )
=> ( ( ord_less_int @ A @ one_one_int )
=> ( ord_less_int @ ( power_power_int @ A @ N4 ) @ ( power_power_int @ A @ N ) ) ) ) ) ).
% power_strict_decreasing
thf(fact_258_power__strict__decreasing,axiom,
! [N: nat,N4: nat,A: real] :
( ( ord_less_nat @ N @ N4 )
=> ( ( ord_less_real @ zero_zero_real @ A )
=> ( ( ord_less_real @ A @ one_one_real )
=> ( ord_less_real @ ( power_power_real @ A @ N4 ) @ ( power_power_real @ A @ N ) ) ) ) ) ).
% power_strict_decreasing
thf(fact_259_one__less__power,axiom,
! [A: nat,N: nat] :
( ( ord_less_nat @ one_one_nat @ A )
=> ( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ord_less_nat @ one_one_nat @ ( power_power_nat @ A @ N ) ) ) ) ).
% one_less_power
thf(fact_260_one__less__power,axiom,
! [A: int,N: nat] :
( ( ord_less_int @ one_one_int @ A )
=> ( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ord_less_int @ one_one_int @ ( power_power_int @ A @ N ) ) ) ) ).
% one_less_power
thf(fact_261_one__less__power,axiom,
! [A: real,N: nat] :
( ( ord_less_real @ one_one_real @ A )
=> ( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ord_less_real @ one_one_real @ ( power_power_real @ A @ N ) ) ) ) ).
% one_less_power
thf(fact_262_dvd__power,axiom,
! [N: nat,X: nat] :
( ( ( ord_less_nat @ zero_zero_nat @ N )
| ( X = one_one_nat ) )
=> ( dvd_dvd_nat @ X @ ( power_power_nat @ X @ N ) ) ) ).
% dvd_power
thf(fact_263_dvd__power,axiom,
! [N: nat,X: int] :
( ( ( ord_less_nat @ zero_zero_nat @ N )
| ( X = one_one_int ) )
=> ( dvd_dvd_int @ X @ ( power_power_int @ X @ N ) ) ) ).
% dvd_power
thf(fact_264_dvd__power,axiom,
! [N: nat,X: real] :
( ( ( ord_less_nat @ zero_zero_nat @ N )
| ( X = one_one_real ) )
=> ( dvd_dvd_real @ X @ ( power_power_real @ X @ N ) ) ) ).
% dvd_power
thf(fact_265_dvd__0__right,axiom,
! [A: nat] : ( dvd_dvd_nat @ A @ zero_zero_nat ) ).
% dvd_0_right
thf(fact_266_dvd__0__right,axiom,
! [A: int] : ( dvd_dvd_int @ A @ zero_zero_int ) ).
% dvd_0_right
thf(fact_267_dvd__0__right,axiom,
! [A: real] : ( dvd_dvd_real @ A @ zero_zero_real ) ).
% dvd_0_right
thf(fact_268_dvd__0__left__iff,axiom,
! [A: nat] :
( ( dvd_dvd_nat @ zero_zero_nat @ A )
= ( A = zero_zero_nat ) ) ).
% dvd_0_left_iff
thf(fact_269_dvd__0__left__iff,axiom,
! [A: int] :
( ( dvd_dvd_int @ zero_zero_int @ A )
= ( A = zero_zero_int ) ) ).
% dvd_0_left_iff
thf(fact_270_dvd__0__left__iff,axiom,
! [A: real] :
( ( dvd_dvd_real @ zero_zero_real @ A )
= ( A = zero_zero_real ) ) ).
% dvd_0_left_iff
thf(fact_271_prime__power__eq__imp__eq,axiom,
! [P: nat,Q: nat,M: nat,N: nat] :
( ( factor1801147406995305544me_nat @ P )
=> ( ( factor1801147406995305544me_nat @ Q )
=> ( ( ord_less_nat @ zero_zero_nat @ M )
=> ( ( ( power_power_nat @ P @ M )
= ( power_power_nat @ Q @ N ) )
=> ( P = Q ) ) ) ) ) ).
% prime_power_eq_imp_eq
thf(fact_272_prime__power__eq__imp__eq,axiom,
! [P: int,Q: int,M: nat,N: nat] :
( ( factor1798656936486255268me_int @ P )
=> ( ( factor1798656936486255268me_int @ Q )
=> ( ( ord_less_nat @ zero_zero_nat @ M )
=> ( ( ( power_power_int @ P @ M )
= ( power_power_int @ Q @ N ) )
=> ( P = Q ) ) ) ) ) ).
% prime_power_eq_imp_eq
thf(fact_273_totatives__less,axiom,
! [X: nat,N: nat] :
( ( member_nat @ X @ ( totatives @ N ) )
=> ( ( ord_less_nat @ one_one_nat @ N )
=> ( ord_less_nat @ X @ N ) ) ) ).
% totatives_less
thf(fact_274_not__is__unit__0,axiom,
~ ( dvd_dvd_nat @ zero_zero_nat @ one_one_nat ) ).
% not_is_unit_0
thf(fact_275_not__is__unit__0,axiom,
~ ( dvd_dvd_int @ zero_zero_int @ one_one_int ) ).
% not_is_unit_0
thf(fact_276_not__one__less__zero,axiom,
~ ( ord_less_nat @ one_one_nat @ zero_zero_nat ) ).
% not_one_less_zero
thf(fact_277_not__one__less__zero,axiom,
~ ( ord_less_int @ one_one_int @ zero_zero_int ) ).
% not_one_less_zero
thf(fact_278_not__one__less__zero,axiom,
~ ( ord_less_real @ one_one_real @ zero_zero_real ) ).
% not_one_less_zero
thf(fact_279_zero__less__one,axiom,
ord_less_nat @ zero_zero_nat @ one_one_nat ).
% zero_less_one
thf(fact_280_zero__less__one,axiom,
ord_less_int @ zero_zero_int @ one_one_int ).
% zero_less_one
thf(fact_281_zero__less__one,axiom,
ord_less_real @ zero_zero_real @ one_one_real ).
% zero_less_one
thf(fact_282_zero__not__in__totatives,axiom,
! [N: nat] :
~ ( member_nat @ zero_zero_nat @ ( totatives @ N ) ) ).
% zero_not_in_totatives
thf(fact_283_primepow__prime__power,axiom,
! [P: nat,N: nat] :
( ( factor1801147406995305544me_nat @ P )
=> ( ( prime_primepow_nat @ ( power_power_nat @ P @ N ) )
= ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% primepow_prime_power
thf(fact_284_primepow__prime__power,axiom,
! [P: int,N: nat] :
( ( factor1798656936486255268me_int @ P )
=> ( ( prime_primepow_int @ ( power_power_int @ P @ N ) )
= ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% primepow_prime_power
thf(fact_285_prime__gt__0__int,axiom,
! [P: int] :
( ( factor1798656936486255268me_int @ P )
=> ( ord_less_int @ zero_zero_int @ P ) ) ).
% prime_gt_0_int
thf(fact_286_prime__gt__1__int,axiom,
! [P: int] :
( ( factor1798656936486255268me_int @ P )
=> ( ord_less_int @ one_one_int @ P ) ) ).
% prime_gt_1_int
thf(fact_287_prime__int__not__dvd,axiom,
! [P: int,N: int] :
( ( factor1798656936486255268me_int @ P )
=> ( ( ord_less_int @ N @ P )
=> ( ( ord_less_int @ one_one_int @ N )
=> ~ ( dvd_dvd_int @ N @ P ) ) ) ) ).
% prime_int_not_dvd
thf(fact_288_prime__dvd__power__int,axiom,
! [P: int,X: int,N: nat] :
( ( factor1798656936486255268me_int @ P )
=> ( ( dvd_dvd_int @ P @ ( power_power_int @ X @ N ) )
=> ( dvd_dvd_int @ P @ X ) ) ) ).
% prime_dvd_power_int
thf(fact_289_zero__not__primepow,axiom,
~ ( prime_primepow_nat @ zero_zero_nat ) ).
% zero_not_primepow
thf(fact_290_zero__not__primepow,axiom,
~ ( prime_primepow_int @ zero_zero_int ) ).
% zero_not_primepow
thf(fact_291_one__not__primepow,axiom,
~ ( prime_primepow_int @ one_one_int ) ).
% one_not_primepow
thf(fact_292_one__not__primepow,axiom,
~ ( prime_primepow_nat @ one_one_nat ) ).
% one_not_primepow
thf(fact_293_primepow__prime,axiom,
! [N: nat] :
( ( factor1801147406995305544me_nat @ N )
=> ( prime_primepow_nat @ N ) ) ).
% primepow_prime
thf(fact_294_primepow__prime,axiom,
! [N: int] :
( ( factor1798656936486255268me_int @ N )
=> ( prime_primepow_int @ N ) ) ).
% primepow_prime
thf(fact_295_primepow__gt__0__nat,axiom,
! [N: nat] :
( ( prime_primepow_nat @ N )
=> ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% primepow_gt_0_nat
thf(fact_296_primepow__not__unit,axiom,
! [P: int] :
( ( prime_primepow_int @ P )
=> ~ ( dvd_dvd_int @ P @ one_one_int ) ) ).
% primepow_not_unit
thf(fact_297_primepow__not__unit,axiom,
! [P: nat] :
( ( prime_primepow_nat @ P )
=> ~ ( dvd_dvd_nat @ P @ one_one_nat ) ) ).
% primepow_not_unit
thf(fact_298_not__primepowI,axiom,
! [P: nat,Q: nat,N: nat] :
( ( factor1801147406995305544me_nat @ P )
=> ( ( factor1801147406995305544me_nat @ Q )
=> ( ( P != Q )
=> ( ( dvd_dvd_nat @ P @ N )
=> ( ( dvd_dvd_nat @ Q @ N )
=> ~ ( prime_primepow_nat @ N ) ) ) ) ) ) ).
% not_primepowI
thf(fact_299_not__primepowI,axiom,
! [P: int,Q: int,N: int] :
( ( factor1798656936486255268me_int @ P )
=> ( ( factor1798656936486255268me_int @ Q )
=> ( ( P != Q )
=> ( ( dvd_dvd_int @ P @ N )
=> ( ( dvd_dvd_int @ Q @ N )
=> ~ ( prime_primepow_int @ N ) ) ) ) ) ) ).
% not_primepowI
thf(fact_300_linorder__neqE__linordered__idom,axiom,
! [X: int,Y: int] :
( ( X != Y )
=> ( ~ ( ord_less_int @ X @ Y )
=> ( ord_less_int @ Y @ X ) ) ) ).
% linorder_neqE_linordered_idom
thf(fact_301_linorder__neqE__linordered__idom,axiom,
! [X: real,Y: real] :
( ( X != Y )
=> ( ~ ( ord_less_real @ X @ Y )
=> ( ord_less_real @ Y @ X ) ) ) ).
% linorder_neqE_linordered_idom
thf(fact_302_dvd__trans,axiom,
! [A: nat,B: nat,C: nat] :
( ( dvd_dvd_nat @ A @ B )
=> ( ( dvd_dvd_nat @ B @ C )
=> ( dvd_dvd_nat @ A @ C ) ) ) ).
% dvd_trans
thf(fact_303_dvd__trans,axiom,
! [A: int,B: int,C: int] :
( ( dvd_dvd_int @ A @ B )
=> ( ( dvd_dvd_int @ B @ C )
=> ( dvd_dvd_int @ A @ C ) ) ) ).
% dvd_trans
thf(fact_304_dvd__refl,axiom,
! [A: nat] : ( dvd_dvd_nat @ A @ A ) ).
% dvd_refl
thf(fact_305_dvd__refl,axiom,
! [A: int] : ( dvd_dvd_int @ A @ A ) ).
% dvd_refl
thf(fact_306_primepow__power__iff__nat,axiom,
! [P: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ P )
=> ( ( prime_primepow_nat @ ( power_power_nat @ P @ N ) )
= ( ( prime_primepow_nat @ P )
& ( ord_less_nat @ zero_zero_nat @ N ) ) ) ) ).
% primepow_power_iff_nat
thf(fact_307_primepow__def,axiom,
( prime_primepow_nat
= ( ^ [N2: nat] :
? [P6: nat,K3: nat] :
( ( factor1801147406995305544me_nat @ P6 )
& ( ord_less_nat @ zero_zero_nat @ K3 )
& ( N2
= ( power_power_nat @ P6 @ K3 ) ) ) ) ) ).
% primepow_def
thf(fact_308_primepow__def,axiom,
( prime_primepow_int
= ( ^ [N2: int] :
? [P6: int,K3: nat] :
( ( factor1798656936486255268me_int @ P6 )
& ( ord_less_nat @ zero_zero_nat @ K3 )
& ( N2
= ( power_power_int @ P6 @ K3 ) ) ) ) ) ).
% primepow_def
thf(fact_309_zero__neq__one,axiom,
zero_zero_nat != one_one_nat ).
% zero_neq_one
thf(fact_310_zero__neq__one,axiom,
zero_zero_int != one_one_int ).
% zero_neq_one
thf(fact_311_zero__neq__one,axiom,
zero_zero_real != one_one_real ).
% zero_neq_one
thf(fact_312_dvd__0__left,axiom,
! [A: nat] :
( ( dvd_dvd_nat @ zero_zero_nat @ A )
=> ( A = zero_zero_nat ) ) ).
% dvd_0_left
thf(fact_313_dvd__0__left,axiom,
! [A: int] :
( ( dvd_dvd_int @ zero_zero_int @ A )
=> ( A = zero_zero_int ) ) ).
% dvd_0_left
thf(fact_314_dvd__0__left,axiom,
! [A: real] :
( ( dvd_dvd_real @ zero_zero_real @ A )
=> ( A = zero_zero_real ) ) ).
% dvd_0_left
thf(fact_315_one__dvd,axiom,
! [A: nat] : ( dvd_dvd_nat @ one_one_nat @ A ) ).
% one_dvd
thf(fact_316_one__dvd,axiom,
! [A: int] : ( dvd_dvd_int @ one_one_int @ A ) ).
% one_dvd
thf(fact_317_one__dvd,axiom,
! [A: real] : ( dvd_dvd_real @ one_one_real @ A ) ).
% one_dvd
thf(fact_318_algebraic__semidom__class_Ounit__imp__dvd,axiom,
! [B: nat,A: nat] :
( ( dvd_dvd_nat @ B @ one_one_nat )
=> ( dvd_dvd_nat @ B @ A ) ) ).
% algebraic_semidom_class.unit_imp_dvd
thf(fact_319_algebraic__semidom__class_Ounit__imp__dvd,axiom,
! [B: int,A: int] :
( ( dvd_dvd_int @ B @ one_one_int )
=> ( dvd_dvd_int @ B @ A ) ) ).
% algebraic_semidom_class.unit_imp_dvd
thf(fact_320_dvd__unit__imp__unit,axiom,
! [A: nat,B: nat] :
( ( dvd_dvd_nat @ A @ B )
=> ( ( dvd_dvd_nat @ B @ one_one_nat )
=> ( dvd_dvd_nat @ A @ one_one_nat ) ) ) ).
% dvd_unit_imp_unit
thf(fact_321_dvd__unit__imp__unit,axiom,
! [A: int,B: int] :
( ( dvd_dvd_int @ A @ B )
=> ( ( dvd_dvd_int @ B @ one_one_int )
=> ( dvd_dvd_int @ A @ one_one_int ) ) ) ).
% dvd_unit_imp_unit
thf(fact_322_dvd__diff,axiom,
! [X: int,Y: int,Z2: int] :
( ( dvd_dvd_int @ X @ Y )
=> ( ( dvd_dvd_int @ X @ Z2 )
=> ( dvd_dvd_int @ X @ ( minus_minus_int @ Y @ Z2 ) ) ) ) ).
% dvd_diff
thf(fact_323_primepowI,axiom,
! [P: nat,K: nat,N: nat] :
( ( factor1801147406995305544me_nat @ P )
=> ( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ( ( power_power_nat @ P @ K )
= N )
=> ( ( prime_primepow_nat @ N )
& ( ( prime_1889911587691200368or_nat @ N )
= P ) ) ) ) ) ).
% primepowI
thf(fact_324_primepowI,axiom,
! [P: int,K: nat,N: int] :
( ( factor1798656936486255268me_int @ P )
=> ( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ( ( power_power_int @ P @ K )
= N )
=> ( ( prime_primepow_int @ N )
& ( ( prime_1887421117182150092or_int @ N )
= P ) ) ) ) ) ).
% primepowI
thf(fact_325_divide__out__primepow,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
=> ( ~ ( dvd_dvd_nat @ N @ one_one_nat )
=> ~ ! [P5: nat] :
( ( factor1801147406995305544me_nat @ P5 )
=> ( ( dvd_dvd_nat @ P5 @ N )
=> ! [K4: nat,N5: nat] :
( ~ ( dvd_dvd_nat @ P5 @ N5 )
=> ( ( ord_less_nat @ zero_zero_nat @ K4 )
=> ( N
!= ( times_times_nat @ ( power_power_nat @ P5 @ K4 ) @ N5 ) ) ) ) ) ) ) ) ).
% divide_out_primepow
thf(fact_326_divide__out__primepow,axiom,
! [N: int] :
( ( N != zero_zero_int )
=> ( ~ ( dvd_dvd_int @ N @ one_one_int )
=> ~ ! [P5: int] :
( ( factor1798656936486255268me_int @ P5 )
=> ( ( dvd_dvd_int @ P5 @ N )
=> ! [K4: nat,N5: int] :
( ~ ( dvd_dvd_int @ P5 @ N5 )
=> ( ( ord_less_nat @ zero_zero_nat @ K4 )
=> ( N
!= ( times_times_int @ ( power_power_int @ P5 @ K4 ) @ N5 ) ) ) ) ) ) ) ) ).
% divide_out_primepow
thf(fact_327_realpow__pos__nth__unique,axiom,
! [N: nat,A: real] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_real @ zero_zero_real @ A )
=> ? [X3: real] :
( ( ord_less_real @ zero_zero_real @ X3 )
& ( ( power_power_real @ X3 @ N )
= A )
& ! [Y3: real] :
( ( ( ord_less_real @ zero_zero_real @ Y3 )
& ( ( power_power_real @ Y3 @ N )
= A ) )
=> ( Y3 = X3 ) ) ) ) ) ).
% realpow_pos_nth_unique
thf(fact_328_realpow__pos__nth,axiom,
! [N: nat,A: real] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_real @ zero_zero_real @ A )
=> ? [R: real] :
( ( ord_less_real @ zero_zero_real @ R )
& ( ( power_power_real @ R @ N )
= A ) ) ) ) ).
% realpow_pos_nth
thf(fact_329_dvd__diff__commute,axiom,
! [A: int,C: int,B: int] :
( ( dvd_dvd_int @ A @ ( minus_minus_int @ C @ B ) )
= ( dvd_dvd_int @ A @ ( minus_minus_int @ B @ C ) ) ) ).
% dvd_diff_commute
thf(fact_330_idom__class_Ounit__imp__dvd,axiom,
! [B: int,A: int] :
( ( dvd_dvd_int @ B @ one_one_int )
=> ( dvd_dvd_int @ B @ A ) ) ).
% idom_class.unit_imp_dvd
thf(fact_331_idom__class_Ounit__imp__dvd,axiom,
! [B: real,A: real] :
( ( dvd_dvd_real @ B @ one_one_real )
=> ( dvd_dvd_real @ B @ A ) ) ).
% idom_class.unit_imp_dvd
thf(fact_332_dvd__field__iff,axiom,
( dvd_dvd_real
= ( ^ [A3: real,B2: real] :
( ( A3 = zero_zero_real )
=> ( B2 = zero_zero_real ) ) ) ) ).
% dvd_field_iff
thf(fact_333_field__lbound__gt__zero,axiom,
! [D1: real,D2: real] :
( ( ord_less_real @ zero_zero_real @ D1 )
=> ( ( ord_less_real @ zero_zero_real @ D2 )
=> ? [E2: real] :
( ( ord_less_real @ zero_zero_real @ E2 )
& ( ord_less_real @ E2 @ D1 )
& ( ord_less_real @ E2 @ D2 ) ) ) ) ).
% field_lbound_gt_zero
thf(fact_334_mult__is__0,axiom,
! [M: nat,N: nat] :
( ( ( times_times_nat @ M @ N )
= zero_zero_nat )
= ( ( M = zero_zero_nat )
| ( N = zero_zero_nat ) ) ) ).
% mult_is_0
thf(fact_335_mult__0__right,axiom,
! [M: nat] :
( ( times_times_nat @ M @ zero_zero_nat )
= zero_zero_nat ) ).
% mult_0_right
thf(fact_336_mult__cancel1,axiom,
! [K: nat,M: nat,N: nat] :
( ( ( times_times_nat @ K @ M )
= ( times_times_nat @ K @ N ) )
= ( ( M = N )
| ( K = zero_zero_nat ) ) ) ).
% mult_cancel1
thf(fact_337_mult__cancel2,axiom,
! [M: nat,K: nat,N: nat] :
( ( ( times_times_nat @ M @ K )
= ( times_times_nat @ N @ K ) )
= ( ( M = N )
| ( K = zero_zero_nat ) ) ) ).
% mult_cancel2
thf(fact_338_nat__mult__eq__1__iff,axiom,
! [M: nat,N: nat] :
( ( ( times_times_nat @ M @ N )
= one_one_nat )
= ( ( M = one_one_nat )
& ( N = one_one_nat ) ) ) ).
% nat_mult_eq_1_iff
thf(fact_339_nat__1__eq__mult__iff,axiom,
! [M: nat,N: nat] :
( ( one_one_nat
= ( times_times_nat @ M @ N ) )
= ( ( M = one_one_nat )
& ( N = one_one_nat ) ) ) ).
% nat_1_eq_mult_iff
thf(fact_340_mult__zero__left,axiom,
! [A: nat] :
( ( times_times_nat @ zero_zero_nat @ A )
= zero_zero_nat ) ).
% mult_zero_left
thf(fact_341_mult__zero__left,axiom,
! [A: real] :
( ( times_times_real @ zero_zero_real @ A )
= zero_zero_real ) ).
% mult_zero_left
thf(fact_342_mult__zero__left,axiom,
! [A: int] :
( ( times_times_int @ zero_zero_int @ A )
= zero_zero_int ) ).
% mult_zero_left
thf(fact_343_mult__zero__right,axiom,
! [A: nat] :
( ( times_times_nat @ A @ zero_zero_nat )
= zero_zero_nat ) ).
% mult_zero_right
thf(fact_344_mult__zero__right,axiom,
! [A: real] :
( ( times_times_real @ A @ zero_zero_real )
= zero_zero_real ) ).
% mult_zero_right
thf(fact_345_mult__zero__right,axiom,
! [A: int] :
( ( times_times_int @ A @ zero_zero_int )
= zero_zero_int ) ).
% mult_zero_right
thf(fact_346_mult__eq__0__iff,axiom,
! [A: nat,B: nat] :
( ( ( times_times_nat @ A @ B )
= zero_zero_nat )
= ( ( A = zero_zero_nat )
| ( B = zero_zero_nat ) ) ) ).
% mult_eq_0_iff
thf(fact_347_mult__eq__0__iff,axiom,
! [A: real,B: real] :
( ( ( times_times_real @ A @ B )
= zero_zero_real )
= ( ( A = zero_zero_real )
| ( B = zero_zero_real ) ) ) ).
% mult_eq_0_iff
thf(fact_348_mult__eq__0__iff,axiom,
! [A: int,B: int] :
( ( ( times_times_int @ A @ B )
= zero_zero_int )
= ( ( A = zero_zero_int )
| ( B = zero_zero_int ) ) ) ).
% mult_eq_0_iff
thf(fact_349_mult__cancel__left,axiom,
! [C: nat,A: nat,B: nat] :
( ( ( times_times_nat @ C @ A )
= ( times_times_nat @ C @ B ) )
= ( ( C = zero_zero_nat )
| ( A = B ) ) ) ).
% mult_cancel_left
thf(fact_350_mult__cancel__left,axiom,
! [C: real,A: real,B: real] :
( ( ( times_times_real @ C @ A )
= ( times_times_real @ C @ B ) )
= ( ( C = zero_zero_real )
| ( A = B ) ) ) ).
% mult_cancel_left
thf(fact_351_mult__cancel__left,axiom,
! [C: int,A: int,B: int] :
( ( ( times_times_int @ C @ A )
= ( times_times_int @ C @ B ) )
= ( ( C = zero_zero_int )
| ( A = B ) ) ) ).
% mult_cancel_left
thf(fact_352_mult__cancel__right,axiom,
! [A: nat,C: nat,B: nat] :
( ( ( times_times_nat @ A @ C )
= ( times_times_nat @ B @ C ) )
= ( ( C = zero_zero_nat )
| ( A = B ) ) ) ).
% mult_cancel_right
thf(fact_353_mult__cancel__right,axiom,
! [A: real,C: real,B: real] :
( ( ( times_times_real @ A @ C )
= ( times_times_real @ B @ C ) )
= ( ( C = zero_zero_real )
| ( A = B ) ) ) ).
% mult_cancel_right
thf(fact_354_mult__cancel__right,axiom,
! [A: int,C: int,B: int] :
( ( ( times_times_int @ A @ C )
= ( times_times_int @ B @ C ) )
= ( ( C = zero_zero_int )
| ( A = B ) ) ) ).
% mult_cancel_right
thf(fact_355_mult__1,axiom,
! [A: nat] :
( ( times_times_nat @ one_one_nat @ A )
= A ) ).
% mult_1
thf(fact_356_mult__1,axiom,
! [A: real] :
( ( times_times_real @ one_one_real @ A )
= A ) ).
% mult_1
thf(fact_357_mult__1,axiom,
! [A: int] :
( ( times_times_int @ one_one_int @ A )
= A ) ).
% mult_1
thf(fact_358_mult_Oright__neutral,axiom,
! [A: nat] :
( ( times_times_nat @ A @ one_one_nat )
= A ) ).
% mult.right_neutral
thf(fact_359_mult_Oright__neutral,axiom,
! [A: real] :
( ( times_times_real @ A @ one_one_real )
= A ) ).
% mult.right_neutral
thf(fact_360_mult_Oright__neutral,axiom,
! [A: int] :
( ( times_times_int @ A @ one_one_int )
= A ) ).
% mult.right_neutral
thf(fact_361_mult__less__cancel2,axiom,
! [M: nat,K: nat,N: nat] :
( ( ord_less_nat @ ( times_times_nat @ M @ K ) @ ( times_times_nat @ N @ K ) )
= ( ( ord_less_nat @ zero_zero_nat @ K )
& ( ord_less_nat @ M @ N ) ) ) ).
% mult_less_cancel2
thf(fact_362_nat__0__less__mult__iff,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( times_times_nat @ M @ N ) )
= ( ( ord_less_nat @ zero_zero_nat @ M )
& ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% nat_0_less_mult_iff
thf(fact_363_not__real__square__gt__zero,axiom,
! [X: real] :
( ( ~ ( ord_less_real @ zero_zero_real @ ( times_times_real @ X @ X ) ) )
= ( X = zero_zero_real ) ) ).
% not_real_square_gt_zero
thf(fact_364_aprimedivisor__of__prime,axiom,
! [P: nat] :
( ( factor1801147406995305544me_nat @ P )
=> ( ( prime_1889911587691200368or_nat @ P )
= P ) ) ).
% aprimedivisor_of_prime
thf(fact_365_aprimedivisor__of__prime,axiom,
! [P: int] :
( ( factor1798656936486255268me_int @ P )
=> ( ( prime_1887421117182150092or_int @ P )
= P ) ) ).
% aprimedivisor_of_prime
thf(fact_366_mult__cancel__right2,axiom,
! [A: real,C: real] :
( ( ( times_times_real @ A @ C )
= C )
= ( ( C = zero_zero_real )
| ( A = one_one_real ) ) ) ).
% mult_cancel_right2
thf(fact_367_mult__cancel__right2,axiom,
! [A: int,C: int] :
( ( ( times_times_int @ A @ C )
= C )
= ( ( C = zero_zero_int )
| ( A = one_one_int ) ) ) ).
% mult_cancel_right2
thf(fact_368_mult__cancel__right1,axiom,
! [C: real,B: real] :
( ( C
= ( times_times_real @ B @ C ) )
= ( ( C = zero_zero_real )
| ( B = one_one_real ) ) ) ).
% mult_cancel_right1
thf(fact_369_mult__cancel__right1,axiom,
! [C: int,B: int] :
( ( C
= ( times_times_int @ B @ C ) )
= ( ( C = zero_zero_int )
| ( B = one_one_int ) ) ) ).
% mult_cancel_right1
thf(fact_370_mult__cancel__left2,axiom,
! [C: real,A: real] :
( ( ( times_times_real @ C @ A )
= C )
= ( ( C = zero_zero_real )
| ( A = one_one_real ) ) ) ).
% mult_cancel_left2
thf(fact_371_mult__cancel__left2,axiom,
! [C: int,A: int] :
( ( ( times_times_int @ C @ A )
= C )
= ( ( C = zero_zero_int )
| ( A = one_one_int ) ) ) ).
% mult_cancel_left2
thf(fact_372_mult__cancel__left1,axiom,
! [C: real,B: real] :
( ( C
= ( times_times_real @ C @ B ) )
= ( ( C = zero_zero_real )
| ( B = one_one_real ) ) ) ).
% mult_cancel_left1
thf(fact_373_mult__cancel__left1,axiom,
! [C: int,B: int] :
( ( C
= ( times_times_int @ C @ B ) )
= ( ( C = zero_zero_int )
| ( B = one_one_int ) ) ) ).
% mult_cancel_left1
thf(fact_374_idom__class_Odvd__times__left__cancel__iff,axiom,
! [A: real,B: real,C: real] :
( ( A != zero_zero_real )
=> ( ( dvd_dvd_real @ ( times_times_real @ A @ B ) @ ( times_times_real @ A @ C ) )
= ( dvd_dvd_real @ B @ C ) ) ) ).
% idom_class.dvd_times_left_cancel_iff
thf(fact_375_idom__class_Odvd__times__left__cancel__iff,axiom,
! [A: int,B: int,C: int] :
( ( A != zero_zero_int )
=> ( ( dvd_dvd_int @ ( times_times_int @ A @ B ) @ ( times_times_int @ A @ C ) )
= ( dvd_dvd_int @ B @ C ) ) ) ).
% idom_class.dvd_times_left_cancel_iff
thf(fact_376_idom__class_Odvd__times__right__cancel__iff,axiom,
! [A: real,B: real,C: real] :
( ( A != zero_zero_real )
=> ( ( dvd_dvd_real @ ( times_times_real @ B @ A ) @ ( times_times_real @ C @ A ) )
= ( dvd_dvd_real @ B @ C ) ) ) ).
% idom_class.dvd_times_right_cancel_iff
thf(fact_377_idom__class_Odvd__times__right__cancel__iff,axiom,
! [A: int,B: int,C: int] :
( ( A != zero_zero_int )
=> ( ( dvd_dvd_int @ ( times_times_int @ B @ A ) @ ( times_times_int @ C @ A ) )
= ( dvd_dvd_int @ B @ C ) ) ) ).
% idom_class.dvd_times_right_cancel_iff
thf(fact_378_dvd__mult__cancel__left,axiom,
! [C: real,A: real,B: real] :
( ( dvd_dvd_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B ) )
= ( ( C = zero_zero_real )
| ( dvd_dvd_real @ A @ B ) ) ) ).
% dvd_mult_cancel_left
thf(fact_379_dvd__mult__cancel__left,axiom,
! [C: int,A: int,B: int] :
( ( dvd_dvd_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) )
= ( ( C = zero_zero_int )
| ( dvd_dvd_int @ A @ B ) ) ) ).
% dvd_mult_cancel_left
thf(fact_380_dvd__mult__cancel__right,axiom,
! [A: real,C: real,B: real] :
( ( dvd_dvd_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ C ) )
= ( ( C = zero_zero_real )
| ( dvd_dvd_real @ A @ B ) ) ) ).
% dvd_mult_cancel_right
thf(fact_381_dvd__mult__cancel__right,axiom,
! [A: int,C: int,B: int] :
( ( dvd_dvd_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) )
= ( ( C = zero_zero_int )
| ( dvd_dvd_int @ A @ B ) ) ) ).
% dvd_mult_cancel_right
thf(fact_382_algebraic__semidom__class_Odvd__times__left__cancel__iff,axiom,
! [A: nat,B: nat,C: nat] :
( ( A != zero_zero_nat )
=> ( ( dvd_dvd_nat @ ( times_times_nat @ A @ B ) @ ( times_times_nat @ A @ C ) )
= ( dvd_dvd_nat @ B @ C ) ) ) ).
% algebraic_semidom_class.dvd_times_left_cancel_iff
thf(fact_383_algebraic__semidom__class_Odvd__times__left__cancel__iff,axiom,
! [A: int,B: int,C: int] :
( ( A != zero_zero_int )
=> ( ( dvd_dvd_int @ ( times_times_int @ A @ B ) @ ( times_times_int @ A @ C ) )
= ( dvd_dvd_int @ B @ C ) ) ) ).
% algebraic_semidom_class.dvd_times_left_cancel_iff
thf(fact_384_algebraic__semidom__class_Odvd__times__right__cancel__iff,axiom,
! [A: nat,B: nat,C: nat] :
( ( A != zero_zero_nat )
=> ( ( dvd_dvd_nat @ ( times_times_nat @ B @ A ) @ ( times_times_nat @ C @ A ) )
= ( dvd_dvd_nat @ B @ C ) ) ) ).
% algebraic_semidom_class.dvd_times_right_cancel_iff
thf(fact_385_algebraic__semidom__class_Odvd__times__right__cancel__iff,axiom,
! [A: int,B: int,C: int] :
( ( A != zero_zero_int )
=> ( ( dvd_dvd_int @ ( times_times_int @ B @ A ) @ ( times_times_int @ C @ A ) )
= ( dvd_dvd_int @ B @ C ) ) ) ).
% algebraic_semidom_class.dvd_times_right_cancel_iff
thf(fact_386_comm__semiring__1__class_Omult__unit__dvd__iff_H,axiom,
! [A: nat,B: nat,C: nat] :
( ( dvd_dvd_nat @ A @ one_one_nat )
=> ( ( dvd_dvd_nat @ ( times_times_nat @ A @ B ) @ C )
= ( dvd_dvd_nat @ B @ C ) ) ) ).
% comm_semiring_1_class.mult_unit_dvd_iff'
thf(fact_387_comm__semiring__1__class_Omult__unit__dvd__iff_H,axiom,
! [A: real,B: real,C: real] :
( ( dvd_dvd_real @ A @ one_one_real )
=> ( ( dvd_dvd_real @ ( times_times_real @ A @ B ) @ C )
= ( dvd_dvd_real @ B @ C ) ) ) ).
% comm_semiring_1_class.mult_unit_dvd_iff'
thf(fact_388_comm__semiring__1__class_Omult__unit__dvd__iff_H,axiom,
! [A: int,B: int,C: int] :
( ( dvd_dvd_int @ A @ one_one_int )
=> ( ( dvd_dvd_int @ ( times_times_int @ A @ B ) @ C )
= ( dvd_dvd_int @ B @ C ) ) ) ).
% comm_semiring_1_class.mult_unit_dvd_iff'
thf(fact_389_comm__semiring__1__class_Omult__unit__dvd__iff,axiom,
! [B: nat,A: nat,C: nat] :
( ( dvd_dvd_nat @ B @ one_one_nat )
=> ( ( dvd_dvd_nat @ ( times_times_nat @ A @ B ) @ C )
= ( dvd_dvd_nat @ A @ C ) ) ) ).
% comm_semiring_1_class.mult_unit_dvd_iff
thf(fact_390_comm__semiring__1__class_Omult__unit__dvd__iff,axiom,
! [B: real,A: real,C: real] :
( ( dvd_dvd_real @ B @ one_one_real )
=> ( ( dvd_dvd_real @ ( times_times_real @ A @ B ) @ C )
= ( dvd_dvd_real @ A @ C ) ) ) ).
% comm_semiring_1_class.mult_unit_dvd_iff
thf(fact_391_comm__semiring__1__class_Omult__unit__dvd__iff,axiom,
! [B: int,A: int,C: int] :
( ( dvd_dvd_int @ B @ one_one_int )
=> ( ( dvd_dvd_int @ ( times_times_int @ A @ B ) @ C )
= ( dvd_dvd_int @ A @ C ) ) ) ).
% comm_semiring_1_class.mult_unit_dvd_iff
thf(fact_392_comm__monoid__mult__class_Ois__unit__mult__iff,axiom,
! [A: nat,B: nat] :
( ( dvd_dvd_nat @ ( times_times_nat @ A @ B ) @ one_one_nat )
= ( ( dvd_dvd_nat @ A @ one_one_nat )
& ( dvd_dvd_nat @ B @ one_one_nat ) ) ) ).
% comm_monoid_mult_class.is_unit_mult_iff
thf(fact_393_comm__monoid__mult__class_Ois__unit__mult__iff,axiom,
! [A: real,B: real] :
( ( dvd_dvd_real @ ( times_times_real @ A @ B ) @ one_one_real )
= ( ( dvd_dvd_real @ A @ one_one_real )
& ( dvd_dvd_real @ B @ one_one_real ) ) ) ).
% comm_monoid_mult_class.is_unit_mult_iff
thf(fact_394_comm__monoid__mult__class_Ois__unit__mult__iff,axiom,
! [A: int,B: int] :
( ( dvd_dvd_int @ ( times_times_int @ A @ B ) @ one_one_int )
= ( ( dvd_dvd_int @ A @ one_one_int )
& ( dvd_dvd_int @ B @ one_one_int ) ) ) ).
% comm_monoid_mult_class.is_unit_mult_iff
thf(fact_395_comm__monoid__mult__class_Ounit__prod,axiom,
! [A: nat,B: nat] :
( ( dvd_dvd_nat @ A @ one_one_nat )
=> ( ( dvd_dvd_nat @ B @ one_one_nat )
=> ( dvd_dvd_nat @ ( times_times_nat @ A @ B ) @ one_one_nat ) ) ) ).
% comm_monoid_mult_class.unit_prod
thf(fact_396_comm__monoid__mult__class_Ounit__prod,axiom,
! [A: real,B: real] :
( ( dvd_dvd_real @ A @ one_one_real )
=> ( ( dvd_dvd_real @ B @ one_one_real )
=> ( dvd_dvd_real @ ( times_times_real @ A @ B ) @ one_one_real ) ) ) ).
% comm_monoid_mult_class.unit_prod
thf(fact_397_comm__monoid__mult__class_Ounit__prod,axiom,
! [A: int,B: int] :
( ( dvd_dvd_int @ A @ one_one_int )
=> ( ( dvd_dvd_int @ B @ one_one_int )
=> ( dvd_dvd_int @ ( times_times_int @ A @ B ) @ one_one_int ) ) ) ).
% comm_monoid_mult_class.unit_prod
thf(fact_398_algebraic__semidom__class_Ounit__prod,axiom,
! [A: nat,B: nat] :
( ( dvd_dvd_nat @ A @ one_one_nat )
=> ( ( dvd_dvd_nat @ B @ one_one_nat )
=> ( dvd_dvd_nat @ ( times_times_nat @ A @ B ) @ one_one_nat ) ) ) ).
% algebraic_semidom_class.unit_prod
thf(fact_399_algebraic__semidom__class_Ounit__prod,axiom,
! [A: int,B: int] :
( ( dvd_dvd_int @ A @ one_one_int )
=> ( ( dvd_dvd_int @ B @ one_one_int )
=> ( dvd_dvd_int @ ( times_times_int @ A @ B ) @ one_one_int ) ) ) ).
% algebraic_semidom_class.unit_prod
thf(fact_400_real__arch__pow,axiom,
! [X: real,Y: real] :
( ( ord_less_real @ one_one_real @ X )
=> ? [N3: nat] : ( ord_less_real @ Y @ ( power_power_real @ X @ N3 ) ) ) ).
% real_arch_pow
thf(fact_401_real__arch__pow__inv,axiom,
! [Y: real,X: real] :
( ( ord_less_real @ zero_zero_real @ Y )
=> ( ( ord_less_real @ X @ one_one_real )
=> ? [N3: nat] : ( ord_less_real @ ( power_power_real @ X @ N3 ) @ Y ) ) ) ).
% real_arch_pow_inv
thf(fact_402_mult_Oleft__commute,axiom,
! [B: nat,A: nat,C: nat] :
( ( times_times_nat @ B @ ( times_times_nat @ A @ C ) )
= ( times_times_nat @ A @ ( times_times_nat @ B @ C ) ) ) ).
% mult.left_commute
thf(fact_403_mult_Oleft__commute,axiom,
! [B: real,A: real,C: real] :
( ( times_times_real @ B @ ( times_times_real @ A @ C ) )
= ( times_times_real @ A @ ( times_times_real @ B @ C ) ) ) ).
% mult.left_commute
thf(fact_404_mult_Oleft__commute,axiom,
! [B: int,A: int,C: int] :
( ( times_times_int @ B @ ( times_times_int @ A @ C ) )
= ( times_times_int @ A @ ( times_times_int @ B @ C ) ) ) ).
% mult.left_commute
thf(fact_405_mult_Ocommute,axiom,
( times_times_nat
= ( ^ [A3: nat,B2: nat] : ( times_times_nat @ B2 @ A3 ) ) ) ).
% mult.commute
thf(fact_406_mult_Ocommute,axiom,
( times_times_real
= ( ^ [A3: real,B2: real] : ( times_times_real @ B2 @ A3 ) ) ) ).
% mult.commute
thf(fact_407_mult_Ocommute,axiom,
( times_times_int
= ( ^ [A3: int,B2: int] : ( times_times_int @ B2 @ A3 ) ) ) ).
% mult.commute
thf(fact_408_mult_Oassoc,axiom,
! [A: nat,B: nat,C: nat] :
( ( times_times_nat @ ( times_times_nat @ A @ B ) @ C )
= ( times_times_nat @ A @ ( times_times_nat @ B @ C ) ) ) ).
% mult.assoc
thf(fact_409_mult_Oassoc,axiom,
! [A: real,B: real,C: real] :
( ( times_times_real @ ( times_times_real @ A @ B ) @ C )
= ( times_times_real @ A @ ( times_times_real @ B @ C ) ) ) ).
% mult.assoc
thf(fact_410_mult_Oassoc,axiom,
! [A: int,B: int,C: int] :
( ( times_times_int @ ( times_times_int @ A @ B ) @ C )
= ( times_times_int @ A @ ( times_times_int @ B @ C ) ) ) ).
% mult.assoc
thf(fact_411_one__integer_Orsp,axiom,
one_one_int = one_one_int ).
% one_integer.rsp
thf(fact_412_ab__semigroup__mult__class_Omult__ac_I1_J,axiom,
! [A: nat,B: nat,C: nat] :
( ( times_times_nat @ ( times_times_nat @ A @ B ) @ C )
= ( times_times_nat @ A @ ( times_times_nat @ B @ C ) ) ) ).
% ab_semigroup_mult_class.mult_ac(1)
thf(fact_413_ab__semigroup__mult__class_Omult__ac_I1_J,axiom,
! [A: real,B: real,C: real] :
( ( times_times_real @ ( times_times_real @ A @ B ) @ C )
= ( times_times_real @ A @ ( times_times_real @ B @ C ) ) ) ).
% ab_semigroup_mult_class.mult_ac(1)
thf(fact_414_ab__semigroup__mult__class_Omult__ac_I1_J,axiom,
! [A: int,B: int,C: int] :
( ( times_times_int @ ( times_times_int @ A @ B ) @ C )
= ( times_times_int @ A @ ( times_times_int @ B @ C ) ) ) ).
% ab_semigroup_mult_class.mult_ac(1)
thf(fact_415_zero__integer_Orsp,axiom,
zero_zero_int = zero_zero_int ).
% zero_integer.rsp
thf(fact_416_primepow__mult__aprimedivisorI,axiom,
! [N: int] :
( ( prime_primepow_int @ N )
=> ( prime_primepow_int @ ( times_times_int @ ( prime_1887421117182150092or_int @ N ) @ N ) ) ) ).
% primepow_mult_aprimedivisorI
thf(fact_417_primepow__mult__aprimedivisorI,axiom,
! [N: nat] :
( ( prime_primepow_nat @ N )
=> ( prime_primepow_nat @ ( times_times_nat @ ( prime_1889911587691200368or_nat @ N ) @ N ) ) ) ).
% primepow_mult_aprimedivisorI
thf(fact_418_mult__right__cancel,axiom,
! [C: nat,A: nat,B: nat] :
( ( C != zero_zero_nat )
=> ( ( ( times_times_nat @ A @ C )
= ( times_times_nat @ B @ C ) )
= ( A = B ) ) ) ).
% mult_right_cancel
thf(fact_419_mult__right__cancel,axiom,
! [C: real,A: real,B: real] :
( ( C != zero_zero_real )
=> ( ( ( times_times_real @ A @ C )
= ( times_times_real @ B @ C ) )
= ( A = B ) ) ) ).
% mult_right_cancel
thf(fact_420_mult__right__cancel,axiom,
! [C: int,A: int,B: int] :
( ( C != zero_zero_int )
=> ( ( ( times_times_int @ A @ C )
= ( times_times_int @ B @ C ) )
= ( A = B ) ) ) ).
% mult_right_cancel
thf(fact_421_mult__left__cancel,axiom,
! [C: nat,A: nat,B: nat] :
( ( C != zero_zero_nat )
=> ( ( ( times_times_nat @ C @ A )
= ( times_times_nat @ C @ B ) )
= ( A = B ) ) ) ).
% mult_left_cancel
thf(fact_422_mult__left__cancel,axiom,
! [C: real,A: real,B: real] :
( ( C != zero_zero_real )
=> ( ( ( times_times_real @ C @ A )
= ( times_times_real @ C @ B ) )
= ( A = B ) ) ) ).
% mult_left_cancel
thf(fact_423_mult__left__cancel,axiom,
! [C: int,A: int,B: int] :
( ( C != zero_zero_int )
=> ( ( ( times_times_int @ C @ A )
= ( times_times_int @ C @ B ) )
= ( A = B ) ) ) ).
% mult_left_cancel
thf(fact_424_no__zero__divisors,axiom,
! [A: nat,B: nat] :
( ( A != zero_zero_nat )
=> ( ( B != zero_zero_nat )
=> ( ( times_times_nat @ A @ B )
!= zero_zero_nat ) ) ) ).
% no_zero_divisors
thf(fact_425_no__zero__divisors,axiom,
! [A: real,B: real] :
( ( A != zero_zero_real )
=> ( ( B != zero_zero_real )
=> ( ( times_times_real @ A @ B )
!= zero_zero_real ) ) ) ).
% no_zero_divisors
thf(fact_426_no__zero__divisors,axiom,
! [A: int,B: int] :
( ( A != zero_zero_int )
=> ( ( B != zero_zero_int )
=> ( ( times_times_int @ A @ B )
!= zero_zero_int ) ) ) ).
% no_zero_divisors
thf(fact_427_divisors__zero,axiom,
! [A: nat,B: nat] :
( ( ( times_times_nat @ A @ B )
= zero_zero_nat )
=> ( ( A = zero_zero_nat )
| ( B = zero_zero_nat ) ) ) ).
% divisors_zero
thf(fact_428_divisors__zero,axiom,
! [A: real,B: real] :
( ( ( times_times_real @ A @ B )
= zero_zero_real )
=> ( ( A = zero_zero_real )
| ( B = zero_zero_real ) ) ) ).
% divisors_zero
thf(fact_429_divisors__zero,axiom,
! [A: int,B: int] :
( ( ( times_times_int @ A @ B )
= zero_zero_int )
=> ( ( A = zero_zero_int )
| ( B = zero_zero_int ) ) ) ).
% divisors_zero
thf(fact_430_mult__not__zero,axiom,
! [A: nat,B: nat] :
( ( ( times_times_nat @ A @ B )
!= zero_zero_nat )
=> ( ( A != zero_zero_nat )
& ( B != zero_zero_nat ) ) ) ).
% mult_not_zero
thf(fact_431_mult__not__zero,axiom,
! [A: real,B: real] :
( ( ( times_times_real @ A @ B )
!= zero_zero_real )
=> ( ( A != zero_zero_real )
& ( B != zero_zero_real ) ) ) ).
% mult_not_zero
thf(fact_432_mult__not__zero,axiom,
! [A: int,B: int] :
( ( ( times_times_int @ A @ B )
!= zero_zero_int )
=> ( ( A != zero_zero_int )
& ( B != zero_zero_int ) ) ) ).
% mult_not_zero
thf(fact_433_mult_Ocomm__neutral,axiom,
! [A: nat] :
( ( times_times_nat @ A @ one_one_nat )
= A ) ).
% mult.comm_neutral
thf(fact_434_mult_Ocomm__neutral,axiom,
! [A: real] :
( ( times_times_real @ A @ one_one_real )
= A ) ).
% mult.comm_neutral
thf(fact_435_mult_Ocomm__neutral,axiom,
! [A: int] :
( ( times_times_int @ A @ one_one_int )
= A ) ).
% mult.comm_neutral
thf(fact_436_comm__monoid__mult__class_Omult__1,axiom,
! [A: nat] :
( ( times_times_nat @ one_one_nat @ A )
= A ) ).
% comm_monoid_mult_class.mult_1
thf(fact_437_comm__monoid__mult__class_Omult__1,axiom,
! [A: real] :
( ( times_times_real @ one_one_real @ A )
= A ) ).
% comm_monoid_mult_class.mult_1
thf(fact_438_comm__monoid__mult__class_Omult__1,axiom,
! [A: int] :
( ( times_times_int @ one_one_int @ A )
= A ) ).
% comm_monoid_mult_class.mult_1
thf(fact_439_right__diff__distrib_H,axiom,
! [A: nat,B: nat,C: nat] :
( ( times_times_nat @ A @ ( minus_minus_nat @ B @ C ) )
= ( minus_minus_nat @ ( times_times_nat @ A @ B ) @ ( times_times_nat @ A @ C ) ) ) ).
% right_diff_distrib'
thf(fact_440_right__diff__distrib_H,axiom,
! [A: real,B: real,C: real] :
( ( times_times_real @ A @ ( minus_minus_real @ B @ C ) )
= ( minus_minus_real @ ( times_times_real @ A @ B ) @ ( times_times_real @ A @ C ) ) ) ).
% right_diff_distrib'
thf(fact_441_right__diff__distrib_H,axiom,
! [A: int,B: int,C: int] :
( ( times_times_int @ A @ ( minus_minus_int @ B @ C ) )
= ( minus_minus_int @ ( times_times_int @ A @ B ) @ ( times_times_int @ A @ C ) ) ) ).
% right_diff_distrib'
thf(fact_442_left__diff__distrib_H,axiom,
! [B: nat,C: nat,A: nat] :
( ( times_times_nat @ ( minus_minus_nat @ B @ C ) @ A )
= ( minus_minus_nat @ ( times_times_nat @ B @ A ) @ ( times_times_nat @ C @ A ) ) ) ).
% left_diff_distrib'
thf(fact_443_left__diff__distrib_H,axiom,
! [B: real,C: real,A: real] :
( ( times_times_real @ ( minus_minus_real @ B @ C ) @ A )
= ( minus_minus_real @ ( times_times_real @ B @ A ) @ ( times_times_real @ C @ A ) ) ) ).
% left_diff_distrib'
thf(fact_444_left__diff__distrib_H,axiom,
! [B: int,C: int,A: int] :
( ( times_times_int @ ( minus_minus_int @ B @ C ) @ A )
= ( minus_minus_int @ ( times_times_int @ B @ A ) @ ( times_times_int @ C @ A ) ) ) ).
% left_diff_distrib'
thf(fact_445_right__diff__distrib,axiom,
! [A: real,B: real,C: real] :
( ( times_times_real @ A @ ( minus_minus_real @ B @ C ) )
= ( minus_minus_real @ ( times_times_real @ A @ B ) @ ( times_times_real @ A @ C ) ) ) ).
% right_diff_distrib
thf(fact_446_right__diff__distrib,axiom,
! [A: int,B: int,C: int] :
( ( times_times_int @ A @ ( minus_minus_int @ B @ C ) )
= ( minus_minus_int @ ( times_times_int @ A @ B ) @ ( times_times_int @ A @ C ) ) ) ).
% right_diff_distrib
thf(fact_447_left__diff__distrib,axiom,
! [A: real,B: real,C: real] :
( ( times_times_real @ ( minus_minus_real @ A @ B ) @ C )
= ( minus_minus_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ C ) ) ) ).
% left_diff_distrib
thf(fact_448_left__diff__distrib,axiom,
! [A: int,B: int,C: int] :
( ( times_times_int @ ( minus_minus_int @ A @ B ) @ C )
= ( minus_minus_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) ) ) ).
% left_diff_distrib
thf(fact_449_power__commutes,axiom,
! [A: nat,N: nat] :
( ( times_times_nat @ ( power_power_nat @ A @ N ) @ A )
= ( times_times_nat @ A @ ( power_power_nat @ A @ N ) ) ) ).
% power_commutes
thf(fact_450_power__commutes,axiom,
! [A: real,N: nat] :
( ( times_times_real @ ( power_power_real @ A @ N ) @ A )
= ( times_times_real @ A @ ( power_power_real @ A @ N ) ) ) ).
% power_commutes
thf(fact_451_power__commutes,axiom,
! [A: int,N: nat] :
( ( times_times_int @ ( power_power_int @ A @ N ) @ A )
= ( times_times_int @ A @ ( power_power_int @ A @ N ) ) ) ).
% power_commutes
thf(fact_452_power__mult__distrib,axiom,
! [A: nat,B: nat,N: nat] :
( ( power_power_nat @ ( times_times_nat @ A @ B ) @ N )
= ( times_times_nat @ ( power_power_nat @ A @ N ) @ ( power_power_nat @ B @ N ) ) ) ).
% power_mult_distrib
thf(fact_453_power__mult__distrib,axiom,
! [A: real,B: real,N: nat] :
( ( power_power_real @ ( times_times_real @ A @ B ) @ N )
= ( times_times_real @ ( power_power_real @ A @ N ) @ ( power_power_real @ B @ N ) ) ) ).
% power_mult_distrib
thf(fact_454_power__mult__distrib,axiom,
! [A: int,B: int,N: nat] :
( ( power_power_int @ ( times_times_int @ A @ B ) @ N )
= ( times_times_int @ ( power_power_int @ A @ N ) @ ( power_power_int @ B @ N ) ) ) ).
% power_mult_distrib
thf(fact_455_power__commuting__commutes,axiom,
! [X: nat,Y: nat,N: nat] :
( ( ( times_times_nat @ X @ Y )
= ( times_times_nat @ Y @ X ) )
=> ( ( times_times_nat @ ( power_power_nat @ X @ N ) @ Y )
= ( times_times_nat @ Y @ ( power_power_nat @ X @ N ) ) ) ) ).
% power_commuting_commutes
thf(fact_456_power__commuting__commutes,axiom,
! [X: real,Y: real,N: nat] :
( ( ( times_times_real @ X @ Y )
= ( times_times_real @ Y @ X ) )
=> ( ( times_times_real @ ( power_power_real @ X @ N ) @ Y )
= ( times_times_real @ Y @ ( power_power_real @ X @ N ) ) ) ) ).
% power_commuting_commutes
thf(fact_457_power__commuting__commutes,axiom,
! [X: int,Y: int,N: nat] :
( ( ( times_times_int @ X @ Y )
= ( times_times_int @ Y @ X ) )
=> ( ( times_times_int @ ( power_power_int @ X @ N ) @ Y )
= ( times_times_int @ Y @ ( power_power_int @ X @ N ) ) ) ) ).
% power_commuting_commutes
thf(fact_458_dvd__productE,axiom,
! [P: nat,A: nat,B: nat] :
( ( dvd_dvd_nat @ P @ ( times_times_nat @ A @ B ) )
=> ~ ! [X3: nat,Y4: nat] :
( ( P
= ( times_times_nat @ X3 @ Y4 ) )
=> ( ( dvd_dvd_nat @ X3 @ A )
=> ~ ( dvd_dvd_nat @ Y4 @ B ) ) ) ) ).
% dvd_productE
thf(fact_459_dvd__productE,axiom,
! [P: int,A: int,B: int] :
( ( dvd_dvd_int @ P @ ( times_times_int @ A @ B ) )
=> ~ ! [X3: int,Y4: int] :
( ( P
= ( times_times_int @ X3 @ Y4 ) )
=> ( ( dvd_dvd_int @ X3 @ A )
=> ~ ( dvd_dvd_int @ Y4 @ B ) ) ) ) ).
% dvd_productE
thf(fact_460_division__decomp,axiom,
! [A: nat,B: nat,C: nat] :
( ( dvd_dvd_nat @ A @ ( times_times_nat @ B @ C ) )
=> ? [B4: nat,C3: nat] :
( ( A
= ( times_times_nat @ B4 @ C3 ) )
& ( dvd_dvd_nat @ B4 @ B )
& ( dvd_dvd_nat @ C3 @ C ) ) ) ).
% division_decomp
thf(fact_461_division__decomp,axiom,
! [A: int,B: int,C: int] :
( ( dvd_dvd_int @ A @ ( times_times_int @ B @ C ) )
=> ? [B4: int,C3: int] :
( ( A
= ( times_times_int @ B4 @ C3 ) )
& ( dvd_dvd_int @ B4 @ B )
& ( dvd_dvd_int @ C3 @ C ) ) ) ).
% division_decomp
thf(fact_462_dvd__triv__right,axiom,
! [A: nat,B: nat] : ( dvd_dvd_nat @ A @ ( times_times_nat @ B @ A ) ) ).
% dvd_triv_right
thf(fact_463_dvd__triv__right,axiom,
! [A: real,B: real] : ( dvd_dvd_real @ A @ ( times_times_real @ B @ A ) ) ).
% dvd_triv_right
thf(fact_464_dvd__triv__right,axiom,
! [A: int,B: int] : ( dvd_dvd_int @ A @ ( times_times_int @ B @ A ) ) ).
% dvd_triv_right
thf(fact_465_dvd__mult__right,axiom,
! [A: nat,B: nat,C: nat] :
( ( dvd_dvd_nat @ ( times_times_nat @ A @ B ) @ C )
=> ( dvd_dvd_nat @ B @ C ) ) ).
% dvd_mult_right
thf(fact_466_dvd__mult__right,axiom,
! [A: real,B: real,C: real] :
( ( dvd_dvd_real @ ( times_times_real @ A @ B ) @ C )
=> ( dvd_dvd_real @ B @ C ) ) ).
% dvd_mult_right
thf(fact_467_dvd__mult__right,axiom,
! [A: int,B: int,C: int] :
( ( dvd_dvd_int @ ( times_times_int @ A @ B ) @ C )
=> ( dvd_dvd_int @ B @ C ) ) ).
% dvd_mult_right
thf(fact_468_mult__dvd__mono,axiom,
! [A: nat,B: nat,C: nat,D: nat] :
( ( dvd_dvd_nat @ A @ B )
=> ( ( dvd_dvd_nat @ C @ D )
=> ( dvd_dvd_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ D ) ) ) ) ).
% mult_dvd_mono
thf(fact_469_mult__dvd__mono,axiom,
! [A: real,B: real,C: real,D: real] :
( ( dvd_dvd_real @ A @ B )
=> ( ( dvd_dvd_real @ C @ D )
=> ( dvd_dvd_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ D ) ) ) ) ).
% mult_dvd_mono
thf(fact_470_mult__dvd__mono,axiom,
! [A: int,B: int,C: int,D: int] :
( ( dvd_dvd_int @ A @ B )
=> ( ( dvd_dvd_int @ C @ D )
=> ( dvd_dvd_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ D ) ) ) ) ).
% mult_dvd_mono
thf(fact_471_dvd__triv__left,axiom,
! [A: nat,B: nat] : ( dvd_dvd_nat @ A @ ( times_times_nat @ A @ B ) ) ).
% dvd_triv_left
thf(fact_472_dvd__triv__left,axiom,
! [A: real,B: real] : ( dvd_dvd_real @ A @ ( times_times_real @ A @ B ) ) ).
% dvd_triv_left
thf(fact_473_dvd__triv__left,axiom,
! [A: int,B: int] : ( dvd_dvd_int @ A @ ( times_times_int @ A @ B ) ) ).
% dvd_triv_left
thf(fact_474_dvd__mult__left,axiom,
! [A: nat,B: nat,C: nat] :
( ( dvd_dvd_nat @ ( times_times_nat @ A @ B ) @ C )
=> ( dvd_dvd_nat @ A @ C ) ) ).
% dvd_mult_left
thf(fact_475_dvd__mult__left,axiom,
! [A: real,B: real,C: real] :
( ( dvd_dvd_real @ ( times_times_real @ A @ B ) @ C )
=> ( dvd_dvd_real @ A @ C ) ) ).
% dvd_mult_left
thf(fact_476_dvd__mult__left,axiom,
! [A: int,B: int,C: int] :
( ( dvd_dvd_int @ ( times_times_int @ A @ B ) @ C )
=> ( dvd_dvd_int @ A @ C ) ) ).
% dvd_mult_left
thf(fact_477_dvd__mult2,axiom,
! [A: nat,B: nat,C: nat] :
( ( dvd_dvd_nat @ A @ B )
=> ( dvd_dvd_nat @ A @ ( times_times_nat @ B @ C ) ) ) ).
% dvd_mult2
thf(fact_478_dvd__mult2,axiom,
! [A: real,B: real,C: real] :
( ( dvd_dvd_real @ A @ B )
=> ( dvd_dvd_real @ A @ ( times_times_real @ B @ C ) ) ) ).
% dvd_mult2
thf(fact_479_dvd__mult2,axiom,
! [A: int,B: int,C: int] :
( ( dvd_dvd_int @ A @ B )
=> ( dvd_dvd_int @ A @ ( times_times_int @ B @ C ) ) ) ).
% dvd_mult2
thf(fact_480_dvd__mult,axiom,
! [A: nat,C: nat,B: nat] :
( ( dvd_dvd_nat @ A @ C )
=> ( dvd_dvd_nat @ A @ ( times_times_nat @ B @ C ) ) ) ).
% dvd_mult
thf(fact_481_dvd__mult,axiom,
! [A: real,C: real,B: real] :
( ( dvd_dvd_real @ A @ C )
=> ( dvd_dvd_real @ A @ ( times_times_real @ B @ C ) ) ) ).
% dvd_mult
thf(fact_482_dvd__mult,axiom,
! [A: int,C: int,B: int] :
( ( dvd_dvd_int @ A @ C )
=> ( dvd_dvd_int @ A @ ( times_times_int @ B @ C ) ) ) ).
% dvd_mult
thf(fact_483_dvd__def,axiom,
( dvd_dvd_nat
= ( ^ [B2: nat,A3: nat] :
? [K3: nat] :
( A3
= ( times_times_nat @ B2 @ K3 ) ) ) ) ).
% dvd_def
thf(fact_484_dvd__def,axiom,
( dvd_dvd_real
= ( ^ [B2: real,A3: real] :
? [K3: real] :
( A3
= ( times_times_real @ B2 @ K3 ) ) ) ) ).
% dvd_def
thf(fact_485_dvd__def,axiom,
( dvd_dvd_int
= ( ^ [B2: int,A3: int] :
? [K3: int] :
( A3
= ( times_times_int @ B2 @ K3 ) ) ) ) ).
% dvd_def
thf(fact_486_dvdI,axiom,
! [A: nat,B: nat,K: nat] :
( ( A
= ( times_times_nat @ B @ K ) )
=> ( dvd_dvd_nat @ B @ A ) ) ).
% dvdI
thf(fact_487_dvdI,axiom,
! [A: real,B: real,K: real] :
( ( A
= ( times_times_real @ B @ K ) )
=> ( dvd_dvd_real @ B @ A ) ) ).
% dvdI
thf(fact_488_dvdI,axiom,
! [A: int,B: int,K: int] :
( ( A
= ( times_times_int @ B @ K ) )
=> ( dvd_dvd_int @ B @ A ) ) ).
% dvdI
thf(fact_489_dvdE,axiom,
! [B: nat,A: nat] :
( ( dvd_dvd_nat @ B @ A )
=> ~ ! [K4: nat] :
( A
!= ( times_times_nat @ B @ K4 ) ) ) ).
% dvdE
thf(fact_490_dvdE,axiom,
! [B: real,A: real] :
( ( dvd_dvd_real @ B @ A )
=> ~ ! [K4: real] :
( A
!= ( times_times_real @ B @ K4 ) ) ) ).
% dvdE
thf(fact_491_dvdE,axiom,
! [B: int,A: int] :
( ( dvd_dvd_int @ B @ A )
=> ~ ! [K4: int] :
( A
!= ( times_times_int @ B @ K4 ) ) ) ).
% dvdE
thf(fact_492_mult__0,axiom,
! [N: nat] :
( ( times_times_nat @ zero_zero_nat @ N )
= zero_zero_nat ) ).
% mult_0
thf(fact_493_power__mult,axiom,
! [A: nat,M: nat,N: nat] :
( ( power_power_nat @ A @ ( times_times_nat @ M @ N ) )
= ( power_power_nat @ ( power_power_nat @ A @ M ) @ N ) ) ).
% power_mult
thf(fact_494_power__mult,axiom,
! [A: int,M: nat,N: nat] :
( ( power_power_int @ A @ ( times_times_nat @ M @ N ) )
= ( power_power_int @ ( power_power_int @ A @ M ) @ N ) ) ).
% power_mult
thf(fact_495_power__mult,axiom,
! [A: real,M: nat,N: nat] :
( ( power_power_real @ A @ ( times_times_nat @ M @ N ) )
= ( power_power_real @ ( power_power_real @ A @ M ) @ N ) ) ).
% power_mult
thf(fact_496_nat__mult__1__right,axiom,
! [N: nat] :
( ( times_times_nat @ N @ one_one_nat )
= N ) ).
% nat_mult_1_right
thf(fact_497_nat__mult__1,axiom,
! [N: nat] :
( ( times_times_nat @ one_one_nat @ N )
= N ) ).
% nat_mult_1
thf(fact_498_primepow__multD_I2_J,axiom,
! [A: nat,B: nat] :
( ( prime_primepow_nat @ ( times_times_nat @ A @ B ) )
=> ( ( B = one_one_nat )
| ( prime_primepow_nat @ B ) ) ) ).
% primepow_multD(2)
thf(fact_499_primepow__multD_I1_J,axiom,
! [A: nat,B: nat] :
( ( prime_primepow_nat @ ( times_times_nat @ A @ B ) )
=> ( ( A = one_one_nat )
| ( prime_primepow_nat @ A ) ) ) ).
% primepow_multD(1)
thf(fact_500_diff__mult__distrib,axiom,
! [M: nat,N: nat,K: nat] :
( ( times_times_nat @ ( minus_minus_nat @ M @ N ) @ K )
= ( minus_minus_nat @ ( times_times_nat @ M @ K ) @ ( times_times_nat @ N @ K ) ) ) ).
% diff_mult_distrib
thf(fact_501_diff__mult__distrib2,axiom,
! [K: nat,M: nat,N: nat] :
( ( times_times_nat @ K @ ( minus_minus_nat @ M @ N ) )
= ( minus_minus_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) ) ) ).
% diff_mult_distrib2
thf(fact_502_cong__scalar__right,axiom,
! [B: nat,C: nat,A: nat,D: nat] :
( ( unique653641344996303876ng_nat @ B @ C @ A )
=> ( unique653641344996303876ng_nat @ ( times_times_nat @ B @ D ) @ ( times_times_nat @ C @ D ) @ A ) ) ).
% cong_scalar_right
thf(fact_503_cong__scalar__right,axiom,
! [B: int,C: int,A: int,D: int] :
( ( unique651150874487253600ng_int @ B @ C @ A )
=> ( unique651150874487253600ng_int @ ( times_times_int @ B @ D ) @ ( times_times_int @ C @ D ) @ A ) ) ).
% cong_scalar_right
thf(fact_504_cong__scalar__left,axiom,
! [B: nat,C: nat,A: nat,D: nat] :
( ( unique653641344996303876ng_nat @ B @ C @ A )
=> ( unique653641344996303876ng_nat @ ( times_times_nat @ D @ B ) @ ( times_times_nat @ D @ C ) @ A ) ) ).
% cong_scalar_left
thf(fact_505_cong__scalar__left,axiom,
! [B: int,C: int,A: int,D: int] :
( ( unique651150874487253600ng_int @ B @ C @ A )
=> ( unique651150874487253600ng_int @ ( times_times_int @ D @ B ) @ ( times_times_int @ D @ C ) @ A ) ) ).
% cong_scalar_left
thf(fact_506_cong__modulus__mult,axiom,
! [X: int,Y: int,M: int,N: int] :
( ( unique651150874487253600ng_int @ X @ Y @ ( times_times_int @ M @ N ) )
=> ( unique651150874487253600ng_int @ X @ Y @ M ) ) ).
% cong_modulus_mult
thf(fact_507_cong__mult,axiom,
! [B: nat,C: nat,A: nat,D: nat,E: nat] :
( ( unique653641344996303876ng_nat @ B @ C @ A )
=> ( ( unique653641344996303876ng_nat @ D @ E @ A )
=> ( unique653641344996303876ng_nat @ ( times_times_nat @ B @ D ) @ ( times_times_nat @ C @ E ) @ A ) ) ) ).
% cong_mult
thf(fact_508_cong__mult,axiom,
! [B: int,C: int,A: int,D: int,E: int] :
( ( unique651150874487253600ng_int @ B @ C @ A )
=> ( ( unique651150874487253600ng_int @ D @ E @ A )
=> ( unique651150874487253600ng_int @ ( times_times_int @ B @ D ) @ ( times_times_int @ C @ E ) @ A ) ) ) ).
% cong_mult
thf(fact_509_cong__modulus__mult__nat,axiom,
! [X: nat,Y: nat,M: nat,N: nat] :
( ( unique653641344996303876ng_nat @ X @ Y @ ( times_times_nat @ M @ N ) )
=> ( unique653641344996303876ng_nat @ X @ Y @ M ) ) ).
% cong_modulus_mult_nat
thf(fact_510_idom__class_Ounit__mult__right__cancel,axiom,
! [A: real,B: real,C: real] :
( ( dvd_dvd_real @ A @ one_one_real )
=> ( ( ( times_times_real @ B @ A )
= ( times_times_real @ C @ A ) )
= ( B = C ) ) ) ).
% idom_class.unit_mult_right_cancel
thf(fact_511_idom__class_Ounit__mult__right__cancel,axiom,
! [A: int,B: int,C: int] :
( ( dvd_dvd_int @ A @ one_one_int )
=> ( ( ( times_times_int @ B @ A )
= ( times_times_int @ C @ A ) )
= ( B = C ) ) ) ).
% idom_class.unit_mult_right_cancel
thf(fact_512_idom__class_Ounit__mult__left__cancel,axiom,
! [A: real,B: real,C: real] :
( ( dvd_dvd_real @ A @ one_one_real )
=> ( ( ( times_times_real @ A @ B )
= ( times_times_real @ A @ C ) )
= ( B = C ) ) ) ).
% idom_class.unit_mult_left_cancel
thf(fact_513_idom__class_Ounit__mult__left__cancel,axiom,
! [A: int,B: int,C: int] :
( ( dvd_dvd_int @ A @ one_one_int )
=> ( ( ( times_times_int @ A @ B )
= ( times_times_int @ A @ C ) )
= ( B = C ) ) ) ).
% idom_class.unit_mult_left_cancel
thf(fact_514_idom__class_Odvd__mult__unit__iff_H,axiom,
! [B: real,A: real,C: real] :
( ( dvd_dvd_real @ B @ one_one_real )
=> ( ( dvd_dvd_real @ A @ ( times_times_real @ B @ C ) )
= ( dvd_dvd_real @ A @ C ) ) ) ).
% idom_class.dvd_mult_unit_iff'
thf(fact_515_idom__class_Odvd__mult__unit__iff_H,axiom,
! [B: int,A: int,C: int] :
( ( dvd_dvd_int @ B @ one_one_int )
=> ( ( dvd_dvd_int @ A @ ( times_times_int @ B @ C ) )
= ( dvd_dvd_int @ A @ C ) ) ) ).
% idom_class.dvd_mult_unit_iff'
thf(fact_516_idom__class_Odvd__mult__unit__iff,axiom,
! [B: real,A: real,C: real] :
( ( dvd_dvd_real @ B @ one_one_real )
=> ( ( dvd_dvd_real @ A @ ( times_times_real @ C @ B ) )
= ( dvd_dvd_real @ A @ C ) ) ) ).
% idom_class.dvd_mult_unit_iff
thf(fact_517_idom__class_Odvd__mult__unit__iff,axiom,
! [B: int,A: int,C: int] :
( ( dvd_dvd_int @ B @ one_one_int )
=> ( ( dvd_dvd_int @ A @ ( times_times_int @ C @ B ) )
= ( dvd_dvd_int @ A @ C ) ) ) ).
% idom_class.dvd_mult_unit_iff
thf(fact_518_aprimedivisor__primepow_I1_J,axiom,
! [P: nat,N: nat] :
( ( factor1801147406995305544me_nat @ P )
=> ( ( dvd_dvd_nat @ P @ N )
=> ( ( prime_primepow_nat @ N )
=> ( ( prime_1889911587691200368or_nat @ ( times_times_nat @ P @ N ) )
= P ) ) ) ) ).
% aprimedivisor_primepow(1)
thf(fact_519_aprimedivisor__primepow_I1_J,axiom,
! [P: int,N: int] :
( ( factor1798656936486255268me_int @ P )
=> ( ( dvd_dvd_int @ P @ N )
=> ( ( prime_primepow_int @ N )
=> ( ( prime_1887421117182150092or_int @ ( times_times_int @ P @ N ) )
= P ) ) ) ) ).
% aprimedivisor_primepow(1)
thf(fact_520_mult__neg__neg,axiom,
! [A: int,B: int] :
( ( ord_less_int @ A @ zero_zero_int )
=> ( ( ord_less_int @ B @ zero_zero_int )
=> ( ord_less_int @ zero_zero_int @ ( times_times_int @ A @ B ) ) ) ) ).
% mult_neg_neg
thf(fact_521_mult__neg__neg,axiom,
! [A: real,B: real] :
( ( ord_less_real @ A @ zero_zero_real )
=> ( ( ord_less_real @ B @ zero_zero_real )
=> ( ord_less_real @ zero_zero_real @ ( times_times_real @ A @ B ) ) ) ) ).
% mult_neg_neg
thf(fact_522_not__square__less__zero,axiom,
! [A: int] :
~ ( ord_less_int @ ( times_times_int @ A @ A ) @ zero_zero_int ) ).
% not_square_less_zero
thf(fact_523_not__square__less__zero,axiom,
! [A: real] :
~ ( ord_less_real @ ( times_times_real @ A @ A ) @ zero_zero_real ) ).
% not_square_less_zero
thf(fact_524_mult__less__0__iff,axiom,
! [A: int,B: int] :
( ( ord_less_int @ ( times_times_int @ A @ B ) @ zero_zero_int )
= ( ( ( ord_less_int @ zero_zero_int @ A )
& ( ord_less_int @ B @ zero_zero_int ) )
| ( ( ord_less_int @ A @ zero_zero_int )
& ( ord_less_int @ zero_zero_int @ B ) ) ) ) ).
% mult_less_0_iff
thf(fact_525_mult__less__0__iff,axiom,
! [A: real,B: real] :
( ( ord_less_real @ ( times_times_real @ A @ B ) @ zero_zero_real )
= ( ( ( ord_less_real @ zero_zero_real @ A )
& ( ord_less_real @ B @ zero_zero_real ) )
| ( ( ord_less_real @ A @ zero_zero_real )
& ( ord_less_real @ zero_zero_real @ B ) ) ) ) ).
% mult_less_0_iff
thf(fact_526_mult__neg__pos,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ zero_zero_nat )
=> ( ( ord_less_nat @ zero_zero_nat @ B )
=> ( ord_less_nat @ ( times_times_nat @ A @ B ) @ zero_zero_nat ) ) ) ).
% mult_neg_pos
thf(fact_527_mult__neg__pos,axiom,
! [A: int,B: int] :
( ( ord_less_int @ A @ zero_zero_int )
=> ( ( ord_less_int @ zero_zero_int @ B )
=> ( ord_less_int @ ( times_times_int @ A @ B ) @ zero_zero_int ) ) ) ).
% mult_neg_pos
thf(fact_528_mult__neg__pos,axiom,
! [A: real,B: real] :
( ( ord_less_real @ A @ zero_zero_real )
=> ( ( ord_less_real @ zero_zero_real @ B )
=> ( ord_less_real @ ( times_times_real @ A @ B ) @ zero_zero_real ) ) ) ).
% mult_neg_pos
thf(fact_529_mult__pos__neg,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ zero_zero_nat @ A )
=> ( ( ord_less_nat @ B @ zero_zero_nat )
=> ( ord_less_nat @ ( times_times_nat @ A @ B ) @ zero_zero_nat ) ) ) ).
% mult_pos_neg
thf(fact_530_mult__pos__neg,axiom,
! [A: int,B: int] :
( ( ord_less_int @ zero_zero_int @ A )
=> ( ( ord_less_int @ B @ zero_zero_int )
=> ( ord_less_int @ ( times_times_int @ A @ B ) @ zero_zero_int ) ) ) ).
% mult_pos_neg
thf(fact_531_mult__pos__neg,axiom,
! [A: real,B: real] :
( ( ord_less_real @ zero_zero_real @ A )
=> ( ( ord_less_real @ B @ zero_zero_real )
=> ( ord_less_real @ ( times_times_real @ A @ B ) @ zero_zero_real ) ) ) ).
% mult_pos_neg
thf(fact_532_mult__pos__pos,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ zero_zero_nat @ A )
=> ( ( ord_less_nat @ zero_zero_nat @ B )
=> ( ord_less_nat @ zero_zero_nat @ ( times_times_nat @ A @ B ) ) ) ) ).
% mult_pos_pos
thf(fact_533_mult__pos__pos,axiom,
! [A: int,B: int] :
( ( ord_less_int @ zero_zero_int @ A )
=> ( ( ord_less_int @ zero_zero_int @ B )
=> ( ord_less_int @ zero_zero_int @ ( times_times_int @ A @ B ) ) ) ) ).
% mult_pos_pos
thf(fact_534_mult__pos__pos,axiom,
! [A: real,B: real] :
( ( ord_less_real @ zero_zero_real @ A )
=> ( ( ord_less_real @ zero_zero_real @ B )
=> ( ord_less_real @ zero_zero_real @ ( times_times_real @ A @ B ) ) ) ) ).
% mult_pos_pos
thf(fact_535_mult__pos__neg2,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ zero_zero_nat @ A )
=> ( ( ord_less_nat @ B @ zero_zero_nat )
=> ( ord_less_nat @ ( times_times_nat @ B @ A ) @ zero_zero_nat ) ) ) ).
% mult_pos_neg2
thf(fact_536_mult__pos__neg2,axiom,
! [A: int,B: int] :
( ( ord_less_int @ zero_zero_int @ A )
=> ( ( ord_less_int @ B @ zero_zero_int )
=> ( ord_less_int @ ( times_times_int @ B @ A ) @ zero_zero_int ) ) ) ).
% mult_pos_neg2
thf(fact_537_mult__pos__neg2,axiom,
! [A: real,B: real] :
( ( ord_less_real @ zero_zero_real @ A )
=> ( ( ord_less_real @ B @ zero_zero_real )
=> ( ord_less_real @ ( times_times_real @ B @ A ) @ zero_zero_real ) ) ) ).
% mult_pos_neg2
thf(fact_538_zero__less__mult__iff,axiom,
! [A: int,B: int] :
( ( ord_less_int @ zero_zero_int @ ( times_times_int @ A @ B ) )
= ( ( ( ord_less_int @ zero_zero_int @ A )
& ( ord_less_int @ zero_zero_int @ B ) )
| ( ( ord_less_int @ A @ zero_zero_int )
& ( ord_less_int @ B @ zero_zero_int ) ) ) ) ).
% zero_less_mult_iff
thf(fact_539_zero__less__mult__iff,axiom,
! [A: real,B: real] :
( ( ord_less_real @ zero_zero_real @ ( times_times_real @ A @ B ) )
= ( ( ( ord_less_real @ zero_zero_real @ A )
& ( ord_less_real @ zero_zero_real @ B ) )
| ( ( ord_less_real @ A @ zero_zero_real )
& ( ord_less_real @ B @ zero_zero_real ) ) ) ) ).
% zero_less_mult_iff
thf(fact_540_zero__less__mult__pos,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( times_times_nat @ A @ B ) )
=> ( ( ord_less_nat @ zero_zero_nat @ A )
=> ( ord_less_nat @ zero_zero_nat @ B ) ) ) ).
% zero_less_mult_pos
thf(fact_541_zero__less__mult__pos,axiom,
! [A: int,B: int] :
( ( ord_less_int @ zero_zero_int @ ( times_times_int @ A @ B ) )
=> ( ( ord_less_int @ zero_zero_int @ A )
=> ( ord_less_int @ zero_zero_int @ B ) ) ) ).
% zero_less_mult_pos
thf(fact_542_zero__less__mult__pos,axiom,
! [A: real,B: real] :
( ( ord_less_real @ zero_zero_real @ ( times_times_real @ A @ B ) )
=> ( ( ord_less_real @ zero_zero_real @ A )
=> ( ord_less_real @ zero_zero_real @ B ) ) ) ).
% zero_less_mult_pos
thf(fact_543_zero__less__mult__pos2,axiom,
! [B: nat,A: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( times_times_nat @ B @ A ) )
=> ( ( ord_less_nat @ zero_zero_nat @ A )
=> ( ord_less_nat @ zero_zero_nat @ B ) ) ) ).
% zero_less_mult_pos2
thf(fact_544_zero__less__mult__pos2,axiom,
! [B: int,A: int] :
( ( ord_less_int @ zero_zero_int @ ( times_times_int @ B @ A ) )
=> ( ( ord_less_int @ zero_zero_int @ A )
=> ( ord_less_int @ zero_zero_int @ B ) ) ) ).
% zero_less_mult_pos2
thf(fact_545_zero__less__mult__pos2,axiom,
! [B: real,A: real] :
( ( ord_less_real @ zero_zero_real @ ( times_times_real @ B @ A ) )
=> ( ( ord_less_real @ zero_zero_real @ A )
=> ( ord_less_real @ zero_zero_real @ B ) ) ) ).
% zero_less_mult_pos2
thf(fact_546_mult__less__cancel__left__neg,axiom,
! [C: int,A: int,B: int] :
( ( ord_less_int @ C @ zero_zero_int )
=> ( ( ord_less_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) )
= ( ord_less_int @ B @ A ) ) ) ).
% mult_less_cancel_left_neg
thf(fact_547_mult__less__cancel__left__neg,axiom,
! [C: real,A: real,B: real] :
( ( ord_less_real @ C @ zero_zero_real )
=> ( ( ord_less_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B ) )
= ( ord_less_real @ B @ A ) ) ) ).
% mult_less_cancel_left_neg
thf(fact_548_mult__less__cancel__left__pos,axiom,
! [C: int,A: int,B: int] :
( ( ord_less_int @ zero_zero_int @ C )
=> ( ( ord_less_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) )
= ( ord_less_int @ A @ B ) ) ) ).
% mult_less_cancel_left_pos
thf(fact_549_mult__less__cancel__left__pos,axiom,
! [C: real,A: real,B: real] :
( ( ord_less_real @ zero_zero_real @ C )
=> ( ( ord_less_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B ) )
= ( ord_less_real @ A @ B ) ) ) ).
% mult_less_cancel_left_pos
thf(fact_550_mult__strict__left__mono__neg,axiom,
! [B: int,A: int,C: int] :
( ( ord_less_int @ B @ A )
=> ( ( ord_less_int @ C @ zero_zero_int )
=> ( ord_less_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) ) ) ) ).
% mult_strict_left_mono_neg
thf(fact_551_mult__strict__left__mono__neg,axiom,
! [B: real,A: real,C: real] :
( ( ord_less_real @ B @ A )
=> ( ( ord_less_real @ C @ zero_zero_real )
=> ( ord_less_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B ) ) ) ) ).
% mult_strict_left_mono_neg
thf(fact_552_mult__strict__left__mono,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_nat @ zero_zero_nat @ C )
=> ( ord_less_nat @ ( times_times_nat @ C @ A ) @ ( times_times_nat @ C @ B ) ) ) ) ).
% mult_strict_left_mono
thf(fact_553_mult__strict__left__mono,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_int @ A @ B )
=> ( ( ord_less_int @ zero_zero_int @ C )
=> ( ord_less_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) ) ) ) ).
% mult_strict_left_mono
thf(fact_554_mult__strict__left__mono,axiom,
! [A: real,B: real,C: real] :
( ( ord_less_real @ A @ B )
=> ( ( ord_less_real @ zero_zero_real @ C )
=> ( ord_less_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B ) ) ) ) ).
% mult_strict_left_mono
thf(fact_555_mult__less__cancel__left__disj,axiom,
! [C: int,A: int,B: int] :
( ( ord_less_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) )
= ( ( ( ord_less_int @ zero_zero_int @ C )
& ( ord_less_int @ A @ B ) )
| ( ( ord_less_int @ C @ zero_zero_int )
& ( ord_less_int @ B @ A ) ) ) ) ).
% mult_less_cancel_left_disj
thf(fact_556_mult__less__cancel__left__disj,axiom,
! [C: real,A: real,B: real] :
( ( ord_less_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B ) )
= ( ( ( ord_less_real @ zero_zero_real @ C )
& ( ord_less_real @ A @ B ) )
| ( ( ord_less_real @ C @ zero_zero_real )
& ( ord_less_real @ B @ A ) ) ) ) ).
% mult_less_cancel_left_disj
thf(fact_557_mult__strict__right__mono__neg,axiom,
! [B: int,A: int,C: int] :
( ( ord_less_int @ B @ A )
=> ( ( ord_less_int @ C @ zero_zero_int )
=> ( ord_less_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) ) ) ) ).
% mult_strict_right_mono_neg
thf(fact_558_mult__strict__right__mono__neg,axiom,
! [B: real,A: real,C: real] :
( ( ord_less_real @ B @ A )
=> ( ( ord_less_real @ C @ zero_zero_real )
=> ( ord_less_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ C ) ) ) ) ).
% mult_strict_right_mono_neg
thf(fact_559_mult__strict__right__mono,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_nat @ zero_zero_nat @ C )
=> ( ord_less_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ C ) ) ) ) ).
% mult_strict_right_mono
thf(fact_560_mult__strict__right__mono,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_int @ A @ B )
=> ( ( ord_less_int @ zero_zero_int @ C )
=> ( ord_less_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) ) ) ) ).
% mult_strict_right_mono
thf(fact_561_mult__strict__right__mono,axiom,
! [A: real,B: real,C: real] :
( ( ord_less_real @ A @ B )
=> ( ( ord_less_real @ zero_zero_real @ C )
=> ( ord_less_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ C ) ) ) ) ).
% mult_strict_right_mono
thf(fact_562_mult__less__cancel__right__disj,axiom,
! [A: int,C: int,B: int] :
( ( ord_less_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) )
= ( ( ( ord_less_int @ zero_zero_int @ C )
& ( ord_less_int @ A @ B ) )
| ( ( ord_less_int @ C @ zero_zero_int )
& ( ord_less_int @ B @ A ) ) ) ) ).
% mult_less_cancel_right_disj
thf(fact_563_mult__less__cancel__right__disj,axiom,
! [A: real,C: real,B: real] :
( ( ord_less_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ C ) )
= ( ( ( ord_less_real @ zero_zero_real @ C )
& ( ord_less_real @ A @ B ) )
| ( ( ord_less_real @ C @ zero_zero_real )
& ( ord_less_real @ B @ A ) ) ) ) ).
% mult_less_cancel_right_disj
thf(fact_564_linordered__comm__semiring__strict__class_Ocomm__mult__strict__left__mono,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_nat @ zero_zero_nat @ C )
=> ( ord_less_nat @ ( times_times_nat @ C @ A ) @ ( times_times_nat @ C @ B ) ) ) ) ).
% linordered_comm_semiring_strict_class.comm_mult_strict_left_mono
thf(fact_565_linordered__comm__semiring__strict__class_Ocomm__mult__strict__left__mono,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_int @ A @ B )
=> ( ( ord_less_int @ zero_zero_int @ C )
=> ( ord_less_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) ) ) ) ).
% linordered_comm_semiring_strict_class.comm_mult_strict_left_mono
thf(fact_566_linordered__comm__semiring__strict__class_Ocomm__mult__strict__left__mono,axiom,
! [A: real,B: real,C: real] :
( ( ord_less_real @ A @ B )
=> ( ( ord_less_real @ zero_zero_real @ C )
=> ( ord_less_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B ) ) ) ) ).
% linordered_comm_semiring_strict_class.comm_mult_strict_left_mono
thf(fact_567_less__1__mult,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ one_one_nat @ M )
=> ( ( ord_less_nat @ one_one_nat @ N )
=> ( ord_less_nat @ one_one_nat @ ( times_times_nat @ M @ N ) ) ) ) ).
% less_1_mult
thf(fact_568_less__1__mult,axiom,
! [M: int,N: int] :
( ( ord_less_int @ one_one_int @ M )
=> ( ( ord_less_int @ one_one_int @ N )
=> ( ord_less_int @ one_one_int @ ( times_times_int @ M @ N ) ) ) ) ).
% less_1_mult
thf(fact_569_less__1__mult,axiom,
! [M: real,N: real] :
( ( ord_less_real @ one_one_real @ M )
=> ( ( ord_less_real @ one_one_real @ N )
=> ( ord_less_real @ one_one_real @ ( times_times_real @ M @ N ) ) ) ) ).
% less_1_mult
thf(fact_570_left__right__inverse__power,axiom,
! [X: nat,Y: nat,N: nat] :
( ( ( times_times_nat @ X @ Y )
= one_one_nat )
=> ( ( times_times_nat @ ( power_power_nat @ X @ N ) @ ( power_power_nat @ Y @ N ) )
= one_one_nat ) ) ).
% left_right_inverse_power
thf(fact_571_left__right__inverse__power,axiom,
! [X: real,Y: real,N: nat] :
( ( ( times_times_real @ X @ Y )
= one_one_real )
=> ( ( times_times_real @ ( power_power_real @ X @ N ) @ ( power_power_real @ Y @ N ) )
= one_one_real ) ) ).
% left_right_inverse_power
thf(fact_572_left__right__inverse__power,axiom,
! [X: int,Y: int,N: nat] :
( ( ( times_times_int @ X @ Y )
= one_one_int )
=> ( ( times_times_int @ ( power_power_int @ X @ N ) @ ( power_power_int @ Y @ N ) )
= one_one_int ) ) ).
% left_right_inverse_power
thf(fact_573_algebraic__semidom__class_Ounit__mult__right__cancel,axiom,
! [A: nat,B: nat,C: nat] :
( ( dvd_dvd_nat @ A @ one_one_nat )
=> ( ( ( times_times_nat @ B @ A )
= ( times_times_nat @ C @ A ) )
= ( B = C ) ) ) ).
% algebraic_semidom_class.unit_mult_right_cancel
thf(fact_574_algebraic__semidom__class_Ounit__mult__right__cancel,axiom,
! [A: int,B: int,C: int] :
( ( dvd_dvd_int @ A @ one_one_int )
=> ( ( ( times_times_int @ B @ A )
= ( times_times_int @ C @ A ) )
= ( B = C ) ) ) ).
% algebraic_semidom_class.unit_mult_right_cancel
thf(fact_575_algebraic__semidom__class_Ounit__mult__left__cancel,axiom,
! [A: nat,B: nat,C: nat] :
( ( dvd_dvd_nat @ A @ one_one_nat )
=> ( ( ( times_times_nat @ A @ B )
= ( times_times_nat @ A @ C ) )
= ( B = C ) ) ) ).
% algebraic_semidom_class.unit_mult_left_cancel
thf(fact_576_algebraic__semidom__class_Ounit__mult__left__cancel,axiom,
! [A: int,B: int,C: int] :
( ( dvd_dvd_int @ A @ one_one_int )
=> ( ( ( times_times_int @ A @ B )
= ( times_times_int @ A @ C ) )
= ( B = C ) ) ) ).
% algebraic_semidom_class.unit_mult_left_cancel
thf(fact_577_algebraic__semidom__class_Omult__unit__dvd__iff_H,axiom,
! [A: nat,B: nat,C: nat] :
( ( dvd_dvd_nat @ A @ one_one_nat )
=> ( ( dvd_dvd_nat @ ( times_times_nat @ A @ B ) @ C )
= ( dvd_dvd_nat @ B @ C ) ) ) ).
% algebraic_semidom_class.mult_unit_dvd_iff'
thf(fact_578_algebraic__semidom__class_Omult__unit__dvd__iff_H,axiom,
! [A: int,B: int,C: int] :
( ( dvd_dvd_int @ A @ one_one_int )
=> ( ( dvd_dvd_int @ ( times_times_int @ A @ B ) @ C )
= ( dvd_dvd_int @ B @ C ) ) ) ).
% algebraic_semidom_class.mult_unit_dvd_iff'
thf(fact_579_algebraic__semidom__class_Odvd__mult__unit__iff_H,axiom,
! [B: nat,A: nat,C: nat] :
( ( dvd_dvd_nat @ B @ one_one_nat )
=> ( ( dvd_dvd_nat @ A @ ( times_times_nat @ B @ C ) )
= ( dvd_dvd_nat @ A @ C ) ) ) ).
% algebraic_semidom_class.dvd_mult_unit_iff'
thf(fact_580_algebraic__semidom__class_Odvd__mult__unit__iff_H,axiom,
! [B: int,A: int,C: int] :
( ( dvd_dvd_int @ B @ one_one_int )
=> ( ( dvd_dvd_int @ A @ ( times_times_int @ B @ C ) )
= ( dvd_dvd_int @ A @ C ) ) ) ).
% algebraic_semidom_class.dvd_mult_unit_iff'
thf(fact_581_algebraic__semidom__class_Omult__unit__dvd__iff,axiom,
! [B: nat,A: nat,C: nat] :
( ( dvd_dvd_nat @ B @ one_one_nat )
=> ( ( dvd_dvd_nat @ ( times_times_nat @ A @ B ) @ C )
= ( dvd_dvd_nat @ A @ C ) ) ) ).
% algebraic_semidom_class.mult_unit_dvd_iff
thf(fact_582_algebraic__semidom__class_Omult__unit__dvd__iff,axiom,
! [B: int,A: int,C: int] :
( ( dvd_dvd_int @ B @ one_one_int )
=> ( ( dvd_dvd_int @ ( times_times_int @ A @ B ) @ C )
= ( dvd_dvd_int @ A @ C ) ) ) ).
% algebraic_semidom_class.mult_unit_dvd_iff
thf(fact_583_algebraic__semidom__class_Odvd__mult__unit__iff,axiom,
! [B: nat,A: nat,C: nat] :
( ( dvd_dvd_nat @ B @ one_one_nat )
=> ( ( dvd_dvd_nat @ A @ ( times_times_nat @ C @ B ) )
= ( dvd_dvd_nat @ A @ C ) ) ) ).
% algebraic_semidom_class.dvd_mult_unit_iff
thf(fact_584_algebraic__semidom__class_Odvd__mult__unit__iff,axiom,
! [B: int,A: int,C: int] :
( ( dvd_dvd_int @ B @ one_one_int )
=> ( ( dvd_dvd_int @ A @ ( times_times_int @ C @ B ) )
= ( dvd_dvd_int @ A @ C ) ) ) ).
% algebraic_semidom_class.dvd_mult_unit_iff
thf(fact_585_algebraic__semidom__class_Ois__unit__mult__iff,axiom,
! [A: nat,B: nat] :
( ( dvd_dvd_nat @ ( times_times_nat @ A @ B ) @ one_one_nat )
= ( ( dvd_dvd_nat @ A @ one_one_nat )
& ( dvd_dvd_nat @ B @ one_one_nat ) ) ) ).
% algebraic_semidom_class.is_unit_mult_iff
thf(fact_586_algebraic__semidom__class_Ois__unit__mult__iff,axiom,
! [A: int,B: int] :
( ( dvd_dvd_int @ ( times_times_int @ A @ B ) @ one_one_int )
= ( ( dvd_dvd_int @ A @ one_one_int )
& ( dvd_dvd_int @ B @ one_one_int ) ) ) ).
% algebraic_semidom_class.is_unit_mult_iff
thf(fact_587_mult__less__mono1,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_nat @ I @ J )
=> ( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ord_less_nat @ ( times_times_nat @ I @ K ) @ ( times_times_nat @ J @ K ) ) ) ) ).
% mult_less_mono1
thf(fact_588_mult__less__mono2,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_nat @ I @ J )
=> ( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ord_less_nat @ ( times_times_nat @ K @ I ) @ ( times_times_nat @ K @ J ) ) ) ) ).
% mult_less_mono2
thf(fact_589_cong__mult__self__left,axiom,
! [A: nat,B: nat] : ( unique653641344996303876ng_nat @ ( times_times_nat @ A @ B ) @ zero_zero_nat @ A ) ).
% cong_mult_self_left
thf(fact_590_cong__mult__self__left,axiom,
! [A: int,B: int] : ( unique651150874487253600ng_int @ ( times_times_int @ A @ B ) @ zero_zero_int @ A ) ).
% cong_mult_self_left
thf(fact_591_cong__mult__self__right,axiom,
! [B: nat,A: nat] : ( unique653641344996303876ng_nat @ ( times_times_nat @ B @ A ) @ zero_zero_nat @ A ) ).
% cong_mult_self_right
thf(fact_592_cong__mult__self__right,axiom,
! [B: int,A: int] : ( unique651150874487253600ng_int @ ( times_times_int @ B @ A ) @ zero_zero_int @ A ) ).
% cong_mult_self_right
thf(fact_593_mult__eq__self__implies__10,axiom,
! [M: nat,N: nat] :
( ( M
= ( times_times_nat @ M @ N ) )
=> ( ( N = one_one_nat )
| ( M = zero_zero_nat ) ) ) ).
% mult_eq_self_implies_10
thf(fact_594_prime__dvd__mult__iff,axiom,
! [P: nat,A: nat,B: nat] :
( ( factor1801147406995305544me_nat @ P )
=> ( ( dvd_dvd_nat @ P @ ( times_times_nat @ A @ B ) )
= ( ( dvd_dvd_nat @ P @ A )
| ( dvd_dvd_nat @ P @ B ) ) ) ) ).
% prime_dvd_mult_iff
thf(fact_595_prime__dvd__mult__iff,axiom,
! [P: int,A: int,B: int] :
( ( factor1798656936486255268me_int @ P )
=> ( ( dvd_dvd_int @ P @ ( times_times_int @ A @ B ) )
= ( ( dvd_dvd_int @ P @ A )
| ( dvd_dvd_int @ P @ B ) ) ) ) ).
% prime_dvd_mult_iff
thf(fact_596_prime__dvd__multD,axiom,
! [P: nat,A: nat,B: nat] :
( ( factor1801147406995305544me_nat @ P )
=> ( ( dvd_dvd_nat @ P @ ( times_times_nat @ A @ B ) )
=> ( ( dvd_dvd_nat @ P @ A )
| ( dvd_dvd_nat @ P @ B ) ) ) ) ).
% prime_dvd_multD
thf(fact_597_prime__dvd__multD,axiom,
! [P: int,A: int,B: int] :
( ( factor1798656936486255268me_int @ P )
=> ( ( dvd_dvd_int @ P @ ( times_times_int @ A @ B ) )
=> ( ( dvd_dvd_int @ P @ A )
| ( dvd_dvd_int @ P @ B ) ) ) ) ).
% prime_dvd_multD
thf(fact_598_bezout1__nat,axiom,
! [A: nat,B: nat] :
? [D3: nat,X3: nat,Y4: nat] :
( ( dvd_dvd_nat @ D3 @ A )
& ( dvd_dvd_nat @ D3 @ B )
& ( ( ( minus_minus_nat @ ( times_times_nat @ A @ X3 ) @ ( times_times_nat @ B @ Y4 ) )
= D3 )
| ( ( minus_minus_nat @ ( times_times_nat @ B @ X3 ) @ ( times_times_nat @ A @ Y4 ) )
= D3 ) ) ) ).
% bezout1_nat
thf(fact_599_prime__product,axiom,
! [P: nat,Q: nat] :
( ( factor1801147406995305544me_nat @ ( times_times_nat @ P @ Q ) )
=> ( ( P = one_one_nat )
| ( Q = one_one_nat ) ) ) ).
% prime_product
thf(fact_600_prime__dvd__mult__eq__nat,axiom,
! [P: nat,A: nat,B: nat] :
( ( factor1801147406995305544me_nat @ P )
=> ( ( dvd_dvd_nat @ P @ ( times_times_nat @ A @ B ) )
= ( ( dvd_dvd_nat @ P @ A )
| ( dvd_dvd_nat @ P @ B ) ) ) ) ).
% prime_dvd_mult_eq_nat
thf(fact_601_prime__power__mult__nat,axiom,
! [P: nat,X: nat,Y: nat,K: nat] :
( ( factor1801147406995305544me_nat @ P )
=> ( ( ( times_times_nat @ X @ Y )
= ( power_power_nat @ P @ K ) )
=> ? [I2: nat,J2: nat] :
( ( X
= ( power_power_nat @ P @ I2 ) )
& ( Y
= ( power_power_nat @ P @ J2 ) ) ) ) ) ).
% prime_power_mult_nat
thf(fact_602_prime__dvd__mult__eq__int,axiom,
! [P: int,A: int,B: int] :
( ( factor1798656936486255268me_int @ P )
=> ( ( dvd_dvd_int @ P @ ( times_times_int @ A @ B ) )
= ( ( dvd_dvd_int @ P @ A )
| ( dvd_dvd_int @ P @ B ) ) ) ) ).
% prime_dvd_mult_eq_int
thf(fact_603_prime__aprimedivisor,axiom,
! [P: nat,N: nat] :
( ( factor1801147406995305544me_nat @ P )
=> ( ( dvd_dvd_nat @ P @ N )
=> ( factor1801147406995305544me_nat @ ( prime_1889911587691200368or_nat @ N ) ) ) ) ).
% prime_aprimedivisor
thf(fact_604_prime__aprimedivisor,axiom,
! [P: int,N: int] :
( ( factor1798656936486255268me_int @ P )
=> ( ( dvd_dvd_int @ P @ N )
=> ( factor1798656936486255268me_int @ ( prime_1887421117182150092or_int @ N ) ) ) ) ).
% prime_aprimedivisor
thf(fact_605_aprimedivisor__dvd,axiom,
! [P: nat,N: nat] :
( ( factor1801147406995305544me_nat @ P )
=> ( ( dvd_dvd_nat @ P @ N )
=> ( dvd_dvd_nat @ ( prime_1889911587691200368or_nat @ N ) @ N ) ) ) ).
% aprimedivisor_dvd
thf(fact_606_aprimedivisor__dvd,axiom,
! [P: int,N: int] :
( ( factor1798656936486255268me_int @ P )
=> ( ( dvd_dvd_int @ P @ N )
=> ( dvd_dvd_int @ ( prime_1887421117182150092or_int @ N ) @ N ) ) ) ).
% aprimedivisor_dvd
thf(fact_607_unit__dvdE,axiom,
! [A: nat,B: nat] :
( ( dvd_dvd_nat @ A @ one_one_nat )
=> ~ ( ( A != zero_zero_nat )
=> ! [C4: nat] :
( B
!= ( times_times_nat @ A @ C4 ) ) ) ) ).
% unit_dvdE
thf(fact_608_unit__dvdE,axiom,
! [A: int,B: int] :
( ( dvd_dvd_int @ A @ one_one_int )
=> ~ ( ( A != zero_zero_int )
=> ! [C4: int] :
( B
!= ( times_times_int @ A @ C4 ) ) ) ) ).
% unit_dvdE
thf(fact_609_power__less__power__Suc,axiom,
! [A: nat,N: nat] :
( ( ord_less_nat @ one_one_nat @ A )
=> ( ord_less_nat @ ( power_power_nat @ A @ N ) @ ( times_times_nat @ A @ ( power_power_nat @ A @ N ) ) ) ) ).
% power_less_power_Suc
thf(fact_610_power__less__power__Suc,axiom,
! [A: int,N: nat] :
( ( ord_less_int @ one_one_int @ A )
=> ( ord_less_int @ ( power_power_int @ A @ N ) @ ( times_times_int @ A @ ( power_power_int @ A @ N ) ) ) ) ).
% power_less_power_Suc
thf(fact_611_power__less__power__Suc,axiom,
! [A: real,N: nat] :
( ( ord_less_real @ one_one_real @ A )
=> ( ord_less_real @ ( power_power_real @ A @ N ) @ ( times_times_real @ A @ ( power_power_real @ A @ N ) ) ) ) ).
% power_less_power_Suc
thf(fact_612_power__gt1__lemma,axiom,
! [A: nat,N: nat] :
( ( ord_less_nat @ one_one_nat @ A )
=> ( ord_less_nat @ one_one_nat @ ( times_times_nat @ A @ ( power_power_nat @ A @ N ) ) ) ) ).
% power_gt1_lemma
thf(fact_613_power__gt1__lemma,axiom,
! [A: int,N: nat] :
( ( ord_less_int @ one_one_int @ A )
=> ( ord_less_int @ one_one_int @ ( times_times_int @ A @ ( power_power_int @ A @ N ) ) ) ) ).
% power_gt1_lemma
thf(fact_614_power__gt1__lemma,axiom,
! [A: real,N: nat] :
( ( ord_less_real @ one_one_real @ A )
=> ( ord_less_real @ one_one_real @ ( times_times_real @ A @ ( power_power_real @ A @ N ) ) ) ) ).
% power_gt1_lemma
thf(fact_615_linordered__field__no__lb,axiom,
! [X7: real] :
? [Y4: real] : ( ord_less_real @ Y4 @ X7 ) ).
% linordered_field_no_lb
thf(fact_616_linordered__field__no__ub,axiom,
! [X7: real] :
? [X_1: real] : ( ord_less_real @ X7 @ X_1 ) ).
% linordered_field_no_ub
thf(fact_617_prime__power__cancel2,axiom,
! [P: nat,M: nat,K: nat,M4: nat,K5: nat] :
( ( factor1801147406995305544me_nat @ P )
=> ( ( ( times_times_nat @ M @ ( power_power_nat @ P @ K ) )
= ( times_times_nat @ M4 @ ( power_power_nat @ P @ K5 ) ) )
=> ( ~ ( dvd_dvd_nat @ P @ M )
=> ( ~ ( dvd_dvd_nat @ P @ M4 )
=> ~ ( ( M = M4 )
=> ( K != K5 ) ) ) ) ) ) ).
% prime_power_cancel2
thf(fact_618_prime__power__cancel2,axiom,
! [P: int,M: int,K: nat,M4: int,K5: nat] :
( ( factor1798656936486255268me_int @ P )
=> ( ( ( times_times_int @ M @ ( power_power_int @ P @ K ) )
= ( times_times_int @ M4 @ ( power_power_int @ P @ K5 ) ) )
=> ( ~ ( dvd_dvd_int @ P @ M )
=> ( ~ ( dvd_dvd_int @ P @ M4 )
=> ~ ( ( M = M4 )
=> ( K != K5 ) ) ) ) ) ) ).
% prime_power_cancel2
thf(fact_619_prime__power__cancel,axiom,
! [P: nat,M: nat,K: nat,M4: nat,K5: nat] :
( ( factor1801147406995305544me_nat @ P )
=> ( ( ( times_times_nat @ M @ ( power_power_nat @ P @ K ) )
= ( times_times_nat @ M4 @ ( power_power_nat @ P @ K5 ) ) )
=> ( ~ ( dvd_dvd_nat @ P @ M )
=> ( ~ ( dvd_dvd_nat @ P @ M4 )
=> ( K = K5 ) ) ) ) ) ).
% prime_power_cancel
thf(fact_620_prime__power__cancel,axiom,
! [P: int,M: int,K: nat,M4: int,K5: nat] :
( ( factor1798656936486255268me_int @ P )
=> ( ( ( times_times_int @ M @ ( power_power_int @ P @ K ) )
= ( times_times_int @ M4 @ ( power_power_int @ P @ K5 ) ) )
=> ( ~ ( dvd_dvd_int @ P @ M )
=> ( ~ ( dvd_dvd_int @ P @ M4 )
=> ( K = K5 ) ) ) ) ) ).
% prime_power_cancel
thf(fact_621_dvd__mult__cancel,axiom,
! [K: nat,M: nat,N: nat] :
( ( dvd_dvd_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) )
=> ( ( ord_less_nat @ zero_zero_nat @ K )
=> ( dvd_dvd_nat @ M @ N ) ) ) ).
% dvd_mult_cancel
thf(fact_622_not__prime__eq__prod__nat,axiom,
! [M: nat] :
( ( ord_less_nat @ one_one_nat @ M )
=> ( ~ ( factor1801147406995305544me_nat @ M )
=> ? [N3: nat,K4: nat] :
( ( N3
= ( times_times_nat @ M @ K4 ) )
& ( ord_less_nat @ one_one_nat @ M )
& ( ord_less_nat @ M @ N3 )
& ( ord_less_nat @ one_one_nat @ K4 )
& ( ord_less_nat @ K4 @ N3 ) ) ) ) ).
% not_prime_eq_prod_nat
thf(fact_623_aprimedivisor__pos__nat,axiom,
! [N: nat] :
( ( ord_less_nat @ one_one_nat @ N )
=> ( ord_less_nat @ zero_zero_nat @ ( prime_1889911587691200368or_nat @ N ) ) ) ).
% aprimedivisor_pos_nat
thf(fact_624_power__Suc__less,axiom,
! [A: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ A )
=> ( ( ord_less_nat @ A @ one_one_nat )
=> ( ord_less_nat @ ( times_times_nat @ A @ ( power_power_nat @ A @ N ) ) @ ( power_power_nat @ A @ N ) ) ) ) ).
% power_Suc_less
thf(fact_625_power__Suc__less,axiom,
! [A: int,N: nat] :
( ( ord_less_int @ zero_zero_int @ A )
=> ( ( ord_less_int @ A @ one_one_int )
=> ( ord_less_int @ ( times_times_int @ A @ ( power_power_int @ A @ N ) ) @ ( power_power_int @ A @ N ) ) ) ) ).
% power_Suc_less
thf(fact_626_power__Suc__less,axiom,
! [A: real,N: nat] :
( ( ord_less_real @ zero_zero_real @ A )
=> ( ( ord_less_real @ A @ one_one_real )
=> ( ord_less_real @ ( times_times_real @ A @ ( power_power_real @ A @ N ) ) @ ( power_power_real @ A @ N ) ) ) ) ).
% power_Suc_less
thf(fact_627_aprimedivisor__dvd_H,axiom,
! [N: int] :
( ( N != zero_zero_int )
=> ( ~ ( dvd_dvd_int @ N @ one_one_int )
=> ( dvd_dvd_int @ ( prime_1887421117182150092or_int @ N ) @ N ) ) ) ).
% aprimedivisor_dvd'
thf(fact_628_aprimedivisor__dvd_H,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
=> ( ~ ( dvd_dvd_nat @ N @ one_one_nat )
=> ( dvd_dvd_nat @ ( prime_1889911587691200368or_nat @ N ) @ N ) ) ) ).
% aprimedivisor_dvd'
thf(fact_629_prime__divisors__induct,axiom,
! [P2: nat > $o,X: nat] :
( ( P2 @ zero_zero_nat )
=> ( ! [X3: nat] :
( ( dvd_dvd_nat @ X3 @ one_one_nat )
=> ( P2 @ X3 ) )
=> ( ! [P5: nat,X3: nat] :
( ( factor1801147406995305544me_nat @ P5 )
=> ( ( P2 @ X3 )
=> ( P2 @ ( times_times_nat @ P5 @ X3 ) ) ) )
=> ( P2 @ X ) ) ) ) ).
% prime_divisors_induct
thf(fact_630_prime__divisors__induct,axiom,
! [P2: int > $o,X: int] :
( ( P2 @ zero_zero_int )
=> ( ! [X3: int] :
( ( dvd_dvd_int @ X3 @ one_one_int )
=> ( P2 @ X3 ) )
=> ( ! [P5: int,X3: int] :
( ( factor1798656936486255268me_int @ P5 )
=> ( ( P2 @ X3 )
=> ( P2 @ ( times_times_int @ P5 @ X3 ) ) ) )
=> ( P2 @ X ) ) ) ) ).
% prime_divisors_induct
thf(fact_631_prime__power__cancel__less,axiom,
! [P: nat,M: nat,K: nat,M4: nat,K5: nat] :
( ( factor1801147406995305544me_nat @ P )
=> ( ( ( times_times_nat @ M @ ( power_power_nat @ P @ K ) )
= ( times_times_nat @ M4 @ ( power_power_nat @ P @ K5 ) ) )
=> ( ( ord_less_nat @ K @ K5 )
=> ( dvd_dvd_nat @ P @ M ) ) ) ) ).
% prime_power_cancel_less
thf(fact_632_prime__power__cancel__less,axiom,
! [P: int,M: int,K: nat,M4: int,K5: nat] :
( ( factor1798656936486255268me_int @ P )
=> ( ( ( times_times_int @ M @ ( power_power_int @ P @ K ) )
= ( times_times_int @ M4 @ ( power_power_int @ P @ K5 ) ) )
=> ( ( ord_less_nat @ K @ K5 )
=> ( dvd_dvd_int @ P @ M ) ) ) ) ).
% prime_power_cancel_less
thf(fact_633_dvd__mult__cancel2,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ M )
=> ( ( dvd_dvd_nat @ ( times_times_nat @ N @ M ) @ M )
= ( N = one_one_nat ) ) ) ).
% dvd_mult_cancel2
thf(fact_634_dvd__mult__cancel1,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ M )
=> ( ( dvd_dvd_nat @ ( times_times_nat @ M @ N ) @ M )
= ( N = one_one_nat ) ) ) ).
% dvd_mult_cancel1
thf(fact_635_aprimedivisor__primepow_I2_J,axiom,
! [P: nat,N: nat] :
( ( factor1801147406995305544me_nat @ P )
=> ( ( dvd_dvd_nat @ P @ N )
=> ( ( prime_primepow_nat @ N )
=> ( ( prime_1889911587691200368or_nat @ N )
= P ) ) ) ) ).
% aprimedivisor_primepow(2)
thf(fact_636_aprimedivisor__primepow_I2_J,axiom,
! [P: int,N: int] :
( ( factor1798656936486255268me_int @ P )
=> ( ( dvd_dvd_int @ P @ N )
=> ( ( prime_primepow_int @ N )
=> ( ( prime_1887421117182150092or_int @ N )
= P ) ) ) ) ).
% aprimedivisor_primepow(2)
thf(fact_637_power__eq__if,axiom,
( power_power_nat
= ( ^ [P6: nat,M2: nat] : ( if_nat @ ( M2 = zero_zero_nat ) @ one_one_nat @ ( times_times_nat @ P6 @ ( power_power_nat @ P6 @ ( minus_minus_nat @ M2 @ one_one_nat ) ) ) ) ) ) ).
% power_eq_if
thf(fact_638_power__eq__if,axiom,
( power_power_real
= ( ^ [P6: real,M2: nat] : ( if_real @ ( M2 = zero_zero_nat ) @ one_one_real @ ( times_times_real @ P6 @ ( power_power_real @ P6 @ ( minus_minus_nat @ M2 @ one_one_nat ) ) ) ) ) ) ).
% power_eq_if
thf(fact_639_power__eq__if,axiom,
( power_power_int
= ( ^ [P6: int,M2: nat] : ( if_int @ ( M2 = zero_zero_nat ) @ one_one_int @ ( times_times_int @ P6 @ ( power_power_int @ P6 @ ( minus_minus_nat @ M2 @ one_one_nat ) ) ) ) ) ) ).
% power_eq_if
thf(fact_640_power__minus__mult,axiom,
! [N: nat,A: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( times_times_nat @ ( power_power_nat @ A @ ( minus_minus_nat @ N @ one_one_nat ) ) @ A )
= ( power_power_nat @ A @ N ) ) ) ).
% power_minus_mult
thf(fact_641_power__minus__mult,axiom,
! [N: nat,A: real] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( times_times_real @ ( power_power_real @ A @ ( minus_minus_nat @ N @ one_one_nat ) ) @ A )
= ( power_power_real @ A @ N ) ) ) ).
% power_minus_mult
thf(fact_642_power__minus__mult,axiom,
! [N: nat,A: int] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( times_times_int @ ( power_power_int @ A @ ( minus_minus_nat @ N @ one_one_nat ) ) @ A )
= ( power_power_int @ A @ N ) ) ) ).
% power_minus_mult
thf(fact_643_prime__power__canonical,axiom,
! [P: nat,M: nat] :
( ( factor1801147406995305544me_nat @ P )
=> ( ( ord_less_nat @ zero_zero_nat @ M )
=> ? [K4: nat,N3: nat] :
( ~ ( dvd_dvd_nat @ P @ N3 )
& ( M
= ( times_times_nat @ N3 @ ( power_power_nat @ P @ K4 ) ) ) ) ) ) ).
% prime_power_canonical
thf(fact_644_cong__unique__inverse__prime,axiom,
! [P: nat,X: nat] :
( ( factor1801147406995305544me_nat @ P )
=> ( ( ord_less_nat @ zero_zero_nat @ X )
=> ( ( ord_less_nat @ X @ P )
=> ? [X3: nat] :
( ( ord_less_nat @ zero_zero_nat @ X3 )
& ( ord_less_nat @ X3 @ P )
& ( unique653641344996303876ng_nat @ ( times_times_nat @ X @ X3 ) @ one_one_nat @ P )
& ! [Y3: nat] :
( ( ( ord_less_nat @ zero_zero_nat @ Y3 )
& ( ord_less_nat @ Y3 @ P )
& ( unique653641344996303876ng_nat @ ( times_times_nat @ X @ Y3 ) @ one_one_nat @ P ) )
=> ( Y3 = X3 ) ) ) ) ) ) ).
% cong_unique_inverse_prime
thf(fact_645_prime__aprimedivisor_H,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
=> ( ~ ( dvd_dvd_nat @ N @ one_one_nat )
=> ( factor1801147406995305544me_nat @ ( prime_1889911587691200368or_nat @ N ) ) ) ) ).
% prime_aprimedivisor'
thf(fact_646_prime__aprimedivisor_H,axiom,
! [N: int] :
( ( N != zero_zero_int )
=> ( ~ ( dvd_dvd_int @ N @ one_one_int )
=> ( factor1798656936486255268me_int @ ( prime_1887421117182150092or_int @ N ) ) ) ) ).
% prime_aprimedivisor'
thf(fact_647_aprimedivisor__prime__power,axiom,
! [P: nat,K: nat] :
( ( factor1801147406995305544me_nat @ P )
=> ( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ( prime_1889911587691200368or_nat @ ( power_power_nat @ P @ K ) )
= P ) ) ) ).
% aprimedivisor_prime_power
thf(fact_648_aprimedivisor__prime__power,axiom,
! [P: int,K: nat] :
( ( factor1798656936486255268me_int @ P )
=> ( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ( prime_1887421117182150092or_int @ ( power_power_int @ P @ K ) )
= P ) ) ) ).
% aprimedivisor_prime_power
thf(fact_649_aprimedivisor__primepow__power,axiom,
! [N: int,K: nat] :
( ( prime_primepow_int @ N )
=> ( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ( prime_1887421117182150092or_int @ ( power_power_int @ N @ K ) )
= ( prime_1887421117182150092or_int @ N ) ) ) ) ).
% aprimedivisor_primepow_power
thf(fact_650_aprimedivisor__primepow__power,axiom,
! [N: nat,K: nat] :
( ( prime_primepow_nat @ N )
=> ( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ( prime_1889911587691200368or_nat @ ( power_power_nat @ N @ K ) )
= ( prime_1889911587691200368or_nat @ N ) ) ) ) ).
% aprimedivisor_primepow_power
thf(fact_651_nat__mult__dvd__cancel__disj,axiom,
! [K: nat,M: nat,N: nat] :
( ( dvd_dvd_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) )
= ( ( K = zero_zero_nat )
| ( dvd_dvd_nat @ M @ N ) ) ) ).
% nat_mult_dvd_cancel_disj
thf(fact_652_nat__mult__less__cancel__disj,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) )
= ( ( ord_less_nat @ zero_zero_nat @ K )
& ( ord_less_nat @ M @ N ) ) ) ).
% nat_mult_less_cancel_disj
thf(fact_653_mult__hom_Ohom__zero,axiom,
! [C: nat] :
( ( times_times_nat @ C @ zero_zero_nat )
= zero_zero_nat ) ).
% mult_hom.hom_zero
thf(fact_654_mult__hom_Ohom__zero,axiom,
! [C: real] :
( ( times_times_real @ C @ zero_zero_real )
= zero_zero_real ) ).
% mult_hom.hom_zero
thf(fact_655_mult__hom_Ohom__zero,axiom,
! [C: int] :
( ( times_times_int @ C @ zero_zero_int )
= zero_zero_int ) ).
% mult_hom.hom_zero
thf(fact_656_cong__prime__prod__zero__nat,axiom,
! [A: nat,B: nat,P: nat] :
( ( unique653641344996303876ng_nat @ ( times_times_nat @ A @ B ) @ zero_zero_nat @ P )
=> ( ( factor1801147406995305544me_nat @ P )
=> ( ( unique653641344996303876ng_nat @ A @ zero_zero_nat @ P )
| ( unique653641344996303876ng_nat @ B @ zero_zero_nat @ P ) ) ) ) ).
% cong_prime_prod_zero_nat
thf(fact_657_zdvd__not__zless,axiom,
! [M: int,N: int] :
( ( ord_less_int @ zero_zero_int @ M )
=> ( ( ord_less_int @ M @ N )
=> ~ ( dvd_dvd_int @ N @ M ) ) ) ).
% zdvd_not_zless
thf(fact_658_pos__zmult__eq__1__iff,axiom,
! [M: int,N: int] :
( ( ord_less_int @ zero_zero_int @ M )
=> ( ( ( times_times_int @ M @ N )
= one_one_int )
= ( ( M = one_one_int )
& ( N = one_one_int ) ) ) ) ).
% pos_zmult_eq_1_iff
thf(fact_659_int__distrib_I4_J,axiom,
! [W: int,Z1: int,Z22: int] :
( ( times_times_int @ W @ ( minus_minus_int @ Z1 @ Z22 ) )
= ( minus_minus_int @ ( times_times_int @ W @ Z1 ) @ ( times_times_int @ W @ Z22 ) ) ) ).
% int_distrib(4)
thf(fact_660_int__distrib_I3_J,axiom,
! [Z1: int,Z22: int,W: int] :
( ( times_times_int @ ( minus_minus_int @ Z1 @ Z22 ) @ W )
= ( minus_minus_int @ ( times_times_int @ Z1 @ W ) @ ( times_times_int @ Z22 @ W ) ) ) ).
% int_distrib(3)
thf(fact_661_nat__mult__eq__cancel__disj,axiom,
! [K: nat,M: nat,N: nat] :
( ( ( times_times_nat @ K @ M )
= ( times_times_nat @ K @ N ) )
= ( ( K = zero_zero_nat )
| ( M = N ) ) ) ).
% nat_mult_eq_cancel_disj
thf(fact_662_minus__int__code_I1_J,axiom,
! [K: int] :
( ( minus_minus_int @ K @ zero_zero_int )
= K ) ).
% minus_int_code(1)
thf(fact_663_times__int__code_I1_J,axiom,
! [K: int] :
( ( times_times_int @ K @ zero_zero_int )
= zero_zero_int ) ).
% times_int_code(1)
thf(fact_664_times__int__code_I2_J,axiom,
! [L: int] :
( ( times_times_int @ zero_zero_int @ L )
= zero_zero_int ) ).
% times_int_code(2)
thf(fact_665_zdvd__zdiffD,axiom,
! [K: int,M: int,N: int] :
( ( dvd_dvd_int @ K @ ( minus_minus_int @ M @ N ) )
=> ( ( dvd_dvd_int @ K @ N )
=> ( dvd_dvd_int @ K @ M ) ) ) ).
% zdvd_zdiffD
thf(fact_666_nat__mult__less__cancel1,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ( ord_less_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) )
= ( ord_less_nat @ M @ N ) ) ) ).
% nat_mult_less_cancel1
thf(fact_667_nat__mult__eq__cancel1,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ( ( times_times_nat @ K @ M )
= ( times_times_nat @ K @ N ) )
= ( M = N ) ) ) ).
% nat_mult_eq_cancel1
thf(fact_668_zmult__zless__mono2,axiom,
! [I: int,J: int,K: int] :
( ( ord_less_int @ I @ J )
=> ( ( ord_less_int @ zero_zero_int @ K )
=> ( ord_less_int @ ( times_times_int @ K @ I ) @ ( times_times_int @ K @ J ) ) ) ) ).
% zmult_zless_mono2
thf(fact_669_less__int__code_I1_J,axiom,
~ ( ord_less_int @ zero_zero_int @ zero_zero_int ) ).
% less_int_code(1)
thf(fact_670_int__less__induct,axiom,
! [I: int,K: int,P2: int > $o] :
( ( ord_less_int @ I @ K )
=> ( ( P2 @ ( minus_minus_int @ K @ one_one_int ) )
=> ( ! [I2: int] :
( ( ord_less_int @ I2 @ K )
=> ( ( P2 @ I2 )
=> ( P2 @ ( minus_minus_int @ I2 @ one_one_int ) ) ) )
=> ( P2 @ I ) ) ) ) ).
% int_less_induct
thf(fact_671_zdvd__mult__cancel,axiom,
! [K: int,M: int,N: int] :
( ( dvd_dvd_int @ ( times_times_int @ K @ M ) @ ( times_times_int @ K @ N ) )
=> ( ( K != zero_zero_int )
=> ( dvd_dvd_int @ M @ N ) ) ) ).
% zdvd_mult_cancel
thf(fact_672_cong__prime__prod__zero__int,axiom,
! [A: int,B: int,P: int] :
( ( unique651150874487253600ng_int @ ( times_times_int @ A @ B ) @ zero_zero_int @ P )
=> ( ( factor1798656936486255268me_int @ P )
=> ( ( unique651150874487253600ng_int @ A @ zero_zero_int @ P )
| ( unique651150874487253600ng_int @ B @ zero_zero_int @ P ) ) ) ) ).
% cong_prime_prod_zero_int
thf(fact_673_nat__mult__dvd__cancel1,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ( dvd_dvd_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) )
= ( dvd_dvd_nat @ M @ N ) ) ) ).
% nat_mult_dvd_cancel1
thf(fact_674_plusinfinity,axiom,
! [D: int,P7: int > $o,P2: int > $o] :
( ( ord_less_int @ zero_zero_int @ D )
=> ( ! [X3: int,K4: int] :
( ( P7 @ X3 )
= ( P7 @ ( minus_minus_int @ X3 @ ( times_times_int @ K4 @ D ) ) ) )
=> ( ? [Z3: int] :
! [X3: int] :
( ( ord_less_int @ Z3 @ X3 )
=> ( ( P2 @ X3 )
= ( P7 @ X3 ) ) )
=> ( ? [X_12: int] : ( P7 @ X_12 )
=> ? [X_1: int] : ( P2 @ X_1 ) ) ) ) ) ).
% plusinfinity
thf(fact_675_minusinfinity,axiom,
! [D: int,P1: int > $o,P2: int > $o] :
( ( ord_less_int @ zero_zero_int @ D )
=> ( ! [X3: int,K4: int] :
( ( P1 @ X3 )
= ( P1 @ ( minus_minus_int @ X3 @ ( times_times_int @ K4 @ D ) ) ) )
=> ( ? [Z3: int] :
! [X3: int] :
( ( ord_less_int @ X3 @ Z3 )
=> ( ( P2 @ X3 )
= ( P1 @ X3 ) ) )
=> ( ? [X_12: int] : ( P1 @ X_12 )
=> ? [X_1: int] : ( P2 @ X_1 ) ) ) ) ) ).
% minusinfinity
thf(fact_676_zdvd__mono,axiom,
! [K: int,M: int,T: int] :
( ( K != zero_zero_int )
=> ( ( dvd_dvd_int @ M @ T )
= ( dvd_dvd_int @ ( times_times_int @ K @ M ) @ ( times_times_int @ K @ T ) ) ) ) ).
% zdvd_mono
thf(fact_677_mult__less__iff1,axiom,
! [Z2: int,X: int,Y: int] :
( ( ord_less_int @ zero_zero_int @ Z2 )
=> ( ( ord_less_int @ ( times_times_int @ X @ Z2 ) @ ( times_times_int @ Y @ Z2 ) )
= ( ord_less_int @ X @ Y ) ) ) ).
% mult_less_iff1
thf(fact_678_mult__less__iff1,axiom,
! [Z2: real,X: real,Y: real] :
( ( ord_less_real @ zero_zero_real @ Z2 )
=> ( ( ord_less_real @ ( times_times_real @ X @ Z2 ) @ ( times_times_real @ Y @ Z2 ) )
= ( ord_less_real @ X @ Y ) ) ) ).
% mult_less_iff1
thf(fact_679_p__fact,axiom,
( p
= ( plus_plus_nat @ ( times_times_nat @ k @ n ) @ one_one_nat ) ) ).
% p_fact
thf(fact_680_add__right__cancel,axiom,
! [B: nat,A: nat,C: nat] :
( ( ( plus_plus_nat @ B @ A )
= ( plus_plus_nat @ C @ A ) )
= ( B = C ) ) ).
% add_right_cancel
thf(fact_681_add__right__cancel,axiom,
! [B: int,A: int,C: int] :
( ( ( plus_plus_int @ B @ A )
= ( plus_plus_int @ C @ A ) )
= ( B = C ) ) ).
% add_right_cancel
thf(fact_682_add__left__cancel,axiom,
! [A: nat,B: nat,C: nat] :
( ( ( plus_plus_nat @ A @ B )
= ( plus_plus_nat @ A @ C ) )
= ( B = C ) ) ).
% add_left_cancel
thf(fact_683_add__left__cancel,axiom,
! [A: int,B: int,C: int] :
( ( ( plus_plus_int @ A @ B )
= ( plus_plus_int @ A @ C ) )
= ( B = C ) ) ).
% add_left_cancel
thf(fact_684_add__0,axiom,
! [A: nat] :
( ( plus_plus_nat @ zero_zero_nat @ A )
= A ) ).
% add_0
thf(fact_685_add__0,axiom,
! [A: int] :
( ( plus_plus_int @ zero_zero_int @ A )
= A ) ).
% add_0
thf(fact_686_add__0,axiom,
! [A: real] :
( ( plus_plus_real @ zero_zero_real @ A )
= A ) ).
% add_0
thf(fact_687_zero__eq__add__iff__both__eq__0,axiom,
! [X: nat,Y: nat] :
( ( zero_zero_nat
= ( plus_plus_nat @ X @ Y ) )
= ( ( X = zero_zero_nat )
& ( Y = zero_zero_nat ) ) ) ).
% zero_eq_add_iff_both_eq_0
thf(fact_688_add__eq__0__iff__both__eq__0,axiom,
! [X: nat,Y: nat] :
( ( ( plus_plus_nat @ X @ Y )
= zero_zero_nat )
= ( ( X = zero_zero_nat )
& ( Y = zero_zero_nat ) ) ) ).
% add_eq_0_iff_both_eq_0
thf(fact_689_add__cancel__right__right,axiom,
! [A: nat,B: nat] :
( ( A
= ( plus_plus_nat @ A @ B ) )
= ( B = zero_zero_nat ) ) ).
% add_cancel_right_right
thf(fact_690_add__cancel__right__right,axiom,
! [A: int,B: int] :
( ( A
= ( plus_plus_int @ A @ B ) )
= ( B = zero_zero_int ) ) ).
% add_cancel_right_right
thf(fact_691_add__cancel__right__right,axiom,
! [A: real,B: real] :
( ( A
= ( plus_plus_real @ A @ B ) )
= ( B = zero_zero_real ) ) ).
% add_cancel_right_right
thf(fact_692_add__cancel__right__left,axiom,
! [A: nat,B: nat] :
( ( A
= ( plus_plus_nat @ B @ A ) )
= ( B = zero_zero_nat ) ) ).
% add_cancel_right_left
thf(fact_693_add__cancel__right__left,axiom,
! [A: int,B: int] :
( ( A
= ( plus_plus_int @ B @ A ) )
= ( B = zero_zero_int ) ) ).
% add_cancel_right_left
thf(fact_694_add__cancel__right__left,axiom,
! [A: real,B: real] :
( ( A
= ( plus_plus_real @ B @ A ) )
= ( B = zero_zero_real ) ) ).
% add_cancel_right_left
thf(fact_695_add__cancel__left__right,axiom,
! [A: nat,B: nat] :
( ( ( plus_plus_nat @ A @ B )
= A )
= ( B = zero_zero_nat ) ) ).
% add_cancel_left_right
thf(fact_696_add__cancel__left__right,axiom,
! [A: int,B: int] :
( ( ( plus_plus_int @ A @ B )
= A )
= ( B = zero_zero_int ) ) ).
% add_cancel_left_right
thf(fact_697_add__cancel__left__right,axiom,
! [A: real,B: real] :
( ( ( plus_plus_real @ A @ B )
= A )
= ( B = zero_zero_real ) ) ).
% add_cancel_left_right
thf(fact_698_add__cancel__left__left,axiom,
! [B: nat,A: nat] :
( ( ( plus_plus_nat @ B @ A )
= A )
= ( B = zero_zero_nat ) ) ).
% add_cancel_left_left
thf(fact_699_add__cancel__left__left,axiom,
! [B: int,A: int] :
( ( ( plus_plus_int @ B @ A )
= A )
= ( B = zero_zero_int ) ) ).
% add_cancel_left_left
thf(fact_700_add__cancel__left__left,axiom,
! [B: real,A: real] :
( ( ( plus_plus_real @ B @ A )
= A )
= ( B = zero_zero_real ) ) ).
% add_cancel_left_left
thf(fact_701_double__zero__sym,axiom,
! [A: int] :
( ( zero_zero_int
= ( plus_plus_int @ A @ A ) )
= ( A = zero_zero_int ) ) ).
% double_zero_sym
thf(fact_702_double__zero__sym,axiom,
! [A: real] :
( ( zero_zero_real
= ( plus_plus_real @ A @ A ) )
= ( A = zero_zero_real ) ) ).
% double_zero_sym
thf(fact_703_add_Oright__neutral,axiom,
! [A: nat] :
( ( plus_plus_nat @ A @ zero_zero_nat )
= A ) ).
% add.right_neutral
thf(fact_704_add_Oright__neutral,axiom,
! [A: int] :
( ( plus_plus_int @ A @ zero_zero_int )
= A ) ).
% add.right_neutral
thf(fact_705_add_Oright__neutral,axiom,
! [A: real] :
( ( plus_plus_real @ A @ zero_zero_real )
= A ) ).
% add.right_neutral
thf(fact_706_double__eq__0__iff,axiom,
! [A: int] :
( ( ( plus_plus_int @ A @ A )
= zero_zero_int )
= ( A = zero_zero_int ) ) ).
% double_eq_0_iff
thf(fact_707_double__eq__0__iff,axiom,
! [A: real] :
( ( ( plus_plus_real @ A @ A )
= zero_zero_real )
= ( A = zero_zero_real ) ) ).
% double_eq_0_iff
thf(fact_708_add__less__cancel__right,axiom,
! [A: nat,C: nat,B: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ C ) )
= ( ord_less_nat @ A @ B ) ) ).
% add_less_cancel_right
thf(fact_709_add__less__cancel__right,axiom,
! [A: int,C: int,B: int] :
( ( ord_less_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ C ) )
= ( ord_less_int @ A @ B ) ) ).
% add_less_cancel_right
thf(fact_710_add__less__cancel__right,axiom,
! [A: real,C: real,B: real] :
( ( ord_less_real @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ B @ C ) )
= ( ord_less_real @ A @ B ) ) ).
% add_less_cancel_right
thf(fact_711_add__less__cancel__left,axiom,
! [C: nat,A: nat,B: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B ) )
= ( ord_less_nat @ A @ B ) ) ).
% add_less_cancel_left
thf(fact_712_add__less__cancel__left,axiom,
! [C: int,A: int,B: int] :
( ( ord_less_int @ ( plus_plus_int @ C @ A ) @ ( plus_plus_int @ C @ B ) )
= ( ord_less_int @ A @ B ) ) ).
% add_less_cancel_left
thf(fact_713_add__less__cancel__left,axiom,
! [C: real,A: real,B: real] :
( ( ord_less_real @ ( plus_plus_real @ C @ A ) @ ( plus_plus_real @ C @ B ) )
= ( ord_less_real @ A @ B ) ) ).
% add_less_cancel_left
thf(fact_714_add__diff__cancel,axiom,
! [A: int,B: int] :
( ( minus_minus_int @ ( plus_plus_int @ A @ B ) @ B )
= A ) ).
% add_diff_cancel
thf(fact_715_diff__add__cancel,axiom,
! [A: int,B: int] :
( ( plus_plus_int @ ( minus_minus_int @ A @ B ) @ B )
= A ) ).
% diff_add_cancel
thf(fact_716_add__diff__cancel__left,axiom,
! [C: nat,A: nat,B: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B ) )
= ( minus_minus_nat @ A @ B ) ) ).
% add_diff_cancel_left
thf(fact_717_add__diff__cancel__left,axiom,
! [C: int,A: int,B: int] :
( ( minus_minus_int @ ( plus_plus_int @ C @ A ) @ ( plus_plus_int @ C @ B ) )
= ( minus_minus_int @ A @ B ) ) ).
% add_diff_cancel_left
thf(fact_718_add__diff__cancel__left_H,axiom,
! [A: nat,B: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ A @ B ) @ A )
= B ) ).
% add_diff_cancel_left'
thf(fact_719_add__diff__cancel__left_H,axiom,
! [A: int,B: int] :
( ( minus_minus_int @ ( plus_plus_int @ A @ B ) @ A )
= B ) ).
% add_diff_cancel_left'
thf(fact_720_add__diff__cancel__right,axiom,
! [A: nat,C: nat,B: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ C ) )
= ( minus_minus_nat @ A @ B ) ) ).
% add_diff_cancel_right
thf(fact_721_add__diff__cancel__right,axiom,
! [A: int,C: int,B: int] :
( ( minus_minus_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ C ) )
= ( minus_minus_int @ A @ B ) ) ).
% add_diff_cancel_right
thf(fact_722_add__diff__cancel__right_H,axiom,
! [A: nat,B: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ A @ B ) @ B )
= A ) ).
% add_diff_cancel_right'
thf(fact_723_add__diff__cancel__right_H,axiom,
! [A: int,B: int] :
( ( minus_minus_int @ ( plus_plus_int @ A @ B ) @ B )
= A ) ).
% add_diff_cancel_right'
thf(fact_724_dvd__add__triv__left__iff,axiom,
! [A: nat,B: nat] :
( ( dvd_dvd_nat @ A @ ( plus_plus_nat @ A @ B ) )
= ( dvd_dvd_nat @ A @ B ) ) ).
% dvd_add_triv_left_iff
thf(fact_725_dvd__add__triv__left__iff,axiom,
! [A: int,B: int] :
( ( dvd_dvd_int @ A @ ( plus_plus_int @ A @ B ) )
= ( dvd_dvd_int @ A @ B ) ) ).
% dvd_add_triv_left_iff
thf(fact_726_dvd__add__triv__right__iff,axiom,
! [A: nat,B: nat] :
( ( dvd_dvd_nat @ A @ ( plus_plus_nat @ B @ A ) )
= ( dvd_dvd_nat @ A @ B ) ) ).
% dvd_add_triv_right_iff
thf(fact_727_dvd__add__triv__right__iff,axiom,
! [A: int,B: int] :
( ( dvd_dvd_int @ A @ ( plus_plus_int @ B @ A ) )
= ( dvd_dvd_int @ A @ B ) ) ).
% dvd_add_triv_right_iff
thf(fact_728_add__is__0,axiom,
! [M: nat,N: nat] :
( ( ( plus_plus_nat @ M @ N )
= zero_zero_nat )
= ( ( M = zero_zero_nat )
& ( N = zero_zero_nat ) ) ) ).
% add_is_0
thf(fact_729_Nat_Oadd__0__right,axiom,
! [M: nat] :
( ( plus_plus_nat @ M @ zero_zero_nat )
= M ) ).
% Nat.add_0_right
thf(fact_730_nat__add__left__cancel__less,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ K @ M ) @ ( plus_plus_nat @ K @ N ) )
= ( ord_less_nat @ M @ N ) ) ).
% nat_add_left_cancel_less
thf(fact_731_diff__diff__left,axiom,
! [I: nat,J: nat,K: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ I @ J ) @ K )
= ( minus_minus_nat @ I @ ( plus_plus_nat @ J @ K ) ) ) ).
% diff_diff_left
thf(fact_732_zero__less__double__add__iff__zero__less__single__add,axiom,
! [A: int] :
( ( ord_less_int @ zero_zero_int @ ( plus_plus_int @ A @ A ) )
= ( ord_less_int @ zero_zero_int @ A ) ) ).
% zero_less_double_add_iff_zero_less_single_add
thf(fact_733_zero__less__double__add__iff__zero__less__single__add,axiom,
! [A: real] :
( ( ord_less_real @ zero_zero_real @ ( plus_plus_real @ A @ A ) )
= ( ord_less_real @ zero_zero_real @ A ) ) ).
% zero_less_double_add_iff_zero_less_single_add
thf(fact_734_double__add__less__zero__iff__single__add__less__zero,axiom,
! [A: int] :
( ( ord_less_int @ ( plus_plus_int @ A @ A ) @ zero_zero_int )
= ( ord_less_int @ A @ zero_zero_int ) ) ).
% double_add_less_zero_iff_single_add_less_zero
thf(fact_735_double__add__less__zero__iff__single__add__less__zero,axiom,
! [A: real] :
( ( ord_less_real @ ( plus_plus_real @ A @ A ) @ zero_zero_real )
= ( ord_less_real @ A @ zero_zero_real ) ) ).
% double_add_less_zero_iff_single_add_less_zero
thf(fact_736_less__add__same__cancel2,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ ( plus_plus_nat @ B @ A ) )
= ( ord_less_nat @ zero_zero_nat @ B ) ) ).
% less_add_same_cancel2
thf(fact_737_less__add__same__cancel2,axiom,
! [A: int,B: int] :
( ( ord_less_int @ A @ ( plus_plus_int @ B @ A ) )
= ( ord_less_int @ zero_zero_int @ B ) ) ).
% less_add_same_cancel2
thf(fact_738_less__add__same__cancel2,axiom,
! [A: real,B: real] :
( ( ord_less_real @ A @ ( plus_plus_real @ B @ A ) )
= ( ord_less_real @ zero_zero_real @ B ) ) ).
% less_add_same_cancel2
thf(fact_739_less__add__same__cancel1,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ ( plus_plus_nat @ A @ B ) )
= ( ord_less_nat @ zero_zero_nat @ B ) ) ).
% less_add_same_cancel1
thf(fact_740_less__add__same__cancel1,axiom,
! [A: int,B: int] :
( ( ord_less_int @ A @ ( plus_plus_int @ A @ B ) )
= ( ord_less_int @ zero_zero_int @ B ) ) ).
% less_add_same_cancel1
thf(fact_741_less__add__same__cancel1,axiom,
! [A: real,B: real] :
( ( ord_less_real @ A @ ( plus_plus_real @ A @ B ) )
= ( ord_less_real @ zero_zero_real @ B ) ) ).
% less_add_same_cancel1
thf(fact_742_add__less__same__cancel2,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ A @ B ) @ B )
= ( ord_less_nat @ A @ zero_zero_nat ) ) ).
% add_less_same_cancel2
thf(fact_743_add__less__same__cancel2,axiom,
! [A: int,B: int] :
( ( ord_less_int @ ( plus_plus_int @ A @ B ) @ B )
= ( ord_less_int @ A @ zero_zero_int ) ) ).
% add_less_same_cancel2
thf(fact_744_add__less__same__cancel2,axiom,
! [A: real,B: real] :
( ( ord_less_real @ ( plus_plus_real @ A @ B ) @ B )
= ( ord_less_real @ A @ zero_zero_real ) ) ).
% add_less_same_cancel2
thf(fact_745_add__less__same__cancel1,axiom,
! [B: nat,A: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ B @ A ) @ B )
= ( ord_less_nat @ A @ zero_zero_nat ) ) ).
% add_less_same_cancel1
thf(fact_746_add__less__same__cancel1,axiom,
! [B: int,A: int] :
( ( ord_less_int @ ( plus_plus_int @ B @ A ) @ B )
= ( ord_less_int @ A @ zero_zero_int ) ) ).
% add_less_same_cancel1
thf(fact_747_add__less__same__cancel1,axiom,
! [B: real,A: real] :
( ( ord_less_real @ ( plus_plus_real @ B @ A ) @ B )
= ( ord_less_real @ A @ zero_zero_real ) ) ).
% add_less_same_cancel1
thf(fact_748_sum__squares__eq__zero__iff,axiom,
! [X: real,Y: real] :
( ( ( plus_plus_real @ ( times_times_real @ X @ X ) @ ( times_times_real @ Y @ Y ) )
= zero_zero_real )
= ( ( X = zero_zero_real )
& ( Y = zero_zero_real ) ) ) ).
% sum_squares_eq_zero_iff
thf(fact_749_sum__squares__eq__zero__iff,axiom,
! [X: int,Y: int] :
( ( ( plus_plus_int @ ( times_times_int @ X @ X ) @ ( times_times_int @ Y @ Y ) )
= zero_zero_int )
= ( ( X = zero_zero_int )
& ( Y = zero_zero_int ) ) ) ).
% sum_squares_eq_zero_iff
thf(fact_750_diff__add__zero,axiom,
! [A: nat,B: nat] :
( ( minus_minus_nat @ A @ ( plus_plus_nat @ A @ B ) )
= zero_zero_nat ) ).
% diff_add_zero
thf(fact_751_dvd__add__times__triv__right__iff,axiom,
! [A: nat,B: nat,C: nat] :
( ( dvd_dvd_nat @ A @ ( plus_plus_nat @ B @ ( times_times_nat @ C @ A ) ) )
= ( dvd_dvd_nat @ A @ B ) ) ).
% dvd_add_times_triv_right_iff
thf(fact_752_dvd__add__times__triv__right__iff,axiom,
! [A: real,B: real,C: real] :
( ( dvd_dvd_real @ A @ ( plus_plus_real @ B @ ( times_times_real @ C @ A ) ) )
= ( dvd_dvd_real @ A @ B ) ) ).
% dvd_add_times_triv_right_iff
thf(fact_753_dvd__add__times__triv__right__iff,axiom,
! [A: int,B: int,C: int] :
( ( dvd_dvd_int @ A @ ( plus_plus_int @ B @ ( times_times_int @ C @ A ) ) )
= ( dvd_dvd_int @ A @ B ) ) ).
% dvd_add_times_triv_right_iff
thf(fact_754_dvd__add__times__triv__left__iff,axiom,
! [A: nat,C: nat,B: nat] :
( ( dvd_dvd_nat @ A @ ( plus_plus_nat @ ( times_times_nat @ C @ A ) @ B ) )
= ( dvd_dvd_nat @ A @ B ) ) ).
% dvd_add_times_triv_left_iff
thf(fact_755_dvd__add__times__triv__left__iff,axiom,
! [A: real,C: real,B: real] :
( ( dvd_dvd_real @ A @ ( plus_plus_real @ ( times_times_real @ C @ A ) @ B ) )
= ( dvd_dvd_real @ A @ B ) ) ).
% dvd_add_times_triv_left_iff
thf(fact_756_dvd__add__times__triv__left__iff,axiom,
! [A: int,C: int,B: int] :
( ( dvd_dvd_int @ A @ ( plus_plus_int @ ( times_times_int @ C @ A ) @ B ) )
= ( dvd_dvd_int @ A @ B ) ) ).
% dvd_add_times_triv_left_iff
thf(fact_757_add__gr__0,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ M @ N ) )
= ( ( ord_less_nat @ zero_zero_nat @ M )
| ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% add_gr_0
thf(fact_758_add__right__imp__eq,axiom,
! [B: nat,A: nat,C: nat] :
( ( ( plus_plus_nat @ B @ A )
= ( plus_plus_nat @ C @ A ) )
=> ( B = C ) ) ).
% add_right_imp_eq
thf(fact_759_add__right__imp__eq,axiom,
! [B: int,A: int,C: int] :
( ( ( plus_plus_int @ B @ A )
= ( plus_plus_int @ C @ A ) )
=> ( B = C ) ) ).
% add_right_imp_eq
thf(fact_760_add__left__imp__eq,axiom,
! [A: nat,B: nat,C: nat] :
( ( ( plus_plus_nat @ A @ B )
= ( plus_plus_nat @ A @ C ) )
=> ( B = C ) ) ).
% add_left_imp_eq
thf(fact_761_add__left__imp__eq,axiom,
! [A: int,B: int,C: int] :
( ( ( plus_plus_int @ A @ B )
= ( plus_plus_int @ A @ C ) )
=> ( B = C ) ) ).
% add_left_imp_eq
thf(fact_762_add_Oleft__commute,axiom,
! [B: nat,A: nat,C: nat] :
( ( plus_plus_nat @ B @ ( plus_plus_nat @ A @ C ) )
= ( plus_plus_nat @ A @ ( plus_plus_nat @ B @ C ) ) ) ).
% add.left_commute
thf(fact_763_add_Oleft__commute,axiom,
! [B: int,A: int,C: int] :
( ( plus_plus_int @ B @ ( plus_plus_int @ A @ C ) )
= ( plus_plus_int @ A @ ( plus_plus_int @ B @ C ) ) ) ).
% add.left_commute
thf(fact_764_add_Ocommute,axiom,
( plus_plus_nat
= ( ^ [A3: nat,B2: nat] : ( plus_plus_nat @ B2 @ A3 ) ) ) ).
% add.commute
thf(fact_765_add_Ocommute,axiom,
( plus_plus_int
= ( ^ [A3: int,B2: int] : ( plus_plus_int @ B2 @ A3 ) ) ) ).
% add.commute
thf(fact_766_add_Oright__cancel,axiom,
! [B: int,A: int,C: int] :
( ( ( plus_plus_int @ B @ A )
= ( plus_plus_int @ C @ A ) )
= ( B = C ) ) ).
% add.right_cancel
thf(fact_767_add_Oleft__cancel,axiom,
! [A: int,B: int,C: int] :
( ( ( plus_plus_int @ A @ B )
= ( plus_plus_int @ A @ C ) )
= ( B = C ) ) ).
% add.left_cancel
thf(fact_768_add_Oassoc,axiom,
! [A: nat,B: nat,C: nat] :
( ( plus_plus_nat @ ( plus_plus_nat @ A @ B ) @ C )
= ( plus_plus_nat @ A @ ( plus_plus_nat @ B @ C ) ) ) ).
% add.assoc
thf(fact_769_add_Oassoc,axiom,
! [A: int,B: int,C: int] :
( ( plus_plus_int @ ( plus_plus_int @ A @ B ) @ C )
= ( plus_plus_int @ A @ ( plus_plus_int @ B @ C ) ) ) ).
% add.assoc
thf(fact_770_group__cancel_Oadd2,axiom,
! [B5: nat,K: nat,B: nat,A: nat] :
( ( B5
= ( plus_plus_nat @ K @ B ) )
=> ( ( plus_plus_nat @ A @ B5 )
= ( plus_plus_nat @ K @ ( plus_plus_nat @ A @ B ) ) ) ) ).
% group_cancel.add2
thf(fact_771_group__cancel_Oadd2,axiom,
! [B5: int,K: int,B: int,A: int] :
( ( B5
= ( plus_plus_int @ K @ B ) )
=> ( ( plus_plus_int @ A @ B5 )
= ( plus_plus_int @ K @ ( plus_plus_int @ A @ B ) ) ) ) ).
% group_cancel.add2
thf(fact_772_group__cancel_Oadd1,axiom,
! [A2: nat,K: nat,A: nat,B: nat] :
( ( A2
= ( plus_plus_nat @ K @ A ) )
=> ( ( plus_plus_nat @ A2 @ B )
= ( plus_plus_nat @ K @ ( plus_plus_nat @ A @ B ) ) ) ) ).
% group_cancel.add1
thf(fact_773_group__cancel_Oadd1,axiom,
! [A2: int,K: int,A: int,B: int] :
( ( A2
= ( plus_plus_int @ K @ A ) )
=> ( ( plus_plus_int @ A2 @ B )
= ( plus_plus_int @ K @ ( plus_plus_int @ A @ B ) ) ) ) ).
% group_cancel.add1
thf(fact_774_add__mono__thms__linordered__semiring_I4_J,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ( I = J )
& ( K = L ) )
=> ( ( plus_plus_nat @ I @ K )
= ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(4)
thf(fact_775_add__mono__thms__linordered__semiring_I4_J,axiom,
! [I: int,J: int,K: int,L: int] :
( ( ( I = J )
& ( K = L ) )
=> ( ( plus_plus_int @ I @ K )
= ( plus_plus_int @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(4)
thf(fact_776_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
! [A: nat,B: nat,C: nat] :
( ( plus_plus_nat @ ( plus_plus_nat @ A @ B ) @ C )
= ( plus_plus_nat @ A @ ( plus_plus_nat @ B @ C ) ) ) ).
% ab_semigroup_add_class.add_ac(1)
thf(fact_777_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
! [A: int,B: int,C: int] :
( ( plus_plus_int @ ( plus_plus_int @ A @ B ) @ C )
= ( plus_plus_int @ A @ ( plus_plus_int @ B @ C ) ) ) ).
% ab_semigroup_add_class.add_ac(1)
thf(fact_778_is__num__normalize_I1_J,axiom,
! [A: int,B: int,C: int] :
( ( plus_plus_int @ ( plus_plus_int @ A @ B ) @ C )
= ( plus_plus_int @ A @ ( plus_plus_int @ B @ C ) ) ) ).
% is_num_normalize(1)
thf(fact_779_add_Ogroup__left__neutral,axiom,
! [A: int] :
( ( plus_plus_int @ zero_zero_int @ A )
= A ) ).
% add.group_left_neutral
thf(fact_780_add_Ogroup__left__neutral,axiom,
! [A: real] :
( ( plus_plus_real @ zero_zero_real @ A )
= A ) ).
% add.group_left_neutral
thf(fact_781_add_Ocomm__neutral,axiom,
! [A: nat] :
( ( plus_plus_nat @ A @ zero_zero_nat )
= A ) ).
% add.comm_neutral
thf(fact_782_add_Ocomm__neutral,axiom,
! [A: int] :
( ( plus_plus_int @ A @ zero_zero_int )
= A ) ).
% add.comm_neutral
thf(fact_783_add_Ocomm__neutral,axiom,
! [A: real] :
( ( plus_plus_real @ A @ zero_zero_real )
= A ) ).
% add.comm_neutral
thf(fact_784_comm__monoid__add__class_Oadd__0,axiom,
! [A: nat] :
( ( plus_plus_nat @ zero_zero_nat @ A )
= A ) ).
% comm_monoid_add_class.add_0
thf(fact_785_comm__monoid__add__class_Oadd__0,axiom,
! [A: int] :
( ( plus_plus_int @ zero_zero_int @ A )
= A ) ).
% comm_monoid_add_class.add_0
thf(fact_786_comm__monoid__add__class_Oadd__0,axiom,
! [A: real] :
( ( plus_plus_real @ zero_zero_real @ A )
= A ) ).
% comm_monoid_add_class.add_0
thf(fact_787_add__less__imp__less__right,axiom,
! [A: nat,C: nat,B: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ C ) )
=> ( ord_less_nat @ A @ B ) ) ).
% add_less_imp_less_right
thf(fact_788_add__less__imp__less__right,axiom,
! [A: int,C: int,B: int] :
( ( ord_less_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ C ) )
=> ( ord_less_int @ A @ B ) ) ).
% add_less_imp_less_right
thf(fact_789_add__less__imp__less__right,axiom,
! [A: real,C: real,B: real] :
( ( ord_less_real @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ B @ C ) )
=> ( ord_less_real @ A @ B ) ) ).
% add_less_imp_less_right
thf(fact_790_add__less__imp__less__left,axiom,
! [C: nat,A: nat,B: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B ) )
=> ( ord_less_nat @ A @ B ) ) ).
% add_less_imp_less_left
thf(fact_791_add__less__imp__less__left,axiom,
! [C: int,A: int,B: int] :
( ( ord_less_int @ ( plus_plus_int @ C @ A ) @ ( plus_plus_int @ C @ B ) )
=> ( ord_less_int @ A @ B ) ) ).
% add_less_imp_less_left
thf(fact_792_add__less__imp__less__left,axiom,
! [C: real,A: real,B: real] :
( ( ord_less_real @ ( plus_plus_real @ C @ A ) @ ( plus_plus_real @ C @ B ) )
=> ( ord_less_real @ A @ B ) ) ).
% add_less_imp_less_left
thf(fact_793_add__strict__right__mono,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ord_less_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ C ) ) ) ).
% add_strict_right_mono
thf(fact_794_add__strict__right__mono,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_int @ A @ B )
=> ( ord_less_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ C ) ) ) ).
% add_strict_right_mono
thf(fact_795_add__strict__right__mono,axiom,
! [A: real,B: real,C: real] :
( ( ord_less_real @ A @ B )
=> ( ord_less_real @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ B @ C ) ) ) ).
% add_strict_right_mono
thf(fact_796_add__strict__left__mono,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ord_less_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B ) ) ) ).
% add_strict_left_mono
thf(fact_797_add__strict__left__mono,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_int @ A @ B )
=> ( ord_less_int @ ( plus_plus_int @ C @ A ) @ ( plus_plus_int @ C @ B ) ) ) ).
% add_strict_left_mono
thf(fact_798_add__strict__left__mono,axiom,
! [A: real,B: real,C: real] :
( ( ord_less_real @ A @ B )
=> ( ord_less_real @ ( plus_plus_real @ C @ A ) @ ( plus_plus_real @ C @ B ) ) ) ).
% add_strict_left_mono
thf(fact_799_add__strict__mono,axiom,
! [A: nat,B: nat,C: nat,D: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_nat @ C @ D )
=> ( ord_less_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ D ) ) ) ) ).
% add_strict_mono
thf(fact_800_add__strict__mono,axiom,
! [A: int,B: int,C: int,D: int] :
( ( ord_less_int @ A @ B )
=> ( ( ord_less_int @ C @ D )
=> ( ord_less_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ D ) ) ) ) ).
% add_strict_mono
thf(fact_801_add__strict__mono,axiom,
! [A: real,B: real,C: real,D: real] :
( ( ord_less_real @ A @ B )
=> ( ( ord_less_real @ C @ D )
=> ( ord_less_real @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ B @ D ) ) ) ) ).
% add_strict_mono
thf(fact_802_add__mono__thms__linordered__field_I1_J,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ( ord_less_nat @ I @ J )
& ( K = L ) )
=> ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(1)
thf(fact_803_add__mono__thms__linordered__field_I1_J,axiom,
! [I: int,J: int,K: int,L: int] :
( ( ( ord_less_int @ I @ J )
& ( K = L ) )
=> ( ord_less_int @ ( plus_plus_int @ I @ K ) @ ( plus_plus_int @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(1)
thf(fact_804_add__mono__thms__linordered__field_I1_J,axiom,
! [I: real,J: real,K: real,L: real] :
( ( ( ord_less_real @ I @ J )
& ( K = L ) )
=> ( ord_less_real @ ( plus_plus_real @ I @ K ) @ ( plus_plus_real @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(1)
thf(fact_805_add__mono__thms__linordered__field_I2_J,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ( I = J )
& ( ord_less_nat @ K @ L ) )
=> ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(2)
thf(fact_806_add__mono__thms__linordered__field_I2_J,axiom,
! [I: int,J: int,K: int,L: int] :
( ( ( I = J )
& ( ord_less_int @ K @ L ) )
=> ( ord_less_int @ ( plus_plus_int @ I @ K ) @ ( plus_plus_int @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(2)
thf(fact_807_add__mono__thms__linordered__field_I2_J,axiom,
! [I: real,J: real,K: real,L: real] :
( ( ( I = J )
& ( ord_less_real @ K @ L ) )
=> ( ord_less_real @ ( plus_plus_real @ I @ K ) @ ( plus_plus_real @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(2)
thf(fact_808_add__mono__thms__linordered__field_I5_J,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ( ord_less_nat @ I @ J )
& ( ord_less_nat @ K @ L ) )
=> ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(5)
thf(fact_809_add__mono__thms__linordered__field_I5_J,axiom,
! [I: int,J: int,K: int,L: int] :
( ( ( ord_less_int @ I @ J )
& ( ord_less_int @ K @ L ) )
=> ( ord_less_int @ ( plus_plus_int @ I @ K ) @ ( plus_plus_int @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(5)
thf(fact_810_add__mono__thms__linordered__field_I5_J,axiom,
! [I: real,J: real,K: real,L: real] :
( ( ( ord_less_real @ I @ J )
& ( ord_less_real @ K @ L ) )
=> ( ord_less_real @ ( plus_plus_real @ I @ K ) @ ( plus_plus_real @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(5)
thf(fact_811_combine__common__factor,axiom,
! [A: nat,E: nat,B: nat,C: nat] :
( ( plus_plus_nat @ ( times_times_nat @ A @ E ) @ ( plus_plus_nat @ ( times_times_nat @ B @ E ) @ C ) )
= ( plus_plus_nat @ ( times_times_nat @ ( plus_plus_nat @ A @ B ) @ E ) @ C ) ) ).
% combine_common_factor
thf(fact_812_combine__common__factor,axiom,
! [A: real,E: real,B: real,C: real] :
( ( plus_plus_real @ ( times_times_real @ A @ E ) @ ( plus_plus_real @ ( times_times_real @ B @ E ) @ C ) )
= ( plus_plus_real @ ( times_times_real @ ( plus_plus_real @ A @ B ) @ E ) @ C ) ) ).
% combine_common_factor
thf(fact_813_combine__common__factor,axiom,
! [A: int,E: int,B: int,C: int] :
( ( plus_plus_int @ ( times_times_int @ A @ E ) @ ( plus_plus_int @ ( times_times_int @ B @ E ) @ C ) )
= ( plus_plus_int @ ( times_times_int @ ( plus_plus_int @ A @ B ) @ E ) @ C ) ) ).
% combine_common_factor
thf(fact_814_distrib__right,axiom,
! [A: nat,B: nat,C: nat] :
( ( times_times_nat @ ( plus_plus_nat @ A @ B ) @ C )
= ( plus_plus_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ C ) ) ) ).
% distrib_right
thf(fact_815_distrib__right,axiom,
! [A: real,B: real,C: real] :
( ( times_times_real @ ( plus_plus_real @ A @ B ) @ C )
= ( plus_plus_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ C ) ) ) ).
% distrib_right
thf(fact_816_distrib__right,axiom,
! [A: int,B: int,C: int] :
( ( times_times_int @ ( plus_plus_int @ A @ B ) @ C )
= ( plus_plus_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) ) ) ).
% distrib_right
thf(fact_817_distrib__left,axiom,
! [A: nat,B: nat,C: nat] :
( ( times_times_nat @ A @ ( plus_plus_nat @ B @ C ) )
= ( plus_plus_nat @ ( times_times_nat @ A @ B ) @ ( times_times_nat @ A @ C ) ) ) ).
% distrib_left
thf(fact_818_distrib__left,axiom,
! [A: real,B: real,C: real] :
( ( times_times_real @ A @ ( plus_plus_real @ B @ C ) )
= ( plus_plus_real @ ( times_times_real @ A @ B ) @ ( times_times_real @ A @ C ) ) ) ).
% distrib_left
thf(fact_819_distrib__left,axiom,
! [A: int,B: int,C: int] :
( ( times_times_int @ A @ ( plus_plus_int @ B @ C ) )
= ( plus_plus_int @ ( times_times_int @ A @ B ) @ ( times_times_int @ A @ C ) ) ) ).
% distrib_left
thf(fact_820_comm__semiring__class_Odistrib,axiom,
! [A: nat,B: nat,C: nat] :
( ( times_times_nat @ ( plus_plus_nat @ A @ B ) @ C )
= ( plus_plus_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ C ) ) ) ).
% comm_semiring_class.distrib
thf(fact_821_comm__semiring__class_Odistrib,axiom,
! [A: real,B: real,C: real] :
( ( times_times_real @ ( plus_plus_real @ A @ B ) @ C )
= ( plus_plus_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ C ) ) ) ).
% comm_semiring_class.distrib
thf(fact_822_comm__semiring__class_Odistrib,axiom,
! [A: int,B: int,C: int] :
( ( times_times_int @ ( plus_plus_int @ A @ B ) @ C )
= ( plus_plus_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) ) ) ).
% comm_semiring_class.distrib
thf(fact_823_ring__class_Oring__distribs_I1_J,axiom,
! [A: real,B: real,C: real] :
( ( times_times_real @ A @ ( plus_plus_real @ B @ C ) )
= ( plus_plus_real @ ( times_times_real @ A @ B ) @ ( times_times_real @ A @ C ) ) ) ).
% ring_class.ring_distribs(1)
thf(fact_824_ring__class_Oring__distribs_I1_J,axiom,
! [A: int,B: int,C: int] :
( ( times_times_int @ A @ ( plus_plus_int @ B @ C ) )
= ( plus_plus_int @ ( times_times_int @ A @ B ) @ ( times_times_int @ A @ C ) ) ) ).
% ring_class.ring_distribs(1)
thf(fact_825_ring__class_Oring__distribs_I2_J,axiom,
! [A: real,B: real,C: real] :
( ( times_times_real @ ( plus_plus_real @ A @ B ) @ C )
= ( plus_plus_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ C ) ) ) ).
% ring_class.ring_distribs(2)
thf(fact_826_ring__class_Oring__distribs_I2_J,axiom,
! [A: int,B: int,C: int] :
( ( times_times_int @ ( plus_plus_int @ A @ B ) @ C )
= ( plus_plus_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) ) ) ).
% ring_class.ring_distribs(2)
thf(fact_827_mult__hom_Ohom__add,axiom,
! [C: nat,X: nat,Y: nat] :
( ( times_times_nat @ C @ ( plus_plus_nat @ X @ Y ) )
= ( plus_plus_nat @ ( times_times_nat @ C @ X ) @ ( times_times_nat @ C @ Y ) ) ) ).
% mult_hom.hom_add
thf(fact_828_mult__hom_Ohom__add,axiom,
! [C: real,X: real,Y: real] :
( ( times_times_real @ C @ ( plus_plus_real @ X @ Y ) )
= ( plus_plus_real @ ( times_times_real @ C @ X ) @ ( times_times_real @ C @ Y ) ) ) ).
% mult_hom.hom_add
thf(fact_829_mult__hom_Ohom__add,axiom,
! [C: int,X: int,Y: int] :
( ( times_times_int @ C @ ( plus_plus_int @ X @ Y ) )
= ( plus_plus_int @ ( times_times_int @ C @ X ) @ ( times_times_int @ C @ Y ) ) ) ).
% mult_hom.hom_add
thf(fact_830_add__diff__add,axiom,
! [A: int,C: int,B: int,D: int] :
( ( minus_minus_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ D ) )
= ( plus_plus_int @ ( minus_minus_int @ A @ B ) @ ( minus_minus_int @ C @ D ) ) ) ).
% add_diff_add
thf(fact_831_group__cancel_Osub1,axiom,
! [A2: int,K: int,A: int,B: int] :
( ( A2
= ( plus_plus_int @ K @ A ) )
=> ( ( minus_minus_int @ A2 @ B )
= ( plus_plus_int @ K @ ( minus_minus_int @ A @ B ) ) ) ) ).
% group_cancel.sub1
thf(fact_832_diff__eq__eq,axiom,
! [A: int,B: int,C: int] :
( ( ( minus_minus_int @ A @ B )
= C )
= ( A
= ( plus_plus_int @ C @ B ) ) ) ).
% diff_eq_eq
thf(fact_833_eq__diff__eq,axiom,
! [A: int,C: int,B: int] :
( ( A
= ( minus_minus_int @ C @ B ) )
= ( ( plus_plus_int @ A @ B )
= C ) ) ).
% eq_diff_eq
thf(fact_834_add__diff__eq,axiom,
! [A: int,B: int,C: int] :
( ( plus_plus_int @ A @ ( minus_minus_int @ B @ C ) )
= ( minus_minus_int @ ( plus_plus_int @ A @ B ) @ C ) ) ).
% add_diff_eq
thf(fact_835_diff__diff__eq2,axiom,
! [A: int,B: int,C: int] :
( ( minus_minus_int @ A @ ( minus_minus_int @ B @ C ) )
= ( minus_minus_int @ ( plus_plus_int @ A @ C ) @ B ) ) ).
% diff_diff_eq2
thf(fact_836_diff__add__eq,axiom,
! [A: int,B: int,C: int] :
( ( plus_plus_int @ ( minus_minus_int @ A @ B ) @ C )
= ( minus_minus_int @ ( plus_plus_int @ A @ C ) @ B ) ) ).
% diff_add_eq
thf(fact_837_diff__add__eq__diff__diff__swap,axiom,
! [A: int,B: int,C: int] :
( ( minus_minus_int @ A @ ( plus_plus_int @ B @ C ) )
= ( minus_minus_int @ ( minus_minus_int @ A @ C ) @ B ) ) ).
% diff_add_eq_diff_diff_swap
thf(fact_838_add__implies__diff,axiom,
! [C: nat,B: nat,A: nat] :
( ( ( plus_plus_nat @ C @ B )
= A )
=> ( C
= ( minus_minus_nat @ A @ B ) ) ) ).
% add_implies_diff
thf(fact_839_add__implies__diff,axiom,
! [C: int,B: int,A: int] :
( ( ( plus_plus_int @ C @ B )
= A )
=> ( C
= ( minus_minus_int @ A @ B ) ) ) ).
% add_implies_diff
thf(fact_840_diff__diff__eq,axiom,
! [A: nat,B: nat,C: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ A @ B ) @ C )
= ( minus_minus_nat @ A @ ( plus_plus_nat @ B @ C ) ) ) ).
% diff_diff_eq
thf(fact_841_diff__diff__eq,axiom,
! [A: int,B: int,C: int] :
( ( minus_minus_int @ ( minus_minus_int @ A @ B ) @ C )
= ( minus_minus_int @ A @ ( plus_plus_int @ B @ C ) ) ) ).
% diff_diff_eq
thf(fact_842_dvd__add__right__iff,axiom,
! [A: nat,B: nat,C: nat] :
( ( dvd_dvd_nat @ A @ B )
=> ( ( dvd_dvd_nat @ A @ ( plus_plus_nat @ B @ C ) )
= ( dvd_dvd_nat @ A @ C ) ) ) ).
% dvd_add_right_iff
thf(fact_843_dvd__add__right__iff,axiom,
! [A: int,B: int,C: int] :
( ( dvd_dvd_int @ A @ B )
=> ( ( dvd_dvd_int @ A @ ( plus_plus_int @ B @ C ) )
= ( dvd_dvd_int @ A @ C ) ) ) ).
% dvd_add_right_iff
thf(fact_844_dvd__add__left__iff,axiom,
! [A: nat,C: nat,B: nat] :
( ( dvd_dvd_nat @ A @ C )
=> ( ( dvd_dvd_nat @ A @ ( plus_plus_nat @ B @ C ) )
= ( dvd_dvd_nat @ A @ B ) ) ) ).
% dvd_add_left_iff
thf(fact_845_dvd__add__left__iff,axiom,
! [A: int,C: int,B: int] :
( ( dvd_dvd_int @ A @ C )
=> ( ( dvd_dvd_int @ A @ ( plus_plus_int @ B @ C ) )
= ( dvd_dvd_int @ A @ B ) ) ) ).
% dvd_add_left_iff
thf(fact_846_dvd__add,axiom,
! [A: nat,B: nat,C: nat] :
( ( dvd_dvd_nat @ A @ B )
=> ( ( dvd_dvd_nat @ A @ C )
=> ( dvd_dvd_nat @ A @ ( plus_plus_nat @ B @ C ) ) ) ) ).
% dvd_add
thf(fact_847_dvd__add,axiom,
! [A: int,B: int,C: int] :
( ( dvd_dvd_int @ A @ B )
=> ( ( dvd_dvd_int @ A @ C )
=> ( dvd_dvd_int @ A @ ( plus_plus_int @ B @ C ) ) ) ) ).
% dvd_add
thf(fact_848_Euclid__induct,axiom,
! [P2: nat > nat > $o,A: nat,B: nat] :
( ! [A4: nat,B3: nat] :
( ( P2 @ A4 @ B3 )
= ( P2 @ B3 @ A4 ) )
=> ( ! [A4: nat] : ( P2 @ A4 @ zero_zero_nat )
=> ( ! [A4: nat,B3: nat] :
( ( P2 @ A4 @ B3 )
=> ( P2 @ A4 @ ( plus_plus_nat @ A4 @ B3 ) ) )
=> ( P2 @ A @ B ) ) ) ) ).
% Euclid_induct
thf(fact_849_plus__nat_Oadd__0,axiom,
! [N: nat] :
( ( plus_plus_nat @ zero_zero_nat @ N )
= N ) ).
% plus_nat.add_0
thf(fact_850_add__eq__self__zero,axiom,
! [M: nat,N: nat] :
( ( ( plus_plus_nat @ M @ N )
= M )
=> ( N = zero_zero_nat ) ) ).
% add_eq_self_zero
thf(fact_851_add__lessD1,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ I @ J ) @ K )
=> ( ord_less_nat @ I @ K ) ) ).
% add_lessD1
thf(fact_852_add__less__mono,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ord_less_nat @ I @ J )
=> ( ( ord_less_nat @ K @ L )
=> ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ) ).
% add_less_mono
thf(fact_853_not__add__less1,axiom,
! [I: nat,J: nat] :
~ ( ord_less_nat @ ( plus_plus_nat @ I @ J ) @ I ) ).
% not_add_less1
thf(fact_854_not__add__less2,axiom,
! [J: nat,I: nat] :
~ ( ord_less_nat @ ( plus_plus_nat @ J @ I ) @ I ) ).
% not_add_less2
thf(fact_855_add__less__mono1,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_nat @ I @ J )
=> ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ K ) ) ) ).
% add_less_mono1
thf(fact_856_trans__less__add1,axiom,
! [I: nat,J: nat,M: nat] :
( ( ord_less_nat @ I @ J )
=> ( ord_less_nat @ I @ ( plus_plus_nat @ J @ M ) ) ) ).
% trans_less_add1
thf(fact_857_trans__less__add2,axiom,
! [I: nat,J: nat,M: nat] :
( ( ord_less_nat @ I @ J )
=> ( ord_less_nat @ I @ ( plus_plus_nat @ M @ J ) ) ) ).
% trans_less_add2
thf(fact_858_less__add__eq__less,axiom,
! [K: nat,L: nat,M: nat,N: nat] :
( ( ord_less_nat @ K @ L )
=> ( ( ( plus_plus_nat @ M @ L )
= ( plus_plus_nat @ K @ N ) )
=> ( ord_less_nat @ M @ N ) ) ) ).
% less_add_eq_less
thf(fact_859_Nat_Odiff__cancel,axiom,
! [K: nat,M: nat,N: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ K @ M ) @ ( plus_plus_nat @ K @ N ) )
= ( minus_minus_nat @ M @ N ) ) ).
% Nat.diff_cancel
thf(fact_860_diff__cancel2,axiom,
! [M: nat,K: nat,N: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ M @ K ) @ ( plus_plus_nat @ N @ K ) )
= ( minus_minus_nat @ M @ N ) ) ).
% diff_cancel2
thf(fact_861_diff__add__inverse,axiom,
! [N: nat,M: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ N @ M ) @ N )
= M ) ).
% diff_add_inverse
thf(fact_862_diff__add__inverse2,axiom,
! [M: nat,N: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ M @ N ) @ N )
= M ) ).
% diff_add_inverse2
thf(fact_863_left__add__mult__distrib,axiom,
! [I: nat,U: nat,J: nat,K: nat] :
( ( plus_plus_nat @ ( times_times_nat @ I @ U ) @ ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ K ) )
= ( plus_plus_nat @ ( times_times_nat @ ( plus_plus_nat @ I @ J ) @ U ) @ K ) ) ).
% left_add_mult_distrib
thf(fact_864_add__mult__distrib,axiom,
! [M: nat,N: nat,K: nat] :
( ( times_times_nat @ ( plus_plus_nat @ M @ N ) @ K )
= ( plus_plus_nat @ ( times_times_nat @ M @ K ) @ ( times_times_nat @ N @ K ) ) ) ).
% add_mult_distrib
thf(fact_865_add__mult__distrib2,axiom,
! [K: nat,M: nat,N: nat] :
( ( times_times_nat @ K @ ( plus_plus_nat @ M @ N ) )
= ( plus_plus_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) ) ) ).
% add_mult_distrib2
thf(fact_866_cong__add__rcancel,axiom,
! [X: int,A: int,Y: int,N: int] :
( ( unique651150874487253600ng_int @ ( plus_plus_int @ X @ A ) @ ( plus_plus_int @ Y @ A ) @ N )
= ( unique651150874487253600ng_int @ X @ Y @ N ) ) ).
% cong_add_rcancel
thf(fact_867_cong__add__lcancel,axiom,
! [A: int,X: int,Y: int,N: int] :
( ( unique651150874487253600ng_int @ ( plus_plus_int @ A @ X ) @ ( plus_plus_int @ A @ Y ) @ N )
= ( unique651150874487253600ng_int @ X @ Y @ N ) ) ).
% cong_add_lcancel
thf(fact_868_cong__add,axiom,
! [B: nat,C: nat,A: nat,D: nat,E: nat] :
( ( unique653641344996303876ng_nat @ B @ C @ A )
=> ( ( unique653641344996303876ng_nat @ D @ E @ A )
=> ( unique653641344996303876ng_nat @ ( plus_plus_nat @ B @ D ) @ ( plus_plus_nat @ C @ E ) @ A ) ) ) ).
% cong_add
thf(fact_869_cong__add,axiom,
! [B: int,C: int,A: int,D: int,E: int] :
( ( unique651150874487253600ng_int @ B @ C @ A )
=> ( ( unique651150874487253600ng_int @ D @ E @ A )
=> ( unique651150874487253600ng_int @ ( plus_plus_int @ B @ D ) @ ( plus_plus_int @ C @ E ) @ A ) ) ) ).
% cong_add
thf(fact_870_cong__add__rcancel__nat,axiom,
! [X: nat,A: nat,Y: nat,N: nat] :
( ( unique653641344996303876ng_nat @ ( plus_plus_nat @ X @ A ) @ ( plus_plus_nat @ Y @ A ) @ N )
= ( unique653641344996303876ng_nat @ X @ Y @ N ) ) ).
% cong_add_rcancel_nat
thf(fact_871_cong__add__lcancel__nat,axiom,
! [A: nat,X: nat,Y: nat,N: nat] :
( ( unique653641344996303876ng_nat @ ( plus_plus_nat @ A @ X ) @ ( plus_plus_nat @ A @ Y ) @ N )
= ( unique653641344996303876ng_nat @ X @ Y @ N ) ) ).
% cong_add_lcancel_nat
thf(fact_872_minf_I10_J,axiom,
! [D: nat,S: nat] :
? [Z4: nat] :
! [X7: nat] :
( ( ord_less_nat @ X7 @ Z4 )
=> ( ( ~ ( dvd_dvd_nat @ D @ ( plus_plus_nat @ X7 @ S ) ) )
= ( ~ ( dvd_dvd_nat @ D @ ( plus_plus_nat @ X7 @ S ) ) ) ) ) ).
% minf(10)
thf(fact_873_minf_I10_J,axiom,
! [D: int,S: int] :
? [Z4: int] :
! [X7: int] :
( ( ord_less_int @ X7 @ Z4 )
=> ( ( ~ ( dvd_dvd_int @ D @ ( plus_plus_int @ X7 @ S ) ) )
= ( ~ ( dvd_dvd_int @ D @ ( plus_plus_int @ X7 @ S ) ) ) ) ) ).
% minf(10)
thf(fact_874_minf_I10_J,axiom,
! [D: real,S: real] :
? [Z4: real] :
! [X7: real] :
( ( ord_less_real @ X7 @ Z4 )
=> ( ( ~ ( dvd_dvd_real @ D @ ( plus_plus_real @ X7 @ S ) ) )
= ( ~ ( dvd_dvd_real @ D @ ( plus_plus_real @ X7 @ S ) ) ) ) ) ).
% minf(10)
thf(fact_875_minf_I9_J,axiom,
! [D: nat,S: nat] :
? [Z4: nat] :
! [X7: nat] :
( ( ord_less_nat @ X7 @ Z4 )
=> ( ( dvd_dvd_nat @ D @ ( plus_plus_nat @ X7 @ S ) )
= ( dvd_dvd_nat @ D @ ( plus_plus_nat @ X7 @ S ) ) ) ) ).
% minf(9)
thf(fact_876_minf_I9_J,axiom,
! [D: int,S: int] :
? [Z4: int] :
! [X7: int] :
( ( ord_less_int @ X7 @ Z4 )
=> ( ( dvd_dvd_int @ D @ ( plus_plus_int @ X7 @ S ) )
= ( dvd_dvd_int @ D @ ( plus_plus_int @ X7 @ S ) ) ) ) ).
% minf(9)
thf(fact_877_minf_I9_J,axiom,
! [D: real,S: real] :
? [Z4: real] :
! [X7: real] :
( ( ord_less_real @ X7 @ Z4 )
=> ( ( dvd_dvd_real @ D @ ( plus_plus_real @ X7 @ S ) )
= ( dvd_dvd_real @ D @ ( plus_plus_real @ X7 @ S ) ) ) ) ).
% minf(9)
thf(fact_878_pinf_I10_J,axiom,
! [D: nat,S: nat] :
? [Z4: nat] :
! [X7: nat] :
( ( ord_less_nat @ Z4 @ X7 )
=> ( ( ~ ( dvd_dvd_nat @ D @ ( plus_plus_nat @ X7 @ S ) ) )
= ( ~ ( dvd_dvd_nat @ D @ ( plus_plus_nat @ X7 @ S ) ) ) ) ) ).
% pinf(10)
thf(fact_879_pinf_I10_J,axiom,
! [D: int,S: int] :
? [Z4: int] :
! [X7: int] :
( ( ord_less_int @ Z4 @ X7 )
=> ( ( ~ ( dvd_dvd_int @ D @ ( plus_plus_int @ X7 @ S ) ) )
= ( ~ ( dvd_dvd_int @ D @ ( plus_plus_int @ X7 @ S ) ) ) ) ) ).
% pinf(10)
thf(fact_880_pinf_I10_J,axiom,
! [D: real,S: real] :
? [Z4: real] :
! [X7: real] :
( ( ord_less_real @ Z4 @ X7 )
=> ( ( ~ ( dvd_dvd_real @ D @ ( plus_plus_real @ X7 @ S ) ) )
= ( ~ ( dvd_dvd_real @ D @ ( plus_plus_real @ X7 @ S ) ) ) ) ) ).
% pinf(10)
thf(fact_881_pinf_I9_J,axiom,
! [D: nat,S: nat] :
? [Z4: nat] :
! [X7: nat] :
( ( ord_less_nat @ Z4 @ X7 )
=> ( ( dvd_dvd_nat @ D @ ( plus_plus_nat @ X7 @ S ) )
= ( dvd_dvd_nat @ D @ ( plus_plus_nat @ X7 @ S ) ) ) ) ).
% pinf(9)
thf(fact_882_pinf_I9_J,axiom,
! [D: int,S: int] :
? [Z4: int] :
! [X7: int] :
( ( ord_less_int @ Z4 @ X7 )
=> ( ( dvd_dvd_int @ D @ ( plus_plus_int @ X7 @ S ) )
= ( dvd_dvd_int @ D @ ( plus_plus_int @ X7 @ S ) ) ) ) ).
% pinf(9)
thf(fact_883_pinf_I9_J,axiom,
! [D: real,S: real] :
? [Z4: real] :
! [X7: real] :
( ( ord_less_real @ Z4 @ X7 )
=> ( ( dvd_dvd_real @ D @ ( plus_plus_real @ X7 @ S ) )
= ( dvd_dvd_real @ D @ ( plus_plus_real @ X7 @ S ) ) ) ) ).
% pinf(9)
thf(fact_884_unity__coeff__ex,axiom,
! [P2: nat > $o,L: nat] :
( ( ? [X5: nat] : ( P2 @ ( times_times_nat @ L @ X5 ) ) )
= ( ? [X5: nat] :
( ( dvd_dvd_nat @ L @ ( plus_plus_nat @ X5 @ zero_zero_nat ) )
& ( P2 @ X5 ) ) ) ) ).
% unity_coeff_ex
thf(fact_885_unity__coeff__ex,axiom,
! [P2: real > $o,L: real] :
( ( ? [X5: real] : ( P2 @ ( times_times_real @ L @ X5 ) ) )
= ( ? [X5: real] :
( ( dvd_dvd_real @ L @ ( plus_plus_real @ X5 @ zero_zero_real ) )
& ( P2 @ X5 ) ) ) ) ).
% unity_coeff_ex
thf(fact_886_unity__coeff__ex,axiom,
! [P2: int > $o,L: int] :
( ( ? [X5: int] : ( P2 @ ( times_times_int @ L @ X5 ) ) )
= ( ? [X5: int] :
( ( dvd_dvd_int @ L @ ( plus_plus_int @ X5 @ zero_zero_int ) )
& ( P2 @ X5 ) ) ) ) ).
% unity_coeff_ex
thf(fact_887_inf__period_I4_J,axiom,
! [D: real,D4: real,T: real] :
( ( dvd_dvd_real @ D @ D4 )
=> ! [X7: real,K6: real] :
( ( ~ ( dvd_dvd_real @ D @ ( plus_plus_real @ X7 @ T ) ) )
= ( ~ ( dvd_dvd_real @ D @ ( plus_plus_real @ ( minus_minus_real @ X7 @ ( times_times_real @ K6 @ D4 ) ) @ T ) ) ) ) ) ).
% inf_period(4)
thf(fact_888_inf__period_I4_J,axiom,
! [D: int,D4: int,T: int] :
( ( dvd_dvd_int @ D @ D4 )
=> ! [X7: int,K6: int] :
( ( ~ ( dvd_dvd_int @ D @ ( plus_plus_int @ X7 @ T ) ) )
= ( ~ ( dvd_dvd_int @ D @ ( plus_plus_int @ ( minus_minus_int @ X7 @ ( times_times_int @ K6 @ D4 ) ) @ T ) ) ) ) ) ).
% inf_period(4)
thf(fact_889_inf__period_I3_J,axiom,
! [D: real,D4: real,T: real] :
( ( dvd_dvd_real @ D @ D4 )
=> ! [X7: real,K6: real] :
( ( dvd_dvd_real @ D @ ( plus_plus_real @ X7 @ T ) )
= ( dvd_dvd_real @ D @ ( plus_plus_real @ ( minus_minus_real @ X7 @ ( times_times_real @ K6 @ D4 ) ) @ T ) ) ) ) ).
% inf_period(3)
thf(fact_890_inf__period_I3_J,axiom,
! [D: int,D4: int,T: int] :
( ( dvd_dvd_int @ D @ D4 )
=> ! [X7: int,K6: int] :
( ( dvd_dvd_int @ D @ ( plus_plus_int @ X7 @ T ) )
= ( dvd_dvd_int @ D @ ( plus_plus_int @ ( minus_minus_int @ X7 @ ( times_times_int @ K6 @ D4 ) ) @ T ) ) ) ) ).
% inf_period(3)
thf(fact_891_pos__add__strict,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_nat @ zero_zero_nat @ A )
=> ( ( ord_less_nat @ B @ C )
=> ( ord_less_nat @ B @ ( plus_plus_nat @ A @ C ) ) ) ) ).
% pos_add_strict
thf(fact_892_pos__add__strict,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_int @ zero_zero_int @ A )
=> ( ( ord_less_int @ B @ C )
=> ( ord_less_int @ B @ ( plus_plus_int @ A @ C ) ) ) ) ).
% pos_add_strict
thf(fact_893_pos__add__strict,axiom,
! [A: real,B: real,C: real] :
( ( ord_less_real @ zero_zero_real @ A )
=> ( ( ord_less_real @ B @ C )
=> ( ord_less_real @ B @ ( plus_plus_real @ A @ C ) ) ) ) ).
% pos_add_strict
thf(fact_894_canonically__ordered__monoid__add__class_OlessE,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ B )
=> ~ ! [C4: nat] :
( ( B
= ( plus_plus_nat @ A @ C4 ) )
=> ( C4 = zero_zero_nat ) ) ) ).
% canonically_ordered_monoid_add_class.lessE
thf(fact_895_add__pos__pos,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ zero_zero_nat @ A )
=> ( ( ord_less_nat @ zero_zero_nat @ B )
=> ( ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ A @ B ) ) ) ) ).
% add_pos_pos
thf(fact_896_add__pos__pos,axiom,
! [A: int,B: int] :
( ( ord_less_int @ zero_zero_int @ A )
=> ( ( ord_less_int @ zero_zero_int @ B )
=> ( ord_less_int @ zero_zero_int @ ( plus_plus_int @ A @ B ) ) ) ) ).
% add_pos_pos
thf(fact_897_add__pos__pos,axiom,
! [A: real,B: real] :
( ( ord_less_real @ zero_zero_real @ A )
=> ( ( ord_less_real @ zero_zero_real @ B )
=> ( ord_less_real @ zero_zero_real @ ( plus_plus_real @ A @ B ) ) ) ) ).
% add_pos_pos
thf(fact_898_add__neg__neg,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ zero_zero_nat )
=> ( ( ord_less_nat @ B @ zero_zero_nat )
=> ( ord_less_nat @ ( plus_plus_nat @ A @ B ) @ zero_zero_nat ) ) ) ).
% add_neg_neg
thf(fact_899_add__neg__neg,axiom,
! [A: int,B: int] :
( ( ord_less_int @ A @ zero_zero_int )
=> ( ( ord_less_int @ B @ zero_zero_int )
=> ( ord_less_int @ ( plus_plus_int @ A @ B ) @ zero_zero_int ) ) ) ).
% add_neg_neg
thf(fact_900_add__neg__neg,axiom,
! [A: real,B: real] :
( ( ord_less_real @ A @ zero_zero_real )
=> ( ( ord_less_real @ B @ zero_zero_real )
=> ( ord_less_real @ ( plus_plus_real @ A @ B ) @ zero_zero_real ) ) ) ).
% add_neg_neg
thf(fact_901_add__less__zeroD,axiom,
! [X: int,Y: int] :
( ( ord_less_int @ ( plus_plus_int @ X @ Y ) @ zero_zero_int )
=> ( ( ord_less_int @ X @ zero_zero_int )
| ( ord_less_int @ Y @ zero_zero_int ) ) ) ).
% add_less_zeroD
thf(fact_902_add__less__zeroD,axiom,
! [X: real,Y: real] :
( ( ord_less_real @ ( plus_plus_real @ X @ Y ) @ zero_zero_real )
=> ( ( ord_less_real @ X @ zero_zero_real )
| ( ord_less_real @ Y @ zero_zero_real ) ) ) ).
% add_less_zeroD
thf(fact_903_mult__hom_Ohom__add__eq__zero,axiom,
! [X: nat,Y: nat,C: nat] :
( ( ( plus_plus_nat @ X @ Y )
= zero_zero_nat )
=> ( ( plus_plus_nat @ ( times_times_nat @ C @ X ) @ ( times_times_nat @ C @ Y ) )
= zero_zero_nat ) ) ).
% mult_hom.hom_add_eq_zero
thf(fact_904_mult__hom_Ohom__add__eq__zero,axiom,
! [X: real,Y: real,C: real] :
( ( ( plus_plus_real @ X @ Y )
= zero_zero_real )
=> ( ( plus_plus_real @ ( times_times_real @ C @ X ) @ ( times_times_real @ C @ Y ) )
= zero_zero_real ) ) ).
% mult_hom.hom_add_eq_zero
thf(fact_905_mult__hom_Ohom__add__eq__zero,axiom,
! [X: int,Y: int,C: int] :
( ( ( plus_plus_int @ X @ Y )
= zero_zero_int )
=> ( ( plus_plus_int @ ( times_times_int @ C @ X ) @ ( times_times_int @ C @ Y ) )
= zero_zero_int ) ) ).
% mult_hom.hom_add_eq_zero
thf(fact_906_add__mono1,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ B )
=> ( ord_less_nat @ ( plus_plus_nat @ A @ one_one_nat ) @ ( plus_plus_nat @ B @ one_one_nat ) ) ) ).
% add_mono1
thf(fact_907_add__mono1,axiom,
! [A: int,B: int] :
( ( ord_less_int @ A @ B )
=> ( ord_less_int @ ( plus_plus_int @ A @ one_one_int ) @ ( plus_plus_int @ B @ one_one_int ) ) ) ).
% add_mono1
thf(fact_908_add__mono1,axiom,
! [A: real,B: real] :
( ( ord_less_real @ A @ B )
=> ( ord_less_real @ ( plus_plus_real @ A @ one_one_real ) @ ( plus_plus_real @ B @ one_one_real ) ) ) ).
% add_mono1
thf(fact_909_less__add__one,axiom,
! [A: nat] : ( ord_less_nat @ A @ ( plus_plus_nat @ A @ one_one_nat ) ) ).
% less_add_one
thf(fact_910_less__add__one,axiom,
! [A: int] : ( ord_less_int @ A @ ( plus_plus_int @ A @ one_one_int ) ) ).
% less_add_one
thf(fact_911_less__add__one,axiom,
! [A: real] : ( ord_less_real @ A @ ( plus_plus_real @ A @ one_one_real ) ) ).
% less_add_one
thf(fact_912_diff__less__eq,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_int @ ( minus_minus_int @ A @ B ) @ C )
= ( ord_less_int @ A @ ( plus_plus_int @ C @ B ) ) ) ).
% diff_less_eq
thf(fact_913_diff__less__eq,axiom,
! [A: real,B: real,C: real] :
( ( ord_less_real @ ( minus_minus_real @ A @ B ) @ C )
= ( ord_less_real @ A @ ( plus_plus_real @ C @ B ) ) ) ).
% diff_less_eq
thf(fact_914_less__diff__eq,axiom,
! [A: int,C: int,B: int] :
( ( ord_less_int @ A @ ( minus_minus_int @ C @ B ) )
= ( ord_less_int @ ( plus_plus_int @ A @ B ) @ C ) ) ).
% less_diff_eq
thf(fact_915_less__diff__eq,axiom,
! [A: real,C: real,B: real] :
( ( ord_less_real @ A @ ( minus_minus_real @ C @ B ) )
= ( ord_less_real @ ( plus_plus_real @ A @ B ) @ C ) ) ).
% less_diff_eq
thf(fact_916_linordered__semidom__class_Oadd__diff__inverse,axiom,
! [A: nat,B: nat] :
( ~ ( ord_less_nat @ A @ B )
=> ( ( plus_plus_nat @ B @ ( minus_minus_nat @ A @ B ) )
= A ) ) ).
% linordered_semidom_class.add_diff_inverse
thf(fact_917_linordered__semidom__class_Oadd__diff__inverse,axiom,
! [A: int,B: int] :
( ~ ( ord_less_int @ A @ B )
=> ( ( plus_plus_int @ B @ ( minus_minus_int @ A @ B ) )
= A ) ) ).
% linordered_semidom_class.add_diff_inverse
thf(fact_918_linordered__semidom__class_Oadd__diff__inverse,axiom,
! [A: real,B: real] :
( ~ ( ord_less_real @ A @ B )
=> ( ( plus_plus_real @ B @ ( minus_minus_real @ A @ B ) )
= A ) ) ).
% linordered_semidom_class.add_diff_inverse
thf(fact_919_square__diff__square__factored,axiom,
! [X: real,Y: real] :
( ( minus_minus_real @ ( times_times_real @ X @ X ) @ ( times_times_real @ Y @ Y ) )
= ( times_times_real @ ( plus_plus_real @ X @ Y ) @ ( minus_minus_real @ X @ Y ) ) ) ).
% square_diff_square_factored
thf(fact_920_square__diff__square__factored,axiom,
! [X: int,Y: int] :
( ( minus_minus_int @ ( times_times_int @ X @ X ) @ ( times_times_int @ Y @ Y ) )
= ( times_times_int @ ( plus_plus_int @ X @ Y ) @ ( minus_minus_int @ X @ Y ) ) ) ).
% square_diff_square_factored
thf(fact_921_eq__add__iff2,axiom,
! [A: real,E: real,C: real,B: real,D: real] :
( ( ( plus_plus_real @ ( times_times_real @ A @ E ) @ C )
= ( plus_plus_real @ ( times_times_real @ B @ E ) @ D ) )
= ( C
= ( plus_plus_real @ ( times_times_real @ ( minus_minus_real @ B @ A ) @ E ) @ D ) ) ) ).
% eq_add_iff2
thf(fact_922_eq__add__iff2,axiom,
! [A: int,E: int,C: int,B: int,D: int] :
( ( ( plus_plus_int @ ( times_times_int @ A @ E ) @ C )
= ( plus_plus_int @ ( times_times_int @ B @ E ) @ D ) )
= ( C
= ( plus_plus_int @ ( times_times_int @ ( minus_minus_int @ B @ A ) @ E ) @ D ) ) ) ).
% eq_add_iff2
thf(fact_923_eq__add__iff1,axiom,
! [A: real,E: real,C: real,B: real,D: real] :
( ( ( plus_plus_real @ ( times_times_real @ A @ E ) @ C )
= ( plus_plus_real @ ( times_times_real @ B @ E ) @ D ) )
= ( ( plus_plus_real @ ( times_times_real @ ( minus_minus_real @ A @ B ) @ E ) @ C )
= D ) ) ).
% eq_add_iff1
thf(fact_924_eq__add__iff1,axiom,
! [A: int,E: int,C: int,B: int,D: int] :
( ( ( plus_plus_int @ ( times_times_int @ A @ E ) @ C )
= ( plus_plus_int @ ( times_times_int @ B @ E ) @ D ) )
= ( ( plus_plus_int @ ( times_times_int @ ( minus_minus_int @ A @ B ) @ E ) @ C )
= D ) ) ).
% eq_add_iff1
thf(fact_925_mult__diff__mult,axiom,
! [X: real,Y: real,A: real,B: real] :
( ( minus_minus_real @ ( times_times_real @ X @ Y ) @ ( times_times_real @ A @ B ) )
= ( plus_plus_real @ ( times_times_real @ X @ ( minus_minus_real @ Y @ B ) ) @ ( times_times_real @ ( minus_minus_real @ X @ A ) @ B ) ) ) ).
% mult_diff_mult
thf(fact_926_mult__diff__mult,axiom,
! [X: int,Y: int,A: int,B: int] :
( ( minus_minus_int @ ( times_times_int @ X @ Y ) @ ( times_times_int @ A @ B ) )
= ( plus_plus_int @ ( times_times_int @ X @ ( minus_minus_int @ Y @ B ) ) @ ( times_times_int @ ( minus_minus_int @ X @ A ) @ B ) ) ) ).
% mult_diff_mult
thf(fact_927_power__add,axiom,
! [A: nat,M: nat,N: nat] :
( ( power_power_nat @ A @ ( plus_plus_nat @ M @ N ) )
= ( times_times_nat @ ( power_power_nat @ A @ M ) @ ( power_power_nat @ A @ N ) ) ) ).
% power_add
thf(fact_928_power__add,axiom,
! [A: real,M: nat,N: nat] :
( ( power_power_real @ A @ ( plus_plus_nat @ M @ N ) )
= ( times_times_real @ ( power_power_real @ A @ M ) @ ( power_power_real @ A @ N ) ) ) ).
% power_add
thf(fact_929_power__add,axiom,
! [A: int,M: nat,N: nat] :
( ( power_power_int @ A @ ( plus_plus_nat @ M @ N ) )
= ( times_times_int @ ( power_power_int @ A @ M ) @ ( power_power_int @ A @ N ) ) ) ).
% power_add
thf(fact_930_less__imp__add__positive,axiom,
! [I: nat,J: nat] :
( ( ord_less_nat @ I @ J )
=> ? [K4: nat] :
( ( ord_less_nat @ zero_zero_nat @ K4 )
& ( ( plus_plus_nat @ I @ K4 )
= J ) ) ) ).
% less_imp_add_positive
thf(fact_931_cong__add__lcancel__0,axiom,
! [A: int,X: int,N: int] :
( ( unique651150874487253600ng_int @ ( plus_plus_int @ A @ X ) @ A @ N )
= ( unique651150874487253600ng_int @ X @ zero_zero_int @ N ) ) ).
% cong_add_lcancel_0
thf(fact_932_cong__add__rcancel__0,axiom,
! [X: int,A: int,N: int] :
( ( unique651150874487253600ng_int @ ( plus_plus_int @ X @ A ) @ A @ N )
= ( unique651150874487253600ng_int @ X @ zero_zero_int @ N ) ) ).
% cong_add_rcancel_0
thf(fact_933_cong__iff__lin,axiom,
( unique651150874487253600ng_int
= ( ^ [A3: int,B2: int,M2: int] :
? [K3: int] :
( B2
= ( plus_plus_int @ A3 @ ( times_times_int @ M2 @ K3 ) ) ) ) ) ).
% cong_iff_lin
thf(fact_934_diff__add__0,axiom,
! [N: nat,M: nat] :
( ( minus_minus_nat @ N @ ( plus_plus_nat @ N @ M ) )
= zero_zero_nat ) ).
% diff_add_0
thf(fact_935_add__diff__inverse__nat,axiom,
! [M: nat,N: nat] :
( ~ ( ord_less_nat @ M @ N )
=> ( ( plus_plus_nat @ N @ ( minus_minus_nat @ M @ N ) )
= M ) ) ).
% add_diff_inverse_nat
thf(fact_936_less__diff__conv,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_nat @ I @ ( minus_minus_nat @ J @ K ) )
= ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ J ) ) ).
% less_diff_conv
thf(fact_937_bezout__lemma__nat,axiom,
! [D: nat,A: nat,B: nat,X: nat,Y: nat] :
( ( dvd_dvd_nat @ D @ A )
=> ( ( dvd_dvd_nat @ D @ B )
=> ( ( ( ( times_times_nat @ A @ X )
= ( plus_plus_nat @ ( times_times_nat @ B @ Y ) @ D ) )
| ( ( times_times_nat @ B @ X )
= ( plus_plus_nat @ ( times_times_nat @ A @ Y ) @ D ) ) )
=> ? [X3: nat,Y4: nat] :
( ( dvd_dvd_nat @ D @ A )
& ( dvd_dvd_nat @ D @ ( plus_plus_nat @ A @ B ) )
& ( ( ( times_times_nat @ A @ X3 )
= ( plus_plus_nat @ ( times_times_nat @ ( plus_plus_nat @ A @ B ) @ Y4 ) @ D ) )
| ( ( times_times_nat @ ( plus_plus_nat @ A @ B ) @ X3 )
= ( plus_plus_nat @ ( times_times_nat @ A @ Y4 ) @ D ) ) ) ) ) ) ) ).
% bezout_lemma_nat
thf(fact_938_bezout__add__nat,axiom,
! [A: nat,B: nat] :
? [D3: nat,X3: nat,Y4: nat] :
( ( dvd_dvd_nat @ D3 @ A )
& ( dvd_dvd_nat @ D3 @ B )
& ( ( ( times_times_nat @ A @ X3 )
= ( plus_plus_nat @ ( times_times_nat @ B @ Y4 ) @ D3 ) )
| ( ( times_times_nat @ B @ X3 )
= ( plus_plus_nat @ ( times_times_nat @ A @ Y4 ) @ D3 ) ) ) ) ).
% bezout_add_nat
thf(fact_939_cong__add__lcancel__0__nat,axiom,
! [A: nat,X: nat,N: nat] :
( ( unique653641344996303876ng_nat @ ( plus_plus_nat @ A @ X ) @ A @ N )
= ( unique653641344996303876ng_nat @ X @ zero_zero_nat @ N ) ) ).
% cong_add_lcancel_0_nat
thf(fact_940_cong__add__rcancel__0__nat,axiom,
! [X: nat,A: nat,N: nat] :
( ( unique653641344996303876ng_nat @ ( plus_plus_nat @ X @ A ) @ A @ N )
= ( unique653641344996303876ng_nat @ X @ zero_zero_nat @ N ) ) ).
% cong_add_rcancel_0_nat
thf(fact_941_cong__iff__lin__nat,axiom,
( unique653641344996303876ng_nat
= ( ^ [A3: nat,B2: nat,M2: nat] :
? [K12: nat,K22: nat] :
( ( plus_plus_nat @ B2 @ ( times_times_nat @ K12 @ M2 ) )
= ( plus_plus_nat @ A3 @ ( times_times_nat @ K22 @ M2 ) ) ) ) ) ).
% cong_iff_lin_nat
thf(fact_942_sum__squares__gt__zero__iff,axiom,
! [X: int,Y: int] :
( ( ord_less_int @ zero_zero_int @ ( plus_plus_int @ ( times_times_int @ X @ X ) @ ( times_times_int @ Y @ Y ) ) )
= ( ( X != zero_zero_int )
| ( Y != zero_zero_int ) ) ) ).
% sum_squares_gt_zero_iff
thf(fact_943_sum__squares__gt__zero__iff,axiom,
! [X: real,Y: real] :
( ( ord_less_real @ zero_zero_real @ ( plus_plus_real @ ( times_times_real @ X @ X ) @ ( times_times_real @ Y @ Y ) ) )
= ( ( X != zero_zero_real )
| ( Y != zero_zero_real ) ) ) ).
% sum_squares_gt_zero_iff
thf(fact_944_not__sum__squares__lt__zero,axiom,
! [X: int,Y: int] :
~ ( ord_less_int @ ( plus_plus_int @ ( times_times_int @ X @ X ) @ ( times_times_int @ Y @ Y ) ) @ zero_zero_int ) ).
% not_sum_squares_lt_zero
thf(fact_945_not__sum__squares__lt__zero,axiom,
! [X: real,Y: real] :
~ ( ord_less_real @ ( plus_plus_real @ ( times_times_real @ X @ X ) @ ( times_times_real @ Y @ Y ) ) @ zero_zero_real ) ).
% not_sum_squares_lt_zero
thf(fact_946_zero__less__two,axiom,
ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ one_one_nat @ one_one_nat ) ).
% zero_less_two
thf(fact_947_zero__less__two,axiom,
ord_less_int @ zero_zero_int @ ( plus_plus_int @ one_one_int @ one_one_int ) ).
% zero_less_two
thf(fact_948_zero__less__two,axiom,
ord_less_real @ zero_zero_real @ ( plus_plus_real @ one_one_real @ one_one_real ) ).
% zero_less_two
thf(fact_949_less__add__iff2,axiom,
! [A: int,E: int,C: int,B: int,D: int] :
( ( ord_less_int @ ( plus_plus_int @ ( times_times_int @ A @ E ) @ C ) @ ( plus_plus_int @ ( times_times_int @ B @ E ) @ D ) )
= ( ord_less_int @ C @ ( plus_plus_int @ ( times_times_int @ ( minus_minus_int @ B @ A ) @ E ) @ D ) ) ) ).
% less_add_iff2
thf(fact_950_less__add__iff2,axiom,
! [A: real,E: real,C: real,B: real,D: real] :
( ( ord_less_real @ ( plus_plus_real @ ( times_times_real @ A @ E ) @ C ) @ ( plus_plus_real @ ( times_times_real @ B @ E ) @ D ) )
= ( ord_less_real @ C @ ( plus_plus_real @ ( times_times_real @ ( minus_minus_real @ B @ A ) @ E ) @ D ) ) ) ).
% less_add_iff2
thf(fact_951_less__add__iff1,axiom,
! [A: int,E: int,C: int,B: int,D: int] :
( ( ord_less_int @ ( plus_plus_int @ ( times_times_int @ A @ E ) @ C ) @ ( plus_plus_int @ ( times_times_int @ B @ E ) @ D ) )
= ( ord_less_int @ ( plus_plus_int @ ( times_times_int @ ( minus_minus_int @ A @ B ) @ E ) @ C ) @ D ) ) ).
% less_add_iff1
thf(fact_952_less__add__iff1,axiom,
! [A: real,E: real,C: real,B: real,D: real] :
( ( ord_less_real @ ( plus_plus_real @ ( times_times_real @ A @ E ) @ C ) @ ( plus_plus_real @ ( times_times_real @ B @ E ) @ D ) )
= ( ord_less_real @ ( plus_plus_real @ ( times_times_real @ ( minus_minus_real @ A @ B ) @ E ) @ C ) @ D ) ) ).
% less_add_iff1
thf(fact_953_square__diff__one__factored,axiom,
! [X: real] :
( ( minus_minus_real @ ( times_times_real @ X @ X ) @ one_one_real )
= ( times_times_real @ ( plus_plus_real @ X @ one_one_real ) @ ( minus_minus_real @ X @ one_one_real ) ) ) ).
% square_diff_one_factored
thf(fact_954_square__diff__one__factored,axiom,
! [X: int] :
( ( minus_minus_int @ ( times_times_int @ X @ X ) @ one_one_int )
= ( times_times_int @ ( plus_plus_int @ X @ one_one_int ) @ ( minus_minus_int @ X @ one_one_int ) ) ) ).
% square_diff_one_factored
thf(fact_955_nat__diff__split__asm,axiom,
! [P2: nat > $o,A: nat,B: nat] :
( ( P2 @ ( minus_minus_nat @ A @ B ) )
= ( ~ ( ( ( ord_less_nat @ A @ B )
& ~ ( P2 @ zero_zero_nat ) )
| ? [D5: nat] :
( ( A
= ( plus_plus_nat @ B @ D5 ) )
& ~ ( P2 @ D5 ) ) ) ) ) ).
% nat_diff_split_asm
thf(fact_956_nat__diff__split,axiom,
! [P2: nat > $o,A: nat,B: nat] :
( ( P2 @ ( minus_minus_nat @ A @ B ) )
= ( ( ( ord_less_nat @ A @ B )
=> ( P2 @ zero_zero_nat ) )
& ! [D5: nat] :
( ( A
= ( plus_plus_nat @ B @ D5 ) )
=> ( P2 @ D5 ) ) ) ) ).
% nat_diff_split
thf(fact_957_minf_I7_J,axiom,
! [T: nat] :
? [Z4: nat] :
! [X7: nat] :
( ( ord_less_nat @ X7 @ Z4 )
=> ~ ( ord_less_nat @ T @ X7 ) ) ).
% minf(7)
thf(fact_958_minf_I7_J,axiom,
! [T: int] :
? [Z4: int] :
! [X7: int] :
( ( ord_less_int @ X7 @ Z4 )
=> ~ ( ord_less_int @ T @ X7 ) ) ).
% minf(7)
thf(fact_959_minf_I7_J,axiom,
! [T: real] :
? [Z4: real] :
! [X7: real] :
( ( ord_less_real @ X7 @ Z4 )
=> ~ ( ord_less_real @ T @ X7 ) ) ).
% minf(7)
thf(fact_960_minf_I5_J,axiom,
! [T: nat] :
? [Z4: nat] :
! [X7: nat] :
( ( ord_less_nat @ X7 @ Z4 )
=> ( ord_less_nat @ X7 @ T ) ) ).
% minf(5)
thf(fact_961_minf_I5_J,axiom,
! [T: int] :
? [Z4: int] :
! [X7: int] :
( ( ord_less_int @ X7 @ Z4 )
=> ( ord_less_int @ X7 @ T ) ) ).
% minf(5)
thf(fact_962_minf_I5_J,axiom,
! [T: real] :
? [Z4: real] :
! [X7: real] :
( ( ord_less_real @ X7 @ Z4 )
=> ( ord_less_real @ X7 @ T ) ) ).
% minf(5)
thf(fact_963_minf_I4_J,axiom,
! [T: nat] :
? [Z4: nat] :
! [X7: nat] :
( ( ord_less_nat @ X7 @ Z4 )
=> ( X7 != T ) ) ).
% minf(4)
thf(fact_964_minf_I4_J,axiom,
! [T: int] :
? [Z4: int] :
! [X7: int] :
( ( ord_less_int @ X7 @ Z4 )
=> ( X7 != T ) ) ).
% minf(4)
thf(fact_965_minf_I4_J,axiom,
! [T: real] :
? [Z4: real] :
! [X7: real] :
( ( ord_less_real @ X7 @ Z4 )
=> ( X7 != T ) ) ).
% minf(4)
thf(fact_966_minf_I3_J,axiom,
! [T: nat] :
? [Z4: nat] :
! [X7: nat] :
( ( ord_less_nat @ X7 @ Z4 )
=> ( X7 != T ) ) ).
% minf(3)
thf(fact_967_minf_I3_J,axiom,
! [T: int] :
? [Z4: int] :
! [X7: int] :
( ( ord_less_int @ X7 @ Z4 )
=> ( X7 != T ) ) ).
% minf(3)
thf(fact_968_minf_I3_J,axiom,
! [T: real] :
? [Z4: real] :
! [X7: real] :
( ( ord_less_real @ X7 @ Z4 )
=> ( X7 != T ) ) ).
% minf(3)
thf(fact_969_minf_I2_J,axiom,
! [P2: nat > $o,P7: nat > $o,Q2: nat > $o,Q3: nat > $o] :
( ? [Z3: nat] :
! [X3: nat] :
( ( ord_less_nat @ X3 @ Z3 )
=> ( ( P2 @ X3 )
= ( P7 @ X3 ) ) )
=> ( ? [Z3: nat] :
! [X3: nat] :
( ( ord_less_nat @ X3 @ Z3 )
=> ( ( Q2 @ X3 )
= ( Q3 @ X3 ) ) )
=> ? [Z4: nat] :
! [X7: nat] :
( ( ord_less_nat @ X7 @ Z4 )
=> ( ( ( P2 @ X7 )
| ( Q2 @ X7 ) )
= ( ( P7 @ X7 )
| ( Q3 @ X7 ) ) ) ) ) ) ).
% minf(2)
thf(fact_970_minf_I2_J,axiom,
! [P2: int > $o,P7: int > $o,Q2: int > $o,Q3: int > $o] :
( ? [Z3: int] :
! [X3: int] :
( ( ord_less_int @ X3 @ Z3 )
=> ( ( P2 @ X3 )
= ( P7 @ X3 ) ) )
=> ( ? [Z3: int] :
! [X3: int] :
( ( ord_less_int @ X3 @ Z3 )
=> ( ( Q2 @ X3 )
= ( Q3 @ X3 ) ) )
=> ? [Z4: int] :
! [X7: int] :
( ( ord_less_int @ X7 @ Z4 )
=> ( ( ( P2 @ X7 )
| ( Q2 @ X7 ) )
= ( ( P7 @ X7 )
| ( Q3 @ X7 ) ) ) ) ) ) ).
% minf(2)
thf(fact_971_minf_I2_J,axiom,
! [P2: real > $o,P7: real > $o,Q2: real > $o,Q3: real > $o] :
( ? [Z3: real] :
! [X3: real] :
( ( ord_less_real @ X3 @ Z3 )
=> ( ( P2 @ X3 )
= ( P7 @ X3 ) ) )
=> ( ? [Z3: real] :
! [X3: real] :
( ( ord_less_real @ X3 @ Z3 )
=> ( ( Q2 @ X3 )
= ( Q3 @ X3 ) ) )
=> ? [Z4: real] :
! [X7: real] :
( ( ord_less_real @ X7 @ Z4 )
=> ( ( ( P2 @ X7 )
| ( Q2 @ X7 ) )
= ( ( P7 @ X7 )
| ( Q3 @ X7 ) ) ) ) ) ) ).
% minf(2)
thf(fact_972_minf_I1_J,axiom,
! [P2: nat > $o,P7: nat > $o,Q2: nat > $o,Q3: nat > $o] :
( ? [Z3: nat] :
! [X3: nat] :
( ( ord_less_nat @ X3 @ Z3 )
=> ( ( P2 @ X3 )
= ( P7 @ X3 ) ) )
=> ( ? [Z3: nat] :
! [X3: nat] :
( ( ord_less_nat @ X3 @ Z3 )
=> ( ( Q2 @ X3 )
= ( Q3 @ X3 ) ) )
=> ? [Z4: nat] :
! [X7: nat] :
( ( ord_less_nat @ X7 @ Z4 )
=> ( ( ( P2 @ X7 )
& ( Q2 @ X7 ) )
= ( ( P7 @ X7 )
& ( Q3 @ X7 ) ) ) ) ) ) ).
% minf(1)
thf(fact_973_minf_I1_J,axiom,
! [P2: int > $o,P7: int > $o,Q2: int > $o,Q3: int > $o] :
( ? [Z3: int] :
! [X3: int] :
( ( ord_less_int @ X3 @ Z3 )
=> ( ( P2 @ X3 )
= ( P7 @ X3 ) ) )
=> ( ? [Z3: int] :
! [X3: int] :
( ( ord_less_int @ X3 @ Z3 )
=> ( ( Q2 @ X3 )
= ( Q3 @ X3 ) ) )
=> ? [Z4: int] :
! [X7: int] :
( ( ord_less_int @ X7 @ Z4 )
=> ( ( ( P2 @ X7 )
& ( Q2 @ X7 ) )
= ( ( P7 @ X7 )
& ( Q3 @ X7 ) ) ) ) ) ) ).
% minf(1)
thf(fact_974_minf_I1_J,axiom,
! [P2: real > $o,P7: real > $o,Q2: real > $o,Q3: real > $o] :
( ? [Z3: real] :
! [X3: real] :
( ( ord_less_real @ X3 @ Z3 )
=> ( ( P2 @ X3 )
= ( P7 @ X3 ) ) )
=> ( ? [Z3: real] :
! [X3: real] :
( ( ord_less_real @ X3 @ Z3 )
=> ( ( Q2 @ X3 )
= ( Q3 @ X3 ) ) )
=> ? [Z4: real] :
! [X7: real] :
( ( ord_less_real @ X7 @ Z4 )
=> ( ( ( P2 @ X7 )
& ( Q2 @ X7 ) )
= ( ( P7 @ X7 )
& ( Q3 @ X7 ) ) ) ) ) ) ).
% minf(1)
thf(fact_975_pinf_I7_J,axiom,
! [T: nat] :
? [Z4: nat] :
! [X7: nat] :
( ( ord_less_nat @ Z4 @ X7 )
=> ( ord_less_nat @ T @ X7 ) ) ).
% pinf(7)
thf(fact_976_pinf_I7_J,axiom,
! [T: int] :
? [Z4: int] :
! [X7: int] :
( ( ord_less_int @ Z4 @ X7 )
=> ( ord_less_int @ T @ X7 ) ) ).
% pinf(7)
thf(fact_977_pinf_I7_J,axiom,
! [T: real] :
? [Z4: real] :
! [X7: real] :
( ( ord_less_real @ Z4 @ X7 )
=> ( ord_less_real @ T @ X7 ) ) ).
% pinf(7)
thf(fact_978_pinf_I5_J,axiom,
! [T: nat] :
? [Z4: nat] :
! [X7: nat] :
( ( ord_less_nat @ Z4 @ X7 )
=> ~ ( ord_less_nat @ X7 @ T ) ) ).
% pinf(5)
thf(fact_979_pinf_I5_J,axiom,
! [T: int] :
? [Z4: int] :
! [X7: int] :
( ( ord_less_int @ Z4 @ X7 )
=> ~ ( ord_less_int @ X7 @ T ) ) ).
% pinf(5)
thf(fact_980_pinf_I5_J,axiom,
! [T: real] :
? [Z4: real] :
! [X7: real] :
( ( ord_less_real @ Z4 @ X7 )
=> ~ ( ord_less_real @ X7 @ T ) ) ).
% pinf(5)
thf(fact_981_pinf_I4_J,axiom,
! [T: nat] :
? [Z4: nat] :
! [X7: nat] :
( ( ord_less_nat @ Z4 @ X7 )
=> ( X7 != T ) ) ).
% pinf(4)
thf(fact_982_pinf_I4_J,axiom,
! [T: int] :
? [Z4: int] :
! [X7: int] :
( ( ord_less_int @ Z4 @ X7 )
=> ( X7 != T ) ) ).
% pinf(4)
thf(fact_983_pinf_I4_J,axiom,
! [T: real] :
? [Z4: real] :
! [X7: real] :
( ( ord_less_real @ Z4 @ X7 )
=> ( X7 != T ) ) ).
% pinf(4)
thf(fact_984_pinf_I3_J,axiom,
! [T: nat] :
? [Z4: nat] :
! [X7: nat] :
( ( ord_less_nat @ Z4 @ X7 )
=> ( X7 != T ) ) ).
% pinf(3)
thf(fact_985_pinf_I3_J,axiom,
! [T: int] :
? [Z4: int] :
! [X7: int] :
( ( ord_less_int @ Z4 @ X7 )
=> ( X7 != T ) ) ).
% pinf(3)
thf(fact_986_pinf_I3_J,axiom,
! [T: real] :
? [Z4: real] :
! [X7: real] :
( ( ord_less_real @ Z4 @ X7 )
=> ( X7 != T ) ) ).
% pinf(3)
thf(fact_987_pinf_I2_J,axiom,
! [P2: nat > $o,P7: nat > $o,Q2: nat > $o,Q3: nat > $o] :
( ? [Z3: nat] :
! [X3: nat] :
( ( ord_less_nat @ Z3 @ X3 )
=> ( ( P2 @ X3 )
= ( P7 @ X3 ) ) )
=> ( ? [Z3: nat] :
! [X3: nat] :
( ( ord_less_nat @ Z3 @ X3 )
=> ( ( Q2 @ X3 )
= ( Q3 @ X3 ) ) )
=> ? [Z4: nat] :
! [X7: nat] :
( ( ord_less_nat @ Z4 @ X7 )
=> ( ( ( P2 @ X7 )
| ( Q2 @ X7 ) )
= ( ( P7 @ X7 )
| ( Q3 @ X7 ) ) ) ) ) ) ).
% pinf(2)
thf(fact_988_pinf_I2_J,axiom,
! [P2: int > $o,P7: int > $o,Q2: int > $o,Q3: int > $o] :
( ? [Z3: int] :
! [X3: int] :
( ( ord_less_int @ Z3 @ X3 )
=> ( ( P2 @ X3 )
= ( P7 @ X3 ) ) )
=> ( ? [Z3: int] :
! [X3: int] :
( ( ord_less_int @ Z3 @ X3 )
=> ( ( Q2 @ X3 )
= ( Q3 @ X3 ) ) )
=> ? [Z4: int] :
! [X7: int] :
( ( ord_less_int @ Z4 @ X7 )
=> ( ( ( P2 @ X7 )
| ( Q2 @ X7 ) )
= ( ( P7 @ X7 )
| ( Q3 @ X7 ) ) ) ) ) ) ).
% pinf(2)
thf(fact_989_pinf_I2_J,axiom,
! [P2: real > $o,P7: real > $o,Q2: real > $o,Q3: real > $o] :
( ? [Z3: real] :
! [X3: real] :
( ( ord_less_real @ Z3 @ X3 )
=> ( ( P2 @ X3 )
= ( P7 @ X3 ) ) )
=> ( ? [Z3: real] :
! [X3: real] :
( ( ord_less_real @ Z3 @ X3 )
=> ( ( Q2 @ X3 )
= ( Q3 @ X3 ) ) )
=> ? [Z4: real] :
! [X7: real] :
( ( ord_less_real @ Z4 @ X7 )
=> ( ( ( P2 @ X7 )
| ( Q2 @ X7 ) )
= ( ( P7 @ X7 )
| ( Q3 @ X7 ) ) ) ) ) ) ).
% pinf(2)
thf(fact_990_pinf_I1_J,axiom,
! [P2: nat > $o,P7: nat > $o,Q2: nat > $o,Q3: nat > $o] :
( ? [Z3: nat] :
! [X3: nat] :
( ( ord_less_nat @ Z3 @ X3 )
=> ( ( P2 @ X3 )
= ( P7 @ X3 ) ) )
=> ( ? [Z3: nat] :
! [X3: nat] :
( ( ord_less_nat @ Z3 @ X3 )
=> ( ( Q2 @ X3 )
= ( Q3 @ X3 ) ) )
=> ? [Z4: nat] :
! [X7: nat] :
( ( ord_less_nat @ Z4 @ X7 )
=> ( ( ( P2 @ X7 )
& ( Q2 @ X7 ) )
= ( ( P7 @ X7 )
& ( Q3 @ X7 ) ) ) ) ) ) ).
% pinf(1)
thf(fact_991_pinf_I1_J,axiom,
! [P2: int > $o,P7: int > $o,Q2: int > $o,Q3: int > $o] :
( ? [Z3: int] :
! [X3: int] :
( ( ord_less_int @ Z3 @ X3 )
=> ( ( P2 @ X3 )
= ( P7 @ X3 ) ) )
=> ( ? [Z3: int] :
! [X3: int] :
( ( ord_less_int @ Z3 @ X3 )
=> ( ( Q2 @ X3 )
= ( Q3 @ X3 ) ) )
=> ? [Z4: int] :
! [X7: int] :
( ( ord_less_int @ Z4 @ X7 )
=> ( ( ( P2 @ X7 )
& ( Q2 @ X7 ) )
= ( ( P7 @ X7 )
& ( Q3 @ X7 ) ) ) ) ) ) ).
% pinf(1)
thf(fact_992_pinf_I1_J,axiom,
! [P2: real > $o,P7: real > $o,Q2: real > $o,Q3: real > $o] :
( ? [Z3: real] :
! [X3: real] :
( ( ord_less_real @ Z3 @ X3 )
=> ( ( P2 @ X3 )
= ( P7 @ X3 ) ) )
=> ( ? [Z3: real] :
! [X3: real] :
( ( ord_less_real @ Z3 @ X3 )
=> ( ( Q2 @ X3 )
= ( Q3 @ X3 ) ) )
=> ? [Z4: real] :
! [X7: real] :
( ( ord_less_real @ Z4 @ X7 )
=> ( ( ( P2 @ X7 )
& ( Q2 @ X7 ) )
= ( ( P7 @ X7 )
& ( Q3 @ X7 ) ) ) ) ) ) ).
% pinf(1)
thf(fact_993_bezout__add__strong__nat,axiom,
! [A: nat,B: nat] :
( ( A != zero_zero_nat )
=> ? [D3: nat,X3: nat,Y4: nat] :
( ( dvd_dvd_nat @ D3 @ A )
& ( dvd_dvd_nat @ D3 @ B )
& ( ( times_times_nat @ A @ X3 )
= ( plus_plus_nat @ ( times_times_nat @ B @ Y4 ) @ D3 ) ) ) ) ).
% bezout_add_strong_nat
thf(fact_994_mult__eq__if,axiom,
( times_times_nat
= ( ^ [M2: nat,N2: nat] : ( if_nat @ ( M2 = zero_zero_nat ) @ zero_zero_nat @ ( plus_plus_nat @ N2 @ ( times_times_nat @ ( minus_minus_nat @ M2 @ one_one_nat ) @ N2 ) ) ) ) ) ).
% mult_eq_if
thf(fact_995_cong__to__1_H__nat,axiom,
! [A: nat,N: nat] :
( ( unique653641344996303876ng_nat @ A @ one_one_nat @ N )
= ( ( ( A = zero_zero_nat )
& ( N = one_one_nat ) )
| ? [M2: nat] :
( A
= ( plus_plus_nat @ one_one_nat @ ( times_times_nat @ M2 @ N ) ) ) ) ) ).
% cong_to_1'_nat
thf(fact_996_inf__period_I1_J,axiom,
! [P2: real > $o,D4: real,Q2: real > $o] :
( ! [X3: real,K4: real] :
( ( P2 @ X3 )
= ( P2 @ ( minus_minus_real @ X3 @ ( times_times_real @ K4 @ D4 ) ) ) )
=> ( ! [X3: real,K4: real] :
( ( Q2 @ X3 )
= ( Q2 @ ( minus_minus_real @ X3 @ ( times_times_real @ K4 @ D4 ) ) ) )
=> ! [X7: real,K6: real] :
( ( ( P2 @ X7 )
& ( Q2 @ X7 ) )
= ( ( P2 @ ( minus_minus_real @ X7 @ ( times_times_real @ K6 @ D4 ) ) )
& ( Q2 @ ( minus_minus_real @ X7 @ ( times_times_real @ K6 @ D4 ) ) ) ) ) ) ) ).
% inf_period(1)
thf(fact_997_inf__period_I1_J,axiom,
! [P2: int > $o,D4: int,Q2: int > $o] :
( ! [X3: int,K4: int] :
( ( P2 @ X3 )
= ( P2 @ ( minus_minus_int @ X3 @ ( times_times_int @ K4 @ D4 ) ) ) )
=> ( ! [X3: int,K4: int] :
( ( Q2 @ X3 )
= ( Q2 @ ( minus_minus_int @ X3 @ ( times_times_int @ K4 @ D4 ) ) ) )
=> ! [X7: int,K6: int] :
( ( ( P2 @ X7 )
& ( Q2 @ X7 ) )
= ( ( P2 @ ( minus_minus_int @ X7 @ ( times_times_int @ K6 @ D4 ) ) )
& ( Q2 @ ( minus_minus_int @ X7 @ ( times_times_int @ K6 @ D4 ) ) ) ) ) ) ) ).
% inf_period(1)
thf(fact_998_inf__period_I2_J,axiom,
! [P2: real > $o,D4: real,Q2: real > $o] :
( ! [X3: real,K4: real] :
( ( P2 @ X3 )
= ( P2 @ ( minus_minus_real @ X3 @ ( times_times_real @ K4 @ D4 ) ) ) )
=> ( ! [X3: real,K4: real] :
( ( Q2 @ X3 )
= ( Q2 @ ( minus_minus_real @ X3 @ ( times_times_real @ K4 @ D4 ) ) ) )
=> ! [X7: real,K6: real] :
( ( ( P2 @ X7 )
| ( Q2 @ X7 ) )
= ( ( P2 @ ( minus_minus_real @ X7 @ ( times_times_real @ K6 @ D4 ) ) )
| ( Q2 @ ( minus_minus_real @ X7 @ ( times_times_real @ K6 @ D4 ) ) ) ) ) ) ) ).
% inf_period(2)
thf(fact_999_inf__period_I2_J,axiom,
! [P2: int > $o,D4: int,Q2: int > $o] :
( ! [X3: int,K4: int] :
( ( P2 @ X3 )
= ( P2 @ ( minus_minus_int @ X3 @ ( times_times_int @ K4 @ D4 ) ) ) )
=> ( ! [X3: int,K4: int] :
( ( Q2 @ X3 )
= ( Q2 @ ( minus_minus_int @ X3 @ ( times_times_int @ K4 @ D4 ) ) ) )
=> ! [X7: int,K6: int] :
( ( ( P2 @ X7 )
| ( Q2 @ X7 ) )
= ( ( P2 @ ( minus_minus_int @ X7 @ ( times_times_int @ K6 @ D4 ) ) )
| ( Q2 @ ( minus_minus_int @ X7 @ ( times_times_int @ K6 @ D4 ) ) ) ) ) ) ) ).
% inf_period(2)
thf(fact_1000_preliminary__axioms,axiom,
prelim7757304714281691100nary_a @ type_a @ p @ n @ k ).
% preliminary_axioms
thf(fact_1001_add__scale__eq__noteq,axiom,
! [R2: nat,A: nat,B: nat,C: nat,D: nat] :
( ( R2 != zero_zero_nat )
=> ( ( ( A = B )
& ( C != D ) )
=> ( ( plus_plus_nat @ A @ ( times_times_nat @ R2 @ C ) )
!= ( plus_plus_nat @ B @ ( times_times_nat @ R2 @ D ) ) ) ) ) ).
% add_scale_eq_noteq
thf(fact_1002_add__scale__eq__noteq,axiom,
! [R2: real,A: real,B: real,C: real,D: real] :
( ( R2 != zero_zero_real )
=> ( ( ( A = B )
& ( C != D ) )
=> ( ( plus_plus_real @ A @ ( times_times_real @ R2 @ C ) )
!= ( plus_plus_real @ B @ ( times_times_real @ R2 @ D ) ) ) ) ) ).
% add_scale_eq_noteq
thf(fact_1003_add__scale__eq__noteq,axiom,
! [R2: int,A: int,B: int,C: int,D: int] :
( ( R2 != zero_zero_int )
=> ( ( ( A = B )
& ( C != D ) )
=> ( ( plus_plus_int @ A @ ( times_times_int @ R2 @ C ) )
!= ( plus_plus_int @ B @ ( times_times_int @ R2 @ D ) ) ) ) ) ).
% add_scale_eq_noteq
thf(fact_1004_totient__prime__power,axiom,
! [P: nat,N: nat] :
( ( factor1801147406995305544me_nat @ P )
=> ( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( totient @ ( power_power_nat @ P @ N ) )
= ( times_times_nat @ ( power_power_nat @ P @ ( minus_minus_nat @ N @ one_one_nat ) ) @ ( minus_minus_nat @ P @ one_one_nat ) ) ) ) ) ).
% totient_prime_power
thf(fact_1005_crossproduct__eq,axiom,
! [W: nat,Y: nat,X: nat,Z2: nat] :
( ( ( plus_plus_nat @ ( times_times_nat @ W @ Y ) @ ( times_times_nat @ X @ Z2 ) )
= ( plus_plus_nat @ ( times_times_nat @ W @ Z2 ) @ ( times_times_nat @ X @ Y ) ) )
= ( ( W = X )
| ( Y = Z2 ) ) ) ).
% crossproduct_eq
thf(fact_1006_crossproduct__eq,axiom,
! [W: real,Y: real,X: real,Z2: real] :
( ( ( plus_plus_real @ ( times_times_real @ W @ Y ) @ ( times_times_real @ X @ Z2 ) )
= ( plus_plus_real @ ( times_times_real @ W @ Z2 ) @ ( times_times_real @ X @ Y ) ) )
= ( ( W = X )
| ( Y = Z2 ) ) ) ).
% crossproduct_eq
thf(fact_1007_crossproduct__eq,axiom,
! [W: int,Y: int,X: int,Z2: int] :
( ( ( plus_plus_int @ ( times_times_int @ W @ Y ) @ ( times_times_int @ X @ Z2 ) )
= ( plus_plus_int @ ( times_times_int @ W @ Z2 ) @ ( times_times_int @ X @ Y ) ) )
= ( ( W = X )
| ( Y = Z2 ) ) ) ).
% crossproduct_eq
thf(fact_1008_totient__0__iff,axiom,
! [N: nat] :
( ( ( totient @ N )
= zero_zero_nat )
= ( N = zero_zero_nat ) ) ).
% totient_0_iff
thf(fact_1009_totient__0,axiom,
( ( totient @ zero_zero_nat )
= zero_zero_nat ) ).
% totient_0
thf(fact_1010_totient__gt__0__iff,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( totient @ N ) )
= ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% totient_gt_0_iff
thf(fact_1011_preliminary_Ok__bound,axiom,
! [P: nat,N: nat,K: nat] :
( ( prelim7757304714281691100nary_a @ type_a @ P @ N @ K )
=> ( ord_less_nat @ zero_zero_nat @ K ) ) ).
% preliminary.k_bound
thf(fact_1012_preliminary_Otest,axiom,
! [P: nat,N: nat,K: nat] :
( ( prelim7757304714281691100nary_a @ type_a @ P @ N @ K )
=> ( factor1801147406995305544me_nat @ P ) ) ).
% preliminary.test
thf(fact_1013_totient__dvd,axiom,
! [M: nat,N: nat] :
( ( dvd_dvd_nat @ M @ N )
=> ( dvd_dvd_nat @ ( totient @ M ) @ ( totient @ N ) ) ) ).
% totient_dvd
thf(fact_1014_preliminary_Oexp__rule,axiom,
! [P: nat,N: nat,K: nat,C: finite_mod_ring_a,D: finite_mod_ring_a,E: nat] :
( ( prelim7757304714281691100nary_a @ type_a @ P @ N @ K )
=> ( ( power_6826135765519566523ring_a @ ( times_5121417576591743744ring_a @ C @ D ) @ E )
= ( times_5121417576591743744ring_a @ ( power_6826135765519566523ring_a @ C @ E ) @ ( power_6826135765519566523ring_a @ D @ E ) ) ) ) ).
% preliminary.exp_rule
thf(fact_1015_plus__int__code_I2_J,axiom,
! [L: int] :
( ( plus_plus_int @ zero_zero_int @ L )
= L ) ).
% plus_int_code(2)
thf(fact_1016_plus__int__code_I1_J,axiom,
! [K: int] :
( ( plus_plus_int @ K @ zero_zero_int )
= K ) ).
% plus_int_code(1)
thf(fact_1017_int__distrib_I2_J,axiom,
! [W: int,Z1: int,Z22: int] :
( ( times_times_int @ W @ ( plus_plus_int @ Z1 @ Z22 ) )
= ( plus_plus_int @ ( times_times_int @ W @ Z1 ) @ ( times_times_int @ W @ Z22 ) ) ) ).
% int_distrib(2)
thf(fact_1018_int__distrib_I1_J,axiom,
! [Z1: int,Z22: int,W: int] :
( ( times_times_int @ ( plus_plus_int @ Z1 @ Z22 ) @ W )
= ( plus_plus_int @ ( times_times_int @ Z1 @ W ) @ ( times_times_int @ Z22 @ W ) ) ) ).
% int_distrib(1)
thf(fact_1019_preliminary_Op__fact,axiom,
! [P: nat,N: nat,K: nat] :
( ( prelim7757304714281691100nary_a @ type_a @ P @ N @ K )
=> ( P
= ( plus_plus_nat @ ( times_times_nat @ K @ N ) @ one_one_nat ) ) ) ).
% preliminary.p_fact
thf(fact_1020_totient__less,axiom,
! [N: nat] :
( ( ord_less_nat @ one_one_nat @ N )
=> ( ord_less_nat @ ( totient @ N ) @ N ) ) ).
% totient_less
thf(fact_1021_odd__nonzero,axiom,
! [Z2: int] :
( ( plus_plus_int @ ( plus_plus_int @ one_one_int @ Z2 ) @ Z2 )
!= zero_zero_int ) ).
% odd_nonzero
thf(fact_1022_Carmichael__dvd__totient,axiom,
! [N: nat] : ( dvd_dvd_nat @ ( residu178308219970301372ichael @ N ) @ ( totient @ N ) ) ).
% Carmichael_dvd_totient
thf(fact_1023_int__gr__induct,axiom,
! [K: int,I: int,P2: int > $o] :
( ( ord_less_int @ K @ I )
=> ( ( P2 @ ( plus_plus_int @ K @ one_one_int ) )
=> ( ! [I2: int] :
( ( ord_less_int @ K @ I2 )
=> ( ( P2 @ I2 )
=> ( P2 @ ( plus_plus_int @ I2 @ one_one_int ) ) ) )
=> ( P2 @ I ) ) ) ) ).
% int_gr_induct
thf(fact_1024_zless__add1__eq,axiom,
! [W: int,Z2: int] :
( ( ord_less_int @ W @ ( plus_plus_int @ Z2 @ one_one_int ) )
= ( ( ord_less_int @ W @ Z2 )
| ( W = Z2 ) ) ) ).
% zless_add1_eq
thf(fact_1025_zdvd__reduce,axiom,
! [K: int,N: int,M: int] :
( ( dvd_dvd_int @ K @ ( plus_plus_int @ N @ ( times_times_int @ K @ M ) ) )
= ( dvd_dvd_int @ K @ N ) ) ).
% zdvd_reduce
thf(fact_1026_zdvd__period,axiom,
! [A: int,D: int,X: int,T: int,C: int] :
( ( dvd_dvd_int @ A @ D )
=> ( ( dvd_dvd_int @ A @ ( plus_plus_int @ X @ T ) )
= ( dvd_dvd_int @ A @ ( plus_plus_int @ ( plus_plus_int @ X @ ( times_times_int @ C @ D ) ) @ T ) ) ) ) ).
% zdvd_period
thf(fact_1027_residue__primroot__Carmichael,axiom,
! [N: nat,G2: nat] :
( ( residu2993863765933214154imroot @ N @ G2 )
=> ( ( residu178308219970301372ichael @ N )
= ( totient @ N ) ) ) ).
% residue_primroot_Carmichael
thf(fact_1028_preliminary_Oprimroot__ord,axiom,
! [P: nat,N: nat,K: nat,G2: nat] :
( ( prelim7757304714281691100nary_a @ type_a @ P @ N @ K )
=> ( ( residu2993863765933214154imroot @ P @ G2 )
=> ( ( ord_nat @ P @ G2 )
= ( minus_minus_nat @ P @ one_one_nat ) ) ) ) ).
% preliminary.primroot_ord
thf(fact_1029_totient__prime,axiom,
! [P: nat] :
( ( factor1801147406995305544me_nat @ P )
=> ( ( totient @ P )
= ( minus_minus_nat @ P @ one_one_nat ) ) ) ).
% totient_prime
thf(fact_1030_odd__less__0__iff,axiom,
! [Z2: int] :
( ( ord_less_int @ ( plus_plus_int @ ( plus_plus_int @ one_one_int @ Z2 ) @ Z2 ) @ zero_zero_int )
= ( ord_less_int @ Z2 @ zero_zero_int ) ) ).
% odd_less_0_iff
thf(fact_1031_residue__primroot__iff__Carmichael,axiom,
! [N: nat] :
( ( ? [X4: nat] : ( residu2993863765933214154imroot @ N @ X4 ) )
= ( ( ( residu178308219970301372ichael @ N )
= ( totient @ N ) )
& ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% residue_primroot_iff_Carmichael
thf(fact_1032_Carmichael__eq__totient__imp__primroot,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ( residu178308219970301372ichael @ N )
= ( totient @ N ) )
=> ? [X_1: nat] : ( residu2993863765933214154imroot @ N @ X_1 ) ) ) ).
% Carmichael_eq_totient_imp_primroot
thf(fact_1033_totient__imp__prime,axiom,
! [P: nat] :
( ( ( totient @ P )
= ( minus_minus_nat @ P @ one_one_nat ) )
=> ( ( ord_less_nat @ zero_zero_nat @ P )
=> ( factor1801147406995305544me_nat @ P ) ) ) ).
% totient_imp_prime
thf(fact_1034_totient__power,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ M )
=> ( ( totient @ ( power_power_nat @ N @ M ) )
= ( times_times_nat @ ( power_power_nat @ N @ ( minus_minus_nat @ M @ one_one_nat ) ) @ ( totient @ N ) ) ) ) ).
% totient_power
thf(fact_1035_add__0__iff,axiom,
! [B: nat,A: nat] :
( ( B
= ( plus_plus_nat @ B @ A ) )
= ( A = zero_zero_nat ) ) ).
% add_0_iff
thf(fact_1036_add__0__iff,axiom,
! [B: int,A: int] :
( ( B
= ( plus_plus_int @ B @ A ) )
= ( A = zero_zero_int ) ) ).
% add_0_iff
thf(fact_1037_add__0__iff,axiom,
! [B: real,A: real] :
( ( B
= ( plus_plus_real @ B @ A ) )
= ( A = zero_zero_real ) ) ).
% add_0_iff
thf(fact_1038_crossproduct__noteq,axiom,
! [A: nat,B: nat,C: nat,D: nat] :
( ( ( A != B )
& ( C != D ) )
= ( ( plus_plus_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ D ) )
!= ( plus_plus_nat @ ( times_times_nat @ A @ D ) @ ( times_times_nat @ B @ C ) ) ) ) ).
% crossproduct_noteq
thf(fact_1039_crossproduct__noteq,axiom,
! [A: real,B: real,C: real,D: real] :
( ( ( A != B )
& ( C != D ) )
= ( ( plus_plus_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ D ) )
!= ( plus_plus_real @ ( times_times_real @ A @ D ) @ ( times_times_real @ B @ C ) ) ) ) ).
% crossproduct_noteq
thf(fact_1040_crossproduct__noteq,axiom,
! [A: int,B: int,C: int,D: int] :
( ( ( A != B )
& ( C != D ) )
= ( ( plus_plus_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ D ) )
!= ( plus_plus_int @ ( times_times_int @ A @ D ) @ ( times_times_int @ B @ C ) ) ) ) ).
% crossproduct_noteq
thf(fact_1041_verit__sum__simplify,axiom,
! [A: nat] :
( ( plus_plus_nat @ A @ zero_zero_nat )
= A ) ).
% verit_sum_simplify
thf(fact_1042_verit__sum__simplify,axiom,
! [A: int] :
( ( plus_plus_int @ A @ zero_zero_int )
= A ) ).
% verit_sum_simplify
thf(fact_1043_verit__sum__simplify,axiom,
! [A: real] :
( ( plus_plus_real @ A @ zero_zero_real )
= A ) ).
% verit_sum_simplify
thf(fact_1044_totient__prime__power__Suc,axiom,
! [P: nat,N: nat] :
( ( factor1801147406995305544me_nat @ P )
=> ( ( totient @ ( power_power_nat @ P @ ( suc @ N ) ) )
= ( times_times_nat @ ( power_power_nat @ P @ N ) @ ( minus_minus_nat @ P @ one_one_nat ) ) ) ) ).
% totient_prime_power_Suc
thf(fact_1045_old_Onat_Oinject,axiom,
! [Nat: nat,Nat2: nat] :
( ( ( suc @ Nat )
= ( suc @ Nat2 ) )
= ( Nat = Nat2 ) ) ).
% old.nat.inject
thf(fact_1046_nat_Oinject,axiom,
! [X22: nat,Y22: nat] :
( ( ( suc @ X22 )
= ( suc @ Y22 ) )
= ( X22 = Y22 ) ) ).
% nat.inject
thf(fact_1047_Suc__less__eq,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ ( suc @ M ) @ ( suc @ N ) )
= ( ord_less_nat @ M @ N ) ) ).
% Suc_less_eq
thf(fact_1048_Suc__mono,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( ord_less_nat @ ( suc @ M ) @ ( suc @ N ) ) ) ).
% Suc_mono
thf(fact_1049_lessI,axiom,
! [N: nat] : ( ord_less_nat @ N @ ( suc @ N ) ) ).
% lessI
thf(fact_1050_add__Suc__right,axiom,
! [M: nat,N: nat] :
( ( plus_plus_nat @ M @ ( suc @ N ) )
= ( suc @ ( plus_plus_nat @ M @ N ) ) ) ).
% add_Suc_right
thf(fact_1051_Suc__diff__diff,axiom,
! [M: nat,N: nat,K: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ ( suc @ M ) @ N ) @ ( suc @ K ) )
= ( minus_minus_nat @ ( minus_minus_nat @ M @ N ) @ K ) ) ).
% Suc_diff_diff
thf(fact_1052_diff__Suc__Suc,axiom,
! [M: nat,N: nat] :
( ( minus_minus_nat @ ( suc @ M ) @ ( suc @ N ) )
= ( minus_minus_nat @ M @ N ) ) ).
% diff_Suc_Suc
thf(fact_1053_power__0__Suc,axiom,
! [N: nat] :
( ( power_power_nat @ zero_zero_nat @ ( suc @ N ) )
= zero_zero_nat ) ).
% power_0_Suc
thf(fact_1054_power__0__Suc,axiom,
! [N: nat] :
( ( power_power_int @ zero_zero_int @ ( suc @ N ) )
= zero_zero_int ) ).
% power_0_Suc
thf(fact_1055_power__0__Suc,axiom,
! [N: nat] :
( ( power_power_real @ zero_zero_real @ ( suc @ N ) )
= zero_zero_real ) ).
% power_0_Suc
thf(fact_1056_power__Suc0__right,axiom,
! [A: nat] :
( ( power_power_nat @ A @ ( suc @ zero_zero_nat ) )
= A ) ).
% power_Suc0_right
thf(fact_1057_power__Suc0__right,axiom,
! [A: int] :
( ( power_power_int @ A @ ( suc @ zero_zero_nat ) )
= A ) ).
% power_Suc0_right
thf(fact_1058_power__Suc0__right,axiom,
! [A: real] :
( ( power_power_real @ A @ ( suc @ zero_zero_nat ) )
= A ) ).
% power_Suc0_right
thf(fact_1059_less__Suc0,axiom,
! [N: nat] :
( ( ord_less_nat @ N @ ( suc @ zero_zero_nat ) )
= ( N = zero_zero_nat ) ) ).
% less_Suc0
thf(fact_1060_zero__less__Suc,axiom,
! [N: nat] : ( ord_less_nat @ zero_zero_nat @ ( suc @ N ) ) ).
% zero_less_Suc
thf(fact_1061_one__eq__mult__iff,axiom,
! [M: nat,N: nat] :
( ( ( suc @ zero_zero_nat )
= ( times_times_nat @ M @ N ) )
= ( ( M
= ( suc @ zero_zero_nat ) )
& ( N
= ( suc @ zero_zero_nat ) ) ) ) ).
% one_eq_mult_iff
thf(fact_1062_mult__eq__1__iff,axiom,
! [M: nat,N: nat] :
( ( ( times_times_nat @ M @ N )
= ( suc @ zero_zero_nat ) )
= ( ( M
= ( suc @ zero_zero_nat ) )
& ( N
= ( suc @ zero_zero_nat ) ) ) ) ).
% mult_eq_1_iff
thf(fact_1063_dvd__1__left,axiom,
! [K: nat] : ( dvd_dvd_nat @ ( suc @ zero_zero_nat ) @ K ) ).
% dvd_1_left
thf(fact_1064_dvd__1__iff__1,axiom,
! [M: nat] :
( ( dvd_dvd_nat @ M @ ( suc @ zero_zero_nat ) )
= ( M
= ( suc @ zero_zero_nat ) ) ) ).
% dvd_1_iff_1
thf(fact_1065_diff__Suc__1,axiom,
! [N: nat] :
( ( minus_minus_nat @ ( suc @ N ) @ one_one_nat )
= N ) ).
% diff_Suc_1
thf(fact_1066_mult__Suc__right,axiom,
! [M: nat,N: nat] :
( ( times_times_nat @ M @ ( suc @ N ) )
= ( plus_plus_nat @ M @ ( times_times_nat @ M @ N ) ) ) ).
% mult_Suc_right
thf(fact_1067_power__Suc__0,axiom,
! [N: nat] :
( ( power_power_nat @ ( suc @ zero_zero_nat ) @ N )
= ( suc @ zero_zero_nat ) ) ).
% power_Suc_0
thf(fact_1068_nat__power__eq__Suc__0__iff,axiom,
! [X: nat,M: nat] :
( ( ( power_power_nat @ X @ M )
= ( suc @ zero_zero_nat ) )
= ( ( M = zero_zero_nat )
| ( X
= ( suc @ zero_zero_nat ) ) ) ) ).
% nat_power_eq_Suc_0_iff
thf(fact_1069_totient__Suc__0,axiom,
( ( totient @ ( suc @ zero_zero_nat ) )
= ( suc @ zero_zero_nat ) ) ).
% totient_Suc_0
thf(fact_1070_Suc__pred,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( suc @ ( minus_minus_nat @ N @ ( suc @ zero_zero_nat ) ) )
= N ) ) ).
% Suc_pred
thf(fact_1071_totient__1,axiom,
( ( totient @ one_one_nat )
= ( suc @ zero_zero_nat ) ) ).
% totient_1
thf(fact_1072_ord__Suc__0,axiom,
! [N: nat] :
( ( ord_nat @ ( suc @ zero_zero_nat ) @ N )
= one_one_nat ) ).
% ord_Suc_0
thf(fact_1073_ord__Suc__0__right,axiom,
! [N: nat] :
( ( ord_nat @ N @ ( suc @ zero_zero_nat ) )
= one_one_nat ) ).
% ord_Suc_0_right
thf(fact_1074_Carmichael__Suc__0,axiom,
( ( residu178308219970301372ichael @ ( suc @ zero_zero_nat ) )
= one_one_nat ) ).
% Carmichael_Suc_0
thf(fact_1075_Suc__diff__1,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( suc @ ( minus_minus_nat @ N @ one_one_nat ) )
= N ) ) ).
% Suc_diff_1
thf(fact_1076_not__less__less__Suc__eq,axiom,
! [N: nat,M: nat] :
( ~ ( ord_less_nat @ N @ M )
=> ( ( ord_less_nat @ N @ ( suc @ M ) )
= ( N = M ) ) ) ).
% not_less_less_Suc_eq
thf(fact_1077_strict__inc__induct,axiom,
! [I: nat,J: nat,P2: nat > $o] :
( ( ord_less_nat @ I @ J )
=> ( ! [I2: nat] :
( ( J
= ( suc @ I2 ) )
=> ( P2 @ I2 ) )
=> ( ! [I2: nat] :
( ( ord_less_nat @ I2 @ J )
=> ( ( P2 @ ( suc @ I2 ) )
=> ( P2 @ I2 ) ) )
=> ( P2 @ I ) ) ) ) ).
% strict_inc_induct
thf(fact_1078_less__Suc__induct,axiom,
! [I: nat,J: nat,P2: nat > nat > $o] :
( ( ord_less_nat @ I @ J )
=> ( ! [I2: nat] : ( P2 @ I2 @ ( suc @ I2 ) )
=> ( ! [I2: nat,J2: nat,K4: nat] :
( ( ord_less_nat @ I2 @ J2 )
=> ( ( ord_less_nat @ J2 @ K4 )
=> ( ( P2 @ I2 @ J2 )
=> ( ( P2 @ J2 @ K4 )
=> ( P2 @ I2 @ K4 ) ) ) ) )
=> ( P2 @ I @ J ) ) ) ) ).
% less_Suc_induct
thf(fact_1079_less__trans__Suc,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_nat @ I @ J )
=> ( ( ord_less_nat @ J @ K )
=> ( ord_less_nat @ ( suc @ I ) @ K ) ) ) ).
% less_trans_Suc
thf(fact_1080_Suc__less__SucD,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ ( suc @ M ) @ ( suc @ N ) )
=> ( ord_less_nat @ M @ N ) ) ).
% Suc_less_SucD
thf(fact_1081_less__antisym,axiom,
! [N: nat,M: nat] :
( ~ ( ord_less_nat @ N @ M )
=> ( ( ord_less_nat @ N @ ( suc @ M ) )
=> ( M = N ) ) ) ).
% less_antisym
thf(fact_1082_Suc__less__eq2,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ ( suc @ N ) @ M )
= ( ? [M5: nat] :
( ( M
= ( suc @ M5 ) )
& ( ord_less_nat @ N @ M5 ) ) ) ) ).
% Suc_less_eq2
thf(fact_1083_All__less__Suc,axiom,
! [N: nat,P2: nat > $o] :
( ( ! [I3: nat] :
( ( ord_less_nat @ I3 @ ( suc @ N ) )
=> ( P2 @ I3 ) ) )
= ( ( P2 @ N )
& ! [I3: nat] :
( ( ord_less_nat @ I3 @ N )
=> ( P2 @ I3 ) ) ) ) ).
% All_less_Suc
thf(fact_1084_not__less__eq,axiom,
! [M: nat,N: nat] :
( ( ~ ( ord_less_nat @ M @ N ) )
= ( ord_less_nat @ N @ ( suc @ M ) ) ) ).
% not_less_eq
thf(fact_1085_less__Suc__eq,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ ( suc @ N ) )
= ( ( ord_less_nat @ M @ N )
| ( M = N ) ) ) ).
% less_Suc_eq
thf(fact_1086_Ex__less__Suc,axiom,
! [N: nat,P2: nat > $o] :
( ( ? [I3: nat] :
( ( ord_less_nat @ I3 @ ( suc @ N ) )
& ( P2 @ I3 ) ) )
= ( ( P2 @ N )
| ? [I3: nat] :
( ( ord_less_nat @ I3 @ N )
& ( P2 @ I3 ) ) ) ) ).
% Ex_less_Suc
thf(fact_1087_less__SucI,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( ord_less_nat @ M @ ( suc @ N ) ) ) ).
% less_SucI
thf(fact_1088_less__SucE,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ ( suc @ N ) )
=> ( ~ ( ord_less_nat @ M @ N )
=> ( M = N ) ) ) ).
% less_SucE
thf(fact_1089_Suc__lessI,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( ( ( suc @ M )
!= N )
=> ( ord_less_nat @ ( suc @ M ) @ N ) ) ) ).
% Suc_lessI
thf(fact_1090_Suc__lessE,axiom,
! [I: nat,K: nat] :
( ( ord_less_nat @ ( suc @ I ) @ K )
=> ~ ! [J2: nat] :
( ( ord_less_nat @ I @ J2 )
=> ( K
!= ( suc @ J2 ) ) ) ) ).
% Suc_lessE
thf(fact_1091_Suc__lessD,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ ( suc @ M ) @ N )
=> ( ord_less_nat @ M @ N ) ) ).
% Suc_lessD
thf(fact_1092_Nat_OlessE,axiom,
! [I: nat,K: nat] :
( ( ord_less_nat @ I @ K )
=> ( ( K
!= ( suc @ I ) )
=> ~ ! [J2: nat] :
( ( ord_less_nat @ I @ J2 )
=> ( K
!= ( suc @ J2 ) ) ) ) ) ).
% Nat.lessE
thf(fact_1093_nat__arith_Osuc1,axiom,
! [A2: nat,K: nat,A: nat] :
( ( A2
= ( plus_plus_nat @ K @ A ) )
=> ( ( suc @ A2 )
= ( plus_plus_nat @ K @ ( suc @ A ) ) ) ) ).
% nat_arith.suc1
thf(fact_1094_add__Suc,axiom,
! [M: nat,N: nat] :
( ( plus_plus_nat @ ( suc @ M ) @ N )
= ( suc @ ( plus_plus_nat @ M @ N ) ) ) ).
% add_Suc
thf(fact_1095_add__Suc__shift,axiom,
! [M: nat,N: nat] :
( ( plus_plus_nat @ ( suc @ M ) @ N )
= ( plus_plus_nat @ M @ ( suc @ N ) ) ) ).
% add_Suc_shift
thf(fact_1096_zero__induct__lemma,axiom,
! [P2: nat > $o,K: nat,I: nat] :
( ( P2 @ K )
=> ( ! [N3: nat] :
( ( P2 @ ( suc @ N3 ) )
=> ( P2 @ N3 ) )
=> ( P2 @ ( minus_minus_nat @ K @ I ) ) ) ) ).
% zero_induct_lemma
thf(fact_1097_Suc__mult__cancel1,axiom,
! [K: nat,M: nat,N: nat] :
( ( ( times_times_nat @ ( suc @ K ) @ M )
= ( times_times_nat @ ( suc @ K ) @ N ) )
= ( M = N ) ) ).
% Suc_mult_cancel1
thf(fact_1098_n__not__Suc__n,axiom,
! [N: nat] :
( N
!= ( suc @ N ) ) ).
% n_not_Suc_n
thf(fact_1099_Suc__inject,axiom,
! [X: nat,Y: nat] :
( ( ( suc @ X )
= ( suc @ Y ) )
=> ( X = Y ) ) ).
% Suc_inject
thf(fact_1100_not0__implies__Suc,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
=> ? [M6: nat] :
( N
= ( suc @ M6 ) ) ) ).
% not0_implies_Suc
thf(fact_1101_Zero__not__Suc,axiom,
! [M: nat] :
( zero_zero_nat
!= ( suc @ M ) ) ).
% Zero_not_Suc
thf(fact_1102_Zero__neq__Suc,axiom,
! [M: nat] :
( zero_zero_nat
!= ( suc @ M ) ) ).
% Zero_neq_Suc
thf(fact_1103_Suc__neq__Zero,axiom,
! [M: nat] :
( ( suc @ M )
!= zero_zero_nat ) ).
% Suc_neq_Zero
thf(fact_1104_zero__induct,axiom,
! [P2: nat > $o,K: nat] :
( ( P2 @ K )
=> ( ! [N3: nat] :
( ( P2 @ ( suc @ N3 ) )
=> ( P2 @ N3 ) )
=> ( P2 @ zero_zero_nat ) ) ) ).
% zero_induct
thf(fact_1105_diff__induct,axiom,
! [P2: nat > nat > $o,M: nat,N: nat] :
( ! [X3: nat] : ( P2 @ X3 @ zero_zero_nat )
=> ( ! [Y4: nat] : ( P2 @ zero_zero_nat @ ( suc @ Y4 ) )
=> ( ! [X3: nat,Y4: nat] :
( ( P2 @ X3 @ Y4 )
=> ( P2 @ ( suc @ X3 ) @ ( suc @ Y4 ) ) )
=> ( P2 @ M @ N ) ) ) ) ).
% diff_induct
thf(fact_1106_nat__induct,axiom,
! [P2: nat > $o,N: nat] :
( ( P2 @ zero_zero_nat )
=> ( ! [N3: nat] :
( ( P2 @ N3 )
=> ( P2 @ ( suc @ N3 ) ) )
=> ( P2 @ N ) ) ) ).
% nat_induct
thf(fact_1107_old_Onat_Oexhaust,axiom,
! [Y: nat] :
( ( Y != zero_zero_nat )
=> ~ ! [Nat3: nat] :
( Y
!= ( suc @ Nat3 ) ) ) ).
% old.nat.exhaust
thf(fact_1108_nat_OdiscI,axiom,
! [Nat: nat,X22: nat] :
( ( Nat
= ( suc @ X22 ) )
=> ( Nat != zero_zero_nat ) ) ).
% nat.discI
thf(fact_1109_old_Onat_Odistinct_I1_J,axiom,
! [Nat2: nat] :
( zero_zero_nat
!= ( suc @ Nat2 ) ) ).
% old.nat.distinct(1)
thf(fact_1110_old_Onat_Odistinct_I2_J,axiom,
! [Nat2: nat] :
( ( suc @ Nat2 )
!= zero_zero_nat ) ).
% old.nat.distinct(2)
thf(fact_1111_nat_Odistinct_I1_J,axiom,
! [X22: nat] :
( zero_zero_nat
!= ( suc @ X22 ) ) ).
% nat.distinct(1)
thf(fact_1112_power__Suc2,axiom,
! [A: nat,N: nat] :
( ( power_power_nat @ A @ ( suc @ N ) )
= ( times_times_nat @ ( power_power_nat @ A @ N ) @ A ) ) ).
% power_Suc2
thf(fact_1113_power__Suc2,axiom,
! [A: real,N: nat] :
( ( power_power_real @ A @ ( suc @ N ) )
= ( times_times_real @ ( power_power_real @ A @ N ) @ A ) ) ).
% power_Suc2
thf(fact_1114_power__Suc2,axiom,
! [A: int,N: nat] :
( ( power_power_int @ A @ ( suc @ N ) )
= ( times_times_int @ ( power_power_int @ A @ N ) @ A ) ) ).
% power_Suc2
thf(fact_1115_power__Suc,axiom,
! [A: nat,N: nat] :
( ( power_power_nat @ A @ ( suc @ N ) )
= ( times_times_nat @ A @ ( power_power_nat @ A @ N ) ) ) ).
% power_Suc
thf(fact_1116_power__Suc,axiom,
! [A: real,N: nat] :
( ( power_power_real @ A @ ( suc @ N ) )
= ( times_times_real @ A @ ( power_power_real @ A @ N ) ) ) ).
% power_Suc
thf(fact_1117_power__Suc,axiom,
! [A: int,N: nat] :
( ( power_power_int @ A @ ( suc @ N ) )
= ( times_times_int @ A @ ( power_power_int @ A @ N ) ) ) ).
% power_Suc
thf(fact_1118_lift__Suc__mono__less,axiom,
! [F: nat > nat,N: nat,N6: nat] :
( ! [N3: nat] : ( ord_less_nat @ ( F @ N3 ) @ ( F @ ( suc @ N3 ) ) )
=> ( ( ord_less_nat @ N @ N6 )
=> ( ord_less_nat @ ( F @ N ) @ ( F @ N6 ) ) ) ) ).
% lift_Suc_mono_less
thf(fact_1119_lift__Suc__mono__less,axiom,
! [F: nat > int,N: nat,N6: nat] :
( ! [N3: nat] : ( ord_less_int @ ( F @ N3 ) @ ( F @ ( suc @ N3 ) ) )
=> ( ( ord_less_nat @ N @ N6 )
=> ( ord_less_int @ ( F @ N ) @ ( F @ N6 ) ) ) ) ).
% lift_Suc_mono_less
thf(fact_1120_lift__Suc__mono__less,axiom,
! [F: nat > real,N: nat,N6: nat] :
( ! [N3: nat] : ( ord_less_real @ ( F @ N3 ) @ ( F @ ( suc @ N3 ) ) )
=> ( ( ord_less_nat @ N @ N6 )
=> ( ord_less_real @ ( F @ N ) @ ( F @ N6 ) ) ) ) ).
% lift_Suc_mono_less
thf(fact_1121_lift__Suc__mono__less__iff,axiom,
! [F: nat > nat,N: nat,M: nat] :
( ! [N3: nat] : ( ord_less_nat @ ( F @ N3 ) @ ( F @ ( suc @ N3 ) ) )
=> ( ( ord_less_nat @ ( F @ N ) @ ( F @ M ) )
= ( ord_less_nat @ N @ M ) ) ) ).
% lift_Suc_mono_less_iff
thf(fact_1122_lift__Suc__mono__less__iff,axiom,
! [F: nat > int,N: nat,M: nat] :
( ! [N3: nat] : ( ord_less_int @ ( F @ N3 ) @ ( F @ ( suc @ N3 ) ) )
=> ( ( ord_less_int @ ( F @ N ) @ ( F @ M ) )
= ( ord_less_nat @ N @ M ) ) ) ).
% lift_Suc_mono_less_iff
thf(fact_1123_lift__Suc__mono__less__iff,axiom,
! [F: nat > real,N: nat,M: nat] :
( ! [N3: nat] : ( ord_less_real @ ( F @ N3 ) @ ( F @ ( suc @ N3 ) ) )
=> ( ( ord_less_real @ ( F @ N ) @ ( F @ M ) )
= ( ord_less_nat @ N @ M ) ) ) ).
% lift_Suc_mono_less_iff
thf(fact_1124_Ex__less__Suc2,axiom,
! [N: nat,P2: nat > $o] :
( ( ? [I3: nat] :
( ( ord_less_nat @ I3 @ ( suc @ N ) )
& ( P2 @ I3 ) ) )
= ( ( P2 @ zero_zero_nat )
| ? [I3: nat] :
( ( ord_less_nat @ I3 @ N )
& ( P2 @ ( suc @ I3 ) ) ) ) ) ).
% Ex_less_Suc2
thf(fact_1125_gr0__conv__Suc,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
= ( ? [M2: nat] :
( N
= ( suc @ M2 ) ) ) ) ).
% gr0_conv_Suc
thf(fact_1126_All__less__Suc2,axiom,
! [N: nat,P2: nat > $o] :
( ( ! [I3: nat] :
( ( ord_less_nat @ I3 @ ( suc @ N ) )
=> ( P2 @ I3 ) ) )
= ( ( P2 @ zero_zero_nat )
& ! [I3: nat] :
( ( ord_less_nat @ I3 @ N )
=> ( P2 @ ( suc @ I3 ) ) ) ) ) ).
% All_less_Suc2
thf(fact_1127_gr0__implies__Suc,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ? [M6: nat] :
( N
= ( suc @ M6 ) ) ) ).
% gr0_implies_Suc
thf(fact_1128_less__Suc__eq__0__disj,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ ( suc @ N ) )
= ( ( M = zero_zero_nat )
| ? [J3: nat] :
( ( M
= ( suc @ J3 ) )
& ( ord_less_nat @ J3 @ N ) ) ) ) ).
% less_Suc_eq_0_disj
thf(fact_1129_add__is__1,axiom,
! [M: nat,N: nat] :
( ( ( plus_plus_nat @ M @ N )
= ( suc @ zero_zero_nat ) )
= ( ( ( M
= ( suc @ zero_zero_nat ) )
& ( N = zero_zero_nat ) )
| ( ( M = zero_zero_nat )
& ( N
= ( suc @ zero_zero_nat ) ) ) ) ) ).
% add_is_1
thf(fact_1130_one__is__add,axiom,
! [M: nat,N: nat] :
( ( ( suc @ zero_zero_nat )
= ( plus_plus_nat @ M @ N ) )
= ( ( ( M
= ( suc @ zero_zero_nat ) )
& ( N = zero_zero_nat ) )
| ( ( M = zero_zero_nat )
& ( N
= ( suc @ zero_zero_nat ) ) ) ) ) ).
% one_is_add
thf(fact_1131_One__nat__def,axiom,
( one_one_nat
= ( suc @ zero_zero_nat ) ) ).
% One_nat_def
thf(fact_1132_less__natE,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ~ ! [Q4: nat] :
( N
!= ( suc @ ( plus_plus_nat @ M @ Q4 ) ) ) ) ).
% less_natE
thf(fact_1133_less__add__Suc1,axiom,
! [I: nat,M: nat] : ( ord_less_nat @ I @ ( suc @ ( plus_plus_nat @ I @ M ) ) ) ).
% less_add_Suc1
thf(fact_1134_less__add__Suc2,axiom,
! [I: nat,M: nat] : ( ord_less_nat @ I @ ( suc @ ( plus_plus_nat @ M @ I ) ) ) ).
% less_add_Suc2
thf(fact_1135_less__iff__Suc__add,axiom,
( ord_less_nat
= ( ^ [M2: nat,N2: nat] :
? [K3: nat] :
( N2
= ( suc @ ( plus_plus_nat @ M2 @ K3 ) ) ) ) ) ).
% less_iff_Suc_add
thf(fact_1136_less__imp__Suc__add,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ? [K4: nat] :
( N
= ( suc @ ( plus_plus_nat @ M @ K4 ) ) ) ) ).
% less_imp_Suc_add
thf(fact_1137_diff__less__Suc,axiom,
! [M: nat,N: nat] : ( ord_less_nat @ ( minus_minus_nat @ M @ N ) @ ( suc @ M ) ) ).
% diff_less_Suc
thf(fact_1138_Suc__diff__Suc,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ N @ M )
=> ( ( suc @ ( minus_minus_nat @ M @ ( suc @ N ) ) )
= ( minus_minus_nat @ M @ N ) ) ) ).
% Suc_diff_Suc
thf(fact_1139_Suc__mult__less__cancel1,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_nat @ ( times_times_nat @ ( suc @ K ) @ M ) @ ( times_times_nat @ ( suc @ K ) @ N ) )
= ( ord_less_nat @ M @ N ) ) ).
% Suc_mult_less_cancel1
thf(fact_1140_Suc__eq__plus1,axiom,
( suc
= ( ^ [N2: nat] : ( plus_plus_nat @ N2 @ one_one_nat ) ) ) ).
% Suc_eq_plus1
thf(fact_1141_plus__1__eq__Suc,axiom,
( ( plus_plus_nat @ one_one_nat )
= suc ) ).
% plus_1_eq_Suc
thf(fact_1142_Suc__eq__plus1__left,axiom,
( suc
= ( plus_plus_nat @ one_one_nat ) ) ).
% Suc_eq_plus1_left
thf(fact_1143_diff__Suc__eq__diff__pred,axiom,
! [M: nat,N: nat] :
( ( minus_minus_nat @ M @ ( suc @ N ) )
= ( minus_minus_nat @ ( minus_minus_nat @ M @ one_one_nat ) @ N ) ) ).
% diff_Suc_eq_diff_pred
thf(fact_1144_mult__Suc,axiom,
! [M: nat,N: nat] :
( ( times_times_nat @ ( suc @ M ) @ N )
= ( plus_plus_nat @ N @ ( times_times_nat @ M @ N ) ) ) ).
% mult_Suc
thf(fact_1145_Suc__0__not__prime__nat,axiom,
~ ( factor1801147406995305544me_nat @ ( suc @ zero_zero_nat ) ) ).
% Suc_0_not_prime_nat
thf(fact_1146_cong__Suc__0,axiom,
! [M: nat,N: nat] : ( unique653641344996303876ng_nat @ M @ N @ ( suc @ zero_zero_nat ) ) ).
% cong_Suc_0
thf(fact_1147_cong__0__1__nat_H,axiom,
! [N: nat] :
( ( unique653641344996303876ng_nat @ zero_zero_nat @ ( suc @ zero_zero_nat ) @ N )
= ( N
= ( suc @ zero_zero_nat ) ) ) ).
% cong_0_1_nat'
thf(fact_1148_divides__rexp,axiom,
! [X: nat,Y: nat,N: nat] :
( ( dvd_dvd_nat @ X @ Y )
=> ( dvd_dvd_nat @ X @ ( power_power_nat @ Y @ ( suc @ N ) ) ) ) ).
% divides_rexp
thf(fact_1149_not__primepow__Suc__0__nat,axiom,
~ ( prime_primepow_nat @ ( suc @ zero_zero_nat ) ) ).
% not_primepow_Suc_0_nat
thf(fact_1150_power__gt1,axiom,
! [A: nat,N: nat] :
( ( ord_less_nat @ one_one_nat @ A )
=> ( ord_less_nat @ one_one_nat @ ( power_power_nat @ A @ ( suc @ N ) ) ) ) ).
% power_gt1
thf(fact_1151_power__gt1,axiom,
! [A: int,N: nat] :
( ( ord_less_int @ one_one_int @ A )
=> ( ord_less_int @ one_one_int @ ( power_power_int @ A @ ( suc @ N ) ) ) ) ).
% power_gt1
thf(fact_1152_power__gt1,axiom,
! [A: real,N: nat] :
( ( ord_less_real @ one_one_real @ A )
=> ( ord_less_real @ one_one_real @ ( power_power_real @ A @ ( suc @ N ) ) ) ) ).
% power_gt1
thf(fact_1153_nat__induct__non__zero,axiom,
! [N: nat,P2: nat > $o] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( P2 @ one_one_nat )
=> ( ! [N3: nat] :
( ( ord_less_nat @ zero_zero_nat @ N3 )
=> ( ( P2 @ N3 )
=> ( P2 @ ( suc @ N3 ) ) ) )
=> ( P2 @ N ) ) ) ) ).
% nat_induct_non_zero
thf(fact_1154_diff__Suc__less,axiom,
! [N: nat,I: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ord_less_nat @ ( minus_minus_nat @ N @ ( suc @ I ) ) @ N ) ) ).
% diff_Suc_less
thf(fact_1155_one__less__mult,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ N )
=> ( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ M )
=> ( ord_less_nat @ ( suc @ zero_zero_nat ) @ ( times_times_nat @ M @ N ) ) ) ) ).
% one_less_mult
thf(fact_1156_n__less__m__mult__n,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ M )
=> ( ord_less_nat @ N @ ( times_times_nat @ M @ N ) ) ) ) ).
% n_less_m_mult_n
thf(fact_1157_n__less__n__mult__m,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ M )
=> ( ord_less_nat @ N @ ( times_times_nat @ N @ M ) ) ) ) ).
% n_less_n_mult_m
thf(fact_1158_verit__comp__simplify1_I1_J,axiom,
! [A: nat] :
~ ( ord_less_nat @ A @ A ) ).
% verit_comp_simplify1(1)
thf(fact_1159_verit__comp__simplify1_I1_J,axiom,
! [A: int] :
~ ( ord_less_int @ A @ A ) ).
% verit_comp_simplify1(1)
thf(fact_1160_verit__comp__simplify1_I1_J,axiom,
! [A: real] :
~ ( ord_less_real @ A @ A ) ).
% verit_comp_simplify1(1)
thf(fact_1161_power__gt__expt,axiom,
! [N: nat,K: nat] :
( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ N )
=> ( ord_less_nat @ K @ ( power_power_nat @ N @ K ) ) ) ).
% power_gt_expt
thf(fact_1162_realpow__pos__nth2,axiom,
! [A: real,N: nat] :
( ( ord_less_real @ zero_zero_real @ A )
=> ? [R: real] :
( ( ord_less_real @ zero_zero_real @ R )
& ( ( power_power_real @ R @ ( suc @ N ) )
= A ) ) ) ).
% realpow_pos_nth2
thf(fact_1163_prime__gt__Suc__0__nat,axiom,
! [P: nat] :
( ( factor1801147406995305544me_nat @ P )
=> ( ord_less_nat @ ( suc @ zero_zero_nat ) @ P ) ) ).
% prime_gt_Suc_0_nat
thf(fact_1164_bezout__prime,axiom,
! [P: nat,A: nat] :
( ( factor1801147406995305544me_nat @ P )
=> ( ~ ( dvd_dvd_nat @ P @ A )
=> ? [X3: nat,Y4: nat] :
( ( times_times_nat @ A @ X3 )
= ( suc @ ( times_times_nat @ P @ Y4 ) ) ) ) ) ).
% bezout_prime
thf(fact_1165_one__in__totatives,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( member_nat @ ( suc @ zero_zero_nat ) @ ( totatives @ N ) ) ) ).
% one_in_totatives
thf(fact_1166_totient__power__Suc,axiom,
! [N: nat,M: nat] :
( ( totient @ ( power_power_nat @ N @ ( suc @ M ) ) )
= ( times_times_nat @ ( power_power_nat @ N @ M ) @ ( totient @ N ) ) ) ).
% totient_power_Suc
thf(fact_1167_aprimedivisor__gt__0__nat,axiom,
! [D: nat] :
( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ D )
=> ( ord_less_nat @ zero_zero_nat @ ( prime_1889911587691200368or_nat @ D ) ) ) ).
% aprimedivisor_gt_0_nat
thf(fact_1168_aprimedivisor__gt__Suc__0,axiom,
! [N: nat] :
( ( N
!= ( suc @ zero_zero_nat ) )
=> ( ord_less_nat @ ( suc @ zero_zero_nat ) @ ( prime_1889911587691200368or_nat @ N ) ) ) ).
% aprimedivisor_gt_Suc_0
thf(fact_1169_aprimedivisor__gt__Suc__0__nat,axiom,
! [D: nat] :
( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ D )
=> ( ord_less_nat @ ( suc @ zero_zero_nat ) @ ( prime_1889911587691200368or_nat @ D ) ) ) ).
% aprimedivisor_gt_Suc_0_nat
thf(fact_1170_aprimedivisor__not__Suc__0__nat,axiom,
! [D: nat] :
( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ D )
=> ( ( prime_1889911587691200368or_nat @ D )
!= ( suc @ zero_zero_nat ) ) ) ).
% aprimedivisor_not_Suc_0_nat
thf(fact_1171_primepow__gt__Suc__0,axiom,
! [N: nat] :
( ( prime_primepow_nat @ N )
=> ( ord_less_nat @ ( suc @ zero_zero_nat ) @ N ) ) ).
% primepow_gt_Suc_0
thf(fact_1172_aprimedivisor__nat_I2_J,axiom,
! [N: nat] :
( ( N
!= ( suc @ zero_zero_nat ) )
=> ( dvd_dvd_nat @ ( prime_1889911587691200368or_nat @ N ) @ N ) ) ).
% aprimedivisor_nat(2)
thf(fact_1173_aprimedivisor__nat_I1_J,axiom,
! [N: nat] :
( ( N
!= ( suc @ zero_zero_nat ) )
=> ( factor1801147406995305544me_nat @ ( prime_1889911587691200368or_nat @ N ) ) ) ).
% aprimedivisor_nat(1)
thf(fact_1174_power__Suc__less__one,axiom,
! [A: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ A )
=> ( ( ord_less_nat @ A @ one_one_nat )
=> ( ord_less_nat @ ( power_power_nat @ A @ ( suc @ N ) ) @ one_one_nat ) ) ) ).
% power_Suc_less_one
thf(fact_1175_power__Suc__less__one,axiom,
! [A: int,N: nat] :
( ( ord_less_int @ zero_zero_int @ A )
=> ( ( ord_less_int @ A @ one_one_int )
=> ( ord_less_int @ ( power_power_int @ A @ ( suc @ N ) ) @ one_one_int ) ) ) ).
% power_Suc_less_one
thf(fact_1176_power__Suc__less__one,axiom,
! [A: real,N: nat] :
( ( ord_less_real @ zero_zero_real @ A )
=> ( ( ord_less_real @ A @ one_one_real )
=> ( ord_less_real @ ( power_power_real @ A @ ( suc @ N ) ) @ one_one_real ) ) ) ).
% power_Suc_less_one
thf(fact_1177_Suc__pred_H,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( N
= ( suc @ ( minus_minus_nat @ N @ one_one_nat ) ) ) ) ).
% Suc_pred'
thf(fact_1178_Suc__diff__eq__diff__pred,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( minus_minus_nat @ ( suc @ M ) @ N )
= ( minus_minus_nat @ M @ ( minus_minus_nat @ N @ one_one_nat ) ) ) ) ).
% Suc_diff_eq_diff_pred
thf(fact_1179_add__eq__if,axiom,
( plus_plus_nat
= ( ^ [M2: nat,N2: nat] : ( if_nat @ ( M2 = zero_zero_nat ) @ N2 @ ( suc @ ( plus_plus_nat @ ( minus_minus_nat @ M2 @ one_one_nat ) @ N2 ) ) ) ) ) ).
% add_eq_if
thf(fact_1180_ord__eq__Suc__0__iff,axiom,
! [N: nat,X: nat] :
( ( ( ord_nat @ N @ X )
= ( suc @ zero_zero_nat ) )
= ( unique653641344996303876ng_nat @ X @ one_one_nat @ N ) ) ).
% ord_eq_Suc_0_iff
thf(fact_1181_ord__divides_H,axiom,
! [A: nat,D: nat,N: nat] :
( ( unique653641344996303876ng_nat @ ( power_power_nat @ A @ D ) @ ( suc @ zero_zero_nat ) @ N )
= ( dvd_dvd_nat @ ( ord_nat @ N @ A ) @ D ) ) ).
% ord_divides'
thf(fact_1182_mangoldt__Suc__0,axiom,
( ( prime_mangoldt_real @ ( suc @ zero_zero_nat ) )
= zero_zero_real ) ).
% mangoldt_Suc_0
thf(fact_1183_int__power__div__base,axiom,
! [M: nat,K: int] :
( ( ord_less_nat @ zero_zero_nat @ M )
=> ( ( ord_less_int @ zero_zero_int @ K )
=> ( ( divide_divide_int @ ( power_power_int @ K @ M ) @ K )
= ( power_power_int @ K @ ( minus_minus_nat @ M @ ( suc @ zero_zero_nat ) ) ) ) ) ) ).
% int_power_div_base
thf(fact_1184_division__ring__divide__zero,axiom,
! [A: real] :
( ( divide_divide_real @ A @ zero_zero_real )
= zero_zero_real ) ).
% division_ring_divide_zero
thf(fact_1185_divide__cancel__right,axiom,
! [A: real,C: real,B: real] :
( ( ( divide_divide_real @ A @ C )
= ( divide_divide_real @ B @ C ) )
= ( ( C = zero_zero_real )
| ( A = B ) ) ) ).
% divide_cancel_right
thf(fact_1186_divide__cancel__left,axiom,
! [C: real,A: real,B: real] :
( ( ( divide_divide_real @ C @ A )
= ( divide_divide_real @ C @ B ) )
= ( ( C = zero_zero_real )
| ( A = B ) ) ) ).
% divide_cancel_left
thf(fact_1187_divide__eq__0__iff,axiom,
! [A: real,B: real] :
( ( ( divide_divide_real @ A @ B )
= zero_zero_real )
= ( ( A = zero_zero_real )
| ( B = zero_zero_real ) ) ) ).
% divide_eq_0_iff
thf(fact_1188_div__0,axiom,
! [A: int] :
( ( divide_divide_int @ zero_zero_int @ A )
= zero_zero_int ) ).
% div_0
thf(fact_1189_div__0,axiom,
! [A: real] :
( ( divide_divide_real @ zero_zero_real @ A )
= zero_zero_real ) ).
% div_0
thf(fact_1190_div__0,axiom,
! [A: nat] :
( ( divide_divide_nat @ zero_zero_nat @ A )
= zero_zero_nat ) ).
% div_0
thf(fact_1191_div__by__0,axiom,
! [A: int] :
( ( divide_divide_int @ A @ zero_zero_int )
= zero_zero_int ) ).
% div_by_0
thf(fact_1192_div__by__0,axiom,
! [A: real] :
( ( divide_divide_real @ A @ zero_zero_real )
= zero_zero_real ) ).
% div_by_0
thf(fact_1193_div__by__0,axiom,
! [A: nat] :
( ( divide_divide_nat @ A @ zero_zero_nat )
= zero_zero_nat ) ).
% div_by_0
thf(fact_1194_times__divide__eq__right,axiom,
! [A: real,B: real,C: real] :
( ( times_times_real @ A @ ( divide_divide_real @ B @ C ) )
= ( divide_divide_real @ ( times_times_real @ A @ B ) @ C ) ) ).
% times_divide_eq_right
thf(fact_1195_divide__divide__eq__right,axiom,
! [A: real,B: real,C: real] :
( ( divide_divide_real @ A @ ( divide_divide_real @ B @ C ) )
= ( divide_divide_real @ ( times_times_real @ A @ C ) @ B ) ) ).
% divide_divide_eq_right
thf(fact_1196_divide__divide__eq__left,axiom,
! [A: real,B: real,C: real] :
( ( divide_divide_real @ ( divide_divide_real @ A @ B ) @ C )
= ( divide_divide_real @ A @ ( times_times_real @ B @ C ) ) ) ).
% divide_divide_eq_left
thf(fact_1197_times__divide__eq__left,axiom,
! [B: real,C: real,A: real] :
( ( times_times_real @ ( divide_divide_real @ B @ C ) @ A )
= ( divide_divide_real @ ( times_times_real @ B @ A ) @ C ) ) ).
% times_divide_eq_left
thf(fact_1198_div__by__1,axiom,
! [A: int] :
( ( divide_divide_int @ A @ one_one_int )
= A ) ).
% div_by_1
thf(fact_1199_div__by__1,axiom,
! [A: real] :
( ( divide_divide_real @ A @ one_one_real )
= A ) ).
% div_by_1
thf(fact_1200_div__by__1,axiom,
! [A: nat] :
( ( divide_divide_nat @ A @ one_one_nat )
= A ) ).
% div_by_1
thf(fact_1201_div__dvd__div,axiom,
! [A: int,B: int,C: int] :
( ( dvd_dvd_int @ A @ B )
=> ( ( dvd_dvd_int @ A @ C )
=> ( ( dvd_dvd_int @ ( divide_divide_int @ B @ A ) @ ( divide_divide_int @ C @ A ) )
= ( dvd_dvd_int @ B @ C ) ) ) ) ).
% div_dvd_div
thf(fact_1202_div__dvd__div,axiom,
! [A: nat,B: nat,C: nat] :
( ( dvd_dvd_nat @ A @ B )
=> ( ( dvd_dvd_nat @ A @ C )
=> ( ( dvd_dvd_nat @ ( divide_divide_nat @ B @ A ) @ ( divide_divide_nat @ C @ A ) )
= ( dvd_dvd_nat @ B @ C ) ) ) ) ).
% div_dvd_div
thf(fact_1203_div__mult__mult1__if,axiom,
! [C: int,A: int,B: int] :
( ( ( C = zero_zero_int )
=> ( ( divide_divide_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) )
= zero_zero_int ) )
& ( ( C != zero_zero_int )
=> ( ( divide_divide_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) )
= ( divide_divide_int @ A @ B ) ) ) ) ).
% div_mult_mult1_if
thf(fact_1204_div__mult__mult1__if,axiom,
! [C: nat,A: nat,B: nat] :
( ( ( C = zero_zero_nat )
=> ( ( divide_divide_nat @ ( times_times_nat @ C @ A ) @ ( times_times_nat @ C @ B ) )
= zero_zero_nat ) )
& ( ( C != zero_zero_nat )
=> ( ( divide_divide_nat @ ( times_times_nat @ C @ A ) @ ( times_times_nat @ C @ B ) )
= ( divide_divide_nat @ A @ B ) ) ) ) ).
% div_mult_mult1_if
thf(fact_1205_div__mult__mult2,axiom,
! [C: int,A: int,B: int] :
( ( C != zero_zero_int )
=> ( ( divide_divide_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) )
= ( divide_divide_int @ A @ B ) ) ) ).
% div_mult_mult2
thf(fact_1206_div__mult__mult2,axiom,
! [C: nat,A: nat,B: nat] :
( ( C != zero_zero_nat )
=> ( ( divide_divide_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ C ) )
= ( divide_divide_nat @ A @ B ) ) ) ).
% div_mult_mult2
thf(fact_1207_div__mult__mult1,axiom,
! [C: int,A: int,B: int] :
( ( C != zero_zero_int )
=> ( ( divide_divide_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) )
= ( divide_divide_int @ A @ B ) ) ) ).
% div_mult_mult1
thf(fact_1208_div__mult__mult1,axiom,
! [C: nat,A: nat,B: nat] :
( ( C != zero_zero_nat )
=> ( ( divide_divide_nat @ ( times_times_nat @ C @ A ) @ ( times_times_nat @ C @ B ) )
= ( divide_divide_nat @ A @ B ) ) ) ).
% div_mult_mult1
thf(fact_1209_nonzero__mult__divide__mult__cancel__right2,axiom,
! [C: real,A: real,B: real] :
( ( C != zero_zero_real )
=> ( ( divide_divide_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ C @ B ) )
= ( divide_divide_real @ A @ B ) ) ) ).
% nonzero_mult_divide_mult_cancel_right2
thf(fact_1210_nonzero__mult__divide__mult__cancel__right,axiom,
! [C: real,A: real,B: real] :
( ( C != zero_zero_real )
=> ( ( divide_divide_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ C ) )
= ( divide_divide_real @ A @ B ) ) ) ).
% nonzero_mult_divide_mult_cancel_right
thf(fact_1211_nonzero__mult__divide__mult__cancel__left2,axiom,
! [C: real,A: real,B: real] :
( ( C != zero_zero_real )
=> ( ( divide_divide_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ B @ C ) )
= ( divide_divide_real @ A @ B ) ) ) ).
% nonzero_mult_divide_mult_cancel_left2
thf(fact_1212_nonzero__mult__divide__mult__cancel__left,axiom,
! [C: real,A: real,B: real] :
( ( C != zero_zero_real )
=> ( ( divide_divide_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B ) )
= ( divide_divide_real @ A @ B ) ) ) ).
% nonzero_mult_divide_mult_cancel_left
thf(fact_1213_mult__divide__mult__cancel__left__if,axiom,
! [C: real,A: real,B: real] :
( ( ( C = zero_zero_real )
=> ( ( divide_divide_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B ) )
= zero_zero_real ) )
& ( ( C != zero_zero_real )
=> ( ( divide_divide_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B ) )
= ( divide_divide_real @ A @ B ) ) ) ) ).
% mult_divide_mult_cancel_left_if
thf(fact_1214_nonzero__mult__div__cancel__left,axiom,
! [A: int,B: int] :
( ( A != zero_zero_int )
=> ( ( divide_divide_int @ ( times_times_int @ A @ B ) @ A )
= B ) ) ).
% nonzero_mult_div_cancel_left
thf(fact_1215_nonzero__mult__div__cancel__left,axiom,
! [A: real,B: real] :
( ( A != zero_zero_real )
=> ( ( divide_divide_real @ ( times_times_real @ A @ B ) @ A )
= B ) ) ).
% nonzero_mult_div_cancel_left
thf(fact_1216_nonzero__mult__div__cancel__left,axiom,
! [A: nat,B: nat] :
( ( A != zero_zero_nat )
=> ( ( divide_divide_nat @ ( times_times_nat @ A @ B ) @ A )
= B ) ) ).
% nonzero_mult_div_cancel_left
thf(fact_1217_nonzero__mult__div__cancel__right,axiom,
! [B: int,A: int] :
( ( B != zero_zero_int )
=> ( ( divide_divide_int @ ( times_times_int @ A @ B ) @ B )
= A ) ) ).
% nonzero_mult_div_cancel_right
thf(fact_1218_nonzero__mult__div__cancel__right,axiom,
! [B: real,A: real] :
( ( B != zero_zero_real )
=> ( ( divide_divide_real @ ( times_times_real @ A @ B ) @ B )
= A ) ) ).
% nonzero_mult_div_cancel_right
thf(fact_1219_nonzero__mult__div__cancel__right,axiom,
! [B: nat,A: nat] :
( ( B != zero_zero_nat )
=> ( ( divide_divide_nat @ ( times_times_nat @ A @ B ) @ B )
= A ) ) ).
% nonzero_mult_div_cancel_right
thf(fact_1220_div__self,axiom,
! [A: int] :
( ( A != zero_zero_int )
=> ( ( divide_divide_int @ A @ A )
= one_one_int ) ) ).
% div_self
thf(fact_1221_div__self,axiom,
! [A: real] :
( ( A != zero_zero_real )
=> ( ( divide_divide_real @ A @ A )
= one_one_real ) ) ).
% div_self
thf(fact_1222_div__self,axiom,
! [A: nat] :
( ( A != zero_zero_nat )
=> ( ( divide_divide_nat @ A @ A )
= one_one_nat ) ) ).
% div_self
thf(fact_1223_divide__eq__1__iff,axiom,
! [A: real,B: real] :
( ( ( divide_divide_real @ A @ B )
= one_one_real )
= ( ( B != zero_zero_real )
& ( A = B ) ) ) ).
% divide_eq_1_iff
thf(fact_1224_one__eq__divide__iff,axiom,
! [A: real,B: real] :
( ( one_one_real
= ( divide_divide_real @ A @ B ) )
= ( ( B != zero_zero_real )
& ( A = B ) ) ) ).
% one_eq_divide_iff
thf(fact_1225_divide__self,axiom,
! [A: real] :
( ( A != zero_zero_real )
=> ( ( divide_divide_real @ A @ A )
= one_one_real ) ) ).
% divide_self
thf(fact_1226_divide__self__if,axiom,
! [A: real] :
( ( ( A = zero_zero_real )
=> ( ( divide_divide_real @ A @ A )
= zero_zero_real ) )
& ( ( A != zero_zero_real )
=> ( ( divide_divide_real @ A @ A )
= one_one_real ) ) ) ).
% divide_self_if
thf(fact_1227_divide__eq__eq__1,axiom,
! [B: real,A: real] :
( ( ( divide_divide_real @ B @ A )
= one_one_real )
= ( ( A != zero_zero_real )
& ( A = B ) ) ) ).
% divide_eq_eq_1
thf(fact_1228_eq__divide__eq__1,axiom,
! [B: real,A: real] :
( ( one_one_real
= ( divide_divide_real @ B @ A ) )
= ( ( A != zero_zero_real )
& ( A = B ) ) ) ).
% eq_divide_eq_1
thf(fact_1229_one__divide__eq__0__iff,axiom,
! [A: real] :
( ( ( divide_divide_real @ one_one_real @ A )
= zero_zero_real )
= ( A = zero_zero_real ) ) ).
% one_divide_eq_0_iff
thf(fact_1230_pos__imp__zdiv__neg__iff,axiom,
! [B: int,A: int] :
( ( ord_less_int @ zero_zero_int @ B )
=> ( ( ord_less_int @ ( divide_divide_int @ A @ B ) @ zero_zero_int )
= ( ord_less_int @ A @ zero_zero_int ) ) ) ).
% pos_imp_zdiv_neg_iff
thf(fact_1231_neg__imp__zdiv__neg__iff,axiom,
! [B: int,A: int] :
( ( ord_less_int @ B @ zero_zero_int )
=> ( ( ord_less_int @ ( divide_divide_int @ A @ B ) @ zero_zero_int )
= ( ord_less_int @ zero_zero_int @ A ) ) ) ).
% neg_imp_zdiv_neg_iff
thf(fact_1232_div__neg__pos__less0,axiom,
! [A: int,B: int] :
( ( ord_less_int @ A @ zero_zero_int )
=> ( ( ord_less_int @ zero_zero_int @ B )
=> ( ord_less_int @ ( divide_divide_int @ A @ B ) @ zero_zero_int ) ) ) ).
% div_neg_pos_less0
thf(fact_1233_int__div__less__self,axiom,
! [X: int,K: int] :
( ( ord_less_int @ zero_zero_int @ X )
=> ( ( ord_less_int @ one_one_int @ K )
=> ( ord_less_int @ ( divide_divide_int @ X @ K ) @ X ) ) ) ).
% int_div_less_self
thf(fact_1234_fib_Ocases,axiom,
! [X: nat] :
( ( X != zero_zero_nat )
=> ( ( X
!= ( suc @ zero_zero_nat ) )
=> ~ ! [N3: nat] :
( X
!= ( suc @ ( suc @ N3 ) ) ) ) ) ).
% fib.cases
thf(fact_1235_real__divide__square__eq,axiom,
! [R2: real,A: real] :
( ( divide_divide_real @ ( times_times_real @ R2 @ A ) @ ( times_times_real @ R2 @ R2 ) )
= ( divide_divide_real @ A @ R2 ) ) ).
% real_divide_square_eq
thf(fact_1236_div__by__Suc__0,axiom,
! [M: nat] :
( ( divide_divide_nat @ M @ ( suc @ zero_zero_nat ) )
= M ) ).
% div_by_Suc_0
thf(fact_1237_div__less,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( ( divide_divide_nat @ M @ N )
= zero_zero_nat ) ) ).
% div_less
thf(fact_1238_div__mult__self__is__m,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( divide_divide_nat @ ( times_times_nat @ M @ N ) @ N )
= M ) ) ).
% div_mult_self_is_m
thf(fact_1239_div__mult__self1__is__m,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( divide_divide_nat @ ( times_times_nat @ N @ M ) @ N )
= M ) ) ).
% div_mult_self1_is_m
thf(fact_1240_div__mult2__eq,axiom,
! [M: nat,N: nat,Q: nat] :
( ( divide_divide_nat @ M @ ( times_times_nat @ N @ Q ) )
= ( divide_divide_nat @ ( divide_divide_nat @ M @ N ) @ Q ) ) ).
% div_mult2_eq
thf(fact_1241_Euclidean__Division_Odiv__eq__0__iff,axiom,
! [M: nat,N: nat] :
( ( ( divide_divide_nat @ M @ N )
= zero_zero_nat )
= ( ( ord_less_nat @ M @ N )
| ( N = zero_zero_nat ) ) ) ).
% Euclidean_Division.div_eq_0_iff
thf(fact_1242_nat__mult__div__cancel__disj,axiom,
! [K: nat,M: nat,N: nat] :
( ( ( K = zero_zero_nat )
=> ( ( divide_divide_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) )
= zero_zero_nat ) )
& ( ( K != zero_zero_nat )
=> ( ( divide_divide_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) )
= ( divide_divide_nat @ M @ N ) ) ) ) ).
% nat_mult_div_cancel_disj
thf(fact_1243_less__mult__imp__div__less,axiom,
! [M: nat,I: nat,N: nat] :
( ( ord_less_nat @ M @ ( times_times_nat @ I @ N ) )
=> ( ord_less_nat @ ( divide_divide_nat @ M @ N ) @ I ) ) ).
% less_mult_imp_div_less
thf(fact_1244_div__eq__dividend__iff,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ M )
=> ( ( ( divide_divide_nat @ M @ N )
= M )
= ( N = one_one_nat ) ) ) ).
% div_eq_dividend_iff
thf(fact_1245_div__less__dividend,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ one_one_nat @ N )
=> ( ( ord_less_nat @ zero_zero_nat @ M )
=> ( ord_less_nat @ ( divide_divide_nat @ M @ N ) @ M ) ) ) ).
% div_less_dividend
thf(fact_1246_div__less__iff__less__mult,axiom,
! [Q: nat,M: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ Q )
=> ( ( ord_less_nat @ ( divide_divide_nat @ M @ Q ) @ N )
= ( ord_less_nat @ M @ ( times_times_nat @ N @ Q ) ) ) ) ).
% div_less_iff_less_mult
thf(fact_1247_nat__mult__div__cancel1,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ( divide_divide_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) )
= ( divide_divide_nat @ M @ N ) ) ) ).
% nat_mult_div_cancel1
thf(fact_1248_div__if,axiom,
( divide_divide_nat
= ( ^ [M2: nat,N2: nat] :
( if_nat
@ ( ( ord_less_nat @ M2 @ N2 )
| ( N2 = zero_zero_nat ) )
@ zero_zero_nat
@ ( suc @ ( divide_divide_nat @ ( minus_minus_nat @ M2 @ N2 ) @ N2 ) ) ) ) ) ).
% div_if
thf(fact_1249_split__div,axiom,
! [P2: nat > $o,M: nat,N: nat] :
( ( P2 @ ( divide_divide_nat @ M @ N ) )
= ( ( ( N = zero_zero_nat )
=> ( P2 @ zero_zero_nat ) )
& ( ( N != zero_zero_nat )
=> ! [I3: nat,J3: nat] :
( ( ( ord_less_nat @ J3 @ N )
& ( M
= ( plus_plus_nat @ ( times_times_nat @ N @ I3 ) @ J3 ) ) )
=> ( P2 @ I3 ) ) ) ) ) ).
% split_div
thf(fact_1250_dividend__less__div__times,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ord_less_nat @ M @ ( plus_plus_nat @ N @ ( times_times_nat @ ( divide_divide_nat @ M @ N ) @ N ) ) ) ) ).
% dividend_less_div_times
thf(fact_1251_dividend__less__times__div,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ord_less_nat @ M @ ( plus_plus_nat @ N @ ( times_times_nat @ N @ ( divide_divide_nat @ M @ N ) ) ) ) ) ).
% dividend_less_times_div
thf(fact_1252_dvd__div__eq__2,axiom,
! [C: nat,A: nat,B: nat] :
( ( ord_less_nat @ zero_zero_nat @ C )
=> ( ( dvd_dvd_nat @ A @ C )
=> ( ( dvd_dvd_nat @ B @ C )
=> ( ( ( divide_divide_nat @ C @ A )
= ( divide_divide_nat @ C @ B ) )
=> ( A = B ) ) ) ) ) ).
% dvd_div_eq_2
thf(fact_1253_dvd__div__eq__1,axiom,
! [C: nat,A: nat,B: nat] :
( ( dvd_dvd_nat @ C @ A )
=> ( ( dvd_dvd_nat @ C @ B )
=> ( ( ( divide_divide_nat @ A @ C )
= ( divide_divide_nat @ B @ C ) )
=> ( A = B ) ) ) ) ).
% dvd_div_eq_1
thf(fact_1254_exists__least__lemma,axiom,
! [P2: nat > $o] :
( ~ ( P2 @ zero_zero_nat )
=> ( ? [X_12: nat] : ( P2 @ X_12 )
=> ? [N3: nat] :
( ~ ( P2 @ N3 )
& ( P2 @ ( suc @ N3 ) ) ) ) ) ).
% exists_least_lemma
thf(fact_1255_div__pos__geq,axiom,
! [L: int,K: int] :
( ( ord_less_int @ zero_zero_int @ L )
=> ( ( ord_less_eq_int @ L @ K )
=> ( ( divide_divide_int @ K @ L )
= ( plus_plus_int @ ( divide_divide_int @ ( minus_minus_int @ K @ L ) @ L ) @ one_one_int ) ) ) ) ).
% div_pos_geq
thf(fact_1256_div__pos__pos__trivial,axiom,
! [K: int,L: int] :
( ( ord_less_eq_int @ zero_zero_int @ K )
=> ( ( ord_less_int @ K @ L )
=> ( ( divide_divide_int @ K @ L )
= zero_zero_int ) ) ) ).
% div_pos_pos_trivial
thf(fact_1257_div__neg__neg__trivial,axiom,
! [K: int,L: int] :
( ( ord_less_eq_int @ K @ zero_zero_int )
=> ( ( ord_less_int @ L @ K )
=> ( ( divide_divide_int @ K @ L )
= zero_zero_int ) ) ) ).
% div_neg_neg_trivial
thf(fact_1258_zle__diff1__eq,axiom,
! [W: int,Z2: int] :
( ( ord_less_eq_int @ W @ ( minus_minus_int @ Z2 @ one_one_int ) )
= ( ord_less_int @ W @ Z2 ) ) ).
% zle_diff1_eq
thf(fact_1259_zle__add1__eq__le,axiom,
! [W: int,Z2: int] :
( ( ord_less_int @ W @ ( plus_plus_int @ Z2 @ one_one_int ) )
= ( ord_less_eq_int @ W @ Z2 ) ) ).
% zle_add1_eq_le
thf(fact_1260_less__eq__int__code_I1_J,axiom,
ord_less_eq_int @ zero_zero_int @ zero_zero_int ).
% less_eq_int_code(1)
thf(fact_1261_imp__le__cong,axiom,
! [X: int,X2: int,P2: $o,P7: $o] :
( ( X = X2 )
=> ( ( ( ord_less_eq_int @ zero_zero_int @ X2 )
=> ( P2 = P7 ) )
=> ( ( ( ord_less_eq_int @ zero_zero_int @ X )
=> P2 )
= ( ( ord_less_eq_int @ zero_zero_int @ X2 )
=> P7 ) ) ) ) ).
% imp_le_cong
thf(fact_1262_conj__le__cong,axiom,
! [X: int,X2: int,P2: $o,P7: $o] :
( ( X = X2 )
=> ( ( ( ord_less_eq_int @ zero_zero_int @ X2 )
=> ( P2 = P7 ) )
=> ( ( ( ord_less_eq_int @ zero_zero_int @ X )
& P2 )
= ( ( ord_less_eq_int @ zero_zero_int @ X2 )
& P7 ) ) ) ) ).
% conj_le_cong
thf(fact_1263_prime__ge__1__int,axiom,
! [P: int] :
( ( factor1798656936486255268me_int @ P )
=> ( ord_less_eq_int @ one_one_int @ P ) ) ).
% prime_ge_1_int
thf(fact_1264_prime__ge__0__int,axiom,
! [P: int] :
( ( factor1798656936486255268me_int @ P )
=> ( ord_less_eq_int @ zero_zero_int @ P ) ) ).
% prime_ge_0_int
thf(fact_1265_zdvd__antisym__nonneg,axiom,
! [M: int,N: int] :
( ( ord_less_eq_int @ zero_zero_int @ M )
=> ( ( ord_less_eq_int @ zero_zero_int @ N )
=> ( ( dvd_dvd_int @ M @ N )
=> ( ( dvd_dvd_int @ N @ M )
=> ( M = N ) ) ) ) ) ).
% zdvd_antisym_nonneg
thf(fact_1266_int__ge__induct,axiom,
! [K: int,I: int,P2: int > $o] :
( ( ord_less_eq_int @ K @ I )
=> ( ( P2 @ K )
=> ( ! [I2: int] :
( ( ord_less_eq_int @ K @ I2 )
=> ( ( P2 @ I2 )
=> ( P2 @ ( plus_plus_int @ I2 @ one_one_int ) ) ) )
=> ( P2 @ I ) ) ) ) ).
% int_ge_induct
thf(fact_1267_int__le__induct,axiom,
! [I: int,K: int,P2: int > $o] :
( ( ord_less_eq_int @ I @ K )
=> ( ( P2 @ K )
=> ( ! [I2: int] :
( ( ord_less_eq_int @ I2 @ K )
=> ( ( P2 @ I2 )
=> ( P2 @ ( minus_minus_int @ I2 @ one_one_int ) ) ) )
=> ( P2 @ I ) ) ) ) ).
% int_le_induct
% Helper facts (7)
thf(help_If_2_1_If_001t__Int__Oint_T,axiom,
! [X: int,Y: int] :
( ( if_int @ $false @ X @ Y )
= Y ) ).
thf(help_If_1_1_If_001t__Int__Oint_T,axiom,
! [X: int,Y: int] :
( ( if_int @ $true @ X @ Y )
= X ) ).
thf(help_If_2_1_If_001t__Nat__Onat_T,axiom,
! [X: nat,Y: nat] :
( ( if_nat @ $false @ X @ Y )
= Y ) ).
thf(help_If_1_1_If_001t__Nat__Onat_T,axiom,
! [X: nat,Y: nat] :
( ( if_nat @ $true @ X @ Y )
= X ) ).
thf(help_If_3_1_If_001t__Real__Oreal_T,axiom,
! [P2: $o] :
( ( P2 = $true )
| ( P2 = $false ) ) ).
thf(help_If_2_1_If_001t__Real__Oreal_T,axiom,
! [X: real,Y: real] :
( ( if_real @ $false @ X @ Y )
= Y ) ).
thf(help_If_1_1_If_001t__Real__Oreal_T,axiom,
! [X: real,Y: real] :
( ( if_real @ $true @ X @ Y )
= X ) ).
% Conjectures (1)
thf(conj_0,conjecture,
~ ( unique653641344996303876ng_nat @ g @ one_one_nat @ p ) ).
%------------------------------------------------------------------------------