TPTP Problem File: ITP026^2.p
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%------------------------------------------------------------------------------
% File : ITP026^2 : TPTP v8.2.0. Released v7.5.0.
% Domain : Interactive Theorem Proving
% Problem : Sledgehammer Algebra8 problem prob_1404__6432666_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 : Algebra8/prob_1404__6432666_1 [Des21]
% Status : Theorem
% Rating : 0.00 v8.1.0, 0.25 v7.5.0
% Syntax : Number of formulae : 346 ( 118 unt; 60 typ; 0 def)
% Number of atoms : 769 ( 189 equ; 0 cnn)
% Maximal formula atoms : 12 ( 2 avg)
% Number of connectives : 5767 ( 66 ~; 11 |; 40 &;5270 @)
% ( 0 <=>; 380 =>; 0 <=; 0 <~>)
% Maximal formula depth : 28 ( 9 avg)
% Number of types : 9 ( 8 usr)
% Number of type conns : 215 ( 215 >; 0 *; 0 +; 0 <<)
% Number of symbols : 54 ( 52 usr; 10 con; 0-9 aty)
% Number of variables : 978 ( 105 ^; 762 !; 29 ?; 978 :)
% ( 82 !>; 0 ?*; 0 @-; 0 @+)
% SPC : TH1_THM_EQU_NAR
% Comments : This file was generated by Sledgehammer 2021-02-23 16:31:27.197
%------------------------------------------------------------------------------
% Could-be-implicit typings (13)
thf(ty_t_Algebra1_Ocarrier_Ocarrier__ext,type,
carrier_ext: $tType > $tType > $tType ).
thf(ty_t_Algebra7_OModule_OModule__ext,type,
module_ext: $tType > $tType > $tType > $tType ).
thf(ty_t_Algebra4_OaGroup_OaGroup__ext,type,
aGroup_ext: $tType > $tType > $tType ).
thf(ty_t_Algebra4_ORing_ORing__ext,type,
ring_ext: $tType > $tType > $tType ).
thf(ty_t_Product__Type_Ounit,type,
product_unit: $tType ).
thf(ty_t_Set_Oset,type,
set: $tType > $tType ).
thf(ty_t_Nat_Onat,type,
nat: $tType ).
thf(ty_tf_f,type,
f: $tType ).
thf(ty_tf_e,type,
e: $tType ).
thf(ty_tf_d,type,
d: $tType ).
thf(ty_tf_c,type,
c: $tType ).
thf(ty_tf_b,type,
b: $tType ).
thf(ty_tf_a,type,
a: $tType ).
% Explicit typings (47)
thf(sy_cl_HOL_Otype,type,
type:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Ozero,type,
zero:
!>[A: $tType] : $o ).
thf(sy_cl_Orderings_Oorder,type,
order:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Ogroup__add,type,
group_add:
!>[A: $tType] : $o ).
thf(sy_cl_Orderings_Owellorder,type,
wellorder:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Ocomm__monoid__diff,type,
comm_monoid_diff:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Oordered__ab__group__add,type,
ordered_ab_group_add:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Ocancel__comm__monoid__add,type,
cancel1352612707id_add:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Ocanonically__ordered__monoid__add,type,
canoni770627133id_add:
!>[A: $tType] : $o ).
thf(sy_c_Algebra1_Ocarrier_Ocarrier,type,
carrier:
!>[A: $tType,Z: $tType] : ( ( carrier_ext @ A @ Z ) > ( set @ A ) ) ).
thf(sy_c_Algebra4_ORing,type,
ring:
!>[A: $tType,B: $tType] : ( ( carrier_ext @ A @ ( aGroup_ext @ A @ ( ring_ext @ A @ B ) ) ) > $o ) ).
thf(sy_c_Algebra4_OaGroup_Ozero,type,
zero2:
!>[A: $tType,Z: $tType] : ( ( carrier_ext @ A @ ( aGroup_ext @ A @ Z ) ) > A ) ).
thf(sy_c_Algebra4_Oideal,type,
ideal:
!>[A: $tType,B: $tType] : ( ( carrier_ext @ A @ ( aGroup_ext @ A @ ( ring_ext @ A @ B ) ) ) > ( set @ A ) > $o ) ).
thf(sy_c_Algebra7_OAnnihilator,type,
annihilator:
!>[R: $tType,M: $tType,A: $tType,M1: $tType] : ( ( carrier_ext @ R @ ( aGroup_ext @ R @ ( ring_ext @ R @ M ) ) ) > ( carrier_ext @ A @ ( aGroup_ext @ A @ ( module_ext @ A @ R @ M1 ) ) ) > ( set @ R ) ) ).
thf(sy_c_Algebra7_OHOM,type,
hom:
!>[B: $tType,More: $tType,A: $tType,More1: $tType,C: $tType,More2: $tType] : ( ( carrier_ext @ B @ ( aGroup_ext @ B @ ( ring_ext @ B @ More ) ) ) > ( carrier_ext @ A @ ( aGroup_ext @ A @ ( module_ext @ A @ B @ More1 ) ) ) > ( carrier_ext @ C @ ( aGroup_ext @ C @ ( module_ext @ C @ B @ More2 ) ) ) > ( carrier_ext @ ( A > C ) @ ( aGroup_ext @ ( A > C ) @ ( module_ext @ ( A > C ) @ B @ product_unit ) ) ) ) ).
thf(sy_c_Algebra7_OModule,type,
module:
!>[A: $tType,B: $tType,C: $tType,D: $tType] : ( ( carrier_ext @ A @ ( aGroup_ext @ A @ ( module_ext @ A @ B @ C ) ) ) > ( carrier_ext @ B @ ( aGroup_ext @ B @ ( ring_ext @ B @ D ) ) ) > $o ) ).
thf(sy_c_Algebra7_Ofree__generator,type,
free_generator:
!>[R: $tType,M: $tType,A: $tType,M1: $tType] : ( ( carrier_ext @ R @ ( aGroup_ext @ R @ ( ring_ext @ R @ M ) ) ) > ( carrier_ext @ A @ ( aGroup_ext @ A @ ( module_ext @ A @ R @ M1 ) ) ) > ( set @ A ) > $o ) ).
thf(sy_c_Algebra7_Ogenerator,type,
generator:
!>[R: $tType,M: $tType,A: $tType,M1: $tType] : ( ( carrier_ext @ R @ ( aGroup_ext @ R @ ( ring_ext @ R @ M ) ) ) > ( carrier_ext @ A @ ( aGroup_ext @ A @ ( module_ext @ A @ R @ M1 ) ) ) > ( set @ A ) > $o ) ).
thf(sy_c_Algebra7_Ol__comb,type,
l_comb:
!>[R: $tType,M: $tType,A: $tType,M1: $tType] : ( ( carrier_ext @ R @ ( aGroup_ext @ R @ ( ring_ext @ R @ M ) ) ) > ( carrier_ext @ A @ ( aGroup_ext @ A @ ( module_ext @ A @ R @ M1 ) ) ) > nat > ( nat > R ) > ( nat > A ) > A ) ).
thf(sy_c_Algebra7_Olinear__span,type,
linear_span:
!>[R: $tType,M: $tType,A: $tType,M1: $tType] : ( ( carrier_ext @ R @ ( aGroup_ext @ R @ ( ring_ext @ R @ M ) ) ) > ( carrier_ext @ A @ ( aGroup_ext @ A @ ( module_ext @ A @ R @ M1 ) ) ) > ( set @ R ) > ( set @ A ) > ( set @ A ) ) ).
thf(sy_c_Algebra7_Omdl,type,
mdl:
!>[A: $tType,B: $tType,M: $tType] : ( ( carrier_ext @ A @ ( aGroup_ext @ A @ ( module_ext @ A @ B @ M ) ) ) > ( set @ A ) > ( carrier_ext @ A @ ( aGroup_ext @ A @ ( module_ext @ A @ B @ product_unit ) ) ) ) ).
thf(sy_c_Algebra7_Omisomorphic,type,
misomorphic:
!>[B: $tType,M: $tType,A: $tType,M1: $tType,C: $tType,M2: $tType] : ( ( carrier_ext @ B @ ( aGroup_ext @ B @ ( ring_ext @ B @ M ) ) ) > ( carrier_ext @ A @ ( aGroup_ext @ A @ ( module_ext @ A @ B @ M1 ) ) ) > ( carrier_ext @ C @ ( aGroup_ext @ C @ ( module_ext @ C @ B @ M2 ) ) ) > $o ) ).
thf(sy_c_Algebra7_Oquotient__of__submodules,type,
quotie135653060odules:
!>[R: $tType,M: $tType,A: $tType,M1: $tType] : ( ( carrier_ext @ R @ ( aGroup_ext @ R @ ( ring_ext @ R @ M ) ) ) > ( carrier_ext @ A @ ( aGroup_ext @ A @ ( module_ext @ A @ R @ M1 ) ) ) > ( set @ A ) > ( set @ A ) > ( set @ R ) ) ).
thf(sy_c_Algebra7_Osmodule__ideal__coeff,type,
smodule_ideal_coeff:
!>[R: $tType,M: $tType,A: $tType,M1: $tType] : ( ( carrier_ext @ R @ ( aGroup_ext @ R @ ( ring_ext @ R @ M ) ) ) > ( carrier_ext @ A @ ( aGroup_ext @ A @ ( module_ext @ A @ R @ M1 ) ) ) > ( set @ R ) > ( set @ A ) ) ).
thf(sy_c_Algebra7_Osubmodule,type,
submodule:
!>[B: $tType,M: $tType,A: $tType,C: $tType] : ( ( carrier_ext @ B @ ( aGroup_ext @ B @ ( ring_ext @ B @ M ) ) ) > ( carrier_ext @ A @ ( aGroup_ext @ A @ ( module_ext @ A @ B @ C ) ) ) > ( set @ A ) > $o ) ).
thf(sy_c_Algebra8__Mirabelle__lwvimexpoc_Ofgs,type,
algebr833503410le_fgs:
!>[R: $tType,M: $tType,A: $tType,M1: $tType] : ( ( carrier_ext @ R @ ( aGroup_ext @ R @ ( ring_ext @ R @ M ) ) ) > ( carrier_ext @ A @ ( aGroup_ext @ A @ ( module_ext @ A @ R @ M1 ) ) ) > ( set @ A ) > ( set @ A ) ) ).
thf(sy_c_FuncSet_OPi,type,
pi:
!>[A: $tType,B: $tType] : ( ( set @ A ) > ( A > ( set @ B ) ) > ( set @ ( A > B ) ) ) ).
thf(sy_c_Groups_Ominus__class_Ominus,type,
minus_minus:
!>[A: $tType] : ( A > A > A ) ).
thf(sy_c_Groups_Ozero__class_Ozero,type,
zero_zero:
!>[A: $tType] : A ).
thf(sy_c_Nat_OSuc,type,
suc: nat > nat ).
thf(sy_c_Orderings_Obot__class_Obot,type,
bot_bot:
!>[A: $tType] : A ).
thf(sy_c_Orderings_Oord__class_Oless,type,
ord_less:
!>[A: $tType] : ( A > A > $o ) ).
thf(sy_c_Orderings_Oord__class_Oless__eq,type,
ord_less_eq:
!>[A: $tType] : ( A > A > $o ) ).
thf(sy_c_Set_OCollect,type,
collect:
!>[A: $tType] : ( ( A > $o ) > ( set @ A ) ) ).
thf(sy_c_Set_Oinsert,type,
insert:
!>[A: $tType] : ( A > ( set @ A ) > ( set @ A ) ) ).
thf(sy_c_member,type,
member:
!>[A: $tType] : ( A > ( set @ A ) > $o ) ).
thf(sy_v_H,type,
h: set @ a ).
thf(sy_v_H1,type,
h1: set @ a ).
thf(sy_v_M,type,
m: carrier_ext @ a @ ( aGroup_ext @ a @ ( module_ext @ a @ b @ c ) ) ).
thf(sy_v_N,type,
n: carrier_ext @ e @ ( aGroup_ext @ e @ ( module_ext @ e @ b @ f ) ) ).
thf(sy_v_R,type,
r: carrier_ext @ b @ ( aGroup_ext @ b @ ( ring_ext @ b @ d ) ) ).
thf(sy_v_g,type,
g: nat > a ).
thf(sy_v_h,type,
h2: a ).
thf(sy_v_m,type,
m2: nat ).
thf(sy_v_t,type,
t: nat > b ).
thf(sy_v_x,type,
x: a ).
thf(sy_v_xa,type,
xa: nat ).
% Relevant facts (253)
thf(fact_0_sc__Ring,axiom,
ring @ b @ d @ r ).
% sc_Ring
thf(fact_1_submodule__subset1,axiom,
! [H: set @ a,H2: a] :
( ( submodule @ b @ d @ a @ c @ r @ m @ H )
=> ( ( member @ a @ H2 @ H )
=> ( member @ a @ H2 @ ( carrier @ a @ ( aGroup_ext @ a @ ( module_ext @ a @ b @ c ) ) @ m ) ) ) ) ).
% submodule_subset1
thf(fact_2_submodule__whole,axiom,
submodule @ b @ d @ a @ c @ r @ m @ ( carrier @ a @ ( aGroup_ext @ a @ ( module_ext @ a @ b @ c ) ) @ m ) ).
% submodule_whole
thf(fact_3_submodule__subset,axiom,
! [H: set @ a] :
( ( submodule @ b @ d @ a @ c @ r @ m @ H )
=> ( ord_less_eq @ ( set @ a ) @ H @ ( carrier @ a @ ( aGroup_ext @ a @ ( module_ext @ a @ b @ c ) ) @ m ) ) ) ).
% submodule_subset
thf(fact_4_free__generator__sub,axiom,
! [H: set @ a] :
( ( free_generator @ b @ d @ a @ c @ r @ m @ H )
=> ( ord_less_eq @ ( set @ a ) @ H @ ( carrier @ a @ ( aGroup_ext @ a @ ( module_ext @ a @ b @ c ) ) @ m ) ) ) ).
% free_generator_sub
thf(fact_5_elem__fgs,axiom,
! [A2: set @ a,X: a] :
( ( ord_less_eq @ ( set @ a ) @ A2 @ ( carrier @ a @ ( aGroup_ext @ a @ ( module_ext @ a @ b @ c ) ) @ m ) )
=> ( ( member @ a @ X @ A2 )
=> ( member @ a @ X @ ( algebr833503410le_fgs @ b @ d @ a @ c @ r @ m @ A2 ) ) ) ) ).
% elem_fgs
thf(fact_6_fgs__sub__carrier,axiom,
! [A2: set @ a] :
( ( ord_less_eq @ ( set @ a ) @ A2 @ ( carrier @ a @ ( aGroup_ext @ a @ ( module_ext @ a @ b @ c ) ) @ m ) )
=> ( ord_less_eq @ ( set @ a ) @ ( algebr833503410le_fgs @ b @ d @ a @ c @ r @ m @ A2 ) @ ( carrier @ a @ ( aGroup_ext @ a @ ( module_ext @ a @ b @ c ) ) @ m ) ) ) ).
% fgs_sub_carrier
thf(fact_7_fgs__mono,axiom,
! [H: set @ a,J: set @ a,K: set @ a] :
( ( free_generator @ b @ d @ a @ c @ r @ m @ H )
=> ( ( ord_less_eq @ ( set @ a ) @ J @ K )
=> ( ( ord_less_eq @ ( set @ a ) @ K @ H )
=> ( ord_less_eq @ ( set @ a ) @ ( algebr833503410le_fgs @ b @ d @ a @ c @ r @ m @ J ) @ ( algebr833503410le_fgs @ b @ d @ a @ c @ r @ m @ K ) ) ) ) ) ).
% fgs_mono
thf(fact_8_fgs__submodule,axiom,
! [A2: set @ a] :
( ( ord_less_eq @ ( set @ a ) @ A2 @ ( carrier @ a @ ( aGroup_ext @ a @ ( module_ext @ a @ b @ c ) ) @ m ) )
=> ( submodule @ b @ d @ a @ c @ r @ m @ ( algebr833503410le_fgs @ b @ d @ a @ c @ r @ m @ A2 ) ) ) ).
% fgs_submodule
thf(fact_9_linear__comb__eq,axiom,
! [H: set @ a,S: nat > b,N: nat,F: nat > a,G: nat > a] :
( ( ord_less_eq @ ( set @ a ) @ H @ ( carrier @ a @ ( aGroup_ext @ a @ ( module_ext @ a @ b @ c ) ) @ m ) )
=> ( ( member @ ( nat > b ) @ S
@ ( pi @ nat @ b
@ ( collect @ nat
@ ^ [J2: nat] : ( ord_less_eq @ nat @ J2 @ N ) )
@ ^ [Uu: nat] : ( carrier @ b @ ( aGroup_ext @ b @ ( ring_ext @ b @ d ) ) @ r ) ) )
=> ( ( member @ ( nat > a ) @ F
@ ( pi @ nat @ a
@ ( collect @ nat
@ ^ [J2: nat] : ( ord_less_eq @ nat @ J2 @ N ) )
@ ^ [Uu: nat] : H ) )
=> ( ( member @ ( nat > a ) @ G
@ ( pi @ nat @ a
@ ( collect @ nat
@ ^ [J2: nat] : ( ord_less_eq @ nat @ J2 @ N ) )
@ ^ [Uu: nat] : H ) )
=> ( ! [X2: nat] :
( ( member @ nat @ X2
@ ( collect @ nat
@ ^ [J2: nat] : ( ord_less_eq @ nat @ J2 @ N ) ) )
=> ( ( F @ X2 )
= ( G @ X2 ) ) )
=> ( ( l_comb @ b @ d @ a @ c @ r @ m @ N @ S @ F )
= ( l_comb @ b @ d @ a @ c @ r @ m @ N @ S @ G ) ) ) ) ) ) ) ).
% linear_comb_eq
thf(fact_10_linear__comb__eqTr,axiom,
! [H: set @ a,S: nat > b,N: nat,F: nat > a,G: nat > a] :
( ( ord_less_eq @ ( set @ a ) @ H @ ( carrier @ a @ ( aGroup_ext @ a @ ( module_ext @ a @ b @ c ) ) @ m ) )
=> ( ( ( member @ ( nat > b ) @ S
@ ( pi @ nat @ b
@ ( collect @ nat
@ ^ [J2: nat] : ( ord_less_eq @ nat @ J2 @ N ) )
@ ^ [Uu: nat] : ( carrier @ b @ ( aGroup_ext @ b @ ( ring_ext @ b @ d ) ) @ r ) ) )
& ( member @ ( nat > a ) @ F
@ ( pi @ nat @ a
@ ( collect @ nat
@ ^ [J2: nat] : ( ord_less_eq @ nat @ J2 @ N ) )
@ ^ [Uu: nat] : H ) )
& ( member @ ( nat > a ) @ G
@ ( pi @ nat @ a
@ ( collect @ nat
@ ^ [J2: nat] : ( ord_less_eq @ nat @ J2 @ N ) )
@ ^ [Uu: nat] : H ) )
& ! [X2: nat] :
( ( member @ nat @ X2
@ ( collect @ nat
@ ^ [J2: nat] : ( ord_less_eq @ nat @ J2 @ N ) ) )
=> ( ( F @ X2 )
= ( G @ X2 ) ) ) )
=> ( ( l_comb @ b @ d @ a @ c @ r @ m @ N @ S @ F )
= ( l_comb @ b @ d @ a @ c @ r @ m @ N @ S @ G ) ) ) ) ).
% linear_comb_eqTr
thf(fact_11_l__comb__mem,axiom,
! [A3: set @ b,H: set @ a,S: nat > b,N: nat,M3: nat > a] :
( ( ideal @ b @ d @ r @ A3 )
=> ( ( ord_less_eq @ ( set @ a ) @ H @ ( carrier @ a @ ( aGroup_ext @ a @ ( module_ext @ a @ b @ c ) ) @ m ) )
=> ( ( member @ ( nat > b ) @ S
@ ( pi @ nat @ b
@ ( collect @ nat
@ ^ [J2: nat] : ( ord_less_eq @ nat @ J2 @ N ) )
@ ^ [Uu: nat] : A3 ) )
=> ( ( member @ ( nat > a ) @ M3
@ ( pi @ nat @ a
@ ( collect @ nat
@ ^ [J2: nat] : ( ord_less_eq @ nat @ J2 @ N ) )
@ ^ [Uu: nat] : H ) )
=> ( member @ a @ ( l_comb @ b @ d @ a @ c @ r @ m @ N @ S @ M3 ) @ ( carrier @ a @ ( aGroup_ext @ a @ ( module_ext @ a @ b @ c ) ) @ m ) ) ) ) ) ) ).
% l_comb_mem
thf(fact_12_liear__comb__memTr,axiom,
! [A3: set @ b,H: set @ a,N: nat] :
( ( ideal @ b @ d @ r @ A3 )
=> ( ( ord_less_eq @ ( set @ a ) @ H @ ( carrier @ a @ ( aGroup_ext @ a @ ( module_ext @ a @ b @ c ) ) @ m ) )
=> ! [S2: nat > b,M4: nat > a] :
( ( ( member @ ( nat > b ) @ S2
@ ( pi @ nat @ b
@ ( collect @ nat
@ ^ [J2: nat] : ( ord_less_eq @ nat @ J2 @ N ) )
@ ^ [Uu: nat] : A3 ) )
& ( member @ ( nat > a ) @ M4
@ ( pi @ nat @ a
@ ( collect @ nat
@ ^ [J2: nat] : ( ord_less_eq @ nat @ J2 @ N ) )
@ ^ [Uu: nat] : H ) ) )
=> ( member @ a @ ( l_comb @ b @ d @ a @ c @ r @ m @ N @ S2 @ M4 ) @ ( carrier @ a @ ( aGroup_ext @ a @ ( module_ext @ a @ b @ c ) ) @ m ) ) ) ) ) ).
% liear_comb_memTr
thf(fact_13_Ann__is__ideal,axiom,
ideal @ b @ d @ r @ ( annihilator @ b @ d @ a @ c @ r @ m ) ).
% Ann_is_ideal
thf(fact_14_mem__fgs__l__comb,axiom,
! [K: set @ a,X: a] :
( ( K
!= ( bot_bot @ ( set @ a ) ) )
=> ( ( ord_less_eq @ ( set @ a ) @ K @ ( carrier @ a @ ( aGroup_ext @ a @ ( module_ext @ a @ b @ c ) ) @ m ) )
=> ( ( member @ a @ X @ ( algebr833503410le_fgs @ b @ d @ a @ c @ r @ m @ K ) )
=> ? [N2: nat,X2: nat > a] :
( ( member @ ( nat > a ) @ X2
@ ( pi @ nat @ a
@ ( collect @ nat
@ ^ [J2: nat] : ( ord_less_eq @ nat @ J2 @ N2 ) )
@ ^ [Uu: nat] : K ) )
& ? [Xa: nat > b] :
( ( member @ ( nat > b ) @ Xa
@ ( pi @ nat @ b
@ ( collect @ nat
@ ^ [J2: nat] : ( ord_less_eq @ nat @ J2 @ N2 ) )
@ ^ [Uu: nat] : ( carrier @ b @ ( aGroup_ext @ b @ ( ring_ext @ b @ d ) ) @ r ) ) )
& ( X
= ( l_comb @ b @ d @ a @ c @ r @ m @ N2 @ Xa @ X2 ) ) ) ) ) ) ) ).
% mem_fgs_l_comb
thf(fact_15_Module__axioms,axiom,
module @ a @ b @ c @ d @ m @ r ).
% Module_axioms
thf(fact_16_quotient__of__submodules__is__ideal,axiom,
! [P: set @ a,Q: set @ a] :
( ( submodule @ b @ d @ a @ c @ r @ m @ P )
=> ( ( submodule @ b @ d @ a @ c @ r @ m @ Q )
=> ( ideal @ b @ d @ r @ ( quotie135653060odules @ b @ d @ a @ c @ r @ m @ P @ Q ) ) ) ) ).
% quotient_of_submodules_is_ideal
thf(fact_17_smodule__ideal__coeff__is__Submodule,axiom,
! [A3: set @ b] :
( ( ideal @ b @ d @ r @ A3 )
=> ( submodule @ b @ d @ a @ c @ r @ m @ ( smodule_ideal_coeff @ b @ d @ a @ c @ r @ m @ A3 ) ) ) ).
% smodule_ideal_coeff_is_Submodule
thf(fact_18_mem__smodule__ideal__coeff,axiom,
! [A3: set @ b,X: a] :
( ( ideal @ b @ d @ r @ A3 )
=> ( ( member @ a @ X @ ( smodule_ideal_coeff @ b @ d @ a @ c @ r @ m @ A3 ) )
=> ? [N2: nat,X2: nat > b] :
( ( member @ ( nat > b ) @ X2
@ ( pi @ nat @ b
@ ( collect @ nat
@ ^ [J2: nat] : ( ord_less_eq @ nat @ J2 @ N2 ) )
@ ^ [Uu: nat] : A3 ) )
& ? [Xa: nat > a] :
( ( member @ ( nat > a ) @ Xa
@ ( pi @ nat @ a
@ ( collect @ nat
@ ^ [J2: nat] : ( ord_less_eq @ nat @ J2 @ N2 ) )
@ ^ [Uu: nat] : ( carrier @ a @ ( aGroup_ext @ a @ ( module_ext @ a @ b @ c ) ) @ m ) ) )
& ( X
= ( l_comb @ b @ d @ a @ c @ r @ m @ N2 @ X2 @ Xa ) ) ) ) ) ) ).
% mem_smodule_ideal_coeff
thf(fact_19_Module_Oelem__fgs,axiom,
! [B: $tType,C: $tType,D: $tType,A: $tType,M5: carrier_ext @ A @ ( aGroup_ext @ A @ ( module_ext @ A @ B @ C ) ),R2: carrier_ext @ B @ ( aGroup_ext @ B @ ( ring_ext @ B @ D ) ),A2: set @ A,X: A] :
( ( module @ A @ B @ C @ D @ M5 @ R2 )
=> ( ( ord_less_eq @ ( set @ A ) @ A2 @ ( carrier @ A @ ( aGroup_ext @ A @ ( module_ext @ A @ B @ C ) ) @ M5 ) )
=> ( ( member @ A @ X @ A2 )
=> ( member @ A @ X @ ( algebr833503410le_fgs @ B @ D @ A @ C @ R2 @ M5 @ A2 ) ) ) ) ) ).
% Module.elem_fgs
thf(fact_20_Module_Ofgs__mono,axiom,
! [B: $tType,C: $tType,D: $tType,A: $tType,M5: carrier_ext @ A @ ( aGroup_ext @ A @ ( module_ext @ A @ B @ C ) ),R2: carrier_ext @ B @ ( aGroup_ext @ B @ ( ring_ext @ B @ D ) ),H: set @ A,J: set @ A,K: set @ A] :
( ( module @ A @ B @ C @ D @ M5 @ R2 )
=> ( ( free_generator @ B @ D @ A @ C @ R2 @ M5 @ H )
=> ( ( ord_less_eq @ ( set @ A ) @ J @ K )
=> ( ( ord_less_eq @ ( set @ A ) @ K @ H )
=> ( ord_less_eq @ ( set @ A ) @ ( algebr833503410le_fgs @ B @ D @ A @ C @ R2 @ M5 @ J ) @ ( algebr833503410le_fgs @ B @ D @ A @ C @ R2 @ M5 @ K ) ) ) ) ) ) ).
% Module.fgs_mono
thf(fact_21_Module_Ofgs__submodule,axiom,
! [B: $tType,C: $tType,D: $tType,A: $tType,M5: carrier_ext @ A @ ( aGroup_ext @ A @ ( module_ext @ A @ B @ C ) ),R2: carrier_ext @ B @ ( aGroup_ext @ B @ ( ring_ext @ B @ D ) ),A2: set @ A] :
( ( module @ A @ B @ C @ D @ M5 @ R2 )
=> ( ( ord_less_eq @ ( set @ A ) @ A2 @ ( carrier @ A @ ( aGroup_ext @ A @ ( module_ext @ A @ B @ C ) ) @ M5 ) )
=> ( submodule @ B @ D @ A @ C @ R2 @ M5 @ ( algebr833503410le_fgs @ B @ D @ A @ C @ R2 @ M5 @ A2 ) ) ) ) ).
% Module.fgs_submodule
thf(fact_22_Module_Omem__fgs__l__comb,axiom,
! [B: $tType,C: $tType,D: $tType,A: $tType,M5: carrier_ext @ A @ ( aGroup_ext @ A @ ( module_ext @ A @ B @ C ) ),R2: carrier_ext @ B @ ( aGroup_ext @ B @ ( ring_ext @ B @ D ) ),K: set @ A,X: A] :
( ( module @ A @ B @ C @ D @ M5 @ R2 )
=> ( ( K
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( ord_less_eq @ ( set @ A ) @ K @ ( carrier @ A @ ( aGroup_ext @ A @ ( module_ext @ A @ B @ C ) ) @ M5 ) )
=> ( ( member @ A @ X @ ( algebr833503410le_fgs @ B @ D @ A @ C @ R2 @ M5 @ K ) )
=> ? [N2: nat,X2: nat > A] :
( ( member @ ( nat > A ) @ X2
@ ( pi @ nat @ A
@ ( collect @ nat
@ ^ [J2: nat] : ( ord_less_eq @ nat @ J2 @ N2 ) )
@ ^ [Uu: nat] : K ) )
& ? [Xa: nat > B] :
( ( member @ ( nat > B ) @ Xa
@ ( pi @ nat @ B
@ ( collect @ nat
@ ^ [J2: nat] : ( ord_less_eq @ nat @ J2 @ N2 ) )
@ ^ [Uu: nat] : ( carrier @ B @ ( aGroup_ext @ B @ ( ring_ext @ B @ D ) ) @ R2 ) ) )
& ( X
= ( l_comb @ B @ D @ A @ C @ R2 @ M5 @ N2 @ Xa @ X2 ) ) ) ) ) ) ) ) ).
% Module.mem_fgs_l_comb
thf(fact_23_Module_Ofgs__sub__carrier,axiom,
! [D: $tType,C: $tType,B: $tType,A: $tType,M5: carrier_ext @ A @ ( aGroup_ext @ A @ ( module_ext @ A @ B @ C ) ),R2: carrier_ext @ B @ ( aGroup_ext @ B @ ( ring_ext @ B @ D ) ),A2: set @ A] :
( ( module @ A @ B @ C @ D @ M5 @ R2 )
=> ( ( ord_less_eq @ ( set @ A ) @ A2 @ ( carrier @ A @ ( aGroup_ext @ A @ ( module_ext @ A @ B @ C ) ) @ M5 ) )
=> ( ord_less_eq @ ( set @ A ) @ ( algebr833503410le_fgs @ B @ D @ A @ C @ R2 @ M5 @ A2 ) @ ( carrier @ A @ ( aGroup_ext @ A @ ( module_ext @ A @ B @ C ) ) @ M5 ) ) ) ) ).
% Module.fgs_sub_carrier
thf(fact_24_misom__self,axiom,
misomorphic @ b @ d @ a @ c @ a @ c @ r @ m @ m ).
% misom_self
thf(fact_25_HOM__is__module,axiom,
! [E: $tType,F2: $tType,N3: carrier_ext @ E @ ( aGroup_ext @ E @ ( module_ext @ E @ b @ F2 ) )] :
( ( module @ E @ b @ F2 @ d @ N3 @ r )
=> ( module @ ( a > E ) @ b @ product_unit @ d @ ( hom @ b @ d @ a @ c @ E @ F2 @ r @ m @ N3 ) @ r ) ) ).
% HOM_is_module
thf(fact_26_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_27_generator__sub__carrier,axiom,
! [H: set @ a] :
( ( generator @ b @ d @ a @ c @ r @ m @ H )
=> ( ord_less_eq @ ( set @ a ) @ H @ ( carrier @ a @ ( aGroup_ext @ a @ ( module_ext @ a @ b @ c ) ) @ m ) ) ) ).
% generator_sub_carrier
thf(fact_28_free__generator__generator,axiom,
! [H: set @ a] :
( ( free_generator @ b @ d @ a @ c @ r @ m @ H )
=> ( generator @ b @ d @ a @ c @ r @ m @ H ) ) ).
% free_generator_generator
thf(fact_29_Module_Ol__comb__mem,axiom,
! [D: $tType,C: $tType,B: $tType,A: $tType,M5: carrier_ext @ A @ ( aGroup_ext @ A @ ( module_ext @ A @ B @ C ) ),R2: carrier_ext @ B @ ( aGroup_ext @ B @ ( ring_ext @ B @ D ) ),A3: set @ B,H: set @ A,S: nat > B,N: nat,M3: nat > A] :
( ( module @ A @ B @ C @ D @ M5 @ R2 )
=> ( ( ideal @ B @ D @ R2 @ A3 )
=> ( ( ord_less_eq @ ( set @ A ) @ H @ ( carrier @ A @ ( aGroup_ext @ A @ ( module_ext @ A @ B @ C ) ) @ M5 ) )
=> ( ( member @ ( nat > B ) @ S
@ ( pi @ nat @ B
@ ( collect @ nat
@ ^ [J2: nat] : ( ord_less_eq @ nat @ J2 @ N ) )
@ ^ [Uu: nat] : A3 ) )
=> ( ( member @ ( nat > A ) @ M3
@ ( pi @ nat @ A
@ ( collect @ nat
@ ^ [J2: nat] : ( ord_less_eq @ nat @ J2 @ N ) )
@ ^ [Uu: nat] : H ) )
=> ( member @ A @ ( l_comb @ B @ D @ A @ C @ R2 @ M5 @ N @ S @ M3 ) @ ( carrier @ A @ ( aGroup_ext @ A @ ( module_ext @ A @ B @ C ) ) @ M5 ) ) ) ) ) ) ) ).
% Module.l_comb_mem
thf(fact_30_Module_Oliear__comb__memTr,axiom,
! [D: $tType,C: $tType,B: $tType,A: $tType,M5: carrier_ext @ A @ ( aGroup_ext @ A @ ( module_ext @ A @ B @ C ) ),R2: carrier_ext @ B @ ( aGroup_ext @ B @ ( ring_ext @ B @ D ) ),A3: set @ B,H: set @ A,N: nat] :
( ( module @ A @ B @ C @ D @ M5 @ R2 )
=> ( ( ideal @ B @ D @ R2 @ A3 )
=> ( ( ord_less_eq @ ( set @ A ) @ H @ ( carrier @ A @ ( aGroup_ext @ A @ ( module_ext @ A @ B @ C ) ) @ M5 ) )
=> ! [S2: nat > B,M4: nat > A] :
( ( ( member @ ( nat > B ) @ S2
@ ( pi @ nat @ B
@ ( collect @ nat
@ ^ [J2: nat] : ( ord_less_eq @ nat @ J2 @ N ) )
@ ^ [Uu: nat] : A3 ) )
& ( member @ ( nat > A ) @ M4
@ ( pi @ nat @ A
@ ( collect @ nat
@ ^ [J2: nat] : ( ord_less_eq @ nat @ J2 @ N ) )
@ ^ [Uu: nat] : H ) ) )
=> ( member @ A @ ( l_comb @ B @ D @ A @ C @ R2 @ M5 @ N @ S2 @ M4 ) @ ( carrier @ A @ ( aGroup_ext @ A @ ( module_ext @ A @ B @ C ) ) @ M5 ) ) ) ) ) ) ).
% Module.liear_comb_memTr
thf(fact_31_zero__less__diff,axiom,
! [N: nat,M3: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ ( minus_minus @ nat @ N @ M3 ) )
= ( ord_less @ nat @ M3 @ N ) ) ).
% zero_less_diff
thf(fact_32_Module_Olinear__comb__eq,axiom,
! [B: $tType,C: $tType,D: $tType,A: $tType,M5: carrier_ext @ A @ ( aGroup_ext @ A @ ( module_ext @ A @ B @ C ) ),R2: carrier_ext @ B @ ( aGroup_ext @ B @ ( ring_ext @ B @ D ) ),H: set @ A,S: nat > B,N: nat,F: nat > A,G: nat > A] :
( ( module @ A @ B @ C @ D @ M5 @ R2 )
=> ( ( ord_less_eq @ ( set @ A ) @ H @ ( carrier @ A @ ( aGroup_ext @ A @ ( module_ext @ A @ B @ C ) ) @ M5 ) )
=> ( ( member @ ( nat > B ) @ S
@ ( pi @ nat @ B
@ ( collect @ nat
@ ^ [J2: nat] : ( ord_less_eq @ nat @ J2 @ N ) )
@ ^ [Uu: nat] : ( carrier @ B @ ( aGroup_ext @ B @ ( ring_ext @ B @ D ) ) @ R2 ) ) )
=> ( ( member @ ( nat > A ) @ F
@ ( pi @ nat @ A
@ ( collect @ nat
@ ^ [J2: nat] : ( ord_less_eq @ nat @ J2 @ N ) )
@ ^ [Uu: nat] : H ) )
=> ( ( member @ ( nat > A ) @ G
@ ( pi @ nat @ A
@ ( collect @ nat
@ ^ [J2: nat] : ( ord_less_eq @ nat @ J2 @ N ) )
@ ^ [Uu: nat] : H ) )
=> ( ! [X2: nat] :
( ( member @ nat @ X2
@ ( collect @ nat
@ ^ [J2: nat] : ( ord_less_eq @ nat @ J2 @ N ) ) )
=> ( ( F @ X2 )
= ( G @ X2 ) ) )
=> ( ( l_comb @ B @ D @ A @ C @ R2 @ M5 @ N @ S @ F )
= ( l_comb @ B @ D @ A @ C @ R2 @ M5 @ N @ S @ G ) ) ) ) ) ) ) ) ).
% Module.linear_comb_eq
thf(fact_33_Module_Olinear__comb__eqTr,axiom,
! [B: $tType,C: $tType,D: $tType,A: $tType,M5: carrier_ext @ A @ ( aGroup_ext @ A @ ( module_ext @ A @ B @ C ) ),R2: carrier_ext @ B @ ( aGroup_ext @ B @ ( ring_ext @ B @ D ) ),H: set @ A,S: nat > B,N: nat,F: nat > A,G: nat > A] :
( ( module @ A @ B @ C @ D @ M5 @ R2 )
=> ( ( ord_less_eq @ ( set @ A ) @ H @ ( carrier @ A @ ( aGroup_ext @ A @ ( module_ext @ A @ B @ C ) ) @ M5 ) )
=> ( ( ( member @ ( nat > B ) @ S
@ ( pi @ nat @ B
@ ( collect @ nat
@ ^ [J2: nat] : ( ord_less_eq @ nat @ J2 @ N ) )
@ ^ [Uu: nat] : ( carrier @ B @ ( aGroup_ext @ B @ ( ring_ext @ B @ D ) ) @ R2 ) ) )
& ( member @ ( nat > A ) @ F
@ ( pi @ nat @ A
@ ( collect @ nat
@ ^ [J2: nat] : ( ord_less_eq @ nat @ J2 @ N ) )
@ ^ [Uu: nat] : H ) )
& ( member @ ( nat > A ) @ G
@ ( pi @ nat @ A
@ ( collect @ nat
@ ^ [J2: nat] : ( ord_less_eq @ nat @ J2 @ N ) )
@ ^ [Uu: nat] : H ) )
& ! [X2: nat] :
( ( member @ nat @ X2
@ ( collect @ nat
@ ^ [J2: nat] : ( ord_less_eq @ nat @ J2 @ N ) ) )
=> ( ( F @ X2 )
= ( G @ X2 ) ) ) )
=> ( ( l_comb @ B @ D @ A @ C @ R2 @ M5 @ N @ S @ F )
= ( l_comb @ B @ D @ A @ C @ R2 @ M5 @ N @ S @ G ) ) ) ) ) ).
% Module.linear_comb_eqTr
thf(fact_34_diff__is__0__eq,axiom,
! [M3: nat,N: nat] :
( ( ( minus_minus @ nat @ M3 @ N )
= ( zero_zero @ nat ) )
= ( ord_less_eq @ nat @ M3 @ N ) ) ).
% diff_is_0_eq
thf(fact_35_diff__is__0__eq_H,axiom,
! [M3: nat,N: nat] :
( ( ord_less_eq @ nat @ M3 @ N )
=> ( ( minus_minus @ nat @ M3 @ N )
= ( zero_zero @ nat ) ) ) ).
% diff_is_0_eq'
thf(fact_36_old_Onat_Oinject,axiom,
! [Nat: nat,Nat2: nat] :
( ( ( suc @ Nat )
= ( suc @ Nat2 ) )
= ( Nat = Nat2 ) ) ).
% old.nat.inject
thf(fact_37_nat_Oinject,axiom,
! [X22: nat,Y2: nat] :
( ( ( suc @ X22 )
= ( suc @ Y2 ) )
= ( X22 = Y2 ) ) ).
% nat.inject
thf(fact_38_misom__sym,axiom,
! [E: $tType,F2: $tType,N3: carrier_ext @ E @ ( aGroup_ext @ E @ ( module_ext @ E @ b @ F2 ) )] :
( ( module @ E @ b @ F2 @ d @ N3 @ r )
=> ( ( misomorphic @ b @ d @ a @ c @ E @ F2 @ r @ m @ N3 )
=> ( misomorphic @ b @ d @ E @ F2 @ a @ c @ r @ N3 @ m ) ) ) ).
% misom_sym
thf(fact_39_Suc__le__mono,axiom,
! [N: nat,M3: nat] :
( ( ord_less_eq @ nat @ ( suc @ N ) @ ( suc @ M3 ) )
= ( ord_less_eq @ nat @ N @ M3 ) ) ).
% Suc_le_mono
thf(fact_40_bot__nat__0_Oextremum,axiom,
! [A2: nat] : ( ord_less_eq @ nat @ ( zero_zero @ nat ) @ A2 ) ).
% bot_nat_0.extremum
thf(fact_41_le0,axiom,
! [N: nat] : ( ord_less_eq @ nat @ ( zero_zero @ nat ) @ N ) ).
% le0
thf(fact_42_Suc__less__eq,axiom,
! [M3: nat,N: nat] :
( ( ord_less @ nat @ ( suc @ M3 ) @ ( suc @ N ) )
= ( ord_less @ nat @ M3 @ N ) ) ).
% Suc_less_eq
thf(fact_43_Suc__mono,axiom,
! [M3: nat,N: nat] :
( ( ord_less @ nat @ M3 @ N )
=> ( ord_less @ nat @ ( suc @ M3 ) @ ( suc @ N ) ) ) ).
% Suc_mono
thf(fact_44_mem__Collect__eq,axiom,
! [A: $tType,A2: A,P: A > $o] :
( ( member @ A @ A2 @ ( collect @ A @ P ) )
= ( P @ A2 ) ) ).
% mem_Collect_eq
thf(fact_45_Collect__mem__eq,axiom,
! [A: $tType,A3: set @ A] :
( ( collect @ A
@ ^ [X3: A] : ( member @ A @ X3 @ A3 ) )
= A3 ) ).
% Collect_mem_eq
thf(fact_46_Collect__cong,axiom,
! [A: $tType,P: A > $o,Q: A > $o] :
( ! [X2: A] :
( ( P @ X2 )
= ( Q @ X2 ) )
=> ( ( collect @ A @ P )
= ( collect @ A @ Q ) ) ) ).
% Collect_cong
thf(fact_47_bot__nat__0_Onot__eq__extremum,axiom,
! [A2: nat] :
( ( A2
!= ( zero_zero @ nat ) )
= ( ord_less @ nat @ ( zero_zero @ nat ) @ A2 ) ) ).
% bot_nat_0.not_eq_extremum
thf(fact_48_less__nat__zero__code,axiom,
! [N: nat] :
~ ( ord_less @ nat @ N @ ( zero_zero @ nat ) ) ).
% less_nat_zero_code
thf(fact_49_neq0__conv,axiom,
! [N: nat] :
( ( N
!= ( zero_zero @ nat ) )
= ( ord_less @ nat @ ( zero_zero @ nat ) @ N ) ) ).
% neq0_conv
thf(fact_50_diff__diff__cancel,axiom,
! [I: nat,N: nat] :
( ( ord_less_eq @ nat @ I @ N )
=> ( ( minus_minus @ nat @ N @ ( minus_minus @ nat @ N @ I ) )
= I ) ) ).
% diff_diff_cancel
thf(fact_51_Suc__diff__diff,axiom,
! [M3: nat,N: nat,K2: nat] :
( ( minus_minus @ nat @ ( minus_minus @ nat @ ( suc @ M3 ) @ N ) @ ( suc @ K2 ) )
= ( minus_minus @ nat @ ( minus_minus @ nat @ M3 @ N ) @ K2 ) ) ).
% Suc_diff_diff
thf(fact_52_diff__Suc__Suc,axiom,
! [M3: nat,N: nat] :
( ( minus_minus @ nat @ ( suc @ M3 ) @ ( suc @ N ) )
= ( minus_minus @ nat @ M3 @ N ) ) ).
% diff_Suc_Suc
thf(fact_53_diff__self__eq__0,axiom,
! [M3: nat] :
( ( minus_minus @ nat @ M3 @ M3 )
= ( zero_zero @ nat ) ) ).
% diff_self_eq_0
thf(fact_54_diff__0__eq__0,axiom,
! [N: nat] :
( ( minus_minus @ nat @ ( zero_zero @ nat ) @ N )
= ( zero_zero @ nat ) ) ).
% diff_0_eq_0
thf(fact_55_misom__trans,axiom,
! [E: $tType,F2: $tType,H3: $tType,G2: $tType,L: carrier_ext @ E @ ( aGroup_ext @ E @ ( module_ext @ E @ b @ F2 ) ),N3: carrier_ext @ G2 @ ( aGroup_ext @ G2 @ ( module_ext @ G2 @ b @ H3 ) )] :
( ( module @ E @ b @ F2 @ d @ L @ r )
=> ( ( module @ G2 @ b @ H3 @ d @ N3 @ r )
=> ( ( misomorphic @ b @ d @ E @ F2 @ a @ c @ r @ L @ m )
=> ( ( misomorphic @ b @ d @ a @ c @ G2 @ H3 @ r @ m @ N3 )
=> ( misomorphic @ b @ d @ E @ F2 @ G2 @ H3 @ r @ L @ N3 ) ) ) ) ) ).
% misom_trans
thf(fact_56_zero__less__Suc,axiom,
! [N: nat] : ( ord_less @ nat @ ( zero_zero @ nat ) @ ( suc @ N ) ) ).
% zero_less_Suc
thf(fact_57_less__Suc0,axiom,
! [N: nat] :
( ( ord_less @ nat @ N @ ( suc @ ( zero_zero @ nat ) ) )
= ( N
= ( zero_zero @ nat ) ) ) ).
% less_Suc0
thf(fact_58_Module_OHOM__is__module,axiom,
! [E: $tType,F2: $tType,C: $tType,A: $tType,D: $tType,B: $tType,M5: carrier_ext @ A @ ( aGroup_ext @ A @ ( module_ext @ A @ B @ C ) ),R2: carrier_ext @ B @ ( aGroup_ext @ B @ ( ring_ext @ B @ D ) ),N3: carrier_ext @ E @ ( aGroup_ext @ E @ ( module_ext @ E @ B @ F2 ) )] :
( ( module @ A @ B @ C @ D @ M5 @ R2 )
=> ( ( module @ E @ B @ F2 @ D @ N3 @ R2 )
=> ( module @ ( A > E ) @ B @ product_unit @ D @ ( hom @ B @ D @ A @ C @ E @ F2 @ R2 @ M5 @ N3 ) @ R2 ) ) ) ).
% Module.HOM_is_module
thf(fact_59_Module_Omisom__trans,axiom,
! [C: $tType,A: $tType,E: $tType,F2: $tType,D: $tType,H3: $tType,B: $tType,G2: $tType,M5: carrier_ext @ A @ ( aGroup_ext @ A @ ( module_ext @ A @ B @ C ) ),R2: carrier_ext @ B @ ( aGroup_ext @ B @ ( ring_ext @ B @ D ) ),L: carrier_ext @ E @ ( aGroup_ext @ E @ ( module_ext @ E @ B @ F2 ) ),N3: carrier_ext @ G2 @ ( aGroup_ext @ G2 @ ( module_ext @ G2 @ B @ H3 ) )] :
( ( module @ A @ B @ C @ D @ M5 @ R2 )
=> ( ( module @ E @ B @ F2 @ D @ L @ R2 )
=> ( ( module @ G2 @ B @ H3 @ D @ N3 @ R2 )
=> ( ( misomorphic @ B @ D @ E @ F2 @ A @ C @ R2 @ L @ M5 )
=> ( ( misomorphic @ B @ D @ A @ C @ G2 @ H3 @ R2 @ M5 @ N3 )
=> ( misomorphic @ B @ D @ E @ F2 @ G2 @ H3 @ R2 @ L @ N3 ) ) ) ) ) ) ).
% Module.misom_trans
thf(fact_60_Module_Omisom__self,axiom,
! [D: $tType,C: $tType,B: $tType,A: $tType,M5: carrier_ext @ A @ ( aGroup_ext @ A @ ( module_ext @ A @ B @ C ) ),R2: carrier_ext @ B @ ( aGroup_ext @ B @ ( ring_ext @ B @ D ) )] :
( ( module @ A @ B @ C @ D @ M5 @ R2 )
=> ( misomorphic @ B @ D @ A @ C @ A @ C @ R2 @ M5 @ M5 ) ) ).
% Module.misom_self
thf(fact_61_Module_Omisom__sym,axiom,
! [E: $tType,F2: $tType,D: $tType,C: $tType,B: $tType,A: $tType,M5: carrier_ext @ A @ ( aGroup_ext @ A @ ( module_ext @ A @ B @ C ) ),R2: carrier_ext @ B @ ( aGroup_ext @ B @ ( ring_ext @ B @ D ) ),N3: carrier_ext @ E @ ( aGroup_ext @ E @ ( module_ext @ E @ B @ F2 ) )] :
( ( module @ A @ B @ C @ D @ M5 @ R2 )
=> ( ( module @ E @ B @ F2 @ D @ N3 @ R2 )
=> ( ( misomorphic @ B @ D @ A @ C @ E @ F2 @ R2 @ M5 @ N3 )
=> ( misomorphic @ B @ D @ E @ F2 @ A @ C @ R2 @ N3 @ M5 ) ) ) ) ).
% Module.misom_sym
thf(fact_62_Module_Ofree__generator__generator,axiom,
! [B: $tType,C: $tType,D: $tType,A: $tType,M5: carrier_ext @ A @ ( aGroup_ext @ A @ ( module_ext @ A @ B @ C ) ),R2: carrier_ext @ B @ ( aGroup_ext @ B @ ( ring_ext @ B @ D ) ),H: set @ A] :
( ( module @ A @ B @ C @ D @ M5 @ R2 )
=> ( ( free_generator @ B @ D @ A @ C @ R2 @ M5 @ H )
=> ( generator @ B @ D @ A @ C @ R2 @ M5 @ H ) ) ) ).
% Module.free_generator_generator
thf(fact_63_Module_Ogenerator__sub__carrier,axiom,
! [D: $tType,C: $tType,B: $tType,A: $tType,M5: carrier_ext @ A @ ( aGroup_ext @ A @ ( module_ext @ A @ B @ C ) ),R2: carrier_ext @ B @ ( aGroup_ext @ B @ ( ring_ext @ B @ D ) ),H: set @ A] :
( ( module @ A @ B @ C @ D @ M5 @ R2 )
=> ( ( generator @ B @ D @ A @ C @ R2 @ M5 @ H )
=> ( ord_less_eq @ ( set @ A ) @ H @ ( carrier @ A @ ( aGroup_ext @ A @ ( module_ext @ A @ B @ C ) ) @ M5 ) ) ) ) ).
% Module.generator_sub_carrier
thf(fact_64_Module_OAnn__is__ideal,axiom,
! [D: $tType,C: $tType,B: $tType,A: $tType,M5: carrier_ext @ A @ ( aGroup_ext @ A @ ( module_ext @ A @ B @ C ) ),R2: carrier_ext @ B @ ( aGroup_ext @ B @ ( ring_ext @ B @ D ) )] :
( ( module @ A @ B @ C @ D @ M5 @ R2 )
=> ( ideal @ B @ D @ R2 @ ( annihilator @ B @ D @ A @ C @ R2 @ M5 ) ) ) ).
% Module.Ann_is_ideal
thf(fact_65_measure__induct__rule,axiom,
! [B: $tType,A: $tType] :
( ( wellorder @ B )
=> ! [F: A > B,P: A > $o,A2: A] :
( ! [X2: A] :
( ! [Y: A] :
( ( ord_less @ B @ ( F @ Y ) @ ( F @ X2 ) )
=> ( P @ Y ) )
=> ( P @ X2 ) )
=> ( P @ A2 ) ) ) ).
% measure_induct_rule
thf(fact_66_measure__induct,axiom,
! [B: $tType,A: $tType] :
( ( wellorder @ B )
=> ! [F: A > B,P: A > $o,A2: A] :
( ! [X2: A] :
( ! [Y: A] :
( ( ord_less @ B @ ( F @ Y ) @ ( F @ X2 ) )
=> ( P @ Y ) )
=> ( P @ X2 ) )
=> ( P @ A2 ) ) ) ).
% measure_induct
thf(fact_67_Nat_Oex__has__greatest__nat,axiom,
! [P: nat > $o,K2: nat,B2: nat] :
( ( P @ K2 )
=> ( ! [Y3: nat] :
( ( P @ Y3 )
=> ( ord_less_eq @ nat @ Y3 @ B2 ) )
=> ? [X2: nat] :
( ( P @ X2 )
& ! [Y: nat] :
( ( P @ Y )
=> ( ord_less_eq @ nat @ Y @ X2 ) ) ) ) ) ).
% Nat.ex_has_greatest_nat
thf(fact_68_nat__le__linear,axiom,
! [M3: nat,N: nat] :
( ( ord_less_eq @ nat @ M3 @ N )
| ( ord_less_eq @ nat @ N @ M3 ) ) ).
% nat_le_linear
thf(fact_69_le__antisym,axiom,
! [M3: nat,N: nat] :
( ( ord_less_eq @ nat @ M3 @ N )
=> ( ( ord_less_eq @ nat @ N @ M3 )
=> ( M3 = N ) ) ) ).
% le_antisym
thf(fact_70_le__trans,axiom,
! [I: nat,J3: nat,K2: nat] :
( ( ord_less_eq @ nat @ I @ J3 )
=> ( ( ord_less_eq @ nat @ J3 @ K2 )
=> ( ord_less_eq @ nat @ I @ K2 ) ) ) ).
% le_trans
thf(fact_71_n__not__Suc__n,axiom,
! [N: nat] :
( N
!= ( suc @ N ) ) ).
% n_not_Suc_n
thf(fact_72_Suc__inject,axiom,
! [X: nat,Y4: nat] :
( ( ( suc @ X )
= ( suc @ Y4 ) )
=> ( X = Y4 ) ) ).
% Suc_inject
thf(fact_73_infinite__descent__measure,axiom,
! [A: $tType,P: A > $o,V: A > nat,X: A] :
( ! [X2: A] :
( ~ ( P @ X2 )
=> ? [Y: A] :
( ( ord_less @ nat @ ( V @ Y ) @ ( V @ X2 ) )
& ~ ( P @ Y ) ) )
=> ( P @ X ) ) ).
% infinite_descent_measure
thf(fact_74_linorder__neqE__nat,axiom,
! [X: nat,Y4: nat] :
( ( X != Y4 )
=> ( ~ ( ord_less @ nat @ X @ Y4 )
=> ( ord_less @ nat @ Y4 @ X ) ) ) ).
% linorder_neqE_nat
thf(fact_75_infinite__descent,axiom,
! [P: nat > $o,N: nat] :
( ! [N2: nat] :
( ~ ( P @ N2 )
=> ? [M4: nat] :
( ( ord_less @ nat @ M4 @ N2 )
& ~ ( P @ M4 ) ) )
=> ( P @ N ) ) ).
% infinite_descent
thf(fact_76_nat__less__induct,axiom,
! [P: nat > $o,N: nat] :
( ! [N2: nat] :
( ! [M4: nat] :
( ( ord_less @ nat @ M4 @ N2 )
=> ( P @ M4 ) )
=> ( P @ N2 ) )
=> ( P @ N ) ) ).
% nat_less_induct
thf(fact_77_less__irrefl__nat,axiom,
! [N: nat] :
~ ( ord_less @ nat @ N @ N ) ).
% less_irrefl_nat
thf(fact_78_less__not__refl3,axiom,
! [S: nat,T: nat] :
( ( ord_less @ nat @ S @ T )
=> ( S != T ) ) ).
% less_not_refl3
thf(fact_79_less__not__refl2,axiom,
! [N: nat,M3: nat] :
( ( ord_less @ nat @ N @ M3 )
=> ( M3 != N ) ) ).
% less_not_refl2
thf(fact_80_less__not__refl,axiom,
! [N: nat] :
~ ( ord_less @ nat @ N @ N ) ).
% less_not_refl
thf(fact_81_nat__neq__iff,axiom,
! [M3: nat,N: nat] :
( ( M3 != N )
= ( ( ord_less @ nat @ M3 @ N )
| ( ord_less @ nat @ N @ M3 ) ) ) ).
% nat_neq_iff
thf(fact_82_diff__commute,axiom,
! [I: nat,J3: nat,K2: nat] :
( ( minus_minus @ nat @ ( minus_minus @ nat @ I @ J3 ) @ K2 )
= ( minus_minus @ nat @ ( minus_minus @ nat @ I @ K2 ) @ J3 ) ) ).
% diff_commute
thf(fact_83_Module_Osmodule__ideal__coeff__is__Submodule,axiom,
! [A: $tType,C: $tType,D: $tType,B: $tType,M5: carrier_ext @ A @ ( aGroup_ext @ A @ ( module_ext @ A @ B @ C ) ),R2: carrier_ext @ B @ ( aGroup_ext @ B @ ( ring_ext @ B @ D ) ),A3: set @ B] :
( ( module @ A @ B @ C @ D @ M5 @ R2 )
=> ( ( ideal @ B @ D @ R2 @ A3 )
=> ( submodule @ B @ D @ A @ C @ R2 @ M5 @ ( smodule_ideal_coeff @ B @ D @ A @ C @ R2 @ M5 @ A3 ) ) ) ) ).
% Module.smodule_ideal_coeff_is_Submodule
thf(fact_84_Module_Oquotient__of__submodules__is__ideal,axiom,
! [D: $tType,C: $tType,B: $tType,A: $tType,M5: carrier_ext @ A @ ( aGroup_ext @ A @ ( module_ext @ A @ B @ C ) ),R2: carrier_ext @ B @ ( aGroup_ext @ B @ ( ring_ext @ B @ D ) ),P: set @ A,Q: set @ A] :
( ( module @ A @ B @ C @ D @ M5 @ R2 )
=> ( ( submodule @ B @ D @ A @ C @ R2 @ M5 @ P )
=> ( ( submodule @ B @ D @ A @ C @ R2 @ M5 @ Q )
=> ( ideal @ B @ D @ R2 @ ( quotie135653060odules @ B @ D @ A @ C @ R2 @ M5 @ P @ Q ) ) ) ) ) ).
% Module.quotient_of_submodules_is_ideal
thf(fact_85_transitive__stepwise__le,axiom,
! [M3: nat,N: nat,R2: nat > nat > $o] :
( ( ord_less_eq @ nat @ M3 @ N )
=> ( ! [X2: nat] : ( R2 @ X2 @ X2 )
=> ( ! [X2: nat,Y3: nat,Z2: nat] :
( ( R2 @ X2 @ Y3 )
=> ( ( R2 @ Y3 @ Z2 )
=> ( R2 @ X2 @ Z2 ) ) )
=> ( ! [N2: nat] : ( R2 @ N2 @ ( suc @ N2 ) )
=> ( R2 @ M3 @ N ) ) ) ) ) ).
% transitive_stepwise_le
thf(fact_86_nat__induct__at__least,axiom,
! [M3: nat,N: nat,P: nat > $o] :
( ( ord_less_eq @ nat @ M3 @ N )
=> ( ( P @ M3 )
=> ( ! [N2: nat] :
( ( ord_less_eq @ nat @ M3 @ N2 )
=> ( ( P @ N2 )
=> ( P @ ( suc @ N2 ) ) ) )
=> ( P @ N ) ) ) ) ).
% nat_induct_at_least
thf(fact_87_full__nat__induct,axiom,
! [P: nat > $o,N: nat] :
( ! [N2: nat] :
( ! [M4: nat] :
( ( ord_less_eq @ nat @ ( suc @ M4 ) @ N2 )
=> ( P @ M4 ) )
=> ( P @ N2 ) )
=> ( P @ N ) ) ).
% full_nat_induct
thf(fact_88_not__less__eq__eq,axiom,
! [M3: nat,N: nat] :
( ( ~ ( ord_less_eq @ nat @ M3 @ N ) )
= ( ord_less_eq @ nat @ ( suc @ N ) @ M3 ) ) ).
% not_less_eq_eq
thf(fact_89_Suc__n__not__le__n,axiom,
! [N: nat] :
~ ( ord_less_eq @ nat @ ( suc @ N ) @ N ) ).
% Suc_n_not_le_n
thf(fact_90_le__Suc__eq,axiom,
! [M3: nat,N: nat] :
( ( ord_less_eq @ nat @ M3 @ ( suc @ N ) )
= ( ( ord_less_eq @ nat @ M3 @ N )
| ( M3
= ( suc @ N ) ) ) ) ).
% le_Suc_eq
thf(fact_91_Suc__le__D,axiom,
! [N: nat,M6: nat] :
( ( ord_less_eq @ nat @ ( suc @ N ) @ M6 )
=> ? [M7: nat] :
( M6
= ( suc @ M7 ) ) ) ).
% Suc_le_D
thf(fact_92_le__SucI,axiom,
! [M3: nat,N: nat] :
( ( ord_less_eq @ nat @ M3 @ N )
=> ( ord_less_eq @ nat @ M3 @ ( suc @ N ) ) ) ).
% le_SucI
thf(fact_93_le__SucE,axiom,
! [M3: nat,N: nat] :
( ( ord_less_eq @ nat @ M3 @ ( suc @ N ) )
=> ( ~ ( ord_less_eq @ nat @ M3 @ N )
=> ( M3
= ( suc @ N ) ) ) ) ).
% le_SucE
thf(fact_94_Suc__leD,axiom,
! [M3: nat,N: nat] :
( ( ord_less_eq @ nat @ ( suc @ M3 ) @ N )
=> ( ord_less_eq @ nat @ M3 @ N ) ) ).
% Suc_leD
thf(fact_95_bot__nat__0_Oextremum__uniqueI,axiom,
! [A2: nat] :
( ( ord_less_eq @ nat @ A2 @ ( zero_zero @ nat ) )
=> ( A2
= ( zero_zero @ nat ) ) ) ).
% bot_nat_0.extremum_uniqueI
thf(fact_96_bot__nat__0_Oextremum__unique,axiom,
! [A2: nat] :
( ( ord_less_eq @ nat @ A2 @ ( zero_zero @ nat ) )
= ( A2
= ( zero_zero @ nat ) ) ) ).
% bot_nat_0.extremum_unique
thf(fact_97_le__0__eq,axiom,
! [N: nat] :
( ( ord_less_eq @ nat @ N @ ( zero_zero @ nat ) )
= ( N
= ( zero_zero @ nat ) ) ) ).
% le_0_eq
thf(fact_98_less__eq__nat_Osimps_I1_J,axiom,
! [N: nat] : ( ord_less_eq @ nat @ ( zero_zero @ nat ) @ N ) ).
% less_eq_nat.simps(1)
thf(fact_99_not0__implies__Suc,axiom,
! [N: nat] :
( ( N
!= ( zero_zero @ nat ) )
=> ? [M7: nat] :
( N
= ( suc @ M7 ) ) ) ).
% not0_implies_Suc
thf(fact_100_old_Onat_Oinducts,axiom,
! [P: nat > $o,Nat: nat] :
( ( P @ ( zero_zero @ nat ) )
=> ( ! [Nat3: nat] :
( ( P @ Nat3 )
=> ( P @ ( suc @ Nat3 ) ) )
=> ( P @ Nat ) ) ) ).
% old.nat.inducts
thf(fact_101_old_Onat_Oexhaust,axiom,
! [Y4: nat] :
( ( Y4
!= ( zero_zero @ nat ) )
=> ~ ! [Nat3: nat] :
( Y4
!= ( suc @ Nat3 ) ) ) ).
% old.nat.exhaust
thf(fact_102_Zero__not__Suc,axiom,
! [M3: nat] :
( ( zero_zero @ nat )
!= ( suc @ M3 ) ) ).
% Zero_not_Suc
thf(fact_103_Zero__neq__Suc,axiom,
! [M3: nat] :
( ( zero_zero @ nat )
!= ( suc @ M3 ) ) ).
% Zero_neq_Suc
thf(fact_104_Suc__neq__Zero,axiom,
! [M3: nat] :
( ( suc @ M3 )
!= ( zero_zero @ nat ) ) ).
% Suc_neq_Zero
thf(fact_105_zero__induct,axiom,
! [P: nat > $o,K2: nat] :
( ( P @ K2 )
=> ( ! [N2: nat] :
( ( P @ ( suc @ N2 ) )
=> ( P @ N2 ) )
=> ( P @ ( zero_zero @ nat ) ) ) ) ).
% zero_induct
thf(fact_106_diff__induct,axiom,
! [P: nat > nat > $o,M3: nat,N: nat] :
( ! [X2: nat] : ( P @ X2 @ ( zero_zero @ nat ) )
=> ( ! [Y3: nat] : ( P @ ( zero_zero @ nat ) @ ( suc @ Y3 ) )
=> ( ! [X2: nat,Y3: nat] :
( ( P @ X2 @ Y3 )
=> ( P @ ( suc @ X2 ) @ ( suc @ Y3 ) ) )
=> ( P @ M3 @ N ) ) ) ) ).
% diff_induct
thf(fact_107_nat__induct,axiom,
! [P: nat > $o,N: nat] :
( ( P @ ( zero_zero @ nat ) )
=> ( ! [N2: nat] :
( ( P @ N2 )
=> ( P @ ( suc @ N2 ) ) )
=> ( P @ N ) ) ) ).
% nat_induct
thf(fact_108_nat_OdiscI,axiom,
! [Nat: nat,X22: nat] :
( ( Nat
= ( suc @ X22 ) )
=> ( Nat
!= ( zero_zero @ nat ) ) ) ).
% nat.discI
thf(fact_109_old_Onat_Odistinct_I1_J,axiom,
! [Nat2: nat] :
( ( zero_zero @ nat )
!= ( suc @ Nat2 ) ) ).
% old.nat.distinct(1)
thf(fact_110_old_Onat_Odistinct_I2_J,axiom,
! [Nat4: nat] :
( ( suc @ Nat4 )
!= ( zero_zero @ nat ) ) ).
% old.nat.distinct(2)
thf(fact_111_nat_Odistinct_I1_J,axiom,
! [X22: nat] :
( ( zero_zero @ nat )
!= ( suc @ X22 ) ) ).
% nat.distinct(1)
thf(fact_112_less__mono__imp__le__mono,axiom,
! [F: nat > nat,I: nat,J3: nat] :
( ! [I2: nat,J4: nat] :
( ( ord_less @ nat @ I2 @ J4 )
=> ( ord_less @ nat @ ( F @ I2 ) @ ( F @ J4 ) ) )
=> ( ( ord_less_eq @ nat @ I @ J3 )
=> ( ord_less_eq @ nat @ ( F @ I ) @ ( F @ J3 ) ) ) ) ).
% less_mono_imp_le_mono
thf(fact_113_le__neq__implies__less,axiom,
! [M3: nat,N: nat] :
( ( ord_less_eq @ nat @ M3 @ N )
=> ( ( M3 != N )
=> ( ord_less @ nat @ M3 @ N ) ) ) ).
% le_neq_implies_less
thf(fact_114_less__or__eq__imp__le,axiom,
! [M3: nat,N: nat] :
( ( ( ord_less @ nat @ M3 @ N )
| ( M3 = N ) )
=> ( ord_less_eq @ nat @ M3 @ N ) ) ).
% less_or_eq_imp_le
thf(fact_115_le__eq__less__or__eq,axiom,
( ( ord_less_eq @ nat )
= ( ^ [M8: nat,N4: nat] :
( ( ord_less @ nat @ M8 @ N4 )
| ( M8 = N4 ) ) ) ) ).
% le_eq_less_or_eq
thf(fact_116_less__imp__le__nat,axiom,
! [M3: nat,N: nat] :
( ( ord_less @ nat @ M3 @ N )
=> ( ord_less_eq @ nat @ M3 @ N ) ) ).
% less_imp_le_nat
thf(fact_117_nat__less__le,axiom,
( ( ord_less @ nat )
= ( ^ [M8: nat,N4: nat] :
( ( ord_less_eq @ nat @ M8 @ N4 )
& ( M8 != N4 ) ) ) ) ).
% nat_less_le
thf(fact_118_not__less__less__Suc__eq,axiom,
! [N: nat,M3: nat] :
( ~ ( ord_less @ nat @ N @ M3 )
=> ( ( ord_less @ nat @ N @ ( suc @ M3 ) )
= ( N = M3 ) ) ) ).
% not_less_less_Suc_eq
thf(fact_119_strict__inc__induct,axiom,
! [I: nat,J3: nat,P: nat > $o] :
( ( ord_less @ nat @ I @ J3 )
=> ( ! [I2: nat] :
( ( J3
= ( suc @ I2 ) )
=> ( P @ I2 ) )
=> ( ! [I2: nat] :
( ( ord_less @ nat @ I2 @ J3 )
=> ( ( P @ ( suc @ I2 ) )
=> ( P @ I2 ) ) )
=> ( P @ I ) ) ) ) ).
% strict_inc_induct
thf(fact_120_less__Suc__induct,axiom,
! [I: nat,J3: nat,P: nat > nat > $o] :
( ( ord_less @ nat @ I @ J3 )
=> ( ! [I2: nat] : ( P @ I2 @ ( suc @ I2 ) )
=> ( ! [I2: nat,J4: nat,K3: nat] :
( ( ord_less @ nat @ I2 @ J4 )
=> ( ( ord_less @ nat @ J4 @ K3 )
=> ( ( P @ I2 @ J4 )
=> ( ( P @ J4 @ K3 )
=> ( P @ I2 @ K3 ) ) ) ) )
=> ( P @ I @ J3 ) ) ) ) ).
% less_Suc_induct
thf(fact_121_less__trans__Suc,axiom,
! [I: nat,J3: nat,K2: nat] :
( ( ord_less @ nat @ I @ J3 )
=> ( ( ord_less @ nat @ J3 @ K2 )
=> ( ord_less @ nat @ ( suc @ I ) @ K2 ) ) ) ).
% less_trans_Suc
thf(fact_122_Suc__less__SucD,axiom,
! [M3: nat,N: nat] :
( ( ord_less @ nat @ ( suc @ M3 ) @ ( suc @ N ) )
=> ( ord_less @ nat @ M3 @ N ) ) ).
% Suc_less_SucD
thf(fact_123_less__antisym,axiom,
! [N: nat,M3: nat] :
( ~ ( ord_less @ nat @ N @ M3 )
=> ( ( ord_less @ nat @ N @ ( suc @ M3 ) )
=> ( M3 = N ) ) ) ).
% less_antisym
thf(fact_124_Suc__less__eq2,axiom,
! [N: nat,M3: nat] :
( ( ord_less @ nat @ ( suc @ N ) @ M3 )
= ( ? [M9: nat] :
( ( M3
= ( suc @ M9 ) )
& ( ord_less @ nat @ N @ M9 ) ) ) ) ).
% Suc_less_eq2
thf(fact_125_All__less__Suc,axiom,
! [N: nat,P: nat > $o] :
( ( ! [I3: nat] :
( ( ord_less @ nat @ I3 @ ( suc @ N ) )
=> ( P @ I3 ) ) )
= ( ( P @ N )
& ! [I3: nat] :
( ( ord_less @ nat @ I3 @ N )
=> ( P @ I3 ) ) ) ) ).
% All_less_Suc
thf(fact_126_not__less__eq,axiom,
! [M3: nat,N: nat] :
( ( ~ ( ord_less @ nat @ M3 @ N ) )
= ( ord_less @ nat @ N @ ( suc @ M3 ) ) ) ).
% not_less_eq
thf(fact_127_less__Suc__eq,axiom,
! [M3: nat,N: nat] :
( ( ord_less @ nat @ M3 @ ( suc @ N ) )
= ( ( ord_less @ nat @ M3 @ N )
| ( M3 = N ) ) ) ).
% less_Suc_eq
thf(fact_128_Ex__less__Suc,axiom,
! [N: nat,P: nat > $o] :
( ( ? [I3: nat] :
( ( ord_less @ nat @ I3 @ ( suc @ N ) )
& ( P @ I3 ) ) )
= ( ( P @ N )
| ? [I3: nat] :
( ( ord_less @ nat @ I3 @ N )
& ( P @ I3 ) ) ) ) ).
% Ex_less_Suc
thf(fact_129_less__SucI,axiom,
! [M3: nat,N: nat] :
( ( ord_less @ nat @ M3 @ N )
=> ( ord_less @ nat @ M3 @ ( suc @ N ) ) ) ).
% less_SucI
thf(fact_130_less__SucE,axiom,
! [M3: nat,N: nat] :
( ( ord_less @ nat @ M3 @ ( suc @ N ) )
=> ( ~ ( ord_less @ nat @ M3 @ N )
=> ( M3 = N ) ) ) ).
% less_SucE
thf(fact_131_Suc__lessI,axiom,
! [M3: nat,N: nat] :
( ( ord_less @ nat @ M3 @ N )
=> ( ( ( suc @ M3 )
!= N )
=> ( ord_less @ nat @ ( suc @ M3 ) @ N ) ) ) ).
% Suc_lessI
thf(fact_132_Suc__lessE,axiom,
! [I: nat,K2: nat] :
( ( ord_less @ nat @ ( suc @ I ) @ K2 )
=> ~ ! [J4: nat] :
( ( ord_less @ nat @ I @ J4 )
=> ( K2
!= ( suc @ J4 ) ) ) ) ).
% Suc_lessE
thf(fact_133_Suc__lessD,axiom,
! [M3: nat,N: nat] :
( ( ord_less @ nat @ ( suc @ M3 ) @ N )
=> ( ord_less @ nat @ M3 @ N ) ) ).
% Suc_lessD
thf(fact_134_Nat_OlessE,axiom,
! [I: nat,K2: nat] :
( ( ord_less @ nat @ I @ K2 )
=> ( ( K2
!= ( suc @ I ) )
=> ~ ! [J4: nat] :
( ( ord_less @ nat @ I @ J4 )
=> ( K2
!= ( suc @ J4 ) ) ) ) ) ).
% Nat.lessE
thf(fact_135_infinite__descent0__measure,axiom,
! [A: $tType,V: A > nat,P: A > $o,X: A] :
( ! [X2: A] :
( ( ( V @ X2 )
= ( zero_zero @ nat ) )
=> ( P @ X2 ) )
=> ( ! [X2: A] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ ( V @ X2 ) )
=> ( ~ ( P @ X2 )
=> ? [Y: A] :
( ( ord_less @ nat @ ( V @ Y ) @ ( V @ X2 ) )
& ~ ( P @ Y ) ) ) )
=> ( P @ X ) ) ) ).
% infinite_descent0_measure
thf(fact_136_bot__nat__0_Oextremum__strict,axiom,
! [A2: nat] :
~ ( ord_less @ nat @ A2 @ ( zero_zero @ nat ) ) ).
% bot_nat_0.extremum_strict
thf(fact_137_infinite__descent0,axiom,
! [P: nat > $o,N: nat] :
( ( P @ ( zero_zero @ nat ) )
=> ( ! [N2: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 )
=> ( ~ ( P @ N2 )
=> ? [M4: nat] :
( ( ord_less @ nat @ M4 @ N2 )
& ~ ( P @ M4 ) ) ) )
=> ( P @ N ) ) ) ).
% infinite_descent0
thf(fact_138_gr__implies__not0,axiom,
! [M3: nat,N: nat] :
( ( ord_less @ nat @ M3 @ N )
=> ( N
!= ( zero_zero @ nat ) ) ) ).
% gr_implies_not0
thf(fact_139_less__zeroE,axiom,
! [N: nat] :
~ ( ord_less @ nat @ N @ ( zero_zero @ nat ) ) ).
% less_zeroE
thf(fact_140_not__less0,axiom,
! [N: nat] :
~ ( ord_less @ nat @ N @ ( zero_zero @ nat ) ) ).
% not_less0
thf(fact_141_not__gr0,axiom,
! [N: nat] :
( ( ~ ( ord_less @ nat @ ( zero_zero @ nat ) @ N ) )
= ( N
= ( zero_zero @ nat ) ) ) ).
% not_gr0
thf(fact_142_gr0I,axiom,
! [N: nat] :
( ( N
!= ( zero_zero @ nat ) )
=> ( ord_less @ nat @ ( zero_zero @ nat ) @ N ) ) ).
% gr0I
thf(fact_143_diff__le__mono2,axiom,
! [M3: nat,N: nat,L2: nat] :
( ( ord_less_eq @ nat @ M3 @ N )
=> ( ord_less_eq @ nat @ ( minus_minus @ nat @ L2 @ N ) @ ( minus_minus @ nat @ L2 @ M3 ) ) ) ).
% diff_le_mono2
thf(fact_144_le__diff__iff_H,axiom,
! [A2: nat,C2: nat,B2: nat] :
( ( ord_less_eq @ nat @ A2 @ C2 )
=> ( ( ord_less_eq @ nat @ B2 @ C2 )
=> ( ( ord_less_eq @ nat @ ( minus_minus @ nat @ C2 @ A2 ) @ ( minus_minus @ nat @ C2 @ B2 ) )
= ( ord_less_eq @ nat @ B2 @ A2 ) ) ) ) ).
% le_diff_iff'
thf(fact_145_diff__le__self,axiom,
! [M3: nat,N: nat] : ( ord_less_eq @ nat @ ( minus_minus @ nat @ M3 @ N ) @ M3 ) ).
% diff_le_self
thf(fact_146_diff__le__mono,axiom,
! [M3: nat,N: nat,L2: nat] :
( ( ord_less_eq @ nat @ M3 @ N )
=> ( ord_less_eq @ nat @ ( minus_minus @ nat @ M3 @ L2 ) @ ( minus_minus @ nat @ N @ L2 ) ) ) ).
% diff_le_mono
thf(fact_147_Nat_Odiff__diff__eq,axiom,
! [K2: nat,M3: nat,N: nat] :
( ( ord_less_eq @ nat @ K2 @ M3 )
=> ( ( ord_less_eq @ nat @ K2 @ N )
=> ( ( minus_minus @ nat @ ( minus_minus @ nat @ M3 @ K2 ) @ ( minus_minus @ nat @ N @ K2 ) )
= ( minus_minus @ nat @ M3 @ N ) ) ) ) ).
% Nat.diff_diff_eq
thf(fact_148_le__diff__iff,axiom,
! [K2: nat,M3: nat,N: nat] :
( ( ord_less_eq @ nat @ K2 @ M3 )
=> ( ( ord_less_eq @ nat @ K2 @ N )
=> ( ( ord_less_eq @ nat @ ( minus_minus @ nat @ M3 @ K2 ) @ ( minus_minus @ nat @ N @ K2 ) )
= ( ord_less_eq @ nat @ M3 @ N ) ) ) ) ).
% le_diff_iff
thf(fact_149_eq__diff__iff,axiom,
! [K2: nat,M3: nat,N: nat] :
( ( ord_less_eq @ nat @ K2 @ M3 )
=> ( ( ord_less_eq @ nat @ K2 @ N )
=> ( ( ( minus_minus @ nat @ M3 @ K2 )
= ( minus_minus @ nat @ N @ K2 ) )
= ( M3 = N ) ) ) ) ).
% eq_diff_iff
thf(fact_150_zero__induct__lemma,axiom,
! [P: nat > $o,K2: nat,I: nat] :
( ( P @ K2 )
=> ( ! [N2: nat] :
( ( P @ ( suc @ N2 ) )
=> ( P @ N2 ) )
=> ( P @ ( minus_minus @ nat @ K2 @ I ) ) ) ) ).
% zero_induct_lemma
thf(fact_151_minus__nat_Odiff__0,axiom,
! [M3: nat] :
( ( minus_minus @ nat @ M3 @ ( zero_zero @ nat ) )
= M3 ) ).
% minus_nat.diff_0
thf(fact_152_diffs0__imp__equal,axiom,
! [M3: nat,N: nat] :
( ( ( minus_minus @ nat @ M3 @ N )
= ( zero_zero @ nat ) )
=> ( ( ( minus_minus @ nat @ N @ M3 )
= ( zero_zero @ nat ) )
=> ( M3 = N ) ) ) ).
% diffs0_imp_equal
thf(fact_153_less__imp__diff__less,axiom,
! [J3: nat,K2: nat,N: nat] :
( ( ord_less @ nat @ J3 @ K2 )
=> ( ord_less @ nat @ ( minus_minus @ nat @ J3 @ N ) @ K2 ) ) ).
% less_imp_diff_less
thf(fact_154_diff__less__mono2,axiom,
! [M3: nat,N: nat,L2: nat] :
( ( ord_less @ nat @ M3 @ N )
=> ( ( ord_less @ nat @ M3 @ L2 )
=> ( ord_less @ nat @ ( minus_minus @ nat @ L2 @ N ) @ ( minus_minus @ nat @ L2 @ M3 ) ) ) ) ).
% diff_less_mono2
thf(fact_155_Module_Osc__Ring,axiom,
! [C: $tType,A: $tType,D: $tType,B: $tType,M5: carrier_ext @ A @ ( aGroup_ext @ A @ ( module_ext @ A @ B @ C ) ),R2: carrier_ext @ B @ ( aGroup_ext @ B @ ( ring_ext @ B @ D ) )] :
( ( module @ A @ B @ C @ D @ M5 @ R2 )
=> ( ring @ B @ D @ R2 ) ) ).
% Module.sc_Ring
thf(fact_156_Module_Omem__smodule__ideal__coeff,axiom,
! [B: $tType,C: $tType,D: $tType,A: $tType,M5: carrier_ext @ A @ ( aGroup_ext @ A @ ( module_ext @ A @ B @ C ) ),R2: carrier_ext @ B @ ( aGroup_ext @ B @ ( ring_ext @ B @ D ) ),A3: set @ B,X: A] :
( ( module @ A @ B @ C @ D @ M5 @ R2 )
=> ( ( ideal @ B @ D @ R2 @ A3 )
=> ( ( member @ A @ X @ ( smodule_ideal_coeff @ B @ D @ A @ C @ R2 @ M5 @ A3 ) )
=> ? [N2: nat,X2: nat > B] :
( ( member @ ( nat > B ) @ X2
@ ( pi @ nat @ B
@ ( collect @ nat
@ ^ [J2: nat] : ( ord_less_eq @ nat @ J2 @ N2 ) )
@ ^ [Uu: nat] : A3 ) )
& ? [Xa: nat > A] :
( ( member @ ( nat > A ) @ Xa
@ ( pi @ nat @ A
@ ( collect @ nat
@ ^ [J2: nat] : ( ord_less_eq @ nat @ J2 @ N2 ) )
@ ^ [Uu: nat] : ( carrier @ A @ ( aGroup_ext @ A @ ( module_ext @ A @ B @ C ) ) @ M5 ) ) )
& ( X
= ( l_comb @ B @ D @ A @ C @ R2 @ M5 @ N2 @ X2 @ Xa ) ) ) ) ) ) ) ).
% Module.mem_smodule_ideal_coeff
thf(fact_157_lift__Suc__antimono__le,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [F: nat > A,N: nat,N5: nat] :
( ! [N2: nat] : ( ord_less_eq @ A @ ( F @ ( suc @ N2 ) ) @ ( F @ N2 ) )
=> ( ( ord_less_eq @ nat @ N @ N5 )
=> ( ord_less_eq @ A @ ( F @ N5 ) @ ( F @ N ) ) ) ) ) ).
% lift_Suc_antimono_le
thf(fact_158_lift__Suc__mono__le,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [F: nat > A,N: nat,N5: nat] :
( ! [N2: nat] : ( ord_less_eq @ A @ ( F @ N2 ) @ ( F @ ( suc @ N2 ) ) )
=> ( ( ord_less_eq @ nat @ N @ N5 )
=> ( ord_less_eq @ A @ ( F @ N ) @ ( F @ N5 ) ) ) ) ) ).
% lift_Suc_mono_le
thf(fact_159_lift__Suc__mono__less__iff,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [F: nat > A,N: nat,M3: nat] :
( ! [N2: nat] : ( ord_less @ A @ ( F @ N2 ) @ ( F @ ( suc @ N2 ) ) )
=> ( ( ord_less @ A @ ( F @ N ) @ ( F @ M3 ) )
= ( ord_less @ nat @ N @ M3 ) ) ) ) ).
% lift_Suc_mono_less_iff
thf(fact_160_lift__Suc__mono__less,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [F: nat > A,N: nat,N5: nat] :
( ! [N2: nat] : ( ord_less @ A @ ( F @ N2 ) @ ( F @ ( suc @ N2 ) ) )
=> ( ( ord_less @ nat @ N @ N5 )
=> ( ord_less @ A @ ( F @ N ) @ ( F @ N5 ) ) ) ) ) ).
% lift_Suc_mono_less
thf(fact_161_le__imp__less__Suc,axiom,
! [M3: nat,N: nat] :
( ( ord_less_eq @ nat @ M3 @ N )
=> ( ord_less @ nat @ M3 @ ( suc @ N ) ) ) ).
% le_imp_less_Suc
thf(fact_162_less__eq__Suc__le,axiom,
( ( ord_less @ nat )
= ( ^ [N4: nat] : ( ord_less_eq @ nat @ ( suc @ N4 ) ) ) ) ).
% less_eq_Suc_le
thf(fact_163_less__Suc__eq__le,axiom,
! [M3: nat,N: nat] :
( ( ord_less @ nat @ M3 @ ( suc @ N ) )
= ( ord_less_eq @ nat @ M3 @ N ) ) ).
% less_Suc_eq_le
thf(fact_164_le__less__Suc__eq,axiom,
! [M3: nat,N: nat] :
( ( ord_less_eq @ nat @ M3 @ N )
=> ( ( ord_less @ nat @ N @ ( suc @ M3 ) )
= ( N = M3 ) ) ) ).
% le_less_Suc_eq
thf(fact_165_Suc__le__lessD,axiom,
! [M3: nat,N: nat] :
( ( ord_less_eq @ nat @ ( suc @ M3 ) @ N )
=> ( ord_less @ nat @ M3 @ N ) ) ).
% Suc_le_lessD
thf(fact_166_inc__induct,axiom,
! [I: nat,J3: nat,P: nat > $o] :
( ( ord_less_eq @ nat @ I @ J3 )
=> ( ( P @ J3 )
=> ( ! [N2: nat] :
( ( ord_less_eq @ nat @ I @ N2 )
=> ( ( ord_less @ nat @ N2 @ J3 )
=> ( ( P @ ( suc @ N2 ) )
=> ( P @ N2 ) ) ) )
=> ( P @ I ) ) ) ) ).
% inc_induct
thf(fact_167_dec__induct,axiom,
! [I: nat,J3: nat,P: nat > $o] :
( ( ord_less_eq @ nat @ I @ J3 )
=> ( ( P @ I )
=> ( ! [N2: nat] :
( ( ord_less_eq @ nat @ I @ N2 )
=> ( ( ord_less @ nat @ N2 @ J3 )
=> ( ( P @ N2 )
=> ( P @ ( suc @ N2 ) ) ) ) )
=> ( P @ J3 ) ) ) ) ).
% dec_induct
thf(fact_168_Suc__le__eq,axiom,
! [M3: nat,N: nat] :
( ( ord_less_eq @ nat @ ( suc @ M3 ) @ N )
= ( ord_less @ nat @ M3 @ N ) ) ).
% Suc_le_eq
thf(fact_169_Suc__leI,axiom,
! [M3: nat,N: nat] :
( ( ord_less @ nat @ M3 @ N )
=> ( ord_less_eq @ nat @ ( suc @ M3 ) @ N ) ) ).
% Suc_leI
thf(fact_170_ex__least__nat__le,axiom,
! [P: nat > $o,N: nat] :
( ( P @ N )
=> ( ~ ( P @ ( zero_zero @ nat ) )
=> ? [K3: nat] :
( ( ord_less_eq @ nat @ K3 @ N )
& ! [I4: nat] :
( ( ord_less @ nat @ I4 @ K3 )
=> ~ ( P @ I4 ) )
& ( P @ K3 ) ) ) ) ).
% ex_least_nat_le
thf(fact_171_less__Suc__eq__0__disj,axiom,
! [M3: nat,N: nat] :
( ( ord_less @ nat @ M3 @ ( suc @ N ) )
= ( ( M3
= ( zero_zero @ nat ) )
| ? [J2: nat] :
( ( M3
= ( suc @ J2 ) )
& ( ord_less @ nat @ J2 @ N ) ) ) ) ).
% less_Suc_eq_0_disj
thf(fact_172_gr0__implies__Suc,axiom,
! [N: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ? [M7: nat] :
( N
= ( suc @ M7 ) ) ) ).
% gr0_implies_Suc
thf(fact_173_All__less__Suc2,axiom,
! [N: nat,P: nat > $o] :
( ( ! [I3: nat] :
( ( ord_less @ nat @ I3 @ ( suc @ N ) )
=> ( P @ I3 ) ) )
= ( ( P @ ( zero_zero @ nat ) )
& ! [I3: nat] :
( ( ord_less @ nat @ I3 @ N )
=> ( P @ ( suc @ I3 ) ) ) ) ) ).
% All_less_Suc2
thf(fact_174_gr0__conv__Suc,axiom,
! [N: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
= ( ? [M8: nat] :
( N
= ( suc @ M8 ) ) ) ) ).
% gr0_conv_Suc
thf(fact_175_Ex__less__Suc2,axiom,
! [N: nat,P: nat > $o] :
( ( ? [I3: nat] :
( ( ord_less @ nat @ I3 @ ( suc @ N ) )
& ( P @ I3 ) ) )
= ( ( P @ ( zero_zero @ nat ) )
| ? [I3: nat] :
( ( ord_less @ nat @ I3 @ N )
& ( P @ ( suc @ I3 ) ) ) ) ) ).
% Ex_less_Suc2
thf(fact_176_Suc__diff__le,axiom,
! [N: nat,M3: nat] :
( ( ord_less_eq @ nat @ N @ M3 )
=> ( ( minus_minus @ nat @ ( suc @ M3 ) @ N )
= ( suc @ ( minus_minus @ nat @ M3 @ N ) ) ) ) ).
% Suc_diff_le
thf(fact_177_diff__less__mono,axiom,
! [A2: nat,B2: nat,C2: nat] :
( ( ord_less @ nat @ A2 @ B2 )
=> ( ( ord_less_eq @ nat @ C2 @ A2 )
=> ( ord_less @ nat @ ( minus_minus @ nat @ A2 @ C2 ) @ ( minus_minus @ nat @ B2 @ C2 ) ) ) ) ).
% diff_less_mono
thf(fact_178_less__diff__iff,axiom,
! [K2: nat,M3: nat,N: nat] :
( ( ord_less_eq @ nat @ K2 @ M3 )
=> ( ( ord_less_eq @ nat @ K2 @ N )
=> ( ( ord_less @ nat @ ( minus_minus @ nat @ M3 @ K2 ) @ ( minus_minus @ nat @ N @ K2 ) )
= ( ord_less @ nat @ M3 @ N ) ) ) ) ).
% less_diff_iff
thf(fact_179_diff__less__Suc,axiom,
! [M3: nat,N: nat] : ( ord_less @ nat @ ( minus_minus @ nat @ M3 @ N ) @ ( suc @ M3 ) ) ).
% diff_less_Suc
thf(fact_180_Suc__diff__Suc,axiom,
! [N: nat,M3: nat] :
( ( ord_less @ nat @ N @ M3 )
=> ( ( suc @ ( minus_minus @ nat @ M3 @ ( suc @ N ) ) )
= ( minus_minus @ nat @ M3 @ N ) ) ) ).
% Suc_diff_Suc
thf(fact_181_diff__less,axiom,
! [N: nat,M3: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( ord_less @ nat @ ( zero_zero @ nat ) @ M3 )
=> ( ord_less @ nat @ ( minus_minus @ nat @ M3 @ N ) @ M3 ) ) ) ).
% diff_less
thf(fact_182_Module_Osubmodule__subset1,axiom,
! [D: $tType,C: $tType,B: $tType,A: $tType,M5: carrier_ext @ A @ ( aGroup_ext @ A @ ( module_ext @ A @ B @ C ) ),R2: carrier_ext @ B @ ( aGroup_ext @ B @ ( ring_ext @ B @ D ) ),H: set @ A,H2: A] :
( ( module @ A @ B @ C @ D @ M5 @ R2 )
=> ( ( submodule @ B @ D @ A @ C @ R2 @ M5 @ H )
=> ( ( member @ A @ H2 @ H )
=> ( member @ A @ H2 @ ( carrier @ A @ ( aGroup_ext @ A @ ( module_ext @ A @ B @ C ) ) @ M5 ) ) ) ) ) ).
% Module.submodule_subset1
thf(fact_183_Module_Osubmodule__whole,axiom,
! [D: $tType,C: $tType,B: $tType,A: $tType,M5: carrier_ext @ A @ ( aGroup_ext @ A @ ( module_ext @ A @ B @ C ) ),R2: carrier_ext @ B @ ( aGroup_ext @ B @ ( ring_ext @ B @ D ) )] :
( ( module @ A @ B @ C @ D @ M5 @ R2 )
=> ( submodule @ B @ D @ A @ C @ R2 @ M5 @ ( carrier @ A @ ( aGroup_ext @ A @ ( module_ext @ A @ B @ C ) ) @ M5 ) ) ) ).
% Module.submodule_whole
thf(fact_184_ex__least__nat__less,axiom,
! [P: nat > $o,N: nat] :
( ( P @ N )
=> ( ~ ( P @ ( zero_zero @ nat ) )
=> ? [K3: nat] :
( ( ord_less @ nat @ K3 @ N )
& ! [I4: nat] :
( ( ord_less_eq @ nat @ I4 @ K3 )
=> ~ ( P @ I4 ) )
& ( P @ ( suc @ K3 ) ) ) ) ) ).
% ex_least_nat_less
thf(fact_185_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_186_Module_Osubmodule__subset,axiom,
! [D: $tType,C: $tType,B: $tType,A: $tType,M5: carrier_ext @ A @ ( aGroup_ext @ A @ ( module_ext @ A @ B @ C ) ),R2: carrier_ext @ B @ ( aGroup_ext @ B @ ( ring_ext @ B @ D ) ),H: set @ A] :
( ( module @ A @ B @ C @ D @ M5 @ R2 )
=> ( ( submodule @ B @ D @ A @ C @ R2 @ M5 @ H )
=> ( ord_less_eq @ ( set @ A ) @ H @ ( carrier @ A @ ( aGroup_ext @ A @ ( module_ext @ A @ B @ C ) ) @ M5 ) ) ) ) ).
% Module.submodule_subset
thf(fact_187_Module_Ofree__generator__sub,axiom,
! [D: $tType,C: $tType,B: $tType,A: $tType,M5: carrier_ext @ A @ ( aGroup_ext @ A @ ( module_ext @ A @ B @ C ) ),R2: carrier_ext @ B @ ( aGroup_ext @ B @ ( ring_ext @ B @ D ) ),H: set @ A] :
( ( module @ A @ B @ C @ D @ M5 @ R2 )
=> ( ( free_generator @ B @ D @ A @ C @ R2 @ M5 @ H )
=> ( ord_less_eq @ ( set @ A ) @ H @ ( carrier @ A @ ( aGroup_ext @ A @ ( module_ext @ A @ B @ C ) ) @ M5 ) ) ) ) ).
% Module.free_generator_sub
thf(fact_188_eSum__in__SubmoduleTr,axiom,
! [H: set @ a,K: set @ a,F: nat > a,N: nat,S: nat > b] :
( ( ord_less_eq @ ( set @ a ) @ H @ ( carrier @ a @ ( aGroup_ext @ a @ ( module_ext @ a @ b @ c ) ) @ m ) )
=> ( ( ord_less_eq @ ( set @ a ) @ K @ H )
=> ( ( ( member @ ( nat > a ) @ F
@ ( pi @ nat @ a
@ ( collect @ nat
@ ^ [J2: nat] : ( ord_less_eq @ nat @ J2 @ N ) )
@ ^ [Uu: nat] : K ) )
& ( member @ ( nat > b ) @ S
@ ( pi @ nat @ b
@ ( collect @ nat
@ ^ [J2: nat] : ( ord_less_eq @ nat @ J2 @ N ) )
@ ^ [Uu: nat] : ( carrier @ b @ ( aGroup_ext @ b @ ( ring_ext @ b @ d ) ) @ r ) ) ) )
=> ( ( l_comb @ b @ d @ a @ product_unit @ r @ ( mdl @ a @ b @ c @ m @ ( algebr833503410le_fgs @ b @ d @ a @ c @ r @ m @ K ) ) @ N @ S @ F )
= ( l_comb @ b @ d @ a @ c @ r @ m @ N @ S @ F ) ) ) ) ) ).
% eSum_in_SubmoduleTr
thf(fact_189_eSum__in__Submodule,axiom,
! [H: set @ a,K: set @ a,F: nat > a,N: nat,S: nat > b] :
( ( ord_less_eq @ ( set @ a ) @ H @ ( carrier @ a @ ( aGroup_ext @ a @ ( module_ext @ a @ b @ c ) ) @ m ) )
=> ( ( ord_less_eq @ ( set @ a ) @ K @ H )
=> ( ( member @ ( nat > a ) @ F
@ ( pi @ nat @ a
@ ( collect @ nat
@ ^ [J2: nat] : ( ord_less_eq @ nat @ J2 @ N ) )
@ ^ [Uu: nat] : K ) )
=> ( ( member @ ( nat > b ) @ S
@ ( pi @ nat @ b
@ ( collect @ nat
@ ^ [J2: nat] : ( ord_less_eq @ nat @ J2 @ N ) )
@ ^ [Uu: nat] : ( carrier @ b @ ( aGroup_ext @ b @ ( ring_ext @ b @ d ) ) @ r ) ) )
=> ( ( l_comb @ b @ d @ a @ product_unit @ r @ ( mdl @ a @ b @ c @ m @ ( algebr833503410le_fgs @ b @ d @ a @ c @ r @ m @ K ) ) @ N @ S @ F )
= ( l_comb @ b @ d @ a @ c @ r @ m @ N @ S @ F ) ) ) ) ) ) ).
% eSum_in_Submodule
thf(fact_190_singleton__insert__inj__eq_H,axiom,
! [A: $tType,A2: A,A3: set @ A,B2: A] :
( ( ( insert @ A @ A2 @ A3 )
= ( insert @ A @ B2 @ ( bot_bot @ ( set @ A ) ) ) )
= ( ( A2 = B2 )
& ( ord_less_eq @ ( set @ A ) @ A3 @ ( insert @ A @ B2 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% singleton_insert_inj_eq'
thf(fact_191_singleton__insert__inj__eq,axiom,
! [A: $tType,B2: A,A2: A,A3: set @ A] :
( ( ( insert @ A @ B2 @ ( bot_bot @ ( set @ A ) ) )
= ( insert @ A @ A2 @ A3 ) )
= ( ( A2 = B2 )
& ( ord_less_eq @ ( set @ A ) @ A3 @ ( insert @ A @ B2 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% singleton_insert_inj_eq
thf(fact_192_diff__gt__0__iff__gt,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [A2: A,B2: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ ( minus_minus @ A @ A2 @ B2 ) )
= ( ord_less @ A @ B2 @ A2 ) ) ) ).
% diff_gt_0_iff_gt
thf(fact_193_diff__ge__0__iff__ge,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [A2: A,B2: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( minus_minus @ A @ A2 @ B2 ) )
= ( ord_less_eq @ A @ B2 @ A2 ) ) ) ).
% diff_ge_0_iff_ge
thf(fact_194_l__span__closed2Tr,axiom,
! [A3: set @ b,H: set @ a,S: nat > b,N: nat,F: nat > a] :
( ( ideal @ b @ d @ r @ A3 )
=> ( ( ord_less_eq @ ( set @ a ) @ H @ ( carrier @ a @ ( aGroup_ext @ a @ ( module_ext @ a @ b @ c ) ) @ m ) )
=> ( ( ( member @ ( nat > b ) @ S
@ ( pi @ nat @ b
@ ( collect @ nat
@ ^ [J2: nat] : ( ord_less_eq @ nat @ J2 @ N ) )
@ ^ [Uu: nat] : A3 ) )
& ( member @ ( nat > a ) @ F
@ ( pi @ nat @ a
@ ( collect @ nat
@ ^ [J2: nat] : ( ord_less_eq @ nat @ J2 @ N ) )
@ ^ [Uu: nat] : ( linear_span @ b @ d @ a @ c @ r @ m @ ( carrier @ b @ ( aGroup_ext @ b @ ( ring_ext @ b @ d ) ) @ r ) @ H ) ) ) )
=> ( member @ a @ ( l_comb @ b @ d @ a @ c @ r @ m @ N @ S @ F ) @ ( linear_span @ b @ d @ a @ c @ r @ m @ A3 @ H ) ) ) ) ) ).
% l_span_closed2Tr
thf(fact_195_l__span__closed2,axiom,
! [A3: set @ b,H: set @ a,S: nat > b,N: nat,F: nat > a] :
( ( ideal @ b @ d @ r @ A3 )
=> ( ( ord_less_eq @ ( set @ a ) @ H @ ( carrier @ a @ ( aGroup_ext @ a @ ( module_ext @ a @ b @ c ) ) @ m ) )
=> ( ( member @ ( nat > b ) @ S
@ ( pi @ nat @ b
@ ( collect @ nat
@ ^ [J2: nat] : ( ord_less_eq @ nat @ J2 @ N ) )
@ ^ [Uu: nat] : A3 ) )
=> ( ( member @ ( nat > a ) @ F
@ ( pi @ nat @ a
@ ( collect @ nat
@ ^ [J2: nat] : ( ord_less_eq @ nat @ J2 @ N ) )
@ ^ [Uu: nat] : ( linear_span @ b @ d @ a @ c @ r @ m @ ( carrier @ b @ ( aGroup_ext @ b @ ( ring_ext @ b @ d ) ) @ r ) @ H ) ) )
=> ( member @ a @ ( l_comb @ b @ d @ a @ c @ r @ m @ N @ S @ F ) @ ( linear_span @ b @ d @ a @ c @ r @ m @ A3 @ H ) ) ) ) ) ) ).
% l_span_closed2
thf(fact_196_linear__comb0__2Tr,axiom,
! [A3: set @ b,S: nat > b,N: nat,M3: nat > a] :
( ( ideal @ b @ d @ r @ A3 )
=> ( ( ( member @ ( nat > b ) @ S
@ ( pi @ nat @ b
@ ( collect @ nat
@ ^ [J2: nat] : ( ord_less_eq @ nat @ J2 @ N ) )
@ ^ [Uu: nat] : A3 ) )
& ( member @ ( nat > a ) @ M3
@ ( pi @ nat @ a
@ ( collect @ nat
@ ^ [J2: nat] : ( ord_less_eq @ nat @ J2 @ N ) )
@ ^ [Uu: nat] : ( insert @ a @ ( zero2 @ a @ ( module_ext @ a @ b @ c ) @ m ) @ ( bot_bot @ ( set @ a ) ) ) ) ) )
=> ( ( l_comb @ b @ d @ a @ c @ r @ m @ N @ S @ M3 )
= ( zero2 @ a @ ( module_ext @ a @ b @ c ) @ m ) ) ) ) ).
% linear_comb0_2Tr
thf(fact_197_module__inc__zero,axiom,
member @ a @ ( zero2 @ a @ ( module_ext @ a @ b @ c ) @ m ) @ ( carrier @ a @ ( aGroup_ext @ a @ ( module_ext @ a @ b @ c ) ) @ m ) ).
% module_inc_zero
thf(fact_198_psubsetI,axiom,
! [A: $tType,A3: set @ A,B3: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A3 @ B3 )
=> ( ( A3 != B3 )
=> ( ord_less @ ( set @ A ) @ A3 @ B3 ) ) ) ).
% psubsetI
thf(fact_199_subsetI,axiom,
! [A: $tType,A3: set @ A,B3: set @ A] :
( ! [X2: A] :
( ( member @ A @ X2 @ A3 )
=> ( member @ A @ X2 @ B3 ) )
=> ( ord_less_eq @ ( set @ A ) @ A3 @ B3 ) ) ).
% subsetI
thf(fact_200_subset__antisym,axiom,
! [A: $tType,A3: set @ A,B3: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A3 @ B3 )
=> ( ( ord_less_eq @ ( set @ A ) @ B3 @ A3 )
=> ( A3 = B3 ) ) ) ).
% subset_antisym
thf(fact_201_Diff__empty,axiom,
! [A: $tType,A3: set @ A] :
( ( minus_minus @ ( set @ A ) @ A3 @ ( bot_bot @ ( set @ A ) ) )
= A3 ) ).
% Diff_empty
thf(fact_202_empty__Diff,axiom,
! [A: $tType,A3: set @ A] :
( ( minus_minus @ ( set @ A ) @ ( bot_bot @ ( set @ A ) ) @ A3 )
= ( bot_bot @ ( set @ A ) ) ) ).
% empty_Diff
thf(fact_203_Diff__cancel,axiom,
! [A: $tType,A3: set @ A] :
( ( minus_minus @ ( set @ A ) @ A3 @ A3 )
= ( bot_bot @ ( set @ A ) ) ) ).
% Diff_cancel
thf(fact_204_empty__iff,axiom,
! [A: $tType,C2: A] :
~ ( member @ A @ C2 @ ( bot_bot @ ( set @ A ) ) ) ).
% empty_iff
thf(fact_205_all__not__in__conv,axiom,
! [A: $tType,A3: set @ A] :
( ( ! [X3: A] :
~ ( member @ A @ X3 @ A3 ) )
= ( A3
= ( bot_bot @ ( set @ A ) ) ) ) ).
% all_not_in_conv
thf(fact_206_Collect__empty__eq,axiom,
! [A: $tType,P: A > $o] :
( ( ( collect @ A @ P )
= ( bot_bot @ ( set @ A ) ) )
= ( ! [X3: A] :
~ ( P @ X3 ) ) ) ).
% Collect_empty_eq
thf(fact_207_empty__Collect__eq,axiom,
! [A: $tType,P: A > $o] :
( ( ( bot_bot @ ( set @ A ) )
= ( collect @ A @ P ) )
= ( ! [X3: A] :
~ ( P @ X3 ) ) ) ).
% empty_Collect_eq
thf(fact_208_Diff__insert0,axiom,
! [A: $tType,X: A,A3: set @ A,B3: set @ A] :
( ~ ( member @ A @ X @ A3 )
=> ( ( minus_minus @ ( set @ A ) @ A3 @ ( insert @ A @ X @ B3 ) )
= ( minus_minus @ ( set @ A ) @ A3 @ B3 ) ) ) ).
% Diff_insert0
thf(fact_209_insert__Diff1,axiom,
! [A: $tType,X: A,B3: set @ A,A3: set @ A] :
( ( member @ A @ X @ B3 )
=> ( ( minus_minus @ ( set @ A ) @ ( insert @ A @ X @ A3 ) @ B3 )
= ( minus_minus @ ( set @ A ) @ A3 @ B3 ) ) ) ).
% insert_Diff1
thf(fact_210_insertCI,axiom,
! [A: $tType,A2: A,B3: set @ A,B2: A] :
( ( ~ ( member @ A @ A2 @ B3 )
=> ( A2 = B2 ) )
=> ( member @ A @ A2 @ ( insert @ A @ B2 @ B3 ) ) ) ).
% insertCI
thf(fact_211_insert__iff,axiom,
! [A: $tType,A2: A,B2: A,A3: set @ A] :
( ( member @ A @ A2 @ ( insert @ A @ B2 @ A3 ) )
= ( ( A2 = B2 )
| ( member @ A @ A2 @ A3 ) ) ) ).
% insert_iff
thf(fact_212_insert__absorb2,axiom,
! [A: $tType,X: A,A3: set @ A] :
( ( insert @ A @ X @ ( insert @ A @ X @ A3 ) )
= ( insert @ A @ X @ A3 ) ) ).
% insert_absorb2
thf(fact_213_submodule__inc__0,axiom,
! [H: set @ a] :
( ( submodule @ b @ d @ a @ c @ r @ m @ H )
=> ( member @ a @ ( zero2 @ a @ ( module_ext @ a @ b @ c ) @ m ) @ H ) ) ).
% submodule_inc_0
thf(fact_214_mdl__carrier,axiom,
! [H: set @ a] :
( ( submodule @ b @ d @ a @ c @ r @ m @ H )
=> ( ( carrier @ a @ ( aGroup_ext @ a @ ( module_ext @ a @ b @ product_unit ) ) @ ( mdl @ a @ b @ c @ m @ H ) )
= H ) ) ).
% mdl_carrier
thf(fact_215_mdl__is__module,axiom,
! [H: set @ a] :
( ( submodule @ b @ d @ a @ c @ r @ m @ H )
=> ( module @ a @ b @ product_unit @ d @ ( mdl @ a @ b @ c @ m @ H ) @ r ) ) ).
% mdl_is_module
thf(fact_216_submodule__of__mdl,axiom,
! [H: set @ a,N3: set @ a] :
( ( submodule @ b @ d @ a @ c @ r @ m @ H )
=> ( ( submodule @ b @ d @ a @ c @ r @ m @ N3 )
=> ( ( ord_less_eq @ ( set @ a ) @ H @ N3 )
=> ( submodule @ b @ d @ a @ product_unit @ r @ ( mdl @ a @ b @ c @ m @ N3 ) @ H ) ) ) ) ).
% submodule_of_mdl
thf(fact_217_h__in__linear__span,axiom,
! [H: set @ a,H2: a] :
( ( ord_less_eq @ ( set @ a ) @ H @ ( carrier @ a @ ( aGroup_ext @ a @ ( module_ext @ a @ b @ c ) ) @ m ) )
=> ( ( member @ a @ H2 @ H )
=> ( member @ a @ H2 @ ( linear_span @ b @ d @ a @ c @ r @ m @ ( carrier @ b @ ( aGroup_ext @ b @ ( ring_ext @ b @ d ) ) @ r ) @ H ) ) ) ) ).
% h_in_linear_span
thf(fact_218_l__span__cont__H,axiom,
! [H: set @ a] :
( ( ord_less_eq @ ( set @ a ) @ H @ ( carrier @ a @ ( aGroup_ext @ a @ ( module_ext @ a @ b @ c ) ) @ m ) )
=> ( ord_less_eq @ ( set @ a ) @ H @ ( linear_span @ b @ d @ a @ c @ r @ m @ ( carrier @ b @ ( aGroup_ext @ b @ ( ring_ext @ b @ d ) ) @ r ) @ H ) ) ) ).
% l_span_cont_H
thf(fact_219_l__span__l__span,axiom,
! [H: set @ a] :
( ( ord_less_eq @ ( set @ a ) @ H @ ( carrier @ a @ ( aGroup_ext @ a @ ( module_ext @ a @ b @ c ) ) @ m ) )
=> ( ( linear_span @ b @ d @ a @ c @ r @ m @ ( carrier @ b @ ( aGroup_ext @ b @ ( ring_ext @ b @ d ) ) @ r ) @ ( linear_span @ b @ d @ a @ c @ r @ m @ ( carrier @ b @ ( aGroup_ext @ b @ ( ring_ext @ b @ d ) ) @ r ) @ H ) )
= ( linear_span @ b @ d @ a @ c @ r @ m @ ( carrier @ b @ ( aGroup_ext @ b @ ( ring_ext @ b @ d ) ) @ r ) @ H ) ) ) ).
% l_span_l_span
thf(fact_220_l__span__gen__mono,axiom,
! [K: set @ a,H: set @ a,A3: set @ b] :
( ( ord_less_eq @ ( set @ a ) @ K @ H )
=> ( ( ord_less_eq @ ( set @ a ) @ H @ ( carrier @ a @ ( aGroup_ext @ a @ ( module_ext @ a @ b @ c ) ) @ m ) )
=> ( ( ideal @ b @ d @ r @ A3 )
=> ( ord_less_eq @ ( set @ a ) @ ( linear_span @ b @ d @ a @ c @ r @ m @ A3 @ K ) @ ( linear_span @ b @ d @ a @ c @ r @ m @ A3 @ H ) ) ) ) ) ).
% l_span_gen_mono
thf(fact_221_linear__span__sub,axiom,
! [A3: set @ b,H: set @ a] :
( ( ideal @ b @ d @ r @ A3 )
=> ( ( ord_less_eq @ ( set @ a ) @ H @ ( carrier @ a @ ( aGroup_ext @ a @ ( module_ext @ a @ b @ c ) ) @ m ) )
=> ( ord_less_eq @ ( set @ a ) @ ( linear_span @ b @ d @ a @ c @ r @ m @ A3 @ H ) @ ( carrier @ a @ ( aGroup_ext @ a @ ( module_ext @ a @ b @ c ) ) @ m ) ) ) ) ).
% linear_span_sub
thf(fact_222_l__span__sub__submodule,axiom,
! [A3: set @ b,N3: set @ a,H: set @ a] :
( ( ideal @ b @ d @ r @ A3 )
=> ( ( submodule @ b @ d @ a @ c @ r @ m @ N3 )
=> ( ( ord_less_eq @ ( set @ a ) @ H @ N3 )
=> ( ord_less_eq @ ( set @ a ) @ ( linear_span @ b @ d @ a @ c @ r @ m @ A3 @ H ) @ N3 ) ) ) ) ).
% l_span_sub_submodule
thf(fact_223_fgs__generator,axiom,
! [H: set @ a] :
( ( ord_less_eq @ ( set @ a ) @ H @ ( carrier @ a @ ( aGroup_ext @ a @ ( module_ext @ a @ b @ c ) ) @ m ) )
=> ( generator @ b @ d @ a @ product_unit @ r @ ( mdl @ a @ b @ c @ m @ ( algebr833503410le_fgs @ b @ d @ a @ c @ r @ m @ H ) ) @ H ) ) ).
% fgs_generator
thf(fact_224_submodule__0,axiom,
submodule @ b @ d @ a @ c @ r @ m @ ( insert @ a @ ( zero2 @ a @ ( module_ext @ a @ b @ c ) @ m ) @ ( bot_bot @ ( set @ a ) ) ) ).
% submodule_0
thf(fact_225_empty__fgs,axiom,
( ( algebr833503410le_fgs @ b @ d @ a @ c @ r @ m @ ( bot_bot @ ( set @ a ) ) )
= ( insert @ a @ ( zero2 @ a @ ( module_ext @ a @ b @ c ) @ m ) @ ( bot_bot @ ( set @ a ) ) ) ) ).
% empty_fgs
thf(fact_226_le__zero__eq,axiom,
! [A: $tType] :
( ( canoni770627133id_add @ A )
=> ! [N: A] :
( ( ord_less_eq @ A @ N @ ( zero_zero @ A ) )
= ( N
= ( zero_zero @ A ) ) ) ) ).
% le_zero_eq
thf(fact_227_not__gr__zero,axiom,
! [A: $tType] :
( ( canoni770627133id_add @ A )
=> ! [N: A] :
( ( ~ ( ord_less @ A @ ( zero_zero @ A ) @ N ) )
= ( N
= ( zero_zero @ A ) ) ) ) ).
% not_gr_zero
thf(fact_228_diff__self,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ! [A2: A] :
( ( minus_minus @ A @ A2 @ A2 )
= ( zero_zero @ A ) ) ) ).
% diff_self
thf(fact_229_diff__0__right,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ! [A2: A] :
( ( minus_minus @ A @ A2 @ ( zero_zero @ A ) )
= A2 ) ) ).
% diff_0_right
thf(fact_230_zero__diff,axiom,
! [A: $tType] :
( ( comm_monoid_diff @ A )
=> ! [A2: A] :
( ( minus_minus @ A @ ( zero_zero @ A ) @ A2 )
= ( zero_zero @ A ) ) ) ).
% zero_diff
thf(fact_231_diff__zero,axiom,
! [A: $tType] :
( ( cancel1352612707id_add @ A )
=> ! [A2: A] :
( ( minus_minus @ A @ A2 @ ( zero_zero @ A ) )
= A2 ) ) ).
% diff_zero
thf(fact_232_cancel__comm__monoid__add__class_Odiff__cancel,axiom,
! [A: $tType] :
( ( cancel1352612707id_add @ A )
=> ! [A2: A] :
( ( minus_minus @ A @ A2 @ A2 )
= ( zero_zero @ A ) ) ) ).
% cancel_comm_monoid_add_class.diff_cancel
thf(fact_233_Diff__eq__empty__iff,axiom,
! [A: $tType,A3: set @ A,B3: set @ A] :
( ( ( minus_minus @ ( set @ A ) @ A3 @ B3 )
= ( bot_bot @ ( set @ A ) ) )
= ( ord_less_eq @ ( set @ A ) @ A3 @ B3 ) ) ).
% Diff_eq_empty_iff
thf(fact_234_subset__empty,axiom,
! [A: $tType,A3: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A3 @ ( bot_bot @ ( set @ A ) ) )
= ( A3
= ( bot_bot @ ( set @ A ) ) ) ) ).
% subset_empty
thf(fact_235_empty__subsetI,axiom,
! [A: $tType,A3: set @ A] : ( ord_less_eq @ ( set @ A ) @ ( bot_bot @ ( set @ A ) ) @ A3 ) ).
% empty_subsetI
thf(fact_236_insert__subset,axiom,
! [A: $tType,X: A,A3: set @ A,B3: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ ( insert @ A @ X @ A3 ) @ B3 )
= ( ( member @ A @ X @ B3 )
& ( ord_less_eq @ ( set @ A ) @ A3 @ B3 ) ) ) ).
% insert_subset
thf(fact_237_insert__Diff__single,axiom,
! [A: $tType,A2: A,A3: set @ A] :
( ( insert @ A @ A2 @ ( minus_minus @ ( set @ A ) @ A3 @ ( insert @ A @ A2 @ ( bot_bot @ ( set @ A ) ) ) ) )
= ( insert @ A @ A2 @ A3 ) ) ).
% insert_Diff_single
thf(fact_238_singletonI,axiom,
! [A: $tType,A2: A] : ( member @ A @ A2 @ ( insert @ A @ A2 @ ( bot_bot @ ( set @ A ) ) ) ) ).
% singletonI
thf(fact_239_linear__span__subModule,axiom,
! [A3: set @ b,H: set @ a] :
( ( ideal @ b @ d @ r @ A3 )
=> ( ( ord_less_eq @ ( set @ a ) @ H @ ( carrier @ a @ ( aGroup_ext @ a @ ( module_ext @ a @ b @ c ) ) @ m ) )
=> ( submodule @ b @ d @ a @ c @ r @ m @ ( linear_span @ b @ d @ a @ c @ r @ m @ A3 @ H ) ) ) ) ).
% linear_span_subModule
thf(fact_240_l__spanA__l__span,axiom,
! [A3: set @ b,H: set @ a] :
( ( ideal @ b @ d @ r @ A3 )
=> ( ( ord_less_eq @ ( set @ a ) @ H @ ( carrier @ a @ ( aGroup_ext @ a @ ( module_ext @ a @ b @ c ) ) @ m ) )
=> ( ( linear_span @ b @ d @ a @ c @ r @ m @ A3 @ ( linear_span @ b @ d @ a @ c @ r @ m @ ( carrier @ b @ ( aGroup_ext @ b @ ( ring_ext @ b @ d ) ) @ r ) @ H ) )
= ( linear_span @ b @ d @ a @ c @ r @ m @ A3 @ H ) ) ) ) ).
% l_spanA_l_span
thf(fact_241_lin__span__coeff__mono,axiom,
! [A3: set @ b,H: set @ a] :
( ( ideal @ b @ d @ r @ A3 )
=> ( ( ord_less_eq @ ( set @ a ) @ H @ ( carrier @ a @ ( aGroup_ext @ a @ ( module_ext @ a @ b @ c ) ) @ m ) )
=> ( ord_less_eq @ ( set @ a ) @ ( linear_span @ b @ d @ a @ c @ r @ m @ A3 @ H ) @ ( linear_span @ b @ d @ a @ c @ r @ m @ ( carrier @ b @ ( aGroup_ext @ b @ ( ring_ext @ b @ d ) ) @ r ) @ H ) ) ) ) ).
% lin_span_coeff_mono
thf(fact_242_linear__span__inc__0,axiom,
! [A3: set @ b,H: set @ a] :
( ( ideal @ b @ d @ r @ A3 )
=> ( ( ord_less_eq @ ( set @ a ) @ H @ ( carrier @ a @ ( aGroup_ext @ a @ ( module_ext @ a @ b @ c ) ) @ m ) )
=> ( member @ a @ ( zero2 @ a @ ( module_ext @ a @ b @ c ) @ m ) @ ( linear_span @ b @ d @ a @ c @ r @ m @ A3 @ H ) ) ) ) ).
% linear_span_inc_0
thf(fact_243_generator__generator,axiom,
! [H: set @ a,H1: set @ a] :
( ( generator @ b @ d @ a @ c @ r @ m @ H )
=> ( ( ord_less_eq @ ( set @ a ) @ H1 @ ( carrier @ a @ ( aGroup_ext @ a @ ( module_ext @ a @ b @ c ) ) @ m ) )
=> ( ( ord_less_eq @ ( set @ a ) @ H @ ( linear_span @ b @ d @ a @ c @ r @ m @ ( carrier @ b @ ( aGroup_ext @ b @ ( ring_ext @ b @ d ) ) @ r ) @ H1 ) )
=> ( generator @ b @ d @ a @ c @ r @ m @ H1 ) ) ) ) ).
% generator_generator
thf(fact_244_l__span__zero,axiom,
! [A3: set @ b] :
( ( ideal @ b @ d @ r @ A3 )
=> ( ( linear_span @ b @ d @ a @ c @ r @ m @ A3 @ ( insert @ a @ ( zero2 @ a @ ( module_ext @ a @ b @ c ) @ m ) @ ( bot_bot @ ( set @ a ) ) ) )
= ( insert @ a @ ( zero2 @ a @ ( module_ext @ a @ b @ c ) @ m ) @ ( bot_bot @ ( set @ a ) ) ) ) ) ).
% l_span_zero
thf(fact_245_l__span__closed3,axiom,
! [A3: set @ b,H: set @ a] :
( ( ideal @ b @ d @ r @ A3 )
=> ( ( generator @ b @ d @ a @ c @ r @ m @ H )
=> ( ( ( smodule_ideal_coeff @ b @ d @ a @ c @ r @ m @ A3 )
= ( carrier @ a @ ( aGroup_ext @ a @ ( module_ext @ a @ b @ c ) ) @ m ) )
=> ( ( linear_span @ b @ d @ a @ c @ r @ m @ A3 @ H )
= ( carrier @ a @ ( aGroup_ext @ a @ ( module_ext @ a @ b @ c ) ) @ m ) ) ) ) ) ).
% l_span_closed3
thf(fact_246_singleton__conv,axiom,
! [A: $tType,A2: A] :
( ( collect @ A
@ ^ [X3: A] : X3 = A2 )
= ( insert @ A @ A2 @ ( bot_bot @ ( set @ A ) ) ) ) ).
% singleton_conv
thf(fact_247_singleton__conv2,axiom,
! [A: $tType,A2: A] :
( ( collect @ A
@ ( ^ [Y5: A,Z3: A] : Y5 = Z3
@ A2 ) )
= ( insert @ A @ A2 @ ( bot_bot @ ( set @ A ) ) ) ) ).
% singleton_conv2
thf(fact_248_l__comb__mem__linear__span,axiom,
! [A3: set @ b,H: set @ a,S: nat > b,N: nat,F: nat > a] :
( ( ideal @ b @ d @ r @ A3 )
=> ( ( ord_less_eq @ ( set @ a ) @ H @ ( carrier @ a @ ( aGroup_ext @ a @ ( module_ext @ a @ b @ c ) ) @ m ) )
=> ( ( member @ ( nat > b ) @ S
@ ( pi @ nat @ b
@ ( collect @ nat
@ ^ [J2: nat] : ( ord_less_eq @ nat @ J2 @ N ) )
@ ^ [Uu: nat] : A3 ) )
=> ( ( member @ ( nat > a ) @ F
@ ( pi @ nat @ a
@ ( collect @ nat
@ ^ [J2: nat] : ( ord_less_eq @ nat @ J2 @ N ) )
@ ^ [Uu: nat] : H ) )
=> ( member @ a @ ( l_comb @ b @ d @ a @ c @ r @ m @ N @ S @ F ) @ ( linear_span @ b @ d @ a @ c @ r @ m @ A3 @ H ) ) ) ) ) ) ).
% l_comb_mem_linear_span
thf(fact_249_l__span__closed,axiom,
! [A3: set @ b,H: set @ a,S: nat > b,N: nat,F: nat > a] :
( ( ideal @ b @ d @ r @ A3 )
=> ( ( ord_less_eq @ ( set @ a ) @ H @ ( carrier @ a @ ( aGroup_ext @ a @ ( module_ext @ a @ b @ c ) ) @ m ) )
=> ( ( member @ ( nat > b ) @ S
@ ( pi @ nat @ b
@ ( collect @ nat
@ ^ [J2: nat] : ( ord_less_eq @ nat @ J2 @ N ) )
@ ^ [Uu: nat] : A3 ) )
=> ( ( member @ ( nat > a ) @ F
@ ( pi @ nat @ a
@ ( collect @ nat
@ ^ [J2: nat] : ( ord_less_eq @ nat @ J2 @ N ) )
@ ^ [Uu: nat] : ( linear_span @ b @ d @ a @ c @ r @ m @ A3 @ H ) ) )
=> ( member @ a @ ( l_comb @ b @ d @ a @ c @ r @ m @ N @ S @ F ) @ ( linear_span @ b @ d @ a @ c @ r @ m @ A3 @ H ) ) ) ) ) ) ).
% l_span_closed
thf(fact_250_linear__comb0__2,axiom,
! [A3: set @ b,S: nat > b,N: nat,M3: nat > a] :
( ( ideal @ b @ d @ r @ A3 )
=> ( ( member @ ( nat > b ) @ S
@ ( pi @ nat @ b
@ ( collect @ nat
@ ^ [J2: nat] : ( ord_less_eq @ nat @ J2 @ N ) )
@ ^ [Uu: nat] : A3 ) )
=> ( ( member @ ( nat > a ) @ M3
@ ( pi @ nat @ a
@ ( collect @ nat
@ ^ [J2: nat] : ( ord_less_eq @ nat @ J2 @ N ) )
@ ^ [Uu: nat] : ( insert @ a @ ( zero2 @ a @ ( module_ext @ a @ b @ c ) @ m ) @ ( bot_bot @ ( set @ a ) ) ) ) )
=> ( ( l_comb @ b @ d @ a @ c @ r @ m @ N @ S @ M3 )
= ( zero2 @ a @ ( module_ext @ a @ b @ c ) @ m ) ) ) ) ) ).
% linear_comb0_2
thf(fact_251_double__diff,axiom,
! [A: $tType,A3: set @ A,B3: set @ A,C3: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A3 @ B3 )
=> ( ( ord_less_eq @ ( set @ A ) @ B3 @ C3 )
=> ( ( minus_minus @ ( set @ A ) @ B3 @ ( minus_minus @ ( set @ A ) @ C3 @ A3 ) )
= A3 ) ) ) ).
% double_diff
thf(fact_252_Diff__subset,axiom,
! [A: $tType,A3: set @ A,B3: set @ A] : ( ord_less_eq @ ( set @ A ) @ ( minus_minus @ ( set @ A ) @ A3 @ B3 ) @ A3 ) ).
% Diff_subset
% Type constructors (11)
thf(tcon_fun___Orderings_Oorder,axiom,
! [A4: $tType,A5: $tType] :
( ( order @ A5 )
=> ( order @ ( A4 > A5 ) ) ) ).
thf(tcon_Nat_Onat___Groups_Ocanonically__ordered__monoid__add,axiom,
canoni770627133id_add @ nat ).
thf(tcon_Nat_Onat___Groups_Ocancel__comm__monoid__add,axiom,
cancel1352612707id_add @ nat ).
thf(tcon_Nat_Onat___Groups_Ocomm__monoid__diff,axiom,
comm_monoid_diff @ nat ).
thf(tcon_Nat_Onat___Orderings_Owellorder,axiom,
wellorder @ nat ).
thf(tcon_Nat_Onat___Orderings_Oorder_1,axiom,
order @ nat ).
thf(tcon_Nat_Onat___Groups_Ozero,axiom,
zero @ nat ).
thf(tcon_Set_Oset___Orderings_Oorder_2,axiom,
! [A4: $tType] : ( order @ ( set @ A4 ) ) ).
thf(tcon_HOL_Obool___Orderings_Oorder_3,axiom,
order @ $o ).
thf(tcon_Product__Type_Ounit___Orderings_Owellorder_4,axiom,
wellorder @ product_unit ).
thf(tcon_Product__Type_Ounit___Orderings_Oorder_5,axiom,
order @ product_unit ).
% Conjectures (22)
thf(conj_0,hypothesis,
( member @ ( nat > a ) @ g
@ ( pi @ nat @ a
@ ( collect @ nat
@ ^ [K4: nat] : ( ord_less_eq @ nat @ K4 @ m2 ) )
@ ^ [Uu: nat] : ( insert @ a @ h2 @ h1 ) ) ) ).
thf(conj_1,hypothesis,
! [X4: nat] :
( ( ord_less_eq @ nat @ X4 @ m2 )
=> ! [Y: nat] :
( ( ord_less_eq @ nat @ Y @ m2 )
=> ( ( ( g @ X4 )
= ( g @ Y ) )
=> ( X4 = Y ) ) ) ) ).
thf(conj_2,hypothesis,
( member @ ( nat > b ) @ t
@ ( pi @ nat @ b
@ ( collect @ nat
@ ^ [K4: nat] : ( ord_less_eq @ nat @ K4 @ m2 ) )
@ ^ [Uu: nat] : ( carrier @ b @ ( aGroup_ext @ b @ ( ring_ext @ b @ d ) ) @ r ) ) ) ).
thf(conj_3,hypothesis,
( ( g @ m2 )
= h2 ) ).
thf(conj_4,hypothesis,
module @ e @ b @ f @ d @ n @ r ).
thf(conj_5,hypothesis,
free_generator @ b @ d @ a @ c @ r @ m @ h ).
thf(conj_6,hypothesis,
ord_less_eq @ ( set @ a ) @ h1 @ h ).
thf(conj_7,hypothesis,
$true ).
thf(conj_8,hypothesis,
ideal @ b @ d @ r @ ( carrier @ b @ ( aGroup_ext @ b @ ( ring_ext @ b @ d ) ) @ r ) ).
thf(conj_9,hypothesis,
ord_less_eq @ ( set @ a ) @ h @ ( carrier @ a @ ( aGroup_ext @ a @ ( module_ext @ a @ b @ c ) ) @ m ) ).
thf(conj_10,hypothesis,
ord_less_eq @ ( set @ a ) @ h1 @ ( carrier @ a @ ( aGroup_ext @ a @ ( module_ext @ a @ b @ c ) ) @ m ) ).
thf(conj_11,hypothesis,
ord_less_eq @ ( set @ a ) @ ( algebr833503410le_fgs @ b @ d @ a @ c @ r @ m @ h1 ) @ ( carrier @ a @ ( aGroup_ext @ a @ ( module_ext @ a @ b @ c ) ) @ m ) ).
thf(conj_12,hypothesis,
member @ a @ ( l_comb @ b @ d @ a @ c @ r @ m @ m2 @ t @ g ) @ ( algebr833503410le_fgs @ b @ d @ a @ c @ r @ m @ ( insert @ a @ h2 @ h1 ) ) ).
thf(conj_13,hypothesis,
( x
= ( l_comb @ b @ d @ a @ c @ r @ m @ m2 @ t @ g ) ) ).
thf(conj_14,hypothesis,
member @ a @ h2 @ h ).
thf(conj_15,hypothesis,
~ ( member @ a @ h2 @ h1 ) ).
thf(conj_16,hypothesis,
ord_less @ nat @ ( zero_zero @ nat ) @ m2 ).
thf(conj_17,hypothesis,
member @ a @ h2 @ ( carrier @ a @ ( aGroup_ext @ a @ ( module_ext @ a @ b @ c ) ) @ m ) ).
thf(conj_18,hypothesis,
submodule @ b @ d @ a @ c @ r @ m @ ( algebr833503410le_fgs @ b @ d @ a @ c @ r @ m @ ( insert @ a @ h2 @ ( bot_bot @ ( set @ a ) ) ) ) ).
thf(conj_19,hypothesis,
ord_less_eq @ nat @ xa @ ( minus_minus @ nat @ m2 @ ( suc @ ( zero_zero @ nat ) ) ) ).
thf(conj_20,hypothesis,
( ( ( g @ xa )
= h2 )
| ( member @ a @ ( g @ xa ) @ h1 ) ) ).
thf(conj_21,conjecture,
member @ a @ ( g @ xa ) @ h1 ).
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