TPTP Problem File: ITP035^1.p
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%------------------------------------------------------------------------------
% File : ITP035^1 : TPTP v8.2.0. Released v7.5.0.
% Domain : Interactive Theorem Proving
% Problem : Sledgehammer Cartan problem prob_32__6530878_1
% Version : Especial.
% English :
% Refs : [BH+15] Blanchette et al. (2015), Mining the Archive of Formal
% : [Des21] Desharnais (2021), Email to Geoff Sutcliffe
% Source : [Des21]
% Names : Cartan/prob_32__6530878_1 [Des21]
% Status : Theorem
% Rating : 0.30 v8.2.0, 0.31 v8.1.0, 0.27 v7.5.0
% Syntax : Number of formulae : 433 ( 205 unt; 73 typ; 0 def)
% Number of atoms : 909 ( 532 equ; 0 cnn)
% Maximal formula atoms : 11 ( 2 avg)
% Number of connectives : 2406 ( 60 ~; 38 |; 72 &;1899 @)
% ( 0 <=>; 337 =>; 0 <=; 0 <~>)
% Maximal formula depth : 18 ( 5 avg)
% Number of types : 8 ( 7 usr)
% Number of type conns : 426 ( 426 >; 0 *; 0 +; 0 <<)
% Number of symbols : 69 ( 66 usr; 11 con; 0-4 aty)
% Number of variables : 832 ( 41 ^; 785 !; 6 ?; 832 :)
% SPC : TH0_THM_EQU_NAR
% Comments : This file was generated by Sledgehammer 2021-02-23 15:38:30.825
%------------------------------------------------------------------------------
% Could-be-implicit typings (7)
thf(ty_n_t__Set__Oset_I_062_It__Complex__Ocomplex_Mt__Complex__Ocomplex_J_J,type,
set_complex_complex: $tType ).
thf(ty_n_t__Set__Oset_It__Complex__Ocomplex_J,type,
set_complex: $tType ).
thf(ty_n_t__Set__Oset_It__Real__Oreal_J,type,
set_real: $tType ).
thf(ty_n_t__Set__Oset_It__Nat__Onat_J,type,
set_nat: $tType ).
thf(ty_n_t__Complex__Ocomplex,type,
complex: $tType ).
thf(ty_n_t__Real__Oreal,type,
real: $tType ).
thf(ty_n_t__Nat__Onat,type,
nat: $tType ).
% Explicit typings (66)
thf(sy_c_Complex__Analysis__Basics_Oholomorphic__on,type,
comple372758642hic_on: ( complex > complex ) > set_complex > $o ).
thf(sy_c_Derivative_Oderiv_001t__Complex__Ocomplex,type,
deriv_complex: ( complex > complex ) > complex > complex ).
thf(sy_c_Derivative_Oderiv_001t__Real__Oreal,type,
deriv_real: ( real > real ) > real > real ).
thf(sy_c_Fun_Ocomp_001_062_It__Complex__Ocomplex_Mt__Complex__Ocomplex_J_001_062_It__Complex__Ocomplex_Mt__Complex__Ocomplex_J_001_062_It__Complex__Ocomplex_Mt__Complex__Ocomplex_J,type,
comp_c1610621014omplex: ( ( complex > complex ) > complex > complex ) > ( ( complex > complex ) > complex > complex ) > ( complex > complex ) > complex > complex ).
thf(sy_c_Fun_Ocomp_001_062_It__Complex__Ocomplex_Mt__Complex__Ocomplex_J_001t__Complex__Ocomplex_001t__Complex__Ocomplex,type,
comp_c881053372omplex: ( ( complex > complex ) > complex ) > ( complex > complex > complex ) > complex > complex ).
thf(sy_c_Fun_Ocomp_001_062_It__Complex__Ocomplex_Mt__Complex__Ocomplex_J_001t__Real__Oreal_001t__Real__Oreal,type,
comp_c1884694328l_real: ( ( complex > complex ) > real ) > ( real > complex > complex ) > real > real ).
thf(sy_c_Fun_Ocomp_001t__Complex__Ocomplex_001_062_It__Complex__Ocomplex_Mt__Complex__Ocomplex_J_001_062_It__Complex__Ocomplex_Mt__Complex__Ocomplex_J,type,
comp_c606622857omplex: ( complex > complex > complex ) > ( ( complex > complex ) > complex ) > ( complex > complex ) > complex > complex ).
thf(sy_c_Fun_Ocomp_001t__Complex__Ocomplex_001t__Complex__Ocomplex_001t__Complex__Ocomplex,type,
comp_c130555887omplex: ( complex > complex ) > ( complex > complex ) > complex > complex ).
thf(sy_c_Fun_Ocomp_001t__Complex__Ocomplex_001t__Real__Oreal_001t__Real__Oreal,type,
comp_c819638635l_real: ( complex > real ) > ( real > complex ) > real > real ).
thf(sy_c_Fun_Ocomp_001t__Real__Oreal_001_062_It__Complex__Ocomplex_Mt__Complex__Ocomplex_J_001_062_It__Complex__Ocomplex_Mt__Complex__Ocomplex_J,type,
comp_r2009261319omplex: ( real > complex > complex ) > ( ( complex > complex ) > real ) > ( complex > complex ) > complex > complex ).
thf(sy_c_Fun_Ocomp_001t__Real__Oreal_001t__Complex__Ocomplex_001t__Complex__Ocomplex,type,
comp_r667767405omplex: ( real > complex ) > ( complex > real ) > complex > complex ).
thf(sy_c_Fun_Ocomp_001t__Real__Oreal_001t__Real__Oreal_001t__Real__Oreal,type,
comp_real_real_real: ( real > real ) > ( real > real ) > real > real ).
thf(sy_c_Fun_Oid_001_062_It__Complex__Ocomplex_Mt__Complex__Ocomplex_J,type,
id_complex_complex: ( complex > complex ) > complex > complex ).
thf(sy_c_Fun_Oid_001_062_It__Real__Oreal_Mt__Real__Oreal_J,type,
id_real_real: ( real > real ) > real > real ).
thf(sy_c_Fun_Oid_001t__Complex__Ocomplex,type,
id_complex: complex > complex ).
thf(sy_c_Fun_Oid_001t__Real__Oreal,type,
id_real: real > real ).
thf(sy_c_Fun_Oid_001t__Set__Oset_I_062_It__Complex__Ocomplex_Mt__Complex__Ocomplex_J_J,type,
id_set1618538368omplex: set_complex_complex > set_complex_complex ).
thf(sy_c_Fun_Oid_001t__Set__Oset_It__Complex__Ocomplex_J,type,
id_set_complex: set_complex > set_complex ).
thf(sy_c_Fun_Oid_001t__Set__Oset_It__Real__Oreal_J,type,
id_set_real: set_real > set_real ).
thf(sy_c_Groups_Oone__class_Oone_001t__Complex__Ocomplex,type,
one_one_complex: complex ).
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_Otimes__class_Otimes_001t__Complex__Ocomplex,type,
times_times_complex: complex > complex > complex ).
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_Otimes__class_Otimes_001t__Set__Oset_It__Complex__Ocomplex_J,type,
times_1316095593omplex: set_complex > set_complex > set_complex ).
thf(sy_c_Groups_Otimes__class_Otimes_001t__Set__Oset_It__Nat__Onat_J,type,
times_times_set_nat: set_nat > set_nat > set_nat ).
thf(sy_c_Groups_Otimes__class_Otimes_001t__Set__Oset_It__Real__Oreal_J,type,
times_times_set_real: set_real > set_real > set_real ).
thf(sy_c_Groups_Ozero__class_Ozero_001t__Complex__Ocomplex,type,
zero_zero_complex: complex ).
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__Complex__Ocomplex,type,
if_complex: $o > complex > complex > complex ).
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_Ocompow_001_062_I_062_It__Complex__Ocomplex_Mt__Complex__Ocomplex_J_M_062_It__Complex__Ocomplex_Mt__Complex__Ocomplex_J_J,type,
compow1098280738omplex: nat > ( ( complex > complex ) > complex > complex ) > ( complex > complex ) > complex > complex ).
thf(sy_c_Nat_Ocompow_001_062_I_062_It__Real__Oreal_Mt__Real__Oreal_J_M_062_It__Real__Oreal_Mt__Real__Oreal_J_J,type,
compow1723822618l_real: nat > ( ( real > real ) > real > real ) > ( real > real ) > real > real ).
thf(sy_c_Nat_Ocompow_001_062_It__Complex__Ocomplex_Mt__Complex__Ocomplex_J,type,
compow1667379464omplex: nat > ( complex > complex ) > complex > complex ).
thf(sy_c_Nat_Ocompow_001_062_It__Real__Oreal_Mt__Real__Oreal_J,type,
compow_real_real: nat > ( real > real ) > real > real ).
thf(sy_c_Nat_Ofunpow_001_062_It__Complex__Ocomplex_Mt__Complex__Ocomplex_J,type,
funpow1854104714omplex: nat > ( ( complex > complex ) > complex > complex ) > ( complex > complex ) > complex > complex ).
thf(sy_c_Nat_Ofunpow_001_062_It__Real__Oreal_Mt__Real__Oreal_J,type,
funpow_real_real: nat > ( ( real > real ) > real > real ) > ( real > real ) > real > real ).
thf(sy_c_Nat_Ofunpow_001t__Complex__Ocomplex,type,
funpow_complex: nat > ( complex > complex ) > complex > complex ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Nat__Onat,type,
ord_less_eq_nat: nat > nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Real__Oreal,type,
ord_less_eq_real: real > real > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Complex__Ocomplex_J,type,
ord_le701908932omplex: set_complex > set_complex > $o ).
thf(sy_c_Set_OCollect_001t__Complex__Ocomplex,type,
collect_complex: ( complex > $o ) > set_complex ).
thf(sy_c_Set_Oimage_001_062_It__Complex__Ocomplex_Mt__Complex__Ocomplex_J_001_062_It__Complex__Ocomplex_Mt__Complex__Ocomplex_J,type,
image_944012797omplex: ( ( complex > complex ) > complex > complex ) > set_complex_complex > set_complex_complex ).
thf(sy_c_Set_Oimage_001t__Complex__Ocomplex_001t__Complex__Ocomplex,type,
image_58037603omplex: ( complex > complex ) > set_complex > set_complex ).
thf(sy_c_Set_Oimage_001t__Real__Oreal_001t__Real__Oreal,type,
image_real_real: ( real > real ) > set_real > set_real ).
thf(sy_c_Topological__Spaces_Oopen__class_Oopen_001t__Complex__Ocomplex,type,
topolo935673511omplex: set_complex > $o ).
thf(sy_c_Topological__Spaces_Otopological__space__class_Oconnected_001t__Complex__Ocomplex,type,
topolo2127351575omplex: set_complex > $o ).
thf(sy_c_Transcendental_Oarcosh_001t__Complex__Ocomplex,type,
arcosh_complex: complex > complex ).
thf(sy_c_Transcendental_Oarcosh_001t__Real__Oreal,type,
arcosh_real: real > real ).
thf(sy_c_Transcendental_Oarsinh_001t__Complex__Ocomplex,type,
arsinh_complex: complex > complex ).
thf(sy_c_Transcendental_Oarsinh_001t__Real__Oreal,type,
arsinh_real: real > real ).
thf(sy_c_Transcendental_Oartanh_001t__Complex__Ocomplex,type,
artanh_complex: complex > complex ).
thf(sy_c_Transcendental_Oartanh_001t__Real__Oreal,type,
artanh_real: real > real ).
thf(sy_c_Transcendental_Oln__class_Oln_001t__Complex__Ocomplex,type,
ln_ln_complex: complex > complex ).
thf(sy_c_Transcendental_Oln__class_Oln_001t__Real__Oreal,type,
ln_ln_real: real > real ).
thf(sy_c_member_001t__Complex__Ocomplex,type,
member_complex: complex > set_complex > $o ).
thf(sy_c_member_001t__Nat__Onat,type,
member_nat: nat > set_nat > $o ).
thf(sy_c_member_001t__Real__Oreal,type,
member_real: real > set_real > $o ).
thf(sy_v_S,type,
s: set_complex ).
thf(sy_v_T,type,
t: set_complex ).
thf(sy_v_f,type,
f: complex > complex ).
thf(sy_v_g,type,
g: complex > complex ).
thf(sy_v_w,type,
w: complex ).
% Relevant facts (352)
thf(fact_0__092_060open_062_092_060And_062z_O_A_Ideriv_A_094_094_A1_J_Aid_Az_A_061_A_Iif_A1_A_061_A0_Athen_Az_Aelse_Aif_A1_A_061_A1_Athen_A1_058_058_063_Ha_Aelse_A_I0_058_058_063_Ha_J_J_092_060close_062,axiom,
! [Z: real] :
( ( ( one_one_nat = zero_zero_nat )
=> ( ( compow1723822618l_real @ one_one_nat @ deriv_real @ id_real @ Z )
= Z ) )
& ( ( one_one_nat != zero_zero_nat )
=> ( ( compow1723822618l_real @ one_one_nat @ deriv_real @ id_real @ Z )
= one_one_real ) ) ) ).
% \<open>\<And>z. (deriv ^^ 1) id z = (if 1 = 0 then z else if 1 = 1 then 1::?'a else (0::?'a))\<close>
thf(fact_1__092_060open_062_092_060And_062z_O_A_Ideriv_A_094_094_A1_J_Aid_Az_A_061_A_Iif_A1_A_061_A0_Athen_Az_Aelse_Aif_A1_A_061_A1_Athen_A1_058_058_063_Ha_Aelse_A_I0_058_058_063_Ha_J_J_092_060close_062,axiom,
! [Z: complex] :
( ( ( one_one_nat = zero_zero_nat )
=> ( ( compow1098280738omplex @ one_one_nat @ deriv_complex @ id_complex @ Z )
= Z ) )
& ( ( one_one_nat != zero_zero_nat )
=> ( ( compow1098280738omplex @ one_one_nat @ deriv_complex @ id_complex @ Z )
= one_one_complex ) ) ) ).
% \<open>\<And>z. (deriv ^^ 1) id z = (if 1 = 0 then z else if 1 = 1 then 1::?'a else (0::?'a))\<close>
thf(fact_2_assms_I7_J,axiom,
member_complex @ w @ s ).
% assms(7)
thf(fact_3_deriv__id,axiom,
( ( deriv_complex @ id_complex )
= ( ^ [Z2: complex] : one_one_complex ) ) ).
% deriv_id
thf(fact_4_deriv__id,axiom,
( ( deriv_real @ id_real )
= ( ^ [Z2: real] : one_one_real ) ) ).
% deriv_id
thf(fact_5_id__apply,axiom,
( id_real
= ( ^ [X: real] : X ) ) ).
% id_apply
thf(fact_6_id__apply,axiom,
( id_complex_complex
= ( ^ [X: complex > complex] : X ) ) ).
% id_apply
thf(fact_7_id__apply,axiom,
( id_complex
= ( ^ [X: complex] : X ) ) ).
% id_apply
thf(fact_8_id__def,axiom,
( id_real
= ( ^ [X: real] : X ) ) ).
% id_def
thf(fact_9_id__def,axiom,
( id_complex_complex
= ( ^ [X: complex > complex] : X ) ) ).
% id_def
thf(fact_10_id__def,axiom,
( id_complex
= ( ^ [X: complex] : X ) ) ).
% id_def
thf(fact_11_eq__id__iff,axiom,
! [F: real > real] :
( ( ! [X: real] :
( ( F @ X )
= X ) )
= ( F = id_real ) ) ).
% eq_id_iff
thf(fact_12_eq__id__iff,axiom,
! [F: ( complex > complex ) > complex > complex] :
( ( ! [X: complex > complex] :
( ( F @ X )
= X ) )
= ( F = id_complex_complex ) ) ).
% eq_id_iff
thf(fact_13_eq__id__iff,axiom,
! [F: complex > complex] :
( ( ! [X: complex] :
( ( F @ X )
= X ) )
= ( F = id_complex ) ) ).
% eq_id_iff
thf(fact_14_higher__deriv__id,axiom,
! [N: nat,Z: real] :
( ( ( N = zero_zero_nat )
=> ( ( compow1723822618l_real @ N @ deriv_real @ id_real @ Z )
= Z ) )
& ( ( N != zero_zero_nat )
=> ( ( ( N = one_one_nat )
=> ( ( compow1723822618l_real @ N @ deriv_real @ id_real @ Z )
= one_one_real ) )
& ( ( N != one_one_nat )
=> ( ( compow1723822618l_real @ N @ deriv_real @ id_real @ Z )
= zero_zero_real ) ) ) ) ) ).
% higher_deriv_id
thf(fact_15_higher__deriv__id,axiom,
! [N: nat,Z: complex] :
( ( ( N = zero_zero_nat )
=> ( ( compow1098280738omplex @ N @ deriv_complex @ id_complex @ Z )
= Z ) )
& ( ( N != zero_zero_nat )
=> ( ( ( N = one_one_nat )
=> ( ( compow1098280738omplex @ N @ deriv_complex @ id_complex @ Z )
= one_one_complex ) )
& ( ( N != one_one_nat )
=> ( ( compow1098280738omplex @ N @ deriv_complex @ id_complex @ Z )
= zero_zero_complex ) ) ) ) ) ).
% higher_deriv_id
thf(fact_16_one__reorient,axiom,
! [X2: complex] :
( ( one_one_complex = X2 )
= ( X2 = one_one_complex ) ) ).
% one_reorient
thf(fact_17_one__reorient,axiom,
! [X2: nat] :
( ( one_one_nat = X2 )
= ( X2 = one_one_nat ) ) ).
% one_reorient
thf(fact_18_one__reorient,axiom,
! [X2: real] :
( ( one_one_real = X2 )
= ( X2 = one_one_real ) ) ).
% one_reorient
thf(fact_19_id__funpow,axiom,
! [N: nat] :
( ( compow_real_real @ N @ id_real )
= id_real ) ).
% id_funpow
thf(fact_20_id__funpow,axiom,
! [N: nat] :
( ( compow1723822618l_real @ N @ id_real_real )
= id_real_real ) ).
% id_funpow
thf(fact_21_id__funpow,axiom,
! [N: nat] :
( ( compow1667379464omplex @ N @ id_complex )
= id_complex ) ).
% id_funpow
thf(fact_22_id__funpow,axiom,
! [N: nat] :
( ( compow1098280738omplex @ N @ id_complex_complex )
= id_complex_complex ) ).
% id_funpow
thf(fact_23_calculation,axiom,
( ( times_times_complex @ ( deriv_complex @ f @ w ) @ ( deriv_complex @ g @ ( f @ w ) ) )
= ( deriv_complex @ id_complex @ w ) ) ).
% calculation
thf(fact_24__092_060open_062deriv_A_Ig_A_092_060circ_062_Af_J_Aw_A_061_Aderiv_Aid_Aw_092_060close_062,axiom,
( ( deriv_complex @ ( comp_c130555887omplex @ g @ f ) @ w )
= ( deriv_complex @ id_complex @ w ) ) ).
% \<open>deriv (g \<circ> f) w = deriv id w\<close>
thf(fact_25_zero__neq__one,axiom,
zero_zero_nat != one_one_nat ).
% zero_neq_one
thf(fact_26_zero__neq__one,axiom,
zero_zero_real != one_one_real ).
% zero_neq_one
thf(fact_27_zero__neq__one,axiom,
zero_zero_complex != one_one_complex ).
% zero_neq_one
thf(fact_28__092_060open_062deriv_Af_Aw_A_K_Aderiv_Ag_A_If_Aw_J_A_061_Aderiv_Ag_A_If_Aw_J_A_K_Aderiv_Af_Aw_092_060close_062,axiom,
( ( times_times_complex @ ( deriv_complex @ f @ w ) @ ( deriv_complex @ g @ ( f @ w ) ) )
= ( times_times_complex @ ( deriv_complex @ g @ ( f @ w ) ) @ ( deriv_complex @ f @ w ) ) ) ).
% \<open>deriv f w * deriv g (f w) = deriv g (f w) * deriv f w\<close>
thf(fact_29_comp__apply,axiom,
( comp_c130555887omplex
= ( ^ [F2: complex > complex,G: complex > complex,X: complex] : ( F2 @ ( G @ X ) ) ) ) ).
% comp_apply
thf(fact_30_mult__zero__left,axiom,
! [A: complex] :
( ( times_times_complex @ zero_zero_complex @ A )
= zero_zero_complex ) ).
% mult_zero_left
thf(fact_31_mult__zero__left,axiom,
! [A: nat] :
( ( times_times_nat @ zero_zero_nat @ A )
= zero_zero_nat ) ).
% mult_zero_left
thf(fact_32_mult__zero__left,axiom,
! [A: real] :
( ( times_times_real @ zero_zero_real @ A )
= zero_zero_real ) ).
% mult_zero_left
thf(fact_33_mult__zero__right,axiom,
! [A: complex] :
( ( times_times_complex @ A @ zero_zero_complex )
= zero_zero_complex ) ).
% mult_zero_right
thf(fact_34_mult__zero__right,axiom,
! [A: nat] :
( ( times_times_nat @ A @ zero_zero_nat )
= zero_zero_nat ) ).
% mult_zero_right
thf(fact_35_mult__zero__right,axiom,
! [A: real] :
( ( times_times_real @ A @ zero_zero_real )
= zero_zero_real ) ).
% mult_zero_right
thf(fact_36_mult__eq__0__iff,axiom,
! [A: complex,B: complex] :
( ( ( times_times_complex @ A @ B )
= zero_zero_complex )
= ( ( A = zero_zero_complex )
| ( B = zero_zero_complex ) ) ) ).
% mult_eq_0_iff
thf(fact_37_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_38_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_39_mult__cancel__left,axiom,
! [C: complex,A: complex,B: complex] :
( ( ( times_times_complex @ C @ A )
= ( times_times_complex @ C @ B ) )
= ( ( C = zero_zero_complex )
| ( A = B ) ) ) ).
% mult_cancel_left
thf(fact_40_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_41_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_42_mult__cancel__right,axiom,
! [A: complex,C: complex,B: complex] :
( ( ( times_times_complex @ A @ C )
= ( times_times_complex @ B @ C ) )
= ( ( C = zero_zero_complex )
| ( A = B ) ) ) ).
% mult_cancel_right
thf(fact_43_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_44_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_45_mult_Oright__neutral,axiom,
! [A: complex] :
( ( times_times_complex @ A @ one_one_complex )
= A ) ).
% mult.right_neutral
thf(fact_46_mult_Oright__neutral,axiom,
! [A: nat] :
( ( times_times_nat @ A @ one_one_nat )
= A ) ).
% mult.right_neutral
thf(fact_47_mult_Oright__neutral,axiom,
! [A: real] :
( ( times_times_real @ A @ one_one_real )
= A ) ).
% mult.right_neutral
thf(fact_48_mult_Oleft__neutral,axiom,
! [A: complex] :
( ( times_times_complex @ one_one_complex @ A )
= A ) ).
% mult.left_neutral
thf(fact_49_mult_Oleft__neutral,axiom,
! [A: nat] :
( ( times_times_nat @ one_one_nat @ A )
= A ) ).
% mult.left_neutral
thf(fact_50_mult_Oleft__neutral,axiom,
! [A: real] :
( ( times_times_real @ one_one_real @ A )
= A ) ).
% mult.left_neutral
thf(fact_51_id__comp,axiom,
! [G2: complex > complex] :
( ( comp_c130555887omplex @ id_complex @ G2 )
= G2 ) ).
% id_comp
thf(fact_52_comp__id,axiom,
! [F: complex > complex] :
( ( comp_c130555887omplex @ F @ id_complex )
= F ) ).
% comp_id
thf(fact_53_funpow__0,axiom,
! [F: ( complex > complex ) > complex > complex,X2: complex > complex] :
( ( compow1098280738omplex @ zero_zero_nat @ F @ X2 )
= X2 ) ).
% funpow_0
thf(fact_54_funpow__0,axiom,
! [F: ( real > real ) > real > real,X2: real > real] :
( ( compow1723822618l_real @ zero_zero_nat @ F @ X2 )
= X2 ) ).
% funpow_0
thf(fact_55_funpow__0,axiom,
! [F: complex > complex,X2: complex] :
( ( compow1667379464omplex @ zero_zero_nat @ F @ X2 )
= X2 ) ).
% funpow_0
thf(fact_56_assms_I3_J,axiom,
topolo935673511omplex @ s ).
% assms(3)
thf(fact_57__092_060open_062deriv_Ag_A_If_Aw_J_A_K_Aderiv_Af_Aw_A_061_Aderiv_A_Ig_A_092_060circ_062_Af_J_Aw_092_060close_062,axiom,
( ( times_times_complex @ ( deriv_complex @ g @ ( f @ w ) ) @ ( deriv_complex @ f @ w ) )
= ( deriv_complex @ ( comp_c130555887omplex @ g @ f ) @ w ) ) ).
% \<open>deriv g (f w) * deriv f w = deriv (g \<circ> f) w\<close>
thf(fact_58_mult__cancel__right2,axiom,
! [A: complex,C: complex] :
( ( ( times_times_complex @ A @ C )
= C )
= ( ( C = zero_zero_complex )
| ( A = one_one_complex ) ) ) ).
% mult_cancel_right2
thf(fact_59_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_60_mult__cancel__right1,axiom,
! [C: complex,B: complex] :
( ( C
= ( times_times_complex @ B @ C ) )
= ( ( C = zero_zero_complex )
| ( B = one_one_complex ) ) ) ).
% mult_cancel_right1
thf(fact_61_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_62_mult__cancel__left2,axiom,
! [C: complex,A: complex] :
( ( ( times_times_complex @ C @ A )
= C )
= ( ( C = zero_zero_complex )
| ( A = one_one_complex ) ) ) ).
% mult_cancel_left2
thf(fact_63_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_64_mult__cancel__left1,axiom,
! [C: complex,B: complex] :
( ( C
= ( times_times_complex @ C @ B ) )
= ( ( C = zero_zero_complex )
| ( B = one_one_complex ) ) ) ).
% mult_cancel_left1
thf(fact_65_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_66_assms_I6_J,axiom,
! [Z: complex] :
( ( member_complex @ Z @ s )
=> ( ( g @ ( f @ Z ) )
= Z ) ) ).
% assms(6)
thf(fact_67_assms_I1_J,axiom,
comple372758642hic_on @ f @ s ).
% assms(1)
thf(fact_68_ab__semigroup__mult__class_Omult__ac_I1_J,axiom,
! [A: complex,B: complex,C: complex] :
( ( times_times_complex @ ( times_times_complex @ A @ B ) @ C )
= ( times_times_complex @ A @ ( times_times_complex @ B @ C ) ) ) ).
% ab_semigroup_mult_class.mult_ac(1)
thf(fact_69_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_70_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_71_comp__def,axiom,
( comp_c130555887omplex
= ( ^ [F2: complex > complex,G: complex > complex,X: complex] : ( F2 @ ( G @ X ) ) ) ) ).
% comp_def
thf(fact_72_comp__assoc,axiom,
! [F: complex > complex,G2: complex > complex,H: complex > complex] :
( ( comp_c130555887omplex @ ( comp_c130555887omplex @ F @ G2 ) @ H )
= ( comp_c130555887omplex @ F @ ( comp_c130555887omplex @ G2 @ H ) ) ) ).
% comp_assoc
thf(fact_73_comp__funpow,axiom,
! [N: nat,F: complex > complex] :
( ( compow1098280738omplex @ N @ ( comp_c130555887omplex @ F ) )
= ( comp_c130555887omplex @ ( compow1667379464omplex @ N @ F ) ) ) ).
% comp_funpow
thf(fact_74_comp__funpow,axiom,
! [N: nat,F: real > real] :
( ( compow1723822618l_real @ N @ ( comp_real_real_real @ F ) )
= ( comp_real_real_real @ ( compow_real_real @ N @ F ) ) ) ).
% comp_funpow
thf(fact_75_comp__eq__dest,axiom,
! [A: complex > complex,B: complex > complex,C: complex > complex,D: complex > complex,V: complex] :
( ( ( comp_c130555887omplex @ A @ B )
= ( comp_c130555887omplex @ C @ D ) )
=> ( ( A @ ( B @ V ) )
= ( C @ ( D @ V ) ) ) ) ).
% comp_eq_dest
thf(fact_76_comp__eq__elim,axiom,
! [A: complex > complex,B: complex > complex,C: complex > complex,D: complex > complex] :
( ( ( comp_c130555887omplex @ A @ B )
= ( comp_c130555887omplex @ C @ D ) )
=> ! [V2: complex] :
( ( A @ ( B @ V2 ) )
= ( C @ ( D @ V2 ) ) ) ) ).
% comp_eq_elim
thf(fact_77_funpow__swap1,axiom,
! [F: ( complex > complex ) > complex > complex,N: nat,X2: complex > complex] :
( ( F @ ( compow1098280738omplex @ N @ F @ X2 ) )
= ( compow1098280738omplex @ N @ F @ ( F @ X2 ) ) ) ).
% funpow_swap1
thf(fact_78_funpow__swap1,axiom,
! [F: ( real > real ) > real > real,N: nat,X2: real > real] :
( ( F @ ( compow1723822618l_real @ N @ F @ X2 ) )
= ( compow1723822618l_real @ N @ F @ ( F @ X2 ) ) ) ).
% funpow_swap1
thf(fact_79_funpow__swap1,axiom,
! [F: complex > complex,N: nat,X2: complex] :
( ( F @ ( compow1667379464omplex @ N @ F @ X2 ) )
= ( compow1667379464omplex @ N @ F @ ( F @ X2 ) ) ) ).
% funpow_swap1
thf(fact_80_comp__eq__dest__lhs,axiom,
! [A: complex > complex,B: complex > complex,C: complex > complex,V: complex] :
( ( ( comp_c130555887omplex @ A @ B )
= C )
=> ( ( A @ ( B @ V ) )
= ( C @ V ) ) ) ).
% comp_eq_dest_lhs
thf(fact_81_zero__reorient,axiom,
! [X2: nat] :
( ( zero_zero_nat = X2 )
= ( X2 = zero_zero_nat ) ) ).
% zero_reorient
thf(fact_82_zero__reorient,axiom,
! [X2: real] :
( ( zero_zero_real = X2 )
= ( X2 = zero_zero_real ) ) ).
% zero_reorient
thf(fact_83_zero__reorient,axiom,
! [X2: complex] :
( ( zero_zero_complex = X2 )
= ( X2 = zero_zero_complex ) ) ).
% zero_reorient
thf(fact_84_mult__not__zero,axiom,
! [A: complex,B: complex] :
( ( ( times_times_complex @ A @ B )
!= zero_zero_complex )
=> ( ( A != zero_zero_complex )
& ( B != zero_zero_complex ) ) ) ).
% mult_not_zero
thf(fact_85_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_86_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_87_mem__Collect__eq,axiom,
! [A: complex,P: complex > $o] :
( ( member_complex @ A @ ( collect_complex @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_88_Collect__mem__eq,axiom,
! [A2: set_complex] :
( ( collect_complex
@ ^ [X: complex] : ( member_complex @ X @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_89_mult_Oassoc,axiom,
! [A: complex,B: complex,C: complex] :
( ( times_times_complex @ ( times_times_complex @ A @ B ) @ C )
= ( times_times_complex @ A @ ( times_times_complex @ B @ C ) ) ) ).
% mult.assoc
thf(fact_90_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_91_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_92_mult_Ocommute,axiom,
( times_times_complex
= ( ^ [A3: complex,B2: complex] : ( times_times_complex @ B2 @ A3 ) ) ) ).
% mult.commute
thf(fact_93_mult_Ocommute,axiom,
( times_times_nat
= ( ^ [A3: nat,B2: nat] : ( times_times_nat @ B2 @ A3 ) ) ) ).
% mult.commute
thf(fact_94_mult_Ocommute,axiom,
( times_times_real
= ( ^ [A3: real,B2: real] : ( times_times_real @ B2 @ A3 ) ) ) ).
% mult.commute
thf(fact_95_mult_Oleft__commute,axiom,
! [B: complex,A: complex,C: complex] :
( ( times_times_complex @ B @ ( times_times_complex @ A @ C ) )
= ( times_times_complex @ A @ ( times_times_complex @ B @ C ) ) ) ).
% mult.left_commute
thf(fact_96_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_97_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_98_divisors__zero,axiom,
! [A: complex,B: complex] :
( ( ( times_times_complex @ A @ B )
= zero_zero_complex )
=> ( ( A = zero_zero_complex )
| ( B = zero_zero_complex ) ) ) ).
% divisors_zero
thf(fact_99_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_100_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_101_no__zero__divisors,axiom,
! [A: complex,B: complex] :
( ( A != zero_zero_complex )
=> ( ( B != zero_zero_complex )
=> ( ( times_times_complex @ A @ B )
!= zero_zero_complex ) ) ) ).
% no_zero_divisors
thf(fact_102_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_103_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_104_mult__left__cancel,axiom,
! [C: complex,A: complex,B: complex] :
( ( C != zero_zero_complex )
=> ( ( ( times_times_complex @ C @ A )
= ( times_times_complex @ C @ B ) )
= ( A = B ) ) ) ).
% mult_left_cancel
thf(fact_105_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_106_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_107_mult__right__cancel,axiom,
! [C: complex,A: complex,B: complex] :
( ( C != zero_zero_complex )
=> ( ( ( times_times_complex @ A @ C )
= ( times_times_complex @ B @ C ) )
= ( A = B ) ) ) ).
% mult_right_cancel
thf(fact_108_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_109_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_110_comp__eq__id__dest,axiom,
! [A: complex > complex,B: complex > complex,C: complex > complex,V: complex] :
( ( ( comp_c130555887omplex @ A @ B )
= ( comp_c130555887omplex @ id_complex @ C ) )
=> ( ( A @ ( B @ V ) )
= ( C @ V ) ) ) ).
% comp_eq_id_dest
thf(fact_111_mult_Ocomm__neutral,axiom,
! [A: complex] :
( ( times_times_complex @ A @ one_one_complex )
= A ) ).
% mult.comm_neutral
thf(fact_112_mult_Ocomm__neutral,axiom,
! [A: nat] :
( ( times_times_nat @ A @ one_one_nat )
= A ) ).
% mult.comm_neutral
thf(fact_113_mult_Ocomm__neutral,axiom,
! [A: real] :
( ( times_times_real @ A @ one_one_real )
= A ) ).
% mult.comm_neutral
thf(fact_114_comm__monoid__mult__class_Omult__1,axiom,
! [A: complex] :
( ( times_times_complex @ one_one_complex @ A )
= A ) ).
% comm_monoid_mult_class.mult_1
thf(fact_115_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_116_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_117_funpow__simps__right_I1_J,axiom,
! [F: real > real] :
( ( compow_real_real @ zero_zero_nat @ F )
= id_real ) ).
% funpow_simps_right(1)
thf(fact_118_funpow__simps__right_I1_J,axiom,
! [F: ( complex > complex ) > complex > complex] :
( ( compow1098280738omplex @ zero_zero_nat @ F )
= id_complex_complex ) ).
% funpow_simps_right(1)
thf(fact_119_funpow__simps__right_I1_J,axiom,
! [F: ( real > real ) > real > real] :
( ( compow1723822618l_real @ zero_zero_nat @ F )
= id_real_real ) ).
% funpow_simps_right(1)
thf(fact_120_funpow__simps__right_I1_J,axiom,
! [F: complex > complex] :
( ( compow1667379464omplex @ zero_zero_nat @ F )
= id_complex ) ).
% funpow_simps_right(1)
thf(fact_121_fun_Omap__id,axiom,
! [T: complex > complex] :
( ( comp_c130555887omplex @ id_complex @ T )
= T ) ).
% fun.map_id
thf(fact_122_vector__space__over__itself_Oscale__one,axiom,
! [X2: complex] :
( ( times_times_complex @ one_one_complex @ X2 )
= X2 ) ).
% vector_space_over_itself.scale_one
thf(fact_123_vector__space__over__itself_Oscale__one,axiom,
! [X2: real] :
( ( times_times_real @ one_one_real @ X2 )
= X2 ) ).
% vector_space_over_itself.scale_one
thf(fact_124_vector__space__over__itself_Oscale__eq__0__iff,axiom,
! [A: complex,X2: complex] :
( ( ( times_times_complex @ A @ X2 )
= zero_zero_complex )
= ( ( A = zero_zero_complex )
| ( X2 = zero_zero_complex ) ) ) ).
% vector_space_over_itself.scale_eq_0_iff
thf(fact_125_vector__space__over__itself_Oscale__eq__0__iff,axiom,
! [A: real,X2: real] :
( ( ( times_times_real @ A @ X2 )
= zero_zero_real )
= ( ( A = zero_zero_real )
| ( X2 = zero_zero_real ) ) ) ).
% vector_space_over_itself.scale_eq_0_iff
thf(fact_126_vector__space__over__itself_Oscale__zero__left,axiom,
! [X2: complex] :
( ( times_times_complex @ zero_zero_complex @ X2 )
= zero_zero_complex ) ).
% vector_space_over_itself.scale_zero_left
thf(fact_127_vector__space__over__itself_Oscale__zero__left,axiom,
! [X2: real] :
( ( times_times_real @ zero_zero_real @ X2 )
= zero_zero_real ) ).
% vector_space_over_itself.scale_zero_left
thf(fact_128_vector__space__over__itself_Oscale__zero__right,axiom,
! [A: complex] :
( ( times_times_complex @ A @ zero_zero_complex )
= zero_zero_complex ) ).
% vector_space_over_itself.scale_zero_right
thf(fact_129_vector__space__over__itself_Oscale__zero__right,axiom,
! [A: real] :
( ( times_times_real @ A @ zero_zero_real )
= zero_zero_real ) ).
% vector_space_over_itself.scale_zero_right
thf(fact_130_vector__space__over__itself_Oscale__cancel__left,axiom,
! [A: complex,X2: complex,Y: complex] :
( ( ( times_times_complex @ A @ X2 )
= ( times_times_complex @ A @ Y ) )
= ( ( X2 = Y )
| ( A = zero_zero_complex ) ) ) ).
% vector_space_over_itself.scale_cancel_left
thf(fact_131_vector__space__over__itself_Oscale__cancel__left,axiom,
! [A: real,X2: real,Y: real] :
( ( ( times_times_real @ A @ X2 )
= ( times_times_real @ A @ Y ) )
= ( ( X2 = Y )
| ( A = zero_zero_real ) ) ) ).
% vector_space_over_itself.scale_cancel_left
thf(fact_132_vector__space__over__itself_Oscale__cancel__right,axiom,
! [A: complex,X2: complex,B: complex] :
( ( ( times_times_complex @ A @ X2 )
= ( times_times_complex @ B @ X2 ) )
= ( ( A = B )
| ( X2 = zero_zero_complex ) ) ) ).
% vector_space_over_itself.scale_cancel_right
thf(fact_133_vector__space__over__itself_Oscale__cancel__right,axiom,
! [A: real,X2: real,B: real] :
( ( ( times_times_real @ A @ X2 )
= ( times_times_real @ B @ X2 ) )
= ( ( A = B )
| ( X2 = zero_zero_real ) ) ) ).
% vector_space_over_itself.scale_cancel_right
thf(fact_134_mult__if__delta,axiom,
! [P: $o,Q: complex] :
( ( P
=> ( ( times_times_complex @ ( if_complex @ P @ one_one_complex @ zero_zero_complex ) @ Q )
= Q ) )
& ( ~ P
=> ( ( times_times_complex @ ( if_complex @ P @ one_one_complex @ zero_zero_complex ) @ Q )
= zero_zero_complex ) ) ) ).
% mult_if_delta
thf(fact_135_mult__if__delta,axiom,
! [P: $o,Q: nat] :
( ( P
=> ( ( times_times_nat @ ( if_nat @ P @ one_one_nat @ zero_zero_nat ) @ Q )
= Q ) )
& ( ~ P
=> ( ( times_times_nat @ ( if_nat @ P @ one_one_nat @ zero_zero_nat ) @ Q )
= zero_zero_nat ) ) ) ).
% mult_if_delta
thf(fact_136_mult__if__delta,axiom,
! [P: $o,Q: real] :
( ( P
=> ( ( times_times_real @ ( if_real @ P @ one_one_real @ zero_zero_real ) @ Q )
= Q ) )
& ( ~ P
=> ( ( times_times_real @ ( if_real @ P @ one_one_real @ zero_zero_real ) @ Q )
= zero_zero_real ) ) ) ).
% mult_if_delta
thf(fact_137_set__times__intro,axiom,
! [A: complex,C2: set_complex,B: complex,D2: set_complex] :
( ( member_complex @ A @ C2 )
=> ( ( member_complex @ B @ D2 )
=> ( member_complex @ ( times_times_complex @ A @ B ) @ ( times_1316095593omplex @ C2 @ D2 ) ) ) ) ).
% set_times_intro
thf(fact_138_set__times__intro,axiom,
! [A: nat,C2: set_nat,B: nat,D2: set_nat] :
( ( member_nat @ A @ C2 )
=> ( ( member_nat @ B @ D2 )
=> ( member_nat @ ( times_times_nat @ A @ B ) @ ( times_times_set_nat @ C2 @ D2 ) ) ) ) ).
% set_times_intro
thf(fact_139_set__times__intro,axiom,
! [A: real,C2: set_real,B: real,D2: set_real] :
( ( member_real @ A @ C2 )
=> ( ( member_real @ B @ D2 )
=> ( member_real @ ( times_times_real @ A @ B ) @ ( times_times_set_real @ C2 @ D2 ) ) ) ) ).
% set_times_intro
thf(fact_140_funpow__code__def,axiom,
funpow1854104714omplex = compow1098280738omplex ).
% funpow_code_def
thf(fact_141_funpow__code__def,axiom,
funpow_real_real = compow1723822618l_real ).
% funpow_code_def
thf(fact_142_funpow__code__def,axiom,
funpow_complex = compow1667379464omplex ).
% funpow_code_def
thf(fact_143_assms_I4_J,axiom,
topolo935673511omplex @ t ).
% assms(4)
thf(fact_144_assms_I2_J,axiom,
comple372758642hic_on @ g @ t ).
% assms(2)
thf(fact_145_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_146_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_147_mult__0__right,axiom,
! [M: nat] :
( ( times_times_nat @ M @ zero_zero_nat )
= zero_zero_nat ) ).
% mult_0_right
thf(fact_148_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_149_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_150_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_151_higher__deriv__transform__within__open,axiom,
! [F: complex > complex,S: set_complex,G2: complex > complex,Z: complex,I: nat] :
( ( comple372758642hic_on @ F @ S )
=> ( ( comple372758642hic_on @ G2 @ S )
=> ( ( topolo935673511omplex @ S )
=> ( ( member_complex @ Z @ S )
=> ( ! [W: complex] :
( ( member_complex @ W @ S )
=> ( ( F @ W )
= ( G2 @ W ) ) )
=> ( ( compow1098280738omplex @ I @ deriv_complex @ F @ Z )
= ( compow1098280738omplex @ I @ deriv_complex @ G2 @ Z ) ) ) ) ) ) ) ).
% higher_deriv_transform_within_open
thf(fact_152_holomorphic__higher__deriv,axiom,
! [F: complex > complex,S: set_complex,N: nat] :
( ( comple372758642hic_on @ F @ S )
=> ( ( topolo935673511omplex @ S )
=> ( comple372758642hic_on @ ( compow1098280738omplex @ N @ deriv_complex @ F ) @ S ) ) ) ).
% holomorphic_higher_deriv
thf(fact_153_holomorphic__deriv,axiom,
! [F: complex > complex,S: set_complex] :
( ( comple372758642hic_on @ F @ S )
=> ( ( topolo935673511omplex @ S )
=> ( comple372758642hic_on @ ( deriv_complex @ F ) @ S ) ) ) ).
% holomorphic_deriv
thf(fact_154_mult__0,axiom,
! [N: nat] :
( ( times_times_nat @ zero_zero_nat @ N )
= zero_zero_nat ) ).
% mult_0
thf(fact_155_nat__mult__1__right,axiom,
! [N: nat] :
( ( times_times_nat @ N @ one_one_nat )
= N ) ).
% nat_mult_1_right
thf(fact_156_nat__mult__1,axiom,
! [N: nat] :
( ( times_times_nat @ one_one_nat @ N )
= N ) ).
% nat_mult_1
thf(fact_157_funpow__mult,axiom,
! [N: nat,M: nat,F: ( complex > complex ) > complex > complex] :
( ( compow1098280738omplex @ N @ ( compow1098280738omplex @ M @ F ) )
= ( compow1098280738omplex @ ( times_times_nat @ M @ N ) @ F ) ) ).
% funpow_mult
thf(fact_158_funpow__mult,axiom,
! [N: nat,M: nat,F: ( real > real ) > real > real] :
( ( compow1723822618l_real @ N @ ( compow1723822618l_real @ M @ F ) )
= ( compow1723822618l_real @ ( times_times_nat @ M @ N ) @ F ) ) ).
% funpow_mult
thf(fact_159_funpow__mult,axiom,
! [N: nat,M: nat,F: complex > complex] :
( ( compow1667379464omplex @ N @ ( compow1667379464omplex @ M @ F ) )
= ( compow1667379464omplex @ ( times_times_nat @ M @ N ) @ F ) ) ).
% funpow_mult
thf(fact_160_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_161_vector__space__over__itself_Oscale__left__commute,axiom,
! [A: complex,B: complex,X2: complex] :
( ( times_times_complex @ A @ ( times_times_complex @ B @ X2 ) )
= ( times_times_complex @ B @ ( times_times_complex @ A @ X2 ) ) ) ).
% vector_space_over_itself.scale_left_commute
thf(fact_162_vector__space__over__itself_Oscale__left__commute,axiom,
! [A: real,B: real,X2: real] :
( ( times_times_real @ A @ ( times_times_real @ B @ X2 ) )
= ( times_times_real @ B @ ( times_times_real @ A @ X2 ) ) ) ).
% vector_space_over_itself.scale_left_commute
thf(fact_163_vector__space__over__itself_Oscale__scale,axiom,
! [A: complex,B: complex,X2: complex] :
( ( times_times_complex @ A @ ( times_times_complex @ B @ X2 ) )
= ( times_times_complex @ ( times_times_complex @ A @ B ) @ X2 ) ) ).
% vector_space_over_itself.scale_scale
thf(fact_164_vector__space__over__itself_Oscale__scale,axiom,
! [A: real,B: real,X2: real] :
( ( times_times_real @ A @ ( times_times_real @ B @ X2 ) )
= ( times_times_real @ ( times_times_real @ A @ B ) @ X2 ) ) ).
% vector_space_over_itself.scale_scale
thf(fact_165_set__times__elim,axiom,
! [X2: complex,A2: set_complex,B3: set_complex] :
( ( member_complex @ X2 @ ( times_1316095593omplex @ A2 @ B3 ) )
=> ~ ! [A4: complex,B4: complex] :
( ( X2
= ( times_times_complex @ A4 @ B4 ) )
=> ( ( member_complex @ A4 @ A2 )
=> ~ ( member_complex @ B4 @ B3 ) ) ) ) ).
% set_times_elim
thf(fact_166_set__times__elim,axiom,
! [X2: nat,A2: set_nat,B3: set_nat] :
( ( member_nat @ X2 @ ( times_times_set_nat @ A2 @ B3 ) )
=> ~ ! [A4: nat,B4: nat] :
( ( X2
= ( times_times_nat @ A4 @ B4 ) )
=> ( ( member_nat @ A4 @ A2 )
=> ~ ( member_nat @ B4 @ B3 ) ) ) ) ).
% set_times_elim
thf(fact_167_set__times__elim,axiom,
! [X2: real,A2: set_real,B3: set_real] :
( ( member_real @ X2 @ ( times_times_set_real @ A2 @ B3 ) )
=> ~ ! [A4: real,B4: real] :
( ( X2
= ( times_times_real @ A4 @ B4 ) )
=> ( ( member_real @ A4 @ A2 )
=> ~ ( member_real @ B4 @ B3 ) ) ) ) ).
% set_times_elim
thf(fact_168_fun_Omap__comp,axiom,
! [G2: complex > complex,F: complex > complex,V: complex > complex] :
( ( comp_c130555887omplex @ G2 @ ( comp_c130555887omplex @ F @ V ) )
= ( comp_c130555887omplex @ ( comp_c130555887omplex @ G2 @ F ) @ V ) ) ).
% fun.map_comp
thf(fact_169_vector__space__over__itself_Oscale__right__imp__eq,axiom,
! [X2: complex,A: complex,B: complex] :
( ( X2 != zero_zero_complex )
=> ( ( ( times_times_complex @ A @ X2 )
= ( times_times_complex @ B @ X2 ) )
=> ( A = B ) ) ) ).
% vector_space_over_itself.scale_right_imp_eq
thf(fact_170_vector__space__over__itself_Oscale__right__imp__eq,axiom,
! [X2: real,A: real,B: real] :
( ( X2 != zero_zero_real )
=> ( ( ( times_times_real @ A @ X2 )
= ( times_times_real @ B @ X2 ) )
=> ( A = B ) ) ) ).
% vector_space_over_itself.scale_right_imp_eq
thf(fact_171_vector__space__over__itself_Oscale__left__imp__eq,axiom,
! [A: complex,X2: complex,Y: complex] :
( ( A != zero_zero_complex )
=> ( ( ( times_times_complex @ A @ X2 )
= ( times_times_complex @ A @ Y ) )
=> ( X2 = Y ) ) ) ).
% vector_space_over_itself.scale_left_imp_eq
thf(fact_172_vector__space__over__itself_Oscale__left__imp__eq,axiom,
! [A: real,X2: real,Y: real] :
( ( A != zero_zero_real )
=> ( ( ( times_times_real @ A @ X2 )
= ( times_times_real @ A @ Y ) )
=> ( X2 = Y ) ) ) ).
% vector_space_over_itself.scale_left_imp_eq
thf(fact_173_fun_Omap__id0,axiom,
( ( comp_c130555887omplex @ id_complex )
= id_complex_complex ) ).
% fun.map_id0
thf(fact_174_complex__derivative__transform__within__open,axiom,
! [F: complex > complex,S2: set_complex,G2: complex > complex,Z: complex] :
( ( comple372758642hic_on @ F @ S2 )
=> ( ( comple372758642hic_on @ G2 @ S2 )
=> ( ( topolo935673511omplex @ S2 )
=> ( ( member_complex @ Z @ S2 )
=> ( ! [W: complex] :
( ( member_complex @ W @ S2 )
=> ( ( F @ W )
= ( G2 @ W ) ) )
=> ( ( deriv_complex @ F @ Z )
= ( deriv_complex @ G2 @ Z ) ) ) ) ) ) ) ).
% complex_derivative_transform_within_open
thf(fact_175_holomorphic__on__id,axiom,
! [S2: set_complex] : ( comple372758642hic_on @ id_complex @ S2 ) ).
% holomorphic_on_id
thf(fact_176_holomorphic__on__linear,axiom,
! [C: complex,S2: set_complex] : ( comple372758642hic_on @ ( times_times_complex @ C ) @ S2 ) ).
% holomorphic_on_linear
thf(fact_177_pointfree__idE,axiom,
! [F: complex > complex,G2: complex > complex,X2: complex] :
( ( ( comp_c130555887omplex @ F @ G2 )
= id_complex )
=> ( ( F @ ( G2 @ X2 ) )
= X2 ) ) ).
% pointfree_idE
thf(fact_178_isomorphism__expand,axiom,
! [F: real > complex,G2: complex > real] :
( ( ( ( comp_r667767405omplex @ F @ G2 )
= id_complex )
& ( ( comp_c819638635l_real @ G2 @ F )
= id_real ) )
= ( ! [X: complex] :
( ( F @ ( G2 @ X ) )
= X )
& ! [X: real] :
( ( G2 @ ( F @ X ) )
= X ) ) ) ).
% isomorphism_expand
thf(fact_179_isomorphism__expand,axiom,
! [F: ( complex > complex ) > complex,G2: complex > complex > complex] :
( ( ( ( comp_c881053372omplex @ F @ G2 )
= id_complex )
& ( ( comp_c606622857omplex @ G2 @ F )
= id_complex_complex ) )
= ( ! [X: complex] :
( ( F @ ( G2 @ X ) )
= X )
& ! [X: complex > complex] :
( ( G2 @ ( F @ X ) )
= X ) ) ) ).
% isomorphism_expand
thf(fact_180_isomorphism__expand,axiom,
! [F: complex > real,G2: real > complex] :
( ( ( ( comp_c819638635l_real @ F @ G2 )
= id_real )
& ( ( comp_r667767405omplex @ G2 @ F )
= id_complex ) )
= ( ! [X: real] :
( ( F @ ( G2 @ X ) )
= X )
& ! [X: complex] :
( ( G2 @ ( F @ X ) )
= X ) ) ) ).
% isomorphism_expand
thf(fact_181_isomorphism__expand,axiom,
! [F: real > real,G2: real > real] :
( ( ( ( comp_real_real_real @ F @ G2 )
= id_real )
& ( ( comp_real_real_real @ G2 @ F )
= id_real ) )
= ( ! [X: real] :
( ( F @ ( G2 @ X ) )
= X )
& ! [X: real] :
( ( G2 @ ( F @ X ) )
= X ) ) ) ).
% isomorphism_expand
thf(fact_182_isomorphism__expand,axiom,
! [F: ( complex > complex ) > real,G2: real > complex > complex] :
( ( ( ( comp_c1884694328l_real @ F @ G2 )
= id_real )
& ( ( comp_r2009261319omplex @ G2 @ F )
= id_complex_complex ) )
= ( ! [X: real] :
( ( F @ ( G2 @ X ) )
= X )
& ! [X: complex > complex] :
( ( G2 @ ( F @ X ) )
= X ) ) ) ).
% isomorphism_expand
thf(fact_183_isomorphism__expand,axiom,
! [F: complex > complex > complex,G2: ( complex > complex ) > complex] :
( ( ( ( comp_c606622857omplex @ F @ G2 )
= id_complex_complex )
& ( ( comp_c881053372omplex @ G2 @ F )
= id_complex ) )
= ( ! [X: complex > complex] :
( ( F @ ( G2 @ X ) )
= X )
& ! [X: complex] :
( ( G2 @ ( F @ X ) )
= X ) ) ) ).
% isomorphism_expand
thf(fact_184_isomorphism__expand,axiom,
! [F: real > complex > complex,G2: ( complex > complex ) > real] :
( ( ( ( comp_r2009261319omplex @ F @ G2 )
= id_complex_complex )
& ( ( comp_c1884694328l_real @ G2 @ F )
= id_real ) )
= ( ! [X: complex > complex] :
( ( F @ ( G2 @ X ) )
= X )
& ! [X: real] :
( ( G2 @ ( F @ X ) )
= X ) ) ) ).
% isomorphism_expand
thf(fact_185_isomorphism__expand,axiom,
! [F: ( complex > complex ) > complex > complex,G2: ( complex > complex ) > complex > complex] :
( ( ( ( comp_c1610621014omplex @ F @ G2 )
= id_complex_complex )
& ( ( comp_c1610621014omplex @ G2 @ F )
= id_complex_complex ) )
= ( ! [X: complex > complex] :
( ( F @ ( G2 @ X ) )
= X )
& ! [X: complex > complex] :
( ( G2 @ ( F @ X ) )
= X ) ) ) ).
% isomorphism_expand
thf(fact_186_isomorphism__expand,axiom,
! [F: complex > complex,G2: complex > complex] :
( ( ( ( comp_c130555887omplex @ F @ G2 )
= id_complex )
& ( ( comp_c130555887omplex @ G2 @ F )
= id_complex ) )
= ( ! [X: complex] :
( ( F @ ( G2 @ X ) )
= X )
& ! [X: complex] :
( ( G2 @ ( F @ X ) )
= X ) ) ) ).
% isomorphism_expand
thf(fact_187_left__right__inverse__eq,axiom,
! [F: real > complex,G2: complex > real,H: real > complex] :
( ( ( comp_r667767405omplex @ F @ G2 )
= id_complex )
=> ( ( ( comp_c819638635l_real @ G2 @ H )
= id_real )
=> ( F = H ) ) ) ).
% left_right_inverse_eq
thf(fact_188_left__right__inverse__eq,axiom,
! [F: ( complex > complex ) > complex,G2: complex > complex > complex,H: ( complex > complex ) > complex] :
( ( ( comp_c881053372omplex @ F @ G2 )
= id_complex )
=> ( ( ( comp_c606622857omplex @ G2 @ H )
= id_complex_complex )
=> ( F = H ) ) ) ).
% left_right_inverse_eq
thf(fact_189_left__right__inverse__eq,axiom,
! [F: complex > real,G2: real > complex,H: complex > real] :
( ( ( comp_c819638635l_real @ F @ G2 )
= id_real )
=> ( ( ( comp_r667767405omplex @ G2 @ H )
= id_complex )
=> ( F = H ) ) ) ).
% left_right_inverse_eq
thf(fact_190_left__right__inverse__eq,axiom,
! [F: real > real,G2: real > real,H: real > real] :
( ( ( comp_real_real_real @ F @ G2 )
= id_real )
=> ( ( ( comp_real_real_real @ G2 @ H )
= id_real )
=> ( F = H ) ) ) ).
% left_right_inverse_eq
thf(fact_191_left__right__inverse__eq,axiom,
! [F: ( complex > complex ) > real,G2: real > complex > complex,H: ( complex > complex ) > real] :
( ( ( comp_c1884694328l_real @ F @ G2 )
= id_real )
=> ( ( ( comp_r2009261319omplex @ G2 @ H )
= id_complex_complex )
=> ( F = H ) ) ) ).
% left_right_inverse_eq
thf(fact_192_left__right__inverse__eq,axiom,
! [F: complex > complex > complex,G2: ( complex > complex ) > complex,H: complex > complex > complex] :
( ( ( comp_c606622857omplex @ F @ G2 )
= id_complex_complex )
=> ( ( ( comp_c881053372omplex @ G2 @ H )
= id_complex )
=> ( F = H ) ) ) ).
% left_right_inverse_eq
thf(fact_193_left__right__inverse__eq,axiom,
! [F: real > complex > complex,G2: ( complex > complex ) > real,H: real > complex > complex] :
( ( ( comp_r2009261319omplex @ F @ G2 )
= id_complex_complex )
=> ( ( ( comp_c1884694328l_real @ G2 @ H )
= id_real )
=> ( F = H ) ) ) ).
% left_right_inverse_eq
thf(fact_194_left__right__inverse__eq,axiom,
! [F: ( complex > complex ) > complex > complex,G2: ( complex > complex ) > complex > complex,H: ( complex > complex ) > complex > complex] :
( ( ( comp_c1610621014omplex @ F @ G2 )
= id_complex_complex )
=> ( ( ( comp_c1610621014omplex @ G2 @ H )
= id_complex_complex )
=> ( F = H ) ) ) ).
% left_right_inverse_eq
thf(fact_195_left__right__inverse__eq,axiom,
! [F: complex > complex,G2: complex > complex,H: complex > complex] :
( ( ( comp_c130555887omplex @ F @ G2 )
= id_complex )
=> ( ( ( comp_c130555887omplex @ G2 @ H )
= id_complex )
=> ( F = H ) ) ) ).
% left_right_inverse_eq
thf(fact_196_rewriteL__comp__comp,axiom,
! [F: complex > complex,G2: complex > complex,L: complex > complex,H: complex > complex] :
( ( ( comp_c130555887omplex @ F @ G2 )
= L )
=> ( ( comp_c130555887omplex @ F @ ( comp_c130555887omplex @ G2 @ H ) )
= ( comp_c130555887omplex @ L @ H ) ) ) ).
% rewriteL_comp_comp
thf(fact_197_rewriteR__comp__comp,axiom,
! [G2: complex > complex,H: complex > complex,R: complex > complex,F: complex > complex] :
( ( ( comp_c130555887omplex @ G2 @ H )
= R )
=> ( ( comp_c130555887omplex @ ( comp_c130555887omplex @ F @ G2 ) @ H )
= ( comp_c130555887omplex @ F @ R ) ) ) ).
% rewriteR_comp_comp
thf(fact_198_rewriteL__comp__comp2,axiom,
! [F: complex > complex,G2: complex > complex,L1: complex > complex,L2: complex > complex,H: complex > complex,R: complex > complex] :
( ( ( comp_c130555887omplex @ F @ G2 )
= ( comp_c130555887omplex @ L1 @ L2 ) )
=> ( ( ( comp_c130555887omplex @ L2 @ H )
= R )
=> ( ( comp_c130555887omplex @ F @ ( comp_c130555887omplex @ G2 @ H ) )
= ( comp_c130555887omplex @ L1 @ R ) ) ) ) ).
% rewriteL_comp_comp2
thf(fact_199_rewriteR__comp__comp2,axiom,
! [G2: complex > complex,H: complex > complex,R1: complex > complex,R2: complex > complex,F: complex > complex,L: complex > complex] :
( ( ( comp_c130555887omplex @ G2 @ H )
= ( comp_c130555887omplex @ R1 @ R2 ) )
=> ( ( ( comp_c130555887omplex @ F @ R1 )
= L )
=> ( ( comp_c130555887omplex @ ( comp_c130555887omplex @ F @ G2 ) @ H )
= ( comp_c130555887omplex @ L @ R2 ) ) ) ) ).
% rewriteR_comp_comp2
thf(fact_200_holomorphic__cong,axiom,
! [S2: set_complex,T: set_complex,F: complex > complex,G2: complex > complex] :
( ( S2 = T )
=> ( ! [X3: complex] :
( ( member_complex @ X3 @ S2 )
=> ( ( F @ X3 )
= ( G2 @ X3 ) ) )
=> ( ( comple372758642hic_on @ F @ S2 )
= ( comple372758642hic_on @ G2 @ T ) ) ) ) ).
% holomorphic_cong
thf(fact_201_holomorphic__transform,axiom,
! [F: complex > complex,S2: set_complex,G2: complex > complex] :
( ( comple372758642hic_on @ F @ S2 )
=> ( ! [X3: complex] :
( ( member_complex @ X3 @ S2 )
=> ( ( F @ X3 )
= ( G2 @ X3 ) ) )
=> ( comple372758642hic_on @ G2 @ S2 ) ) ) ).
% holomorphic_transform
thf(fact_202_assms_I5_J,axiom,
ord_le701908932omplex @ ( image_58037603omplex @ f @ s ) @ t ).
% assms(5)
thf(fact_203_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_204_mult__delta__left,axiom,
! [B: $o,X2: complex,Y: complex] :
( ( B
=> ( ( times_times_complex @ ( if_complex @ B @ X2 @ zero_zero_complex ) @ Y )
= ( times_times_complex @ X2 @ Y ) ) )
& ( ~ B
=> ( ( times_times_complex @ ( if_complex @ B @ X2 @ zero_zero_complex ) @ Y )
= zero_zero_complex ) ) ) ).
% mult_delta_left
thf(fact_205_mult__delta__left,axiom,
! [B: $o,X2: nat,Y: nat] :
( ( B
=> ( ( times_times_nat @ ( if_nat @ B @ X2 @ zero_zero_nat ) @ Y )
= ( times_times_nat @ X2 @ Y ) ) )
& ( ~ B
=> ( ( times_times_nat @ ( if_nat @ B @ X2 @ zero_zero_nat ) @ Y )
= zero_zero_nat ) ) ) ).
% mult_delta_left
thf(fact_206_mult__delta__left,axiom,
! [B: $o,X2: real,Y: real] :
( ( B
=> ( ( times_times_real @ ( if_real @ B @ X2 @ zero_zero_real ) @ Y )
= ( times_times_real @ X2 @ Y ) ) )
& ( ~ B
=> ( ( times_times_real @ ( if_real @ B @ X2 @ zero_zero_real ) @ Y )
= zero_zero_real ) ) ) ).
% mult_delta_left
thf(fact_207_mult__delta__right,axiom,
! [B: $o,X2: complex,Y: complex] :
( ( B
=> ( ( times_times_complex @ X2 @ ( if_complex @ B @ Y @ zero_zero_complex ) )
= ( times_times_complex @ X2 @ Y ) ) )
& ( ~ B
=> ( ( times_times_complex @ X2 @ ( if_complex @ B @ Y @ zero_zero_complex ) )
= zero_zero_complex ) ) ) ).
% mult_delta_right
thf(fact_208_mult__delta__right,axiom,
! [B: $o,X2: nat,Y: nat] :
( ( B
=> ( ( times_times_nat @ X2 @ ( if_nat @ B @ Y @ zero_zero_nat ) )
= ( times_times_nat @ X2 @ Y ) ) )
& ( ~ B
=> ( ( times_times_nat @ X2 @ ( if_nat @ B @ Y @ zero_zero_nat ) )
= zero_zero_nat ) ) ) ).
% mult_delta_right
thf(fact_209_mult__delta__right,axiom,
! [B: $o,X2: real,Y: real] :
( ( B
=> ( ( times_times_real @ X2 @ ( if_real @ B @ Y @ zero_zero_real ) )
= ( times_times_real @ X2 @ Y ) ) )
& ( ~ B
=> ( ( times_times_real @ X2 @ ( if_real @ B @ Y @ zero_zero_real ) )
= zero_zero_real ) ) ) ).
% mult_delta_right
thf(fact_210_arcosh__1,axiom,
( ( arcosh_real @ one_one_real )
= zero_zero_real ) ).
% arcosh_1
thf(fact_211_arcosh__1,axiom,
( ( arcosh_complex @ one_one_complex )
= zero_zero_complex ) ).
% arcosh_1
thf(fact_212_le__zero__eq,axiom,
! [N: nat] :
( ( ord_less_eq_nat @ N @ zero_zero_nat )
= ( N = zero_zero_nat ) ) ).
% le_zero_eq
thf(fact_213_image__id,axiom,
( ( image_real_real @ id_real )
= id_set_real ) ).
% image_id
thf(fact_214_image__id,axiom,
( ( image_944012797omplex @ id_complex_complex )
= id_set1618538368omplex ) ).
% image_id
thf(fact_215_image__id,axiom,
( ( image_58037603omplex @ id_complex )
= id_set_complex ) ).
% image_id
thf(fact_216_set__times__mono2,axiom,
! [C2: set_complex,D2: set_complex,E: set_complex,F3: set_complex] :
( ( ord_le701908932omplex @ C2 @ D2 )
=> ( ( ord_le701908932omplex @ E @ F3 )
=> ( ord_le701908932omplex @ ( times_1316095593omplex @ C2 @ E ) @ ( times_1316095593omplex @ D2 @ F3 ) ) ) ) ).
% set_times_mono2
thf(fact_217_holomorphic__on__compose__gen,axiom,
! [F: complex > complex,S2: set_complex,G2: complex > complex,T: set_complex] :
( ( comple372758642hic_on @ F @ S2 )
=> ( ( comple372758642hic_on @ G2 @ T )
=> ( ( ord_le701908932omplex @ ( image_58037603omplex @ F @ S2 ) @ T )
=> ( comple372758642hic_on @ ( comp_c130555887omplex @ G2 @ F ) @ S2 ) ) ) ) ).
% holomorphic_on_compose_gen
thf(fact_218_image__comp,axiom,
! [F: complex > complex,G2: complex > complex,R: set_complex] :
( ( image_58037603omplex @ F @ ( image_58037603omplex @ G2 @ R ) )
= ( image_58037603omplex @ ( comp_c130555887omplex @ F @ G2 ) @ R ) ) ).
% image_comp
thf(fact_219_image__eq__imp__comp,axiom,
! [F: complex > complex,A2: set_complex,G2: complex > complex,B3: set_complex,H: complex > complex] :
( ( ( image_58037603omplex @ F @ A2 )
= ( image_58037603omplex @ G2 @ B3 ) )
=> ( ( image_58037603omplex @ ( comp_c130555887omplex @ H @ F ) @ A2 )
= ( image_58037603omplex @ ( comp_c130555887omplex @ H @ G2 ) @ B3 ) ) ) ).
% image_eq_imp_comp
thf(fact_220_zero__le,axiom,
! [X2: nat] : ( ord_less_eq_nat @ zero_zero_nat @ X2 ) ).
% zero_le
thf(fact_221_holomorphic__on__subset,axiom,
! [F: complex > complex,S2: set_complex,T: set_complex] :
( ( comple372758642hic_on @ F @ S2 )
=> ( ( ord_le701908932omplex @ T @ S2 )
=> ( comple372758642hic_on @ F @ T ) ) ) ).
% holomorphic_on_subset
thf(fact_222_set__times__mono2__b,axiom,
! [C2: set_complex,D2: set_complex,E: set_complex,F3: set_complex,X2: complex] :
( ( ord_le701908932omplex @ C2 @ D2 )
=> ( ( ord_le701908932omplex @ E @ F3 )
=> ( ( member_complex @ X2 @ ( times_1316095593omplex @ C2 @ E ) )
=> ( member_complex @ X2 @ ( times_1316095593omplex @ D2 @ F3 ) ) ) ) ) ).
% set_times_mono2_b
thf(fact_223_ordered__comm__semiring__class_Ocomm__mult__left__mono,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ord_less_eq_nat @ ( times_times_nat @ C @ A ) @ ( times_times_nat @ C @ B ) ) ) ) ).
% ordered_comm_semiring_class.comm_mult_left_mono
thf(fact_224_ordered__comm__semiring__class_Ocomm__mult__left__mono,axiom,
! [A: real,B: real,C: real] :
( ( ord_less_eq_real @ A @ B )
=> ( ( ord_less_eq_real @ zero_zero_real @ C )
=> ( ord_less_eq_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B ) ) ) ) ).
% ordered_comm_semiring_class.comm_mult_left_mono
thf(fact_225_zero__le__mult__iff,axiom,
! [A: real,B: real] :
( ( ord_less_eq_real @ zero_zero_real @ ( times_times_real @ A @ B ) )
= ( ( ( ord_less_eq_real @ zero_zero_real @ A )
& ( ord_less_eq_real @ zero_zero_real @ B ) )
| ( ( ord_less_eq_real @ A @ zero_zero_real )
& ( ord_less_eq_real @ B @ zero_zero_real ) ) ) ) ).
% zero_le_mult_iff
thf(fact_226_mult__nonneg__nonpos2,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ B @ zero_zero_nat )
=> ( ord_less_eq_nat @ ( times_times_nat @ B @ A ) @ zero_zero_nat ) ) ) ).
% mult_nonneg_nonpos2
thf(fact_227_mult__nonneg__nonpos2,axiom,
! [A: real,B: real] :
( ( ord_less_eq_real @ zero_zero_real @ A )
=> ( ( ord_less_eq_real @ B @ zero_zero_real )
=> ( ord_less_eq_real @ ( times_times_real @ B @ A ) @ zero_zero_real ) ) ) ).
% mult_nonneg_nonpos2
thf(fact_228_mult__nonpos__nonneg,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ zero_zero_nat )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B )
=> ( ord_less_eq_nat @ ( times_times_nat @ A @ B ) @ zero_zero_nat ) ) ) ).
% mult_nonpos_nonneg
thf(fact_229_mult__nonpos__nonneg,axiom,
! [A: real,B: real] :
( ( ord_less_eq_real @ A @ zero_zero_real )
=> ( ( ord_less_eq_real @ zero_zero_real @ B )
=> ( ord_less_eq_real @ ( times_times_real @ A @ B ) @ zero_zero_real ) ) ) ).
% mult_nonpos_nonneg
thf(fact_230_mult__nonneg__nonpos,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ B @ zero_zero_nat )
=> ( ord_less_eq_nat @ ( times_times_nat @ A @ B ) @ zero_zero_nat ) ) ) ).
% mult_nonneg_nonpos
thf(fact_231_mult__nonneg__nonpos,axiom,
! [A: real,B: real] :
( ( ord_less_eq_real @ zero_zero_real @ A )
=> ( ( ord_less_eq_real @ B @ zero_zero_real )
=> ( ord_less_eq_real @ ( times_times_real @ A @ B ) @ zero_zero_real ) ) ) ).
% mult_nonneg_nonpos
thf(fact_232_mult__nonneg__nonneg,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( times_times_nat @ A @ B ) ) ) ) ).
% mult_nonneg_nonneg
thf(fact_233_mult__nonneg__nonneg,axiom,
! [A: real,B: real] :
( ( ord_less_eq_real @ zero_zero_real @ A )
=> ( ( ord_less_eq_real @ zero_zero_real @ B )
=> ( ord_less_eq_real @ zero_zero_real @ ( times_times_real @ A @ B ) ) ) ) ).
% mult_nonneg_nonneg
thf(fact_234_split__mult__neg__le,axiom,
! [A: nat,B: nat] :
( ( ( ( ord_less_eq_nat @ zero_zero_nat @ A )
& ( ord_less_eq_nat @ B @ zero_zero_nat ) )
| ( ( ord_less_eq_nat @ A @ zero_zero_nat )
& ( ord_less_eq_nat @ zero_zero_nat @ B ) ) )
=> ( ord_less_eq_nat @ ( times_times_nat @ A @ B ) @ zero_zero_nat ) ) ).
% split_mult_neg_le
thf(fact_235_split__mult__neg__le,axiom,
! [A: real,B: real] :
( ( ( ( ord_less_eq_real @ zero_zero_real @ A )
& ( ord_less_eq_real @ B @ zero_zero_real ) )
| ( ( ord_less_eq_real @ A @ zero_zero_real )
& ( ord_less_eq_real @ zero_zero_real @ B ) ) )
=> ( ord_less_eq_real @ ( times_times_real @ A @ B ) @ zero_zero_real ) ) ).
% split_mult_neg_le
thf(fact_236_mult__le__0__iff,axiom,
! [A: real,B: real] :
( ( ord_less_eq_real @ ( times_times_real @ A @ B ) @ zero_zero_real )
= ( ( ( ord_less_eq_real @ zero_zero_real @ A )
& ( ord_less_eq_real @ B @ zero_zero_real ) )
| ( ( ord_less_eq_real @ A @ zero_zero_real )
& ( ord_less_eq_real @ zero_zero_real @ B ) ) ) ) ).
% mult_le_0_iff
thf(fact_237_mult__right__mono,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ord_less_eq_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ C ) ) ) ) ).
% mult_right_mono
thf(fact_238_mult__right__mono,axiom,
! [A: real,B: real,C: real] :
( ( ord_less_eq_real @ A @ B )
=> ( ( ord_less_eq_real @ zero_zero_real @ C )
=> ( ord_less_eq_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ C ) ) ) ) ).
% mult_right_mono
thf(fact_239_mult__right__mono__neg,axiom,
! [B: real,A: real,C: real] :
( ( ord_less_eq_real @ B @ A )
=> ( ( ord_less_eq_real @ C @ zero_zero_real )
=> ( ord_less_eq_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ C ) ) ) ) ).
% mult_right_mono_neg
thf(fact_240_mult__left__mono,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ord_less_eq_nat @ ( times_times_nat @ C @ A ) @ ( times_times_nat @ C @ B ) ) ) ) ).
% mult_left_mono
thf(fact_241_mult__left__mono,axiom,
! [A: real,B: real,C: real] :
( ( ord_less_eq_real @ A @ B )
=> ( ( ord_less_eq_real @ zero_zero_real @ C )
=> ( ord_less_eq_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B ) ) ) ) ).
% mult_left_mono
thf(fact_242_mult__nonpos__nonpos,axiom,
! [A: real,B: real] :
( ( ord_less_eq_real @ A @ zero_zero_real )
=> ( ( ord_less_eq_real @ B @ zero_zero_real )
=> ( ord_less_eq_real @ zero_zero_real @ ( times_times_real @ A @ B ) ) ) ) ).
% mult_nonpos_nonpos
thf(fact_243_mult__left__mono__neg,axiom,
! [B: real,A: real,C: real] :
( ( ord_less_eq_real @ B @ A )
=> ( ( ord_less_eq_real @ C @ zero_zero_real )
=> ( ord_less_eq_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B ) ) ) ) ).
% mult_left_mono_neg
thf(fact_244_split__mult__pos__le,axiom,
! [A: real,B: real] :
( ( ( ( ord_less_eq_real @ zero_zero_real @ A )
& ( ord_less_eq_real @ zero_zero_real @ B ) )
| ( ( ord_less_eq_real @ A @ zero_zero_real )
& ( ord_less_eq_real @ B @ zero_zero_real ) ) )
=> ( ord_less_eq_real @ zero_zero_real @ ( times_times_real @ A @ B ) ) ) ).
% split_mult_pos_le
thf(fact_245_zero__le__square,axiom,
! [A: real] : ( ord_less_eq_real @ zero_zero_real @ ( times_times_real @ A @ A ) ) ).
% zero_le_square
thf(fact_246_mult__mono_H,axiom,
! [A: nat,B: nat,C: nat,D: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ C @ D )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ord_less_eq_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ D ) ) ) ) ) ) ).
% mult_mono'
thf(fact_247_mult__mono_H,axiom,
! [A: real,B: real,C: real,D: real] :
( ( ord_less_eq_real @ A @ B )
=> ( ( ord_less_eq_real @ C @ D )
=> ( ( ord_less_eq_real @ zero_zero_real @ A )
=> ( ( ord_less_eq_real @ zero_zero_real @ C )
=> ( ord_less_eq_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ D ) ) ) ) ) ) ).
% mult_mono'
thf(fact_248_mult__mono,axiom,
! [A: nat,B: nat,C: nat,D: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ C @ D )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ord_less_eq_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ D ) ) ) ) ) ) ).
% mult_mono
thf(fact_249_mult__mono,axiom,
! [A: real,B: real,C: real,D: real] :
( ( ord_less_eq_real @ A @ B )
=> ( ( ord_less_eq_real @ C @ D )
=> ( ( ord_less_eq_real @ zero_zero_real @ B )
=> ( ( ord_less_eq_real @ zero_zero_real @ C )
=> ( ord_less_eq_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ D ) ) ) ) ) ) ).
% mult_mono
thf(fact_250_zero__le__one,axiom,
ord_less_eq_nat @ zero_zero_nat @ one_one_nat ).
% zero_le_one
thf(fact_251_zero__le__one,axiom,
ord_less_eq_real @ zero_zero_real @ one_one_real ).
% zero_le_one
thf(fact_252_not__one__le__zero,axiom,
~ ( ord_less_eq_nat @ one_one_nat @ zero_zero_nat ) ).
% not_one_le_zero
thf(fact_253_not__one__le__zero,axiom,
~ ( ord_less_eq_real @ one_one_real @ zero_zero_real ) ).
% not_one_le_zero
thf(fact_254_holomorphic__on__compose,axiom,
! [F: complex > complex,S2: set_complex,G2: complex > complex] :
( ( comple372758642hic_on @ F @ S2 )
=> ( ( comple372758642hic_on @ G2 @ ( image_58037603omplex @ F @ S2 ) )
=> ( comple372758642hic_on @ ( comp_c130555887omplex @ G2 @ F ) @ S2 ) ) ) ).
% holomorphic_on_compose
thf(fact_255_deriv__left__inverse,axiom,
! [F: complex > complex,S: set_complex,G2: complex > complex,T2: set_complex,W2: complex] :
( ( comple372758642hic_on @ F @ S )
=> ( ( comple372758642hic_on @ G2 @ T2 )
=> ( ( topolo935673511omplex @ S )
=> ( ( topolo935673511omplex @ T2 )
=> ( ( ord_le701908932omplex @ ( image_58037603omplex @ F @ S ) @ T2 )
=> ( ! [Z3: complex] :
( ( member_complex @ Z3 @ S )
=> ( ( G2 @ ( F @ Z3 ) )
= Z3 ) )
=> ( ( member_complex @ W2 @ S )
=> ( ( times_times_complex @ ( deriv_complex @ F @ W2 ) @ ( deriv_complex @ G2 @ ( F @ W2 ) ) )
= one_one_complex ) ) ) ) ) ) ) ) ).
% deriv_left_inverse
thf(fact_256_mult__left__le,axiom,
! [C: nat,A: nat] :
( ( ord_less_eq_nat @ C @ one_one_nat )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ord_less_eq_nat @ ( times_times_nat @ A @ C ) @ A ) ) ) ).
% mult_left_le
thf(fact_257_mult__left__le,axiom,
! [C: real,A: real] :
( ( ord_less_eq_real @ C @ one_one_real )
=> ( ( ord_less_eq_real @ zero_zero_real @ A )
=> ( ord_less_eq_real @ ( times_times_real @ A @ C ) @ A ) ) ) ).
% mult_left_le
thf(fact_258_mult__le__one,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ one_one_nat )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B )
=> ( ( ord_less_eq_nat @ B @ one_one_nat )
=> ( ord_less_eq_nat @ ( times_times_nat @ A @ B ) @ one_one_nat ) ) ) ) ).
% mult_le_one
thf(fact_259_mult__le__one,axiom,
! [A: real,B: real] :
( ( ord_less_eq_real @ A @ one_one_real )
=> ( ( ord_less_eq_real @ zero_zero_real @ B )
=> ( ( ord_less_eq_real @ B @ one_one_real )
=> ( ord_less_eq_real @ ( times_times_real @ A @ B ) @ one_one_real ) ) ) ) ).
% mult_le_one
thf(fact_260_mult__right__le__one__le,axiom,
! [X2: real,Y: real] :
( ( ord_less_eq_real @ zero_zero_real @ X2 )
=> ( ( ord_less_eq_real @ zero_zero_real @ Y )
=> ( ( ord_less_eq_real @ Y @ one_one_real )
=> ( ord_less_eq_real @ ( times_times_real @ X2 @ Y ) @ X2 ) ) ) ) ).
% mult_right_le_one_le
thf(fact_261_mult__left__le__one__le,axiom,
! [X2: real,Y: real] :
( ( ord_less_eq_real @ zero_zero_real @ X2 )
=> ( ( ord_less_eq_real @ zero_zero_real @ Y )
=> ( ( ord_less_eq_real @ Y @ one_one_real )
=> ( ord_less_eq_real @ ( times_times_real @ Y @ X2 ) @ X2 ) ) ) ) ).
% mult_left_le_one_le
thf(fact_262_mult__eq__1,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ A @ one_one_nat )
=> ( ( ord_less_eq_nat @ B @ one_one_nat )
=> ( ( ( times_times_nat @ A @ B )
= one_one_nat )
= ( ( A = one_one_nat )
& ( B = one_one_nat ) ) ) ) ) ) ).
% mult_eq_1
thf(fact_263_mult__eq__1,axiom,
! [A: real,B: real] :
( ( ord_less_eq_real @ zero_zero_real @ A )
=> ( ( ord_less_eq_real @ A @ one_one_real )
=> ( ( ord_less_eq_real @ B @ one_one_real )
=> ( ( ( times_times_real @ A @ B )
= one_one_real )
= ( ( A = one_one_real )
& ( B = one_one_real ) ) ) ) ) ) ).
% mult_eq_1
thf(fact_264_artanh__0,axiom,
( ( artanh_real @ zero_zero_real )
= zero_zero_real ) ).
% artanh_0
thf(fact_265_artanh__0,axiom,
( ( artanh_complex @ zero_zero_complex )
= zero_zero_complex ) ).
% artanh_0
thf(fact_266_arsinh__0,axiom,
( ( arsinh_real @ zero_zero_real )
= zero_zero_real ) ).
% arsinh_0
thf(fact_267_arsinh__0,axiom,
( ( arsinh_complex @ zero_zero_complex )
= zero_zero_complex ) ).
% arsinh_0
thf(fact_268_subset__antisym,axiom,
! [A2: set_complex,B3: set_complex] :
( ( ord_le701908932omplex @ A2 @ B3 )
=> ( ( ord_le701908932omplex @ B3 @ A2 )
=> ( A2 = B3 ) ) ) ).
% subset_antisym
thf(fact_269_image__eqI,axiom,
! [B: complex,F: complex > complex,X2: complex,A2: set_complex] :
( ( B
= ( F @ X2 ) )
=> ( ( member_complex @ X2 @ A2 )
=> ( member_complex @ B @ ( image_58037603omplex @ F @ A2 ) ) ) ) ).
% image_eqI
thf(fact_270_subsetI,axiom,
! [A2: set_complex,B3: set_complex] :
( ! [X3: complex] :
( ( member_complex @ X3 @ A2 )
=> ( member_complex @ X3 @ B3 ) )
=> ( ord_le701908932omplex @ A2 @ B3 ) ) ).
% subsetI
thf(fact_271_le0,axiom,
! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N ) ).
% le0
thf(fact_272_bot__nat__0_Oextremum,axiom,
! [A: nat] : ( ord_less_eq_nat @ zero_zero_nat @ A ) ).
% bot_nat_0.extremum
thf(fact_273_less__eq__nat_Osimps_I1_J,axiom,
! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N ) ).
% less_eq_nat.simps(1)
thf(fact_274_le__0__eq,axiom,
! [N: nat] :
( ( ord_less_eq_nat @ N @ zero_zero_nat )
= ( N = zero_zero_nat ) ) ).
% le_0_eq
thf(fact_275_bot__nat__0_Oextremum__unique,axiom,
! [A: nat] :
( ( ord_less_eq_nat @ A @ zero_zero_nat )
= ( A = zero_zero_nat ) ) ).
% bot_nat_0.extremum_unique
thf(fact_276_bot__nat__0_Oextremum__uniqueI,axiom,
! [A: nat] :
( ( ord_less_eq_nat @ A @ zero_zero_nat )
=> ( A = zero_zero_nat ) ) ).
% bot_nat_0.extremum_uniqueI
thf(fact_277_mult__le__mono2,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_nat @ ( times_times_nat @ K @ I ) @ ( times_times_nat @ K @ J ) ) ) ).
% mult_le_mono2
thf(fact_278_mult__le__mono1,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_nat @ ( times_times_nat @ I @ K ) @ ( times_times_nat @ J @ K ) ) ) ).
% mult_le_mono1
thf(fact_279_mult__le__mono,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( ord_less_eq_nat @ K @ L )
=> ( ord_less_eq_nat @ ( times_times_nat @ I @ K ) @ ( times_times_nat @ J @ L ) ) ) ) ).
% mult_le_mono
thf(fact_280_le__square,axiom,
! [M: nat] : ( ord_less_eq_nat @ M @ ( times_times_nat @ M @ M ) ) ).
% le_square
thf(fact_281_le__cube,axiom,
! [M: nat] : ( ord_less_eq_nat @ M @ ( times_times_nat @ M @ ( times_times_nat @ M @ M ) ) ) ).
% le_cube
thf(fact_282_imageI,axiom,
! [X2: complex,A2: set_complex,F: complex > complex] :
( ( member_complex @ X2 @ A2 )
=> ( member_complex @ ( F @ X2 ) @ ( image_58037603omplex @ F @ A2 ) ) ) ).
% imageI
thf(fact_283_image__iff,axiom,
! [Z: complex,F: complex > complex,A2: set_complex] :
( ( member_complex @ Z @ ( image_58037603omplex @ F @ A2 ) )
= ( ? [X: complex] :
( ( member_complex @ X @ A2 )
& ( Z
= ( F @ X ) ) ) ) ) ).
% image_iff
thf(fact_284_bex__imageD,axiom,
! [F: complex > complex,A2: set_complex,P: complex > $o] :
( ? [X4: complex] :
( ( member_complex @ X4 @ ( image_58037603omplex @ F @ A2 ) )
& ( P @ X4 ) )
=> ? [X3: complex] :
( ( member_complex @ X3 @ A2 )
& ( P @ ( F @ X3 ) ) ) ) ).
% bex_imageD
thf(fact_285_image__cong,axiom,
! [M2: set_complex,N2: set_complex,F: complex > complex,G2: complex > complex] :
( ( M2 = N2 )
=> ( ! [X3: complex] :
( ( member_complex @ X3 @ N2 )
=> ( ( F @ X3 )
= ( G2 @ X3 ) ) )
=> ( ( image_58037603omplex @ F @ M2 )
= ( image_58037603omplex @ G2 @ N2 ) ) ) ) ).
% image_cong
thf(fact_286_ball__imageD,axiom,
! [F: complex > complex,A2: set_complex,P: complex > $o] :
( ! [X3: complex] :
( ( member_complex @ X3 @ ( image_58037603omplex @ F @ A2 ) )
=> ( P @ X3 ) )
=> ! [X4: complex] :
( ( member_complex @ X4 @ A2 )
=> ( P @ ( F @ X4 ) ) ) ) ).
% ball_imageD
thf(fact_287_rev__image__eqI,axiom,
! [X2: complex,A2: set_complex,B: complex,F: complex > complex] :
( ( member_complex @ X2 @ A2 )
=> ( ( B
= ( F @ X2 ) )
=> ( member_complex @ B @ ( image_58037603omplex @ F @ A2 ) ) ) ) ).
% rev_image_eqI
thf(fact_288_in__mono,axiom,
! [A2: set_complex,B3: set_complex,X2: complex] :
( ( ord_le701908932omplex @ A2 @ B3 )
=> ( ( member_complex @ X2 @ A2 )
=> ( member_complex @ X2 @ B3 ) ) ) ).
% in_mono
thf(fact_289_subsetD,axiom,
! [A2: set_complex,B3: set_complex,C: complex] :
( ( ord_le701908932omplex @ A2 @ B3 )
=> ( ( member_complex @ C @ A2 )
=> ( member_complex @ C @ B3 ) ) ) ).
% subsetD
thf(fact_290_equalityE,axiom,
! [A2: set_complex,B3: set_complex] :
( ( A2 = B3 )
=> ~ ( ( ord_le701908932omplex @ A2 @ B3 )
=> ~ ( ord_le701908932omplex @ B3 @ A2 ) ) ) ).
% equalityE
thf(fact_291_subset__eq,axiom,
( ord_le701908932omplex
= ( ^ [A5: set_complex,B5: set_complex] :
! [X: complex] :
( ( member_complex @ X @ A5 )
=> ( member_complex @ X @ B5 ) ) ) ) ).
% subset_eq
thf(fact_292_equalityD1,axiom,
! [A2: set_complex,B3: set_complex] :
( ( A2 = B3 )
=> ( ord_le701908932omplex @ A2 @ B3 ) ) ).
% equalityD1
thf(fact_293_equalityD2,axiom,
! [A2: set_complex,B3: set_complex] :
( ( A2 = B3 )
=> ( ord_le701908932omplex @ B3 @ A2 ) ) ).
% equalityD2
thf(fact_294_subset__iff,axiom,
( ord_le701908932omplex
= ( ^ [A5: set_complex,B5: set_complex] :
! [T3: complex] :
( ( member_complex @ T3 @ A5 )
=> ( member_complex @ T3 @ B5 ) ) ) ) ).
% subset_iff
thf(fact_295_subset__refl,axiom,
! [A2: set_complex] : ( ord_le701908932omplex @ A2 @ A2 ) ).
% subset_refl
thf(fact_296_Collect__mono,axiom,
! [P: complex > $o,Q2: complex > $o] :
( ! [X3: complex] :
( ( P @ X3 )
=> ( Q2 @ X3 ) )
=> ( ord_le701908932omplex @ ( collect_complex @ P ) @ ( collect_complex @ Q2 ) ) ) ).
% Collect_mono
thf(fact_297_subset__trans,axiom,
! [A2: set_complex,B3: set_complex,C2: set_complex] :
( ( ord_le701908932omplex @ A2 @ B3 )
=> ( ( ord_le701908932omplex @ B3 @ C2 )
=> ( ord_le701908932omplex @ A2 @ C2 ) ) ) ).
% subset_trans
thf(fact_298_set__eq__subset,axiom,
( ( ^ [Y2: set_complex,Z4: set_complex] : Y2 = Z4 )
= ( ^ [A5: set_complex,B5: set_complex] :
( ( ord_le701908932omplex @ A5 @ B5 )
& ( ord_le701908932omplex @ B5 @ A5 ) ) ) ) ).
% set_eq_subset
thf(fact_299_Collect__mono__iff,axiom,
! [P: complex > $o,Q2: complex > $o] :
( ( ord_le701908932omplex @ ( collect_complex @ P ) @ ( collect_complex @ Q2 ) )
= ( ! [X: complex] :
( ( P @ X )
=> ( Q2 @ X ) ) ) ) ).
% Collect_mono_iff
thf(fact_300_image__mono,axiom,
! [A2: set_complex,B3: set_complex,F: complex > complex] :
( ( ord_le701908932omplex @ A2 @ B3 )
=> ( ord_le701908932omplex @ ( image_58037603omplex @ F @ A2 ) @ ( image_58037603omplex @ F @ B3 ) ) ) ).
% image_mono
thf(fact_301_image__subsetI,axiom,
! [A2: set_complex,F: complex > complex,B3: set_complex] :
( ! [X3: complex] :
( ( member_complex @ X3 @ A2 )
=> ( member_complex @ ( F @ X3 ) @ B3 ) )
=> ( ord_le701908932omplex @ ( image_58037603omplex @ F @ A2 ) @ B3 ) ) ).
% image_subsetI
thf(fact_302_subset__imageE,axiom,
! [B3: set_complex,F: complex > complex,A2: set_complex] :
( ( ord_le701908932omplex @ B3 @ ( image_58037603omplex @ F @ A2 ) )
=> ~ ! [C3: set_complex] :
( ( ord_le701908932omplex @ C3 @ A2 )
=> ( B3
!= ( image_58037603omplex @ F @ C3 ) ) ) ) ).
% subset_imageE
thf(fact_303_image__subset__iff,axiom,
! [F: complex > complex,A2: set_complex,B3: set_complex] :
( ( ord_le701908932omplex @ ( image_58037603omplex @ F @ A2 ) @ B3 )
= ( ! [X: complex] :
( ( member_complex @ X @ A2 )
=> ( member_complex @ ( F @ X ) @ B3 ) ) ) ) ).
% image_subset_iff
thf(fact_304_subset__image__iff,axiom,
! [B3: set_complex,F: complex > complex,A2: set_complex] :
( ( ord_le701908932omplex @ B3 @ ( image_58037603omplex @ F @ A2 ) )
= ( ? [AA: set_complex] :
( ( ord_le701908932omplex @ AA @ A2 )
& ( B3
= ( image_58037603omplex @ F @ AA ) ) ) ) ) ).
% subset_image_iff
thf(fact_305_order__refl,axiom,
! [X2: set_complex] : ( ord_le701908932omplex @ X2 @ X2 ) ).
% order_refl
thf(fact_306_order__refl,axiom,
! [X2: nat] : ( ord_less_eq_nat @ X2 @ X2 ) ).
% order_refl
thf(fact_307_order__refl,axiom,
! [X2: real] : ( ord_less_eq_real @ X2 @ X2 ) ).
% order_refl
thf(fact_308_kuhn__labelling__lemma_H,axiom,
! [P: ( nat > real ) > $o,F: ( nat > real ) > nat > real,Q2: nat > $o] :
( ! [X3: nat > real] :
( ( P @ X3 )
=> ( P @ ( F @ X3 ) ) )
=> ( ! [X3: nat > real] :
( ( P @ X3 )
=> ! [I2: nat] :
( ( Q2 @ I2 )
=> ( ( ord_less_eq_real @ zero_zero_real @ ( X3 @ I2 ) )
& ( ord_less_eq_real @ ( X3 @ I2 ) @ one_one_real ) ) ) )
=> ? [L3: ( nat > real ) > nat > nat] :
( ! [X4: nat > real,I3: nat] : ( ord_less_eq_nat @ ( L3 @ X4 @ I3 ) @ one_one_nat )
& ! [X4: nat > real,I3: nat] :
( ( ( P @ X4 )
& ( Q2 @ I3 )
& ( ( X4 @ I3 )
= zero_zero_real ) )
=> ( ( L3 @ X4 @ I3 )
= zero_zero_nat ) )
& ! [X4: nat > real,I3: nat] :
( ( ( P @ X4 )
& ( Q2 @ I3 )
& ( ( X4 @ I3 )
= one_one_real ) )
=> ( ( L3 @ X4 @ I3 )
= one_one_nat ) )
& ! [X4: nat > real,I3: nat] :
( ( ( P @ X4 )
& ( Q2 @ I3 )
& ( ( L3 @ X4 @ I3 )
= zero_zero_nat ) )
=> ( ord_less_eq_real @ ( X4 @ I3 ) @ ( F @ X4 @ I3 ) ) )
& ! [X4: nat > real,I3: nat] :
( ( ( P @ X4 )
& ( Q2 @ I3 )
& ( ( L3 @ X4 @ I3 )
= one_one_nat ) )
=> ( ord_less_eq_real @ ( F @ X4 @ I3 ) @ ( X4 @ I3 ) ) ) ) ) ) ).
% kuhn_labelling_lemma'
thf(fact_309_Sup_OSUP__id__eq,axiom,
! [Sup: set_real > real,A2: set_real] :
( ( Sup @ ( image_real_real @ id_real @ A2 ) )
= ( Sup @ A2 ) ) ).
% Sup.SUP_id_eq
thf(fact_310_Sup_OSUP__id__eq,axiom,
! [Sup: set_complex_complex > complex > complex,A2: set_complex_complex] :
( ( Sup @ ( image_944012797omplex @ id_complex_complex @ A2 ) )
= ( Sup @ A2 ) ) ).
% Sup.SUP_id_eq
thf(fact_311_Sup_OSUP__id__eq,axiom,
! [Sup: set_complex > complex,A2: set_complex] :
( ( Sup @ ( image_58037603omplex @ id_complex @ A2 ) )
= ( Sup @ A2 ) ) ).
% Sup.SUP_id_eq
thf(fact_312_le__refl,axiom,
! [N: nat] : ( ord_less_eq_nat @ N @ N ) ).
% le_refl
thf(fact_313_le__trans,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( ord_less_eq_nat @ J @ K )
=> ( ord_less_eq_nat @ I @ K ) ) ) ).
% le_trans
thf(fact_314_eq__imp__le,axiom,
! [M: nat,N: nat] :
( ( M = N )
=> ( ord_less_eq_nat @ M @ N ) ) ).
% eq_imp_le
thf(fact_315_le__antisym,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ( ord_less_eq_nat @ N @ M )
=> ( M = N ) ) ) ).
% le_antisym
thf(fact_316_nat__le__linear,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
| ( ord_less_eq_nat @ N @ M ) ) ).
% nat_le_linear
thf(fact_317_Nat_Oex__has__greatest__nat,axiom,
! [P: nat > $o,K: nat,B: nat] :
( ( P @ K )
=> ( ! [Y3: nat] :
( ( P @ Y3 )
=> ( ord_less_eq_nat @ Y3 @ B ) )
=> ? [X3: nat] :
( ( P @ X3 )
& ! [Y4: nat] :
( ( P @ Y4 )
=> ( ord_less_eq_nat @ Y4 @ X3 ) ) ) ) ) ).
% Nat.ex_has_greatest_nat
thf(fact_318_dual__order_Oantisym,axiom,
! [B: set_complex,A: set_complex] :
( ( ord_le701908932omplex @ B @ A )
=> ( ( ord_le701908932omplex @ A @ B )
=> ( A = B ) ) ) ).
% dual_order.antisym
thf(fact_319_dual__order_Oantisym,axiom,
! [B: nat,A: nat] :
( ( ord_less_eq_nat @ B @ A )
=> ( ( ord_less_eq_nat @ A @ B )
=> ( A = B ) ) ) ).
% dual_order.antisym
thf(fact_320_dual__order_Oantisym,axiom,
! [B: real,A: real] :
( ( ord_less_eq_real @ B @ A )
=> ( ( ord_less_eq_real @ A @ B )
=> ( A = B ) ) ) ).
% dual_order.antisym
thf(fact_321_dual__order_Oeq__iff,axiom,
( ( ^ [Y2: set_complex,Z4: set_complex] : Y2 = Z4 )
= ( ^ [A3: set_complex,B2: set_complex] :
( ( ord_le701908932omplex @ B2 @ A3 )
& ( ord_le701908932omplex @ A3 @ B2 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_322_dual__order_Oeq__iff,axiom,
( ( ^ [Y2: nat,Z4: nat] : Y2 = Z4 )
= ( ^ [A3: nat,B2: nat] :
( ( ord_less_eq_nat @ B2 @ A3 )
& ( ord_less_eq_nat @ A3 @ B2 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_323_dual__order_Oeq__iff,axiom,
( ( ^ [Y2: real,Z4: real] : Y2 = Z4 )
= ( ^ [A3: real,B2: real] :
( ( ord_less_eq_real @ B2 @ A3 )
& ( ord_less_eq_real @ A3 @ B2 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_324_dual__order_Otrans,axiom,
! [B: set_complex,A: set_complex,C: set_complex] :
( ( ord_le701908932omplex @ B @ A )
=> ( ( ord_le701908932omplex @ C @ B )
=> ( ord_le701908932omplex @ C @ A ) ) ) ).
% dual_order.trans
thf(fact_325_dual__order_Otrans,axiom,
! [B: nat,A: nat,C: nat] :
( ( ord_less_eq_nat @ B @ A )
=> ( ( ord_less_eq_nat @ C @ B )
=> ( ord_less_eq_nat @ C @ A ) ) ) ).
% dual_order.trans
thf(fact_326_dual__order_Otrans,axiom,
! [B: real,A: real,C: real] :
( ( ord_less_eq_real @ B @ A )
=> ( ( ord_less_eq_real @ C @ B )
=> ( ord_less_eq_real @ C @ A ) ) ) ).
% dual_order.trans
thf(fact_327_linorder__wlog,axiom,
! [P: nat > nat > $o,A: nat,B: nat] :
( ! [A4: nat,B4: nat] :
( ( ord_less_eq_nat @ A4 @ B4 )
=> ( P @ A4 @ B4 ) )
=> ( ! [A4: nat,B4: nat] :
( ( P @ B4 @ A4 )
=> ( P @ A4 @ B4 ) )
=> ( P @ A @ B ) ) ) ).
% linorder_wlog
thf(fact_328_linorder__wlog,axiom,
! [P: real > real > $o,A: real,B: real] :
( ! [A4: real,B4: real] :
( ( ord_less_eq_real @ A4 @ B4 )
=> ( P @ A4 @ B4 ) )
=> ( ! [A4: real,B4: real] :
( ( P @ B4 @ A4 )
=> ( P @ A4 @ B4 ) )
=> ( P @ A @ B ) ) ) ).
% linorder_wlog
thf(fact_329_dual__order_Orefl,axiom,
! [A: set_complex] : ( ord_le701908932omplex @ A @ A ) ).
% dual_order.refl
thf(fact_330_dual__order_Orefl,axiom,
! [A: nat] : ( ord_less_eq_nat @ A @ A ) ).
% dual_order.refl
thf(fact_331_dual__order_Orefl,axiom,
! [A: real] : ( ord_less_eq_real @ A @ A ) ).
% dual_order.refl
thf(fact_332_order__trans,axiom,
! [X2: set_complex,Y: set_complex,Z: set_complex] :
( ( ord_le701908932omplex @ X2 @ Y )
=> ( ( ord_le701908932omplex @ Y @ Z )
=> ( ord_le701908932omplex @ X2 @ Z ) ) ) ).
% order_trans
thf(fact_333_order__trans,axiom,
! [X2: nat,Y: nat,Z: nat] :
( ( ord_less_eq_nat @ X2 @ Y )
=> ( ( ord_less_eq_nat @ Y @ Z )
=> ( ord_less_eq_nat @ X2 @ Z ) ) ) ).
% order_trans
thf(fact_334_order__trans,axiom,
! [X2: real,Y: real,Z: real] :
( ( ord_less_eq_real @ X2 @ Y )
=> ( ( ord_less_eq_real @ Y @ Z )
=> ( ord_less_eq_real @ X2 @ Z ) ) ) ).
% order_trans
thf(fact_335_order__class_Oorder_Oantisym,axiom,
! [A: set_complex,B: set_complex] :
( ( ord_le701908932omplex @ A @ B )
=> ( ( ord_le701908932omplex @ B @ A )
=> ( A = B ) ) ) ).
% order_class.order.antisym
thf(fact_336_order__class_Oorder_Oantisym,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ B @ A )
=> ( A = B ) ) ) ).
% order_class.order.antisym
thf(fact_337_order__class_Oorder_Oantisym,axiom,
! [A: real,B: real] :
( ( ord_less_eq_real @ A @ B )
=> ( ( ord_less_eq_real @ B @ A )
=> ( A = B ) ) ) ).
% order_class.order.antisym
thf(fact_338_ord__le__eq__trans,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( B = C )
=> ( ord_less_eq_nat @ A @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_339_ord__le__eq__trans,axiom,
! [A: real,B: real,C: real] :
( ( ord_less_eq_real @ A @ B )
=> ( ( B = C )
=> ( ord_less_eq_real @ A @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_340_landau__omega_OR__mult__right__mono,axiom,
! [B: real,A: real,C: real] :
( ( ord_less_eq_real @ B @ A )
=> ( ( ord_less_eq_real @ zero_zero_real @ C )
=> ( ord_less_eq_real @ ( times_times_real @ B @ C ) @ ( times_times_real @ A @ C ) ) ) ) ).
% landau_omega.R_mult_right_mono
thf(fact_341_landau__omega_OR__mult__left__mono,axiom,
! [B: real,A: real,C: real] :
( ( ord_less_eq_real @ B @ A )
=> ( ( ord_less_eq_real @ zero_zero_real @ C )
=> ( ord_less_eq_real @ ( times_times_real @ C @ B ) @ ( times_times_real @ C @ A ) ) ) ) ).
% landau_omega.R_mult_left_mono
thf(fact_342_landau__omega_OR__linear,axiom,
! [Y: real,X2: real] :
( ~ ( ord_less_eq_real @ Y @ X2 )
=> ( ord_less_eq_real @ X2 @ Y ) ) ).
% landau_omega.R_linear
thf(fact_343_landau__omega_OR__trans,axiom,
! [B: real,A: real,C: real] :
( ( ord_less_eq_real @ B @ A )
=> ( ( ord_less_eq_real @ C @ B )
=> ( ord_less_eq_real @ C @ A ) ) ) ).
% landau_omega.R_trans
thf(fact_344_landau__omega_OR__refl,axiom,
! [X2: real] : ( ord_less_eq_real @ X2 @ X2 ) ).
% landau_omega.R_refl
thf(fact_345_landau__o_OR__linear,axiom,
! [X2: real,Y: real] :
( ~ ( ord_less_eq_real @ X2 @ Y )
=> ( ord_less_eq_real @ Y @ X2 ) ) ).
% landau_o.R_linear
thf(fact_346_landau__o_OR__trans,axiom,
! [A: real,B: real,C: real] :
( ( ord_less_eq_real @ A @ B )
=> ( ( ord_less_eq_real @ B @ C )
=> ( ord_less_eq_real @ A @ C ) ) ) ).
% landau_o.R_trans
thf(fact_347_landau__o_OR__mult__left__mono,axiom,
! [A: real,B: real,C: real] :
( ( ord_less_eq_real @ A @ B )
=> ( ( ord_less_eq_real @ zero_zero_real @ C )
=> ( ord_less_eq_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B ) ) ) ) ).
% landau_o.R_mult_left_mono
thf(fact_348_landau__o_OR__mult__right__mono,axiom,
! [A: real,B: real,C: real] :
( ( ord_less_eq_real @ A @ B )
=> ( ( ord_less_eq_real @ zero_zero_real @ C )
=> ( ord_less_eq_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ C ) ) ) ) ).
% landau_o.R_mult_right_mono
thf(fact_349_ln__ge__zero,axiom,
! [X2: real] :
( ( ord_less_eq_real @ one_one_real @ X2 )
=> ( ord_less_eq_real @ zero_zero_real @ ( ln_ln_real @ X2 ) ) ) ).
% ln_ge_zero
thf(fact_350_Ln__1,axiom,
( ( ln_ln_complex @ one_one_complex )
= zero_zero_complex ) ).
% Ln_1
thf(fact_351_holomorphic__fun__eq__0__on__connected,axiom,
! [F: complex > complex,S: set_complex,Z: complex,W2: complex] :
( ( comple372758642hic_on @ F @ S )
=> ( ( topolo935673511omplex @ S )
=> ( ( topolo2127351575omplex @ S )
=> ( ! [N3: nat] :
( ( compow1098280738omplex @ N3 @ deriv_complex @ F @ Z )
= zero_zero_complex )
=> ( ( member_complex @ Z @ S )
=> ( ( member_complex @ W2 @ S )
=> ( ( F @ W2 )
= zero_zero_complex ) ) ) ) ) ) ) ).
% holomorphic_fun_eq_0_on_connected
% Helper facts (7)
thf(help_If_2_1_If_001t__Nat__Onat_T,axiom,
! [X2: nat,Y: nat] :
( ( if_nat @ $false @ X2 @ Y )
= Y ) ).
thf(help_If_1_1_If_001t__Nat__Onat_T,axiom,
! [X2: nat,Y: nat] :
( ( if_nat @ $true @ X2 @ Y )
= X2 ) ).
thf(help_If_2_1_If_001t__Real__Oreal_T,axiom,
! [X2: real,Y: real] :
( ( if_real @ $false @ X2 @ Y )
= Y ) ).
thf(help_If_1_1_If_001t__Real__Oreal_T,axiom,
! [X2: real,Y: real] :
( ( if_real @ $true @ X2 @ Y )
= X2 ) ).
thf(help_If_3_1_If_001t__Complex__Ocomplex_T,axiom,
! [P: $o] :
( ( P = $true )
| ( P = $false ) ) ).
thf(help_If_2_1_If_001t__Complex__Ocomplex_T,axiom,
! [X2: complex,Y: complex] :
( ( if_complex @ $false @ X2 @ Y )
= Y ) ).
thf(help_If_1_1_If_001t__Complex__Ocomplex_T,axiom,
! [X2: complex,Y: complex] :
( ( if_complex @ $true @ X2 @ Y )
= X2 ) ).
% Conjectures (1)
thf(conj_0,conjecture,
( ( deriv_complex @ id_complex @ w )
= one_one_complex ) ).
%------------------------------------------------------------------------------