TPTP Problem File: ITP027^2.p
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%------------------------------------------------------------------------------
% File : ITP027^2 : TPTP v8.2.0. Released v7.5.0.
% Domain : Interactive Theorem Proving
% Problem : Sledgehammer Algebra8 problem prob_2341__6471840_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_2341__6471840_1 [Des21]
% Status : Theorem
% Rating : 0.67 v8.2.0, 0.33 v8.1.0, 0.50 v7.5.0
% Syntax : Number of formulae : 332 ( 27 unt; 61 typ; 0 def)
% Number of atoms : 1000 ( 202 equ; 0 cnn)
% Maximal formula atoms : 14 ( 3 avg)
% Number of connectives : 10303 ( 28 ~; 3 |; 27 &;9574 @)
% ( 0 <=>; 671 =>; 0 <=; 0 <~>)
% Maximal formula depth : 27 ( 13 avg)
% Number of types : 8 ( 7 usr)
% Number of type conns : 400 ( 400 >; 0 *; 0 +; 0 <<)
% Number of symbols : 55 ( 54 usr; 2 con; 0-11 aty)
% Number of variables : 1403 ( 59 ^;1232 !; 13 ?;1403 :)
% ( 99 !>; 0 ?*; 0 @-; 0 @+)
% SPC : TH1_THM_EQU_NAR
% Comments : This file was generated by Sledgehammer 2021-02-23 16:32:56.184
%------------------------------------------------------------------------------
% Could-be-implicit typings (12)
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_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 (49)
thf(sy_cl_HOL_Otype,type,
type:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Oplus,type,
plus:
!>[A: $tType] : $o ).
thf(sy_cl_Orderings_Oord,type,
ord:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Ogroup__add,type,
group_add:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Osemigroup__add,type,
semigroup_add:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Ocomm__monoid__add,type,
comm_monoid_add:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Oab__semigroup__add,type,
ab_semigroup_add:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Ocancel__semigroup__add,type,
cancel_semigroup_add:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Oordered__ab__semigroup__add,type,
ordere779506340up_add:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Oordered__ab__semigroup__add__imp__le,type,
ordere236663937imp_le:
!>[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_OJ__rad,type,
j_rad:
!>[A: $tType,B: $tType] : ( ( carrier_ext @ A @ ( aGroup_ext @ A @ ( ring_ext @ A @ B ) ) ) > ( 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_ORing_Otp,type,
tp:
!>[A: $tType,Z: $tType] : ( ( carrier_ext @ A @ ( aGroup_ext @ A @ ( ring_ext @ A @ Z ) ) ) > A > A > A ) ).
thf(sy_c_Algebra4_ORing_Oun,type,
un:
!>[A: $tType,Z: $tType] : ( ( carrier_ext @ A @ ( aGroup_ext @ A @ ( ring_ext @ A @ Z ) ) ) > A ) ).
thf(sy_c_Algebra4_OSr,type,
sr:
!>[A: $tType,B: $tType] : ( ( carrier_ext @ A @ ( aGroup_ext @ A @ ( ring_ext @ A @ B ) ) ) > ( set @ A ) > ( carrier_ext @ A @ ( aGroup_ext @ A @ ( ring_ext @ A @ B ) ) ) ) ).
thf(sy_c_Algebra4_OSubring,type,
subring:
!>[A: $tType,M: $tType,M1: $tType] : ( ( carrier_ext @ A @ ( aGroup_ext @ A @ ( ring_ext @ A @ M ) ) ) > ( carrier_ext @ A @ ( aGroup_ext @ A @ ( ring_ext @ A @ M1 ) ) ) > $o ) ).
thf(sy_c_Algebra4_OUnit,type,
unit:
!>[A: $tType,B: $tType] : ( ( carrier_ext @ A @ ( aGroup_ext @ A @ ( ring_ext @ A @ B ) ) ) > A > $o ) ).
thf(sy_c_Algebra4_OaGroup,type,
aGroup:
!>[A: $tType,B: $tType] : ( ( carrier_ext @ A @ ( aGroup_ext @ A @ B ) ) > $o ) ).
thf(sy_c_Algebra4_OaGroup_Omop,type,
mop:
!>[A: $tType,Z: $tType] : ( ( carrier_ext @ A @ ( aGroup_ext @ A @ Z ) ) > A > A ) ).
thf(sy_c_Algebra4_OaGroup_Opop,type,
pop:
!>[A: $tType,Z: $tType] : ( ( carrier_ext @ A @ ( aGroup_ext @ A @ Z ) ) > A > A > A ) ).
thf(sy_c_Algebra4_Oaset__sum,type,
aset_sum:
!>[A: $tType,B: $tType] : ( ( carrier_ext @ A @ ( aGroup_ext @ A @ B ) ) > ( set @ A ) > ( set @ A ) > ( set @ A ) ) ).
thf(sy_c_Algebra4_Ocarr__dsumag,type,
carr_dsumag:
!>[I: $tType,A: $tType,More: $tType] : ( ( set @ I ) > ( I > ( carrier_ext @ A @ ( aGroup_ext @ A @ More ) ) ) > ( set @ ( I > A ) ) ) ).
thf(sy_c_Algebra4_Ocarr__prodag,type,
carr_prodag:
!>[I: $tType,A: $tType,More: $tType] : ( ( set @ I ) > ( I > ( carrier_ext @ A @ ( aGroup_ext @ A @ More ) ) ) > ( set @ ( I > A ) ) ) ).
thf(sy_c_Algebra4_Olocal__ring,type,
local_ring:
!>[A: $tType,B: $tType] : ( ( carrier_ext @ A @ ( aGroup_ext @ A @ ( ring_ext @ A @ B ) ) ) > $o ) ).
thf(sy_c_Algebra4_Omaximal__ideal,type,
maximal_ideal:
!>[A: $tType,B: $tType] : ( ( carrier_ext @ A @ ( aGroup_ext @ A @ ( ring_ext @ A @ B ) ) ) > ( set @ A ) > $o ) ).
thf(sy_c_Algebra4_Omul__closed__set,type,
mul_closed_set:
!>[A: $tType,B: $tType] : ( ( carrier_ext @ A @ ( aGroup_ext @ A @ ( ring_ext @ A @ B ) ) ) > ( set @ A ) > $o ) ).
thf(sy_c_Algebra4_Onscal,type,
nscal:
!>[A: $tType,More: $tType] : ( ( carrier_ext @ A @ ( aGroup_ext @ A @ ( ring_ext @ A @ More ) ) ) > A > nat > A ) ).
thf(sy_c_Algebra4_Oprimary__ideal,type,
primary_ideal:
!>[A: $tType,B: $tType] : ( ( carrier_ext @ A @ ( aGroup_ext @ A @ ( ring_ext @ A @ B ) ) ) > ( set @ A ) > $o ) ).
thf(sy_c_Algebra4_Oprod__mOp,type,
prod_mOp:
!>[I: $tType,A: $tType,More: $tType] : ( ( set @ I ) > ( I > ( carrier_ext @ A @ ( aGroup_ext @ A @ More ) ) ) > ( I > A ) > I > A ) ).
thf(sy_c_Algebra4_Oprod__pOp,type,
prod_pOp:
!>[I: $tType,A: $tType,More: $tType] : ( ( set @ I ) > ( I > ( carrier_ext @ A @ ( aGroup_ext @ A @ More ) ) ) > ( I > A ) > ( I > A ) > I > A ) ).
thf(sy_c_Algebra4_Oprod__zero,type,
prod_zero:
!>[I: $tType,A: $tType,More: $tType] : ( ( set @ I ) > ( I > ( carrier_ext @ A @ ( aGroup_ext @ A @ More ) ) ) > I > A ) ).
thf(sy_c_Algebra4_OrInvim,type,
rInvim:
!>[A: $tType,M: $tType,B: $tType,M1: $tType] : ( ( carrier_ext @ A @ ( aGroup_ext @ A @ ( ring_ext @ A @ M ) ) ) > ( carrier_ext @ B @ ( aGroup_ext @ B @ ( ring_ext @ B @ M1 ) ) ) > ( A > B ) > ( set @ B ) > ( set @ A ) ) ).
thf(sy_c_Algebra4_Osr,type,
sr2:
!>[A: $tType,B: $tType] : ( ( carrier_ext @ A @ ( aGroup_ext @ A @ ( ring_ext @ A @ B ) ) ) > ( set @ A ) > $o ) ).
thf(sy_c_Algebra4_Ozeroring,type,
zeroring:
!>[A: $tType,More: $tType] : ( ( carrier_ext @ A @ ( aGroup_ext @ A @ ( ring_ext @ A @ More ) ) ) > $o ) ).
thf(sy_c_Algebra5_Osum__mult,type,
sum_mult:
!>[A: $tType,B: $tType] : ( ( carrier_ext @ A @ ( aGroup_ext @ A @ ( ring_ext @ A @ B ) ) ) > ( set @ A ) > ( set @ A ) > ( set @ A ) ) ).
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_OModule_Osprod,type,
sprod:
!>[A: $tType,B: $tType,Z: $tType] : ( ( carrier_ext @ A @ ( aGroup_ext @ A @ ( module_ext @ A @ B @ Z ) ) ) > B > A > A ) ).
thf(sy_c_Algebra7_OmHom,type,
mHom:
!>[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 ) ) ) > ( set @ ( A > C ) ) ) ).
thf(sy_c_Algebra8__Mirabelle__lwvimexpoc_OprodM,type,
algebr1000837334_prodM:
!>[R: $tType,M: $tType,I: $tType,A: $tType,M1: $tType] : ( ( carrier_ext @ R @ ( aGroup_ext @ R @ ( ring_ext @ R @ M ) ) ) > ( set @ I ) > ( I > ( carrier_ext @ A @ ( aGroup_ext @ A @ ( module_ext @ A @ R @ M1 ) ) ) ) > ( carrier_ext @ ( I > A ) @ ( aGroup_ext @ ( I > A ) @ ( module_ext @ ( I > A ) @ R @ product_unit ) ) ) ) ).
thf(sy_c_Algebra8__Mirabelle__lwvimexpoc_OprodM__sprod,type,
algebr142713364_sprod:
!>[R: $tType,M: $tType,I: $tType,A: $tType,M1: $tType] : ( ( carrier_ext @ R @ ( aGroup_ext @ R @ ( ring_ext @ R @ M ) ) ) > ( set @ I ) > ( I > ( carrier_ext @ A @ ( aGroup_ext @ A @ ( module_ext @ A @ R @ M1 ) ) ) ) > R > ( I > A ) > I > A ) ).
thf(sy_c_FuncSet_OPi,type,
pi:
!>[A: $tType,B: $tType] : ( ( set @ A ) > ( A > ( set @ B ) ) > ( set @ ( A > B ) ) ) ).
thf(sy_c_Groups_Oplus__class_Oplus,type,
plus_plus:
!>[A: $tType] : ( A > A > A ) ).
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_member,type,
member:
!>[A: $tType] : ( A > ( set @ A ) > $o ) ).
thf(sy_v_I,type,
i: set @ c ).
thf(sy_v_M,type,
m: c > ( carrier_ext @ d @ ( aGroup_ext @ d @ ( module_ext @ d @ a @ e ) ) ) ).
thf(sy_v_R,type,
r: carrier_ext @ a @ ( aGroup_ext @ a @ ( ring_ext @ a @ b ) ) ).
% Relevant facts (256)
thf(fact_0_Ring__axioms,axiom,
ring @ a @ b @ r ).
% Ring_axioms
thf(fact_1_bivar__fun,axiom,
! [A: $tType,C: $tType,B: $tType,F: A > B > C,A2: set @ A,B2: set @ B,C2: set @ C,A3: A] :
( ( member @ ( A > B > C ) @ F
@ ( pi @ A @ ( B > C ) @ A2
@ ^ [Uu: A] :
( pi @ B @ C @ B2
@ ^ [Uv: B] : C2 ) ) )
=> ( ( member @ A @ A3 @ A2 )
=> ( member @ ( B > C ) @ ( F @ A3 )
@ ( pi @ B @ C @ B2
@ ^ [Uu: B] : C2 ) ) ) ) ).
% bivar_fun
thf(fact_2_bivar__fun__mem,axiom,
! [B: $tType,A: $tType,C: $tType,F: A > B > C,A2: set @ A,B2: set @ B,C2: set @ C,A3: A,B3: B] :
( ( member @ ( A > B > C ) @ F
@ ( pi @ A @ ( B > C ) @ A2
@ ^ [Uu: A] :
( pi @ B @ C @ B2
@ ^ [Uv: B] : C2 ) ) )
=> ( ( member @ A @ A3 @ A2 )
=> ( ( member @ B @ B3 @ B2 )
=> ( member @ C @ ( F @ A3 @ B3 ) @ C2 ) ) ) ) ).
% bivar_fun_mem
thf(fact_3_Pi__I,axiom,
! [B: $tType,A: $tType,A2: set @ A,F: A > B,B2: A > ( set @ B )] :
( ! [X: A] :
( ( member @ A @ X @ A2 )
=> ( member @ B @ ( F @ X ) @ ( B2 @ X ) ) )
=> ( member @ ( A > B ) @ F @ ( pi @ A @ B @ A2 @ B2 ) ) ) ).
% Pi_I
thf(fact_4_prodM__sprod__mem,axiom,
! [E: $tType,D: $tType,C: $tType,I2: set @ C,M3: C > ( carrier_ext @ D @ ( aGroup_ext @ D @ ( module_ext @ D @ a @ E ) ) ),A3: a,M4: C > D] :
( ! [X: C] :
( ( member @ C @ X @ I2 )
=> ( module @ D @ a @ E @ b @ ( M3 @ X ) @ r ) )
=> ( ( member @ a @ A3 @ ( carrier @ a @ ( aGroup_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
=> ( ( member @ ( C > D ) @ M4 @ ( carr_prodag @ C @ D @ ( module_ext @ D @ a @ E ) @ I2 @ M3 ) )
=> ( member @ ( C > D ) @ ( algebr142713364_sprod @ a @ b @ C @ D @ E @ r @ I2 @ M3 @ A3 @ M4 ) @ ( carr_prodag @ C @ D @ ( module_ext @ D @ a @ E ) @ I2 @ M3 ) ) ) ) ) ).
% prodM_sprod_mem
thf(fact_5_prodM__carr,axiom,
! [E: $tType,D: $tType,C: $tType,I2: set @ C,M3: C > ( carrier_ext @ D @ ( aGroup_ext @ D @ ( module_ext @ D @ a @ E ) ) )] :
( ! [X: C] :
( ( member @ C @ X @ I2 )
=> ( module @ D @ a @ E @ b @ ( M3 @ X ) @ r ) )
=> ( ( carrier @ ( C > D ) @ ( aGroup_ext @ ( C > D ) @ ( module_ext @ ( C > D ) @ a @ product_unit ) ) @ ( algebr1000837334_prodM @ a @ b @ C @ D @ E @ r @ I2 @ M3 ) )
= ( carr_prodag @ C @ D @ ( module_ext @ D @ a @ E ) @ I2 @ M3 ) ) ) ).
% prodM_carr
thf(fact_6_eq__fun,axiom,
! [B: $tType,A: $tType,F: A > B,A2: set @ A,B2: set @ B,G: A > B] :
( ( member @ ( A > B ) @ F
@ ( pi @ A @ B @ A2
@ ^ [Uu: A] : B2 ) )
=> ( ( F = G )
=> ( member @ ( A > B ) @ G
@ ( pi @ A @ B @ A2
@ ^ [Uu: A] : B2 ) ) ) ) ).
% eq_fun
thf(fact_7_funcTr,axiom,
! [B: $tType,A: $tType,F: A > B,A2: set @ A,B2: set @ B,G: A > B,A3: A] :
( ( member @ ( A > B ) @ F
@ ( pi @ A @ B @ A2
@ ^ [Uu: A] : B2 ) )
=> ( ( member @ ( A > B ) @ G
@ ( pi @ A @ B @ A2
@ ^ [Uu: A] : B2 ) )
=> ( ( F = G )
=> ( ( member @ A @ A3 @ A2 )
=> ( ( F @ A3 )
= ( G @ A3 ) ) ) ) ) ) ).
% funcTr
thf(fact_8_funcsetI,axiom,
! [B: $tType,A: $tType,A2: set @ A,F: A > B,B2: set @ B] :
( ! [X: A] :
( ( member @ A @ X @ A2 )
=> ( member @ B @ ( F @ X ) @ B2 ) )
=> ( member @ ( A > B ) @ F
@ ( pi @ A @ B @ A2
@ ^ [Uu: A] : B2 ) ) ) ).
% funcsetI
thf(fact_9_eq__funcs,axiom,
! [B: $tType,A: $tType,F: A > B,A2: set @ A,B2: set @ B,G: A > B,X2: A] :
( ( member @ ( A > B ) @ F
@ ( pi @ A @ B @ A2
@ ^ [Uu: A] : B2 ) )
=> ( ( member @ ( A > B ) @ G
@ ( pi @ A @ B @ A2
@ ^ [Uu: A] : B2 ) )
=> ( ( F = G )
=> ( ( member @ A @ X2 @ A2 )
=> ( ( F @ X2 )
= ( G @ X2 ) ) ) ) ) ) ).
% eq_funcs
thf(fact_10_funcset__id,axiom,
! [A: $tType,A2: set @ A] :
( member @ ( A > A )
@ ^ [X3: A] : X3
@ ( pi @ A @ A @ A2
@ ^ [Uu: A] : A2 ) ) ).
% funcset_id
thf(fact_11_Algebra1_Ofuncset__mem,axiom,
! [A: $tType,B: $tType,F: A > B,A2: set @ A,B2: set @ B,X2: A] :
( ( member @ ( A > B ) @ F
@ ( pi @ A @ B @ A2
@ ^ [Uu: A] : B2 ) )
=> ( ( member @ A @ X2 @ A2 )
=> ( member @ B @ ( F @ X2 ) @ B2 ) ) ) ).
% Algebra1.funcset_mem
thf(fact_12_prod__pOp__func,axiom,
! [C: $tType,B: $tType,A: $tType,I2: set @ A,A2: A > ( carrier_ext @ B @ ( aGroup_ext @ B @ C ) )] :
( ! [X: A] :
( ( member @ A @ X @ I2 )
=> ( aGroup @ B @ C @ ( A2 @ X ) ) )
=> ( member @ ( ( A > B ) > ( A > B ) > A > B ) @ ( prod_pOp @ A @ B @ C @ I2 @ A2 )
@ ( pi @ ( A > B ) @ ( ( A > B ) > A > B ) @ ( carr_prodag @ A @ B @ C @ I2 @ A2 )
@ ^ [Uu: A > B] :
( pi @ ( A > B ) @ ( A > B ) @ ( carr_prodag @ A @ B @ C @ I2 @ A2 )
@ ^ [Uv: A > B] : ( carr_prodag @ A @ B @ C @ I2 @ A2 ) ) ) ) ) ).
% prod_pOp_func
thf(fact_13_ring__is__ag,axiom,
aGroup @ a @ ( ring_ext @ a @ b ) @ r ).
% ring_is_ag
thf(fact_14_prodM__mem__eq,axiom,
! [E: $tType,D: $tType,C: $tType,I2: set @ C,M3: C > ( carrier_ext @ D @ ( aGroup_ext @ D @ ( module_ext @ D @ a @ E ) ) ),M12: C > D,M22: C > D] :
( ! [X: C] :
( ( member @ C @ X @ I2 )
=> ( module @ D @ a @ E @ b @ ( M3 @ X ) @ r ) )
=> ( ( member @ ( C > D ) @ M12 @ ( carrier @ ( C > D ) @ ( aGroup_ext @ ( C > D ) @ ( module_ext @ ( C > D ) @ a @ product_unit ) ) @ ( algebr1000837334_prodM @ a @ b @ C @ D @ E @ r @ I2 @ M3 ) ) )
=> ( ( member @ ( C > D ) @ M22 @ ( carrier @ ( C > D ) @ ( aGroup_ext @ ( C > D ) @ ( module_ext @ ( C > D ) @ a @ product_unit ) ) @ ( algebr1000837334_prodM @ a @ b @ C @ D @ E @ r @ I2 @ M3 ) ) )
=> ( ! [X: C] :
( ( member @ C @ X @ I2 )
=> ( ( M12 @ X )
= ( M22 @ X ) ) )
=> ( M12 = M22 ) ) ) ) ) ).
% prodM_mem_eq
thf(fact_15_J__rad__mem,axiom,
! [X2: a] :
( ( member @ a @ X2 @ ( j_rad @ a @ b @ r ) )
=> ( member @ a @ X2 @ ( carrier @ a @ ( aGroup_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) ) ) ).
% J_rad_mem
thf(fact_16_Ring_ORing,axiom,
! [B: $tType,A: $tType,R2: carrier_ext @ A @ ( aGroup_ext @ A @ ( ring_ext @ A @ B ) )] :
( ( ring @ A @ B @ R2 )
=> ( ring @ A @ B @ R2 ) ) ).
% Ring.Ring
thf(fact_17_Ring_Oring__is__ag,axiom,
! [B: $tType,A: $tType,R2: carrier_ext @ A @ ( aGroup_ext @ A @ ( ring_ext @ A @ B ) )] :
( ( ring @ A @ B @ R2 )
=> ( aGroup @ A @ ( ring_ext @ A @ B ) @ R2 ) ) ).
% Ring.ring_is_ag
thf(fact_18_Ring_OprodM__mem__eq,axiom,
! [A: $tType,E: $tType,B: $tType,D: $tType,C: $tType,R2: carrier_ext @ A @ ( aGroup_ext @ A @ ( ring_ext @ A @ B ) ),I2: set @ C,M3: C > ( carrier_ext @ D @ ( aGroup_ext @ D @ ( module_ext @ D @ A @ E ) ) ),M12: C > D,M22: C > D] :
( ( ring @ A @ B @ R2 )
=> ( ! [X: C] :
( ( member @ C @ X @ I2 )
=> ( module @ D @ A @ E @ B @ ( M3 @ X ) @ R2 ) )
=> ( ( member @ ( C > D ) @ M12 @ ( carrier @ ( C > D ) @ ( aGroup_ext @ ( C > D ) @ ( module_ext @ ( C > D ) @ A @ product_unit ) ) @ ( algebr1000837334_prodM @ A @ B @ C @ D @ E @ R2 @ I2 @ M3 ) ) )
=> ( ( member @ ( C > D ) @ M22 @ ( carrier @ ( C > D ) @ ( aGroup_ext @ ( C > D ) @ ( module_ext @ ( C > D ) @ A @ product_unit ) ) @ ( algebr1000837334_prodM @ A @ B @ C @ D @ E @ R2 @ I2 @ M3 ) ) )
=> ( ! [X: C] :
( ( member @ C @ X @ I2 )
=> ( ( M12 @ X )
= ( M22 @ X ) ) )
=> ( M12 = M22 ) ) ) ) ) ) ).
% Ring.prodM_mem_eq
thf(fact_19_carr__prodag__mem__eq,axiom,
! [C: $tType,B: $tType,A: $tType,I2: set @ A,A2: A > ( carrier_ext @ B @ ( aGroup_ext @ B @ C ) ),X4: A > B,Y: A > B] :
( ! [X: A] :
( ( member @ A @ X @ I2 )
=> ( aGroup @ B @ C @ ( A2 @ X ) ) )
=> ( ( member @ ( A > B ) @ X4 @ ( carr_prodag @ A @ B @ C @ I2 @ A2 ) )
=> ( ( member @ ( A > B ) @ Y @ ( carr_prodag @ A @ B @ C @ I2 @ A2 ) )
=> ( ! [X: A] :
( ( member @ A @ X @ I2 )
=> ( ( X4 @ X )
= ( Y @ X ) ) )
=> ( X4 = Y ) ) ) ) ) ).
% carr_prodag_mem_eq
thf(fact_20_prodag__sameTr1,axiom,
! [C: $tType,B: $tType,A: $tType,I2: set @ A,A2: A > ( carrier_ext @ B @ ( aGroup_ext @ B @ C ) ),B2: A > ( carrier_ext @ B @ ( aGroup_ext @ B @ C ) )] :
( ! [X: A] :
( ( member @ A @ X @ I2 )
=> ( aGroup @ B @ C @ ( A2 @ X ) ) )
=> ( ! [X: A] :
( ( member @ A @ X @ I2 )
=> ( ( A2 @ X )
= ( B2 @ X ) ) )
=> ( ( carr_prodag @ A @ B @ C @ I2 @ A2 )
= ( carr_prodag @ A @ B @ C @ I2 @ B2 ) ) ) ) ).
% prodag_sameTr1
thf(fact_21_prodag__comp__i,axiom,
! [B: $tType,C: $tType,A: $tType,A3: A > B,I2: set @ A,A2: A > ( carrier_ext @ B @ ( aGroup_ext @ B @ C ) ),I3: A] :
( ( member @ ( A > B ) @ A3 @ ( carr_prodag @ A @ B @ C @ I2 @ A2 ) )
=> ( ( member @ A @ I3 @ I2 )
=> ( member @ B @ ( A3 @ I3 ) @ ( carrier @ B @ ( aGroup_ext @ B @ C ) @ ( A2 @ I3 ) ) ) ) ) ).
% prodag_comp_i
thf(fact_22_prodag__sameTr2,axiom,
! [C: $tType,B: $tType,A: $tType,I2: set @ A,A2: A > ( carrier_ext @ B @ ( aGroup_ext @ B @ C ) ),B2: A > ( carrier_ext @ B @ ( aGroup_ext @ B @ C ) )] :
( ! [X: A] :
( ( member @ A @ X @ I2 )
=> ( aGroup @ B @ C @ ( A2 @ X ) ) )
=> ( ! [X: A] :
( ( member @ A @ X @ I2 )
=> ( ( A2 @ X )
= ( B2 @ X ) ) )
=> ( ( prod_pOp @ A @ B @ C @ I2 @ A2 )
= ( prod_pOp @ A @ B @ C @ I2 @ B2 ) ) ) ) ).
% prodag_sameTr2
thf(fact_23_prodag__sameTr3,axiom,
! [C: $tType,B: $tType,A: $tType,I2: set @ A,A2: A > ( carrier_ext @ B @ ( aGroup_ext @ B @ C ) ),B2: A > ( carrier_ext @ B @ ( aGroup_ext @ B @ C ) )] :
( ! [X: A] :
( ( member @ A @ X @ I2 )
=> ( aGroup @ B @ C @ ( A2 @ X ) ) )
=> ( ! [X: A] :
( ( member @ A @ X @ I2 )
=> ( ( A2 @ X )
= ( B2 @ X ) ) )
=> ( ( prod_mOp @ A @ B @ C @ I2 @ A2 )
= ( prod_mOp @ A @ B @ C @ I2 @ B2 ) ) ) ) ).
% prodag_sameTr3
thf(fact_24_prodag__sameTr4,axiom,
! [C: $tType,B: $tType,A: $tType,I2: set @ A,A2: A > ( carrier_ext @ B @ ( aGroup_ext @ B @ C ) ),B2: A > ( carrier_ext @ B @ ( aGroup_ext @ B @ C ) )] :
( ! [X: A] :
( ( member @ A @ X @ I2 )
=> ( aGroup @ B @ C @ ( A2 @ X ) ) )
=> ( ! [X: A] :
( ( member @ A @ X @ I2 )
=> ( ( A2 @ X )
= ( B2 @ X ) ) )
=> ( ( prod_zero @ A @ B @ C @ I2 @ A2 )
= ( prod_zero @ A @ B @ C @ I2 @ B2 ) ) ) ) ).
% prodag_sameTr4
thf(fact_25_Ring_OprodM__carr,axiom,
! [B: $tType,E: $tType,A: $tType,D: $tType,C: $tType,R2: carrier_ext @ A @ ( aGroup_ext @ A @ ( ring_ext @ A @ B ) ),I2: set @ C,M3: C > ( carrier_ext @ D @ ( aGroup_ext @ D @ ( module_ext @ D @ A @ E ) ) )] :
( ( ring @ A @ B @ R2 )
=> ( ! [X: C] :
( ( member @ C @ X @ I2 )
=> ( module @ D @ A @ E @ B @ ( M3 @ X ) @ R2 ) )
=> ( ( carrier @ ( C > D ) @ ( aGroup_ext @ ( C > D ) @ ( module_ext @ ( C > D ) @ A @ product_unit ) ) @ ( algebr1000837334_prodM @ A @ B @ C @ D @ E @ R2 @ I2 @ M3 ) )
= ( carr_prodag @ C @ D @ ( module_ext @ D @ A @ E ) @ I2 @ M3 ) ) ) ) ).
% Ring.prodM_carr
thf(fact_26_Ring_OprodM__sprod__mem,axiom,
! [B: $tType,E: $tType,A: $tType,D: $tType,C: $tType,R2: carrier_ext @ A @ ( aGroup_ext @ A @ ( ring_ext @ A @ B ) ),I2: set @ C,M3: C > ( carrier_ext @ D @ ( aGroup_ext @ D @ ( module_ext @ D @ A @ E ) ) ),A3: A,M4: C > D] :
( ( ring @ A @ B @ R2 )
=> ( ! [X: C] :
( ( member @ C @ X @ I2 )
=> ( module @ D @ A @ E @ B @ ( M3 @ X ) @ R2 ) )
=> ( ( member @ A @ A3 @ ( carrier @ A @ ( aGroup_ext @ A @ ( ring_ext @ A @ B ) ) @ R2 ) )
=> ( ( member @ ( C > D ) @ M4 @ ( carr_prodag @ C @ D @ ( module_ext @ D @ A @ E ) @ I2 @ M3 ) )
=> ( member @ ( C > D ) @ ( algebr142713364_sprod @ A @ B @ C @ D @ E @ R2 @ I2 @ M3 @ A3 @ M4 ) @ ( carr_prodag @ C @ D @ ( module_ext @ D @ A @ E ) @ I2 @ M3 ) ) ) ) ) ) ).
% Ring.prodM_sprod_mem
thf(fact_27_prod__pOp__commute,axiom,
! [C: $tType,B: $tType,A: $tType,I2: set @ A,A2: A > ( carrier_ext @ B @ ( aGroup_ext @ B @ C ) ),A3: A > B,B3: A > B] :
( ! [X: A] :
( ( member @ A @ X @ I2 )
=> ( aGroup @ B @ C @ ( A2 @ X ) ) )
=> ( ( member @ ( A > B ) @ A3 @ ( carr_prodag @ A @ B @ C @ I2 @ A2 ) )
=> ( ( member @ ( A > B ) @ B3 @ ( carr_prodag @ A @ B @ C @ I2 @ A2 ) )
=> ( ( prod_pOp @ A @ B @ C @ I2 @ A2 @ A3 @ B3 )
= ( prod_pOp @ A @ B @ C @ I2 @ A2 @ B3 @ A3 ) ) ) ) ) ).
% prod_pOp_commute
thf(fact_28_prod__pOp__assoc,axiom,
! [C: $tType,B: $tType,A: $tType,I2: set @ A,A2: A > ( carrier_ext @ B @ ( aGroup_ext @ B @ C ) ),A3: A > B,B3: A > B,C3: A > B] :
( ! [X: A] :
( ( member @ A @ X @ I2 )
=> ( aGroup @ B @ C @ ( A2 @ X ) ) )
=> ( ( member @ ( A > B ) @ A3 @ ( carr_prodag @ A @ B @ C @ I2 @ A2 ) )
=> ( ( member @ ( A > B ) @ B3 @ ( carr_prodag @ A @ B @ C @ I2 @ A2 ) )
=> ( ( member @ ( A > B ) @ C3 @ ( carr_prodag @ A @ B @ C @ I2 @ A2 ) )
=> ( ( prod_pOp @ A @ B @ C @ I2 @ A2 @ ( prod_pOp @ A @ B @ C @ I2 @ A2 @ A3 @ B3 ) @ C3 )
= ( prod_pOp @ A @ B @ C @ I2 @ A2 @ A3 @ ( prod_pOp @ A @ B @ C @ I2 @ A2 @ B3 @ C3 ) ) ) ) ) ) ) ).
% prod_pOp_assoc
thf(fact_29_prod__pOp__mem,axiom,
! [C: $tType,B: $tType,A: $tType,I2: set @ A,A2: A > ( carrier_ext @ B @ ( aGroup_ext @ B @ C ) ),X4: A > B,Y: A > B] :
( ! [X: A] :
( ( member @ A @ X @ I2 )
=> ( aGroup @ B @ C @ ( A2 @ X ) ) )
=> ( ( member @ ( A > B ) @ X4 @ ( carr_prodag @ A @ B @ C @ I2 @ A2 ) )
=> ( ( member @ ( A > B ) @ Y @ ( carr_prodag @ A @ B @ C @ I2 @ A2 ) )
=> ( member @ ( A > B ) @ ( prod_pOp @ A @ B @ C @ I2 @ A2 @ X4 @ Y ) @ ( carr_prodag @ A @ B @ C @ I2 @ A2 ) ) ) ) ) ).
% prod_pOp_mem
thf(fact_30_prod__mOp__mem,axiom,
! [C: $tType,B: $tType,A: $tType,I2: set @ A,A2: A > ( carrier_ext @ B @ ( aGroup_ext @ B @ C ) ),X4: A > B] :
( ! [X: A] :
( ( member @ A @ X @ I2 )
=> ( aGroup @ B @ C @ ( A2 @ X ) ) )
=> ( ( member @ ( A > B ) @ X4 @ ( carr_prodag @ A @ B @ C @ I2 @ A2 ) )
=> ( member @ ( A > B ) @ ( prod_mOp @ A @ B @ C @ I2 @ A2 @ X4 ) @ ( carr_prodag @ A @ B @ C @ I2 @ A2 ) ) ) ) ).
% prod_mOp_mem
thf(fact_31_prod__zero__func,axiom,
! [C: $tType,B: $tType,A: $tType,I2: set @ A,A2: A > ( carrier_ext @ B @ ( aGroup_ext @ B @ C ) )] :
( ! [X: A] :
( ( member @ A @ X @ I2 )
=> ( aGroup @ B @ C @ ( A2 @ X ) ) )
=> ( member @ ( A > B ) @ ( prod_zero @ A @ B @ C @ I2 @ A2 ) @ ( carr_prodag @ A @ B @ C @ I2 @ A2 ) ) ) ).
% prod_zero_func
thf(fact_32_prod__mOp__func,axiom,
! [C: $tType,B: $tType,A: $tType,I2: set @ A,A2: A > ( carrier_ext @ B @ ( aGroup_ext @ B @ C ) )] :
( ! [X: A] :
( ( member @ A @ X @ I2 )
=> ( aGroup @ B @ C @ ( A2 @ X ) ) )
=> ( member @ ( ( A > B ) > A > B ) @ ( prod_mOp @ A @ B @ C @ I2 @ A2 )
@ ( pi @ ( A > B ) @ ( A > B ) @ ( carr_prodag @ A @ B @ C @ I2 @ A2 )
@ ^ [Uu: A > B] : ( carr_prodag @ A @ B @ C @ I2 @ A2 ) ) ) ) ).
% prod_mOp_func
thf(fact_33_Pi__cong,axiom,
! [B: $tType,A: $tType,A2: set @ A,F: A > B,G: A > B,B2: A > ( set @ B )] :
( ! [W: A] :
( ( member @ A @ W @ A2 )
=> ( ( F @ W )
= ( G @ W ) ) )
=> ( ( member @ ( A > B ) @ F @ ( pi @ A @ B @ A2 @ B2 ) )
= ( member @ ( A > B ) @ G @ ( pi @ A @ B @ A2 @ B2 ) ) ) ) ).
% Pi_cong
thf(fact_34_Pi__mem,axiom,
! [B: $tType,A: $tType,F: A > B,A2: set @ A,B2: A > ( set @ B ),X2: A] :
( ( member @ ( A > B ) @ F @ ( pi @ A @ B @ A2 @ B2 ) )
=> ( ( member @ A @ X2 @ A2 )
=> ( member @ B @ ( F @ X2 ) @ ( B2 @ X2 ) ) ) ) ).
% Pi_mem
thf(fact_35_Pi__iff,axiom,
! [B: $tType,A: $tType,F: A > B,I2: set @ A,X4: A > ( set @ B )] :
( ( member @ ( A > B ) @ F @ ( pi @ A @ B @ I2 @ X4 ) )
= ( ! [X3: A] :
( ( member @ A @ X3 @ I2 )
=> ( member @ B @ ( F @ X3 ) @ ( X4 @ X3 ) ) ) ) ) ).
% Pi_iff
thf(fact_36_Pi__I_H,axiom,
! [B: $tType,A: $tType,A2: set @ A,F: A > B,B2: A > ( set @ B )] :
( ! [X: A] :
( ( member @ A @ X @ A2 )
=> ( member @ B @ ( F @ X ) @ ( B2 @ X ) ) )
=> ( member @ ( A > B ) @ F @ ( pi @ A @ B @ A2 @ B2 ) ) ) ).
% Pi_I'
thf(fact_37_PiE,axiom,
! [B: $tType,A: $tType,F: A > B,A2: set @ A,B2: A > ( set @ B ),X2: A] :
( ( member @ ( A > B ) @ F @ ( pi @ A @ B @ A2 @ B2 ) )
=> ( ~ ( member @ B @ ( F @ X2 ) @ ( B2 @ X2 ) )
=> ~ ( member @ A @ X2 @ A2 ) ) ) ).
% PiE
thf(fact_38_mem__subring__mem__ring,axiom,
! [C: $tType,S: carrier_ext @ a @ ( aGroup_ext @ a @ ( ring_ext @ a @ C ) ),X2: a] :
( ( subring @ a @ b @ C @ r @ S )
=> ( ( member @ a @ X2 @ ( carrier @ a @ ( aGroup_ext @ a @ ( ring_ext @ a @ C ) ) @ S ) )
=> ( member @ a @ X2 @ ( carrier @ a @ ( aGroup_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) ) ) ) ).
% mem_subring_mem_ring
thf(fact_39_subring__Ring,axiom,
! [C: $tType,S: carrier_ext @ a @ ( aGroup_ext @ a @ ( ring_ext @ a @ C ) )] :
( ( subring @ a @ b @ C @ r @ S )
=> ( ring @ a @ C @ S ) ) ).
% subring_Ring
thf(fact_40_prodM__sprod__val,axiom,
! [E: $tType,D: $tType,C: $tType,I2: set @ C,M3: C > ( carrier_ext @ D @ ( aGroup_ext @ D @ ( module_ext @ D @ a @ E ) ) ),A3: a,M4: C > D,J: C] :
( ! [X: C] :
( ( member @ C @ X @ I2 )
=> ( module @ D @ a @ E @ b @ ( M3 @ X ) @ r ) )
=> ( ( member @ a @ A3 @ ( carrier @ a @ ( aGroup_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
=> ( ( member @ ( C > D ) @ M4 @ ( carr_prodag @ C @ D @ ( module_ext @ D @ a @ E ) @ I2 @ M3 ) )
=> ( ( member @ C @ J @ I2 )
=> ( ( algebr142713364_sprod @ a @ b @ C @ D @ E @ r @ I2 @ M3 @ A3 @ M4 @ J )
= ( sprod @ D @ a @ E @ ( M3 @ J ) @ A3 @ ( M4 @ J ) ) ) ) ) ) ) ).
% prodM_sprod_val
thf(fact_41_Sr__ring,axiom,
! [S: set @ a] :
( ( sr2 @ a @ b @ r @ S )
=> ( ring @ a @ b @ ( sr @ a @ b @ r @ S ) ) ) ).
% Sr_ring
thf(fact_42_mHom__func,axiom,
! [E: $tType,D: $tType,F2: $tType,C: $tType,F: C > D,M3: carrier_ext @ C @ ( aGroup_ext @ C @ ( module_ext @ C @ a @ E ) ),N: carrier_ext @ D @ ( aGroup_ext @ D @ ( module_ext @ D @ a @ F2 ) )] :
( ( member @ ( C > D ) @ F @ ( mHom @ a @ b @ C @ E @ D @ F2 @ r @ M3 @ N ) )
=> ( member @ ( C > D ) @ F
@ ( pi @ C @ D @ ( carrier @ C @ ( aGroup_ext @ C @ ( module_ext @ C @ a @ E ) ) @ M3 )
@ ^ [Uu: C] : ( carrier @ D @ ( aGroup_ext @ D @ ( module_ext @ D @ a @ F2 ) ) @ N ) ) ) ) ).
% mHom_func
thf(fact_43_rInvim__def,axiom,
! [B: $tType,M1: $tType,M: $tType,A: $tType] :
( ( rInvim @ A @ M @ B @ M1 )
= ( ^ [A4: carrier_ext @ A @ ( aGroup_ext @ A @ ( ring_ext @ A @ M ) ),R3: carrier_ext @ B @ ( aGroup_ext @ B @ ( ring_ext @ B @ M1 ) ),F3: A > B,K: set @ B] :
( collect @ A
@ ^ [A5: A] :
( ( member @ A @ A5 @ ( carrier @ A @ ( aGroup_ext @ A @ ( ring_ext @ A @ M ) ) @ A4 ) )
& ( member @ B @ ( F3 @ A5 ) @ K ) ) ) ) ) ).
% rInvim_def
thf(fact_44_dsum__pOp__func,axiom,
! [C: $tType,B: $tType,A: $tType,I2: set @ A,A2: A > ( carrier_ext @ B @ ( aGroup_ext @ B @ C ) )] :
( ! [X: A] :
( ( member @ A @ X @ I2 )
=> ( aGroup @ B @ C @ ( A2 @ X ) ) )
=> ( member @ ( ( A > B ) > ( A > B ) > A > B ) @ ( prod_pOp @ A @ B @ C @ I2 @ A2 )
@ ( pi @ ( A > B ) @ ( ( A > B ) > A > B ) @ ( carr_dsumag @ A @ B @ C @ I2 @ A2 )
@ ^ [Uu: A > B] :
( pi @ ( A > B ) @ ( A > B ) @ ( carr_dsumag @ A @ B @ C @ I2 @ A2 )
@ ^ [Uv: A > B] : ( carr_dsumag @ A @ B @ C @ I2 @ A2 ) ) ) ) ) ).
% dsum_pOp_func
thf(fact_45_mem__Collect__eq,axiom,
! [A: $tType,A3: A,P: A > $o] :
( ( member @ A @ A3 @ ( collect @ A @ P ) )
= ( P @ A3 ) ) ).
% mem_Collect_eq
thf(fact_46_Collect__mem__eq,axiom,
! [A: $tType,A2: set @ A] :
( ( collect @ A
@ ^ [X3: A] : ( member @ A @ X3 @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_47_Collect__cong,axiom,
! [A: $tType,P: A > $o,Q: A > $o] :
( ! [X: A] :
( ( P @ X )
= ( Q @ X ) )
=> ( ( collect @ A @ P )
= ( collect @ A @ Q ) ) ) ).
% Collect_cong
thf(fact_48_ext,axiom,
! [B: $tType,A: $tType,F: A > B,G: A > B] :
( ! [X: A] :
( ( F @ X )
= ( G @ X ) )
=> ( F = G ) ) ).
% ext
thf(fact_49_nsEqElm,axiom,
! [X2: a,Y2: a,N2: nat] :
( ( member @ a @ X2 @ ( carrier @ a @ ( aGroup_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
=> ( ( member @ a @ Y2 @ ( carrier @ a @ ( aGroup_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
=> ( ( X2 = Y2 )
=> ( ( nscal @ a @ b @ r @ X2 @ N2 )
= ( nscal @ a @ b @ r @ Y2 @ N2 ) ) ) ) ) ).
% nsEqElm
thf(fact_50_nsClose,axiom,
! [X2: a,N2: nat] :
( ( member @ a @ X2 @ ( carrier @ a @ ( aGroup_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
=> ( member @ a @ ( nscal @ a @ b @ r @ X2 @ N2 ) @ ( carrier @ a @ ( aGroup_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) ) ) ).
% nsClose
thf(fact_51_tp__closed,axiom,
( member @ ( a > a > a ) @ ( tp @ a @ b @ r )
@ ( pi @ a @ ( a > a ) @ ( carrier @ a @ ( aGroup_ext @ a @ ( ring_ext @ a @ b ) ) @ r )
@ ^ [Uu: a] :
( pi @ a @ a @ ( carrier @ a @ ( aGroup_ext @ a @ ( ring_ext @ a @ b ) ) @ r )
@ ^ [Uv: a] : ( carrier @ a @ ( aGroup_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) ) ) ) ).
% tp_closed
thf(fact_52_pop__closed,axiom,
( member @ ( a > a > a ) @ ( pop @ a @ ( ring_ext @ a @ b ) @ r )
@ ( pi @ a @ ( a > a ) @ ( carrier @ a @ ( aGroup_ext @ a @ ( ring_ext @ a @ b ) ) @ r )
@ ^ [Uu: a] :
( pi @ a @ a @ ( carrier @ a @ ( aGroup_ext @ a @ ( ring_ext @ a @ b ) ) @ r )
@ ^ [Uv: a] : ( carrier @ a @ ( aGroup_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) ) ) ) ).
% pop_closed
thf(fact_53_rEQMulR,axiom,
! [X2: a,Y2: a,Z2: a] :
( ( member @ a @ X2 @ ( carrier @ a @ ( aGroup_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
=> ( ( member @ a @ Y2 @ ( carrier @ a @ ( aGroup_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
=> ( ( member @ a @ Z2 @ ( carrier @ a @ ( aGroup_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
=> ( ( X2 = Y2 )
=> ( ( tp @ a @ b @ r @ X2 @ Z2 )
= ( tp @ a @ b @ r @ Y2 @ Z2 ) ) ) ) ) ) ).
% rEQMulR
thf(fact_54_rMulLC,axiom,
! [X2: a,Y2: a,Z2: a] :
( ( member @ a @ X2 @ ( carrier @ a @ ( aGroup_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
=> ( ( member @ a @ Y2 @ ( carrier @ a @ ( aGroup_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
=> ( ( member @ a @ Z2 @ ( carrier @ a @ ( aGroup_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
=> ( ( tp @ a @ b @ r @ X2 @ ( tp @ a @ b @ r @ Y2 @ Z2 ) )
= ( tp @ a @ b @ r @ Y2 @ ( tp @ a @ b @ r @ X2 @ Z2 ) ) ) ) ) ) ).
% rMulLC
thf(fact_55_ring__tOp__assoc,axiom,
! [X2: a,Y2: a,Z2: a] :
( ( member @ a @ X2 @ ( carrier @ a @ ( aGroup_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
=> ( ( member @ a @ Y2 @ ( carrier @ a @ ( aGroup_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
=> ( ( member @ a @ Z2 @ ( carrier @ a @ ( aGroup_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
=> ( ( tp @ a @ b @ r @ ( tp @ a @ b @ r @ X2 @ Y2 ) @ Z2 )
= ( tp @ a @ b @ r @ X2 @ ( tp @ a @ b @ r @ Y2 @ Z2 ) ) ) ) ) ) ).
% ring_tOp_assoc
thf(fact_56_ring__tOp__closed,axiom,
! [X2: a,Y2: a] :
( ( member @ a @ X2 @ ( carrier @ a @ ( aGroup_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
=> ( ( member @ a @ Y2 @ ( carrier @ a @ ( aGroup_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
=> ( member @ a @ ( tp @ a @ b @ r @ X2 @ Y2 ) @ ( carrier @ a @ ( aGroup_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) ) ) ) ).
% ring_tOp_closed
thf(fact_57_ring__tOp__commute,axiom,
! [X2: a,Y2: a] :
( ( member @ a @ X2 @ ( carrier @ a @ ( aGroup_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
=> ( ( member @ a @ Y2 @ ( carrier @ a @ ( aGroup_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
=> ( ( tp @ a @ b @ r @ X2 @ Y2 )
= ( tp @ a @ b @ r @ Y2 @ X2 ) ) ) ) ).
% ring_tOp_commute
thf(fact_58_ring__tOp__rel,axiom,
! [X2: a,Xa: a,Y2: a,Ya: a] :
( ( member @ a @ X2 @ ( carrier @ a @ ( aGroup_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
=> ( ( member @ a @ Xa @ ( carrier @ a @ ( aGroup_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
=> ( ( member @ a @ Y2 @ ( carrier @ a @ ( aGroup_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
=> ( ( member @ a @ Ya @ ( carrier @ a @ ( aGroup_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
=> ( ( tp @ a @ b @ r @ ( tp @ a @ b @ r @ X2 @ Xa ) @ ( tp @ a @ b @ r @ Y2 @ Ya ) )
= ( tp @ a @ b @ r @ ( tp @ a @ b @ r @ X2 @ Y2 ) @ ( tp @ a @ b @ r @ Xa @ Ya ) ) ) ) ) ) ) ).
% ring_tOp_rel
thf(fact_59_tp__assoc,axiom,
! [A3: a,B3: a,C3: a] :
( ( member @ a @ A3 @ ( carrier @ a @ ( aGroup_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
=> ( ( member @ a @ B3 @ ( carrier @ a @ ( aGroup_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
=> ( ( member @ a @ C3 @ ( carrier @ a @ ( aGroup_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
=> ( ( tp @ a @ b @ r @ ( tp @ a @ b @ r @ A3 @ B3 ) @ C3 )
= ( tp @ a @ b @ r @ A3 @ ( tp @ a @ b @ r @ B3 @ C3 ) ) ) ) ) ) ).
% tp_assoc
thf(fact_60_tp__commute,axiom,
! [A3: a,B3: a] :
( ( member @ a @ A3 @ ( carrier @ a @ ( aGroup_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
=> ( ( member @ a @ B3 @ ( carrier @ a @ ( aGroup_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
=> ( ( tp @ a @ b @ r @ A3 @ B3 )
= ( tp @ a @ b @ r @ B3 @ A3 ) ) ) ) ).
% tp_commute
thf(fact_61_pop__aassoc,axiom,
! [A3: a,B3: a,C3: a] :
( ( member @ a @ A3 @ ( carrier @ a @ ( aGroup_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
=> ( ( member @ a @ B3 @ ( carrier @ a @ ( aGroup_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
=> ( ( member @ a @ C3 @ ( carrier @ a @ ( aGroup_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
=> ( ( pop @ a @ ( ring_ext @ a @ b ) @ r @ ( pop @ a @ ( ring_ext @ a @ b ) @ r @ A3 @ B3 ) @ C3 )
= ( pop @ a @ ( ring_ext @ a @ b ) @ r @ A3 @ ( pop @ a @ ( ring_ext @ a @ b ) @ r @ B3 @ C3 ) ) ) ) ) ) ).
% pop_aassoc
thf(fact_62_pop__commute,axiom,
! [A3: a,B3: a] :
( ( member @ a @ A3 @ ( carrier @ a @ ( aGroup_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
=> ( ( member @ a @ B3 @ ( carrier @ a @ ( aGroup_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
=> ( ( pop @ a @ ( ring_ext @ a @ b ) @ r @ A3 @ B3 )
= ( pop @ a @ ( ring_ext @ a @ b ) @ r @ B3 @ A3 ) ) ) ) ).
% pop_commute
thf(fact_63_Sr__tOp__closed,axiom,
! [S: set @ a,X2: a,Y2: a] :
( ( sr2 @ a @ b @ r @ S )
=> ( ( member @ a @ X2 @ S )
=> ( ( member @ a @ Y2 @ S )
=> ( member @ a @ ( tp @ a @ b @ r @ X2 @ Y2 ) @ S ) ) ) ) ).
% Sr_tOp_closed
thf(fact_64_Sr__pOp__closed,axiom,
! [S: set @ a,X2: a,Y2: a] :
( ( sr2 @ a @ b @ r @ S )
=> ( ( member @ a @ X2 @ S )
=> ( ( member @ a @ Y2 @ S )
=> ( member @ a @ ( pop @ a @ ( ring_ext @ a @ b ) @ r @ X2 @ Y2 ) @ S ) ) ) ) ).
% Sr_pOp_closed
thf(fact_65_rg__distrib,axiom,
! [A3: a,B3: a,C3: a] :
( ( member @ a @ A3 @ ( carrier @ a @ ( aGroup_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
=> ( ( member @ a @ B3 @ ( carrier @ a @ ( aGroup_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
=> ( ( member @ a @ C3 @ ( carrier @ a @ ( aGroup_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
=> ( ( tp @ a @ b @ r @ A3 @ ( pop @ a @ ( ring_ext @ a @ b ) @ r @ B3 @ C3 ) )
= ( pop @ a @ ( ring_ext @ a @ b ) @ r @ ( tp @ a @ b @ r @ A3 @ B3 ) @ ( tp @ a @ b @ r @ A3 @ C3 ) ) ) ) ) ) ).
% rg_distrib
thf(fact_66_ring__distrib1,axiom,
! [X2: a,Y2: a,Z2: a] :
( ( member @ a @ X2 @ ( carrier @ a @ ( aGroup_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
=> ( ( member @ a @ Y2 @ ( carrier @ a @ ( aGroup_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
=> ( ( member @ a @ Z2 @ ( carrier @ a @ ( aGroup_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
=> ( ( tp @ a @ b @ r @ X2 @ ( pop @ a @ ( ring_ext @ a @ b ) @ r @ Y2 @ Z2 ) )
= ( pop @ a @ ( ring_ext @ a @ b ) @ r @ ( tp @ a @ b @ r @ X2 @ Y2 ) @ ( tp @ a @ b @ r @ X2 @ Z2 ) ) ) ) ) ) ).
% ring_distrib1
thf(fact_67_ring__distrib2,axiom,
! [X2: a,Y2: a,Z2: a] :
( ( member @ a @ X2 @ ( carrier @ a @ ( aGroup_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
=> ( ( member @ a @ Y2 @ ( carrier @ a @ ( aGroup_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
=> ( ( member @ a @ Z2 @ ( carrier @ a @ ( aGroup_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
=> ( ( tp @ a @ b @ r @ ( pop @ a @ ( ring_ext @ a @ b ) @ r @ Y2 @ Z2 ) @ X2 )
= ( pop @ a @ ( ring_ext @ a @ b ) @ r @ ( tp @ a @ b @ r @ Y2 @ X2 ) @ ( tp @ a @ b @ r @ Z2 @ X2 ) ) ) ) ) ) ).
% ring_distrib2
thf(fact_68_ring__distrib3,axiom,
! [A3: a,B3: a,X2: a,Y2: a] :
( ( member @ a @ A3 @ ( carrier @ a @ ( aGroup_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
=> ( ( member @ a @ B3 @ ( carrier @ a @ ( aGroup_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
=> ( ( member @ a @ X2 @ ( carrier @ a @ ( aGroup_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
=> ( ( member @ a @ Y2 @ ( carrier @ a @ ( aGroup_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
=> ( ( tp @ a @ b @ r @ ( pop @ a @ ( ring_ext @ a @ b ) @ r @ A3 @ B3 ) @ ( pop @ a @ ( ring_ext @ a @ b ) @ r @ X2 @ Y2 ) )
= ( pop @ a @ ( ring_ext @ a @ b ) @ r @ ( pop @ a @ ( ring_ext @ a @ b ) @ r @ ( pop @ a @ ( ring_ext @ a @ b ) @ r @ ( tp @ a @ b @ r @ A3 @ X2 ) @ ( tp @ a @ b @ r @ A3 @ Y2 ) ) @ ( tp @ a @ b @ r @ B3 @ X2 ) ) @ ( tp @ a @ b @ r @ B3 @ Y2 ) ) ) ) ) ) ) ).
% ring_distrib3
thf(fact_69_nsMulDistrL,axiom,
! [X2: a,Y2: a,N2: nat] :
( ( member @ a @ X2 @ ( carrier @ a @ ( aGroup_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
=> ( ( member @ a @ Y2 @ ( carrier @ a @ ( aGroup_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
=> ( ( tp @ a @ b @ r @ X2 @ ( nscal @ a @ b @ r @ Y2 @ N2 ) )
= ( nscal @ a @ b @ r @ ( tp @ a @ b @ r @ X2 @ Y2 ) @ N2 ) ) ) ) ).
% nsMulDistrL
thf(fact_70_nsMulDistrR,axiom,
! [X2: a,Y2: a,N2: nat] :
( ( member @ a @ X2 @ ( carrier @ a @ ( aGroup_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
=> ( ( member @ a @ Y2 @ ( carrier @ a @ ( aGroup_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
=> ( ( tp @ a @ b @ r @ ( nscal @ a @ b @ r @ Y2 @ N2 ) @ X2 )
= ( nscal @ a @ b @ r @ ( tp @ a @ b @ r @ Y2 @ X2 ) @ N2 ) ) ) ) ).
% nsMulDistrR
thf(fact_71_nsDistrL,axiom,
! [X2: a,Y2: a,N2: nat] :
( ( member @ a @ X2 @ ( carrier @ a @ ( aGroup_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
=> ( ( member @ a @ Y2 @ ( carrier @ a @ ( aGroup_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
=> ( ( pop @ a @ ( ring_ext @ a @ b ) @ r @ ( nscal @ a @ b @ r @ X2 @ N2 ) @ ( nscal @ a @ b @ r @ Y2 @ N2 ) )
= ( nscal @ a @ b @ r @ ( pop @ a @ ( ring_ext @ a @ b ) @ r @ X2 @ Y2 ) @ N2 ) ) ) ) ).
% nsDistrL
thf(fact_72_Subring__tOp__ring__tOp,axiom,
! [C: $tType,S: carrier_ext @ a @ ( aGroup_ext @ a @ ( ring_ext @ a @ C ) ),A3: a,B3: a] :
( ( subring @ a @ b @ C @ r @ S )
=> ( ( member @ a @ A3 @ ( carrier @ a @ ( aGroup_ext @ a @ ( ring_ext @ a @ C ) ) @ S ) )
=> ( ( member @ a @ B3 @ ( carrier @ a @ ( aGroup_ext @ a @ ( ring_ext @ a @ C ) ) @ S ) )
=> ( ( tp @ a @ C @ S @ A3 @ B3 )
= ( tp @ a @ b @ r @ A3 @ B3 ) ) ) ) ) ).
% Subring_tOp_ring_tOp
thf(fact_73_Subring__pOp__ring__pOp,axiom,
! [C: $tType,S: carrier_ext @ a @ ( aGroup_ext @ a @ ( ring_ext @ a @ C ) ),A3: a,B3: a] :
( ( subring @ a @ b @ C @ r @ S )
=> ( ( member @ a @ A3 @ ( carrier @ a @ ( aGroup_ext @ a @ ( ring_ext @ a @ C ) ) @ S ) )
=> ( ( member @ a @ B3 @ ( carrier @ a @ ( aGroup_ext @ a @ ( ring_ext @ a @ C ) ) @ S ) )
=> ( ( pop @ a @ ( ring_ext @ a @ C ) @ S @ A3 @ B3 )
= ( pop @ a @ ( ring_ext @ a @ b ) @ r @ A3 @ B3 ) ) ) ) ) ).
% Subring_pOp_ring_pOp
thf(fact_74_mul__closed__set__tOp__closed,axiom,
! [S: set @ a,S2: a,T: a] :
( ( mul_closed_set @ a @ b @ r @ S )
=> ( ( member @ a @ S2 @ S )
=> ( ( member @ a @ T @ S )
=> ( member @ a @ ( tp @ a @ b @ r @ S2 @ T ) @ S ) ) ) ) ).
% mul_closed_set_tOp_closed
thf(fact_75_Ring_OSr__tOp__closed,axiom,
! [B: $tType,A: $tType,R2: carrier_ext @ A @ ( aGroup_ext @ A @ ( ring_ext @ A @ B ) ),S: set @ A,X2: A,Y2: A] :
( ( ring @ A @ B @ R2 )
=> ( ( sr2 @ A @ B @ R2 @ S )
=> ( ( member @ A @ X2 @ S )
=> ( ( member @ A @ Y2 @ S )
=> ( member @ A @ ( tp @ A @ B @ R2 @ X2 @ Y2 ) @ S ) ) ) ) ) ).
% Ring.Sr_tOp_closed
thf(fact_76_Ring_OSr__pOp__closed,axiom,
! [B: $tType,A: $tType,R2: carrier_ext @ A @ ( aGroup_ext @ A @ ( ring_ext @ A @ B ) ),S: set @ A,X2: A,Y2: A] :
( ( ring @ A @ B @ R2 )
=> ( ( sr2 @ A @ B @ R2 @ S )
=> ( ( member @ A @ X2 @ S )
=> ( ( member @ A @ Y2 @ S )
=> ( member @ A @ ( pop @ A @ ( ring_ext @ A @ B ) @ R2 @ X2 @ Y2 ) @ S ) ) ) ) ) ).
% Ring.Sr_pOp_closed
thf(fact_77_Ring_Oring__distrib3,axiom,
! [B: $tType,A: $tType,R2: carrier_ext @ A @ ( aGroup_ext @ A @ ( ring_ext @ A @ B ) ),A3: A,B3: A,X2: A,Y2: A] :
( ( ring @ A @ B @ R2 )
=> ( ( member @ A @ A3 @ ( carrier @ A @ ( aGroup_ext @ A @ ( ring_ext @ A @ B ) ) @ R2 ) )
=> ( ( member @ A @ B3 @ ( carrier @ A @ ( aGroup_ext @ A @ ( ring_ext @ A @ B ) ) @ R2 ) )
=> ( ( member @ A @ X2 @ ( carrier @ A @ ( aGroup_ext @ A @ ( ring_ext @ A @ B ) ) @ R2 ) )
=> ( ( member @ A @ Y2 @ ( carrier @ A @ ( aGroup_ext @ A @ ( ring_ext @ A @ B ) ) @ R2 ) )
=> ( ( tp @ A @ B @ R2 @ ( pop @ A @ ( ring_ext @ A @ B ) @ R2 @ A3 @ B3 ) @ ( pop @ A @ ( ring_ext @ A @ B ) @ R2 @ X2 @ Y2 ) )
= ( pop @ A @ ( ring_ext @ A @ B ) @ R2 @ ( pop @ A @ ( ring_ext @ A @ B ) @ R2 @ ( pop @ A @ ( ring_ext @ A @ B ) @ R2 @ ( tp @ A @ B @ R2 @ A3 @ X2 ) @ ( tp @ A @ B @ R2 @ A3 @ Y2 ) ) @ ( tp @ A @ B @ R2 @ B3 @ X2 ) ) @ ( tp @ A @ B @ R2 @ B3 @ Y2 ) ) ) ) ) ) ) ) ).
% Ring.ring_distrib3
thf(fact_78_Ring_Oring__distrib2,axiom,
! [B: $tType,A: $tType,R2: carrier_ext @ A @ ( aGroup_ext @ A @ ( ring_ext @ A @ B ) ),X2: A,Y2: A,Z2: A] :
( ( ring @ A @ B @ R2 )
=> ( ( member @ A @ X2 @ ( carrier @ A @ ( aGroup_ext @ A @ ( ring_ext @ A @ B ) ) @ R2 ) )
=> ( ( member @ A @ Y2 @ ( carrier @ A @ ( aGroup_ext @ A @ ( ring_ext @ A @ B ) ) @ R2 ) )
=> ( ( member @ A @ Z2 @ ( carrier @ A @ ( aGroup_ext @ A @ ( ring_ext @ A @ B ) ) @ R2 ) )
=> ( ( tp @ A @ B @ R2 @ ( pop @ A @ ( ring_ext @ A @ B ) @ R2 @ Y2 @ Z2 ) @ X2 )
= ( pop @ A @ ( ring_ext @ A @ B ) @ R2 @ ( tp @ A @ B @ R2 @ Y2 @ X2 ) @ ( tp @ A @ B @ R2 @ Z2 @ X2 ) ) ) ) ) ) ) ).
% Ring.ring_distrib2
thf(fact_79_Ring_Oring__distrib1,axiom,
! [B: $tType,A: $tType,R2: carrier_ext @ A @ ( aGroup_ext @ A @ ( ring_ext @ A @ B ) ),X2: A,Y2: A,Z2: A] :
( ( ring @ A @ B @ R2 )
=> ( ( member @ A @ X2 @ ( carrier @ A @ ( aGroup_ext @ A @ ( ring_ext @ A @ B ) ) @ R2 ) )
=> ( ( member @ A @ Y2 @ ( carrier @ A @ ( aGroup_ext @ A @ ( ring_ext @ A @ B ) ) @ R2 ) )
=> ( ( member @ A @ Z2 @ ( carrier @ A @ ( aGroup_ext @ A @ ( ring_ext @ A @ B ) ) @ R2 ) )
=> ( ( tp @ A @ B @ R2 @ X2 @ ( pop @ A @ ( ring_ext @ A @ B ) @ R2 @ Y2 @ Z2 ) )
= ( pop @ A @ ( ring_ext @ A @ B ) @ R2 @ ( tp @ A @ B @ R2 @ X2 @ Y2 ) @ ( tp @ A @ B @ R2 @ X2 @ Z2 ) ) ) ) ) ) ) ).
% Ring.ring_distrib1
thf(fact_80_Ring_Org__distrib,axiom,
! [B: $tType,A: $tType,R2: carrier_ext @ A @ ( aGroup_ext @ A @ ( ring_ext @ A @ B ) ),A3: A,B3: A,C3: A] :
( ( ring @ A @ B @ R2 )
=> ( ( member @ A @ A3 @ ( carrier @ A @ ( aGroup_ext @ A @ ( ring_ext @ A @ B ) ) @ R2 ) )
=> ( ( member @ A @ B3 @ ( carrier @ A @ ( aGroup_ext @ A @ ( ring_ext @ A @ B ) ) @ R2 ) )
=> ( ( member @ A @ C3 @ ( carrier @ A @ ( aGroup_ext @ A @ ( ring_ext @ A @ B ) ) @ R2 ) )
=> ( ( tp @ A @ B @ R2 @ A3 @ ( pop @ A @ ( ring_ext @ A @ B ) @ R2 @ B3 @ C3 ) )
= ( pop @ A @ ( ring_ext @ A @ B ) @ R2 @ ( tp @ A @ B @ R2 @ A3 @ B3 ) @ ( tp @ A @ B @ R2 @ A3 @ C3 ) ) ) ) ) ) ) ).
% Ring.rg_distrib
thf(fact_81_Ring_OnsMulDistrR,axiom,
! [A: $tType,B: $tType,R2: carrier_ext @ A @ ( aGroup_ext @ A @ ( ring_ext @ A @ B ) ),X2: A,Y2: A,N2: nat] :
( ( ring @ A @ B @ R2 )
=> ( ( member @ A @ X2 @ ( carrier @ A @ ( aGroup_ext @ A @ ( ring_ext @ A @ B ) ) @ R2 ) )
=> ( ( member @ A @ Y2 @ ( carrier @ A @ ( aGroup_ext @ A @ ( ring_ext @ A @ B ) ) @ R2 ) )
=> ( ( tp @ A @ B @ R2 @ ( nscal @ A @ B @ R2 @ Y2 @ N2 ) @ X2 )
= ( nscal @ A @ B @ R2 @ ( tp @ A @ B @ R2 @ Y2 @ X2 ) @ N2 ) ) ) ) ) ).
% Ring.nsMulDistrR
thf(fact_82_Ring_OnsMulDistrL,axiom,
! [A: $tType,B: $tType,R2: carrier_ext @ A @ ( aGroup_ext @ A @ ( ring_ext @ A @ B ) ),X2: A,Y2: A,N2: nat] :
( ( ring @ A @ B @ R2 )
=> ( ( member @ A @ X2 @ ( carrier @ A @ ( aGroup_ext @ A @ ( ring_ext @ A @ B ) ) @ R2 ) )
=> ( ( member @ A @ Y2 @ ( carrier @ A @ ( aGroup_ext @ A @ ( ring_ext @ A @ B ) ) @ R2 ) )
=> ( ( tp @ A @ B @ R2 @ X2 @ ( nscal @ A @ B @ R2 @ Y2 @ N2 ) )
= ( nscal @ A @ B @ R2 @ ( tp @ A @ B @ R2 @ X2 @ Y2 ) @ N2 ) ) ) ) ) ).
% Ring.nsMulDistrL
thf(fact_83_Ring_OnsDistrL,axiom,
! [A: $tType,B: $tType,R2: carrier_ext @ A @ ( aGroup_ext @ A @ ( ring_ext @ A @ B ) ),X2: A,Y2: A,N2: nat] :
( ( ring @ A @ B @ R2 )
=> ( ( member @ A @ X2 @ ( carrier @ A @ ( aGroup_ext @ A @ ( ring_ext @ A @ B ) ) @ R2 ) )
=> ( ( member @ A @ Y2 @ ( carrier @ A @ ( aGroup_ext @ A @ ( ring_ext @ A @ B ) ) @ R2 ) )
=> ( ( pop @ A @ ( ring_ext @ A @ B ) @ R2 @ ( nscal @ A @ B @ R2 @ X2 @ N2 ) @ ( nscal @ A @ B @ R2 @ Y2 @ N2 ) )
= ( nscal @ A @ B @ R2 @ ( pop @ A @ ( ring_ext @ A @ B ) @ R2 @ X2 @ Y2 ) @ N2 ) ) ) ) ) ).
% Ring.nsDistrL
thf(fact_84_Ring_OSr__ring,axiom,
! [B: $tType,A: $tType,R2: carrier_ext @ A @ ( aGroup_ext @ A @ ( ring_ext @ A @ B ) ),S: set @ A] :
( ( ring @ A @ B @ R2 )
=> ( ( sr2 @ A @ B @ R2 @ S )
=> ( ring @ A @ B @ ( sr @ A @ B @ R2 @ S ) ) ) ) ).
% Ring.Sr_ring
thf(fact_85_aGroup_Oag__pOp__commute,axiom,
! [B: $tType,A: $tType,A2: carrier_ext @ A @ ( aGroup_ext @ A @ B ),X2: A,Y2: A] :
( ( aGroup @ A @ B @ A2 )
=> ( ( member @ A @ X2 @ ( carrier @ A @ ( aGroup_ext @ A @ B ) @ A2 ) )
=> ( ( member @ A @ Y2 @ ( carrier @ A @ ( aGroup_ext @ A @ B ) @ A2 ) )
=> ( ( pop @ A @ B @ A2 @ X2 @ Y2 )
= ( pop @ A @ B @ A2 @ Y2 @ X2 ) ) ) ) ) ).
% aGroup.ag_pOp_commute
thf(fact_86_aGroup_Oag__add__commute,axiom,
! [B: $tType,A: $tType,A2: carrier_ext @ A @ ( aGroup_ext @ A @ B ),A3: A,B3: A] :
( ( aGroup @ A @ B @ A2 )
=> ( ( member @ A @ A3 @ ( carrier @ A @ ( aGroup_ext @ A @ B ) @ A2 ) )
=> ( ( member @ A @ B3 @ ( carrier @ A @ ( aGroup_ext @ A @ B ) @ A2 ) )
=> ( ( pop @ A @ B @ A2 @ A3 @ B3 )
= ( pop @ A @ B @ A2 @ B3 @ A3 ) ) ) ) ) ).
% aGroup.ag_add_commute
thf(fact_87_aGroup_OpOp__assocTr43,axiom,
! [B: $tType,A: $tType,A2: carrier_ext @ A @ ( aGroup_ext @ A @ B ),A3: A,B3: A,C3: A,D2: A] :
( ( aGroup @ A @ B @ A2 )
=> ( ( member @ A @ A3 @ ( carrier @ A @ ( aGroup_ext @ A @ B ) @ A2 ) )
=> ( ( member @ A @ B3 @ ( carrier @ A @ ( aGroup_ext @ A @ B ) @ A2 ) )
=> ( ( member @ A @ C3 @ ( carrier @ A @ ( aGroup_ext @ A @ B ) @ A2 ) )
=> ( ( member @ A @ D2 @ ( carrier @ A @ ( aGroup_ext @ A @ B ) @ A2 ) )
=> ( ( pop @ A @ B @ A2 @ ( pop @ A @ B @ A2 @ A3 @ B3 ) @ ( pop @ A @ B @ A2 @ C3 @ D2 ) )
= ( pop @ A @ B @ A2 @ ( pop @ A @ B @ A2 @ A3 @ ( pop @ A @ B @ A2 @ B3 @ C3 ) ) @ D2 ) ) ) ) ) ) ) ).
% aGroup.pOp_assocTr43
thf(fact_88_aGroup_OpOp__assocTr42,axiom,
! [B: $tType,A: $tType,A2: carrier_ext @ A @ ( aGroup_ext @ A @ B ),A3: A,B3: A,C3: A,D2: A] :
( ( aGroup @ A @ B @ A2 )
=> ( ( member @ A @ A3 @ ( carrier @ A @ ( aGroup_ext @ A @ B ) @ A2 ) )
=> ( ( member @ A @ B3 @ ( carrier @ A @ ( aGroup_ext @ A @ B ) @ A2 ) )
=> ( ( member @ A @ C3 @ ( carrier @ A @ ( aGroup_ext @ A @ B ) @ A2 ) )
=> ( ( member @ A @ D2 @ ( carrier @ A @ ( aGroup_ext @ A @ B ) @ A2 ) )
=> ( ( pop @ A @ B @ A2 @ ( pop @ A @ B @ A2 @ ( pop @ A @ B @ A2 @ A3 @ B3 ) @ C3 ) @ D2 )
= ( pop @ A @ B @ A2 @ ( pop @ A @ B @ A2 @ A3 @ ( pop @ A @ B @ A2 @ B3 @ C3 ) ) @ D2 ) ) ) ) ) ) ) ).
% aGroup.pOp_assocTr42
thf(fact_89_aGroup_OpOp__assocTr41,axiom,
! [B: $tType,A: $tType,A2: carrier_ext @ A @ ( aGroup_ext @ A @ B ),A3: A,B3: A,C3: A,D2: A] :
( ( aGroup @ A @ B @ A2 )
=> ( ( member @ A @ A3 @ ( carrier @ A @ ( aGroup_ext @ A @ B ) @ A2 ) )
=> ( ( member @ A @ B3 @ ( carrier @ A @ ( aGroup_ext @ A @ B ) @ A2 ) )
=> ( ( member @ A @ C3 @ ( carrier @ A @ ( aGroup_ext @ A @ B ) @ A2 ) )
=> ( ( member @ A @ D2 @ ( carrier @ A @ ( aGroup_ext @ A @ B ) @ A2 ) )
=> ( ( pop @ A @ B @ A2 @ ( pop @ A @ B @ A2 @ ( pop @ A @ B @ A2 @ A3 @ B3 ) @ C3 ) @ D2 )
= ( pop @ A @ B @ A2 @ ( pop @ A @ B @ A2 @ A3 @ B3 ) @ ( pop @ A @ B @ A2 @ C3 @ D2 ) ) ) ) ) ) ) ) ).
% aGroup.pOp_assocTr41
thf(fact_90_aGroup_Oag__pOp__closed,axiom,
! [B: $tType,A: $tType,A2: carrier_ext @ A @ ( aGroup_ext @ A @ B ),X2: A,Y2: A] :
( ( aGroup @ A @ B @ A2 )
=> ( ( member @ A @ X2 @ ( carrier @ A @ ( aGroup_ext @ A @ B ) @ A2 ) )
=> ( ( member @ A @ Y2 @ ( carrier @ A @ ( aGroup_ext @ A @ B ) @ A2 ) )
=> ( member @ A @ ( pop @ A @ B @ A2 @ X2 @ Y2 ) @ ( carrier @ A @ ( aGroup_ext @ A @ B ) @ A2 ) ) ) ) ) ).
% aGroup.ag_pOp_closed
thf(fact_91_aGroup_OpOp__cancel__r,axiom,
! [B: $tType,A: $tType,A2: carrier_ext @ A @ ( aGroup_ext @ A @ B ),A3: A,B3: A,C3: A] :
( ( aGroup @ A @ B @ A2 )
=> ( ( member @ A @ A3 @ ( carrier @ A @ ( aGroup_ext @ A @ B ) @ A2 ) )
=> ( ( member @ A @ B3 @ ( carrier @ A @ ( aGroup_ext @ A @ B ) @ A2 ) )
=> ( ( member @ A @ C3 @ ( carrier @ A @ ( aGroup_ext @ A @ B ) @ A2 ) )
=> ( ( ( pop @ A @ B @ A2 @ A3 @ C3 )
= ( pop @ A @ B @ A2 @ B3 @ C3 ) )
=> ( A3 = B3 ) ) ) ) ) ) ).
% aGroup.pOp_cancel_r
thf(fact_92_aGroup_OpOp__cancel__l,axiom,
! [B: $tType,A: $tType,A2: carrier_ext @ A @ ( aGroup_ext @ A @ B ),A3: A,B3: A,C3: A] :
( ( aGroup @ A @ B @ A2 )
=> ( ( member @ A @ A3 @ ( carrier @ A @ ( aGroup_ext @ A @ B ) @ A2 ) )
=> ( ( member @ A @ B3 @ ( carrier @ A @ ( aGroup_ext @ A @ B ) @ A2 ) )
=> ( ( member @ A @ C3 @ ( carrier @ A @ ( aGroup_ext @ A @ B ) @ A2 ) )
=> ( ( ( pop @ A @ B @ A2 @ C3 @ A3 )
= ( pop @ A @ B @ A2 @ C3 @ B3 ) )
=> ( A3 = B3 ) ) ) ) ) ) ).
% aGroup.pOp_cancel_l
thf(fact_93_aGroup_Oag__pOp__assoc,axiom,
! [B: $tType,A: $tType,A2: carrier_ext @ A @ ( aGroup_ext @ A @ B ),X2: A,Y2: A,Z2: A] :
( ( aGroup @ A @ B @ A2 )
=> ( ( member @ A @ X2 @ ( carrier @ A @ ( aGroup_ext @ A @ B ) @ A2 ) )
=> ( ( member @ A @ Y2 @ ( carrier @ A @ ( aGroup_ext @ A @ B ) @ A2 ) )
=> ( ( member @ A @ Z2 @ ( carrier @ A @ ( aGroup_ext @ A @ B ) @ A2 ) )
=> ( ( pop @ A @ B @ A2 @ ( pop @ A @ B @ A2 @ X2 @ Y2 ) @ Z2 )
= ( pop @ A @ B @ A2 @ X2 @ ( pop @ A @ B @ A2 @ Y2 @ Z2 ) ) ) ) ) ) ) ).
% aGroup.ag_pOp_assoc
thf(fact_94_aGroup_Oag__pOp__add__r,axiom,
! [B: $tType,A: $tType,A2: carrier_ext @ A @ ( aGroup_ext @ A @ B ),A3: A,B3: A,C3: A] :
( ( aGroup @ A @ B @ A2 )
=> ( ( member @ A @ A3 @ ( carrier @ A @ ( aGroup_ext @ A @ B ) @ A2 ) )
=> ( ( member @ A @ B3 @ ( carrier @ A @ ( aGroup_ext @ A @ B ) @ A2 ) )
=> ( ( member @ A @ C3 @ ( carrier @ A @ ( aGroup_ext @ A @ B ) @ A2 ) )
=> ( ( A3 = B3 )
=> ( ( pop @ A @ B @ A2 @ A3 @ C3 )
= ( pop @ A @ B @ A2 @ B3 @ C3 ) ) ) ) ) ) ) ).
% aGroup.ag_pOp_add_r
thf(fact_95_aGroup_Oag__pOp__add__l,axiom,
! [B: $tType,A: $tType,A2: carrier_ext @ A @ ( aGroup_ext @ A @ B ),A3: A,B3: A,C3: A] :
( ( aGroup @ A @ B @ A2 )
=> ( ( member @ A @ A3 @ ( carrier @ A @ ( aGroup_ext @ A @ B ) @ A2 ) )
=> ( ( member @ A @ B3 @ ( carrier @ A @ ( aGroup_ext @ A @ B ) @ A2 ) )
=> ( ( member @ A @ C3 @ ( carrier @ A @ ( aGroup_ext @ A @ B ) @ A2 ) )
=> ( ( A3 = B3 )
=> ( ( pop @ A @ B @ A2 @ C3 @ A3 )
= ( pop @ A @ B @ A2 @ C3 @ B3 ) ) ) ) ) ) ) ).
% aGroup.ag_pOp_add_l
thf(fact_96_aGroup_OgEQAddcross,axiom,
! [B: $tType,A: $tType,A2: carrier_ext @ A @ ( aGroup_ext @ A @ B ),L1: A,L2: A,R1: A,R22: A] :
( ( aGroup @ A @ B @ A2 )
=> ( ( member @ A @ L1 @ ( carrier @ A @ ( aGroup_ext @ A @ B ) @ A2 ) )
=> ( ( member @ A @ L2 @ ( carrier @ A @ ( aGroup_ext @ A @ B ) @ A2 ) )
=> ( ( member @ A @ R1 @ ( carrier @ A @ ( aGroup_ext @ A @ B ) @ A2 ) )
=> ( ( member @ A @ R1 @ ( carrier @ A @ ( aGroup_ext @ A @ B ) @ A2 ) )
=> ( ( L1 = R22 )
=> ( ( L2 = R1 )
=> ( ( pop @ A @ B @ A2 @ L1 @ L2 )
= ( pop @ A @ B @ A2 @ R1 @ R22 ) ) ) ) ) ) ) ) ) ).
% aGroup.gEQAddcross
thf(fact_97_aGroup_Oag__add4__rel,axiom,
! [B: $tType,A: $tType,A2: carrier_ext @ A @ ( aGroup_ext @ A @ B ),A3: A,B3: A,C3: A,D2: A] :
( ( aGroup @ A @ B @ A2 )
=> ( ( member @ A @ A3 @ ( carrier @ A @ ( aGroup_ext @ A @ B ) @ A2 ) )
=> ( ( member @ A @ B3 @ ( carrier @ A @ ( aGroup_ext @ A @ B ) @ A2 ) )
=> ( ( member @ A @ C3 @ ( carrier @ A @ ( aGroup_ext @ A @ B ) @ A2 ) )
=> ( ( member @ A @ D2 @ ( carrier @ A @ ( aGroup_ext @ A @ B ) @ A2 ) )
=> ( ( pop @ A @ B @ A2 @ ( pop @ A @ B @ A2 @ A3 @ B3 ) @ ( pop @ A @ B @ A2 @ C3 @ D2 ) )
= ( pop @ A @ B @ A2 @ ( pop @ A @ B @ A2 @ A3 @ C3 ) @ ( pop @ A @ B @ A2 @ B3 @ D2 ) ) ) ) ) ) ) ) ).
% aGroup.ag_add4_rel
thf(fact_98_aGroup_Oaassoc,axiom,
! [B: $tType,A: $tType,A2: carrier_ext @ A @ ( aGroup_ext @ A @ B ),A3: A,B3: A,C3: A] :
( ( aGroup @ A @ B @ A2 )
=> ( ( member @ A @ A3 @ ( carrier @ A @ ( aGroup_ext @ A @ B ) @ A2 ) )
=> ( ( member @ A @ B3 @ ( carrier @ A @ ( aGroup_ext @ A @ B ) @ A2 ) )
=> ( ( member @ A @ C3 @ ( carrier @ A @ ( aGroup_ext @ A @ B ) @ A2 ) )
=> ( ( pop @ A @ B @ A2 @ ( pop @ A @ B @ A2 @ A3 @ B3 ) @ C3 )
= ( pop @ A @ B @ A2 @ A3 @ ( pop @ A @ B @ A2 @ B3 @ C3 ) ) ) ) ) ) ) ).
% aGroup.aassoc
thf(fact_99_Ring_Oring__tOp__commute,axiom,
! [B: $tType,A: $tType,R2: carrier_ext @ A @ ( aGroup_ext @ A @ ( ring_ext @ A @ B ) ),X2: A,Y2: A] :
( ( ring @ A @ B @ R2 )
=> ( ( member @ A @ X2 @ ( carrier @ A @ ( aGroup_ext @ A @ ( ring_ext @ A @ B ) ) @ R2 ) )
=> ( ( member @ A @ Y2 @ ( carrier @ A @ ( aGroup_ext @ A @ ( ring_ext @ A @ B ) ) @ R2 ) )
=> ( ( tp @ A @ B @ R2 @ X2 @ Y2 )
= ( tp @ A @ B @ R2 @ Y2 @ X2 ) ) ) ) ) ).
% Ring.ring_tOp_commute
thf(fact_100_Ring_Oring__tOp__closed,axiom,
! [B: $tType,A: $tType,R2: carrier_ext @ A @ ( aGroup_ext @ A @ ( ring_ext @ A @ B ) ),X2: A,Y2: A] :
( ( ring @ A @ B @ R2 )
=> ( ( member @ A @ X2 @ ( carrier @ A @ ( aGroup_ext @ A @ ( ring_ext @ A @ B ) ) @ R2 ) )
=> ( ( member @ A @ Y2 @ ( carrier @ A @ ( aGroup_ext @ A @ ( ring_ext @ A @ B ) ) @ R2 ) )
=> ( member @ A @ ( tp @ A @ B @ R2 @ X2 @ Y2 ) @ ( carrier @ A @ ( aGroup_ext @ A @ ( ring_ext @ A @ B ) ) @ R2 ) ) ) ) ) ).
% Ring.ring_tOp_closed
thf(fact_101_Ring_Oring__tOp__assoc,axiom,
! [B: $tType,A: $tType,R2: carrier_ext @ A @ ( aGroup_ext @ A @ ( ring_ext @ A @ B ) ),X2: A,Y2: A,Z2: A] :
( ( ring @ A @ B @ R2 )
=> ( ( member @ A @ X2 @ ( carrier @ A @ ( aGroup_ext @ A @ ( ring_ext @ A @ B ) ) @ R2 ) )
=> ( ( member @ A @ Y2 @ ( carrier @ A @ ( aGroup_ext @ A @ ( ring_ext @ A @ B ) ) @ R2 ) )
=> ( ( member @ A @ Z2 @ ( carrier @ A @ ( aGroup_ext @ A @ ( ring_ext @ A @ B ) ) @ R2 ) )
=> ( ( tp @ A @ B @ R2 @ ( tp @ A @ B @ R2 @ X2 @ Y2 ) @ Z2 )
= ( tp @ A @ B @ R2 @ X2 @ ( tp @ A @ B @ R2 @ Y2 @ Z2 ) ) ) ) ) ) ) ).
% Ring.ring_tOp_assoc
thf(fact_102_Ring_Oring__tOp__rel,axiom,
! [B: $tType,A: $tType,R2: carrier_ext @ A @ ( aGroup_ext @ A @ ( ring_ext @ A @ B ) ),X2: A,Xa: A,Y2: A,Ya: A] :
( ( ring @ A @ B @ R2 )
=> ( ( member @ A @ X2 @ ( carrier @ A @ ( aGroup_ext @ A @ ( ring_ext @ A @ B ) ) @ R2 ) )
=> ( ( member @ A @ Xa @ ( carrier @ A @ ( aGroup_ext @ A @ ( ring_ext @ A @ B ) ) @ R2 ) )
=> ( ( member @ A @ Y2 @ ( carrier @ A @ ( aGroup_ext @ A @ ( ring_ext @ A @ B ) ) @ R2 ) )
=> ( ( member @ A @ Ya @ ( carrier @ A @ ( aGroup_ext @ A @ ( ring_ext @ A @ B ) ) @ R2 ) )
=> ( ( tp @ A @ B @ R2 @ ( tp @ A @ B @ R2 @ X2 @ Xa ) @ ( tp @ A @ B @ R2 @ Y2 @ Ya ) )
= ( tp @ A @ B @ R2 @ ( tp @ A @ B @ R2 @ X2 @ Y2 ) @ ( tp @ A @ B @ R2 @ Xa @ Ya ) ) ) ) ) ) ) ) ).
% Ring.ring_tOp_rel
thf(fact_103_Ring_Otp__commute,axiom,
! [B: $tType,A: $tType,R2: carrier_ext @ A @ ( aGroup_ext @ A @ ( ring_ext @ A @ B ) ),A3: A,B3: A] :
( ( ring @ A @ B @ R2 )
=> ( ( member @ A @ A3 @ ( carrier @ A @ ( aGroup_ext @ A @ ( ring_ext @ A @ B ) ) @ R2 ) )
=> ( ( member @ A @ B3 @ ( carrier @ A @ ( aGroup_ext @ A @ ( ring_ext @ A @ B ) ) @ R2 ) )
=> ( ( tp @ A @ B @ R2 @ A3 @ B3 )
= ( tp @ A @ B @ R2 @ B3 @ A3 ) ) ) ) ) ).
% Ring.tp_commute
thf(fact_104_Ring_Otp__assoc,axiom,
! [B: $tType,A: $tType,R2: carrier_ext @ A @ ( aGroup_ext @ A @ ( ring_ext @ A @ B ) ),A3: A,B3: A,C3: A] :
( ( ring @ A @ B @ R2 )
=> ( ( member @ A @ A3 @ ( carrier @ A @ ( aGroup_ext @ A @ ( ring_ext @ A @ B ) ) @ R2 ) )
=> ( ( member @ A @ B3 @ ( carrier @ A @ ( aGroup_ext @ A @ ( ring_ext @ A @ B ) ) @ R2 ) )
=> ( ( member @ A @ C3 @ ( carrier @ A @ ( aGroup_ext @ A @ ( ring_ext @ A @ B ) ) @ R2 ) )
=> ( ( tp @ A @ B @ R2 @ ( tp @ A @ B @ R2 @ A3 @ B3 ) @ C3 )
= ( tp @ A @ B @ R2 @ A3 @ ( tp @ A @ B @ R2 @ B3 @ C3 ) ) ) ) ) ) ) ).
% Ring.tp_assoc
thf(fact_105_Ring_OrEQMulR,axiom,
! [B: $tType,A: $tType,R2: carrier_ext @ A @ ( aGroup_ext @ A @ ( ring_ext @ A @ B ) ),X2: A,Y2: A,Z2: A] :
( ( ring @ A @ B @ R2 )
=> ( ( member @ A @ X2 @ ( carrier @ A @ ( aGroup_ext @ A @ ( ring_ext @ A @ B ) ) @ R2 ) )
=> ( ( member @ A @ Y2 @ ( carrier @ A @ ( aGroup_ext @ A @ ( ring_ext @ A @ B ) ) @ R2 ) )
=> ( ( member @ A @ Z2 @ ( carrier @ A @ ( aGroup_ext @ A @ ( ring_ext @ A @ B ) ) @ R2 ) )
=> ( ( X2 = Y2 )
=> ( ( tp @ A @ B @ R2 @ X2 @ Z2 )
= ( tp @ A @ B @ R2 @ Y2 @ Z2 ) ) ) ) ) ) ) ).
% Ring.rEQMulR
thf(fact_106_Ring_OrMulLC,axiom,
! [B: $tType,A: $tType,R2: carrier_ext @ A @ ( aGroup_ext @ A @ ( ring_ext @ A @ B ) ),X2: A,Y2: A,Z2: A] :
( ( ring @ A @ B @ R2 )
=> ( ( member @ A @ X2 @ ( carrier @ A @ ( aGroup_ext @ A @ ( ring_ext @ A @ B ) ) @ R2 ) )
=> ( ( member @ A @ Y2 @ ( carrier @ A @ ( aGroup_ext @ A @ ( ring_ext @ A @ B ) ) @ R2 ) )
=> ( ( member @ A @ Z2 @ ( carrier @ A @ ( aGroup_ext @ A @ ( ring_ext @ A @ B ) ) @ R2 ) )
=> ( ( tp @ A @ B @ R2 @ X2 @ ( tp @ A @ B @ R2 @ Y2 @ Z2 ) )
= ( tp @ A @ B @ R2 @ Y2 @ ( tp @ A @ B @ R2 @ X2 @ Z2 ) ) ) ) ) ) ) ).
% Ring.rMulLC
thf(fact_107_Ring_Opop__commute,axiom,
! [B: $tType,A: $tType,R2: carrier_ext @ A @ ( aGroup_ext @ A @ ( ring_ext @ A @ B ) ),A3: A,B3: A] :
( ( ring @ A @ B @ R2 )
=> ( ( member @ A @ A3 @ ( carrier @ A @ ( aGroup_ext @ A @ ( ring_ext @ A @ B ) ) @ R2 ) )
=> ( ( member @ A @ B3 @ ( carrier @ A @ ( aGroup_ext @ A @ ( ring_ext @ A @ B ) ) @ R2 ) )
=> ( ( pop @ A @ ( ring_ext @ A @ B ) @ R2 @ A3 @ B3 )
= ( pop @ A @ ( ring_ext @ A @ B ) @ R2 @ B3 @ A3 ) ) ) ) ) ).
% Ring.pop_commute
thf(fact_108_Ring_Opop__aassoc,axiom,
! [B: $tType,A: $tType,R2: carrier_ext @ A @ ( aGroup_ext @ A @ ( ring_ext @ A @ B ) ),A3: A,B3: A,C3: A] :
( ( ring @ A @ B @ R2 )
=> ( ( member @ A @ A3 @ ( carrier @ A @ ( aGroup_ext @ A @ ( ring_ext @ A @ B ) ) @ R2 ) )
=> ( ( member @ A @ B3 @ ( carrier @ A @ ( aGroup_ext @ A @ ( ring_ext @ A @ B ) ) @ R2 ) )
=> ( ( member @ A @ C3 @ ( carrier @ A @ ( aGroup_ext @ A @ ( ring_ext @ A @ B ) ) @ R2 ) )
=> ( ( pop @ A @ ( ring_ext @ A @ B ) @ R2 @ ( pop @ A @ ( ring_ext @ A @ B ) @ R2 @ A3 @ B3 ) @ C3 )
= ( pop @ A @ ( ring_ext @ A @ B ) @ R2 @ A3 @ ( pop @ A @ ( ring_ext @ A @ B ) @ R2 @ B3 @ C3 ) ) ) ) ) ) ) ).
% Ring.pop_aassoc
thf(fact_109_Ring_OnsEqElm,axiom,
! [A: $tType,B: $tType,R2: carrier_ext @ A @ ( aGroup_ext @ A @ ( ring_ext @ A @ B ) ),X2: A,Y2: A,N2: nat] :
( ( ring @ A @ B @ R2 )
=> ( ( member @ A @ X2 @ ( carrier @ A @ ( aGroup_ext @ A @ ( ring_ext @ A @ B ) ) @ R2 ) )
=> ( ( member @ A @ Y2 @ ( carrier @ A @ ( aGroup_ext @ A @ ( ring_ext @ A @ B ) ) @ R2 ) )
=> ( ( X2 = Y2 )
=> ( ( nscal @ A @ B @ R2 @ X2 @ N2 )
= ( nscal @ A @ B @ R2 @ Y2 @ N2 ) ) ) ) ) ) ).
% Ring.nsEqElm
thf(fact_110_Ring_OnsClose,axiom,
! [B: $tType,A: $tType,R2: carrier_ext @ A @ ( aGroup_ext @ A @ ( ring_ext @ A @ B ) ),X2: A,N2: nat] :
( ( ring @ A @ B @ R2 )
=> ( ( member @ A @ X2 @ ( carrier @ A @ ( aGroup_ext @ A @ ( ring_ext @ A @ B ) ) @ R2 ) )
=> ( member @ A @ ( nscal @ A @ B @ R2 @ X2 @ N2 ) @ ( carrier @ A @ ( aGroup_ext @ A @ ( ring_ext @ A @ B ) ) @ R2 ) ) ) ) ).
% Ring.nsClose
thf(fact_111_prod__pOp__mem__i,axiom,
! [C: $tType,B: $tType,A: $tType,I2: set @ A,A2: A > ( carrier_ext @ B @ ( aGroup_ext @ B @ C ) ),X4: A > B,Y: A > B,I3: A] :
( ! [X: A] :
( ( member @ A @ X @ I2 )
=> ( aGroup @ B @ C @ ( A2 @ X ) ) )
=> ( ( member @ ( A > B ) @ X4 @ ( carr_prodag @ A @ B @ C @ I2 @ A2 ) )
=> ( ( member @ ( A > B ) @ Y @ ( carr_prodag @ A @ B @ C @ I2 @ A2 ) )
=> ( ( member @ A @ I3 @ I2 )
=> ( ( prod_pOp @ A @ B @ C @ I2 @ A2 @ X4 @ Y @ I3 )
= ( pop @ B @ C @ ( A2 @ I3 ) @ ( X4 @ I3 ) @ ( Y @ I3 ) ) ) ) ) ) ) ).
% prod_pOp_mem_i
thf(fact_112_dsum__pOp__mem,axiom,
! [C: $tType,B: $tType,A: $tType,I2: set @ A,A2: A > ( carrier_ext @ B @ ( aGroup_ext @ B @ C ) ),X4: A > B,Y: A > B] :
( ! [X: A] :
( ( member @ A @ X @ I2 )
=> ( aGroup @ B @ C @ ( A2 @ X ) ) )
=> ( ( member @ ( A > B ) @ X4 @ ( carr_dsumag @ A @ B @ C @ I2 @ A2 ) )
=> ( ( member @ ( A > B ) @ Y @ ( carr_dsumag @ A @ B @ C @ I2 @ A2 ) )
=> ( member @ ( A > B ) @ ( prod_pOp @ A @ B @ C @ I2 @ A2 @ X4 @ Y ) @ ( carr_dsumag @ A @ B @ C @ I2 @ A2 ) ) ) ) ) ).
% dsum_pOp_mem
thf(fact_113_dsum__iOp__mem,axiom,
! [C: $tType,B: $tType,A: $tType,I2: set @ A,A2: A > ( carrier_ext @ B @ ( aGroup_ext @ B @ C ) ),X4: A > B] :
( ! [X: A] :
( ( member @ A @ X @ I2 )
=> ( aGroup @ B @ C @ ( A2 @ X ) ) )
=> ( ( member @ ( A > B ) @ X4 @ ( carr_dsumag @ A @ B @ C @ I2 @ A2 ) )
=> ( member @ ( A > B ) @ ( prod_mOp @ A @ B @ C @ I2 @ A2 @ X4 ) @ ( carr_dsumag @ A @ B @ C @ I2 @ A2 ) ) ) ) ).
% dsum_iOp_mem
thf(fact_114_dsum__zero__func,axiom,
! [C: $tType,B: $tType,A: $tType,I2: set @ A,A2: A > ( carrier_ext @ B @ ( aGroup_ext @ B @ C ) )] :
( ! [X: A] :
( ( member @ A @ X @ I2 )
=> ( aGroup @ B @ C @ ( A2 @ X ) ) )
=> ( member @ ( A > B ) @ ( prod_zero @ A @ B @ C @ I2 @ A2 ) @ ( carr_dsumag @ A @ B @ C @ I2 @ A2 ) ) ) ).
% dsum_zero_func
thf(fact_115_aGroup_Opop__closed,axiom,
! [B: $tType,A: $tType,A2: carrier_ext @ A @ ( aGroup_ext @ A @ B )] :
( ( aGroup @ A @ B @ A2 )
=> ( member @ ( A > A > A ) @ ( pop @ A @ B @ A2 )
@ ( pi @ A @ ( A > A ) @ ( carrier @ A @ ( aGroup_ext @ A @ B ) @ A2 )
@ ^ [Uu: A] :
( pi @ A @ A @ ( carrier @ A @ ( aGroup_ext @ A @ B ) @ A2 )
@ ^ [Uv: A] : ( carrier @ A @ ( aGroup_ext @ A @ B ) @ A2 ) ) ) ) ) ).
% aGroup.pop_closed
thf(fact_116_Ring_OJ__rad__mem,axiom,
! [B: $tType,A: $tType,R2: carrier_ext @ A @ ( aGroup_ext @ A @ ( ring_ext @ A @ B ) ),X2: A] :
( ( ring @ A @ B @ R2 )
=> ( ( member @ A @ X2 @ ( j_rad @ A @ B @ R2 ) )
=> ( member @ A @ X2 @ ( carrier @ A @ ( aGroup_ext @ A @ ( ring_ext @ A @ B ) ) @ R2 ) ) ) ) ).
% Ring.J_rad_mem
thf(fact_117_Ring_Otp__closed,axiom,
! [B: $tType,A: $tType,R2: carrier_ext @ A @ ( aGroup_ext @ A @ ( ring_ext @ A @ B ) )] :
( ( ring @ A @ B @ R2 )
=> ( member @ ( A > A > A ) @ ( tp @ A @ B @ R2 )
@ ( pi @ A @ ( A > A ) @ ( carrier @ A @ ( aGroup_ext @ A @ ( ring_ext @ A @ B ) ) @ R2 )
@ ^ [Uu: A] :
( pi @ A @ A @ ( carrier @ A @ ( aGroup_ext @ A @ ( ring_ext @ A @ B ) ) @ R2 )
@ ^ [Uv: A] : ( carrier @ A @ ( aGroup_ext @ A @ ( ring_ext @ A @ B ) ) @ R2 ) ) ) ) ) ).
% Ring.tp_closed
thf(fact_118_Ring_Opop__closed,axiom,
! [B: $tType,A: $tType,R2: carrier_ext @ A @ ( aGroup_ext @ A @ ( ring_ext @ A @ B ) )] :
( ( ring @ A @ B @ R2 )
=> ( member @ ( A > A > A ) @ ( pop @ A @ ( ring_ext @ A @ B ) @ R2 )
@ ( pi @ A @ ( A > A ) @ ( carrier @ A @ ( aGroup_ext @ A @ ( ring_ext @ A @ B ) ) @ R2 )
@ ^ [Uu: A] :
( pi @ A @ A @ ( carrier @ A @ ( aGroup_ext @ A @ ( ring_ext @ A @ B ) ) @ R2 )
@ ^ [Uv: A] : ( carrier @ A @ ( aGroup_ext @ A @ ( ring_ext @ A @ B ) ) @ R2 ) ) ) ) ) ).
% Ring.pop_closed
thf(fact_119_Ring_OprodM__sprod__val,axiom,
! [B: $tType,A: $tType,E: $tType,D: $tType,C: $tType,R2: carrier_ext @ A @ ( aGroup_ext @ A @ ( ring_ext @ A @ B ) ),I2: set @ C,M3: C > ( carrier_ext @ D @ ( aGroup_ext @ D @ ( module_ext @ D @ A @ E ) ) ),A3: A,M4: C > D,J: C] :
( ( ring @ A @ B @ R2 )
=> ( ! [X: C] :
( ( member @ C @ X @ I2 )
=> ( module @ D @ A @ E @ B @ ( M3 @ X ) @ R2 ) )
=> ( ( member @ A @ A3 @ ( carrier @ A @ ( aGroup_ext @ A @ ( ring_ext @ A @ B ) ) @ R2 ) )
=> ( ( member @ ( C > D ) @ M4 @ ( carr_prodag @ C @ D @ ( module_ext @ D @ A @ E ) @ I2 @ M3 ) )
=> ( ( member @ C @ J @ I2 )
=> ( ( algebr142713364_sprod @ A @ B @ C @ D @ E @ R2 @ I2 @ M3 @ A3 @ M4 @ J )
= ( sprod @ D @ A @ E @ ( M3 @ J ) @ A3 @ ( M4 @ J ) ) ) ) ) ) ) ) ).
% Ring.prodM_sprod_val
thf(fact_120_dsum__iOp__func,axiom,
! [C: $tType,B: $tType,A: $tType,I2: set @ A,A2: A > ( carrier_ext @ B @ ( aGroup_ext @ B @ C ) )] :
( ! [X: A] :
( ( member @ A @ X @ I2 )
=> ( aGroup @ B @ C @ ( A2 @ X ) ) )
=> ( member @ ( ( A > B ) > A > B ) @ ( prod_mOp @ A @ B @ C @ I2 @ A2 )
@ ( pi @ ( A > B ) @ ( A > B ) @ ( carr_dsumag @ A @ B @ C @ I2 @ A2 )
@ ^ [Uu: A > B] : ( carr_dsumag @ A @ B @ C @ I2 @ A2 ) ) ) ) ).
% dsum_iOp_func
thf(fact_121_zeroring__J__rad__empty,axiom,
( ( zeroring @ a @ b @ r )
=> ( ( j_rad @ a @ b @ r )
= ( carrier @ a @ ( aGroup_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) ) ) ).
% zeroring_J_rad_empty
thf(fact_122_Module_OmHom__lin,axiom,
! [D: $tType,C: $tType,B: $tType,F2: $tType,E: $tType,A: $tType,M3: carrier_ext @ A @ ( aGroup_ext @ A @ ( module_ext @ A @ B @ C ) ),R2: carrier_ext @ B @ ( aGroup_ext @ B @ ( ring_ext @ B @ D ) ),N: carrier_ext @ E @ ( aGroup_ext @ E @ ( module_ext @ E @ B @ F2 ) ),M4: A,F: A > E,A3: B] :
( ( module @ A @ B @ C @ D @ M3 @ R2 )
=> ( ( module @ E @ B @ F2 @ D @ N @ R2 )
=> ( ( member @ A @ M4 @ ( carrier @ A @ ( aGroup_ext @ A @ ( module_ext @ A @ B @ C ) ) @ M3 ) )
=> ( ( member @ ( A > E ) @ F @ ( mHom @ B @ D @ A @ C @ E @ F2 @ R2 @ M3 @ N ) )
=> ( ( member @ B @ A3 @ ( carrier @ B @ ( aGroup_ext @ B @ ( ring_ext @ B @ D ) ) @ R2 ) )
=> ( ( F @ ( sprod @ A @ B @ C @ M3 @ A3 @ M4 ) )
= ( sprod @ E @ B @ F2 @ N @ A3 @ ( F @ M4 ) ) ) ) ) ) ) ) ).
% Module.mHom_lin
thf(fact_123_Module_Osprod__assoc,axiom,
! [D: $tType,B: $tType,C: $tType,A: $tType,M3: carrier_ext @ A @ ( aGroup_ext @ A @ ( module_ext @ A @ B @ C ) ),R2: carrier_ext @ B @ ( aGroup_ext @ B @ ( ring_ext @ B @ D ) ),A3: B,B3: B,M4: A] :
( ( module @ A @ B @ C @ D @ M3 @ R2 )
=> ( ( member @ B @ A3 @ ( carrier @ B @ ( aGroup_ext @ B @ ( ring_ext @ B @ D ) ) @ R2 ) )
=> ( ( member @ B @ B3 @ ( carrier @ B @ ( aGroup_ext @ B @ ( ring_ext @ B @ D ) ) @ R2 ) )
=> ( ( member @ A @ M4 @ ( carrier @ A @ ( aGroup_ext @ A @ ( module_ext @ A @ B @ C ) ) @ M3 ) )
=> ( ( sprod @ A @ B @ C @ M3 @ ( tp @ B @ D @ R2 @ A3 @ B3 ) @ M4 )
= ( sprod @ A @ B @ C @ M3 @ A3 @ ( sprod @ A @ B @ C @ M3 @ B3 @ M4 ) ) ) ) ) ) ) ).
% Module.sprod_assoc
thf(fact_124_Ring_OmHom__func,axiom,
! [B: $tType,E: $tType,D: $tType,A: $tType,F2: $tType,C: $tType,R2: carrier_ext @ A @ ( aGroup_ext @ A @ ( ring_ext @ A @ B ) ),F: C > D,M3: carrier_ext @ C @ ( aGroup_ext @ C @ ( module_ext @ C @ A @ E ) ),N: carrier_ext @ D @ ( aGroup_ext @ D @ ( module_ext @ D @ A @ F2 ) )] :
( ( ring @ A @ B @ R2 )
=> ( ( member @ ( C > D ) @ F @ ( mHom @ A @ B @ C @ E @ D @ F2 @ R2 @ M3 @ N ) )
=> ( member @ ( C > D ) @ F
@ ( pi @ C @ D @ ( carrier @ C @ ( aGroup_ext @ C @ ( module_ext @ C @ A @ E ) ) @ M3 )
@ ^ [Uu: C] : ( carrier @ D @ ( aGroup_ext @ D @ ( module_ext @ D @ A @ F2 ) ) @ N ) ) ) ) ) ).
% Ring.mHom_func
thf(fact_125_Ring_OSubring__pOp__ring__pOp,axiom,
! [C: $tType,B: $tType,A: $tType,R2: carrier_ext @ A @ ( aGroup_ext @ A @ ( ring_ext @ A @ B ) ),S: carrier_ext @ A @ ( aGroup_ext @ A @ ( ring_ext @ A @ C ) ),A3: A,B3: A] :
( ( ring @ A @ B @ R2 )
=> ( ( subring @ A @ B @ C @ R2 @ S )
=> ( ( member @ A @ A3 @ ( carrier @ A @ ( aGroup_ext @ A @ ( ring_ext @ A @ C ) ) @ S ) )
=> ( ( member @ A @ B3 @ ( carrier @ A @ ( aGroup_ext @ A @ ( ring_ext @ A @ C ) ) @ S ) )
=> ( ( pop @ A @ ( ring_ext @ A @ C ) @ S @ A3 @ B3 )
= ( pop @ A @ ( ring_ext @ A @ B ) @ R2 @ A3 @ B3 ) ) ) ) ) ) ).
% Ring.Subring_pOp_ring_pOp
thf(fact_126_Ring_OSubring__tOp__ring__tOp,axiom,
! [C: $tType,B: $tType,A: $tType,R2: carrier_ext @ A @ ( aGroup_ext @ A @ ( ring_ext @ A @ B ) ),S: carrier_ext @ A @ ( aGroup_ext @ A @ ( ring_ext @ A @ C ) ),A3: A,B3: A] :
( ( ring @ A @ B @ R2 )
=> ( ( subring @ A @ B @ C @ R2 @ S )
=> ( ( member @ A @ A3 @ ( carrier @ A @ ( aGroup_ext @ A @ ( ring_ext @ A @ C ) ) @ S ) )
=> ( ( member @ A @ B3 @ ( carrier @ A @ ( aGroup_ext @ A @ ( ring_ext @ A @ C ) ) @ S ) )
=> ( ( tp @ A @ C @ S @ A3 @ B3 )
= ( tp @ A @ B @ R2 @ A3 @ B3 ) ) ) ) ) ) ).
% Ring.Subring_tOp_ring_tOp
thf(fact_127_Module_Osprod__closed,axiom,
! [D: $tType,C: $tType,B: $tType,A: $tType,M3: carrier_ext @ A @ ( aGroup_ext @ A @ ( module_ext @ A @ B @ C ) ),R2: carrier_ext @ B @ ( aGroup_ext @ B @ ( ring_ext @ B @ D ) ),A3: B,M4: A] :
( ( module @ A @ B @ C @ D @ M3 @ R2 )
=> ( ( member @ B @ A3 @ ( carrier @ B @ ( aGroup_ext @ B @ ( ring_ext @ B @ D ) ) @ R2 ) )
=> ( ( member @ A @ M4 @ ( carrier @ A @ ( aGroup_ext @ A @ ( module_ext @ A @ B @ C ) ) @ M3 ) )
=> ( member @ A @ ( sprod @ A @ B @ C @ M3 @ A3 @ M4 ) @ ( carrier @ A @ ( aGroup_ext @ A @ ( module_ext @ A @ B @ C ) ) @ M3 ) ) ) ) ) ).
% Module.sprod_closed
thf(fact_128_nsDistr,axiom,
! [X2: a,N2: nat,M4: nat] :
( ( member @ a @ X2 @ ( carrier @ a @ ( aGroup_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
=> ( ( pop @ a @ ( ring_ext @ a @ b ) @ r @ ( nscal @ a @ b @ r @ X2 @ N2 ) @ ( nscal @ a @ b @ r @ X2 @ M4 ) )
= ( nscal @ a @ b @ r @ X2 @ ( plus_plus @ nat @ N2 @ M4 ) ) ) ) ).
% nsDistr
thf(fact_129_add__bothl,axiom,
! [A: $tType] :
( ( plus @ A )
=> ! [A3: A,B3: A,C3: A] :
( ( A3 = B3 )
=> ( ( plus_plus @ A @ C3 @ A3 )
= ( plus_plus @ A @ C3 @ B3 ) ) ) ) ).
% add_bothl
thf(fact_130_add__both,axiom,
! [A3: nat,B3: nat,C3: nat] :
( ( A3 = B3 )
=> ( ( plus_plus @ nat @ A3 @ C3 )
= ( plus_plus @ nat @ B3 @ C3 ) ) ) ).
% add_both
thf(fact_131_Module_OmHom__add,axiom,
! [D: $tType,C: $tType,F2: $tType,B: $tType,E: $tType,A: $tType,M3: carrier_ext @ A @ ( aGroup_ext @ A @ ( module_ext @ A @ B @ C ) ),R2: carrier_ext @ B @ ( aGroup_ext @ B @ ( ring_ext @ B @ D ) ),N: carrier_ext @ E @ ( aGroup_ext @ E @ ( module_ext @ E @ B @ F2 ) ),F: A > E,M4: A,N2: A] :
( ( module @ A @ B @ C @ D @ M3 @ R2 )
=> ( ( module @ E @ B @ F2 @ D @ N @ R2 )
=> ( ( member @ ( A > E ) @ F @ ( mHom @ B @ D @ A @ C @ E @ F2 @ R2 @ M3 @ N ) )
=> ( ( member @ A @ M4 @ ( carrier @ A @ ( aGroup_ext @ A @ ( module_ext @ A @ B @ C ) ) @ M3 ) )
=> ( ( member @ A @ N2 @ ( carrier @ A @ ( aGroup_ext @ A @ ( module_ext @ A @ B @ C ) ) @ M3 ) )
=> ( ( F @ ( pop @ A @ ( module_ext @ A @ B @ C ) @ M3 @ M4 @ N2 ) )
= ( pop @ E @ ( module_ext @ E @ B @ F2 ) @ N @ ( F @ M4 ) @ ( F @ N2 ) ) ) ) ) ) ) ) ).
% Module.mHom_add
thf(fact_132_Module_Osprod__r__distr,axiom,
! [D: $tType,B: $tType,C: $tType,A: $tType,M3: carrier_ext @ A @ ( aGroup_ext @ A @ ( module_ext @ A @ B @ C ) ),R2: carrier_ext @ B @ ( aGroup_ext @ B @ ( ring_ext @ B @ D ) ),A3: B,M4: A,N2: A] :
( ( module @ A @ B @ C @ D @ M3 @ R2 )
=> ( ( member @ B @ A3 @ ( carrier @ B @ ( aGroup_ext @ B @ ( ring_ext @ B @ D ) ) @ R2 ) )
=> ( ( member @ A @ M4 @ ( carrier @ A @ ( aGroup_ext @ A @ ( module_ext @ A @ B @ C ) ) @ M3 ) )
=> ( ( member @ A @ N2 @ ( carrier @ A @ ( aGroup_ext @ A @ ( module_ext @ A @ B @ C ) ) @ M3 ) )
=> ( ( sprod @ A @ B @ C @ M3 @ A3 @ ( pop @ A @ ( module_ext @ A @ B @ C ) @ M3 @ M4 @ N2 ) )
= ( pop @ A @ ( module_ext @ A @ B @ C ) @ M3 @ ( sprod @ A @ B @ C @ M3 @ A3 @ M4 ) @ ( sprod @ A @ B @ C @ M3 @ A3 @ N2 ) ) ) ) ) ) ) ).
% Module.sprod_r_distr
thf(fact_133_Module_Omodule__is__ag,axiom,
! [D: $tType,C: $tType,B: $tType,A: $tType,M3: 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 @ M3 @ R2 )
=> ( aGroup @ A @ ( module_ext @ A @ B @ C ) @ M3 ) ) ).
% Module.module_is_ag
thf(fact_134_Module_Osprod__l__distr,axiom,
! [D: $tType,B: $tType,C: $tType,A: $tType,M3: carrier_ext @ A @ ( aGroup_ext @ A @ ( module_ext @ A @ B @ C ) ),R2: carrier_ext @ B @ ( aGroup_ext @ B @ ( ring_ext @ B @ D ) ),A3: B,B3: B,M4: A] :
( ( module @ A @ B @ C @ D @ M3 @ R2 )
=> ( ( member @ B @ A3 @ ( carrier @ B @ ( aGroup_ext @ B @ ( ring_ext @ B @ D ) ) @ R2 ) )
=> ( ( member @ B @ B3 @ ( carrier @ B @ ( aGroup_ext @ B @ ( ring_ext @ B @ D ) ) @ R2 ) )
=> ( ( member @ A @ M4 @ ( carrier @ A @ ( aGroup_ext @ A @ ( module_ext @ A @ B @ C ) ) @ M3 ) )
=> ( ( sprod @ A @ B @ C @ M3 @ ( pop @ B @ ( ring_ext @ B @ D ) @ R2 @ A3 @ B3 ) @ M4 )
= ( pop @ A @ ( module_ext @ A @ B @ C ) @ M3 @ ( sprod @ A @ B @ C @ M3 @ A3 @ M4 ) @ ( sprod @ A @ B @ C @ M3 @ B3 @ M4 ) ) ) ) ) ) ) ).
% Module.sprod_l_distr
thf(fact_135_Ring_OnsDistr,axiom,
! [A: $tType,B: $tType,R2: carrier_ext @ A @ ( aGroup_ext @ A @ ( ring_ext @ A @ B ) ),X2: A,N2: nat,M4: nat] :
( ( ring @ A @ B @ R2 )
=> ( ( member @ A @ X2 @ ( carrier @ A @ ( aGroup_ext @ A @ ( ring_ext @ A @ B ) ) @ R2 ) )
=> ( ( pop @ A @ ( ring_ext @ A @ B ) @ R2 @ ( nscal @ A @ B @ R2 @ X2 @ N2 ) @ ( nscal @ A @ B @ R2 @ X2 @ M4 ) )
= ( nscal @ A @ B @ R2 @ X2 @ ( plus_plus @ nat @ N2 @ M4 ) ) ) ) ) ).
% Ring.nsDistr
thf(fact_136_Ring_Omul__closed__set__tOp__closed,axiom,
! [B: $tType,A: $tType,R2: carrier_ext @ A @ ( aGroup_ext @ A @ ( ring_ext @ A @ B ) ),S: set @ A,S2: A,T: A] :
( ( ring @ A @ B @ R2 )
=> ( ( mul_closed_set @ A @ B @ R2 @ S )
=> ( ( member @ A @ S2 @ S )
=> ( ( member @ A @ T @ S )
=> ( member @ A @ ( tp @ A @ B @ R2 @ S2 @ T ) @ S ) ) ) ) ) ).
% Ring.mul_closed_set_tOp_closed
thf(fact_137_Ring_Ozeroring__J__rad__empty,axiom,
! [B: $tType,A: $tType,R2: carrier_ext @ A @ ( aGroup_ext @ A @ ( ring_ext @ A @ B ) )] :
( ( ring @ A @ B @ R2 )
=> ( ( zeroring @ A @ B @ R2 )
=> ( ( j_rad @ A @ B @ R2 )
= ( carrier @ A @ ( aGroup_ext @ A @ ( ring_ext @ A @ B ) ) @ R2 ) ) ) ) ).
% Ring.zeroring_J_rad_empty
thf(fact_138_Module_Osc__Ring,axiom,
! [C: $tType,A: $tType,D: $tType,B: $tType,M3: 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 @ M3 @ R2 )
=> ( ring @ B @ D @ R2 ) ) ).
% Module.sc_Ring
thf(fact_139_Ring_Osubring__Ring,axiom,
! [B: $tType,C: $tType,A: $tType,R2: carrier_ext @ A @ ( aGroup_ext @ A @ ( ring_ext @ A @ B ) ),S: carrier_ext @ A @ ( aGroup_ext @ A @ ( ring_ext @ A @ C ) )] :
( ( ring @ A @ B @ R2 )
=> ( ( subring @ A @ B @ C @ R2 @ S )
=> ( ring @ A @ C @ S ) ) ) ).
% Ring.subring_Ring
thf(fact_140_Module_OmHom__eq,axiom,
! [F2: $tType,D: $tType,C: $tType,B: $tType,E: $tType,A: $tType,M3: carrier_ext @ A @ ( aGroup_ext @ A @ ( module_ext @ A @ B @ C ) ),R2: carrier_ext @ B @ ( aGroup_ext @ B @ ( ring_ext @ B @ D ) ),N: carrier_ext @ E @ ( aGroup_ext @ E @ ( module_ext @ E @ B @ F2 ) ),F: A > E,G: A > E] :
( ( module @ A @ B @ C @ D @ M3 @ R2 )
=> ( ( module @ E @ B @ F2 @ D @ N @ R2 )
=> ( ( member @ ( A > E ) @ F @ ( mHom @ B @ D @ A @ C @ E @ F2 @ R2 @ M3 @ N ) )
=> ( ( member @ ( A > E ) @ G @ ( mHom @ B @ D @ A @ C @ E @ F2 @ R2 @ M3 @ N ) )
=> ( ! [X: A] :
( ( member @ A @ X @ ( carrier @ A @ ( aGroup_ext @ A @ ( module_ext @ A @ B @ C ) ) @ M3 ) )
=> ( ( F @ X )
= ( G @ X ) ) )
=> ( F = G ) ) ) ) ) ) ).
% Module.mHom_eq
thf(fact_141_Module_OmHom__mem,axiom,
! [C: $tType,D: $tType,A: $tType,F2: $tType,B: $tType,E: $tType,M3: carrier_ext @ A @ ( aGroup_ext @ A @ ( module_ext @ A @ B @ C ) ),R2: carrier_ext @ B @ ( aGroup_ext @ B @ ( ring_ext @ B @ D ) ),N: carrier_ext @ E @ ( aGroup_ext @ E @ ( module_ext @ E @ B @ F2 ) ),F: A > E,M4: A] :
( ( module @ A @ B @ C @ D @ M3 @ R2 )
=> ( ( module @ E @ B @ F2 @ D @ N @ R2 )
=> ( ( member @ ( A > E ) @ F @ ( mHom @ B @ D @ A @ C @ E @ F2 @ R2 @ M3 @ N ) )
=> ( ( member @ A @ M4 @ ( carrier @ A @ ( aGroup_ext @ A @ ( module_ext @ A @ B @ C ) ) @ M3 ) )
=> ( member @ E @ ( F @ M4 ) @ ( carrier @ E @ ( aGroup_ext @ E @ ( module_ext @ E @ B @ F2 ) ) @ N ) ) ) ) ) ) ).
% Module.mHom_mem
thf(fact_142_Ring_Omem__subring__mem__ring,axiom,
! [C: $tType,B: $tType,A: $tType,R2: carrier_ext @ A @ ( aGroup_ext @ A @ ( ring_ext @ A @ B ) ),S: carrier_ext @ A @ ( aGroup_ext @ A @ ( ring_ext @ A @ C ) ),X2: A] :
( ( ring @ A @ B @ R2 )
=> ( ( subring @ A @ B @ C @ R2 @ S )
=> ( ( member @ A @ X2 @ ( carrier @ A @ ( aGroup_ext @ A @ ( ring_ext @ A @ C ) ) @ S ) )
=> ( member @ A @ X2 @ ( carrier @ A @ ( aGroup_ext @ A @ ( ring_ext @ A @ B ) ) @ R2 ) ) ) ) ) ).
% Ring.mem_subring_mem_ring
thf(fact_143_mul__closed__set__sub,axiom,
! [S: set @ a] :
( ( mul_closed_set @ a @ b @ r @ S )
=> ( ord_less_eq @ ( set @ a ) @ S @ ( carrier @ a @ ( aGroup_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) ) ) ).
% mul_closed_set_sub
thf(fact_144_zeroring__no__maximal,axiom,
( ( zeroring @ a @ b @ r )
=> ~ ? [X_1: set @ a] : ( maximal_ideal @ a @ b @ r @ X_1 ) ) ).
% zeroring_no_maximal
thf(fact_145_id__maximal__Exist,axiom,
( ~ ( zeroring @ a @ b @ r )
=> ? [X_12: set @ a] : ( maximal_ideal @ a @ b @ r @ X_12 ) ) ).
% id_maximal_Exist
thf(fact_146_mop__closed,axiom,
( member @ ( a > a ) @ ( mop @ a @ ( ring_ext @ a @ b ) @ r )
@ ( pi @ a @ a @ ( carrier @ a @ ( aGroup_ext @ a @ ( ring_ext @ a @ b ) ) @ r )
@ ^ [Uu: a] : ( carrier @ a @ ( aGroup_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) ) ) ).
% mop_closed
thf(fact_147_Sr__mOp__closed,axiom,
! [S: set @ a,X2: a] :
( ( sr2 @ a @ b @ r @ S )
=> ( ( member @ a @ X2 @ S )
=> ( member @ a @ ( mop @ a @ ( ring_ext @ a @ b ) @ r @ X2 ) @ S ) ) ) ).
% Sr_mOp_closed
thf(fact_148_ring__inv1,axiom,
! [A3: a,B3: a] :
( ( member @ a @ A3 @ ( carrier @ a @ ( aGroup_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
=> ( ( member @ a @ B3 @ ( carrier @ a @ ( aGroup_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
=> ( ( ( mop @ a @ ( ring_ext @ a @ b ) @ r @ ( tp @ a @ b @ r @ A3 @ B3 ) )
= ( tp @ a @ b @ r @ ( mop @ a @ ( ring_ext @ a @ b ) @ r @ A3 ) @ B3 ) )
& ( ( mop @ a @ ( ring_ext @ a @ b ) @ r @ ( tp @ a @ b @ r @ A3 @ B3 ) )
= ( tp @ a @ b @ r @ A3 @ ( mop @ a @ ( ring_ext @ a @ b ) @ r @ B3 ) ) ) ) ) ) ).
% ring_inv1
thf(fact_149_ring__inv1__1,axiom,
! [A3: a,B3: a] :
( ( member @ a @ A3 @ ( carrier @ a @ ( aGroup_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
=> ( ( member @ a @ B3 @ ( carrier @ a @ ( aGroup_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
=> ( ( mop @ a @ ( ring_ext @ a @ b ) @ r @ ( tp @ a @ b @ r @ A3 @ B3 ) )
= ( tp @ a @ b @ r @ ( mop @ a @ ( ring_ext @ a @ b ) @ r @ A3 ) @ B3 ) ) ) ) ).
% ring_inv1_1
thf(fact_150_ring__inv1__2,axiom,
! [A3: a,B3: a] :
( ( member @ a @ A3 @ ( carrier @ a @ ( aGroup_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
=> ( ( member @ a @ B3 @ ( carrier @ a @ ( aGroup_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
=> ( ( mop @ a @ ( ring_ext @ a @ b ) @ r @ ( tp @ a @ b @ r @ A3 @ B3 ) )
= ( tp @ a @ b @ r @ A3 @ ( mop @ a @ ( ring_ext @ a @ b ) @ r @ B3 ) ) ) ) ) ).
% ring_inv1_2
thf(fact_151_ring__inv1__3,axiom,
! [A3: a,B3: a] :
( ( member @ a @ A3 @ ( carrier @ a @ ( aGroup_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
=> ( ( member @ a @ B3 @ ( carrier @ a @ ( aGroup_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
=> ( ( tp @ a @ b @ r @ A3 @ B3 )
= ( tp @ a @ b @ r @ ( mop @ a @ ( ring_ext @ a @ b ) @ r @ A3 ) @ ( mop @ a @ ( ring_ext @ a @ b ) @ r @ B3 ) ) ) ) ) ).
% ring_inv1_3
thf(fact_152_ring__distrib4,axiom,
! [A3: a,B3: a,X2: a,Y2: a] :
( ( member @ a @ A3 @ ( carrier @ a @ ( aGroup_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
=> ( ( member @ a @ B3 @ ( carrier @ a @ ( aGroup_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
=> ( ( member @ a @ X2 @ ( carrier @ a @ ( aGroup_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
=> ( ( member @ a @ Y2 @ ( carrier @ a @ ( aGroup_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
=> ( ( pop @ a @ ( ring_ext @ a @ b ) @ r @ ( tp @ a @ b @ r @ A3 @ B3 ) @ ( mop @ a @ ( ring_ext @ a @ b ) @ r @ ( tp @ a @ b @ r @ X2 @ Y2 ) ) )
= ( pop @ a @ ( ring_ext @ a @ b ) @ r @ ( tp @ a @ b @ r @ A3 @ ( pop @ a @ ( ring_ext @ a @ b ) @ r @ B3 @ ( mop @ a @ ( ring_ext @ a @ b ) @ r @ Y2 ) ) ) @ ( tp @ a @ b @ r @ ( pop @ a @ ( ring_ext @ a @ b ) @ r @ A3 @ ( mop @ a @ ( ring_ext @ a @ b ) @ r @ X2 ) ) @ Y2 ) ) ) ) ) ) ) ).
% ring_distrib4
thf(fact_153_Subring__minus__ring__minus,axiom,
! [C: $tType,S: carrier_ext @ a @ ( aGroup_ext @ a @ ( ring_ext @ a @ C ) ),X2: a] :
( ( subring @ a @ b @ C @ r @ S )
=> ( ( member @ a @ X2 @ ( carrier @ a @ ( aGroup_ext @ a @ ( ring_ext @ a @ C ) ) @ S ) )
=> ( ( mop @ a @ ( ring_ext @ a @ C ) @ S @ X2 )
= ( mop @ a @ ( ring_ext @ a @ b ) @ r @ X2 ) ) ) ) ).
% Subring_minus_ring_minus
thf(fact_154_not__sub,axiom,
! [A: $tType,A2: set @ A,B2: set @ A] :
( ~ ( ord_less_eq @ ( set @ A ) @ A2 @ B2 )
=> ? [A6: A] :
( ( member @ A @ A6 @ A2 )
& ~ ( member @ A @ A6 @ B2 ) ) ) ).
% not_sub
thf(fact_155_eqsets__sub,axiom,
! [A: $tType,A2: set @ A,B2: set @ A] :
( ( A2 = B2 )
=> ( ord_less_eq @ ( set @ A ) @ A2 @ B2 ) ) ).
% eqsets_sub
thf(fact_156_ge__convert,axiom,
! [A: $tType] :
( ( ord @ A )
=> ! [A3: A,B3: A,C3: A] :
( ( A3 = B3 )
=> ( ( ord_less_eq @ A @ C3 @ A3 )
=> ( ord_less_eq @ A @ C3 @ B3 ) ) ) ) ).
% ge_convert
thf(fact_157_le__convert,axiom,
! [A: $tType] :
( ( ord @ A )
=> ! [A3: A,B3: A,C3: A] :
( ( A3 = B3 )
=> ( ( ord_less_eq @ A @ A3 @ C3 )
=> ( ord_less_eq @ A @ B3 @ C3 ) ) ) ) ).
% le_convert
thf(fact_158_sub__which1,axiom,
! [A: $tType,A2: set @ A,B2: set @ A,X2: A] :
( ( ( ord_less_eq @ ( set @ A ) @ A2 @ B2 )
| ( ord_less_eq @ ( set @ A ) @ B2 @ A2 ) )
=> ( ( member @ A @ X2 @ A2 )
=> ( ~ ( member @ A @ X2 @ B2 )
=> ( ord_less_eq @ ( set @ A ) @ B2 @ A2 ) ) ) ) ).
% sub_which1
thf(fact_159_sub__which2,axiom,
! [A: $tType,A2: set @ A,B2: set @ A,X2: A] :
( ( ( ord_less_eq @ ( set @ A ) @ A2 @ B2 )
| ( ord_less_eq @ ( set @ A ) @ B2 @ A2 ) )
=> ( ~ ( member @ A @ X2 @ A2 )
=> ( ( member @ A @ X2 @ B2 )
=> ( ord_less_eq @ ( set @ A ) @ A2 @ B2 ) ) ) ) ).
% sub_which2
thf(fact_160_sets__not__eq,axiom,
! [A: $tType,A2: set @ A,B2: set @ A] :
( ( A2 != B2 )
=> ( ( ord_less_eq @ ( set @ A ) @ B2 @ A2 )
=> ? [X: A] :
( ( member @ A @ X @ A2 )
& ~ ( member @ A @ X @ B2 ) ) ) ) ).
% sets_not_eq
thf(fact_161_subset__self,axiom,
! [A: $tType,A2: set @ A] : ( ord_less_eq @ ( set @ A ) @ A2 @ A2 ) ).
% subset_self
thf(fact_162_not__subseteq,axiom,
! [A: $tType,A2: set @ A,B2: set @ A] :
( ~ ( ord_less_eq @ ( set @ A ) @ A2 @ B2 )
=> ? [X: A] :
( ( member @ A @ X @ A2 )
& ~ ( member @ A @ X @ B2 ) ) ) ).
% not_subseteq
thf(fact_163_proper__subset,axiom,
! [A: $tType,A2: set @ A,B2: set @ A,X2: A] :
( ( ord_less_eq @ ( set @ A ) @ A2 @ B2 )
=> ( ~ ( member @ A @ X2 @ A2 )
=> ( ( member @ A @ X2 @ B2 )
=> ( A2 != B2 ) ) ) ) ).
% proper_subset
thf(fact_164_conditional__subset,axiom,
! [A: $tType,A2: set @ A,P: A > $o] :
( ord_less_eq @ ( set @ A )
@ ( collect @ A
@ ^ [X3: A] :
( ( member @ A @ X3 @ A2 )
& ( P @ X3 ) ) )
@ A2 ) ).
% conditional_subset
thf(fact_165_Nset__le,axiom,
! [A: $tType] :
( ( ord @ A )
=> ! [X2: A,N2: A] :
( ( member @ A @ X2
@ ( collect @ A
@ ^ [I4: A] : ( ord_less_eq @ A @ I4 @ N2 ) ) )
=> ( ord_less_eq @ A @ X2 @ N2 ) ) ) ).
% Nset_le
thf(fact_166_Pi__mono,axiom,
! [B: $tType,A: $tType,A2: set @ A,B2: A > ( set @ B ),C2: A > ( set @ B )] :
( ! [X: A] :
( ( member @ A @ X @ A2 )
=> ( ord_less_eq @ ( set @ B ) @ ( B2 @ X ) @ ( C2 @ X ) ) )
=> ( ord_less_eq @ ( set @ ( A > B ) ) @ ( pi @ A @ B @ A2 @ B2 ) @ ( pi @ A @ B @ A2 @ C2 ) ) ) ).
% Pi_mono
thf(fact_167_Pi__anti__mono,axiom,
! [B: $tType,A: $tType,A7: set @ A,A2: set @ A,B2: A > ( set @ B )] :
( ( ord_less_eq @ ( set @ A ) @ A7 @ A2 )
=> ( ord_less_eq @ ( set @ ( A > B ) ) @ ( pi @ A @ B @ A2 @ B2 ) @ ( pi @ A @ B @ A7 @ B2 ) ) ) ).
% Pi_anti_mono
thf(fact_168_extend__fun,axiom,
! [B: $tType,A: $tType,F: A > B,A2: set @ A,B2: set @ B,B1: set @ B] :
( ( member @ ( A > B ) @ F
@ ( pi @ A @ B @ A2
@ ^ [Uu: A] : B2 ) )
=> ( ( ord_less_eq @ ( set @ B ) @ B2 @ B1 )
=> ( member @ ( A > B ) @ F
@ ( pi @ A @ B @ A2
@ ^ [Uu: A] : B1 ) ) ) ) ).
% extend_fun
thf(fact_169_aGroup_Oag__mOp__closed,axiom,
! [B: $tType,A: $tType,A2: carrier_ext @ A @ ( aGroup_ext @ A @ B ),X2: A] :
( ( aGroup @ A @ B @ A2 )
=> ( ( member @ A @ X2 @ ( carrier @ A @ ( aGroup_ext @ A @ B ) @ A2 ) )
=> ( member @ A @ ( mop @ A @ B @ A2 @ X2 ) @ ( carrier @ A @ ( aGroup_ext @ A @ B ) @ A2 ) ) ) ) ).
% aGroup.ag_mOp_closed
thf(fact_170_aGroup_Oag__inv__inv,axiom,
! [B: $tType,A: $tType,A2: carrier_ext @ A @ ( aGroup_ext @ A @ B ),X2: A] :
( ( aGroup @ A @ B @ A2 )
=> ( ( member @ A @ X2 @ ( carrier @ A @ ( aGroup_ext @ A @ B ) @ A2 ) )
=> ( ( mop @ A @ B @ A2 @ ( mop @ A @ B @ A2 @ X2 ) )
= X2 ) ) ) ).
% aGroup.ag_inv_inv
thf(fact_171_aGroup_Oag__inv__inj,axiom,
! [B: $tType,A: $tType,A2: carrier_ext @ A @ ( aGroup_ext @ A @ B ),X2: A,Y2: A] :
( ( aGroup @ A @ B @ A2 )
=> ( ( member @ A @ X2 @ ( carrier @ A @ ( aGroup_ext @ A @ B ) @ A2 ) )
=> ( ( member @ A @ Y2 @ ( carrier @ A @ ( aGroup_ext @ A @ B ) @ A2 ) )
=> ( ( X2 != Y2 )
=> ( ( mop @ A @ B @ A2 @ X2 )
!= ( mop @ A @ B @ A2 @ Y2 ) ) ) ) ) ) ).
% aGroup.ag_inv_inj
thf(fact_172_Ring_OSr__mOp__closed,axiom,
! [B: $tType,A: $tType,R2: carrier_ext @ A @ ( aGroup_ext @ A @ ( ring_ext @ A @ B ) ),S: set @ A,X2: A] :
( ( ring @ A @ B @ R2 )
=> ( ( sr2 @ A @ B @ R2 @ S )
=> ( ( member @ A @ X2 @ S )
=> ( member @ A @ ( mop @ A @ ( ring_ext @ A @ B ) @ R2 @ X2 ) @ S ) ) ) ) ).
% Ring.Sr_mOp_closed
thf(fact_173_Subring__subset,axiom,
! [C: $tType,B: $tType,A: $tType,R2: carrier_ext @ A @ ( aGroup_ext @ A @ ( ring_ext @ A @ B ) ),S: carrier_ext @ A @ ( aGroup_ext @ A @ ( ring_ext @ A @ C ) )] :
( ( subring @ A @ B @ C @ R2 @ S )
=> ( ord_less_eq @ ( set @ A ) @ ( carrier @ A @ ( aGroup_ext @ A @ ( ring_ext @ A @ C ) ) @ S ) @ ( carrier @ A @ ( aGroup_ext @ A @ ( ring_ext @ A @ B ) ) @ R2 ) ) ) ).
% Subring_subset
thf(fact_174_Ring_Oring__inv1__3,axiom,
! [B: $tType,A: $tType,R2: carrier_ext @ A @ ( aGroup_ext @ A @ ( ring_ext @ A @ B ) ),A3: A,B3: A] :
( ( ring @ A @ B @ R2 )
=> ( ( member @ A @ A3 @ ( carrier @ A @ ( aGroup_ext @ A @ ( ring_ext @ A @ B ) ) @ R2 ) )
=> ( ( member @ A @ B3 @ ( carrier @ A @ ( aGroup_ext @ A @ ( ring_ext @ A @ B ) ) @ R2 ) )
=> ( ( tp @ A @ B @ R2 @ A3 @ B3 )
= ( tp @ A @ B @ R2 @ ( mop @ A @ ( ring_ext @ A @ B ) @ R2 @ A3 ) @ ( mop @ A @ ( ring_ext @ A @ B ) @ R2 @ B3 ) ) ) ) ) ) ).
% Ring.ring_inv1_3
thf(fact_175_Ring_Oring__inv1__2,axiom,
! [B: $tType,A: $tType,R2: carrier_ext @ A @ ( aGroup_ext @ A @ ( ring_ext @ A @ B ) ),A3: A,B3: A] :
( ( ring @ A @ B @ R2 )
=> ( ( member @ A @ A3 @ ( carrier @ A @ ( aGroup_ext @ A @ ( ring_ext @ A @ B ) ) @ R2 ) )
=> ( ( member @ A @ B3 @ ( carrier @ A @ ( aGroup_ext @ A @ ( ring_ext @ A @ B ) ) @ R2 ) )
=> ( ( mop @ A @ ( ring_ext @ A @ B ) @ R2 @ ( tp @ A @ B @ R2 @ A3 @ B3 ) )
= ( tp @ A @ B @ R2 @ A3 @ ( mop @ A @ ( ring_ext @ A @ B ) @ R2 @ B3 ) ) ) ) ) ) ).
% Ring.ring_inv1_2
thf(fact_176_Ring_Oring__inv1__1,axiom,
! [B: $tType,A: $tType,R2: carrier_ext @ A @ ( aGroup_ext @ A @ ( ring_ext @ A @ B ) ),A3: A,B3: A] :
( ( ring @ A @ B @ R2 )
=> ( ( member @ A @ A3 @ ( carrier @ A @ ( aGroup_ext @ A @ ( ring_ext @ A @ B ) ) @ R2 ) )
=> ( ( member @ A @ B3 @ ( carrier @ A @ ( aGroup_ext @ A @ ( ring_ext @ A @ B ) ) @ R2 ) )
=> ( ( mop @ A @ ( ring_ext @ A @ B ) @ R2 @ ( tp @ A @ B @ R2 @ A3 @ B3 ) )
= ( tp @ A @ B @ R2 @ ( mop @ A @ ( ring_ext @ A @ B ) @ R2 @ A3 ) @ B3 ) ) ) ) ) ).
% Ring.ring_inv1_1
thf(fact_177_Ring_Oring__inv1,axiom,
! [B: $tType,A: $tType,R2: carrier_ext @ A @ ( aGroup_ext @ A @ ( ring_ext @ A @ B ) ),A3: A,B3: A] :
( ( ring @ A @ B @ R2 )
=> ( ( member @ A @ A3 @ ( carrier @ A @ ( aGroup_ext @ A @ ( ring_ext @ A @ B ) ) @ R2 ) )
=> ( ( member @ A @ B3 @ ( carrier @ A @ ( aGroup_ext @ A @ ( ring_ext @ A @ B ) ) @ R2 ) )
=> ( ( ( mop @ A @ ( ring_ext @ A @ B ) @ R2 @ ( tp @ A @ B @ R2 @ A3 @ B3 ) )
= ( tp @ A @ B @ R2 @ ( mop @ A @ ( ring_ext @ A @ B ) @ R2 @ A3 ) @ B3 ) )
& ( ( mop @ A @ ( ring_ext @ A @ B ) @ R2 @ ( tp @ A @ B @ R2 @ A3 @ B3 ) )
= ( tp @ A @ B @ R2 @ A3 @ ( mop @ A @ ( ring_ext @ A @ B ) @ R2 @ B3 ) ) ) ) ) ) ) ).
% Ring.ring_inv1
thf(fact_178_Ring_Oid__maximal__Exist,axiom,
! [B: $tType,A: $tType,R2: carrier_ext @ A @ ( aGroup_ext @ A @ ( ring_ext @ A @ B ) )] :
( ( ring @ A @ B @ R2 )
=> ( ~ ( zeroring @ A @ B @ R2 )
=> ? [X_12: set @ A] : ( maximal_ideal @ A @ B @ R2 @ X_12 ) ) ) ).
% Ring.id_maximal_Exist
thf(fact_179_Ring_Ozeroring__no__maximal,axiom,
! [B: $tType,A: $tType,R2: carrier_ext @ A @ ( aGroup_ext @ A @ ( ring_ext @ A @ B ) )] :
( ( ring @ A @ B @ R2 )
=> ( ( zeroring @ A @ B @ R2 )
=> ~ ? [X_1: set @ A] : ( maximal_ideal @ A @ B @ R2 @ X_1 ) ) ) ).
% Ring.zeroring_no_maximal
thf(fact_180_Ring_OSubring__minus__ring__minus,axiom,
! [C: $tType,B: $tType,A: $tType,R2: carrier_ext @ A @ ( aGroup_ext @ A @ ( ring_ext @ A @ B ) ),S: carrier_ext @ A @ ( aGroup_ext @ A @ ( ring_ext @ A @ C ) ),X2: A] :
( ( ring @ A @ B @ R2 )
=> ( ( subring @ A @ B @ C @ R2 @ S )
=> ( ( member @ A @ X2 @ ( carrier @ A @ ( aGroup_ext @ A @ ( ring_ext @ A @ C ) ) @ S ) )
=> ( ( mop @ A @ ( ring_ext @ A @ C ) @ S @ X2 )
= ( mop @ A @ ( ring_ext @ A @ B ) @ R2 @ X2 ) ) ) ) ) ).
% Ring.Subring_minus_ring_minus
thf(fact_181_Ring_Omop__closed,axiom,
! [B: $tType,A: $tType,R2: carrier_ext @ A @ ( aGroup_ext @ A @ ( ring_ext @ A @ B ) )] :
( ( ring @ A @ B @ R2 )
=> ( member @ ( A > A ) @ ( mop @ A @ ( ring_ext @ A @ B ) @ R2 )
@ ( pi @ A @ A @ ( carrier @ A @ ( aGroup_ext @ A @ ( ring_ext @ A @ B ) ) @ R2 )
@ ^ [Uu: A] : ( carrier @ A @ ( aGroup_ext @ A @ ( ring_ext @ A @ B ) ) @ R2 ) ) ) ) ).
% Ring.mop_closed
thf(fact_182_aGroup_OpOp__assoc__cancel,axiom,
! [B: $tType,A: $tType,A2: carrier_ext @ A @ ( aGroup_ext @ A @ B ),A3: A,B3: A,C3: A] :
( ( aGroup @ A @ B @ A2 )
=> ( ( member @ A @ A3 @ ( carrier @ A @ ( aGroup_ext @ A @ B ) @ A2 ) )
=> ( ( member @ A @ B3 @ ( carrier @ A @ ( aGroup_ext @ A @ B ) @ A2 ) )
=> ( ( member @ A @ C3 @ ( carrier @ A @ ( aGroup_ext @ A @ B ) @ A2 ) )
=> ( ( pop @ A @ B @ A2 @ ( pop @ A @ B @ A2 @ A3 @ ( mop @ A @ B @ A2 @ B3 ) ) @ ( pop @ A @ B @ A2 @ B3 @ ( mop @ A @ B @ A2 @ C3 ) ) )
= ( pop @ A @ B @ A2 @ A3 @ ( mop @ A @ B @ A2 @ C3 ) ) ) ) ) ) ) ).
% aGroup.pOp_assoc_cancel
thf(fact_183_aGroup_Oag__diff__minus,axiom,
! [B: $tType,A: $tType,A2: carrier_ext @ A @ ( aGroup_ext @ A @ B ),A3: A,B3: A,C3: A] :
( ( aGroup @ A @ B @ A2 )
=> ( ( member @ A @ A3 @ ( carrier @ A @ ( aGroup_ext @ A @ B ) @ A2 ) )
=> ( ( member @ A @ B3 @ ( carrier @ A @ ( aGroup_ext @ A @ B ) @ A2 ) )
=> ( ( member @ A @ C3 @ ( carrier @ A @ ( aGroup_ext @ A @ B ) @ A2 ) )
=> ( ( ( pop @ A @ B @ A2 @ A3 @ ( mop @ A @ B @ A2 @ B3 ) )
= C3 )
=> ( ( pop @ A @ B @ A2 @ B3 @ ( mop @ A @ B @ A2 @ A3 ) )
= ( mop @ A @ B @ A2 @ C3 ) ) ) ) ) ) ) ).
% aGroup.ag_diff_minus
thf(fact_184_aGroup_Oag__eq__sol2,axiom,
! [B: $tType,A: $tType,A2: carrier_ext @ A @ ( aGroup_ext @ A @ B ),A3: A,X2: A,B3: A] :
( ( aGroup @ A @ B @ A2 )
=> ( ( member @ A @ A3 @ ( carrier @ A @ ( aGroup_ext @ A @ B ) @ A2 ) )
=> ( ( member @ A @ X2 @ ( carrier @ A @ ( aGroup_ext @ A @ B ) @ A2 ) )
=> ( ( member @ A @ B3 @ ( carrier @ A @ ( aGroup_ext @ A @ B ) @ A2 ) )
=> ( ( ( pop @ A @ B @ A2 @ X2 @ A3 )
= B3 )
=> ( X2
= ( pop @ A @ B @ A2 @ B3 @ ( mop @ A @ B @ A2 @ A3 ) ) ) ) ) ) ) ) ).
% aGroup.ag_eq_sol2
thf(fact_185_aGroup_Oag__eq__sol1,axiom,
! [B: $tType,A: $tType,A2: carrier_ext @ A @ ( aGroup_ext @ A @ B ),A3: A,X2: A,B3: A] :
( ( aGroup @ A @ B @ A2 )
=> ( ( member @ A @ A3 @ ( carrier @ A @ ( aGroup_ext @ A @ B ) @ A2 ) )
=> ( ( member @ A @ X2 @ ( carrier @ A @ ( aGroup_ext @ A @ B ) @ A2 ) )
=> ( ( member @ A @ B3 @ ( carrier @ A @ ( aGroup_ext @ A @ B ) @ A2 ) )
=> ( ( ( pop @ A @ B @ A2 @ A3 @ X2 )
= B3 )
=> ( X2
= ( pop @ A @ B @ A2 @ ( mop @ A @ B @ A2 @ A3 ) @ B3 ) ) ) ) ) ) ) ).
% aGroup.ag_eq_sol1
thf(fact_186_aGroup_Oag__p__inv,axiom,
! [B: $tType,A: $tType,A2: carrier_ext @ A @ ( aGroup_ext @ A @ B ),X2: A,Y2: A] :
( ( aGroup @ A @ B @ A2 )
=> ( ( member @ A @ X2 @ ( carrier @ A @ ( aGroup_ext @ A @ B ) @ A2 ) )
=> ( ( member @ A @ Y2 @ ( carrier @ A @ ( aGroup_ext @ A @ B ) @ A2 ) )
=> ( ( mop @ A @ B @ A2 @ ( pop @ A @ B @ A2 @ X2 @ Y2 ) )
= ( pop @ A @ B @ A2 @ ( mop @ A @ B @ A2 @ X2 ) @ ( mop @ A @ B @ A2 @ Y2 ) ) ) ) ) ) ).
% aGroup.ag_p_inv
thf(fact_187_Ring_Omul__closed__set__sub,axiom,
! [B: $tType,A: $tType,R2: carrier_ext @ A @ ( aGroup_ext @ A @ ( ring_ext @ A @ B ) ),S: set @ A] :
( ( ring @ A @ B @ R2 )
=> ( ( mul_closed_set @ A @ B @ R2 @ S )
=> ( ord_less_eq @ ( set @ A ) @ S @ ( carrier @ A @ ( aGroup_ext @ A @ ( ring_ext @ A @ B ) ) @ R2 ) ) ) ) ).
% Ring.mul_closed_set_sub
thf(fact_188_mul__closed__set__def,axiom,
! [B: $tType,A: $tType] :
( ( mul_closed_set @ A @ B )
= ( ^ [R3: carrier_ext @ A @ ( aGroup_ext @ A @ ( ring_ext @ A @ B ) ),S3: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ S3 @ ( carrier @ A @ ( aGroup_ext @ A @ ( ring_ext @ A @ B ) ) @ R3 ) )
& ! [X3: A] :
( ( member @ A @ X3 @ S3 )
=> ! [Y3: A] :
( ( member @ A @ Y3 @ S3 )
=> ( member @ A @ ( tp @ A @ B @ R3 @ X3 @ Y3 ) @ S3 ) ) ) ) ) ) ).
% mul_closed_set_def
thf(fact_189_aGroup_Omop__closed,axiom,
! [B: $tType,A: $tType,A2: carrier_ext @ A @ ( aGroup_ext @ A @ B )] :
( ( aGroup @ A @ B @ A2 )
=> ( member @ ( A > A ) @ ( mop @ A @ B @ A2 )
@ ( pi @ A @ A @ ( carrier @ A @ ( aGroup_ext @ A @ B ) @ A2 )
@ ^ [Uu: A] : ( carrier @ A @ ( aGroup_ext @ A @ B ) @ A2 ) ) ) ) ).
% aGroup.mop_closed
thf(fact_190_prod__mOp__mem__i,axiom,
! [C: $tType,B: $tType,A: $tType,I2: set @ A,A2: A > ( carrier_ext @ B @ ( aGroup_ext @ B @ C ) ),X4: A > B,I3: A] :
( ! [X: A] :
( ( member @ A @ X @ I2 )
=> ( aGroup @ B @ C @ ( A2 @ X ) ) )
=> ( ( member @ ( A > B ) @ X4 @ ( carr_prodag @ A @ B @ C @ I2 @ A2 ) )
=> ( ( member @ A @ I3 @ I2 )
=> ( ( prod_mOp @ A @ B @ C @ I2 @ A2 @ X4 @ I3 )
= ( mop @ B @ C @ ( A2 @ I3 ) @ ( X4 @ I3 ) ) ) ) ) ) ).
% prod_mOp_mem_i
thf(fact_191_Ring_Oring__distrib4,axiom,
! [B: $tType,A: $tType,R2: carrier_ext @ A @ ( aGroup_ext @ A @ ( ring_ext @ A @ B ) ),A3: A,B3: A,X2: A,Y2: A] :
( ( ring @ A @ B @ R2 )
=> ( ( member @ A @ A3 @ ( carrier @ A @ ( aGroup_ext @ A @ ( ring_ext @ A @ B ) ) @ R2 ) )
=> ( ( member @ A @ B3 @ ( carrier @ A @ ( aGroup_ext @ A @ ( ring_ext @ A @ B ) ) @ R2 ) )
=> ( ( member @ A @ X2 @ ( carrier @ A @ ( aGroup_ext @ A @ ( ring_ext @ A @ B ) ) @ R2 ) )
=> ( ( member @ A @ Y2 @ ( carrier @ A @ ( aGroup_ext @ A @ ( ring_ext @ A @ B ) ) @ R2 ) )
=> ( ( pop @ A @ ( ring_ext @ A @ B ) @ R2 @ ( tp @ A @ B @ R2 @ A3 @ B3 ) @ ( mop @ A @ ( ring_ext @ A @ B ) @ R2 @ ( tp @ A @ B @ R2 @ X2 @ Y2 ) ) )
= ( pop @ A @ ( ring_ext @ A @ B ) @ R2 @ ( tp @ A @ B @ R2 @ A3 @ ( pop @ A @ ( ring_ext @ A @ B ) @ R2 @ B3 @ ( mop @ A @ ( ring_ext @ A @ B ) @ R2 @ Y2 ) ) ) @ ( tp @ A @ B @ R2 @ ( pop @ A @ ( ring_ext @ A @ B ) @ R2 @ A3 @ ( mop @ A @ ( ring_ext @ A @ B ) @ R2 @ X2 ) ) @ Y2 ) ) ) ) ) ) ) ) ).
% Ring.ring_distrib4
thf(fact_192_nonunit__contained__maxid,axiom,
! [A3: a] :
( ~ ( zeroring @ a @ b @ r )
=> ( ( member @ a @ A3 @ ( carrier @ a @ ( aGroup_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
=> ( ~ ( unit @ a @ b @ r @ A3 )
=> ? [Mx: set @ a] :
( ( maximal_ideal @ a @ b @ r @ Mx )
& ( member @ a @ A3 @ Mx ) ) ) ) ) ).
% nonunit_contained_maxid
thf(fact_193_set__sum__mem,axiom,
! [A3: a,I2: set @ a,B3: a,J2: set @ a] :
( ( member @ a @ A3 @ I2 )
=> ( ( member @ a @ B3 @ J2 )
=> ( ( ord_less_eq @ ( set @ a ) @ I2 @ ( carrier @ a @ ( aGroup_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
=> ( ( ord_less_eq @ ( set @ a ) @ J2 @ ( carrier @ a @ ( aGroup_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
=> ( member @ a @ ( pop @ a @ ( ring_ext @ a @ b ) @ r @ A3 @ B3 ) @ ( aset_sum @ a @ ( ring_ext @ a @ b ) @ r @ I2 @ J2 ) ) ) ) ) ) ).
% set_sum_mem
thf(fact_194_sum__mult__pOp__closed,axiom,
! [A2: set @ a,B2: set @ a,A3: a,B3: a] :
( ( ord_less_eq @ ( set @ a ) @ A2 @ ( carrier @ a @ ( aGroup_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
=> ( ( ord_less_eq @ ( set @ a ) @ B2 @ ( carrier @ a @ ( aGroup_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
=> ( ( member @ a @ A3 @ ( sum_mult @ a @ b @ r @ A2 @ B2 ) )
=> ( ( member @ a @ B3 @ ( sum_mult @ a @ b @ r @ A2 @ B2 ) )
=> ( member @ a @ ( pop @ a @ ( ring_ext @ a @ b ) @ r @ A3 @ B3 ) @ ( sum_mult @ a @ b @ r @ A2 @ B2 ) ) ) ) ) ) ).
% sum_mult_pOp_closed
thf(fact_195_sum__mult__subR,axiom,
! [A2: set @ a,B2: set @ a] :
( ( ord_less_eq @ ( set @ a ) @ A2 @ ( carrier @ a @ ( aGroup_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
=> ( ( ord_less_eq @ ( set @ a ) @ B2 @ ( carrier @ a @ ( aGroup_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
=> ( ord_less_eq @ ( set @ a ) @ ( sum_mult @ a @ b @ r @ A2 @ B2 ) @ ( carrier @ a @ ( aGroup_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) ) ) ) ).
% sum_mult_subR
thf(fact_196_times__mem__sum__mult,axiom,
! [A2: set @ a,B2: set @ a,A3: a,B3: a] :
( ( ord_less_eq @ ( set @ a ) @ A2 @ ( carrier @ a @ ( aGroup_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
=> ( ( ord_less_eq @ ( set @ a ) @ B2 @ ( carrier @ a @ ( aGroup_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
=> ( ( member @ a @ A3 @ A2 )
=> ( ( member @ a @ B3 @ B2 )
=> ( member @ a @ ( tp @ a @ b @ r @ A3 @ B3 ) @ ( sum_mult @ a @ b @ r @ A2 @ B2 ) ) ) ) ) ) ).
% times_mem_sum_mult
thf(fact_197_le__imp__add__int,axiom,
! [I3: nat,J: nat] :
( ( ord_less_eq @ nat @ I3 @ J )
=> ? [K2: nat] :
( J
= ( plus_plus @ nat @ I3 @ K2 ) ) ) ).
% le_imp_add_int
thf(fact_198_Module_OmHom__inv,axiom,
! [D: $tType,C: $tType,B: $tType,F2: $tType,E: $tType,A: $tType,M3: carrier_ext @ A @ ( aGroup_ext @ A @ ( module_ext @ A @ B @ C ) ),R2: carrier_ext @ B @ ( aGroup_ext @ B @ ( ring_ext @ B @ D ) ),N: carrier_ext @ E @ ( aGroup_ext @ E @ ( module_ext @ E @ B @ F2 ) ),M4: A,F: A > E] :
( ( module @ A @ B @ C @ D @ M3 @ R2 )
=> ( ( module @ E @ B @ F2 @ D @ N @ R2 )
=> ( ( member @ A @ M4 @ ( carrier @ A @ ( aGroup_ext @ A @ ( module_ext @ A @ B @ C ) ) @ M3 ) )
=> ( ( member @ ( A > E ) @ F @ ( mHom @ B @ D @ A @ C @ E @ F2 @ R2 @ M3 @ N ) )
=> ( ( F @ ( mop @ A @ ( module_ext @ A @ B @ C ) @ M3 @ M4 ) )
= ( mop @ E @ ( module_ext @ E @ B @ F2 ) @ N @ ( F @ M4 ) ) ) ) ) ) ) ).
% Module.mHom_inv
thf(fact_199_dsumag__sub__prodag,axiom,
! [C: $tType,B: $tType,A: $tType,I2: set @ A,A2: A > ( carrier_ext @ B @ ( aGroup_ext @ B @ C ) )] :
( ! [X: A] :
( ( member @ A @ X @ I2 )
=> ( aGroup @ B @ C @ ( A2 @ X ) ) )
=> ( ord_less_eq @ ( set @ ( A > B ) ) @ ( carr_dsumag @ A @ B @ C @ I2 @ A2 ) @ ( carr_prodag @ A @ B @ C @ I2 @ A2 ) ) ) ).
% dsumag_sub_prodag
thf(fact_200_Module_Osc__minus__am,axiom,
! [D: $tType,B: $tType,C: $tType,A: $tType,M3: carrier_ext @ A @ ( aGroup_ext @ A @ ( module_ext @ A @ B @ C ) ),R2: carrier_ext @ B @ ( aGroup_ext @ B @ ( ring_ext @ B @ D ) ),A3: B,M4: A] :
( ( module @ A @ B @ C @ D @ M3 @ R2 )
=> ( ( member @ B @ A3 @ ( carrier @ B @ ( aGroup_ext @ B @ ( ring_ext @ B @ D ) ) @ R2 ) )
=> ( ( member @ A @ M4 @ ( carrier @ A @ ( aGroup_ext @ A @ ( module_ext @ A @ B @ C ) ) @ M3 ) )
=> ( ( mop @ A @ ( module_ext @ A @ B @ C ) @ M3 @ ( sprod @ A @ B @ C @ M3 @ A3 @ M4 ) )
= ( sprod @ A @ B @ C @ M3 @ A3 @ ( mop @ A @ ( module_ext @ A @ B @ C ) @ M3 @ M4 ) ) ) ) ) ) ).
% Module.sc_minus_am
thf(fact_201_Ring_Osum__mult__subR,axiom,
! [B: $tType,A: $tType,R2: carrier_ext @ A @ ( aGroup_ext @ A @ ( ring_ext @ A @ B ) ),A2: set @ A,B2: set @ A] :
( ( ring @ A @ B @ R2 )
=> ( ( ord_less_eq @ ( set @ A ) @ A2 @ ( carrier @ A @ ( aGroup_ext @ A @ ( ring_ext @ A @ B ) ) @ R2 ) )
=> ( ( ord_less_eq @ ( set @ A ) @ B2 @ ( carrier @ A @ ( aGroup_ext @ A @ ( ring_ext @ A @ B ) ) @ R2 ) )
=> ( ord_less_eq @ ( set @ A ) @ ( sum_mult @ A @ B @ R2 @ A2 @ B2 ) @ ( carrier @ A @ ( aGroup_ext @ A @ ( ring_ext @ A @ B ) ) @ R2 ) ) ) ) ) ).
% Ring.sum_mult_subR
thf(fact_202_Module_Osc__minus__am1,axiom,
! [B: $tType,D: $tType,C: $tType,A: $tType,M3: carrier_ext @ A @ ( aGroup_ext @ A @ ( module_ext @ A @ B @ C ) ),R2: carrier_ext @ B @ ( aGroup_ext @ B @ ( ring_ext @ B @ D ) ),A3: B,M4: A] :
( ( module @ A @ B @ C @ D @ M3 @ R2 )
=> ( ( member @ B @ A3 @ ( carrier @ B @ ( aGroup_ext @ B @ ( ring_ext @ B @ D ) ) @ R2 ) )
=> ( ( member @ A @ M4 @ ( carrier @ A @ ( aGroup_ext @ A @ ( module_ext @ A @ B @ C ) ) @ M3 ) )
=> ( ( mop @ A @ ( module_ext @ A @ B @ C ) @ M3 @ ( sprod @ A @ B @ C @ M3 @ A3 @ M4 ) )
= ( sprod @ A @ B @ C @ M3 @ ( mop @ B @ ( ring_ext @ B @ D ) @ R2 @ A3 ) @ M4 ) ) ) ) ) ).
% Module.sc_minus_am1
thf(fact_203_Ring_Otimes__mem__sum__mult,axiom,
! [B: $tType,A: $tType,R2: carrier_ext @ A @ ( aGroup_ext @ A @ ( ring_ext @ A @ B ) ),A2: set @ A,B2: set @ A,A3: A,B3: A] :
( ( ring @ A @ B @ R2 )
=> ( ( ord_less_eq @ ( set @ A ) @ A2 @ ( carrier @ A @ ( aGroup_ext @ A @ ( ring_ext @ A @ B ) ) @ R2 ) )
=> ( ( ord_less_eq @ ( set @ A ) @ B2 @ ( carrier @ A @ ( aGroup_ext @ A @ ( ring_ext @ A @ B ) ) @ R2 ) )
=> ( ( member @ A @ A3 @ A2 )
=> ( ( member @ A @ B3 @ B2 )
=> ( member @ A @ ( tp @ A @ B @ R2 @ A3 @ B3 ) @ ( sum_mult @ A @ B @ R2 @ A2 @ B2 ) ) ) ) ) ) ) ).
% Ring.times_mem_sum_mult
thf(fact_204_Ring_Osum__mult__pOp__closed,axiom,
! [B: $tType,A: $tType,R2: carrier_ext @ A @ ( aGroup_ext @ A @ ( ring_ext @ A @ B ) ),A2: set @ A,B2: set @ A,A3: A,B3: A] :
( ( ring @ A @ B @ R2 )
=> ( ( ord_less_eq @ ( set @ A ) @ A2 @ ( carrier @ A @ ( aGroup_ext @ A @ ( ring_ext @ A @ B ) ) @ R2 ) )
=> ( ( ord_less_eq @ ( set @ A ) @ B2 @ ( carrier @ A @ ( aGroup_ext @ A @ ( ring_ext @ A @ B ) ) @ R2 ) )
=> ( ( member @ A @ A3 @ ( sum_mult @ A @ B @ R2 @ A2 @ B2 ) )
=> ( ( member @ A @ B3 @ ( sum_mult @ A @ B @ R2 @ A2 @ B2 ) )
=> ( member @ A @ ( pop @ A @ ( ring_ext @ A @ B ) @ R2 @ A3 @ B3 ) @ ( sum_mult @ A @ B @ R2 @ A2 @ B2 ) ) ) ) ) ) ) ).
% Ring.sum_mult_pOp_closed
thf(fact_205_Ring_Oset__sum__mem,axiom,
! [B: $tType,A: $tType,R2: carrier_ext @ A @ ( aGroup_ext @ A @ ( ring_ext @ A @ B ) ),A3: A,I2: set @ A,B3: A,J2: set @ A] :
( ( ring @ A @ B @ R2 )
=> ( ( member @ A @ A3 @ I2 )
=> ( ( member @ A @ B3 @ J2 )
=> ( ( ord_less_eq @ ( set @ A ) @ I2 @ ( carrier @ A @ ( aGroup_ext @ A @ ( ring_ext @ A @ B ) ) @ R2 ) )
=> ( ( ord_less_eq @ ( set @ A ) @ J2 @ ( carrier @ A @ ( aGroup_ext @ A @ ( ring_ext @ A @ B ) ) @ R2 ) )
=> ( member @ A @ ( pop @ A @ ( ring_ext @ A @ B ) @ R2 @ A3 @ B3 ) @ ( aset_sum @ A @ ( ring_ext @ A @ B ) @ R2 @ I2 @ J2 ) ) ) ) ) ) ) ).
% Ring.set_sum_mem
thf(fact_206_Ring_Ononunit__contained__maxid,axiom,
! [B: $tType,A: $tType,R2: carrier_ext @ A @ ( aGroup_ext @ A @ ( ring_ext @ A @ B ) ),A3: A] :
( ( ring @ A @ B @ R2 )
=> ( ~ ( zeroring @ A @ B @ R2 )
=> ( ( member @ A @ A3 @ ( carrier @ A @ ( aGroup_ext @ A @ ( ring_ext @ A @ B ) ) @ R2 ) )
=> ( ~ ( unit @ A @ B @ R2 @ A3 )
=> ? [Mx: set @ A] :
( ( maximal_ideal @ A @ B @ R2 @ Mx )
& ( member @ A @ A3 @ Mx ) ) ) ) ) ) ).
% Ring.nonunit_contained_maxid
thf(fact_207_aGroup_Omem__set__sum,axiom,
! [B: $tType,A: $tType,A2: carrier_ext @ A @ ( aGroup_ext @ A @ B ),H: set @ A,K3: set @ A,X2: A] :
( ( aGroup @ A @ B @ A2 )
=> ( ( ord_less_eq @ ( set @ A ) @ H @ ( carrier @ A @ ( aGroup_ext @ A @ B ) @ A2 ) )
=> ( ( ord_less_eq @ ( set @ A ) @ K3 @ ( carrier @ A @ ( aGroup_ext @ A @ B ) @ A2 ) )
=> ( ( member @ A @ X2 @ ( aset_sum @ A @ B @ A2 @ H @ K3 ) )
=> ? [X: A] :
( ( member @ A @ X @ H )
& ? [Xa2: A] :
( ( member @ A @ Xa2 @ K3 )
& ( X2
= ( pop @ A @ B @ A2 @ X @ Xa2 ) ) ) ) ) ) ) ) ).
% aGroup.mem_set_sum
thf(fact_208_J__rad__unit,axiom,
! [X2: a] :
( ~ ( zeroring @ a @ b @ r )
=> ( ( member @ a @ X2 @ ( j_rad @ a @ b @ r ) )
=> ! [Y4: a] :
( ( member @ a @ Y4 @ ( carrier @ a @ ( aGroup_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
=> ( unit @ a @ b @ r @ ( pop @ a @ ( ring_ext @ a @ b ) @ r @ ( un @ a @ b @ r ) @ ( tp @ a @ b @ r @ ( mop @ a @ ( ring_ext @ a @ b ) @ r @ X2 ) @ Y4 ) ) ) ) ) ) ).
% J_rad_unit
thf(fact_209_add__le__cancel__right,axiom,
! [A: $tType] :
( ( ordere236663937imp_le @ A )
=> ! [A3: A,C3: A,B3: A] :
( ( ord_less_eq @ A @ ( plus_plus @ A @ A3 @ C3 ) @ ( plus_plus @ A @ B3 @ C3 ) )
= ( ord_less_eq @ A @ A3 @ B3 ) ) ) ).
% add_le_cancel_right
thf(fact_210_un__closed,axiom,
member @ a @ ( un @ a @ b @ r ) @ ( carrier @ a @ ( aGroup_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) ).
% un_closed
thf(fact_211_maximal__ideal__proper,axiom,
! [Mx2: set @ a] :
( ( maximal_ideal @ a @ b @ r @ Mx2 )
=> ~ ( member @ a @ ( un @ a @ b @ r ) @ Mx2 ) ) ).
% maximal_ideal_proper
thf(fact_212_Sr__one,axiom,
! [S: set @ a] :
( ( sr2 @ a @ b @ r @ S )
=> ( member @ a @ ( un @ a @ b @ r ) @ S ) ) ).
% Sr_one
thf(fact_213_add__right__cancel,axiom,
! [A: $tType] :
( ( cancel_semigroup_add @ A )
=> ! [B3: A,A3: A,C3: A] :
( ( ( plus_plus @ A @ B3 @ A3 )
= ( plus_plus @ A @ C3 @ A3 ) )
= ( B3 = C3 ) ) ) ).
% add_right_cancel
thf(fact_214_add__left__cancel,axiom,
! [A: $tType] :
( ( cancel_semigroup_add @ A )
=> ! [A3: A,B3: A,C3: A] :
( ( ( plus_plus @ A @ A3 @ B3 )
= ( plus_plus @ A @ A3 @ C3 ) )
= ( B3 = C3 ) ) ) ).
% add_left_cancel
thf(fact_215_rg__l__unit,axiom,
! [A3: a] :
( ( member @ a @ A3 @ ( carrier @ a @ ( aGroup_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
=> ( ( tp @ a @ b @ r @ ( un @ a @ b @ r ) @ A3 )
= A3 ) ) ).
% rg_l_unit
thf(fact_216_ring__l__one,axiom,
! [X2: a] :
( ( member @ a @ X2 @ ( carrier @ a @ ( aGroup_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
=> ( ( tp @ a @ b @ r @ ( un @ a @ b @ r ) @ X2 )
= X2 ) ) ).
% ring_l_one
thf(fact_217_ring__r__one,axiom,
! [X2: a] :
( ( member @ a @ X2 @ ( carrier @ a @ ( aGroup_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
=> ( ( tp @ a @ b @ r @ X2 @ ( un @ a @ b @ r ) )
= X2 ) ) ).
% ring_r_one
thf(fact_218_Subring__one__ring__one,axiom,
! [C: $tType,S: carrier_ext @ a @ ( aGroup_ext @ a @ ( ring_ext @ a @ C ) )] :
( ( subring @ a @ b @ C @ r @ S )
=> ( ( un @ a @ C @ S )
= ( un @ a @ b @ r ) ) ) ).
% Subring_one_ring_one
thf(fact_219_ring__times__minusr,axiom,
! [A3: a] :
( ( member @ a @ A3 @ ( carrier @ a @ ( aGroup_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
=> ( ( mop @ a @ ( ring_ext @ a @ b ) @ r @ A3 )
= ( tp @ a @ b @ r @ A3 @ ( mop @ a @ ( ring_ext @ a @ b ) @ r @ ( un @ a @ b @ r ) ) ) ) ) ).
% ring_times_minusr
thf(fact_220_ring__times__minusl,axiom,
! [A3: a] :
( ( member @ a @ A3 @ ( carrier @ a @ ( aGroup_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
=> ( ( mop @ a @ ( ring_ext @ a @ b ) @ r @ A3 )
= ( tp @ a @ b @ r @ ( mop @ a @ ( ring_ext @ a @ b ) @ r @ ( un @ a @ b @ r ) ) @ A3 ) ) ) ).
% ring_times_minusl
thf(fact_221_add__le__cancel__left,axiom,
! [A: $tType] :
( ( ordere236663937imp_le @ A )
=> ! [C3: A,A3: A,B3: A] :
( ( ord_less_eq @ A @ ( plus_plus @ A @ C3 @ A3 ) @ ( plus_plus @ A @ C3 @ B3 ) )
= ( ord_less_eq @ A @ A3 @ B3 ) ) ) ).
% add_le_cancel_left
thf(fact_222_localring__unit,axiom,
! [Mx2: set @ a] :
( ~ ( zeroring @ a @ b @ r )
=> ( ( maximal_ideal @ a @ b @ r @ Mx2 )
=> ( ! [X: a] :
( ( member @ a @ X @ Mx2 )
=> ( unit @ a @ b @ r @ ( pop @ a @ ( ring_ext @ a @ b ) @ r @ X @ ( un @ a @ b @ r ) ) ) )
=> ( local_ring @ a @ b @ r ) ) ) ) ).
% localring_unit
thf(fact_223_primary__ideal__proper1,axiom,
! [Q2: set @ a] :
( ( primary_ideal @ a @ b @ r @ Q2 )
=> ~ ( member @ a @ ( un @ a @ b @ r ) @ Q2 ) ) ).
% primary_ideal_proper1
thf(fact_224_self__le,axiom,
! [N2: nat] : ( ord_less_eq @ nat @ N2 @ N2 ) ).
% self_le
thf(fact_225_nat__eq__le,axiom,
! [M4: nat,N2: nat] :
( ( M4 = N2 )
=> ( ord_less_eq @ nat @ M4 @ N2 ) ) ).
% nat_eq_le
thf(fact_226_n__in__Nsetn,axiom,
! [N2: nat] :
( member @ nat @ N2
@ ( collect @ nat
@ ^ [I4: nat] : ( ord_less_eq @ nat @ I4 @ N2 ) ) ) ).
% n_in_Nsetn
thf(fact_227_mem__of__Nset,axiom,
! [X2: nat,N2: nat] :
( ( ord_less_eq @ nat @ X2 @ N2 )
=> ( member @ nat @ X2
@ ( collect @ nat
@ ^ [I4: nat] : ( ord_less_eq @ nat @ I4 @ N2 ) ) ) ) ).
% mem_of_Nset
thf(fact_228_Ring_Oun__closed,axiom,
! [B: $tType,A: $tType,R2: carrier_ext @ A @ ( aGroup_ext @ A @ ( ring_ext @ A @ B ) )] :
( ( ring @ A @ B @ R2 )
=> ( member @ A @ ( un @ A @ B @ R2 ) @ ( carrier @ A @ ( aGroup_ext @ A @ ( ring_ext @ A @ B ) ) @ R2 ) ) ) ).
% Ring.un_closed
thf(fact_229_Ring_Omaximal__ideal__proper,axiom,
! [B: $tType,A: $tType,R2: carrier_ext @ A @ ( aGroup_ext @ A @ ( ring_ext @ A @ B ) ),Mx2: set @ A] :
( ( ring @ A @ B @ R2 )
=> ( ( maximal_ideal @ A @ B @ R2 @ Mx2 )
=> ~ ( member @ A @ ( un @ A @ B @ R2 ) @ Mx2 ) ) ) ).
% Ring.maximal_ideal_proper
thf(fact_230_Ring_OSubring__one__ring__one,axiom,
! [C: $tType,B: $tType,A: $tType,R2: carrier_ext @ A @ ( aGroup_ext @ A @ ( ring_ext @ A @ B ) ),S: carrier_ext @ A @ ( aGroup_ext @ A @ ( ring_ext @ A @ C ) )] :
( ( ring @ A @ B @ R2 )
=> ( ( subring @ A @ B @ C @ R2 @ S )
=> ( ( un @ A @ C @ S )
= ( un @ A @ B @ R2 ) ) ) ) ).
% Ring.Subring_one_ring_one
thf(fact_231_Ring_OSr__one,axiom,
! [B: $tType,A: $tType,R2: carrier_ext @ A @ ( aGroup_ext @ A @ ( ring_ext @ A @ B ) ),S: set @ A] :
( ( ring @ A @ B @ R2 )
=> ( ( sr2 @ A @ B @ R2 @ S )
=> ( member @ A @ ( un @ A @ B @ R2 ) @ S ) ) ) ).
% Ring.Sr_one
thf(fact_232_Ring_Oring__r__one,axiom,
! [B: $tType,A: $tType,R2: carrier_ext @ A @ ( aGroup_ext @ A @ ( ring_ext @ A @ B ) ),X2: A] :
( ( ring @ A @ B @ R2 )
=> ( ( member @ A @ X2 @ ( carrier @ A @ ( aGroup_ext @ A @ ( ring_ext @ A @ B ) ) @ R2 ) )
=> ( ( tp @ A @ B @ R2 @ X2 @ ( un @ A @ B @ R2 ) )
= X2 ) ) ) ).
% Ring.ring_r_one
thf(fact_233_Ring_Oring__l__one,axiom,
! [B: $tType,A: $tType,R2: carrier_ext @ A @ ( aGroup_ext @ A @ ( ring_ext @ A @ B ) ),X2: A] :
( ( ring @ A @ B @ R2 )
=> ( ( member @ A @ X2 @ ( carrier @ A @ ( aGroup_ext @ A @ ( ring_ext @ A @ B ) ) @ R2 ) )
=> ( ( tp @ A @ B @ R2 @ ( un @ A @ B @ R2 ) @ X2 )
= X2 ) ) ) ).
% Ring.ring_l_one
thf(fact_234_Ring_Org__l__unit,axiom,
! [B: $tType,A: $tType,R2: carrier_ext @ A @ ( aGroup_ext @ A @ ( ring_ext @ A @ B ) ),A3: A] :
( ( ring @ A @ B @ R2 )
=> ( ( member @ A @ A3 @ ( carrier @ A @ ( aGroup_ext @ A @ ( ring_ext @ A @ B ) ) @ R2 ) )
=> ( ( tp @ A @ B @ R2 @ ( un @ A @ B @ R2 ) @ A3 )
= A3 ) ) ) ).
% Ring.rg_l_unit
thf(fact_235_Module_Osprod__one,axiom,
! [C: $tType,D: $tType,B: $tType,A: $tType,M3: carrier_ext @ A @ ( aGroup_ext @ A @ ( module_ext @ A @ B @ C ) ),R2: carrier_ext @ B @ ( aGroup_ext @ B @ ( ring_ext @ B @ D ) ),M4: A] :
( ( module @ A @ B @ C @ D @ M3 @ R2 )
=> ( ( member @ A @ M4 @ ( carrier @ A @ ( aGroup_ext @ A @ ( module_ext @ A @ B @ C ) ) @ M3 ) )
=> ( ( sprod @ A @ B @ C @ M3 @ ( un @ B @ D @ R2 ) @ M4 )
= M4 ) ) ) ).
% Module.sprod_one
thf(fact_236_Unit__def,axiom,
! [B: $tType,A: $tType] :
( ( unit @ A @ B )
= ( ^ [R3: carrier_ext @ A @ ( aGroup_ext @ A @ ( ring_ext @ A @ B ) ),A5: A] :
( ( member @ A @ A5 @ ( carrier @ A @ ( aGroup_ext @ A @ ( ring_ext @ A @ B ) ) @ R3 ) )
& ? [X3: A] :
( ( member @ A @ X3 @ ( carrier @ A @ ( aGroup_ext @ A @ ( ring_ext @ A @ B ) ) @ R3 ) )
& ( ( tp @ A @ B @ R3 @ A5 @ X3 )
= ( un @ A @ B @ R3 ) ) ) ) ) ) ).
% Unit_def
thf(fact_237_add__right__imp__eq,axiom,
! [A: $tType] :
( ( cancel_semigroup_add @ A )
=> ! [B3: A,A3: A,C3: A] :
( ( ( plus_plus @ A @ B3 @ A3 )
= ( plus_plus @ A @ C3 @ A3 ) )
=> ( B3 = C3 ) ) ) ).
% add_right_imp_eq
thf(fact_238_add__left__imp__eq,axiom,
! [A: $tType] :
( ( cancel_semigroup_add @ A )
=> ! [A3: A,B3: A,C3: A] :
( ( ( plus_plus @ A @ A3 @ B3 )
= ( plus_plus @ A @ A3 @ C3 ) )
=> ( B3 = C3 ) ) ) ).
% add_left_imp_eq
thf(fact_239_add_Oleft__commute,axiom,
! [A: $tType] :
( ( ab_semigroup_add @ A )
=> ! [B3: A,A3: A,C3: A] :
( ( plus_plus @ A @ B3 @ ( plus_plus @ A @ A3 @ C3 ) )
= ( plus_plus @ A @ A3 @ ( plus_plus @ A @ B3 @ C3 ) ) ) ) ).
% add.left_commute
thf(fact_240_add_Ocommute,axiom,
! [A: $tType] :
( ( ab_semigroup_add @ A )
=> ( ( plus_plus @ A )
= ( ^ [A5: A,B4: A] : ( plus_plus @ A @ B4 @ A5 ) ) ) ) ).
% add.commute
thf(fact_241_add_Oright__cancel,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ! [B3: A,A3: A,C3: A] :
( ( ( plus_plus @ A @ B3 @ A3 )
= ( plus_plus @ A @ C3 @ A3 ) )
= ( B3 = C3 ) ) ) ).
% add.right_cancel
thf(fact_242_add_Oleft__cancel,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ! [A3: A,B3: A,C3: A] :
( ( ( plus_plus @ A @ A3 @ B3 )
= ( plus_plus @ A @ A3 @ C3 ) )
= ( B3 = C3 ) ) ) ).
% add.left_cancel
thf(fact_243_add_Oassoc,axiom,
! [A: $tType] :
( ( semigroup_add @ A )
=> ! [A3: A,B3: A,C3: A] :
( ( plus_plus @ A @ ( plus_plus @ A @ A3 @ B3 ) @ C3 )
= ( plus_plus @ A @ A3 @ ( plus_plus @ A @ B3 @ C3 ) ) ) ) ).
% add.assoc
thf(fact_244_group__cancel_Oadd2,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [B2: A,K4: A,B3: A,A3: A] :
( ( B2
= ( plus_plus @ A @ K4 @ B3 ) )
=> ( ( plus_plus @ A @ A3 @ B2 )
= ( plus_plus @ A @ K4 @ ( plus_plus @ A @ A3 @ B3 ) ) ) ) ) ).
% group_cancel.add2
thf(fact_245_group__cancel_Oadd1,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [A2: A,K4: A,A3: A,B3: A] :
( ( A2
= ( plus_plus @ A @ K4 @ A3 ) )
=> ( ( plus_plus @ A @ A2 @ B3 )
= ( plus_plus @ A @ K4 @ ( plus_plus @ A @ A3 @ B3 ) ) ) ) ) ).
% group_cancel.add1
thf(fact_246_add__mono__thms__linordered__semiring_I4_J,axiom,
! [A: $tType] :
( ( ordere779506340up_add @ A )
=> ! [I3: A,J: A,K4: A,L: A] :
( ( ( I3 = J )
& ( K4 = L ) )
=> ( ( plus_plus @ A @ I3 @ K4 )
= ( plus_plus @ A @ J @ L ) ) ) ) ).
% add_mono_thms_linordered_semiring(4)
thf(fact_247_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
! [A: $tType] :
( ( ab_semigroup_add @ A )
=> ! [A3: A,B3: A,C3: A] :
( ( plus_plus @ A @ ( plus_plus @ A @ A3 @ B3 ) @ C3 )
= ( plus_plus @ A @ A3 @ ( plus_plus @ A @ B3 @ C3 ) ) ) ) ).
% ab_semigroup_add_class.add_ac(1)
thf(fact_248_Ring_Oring__times__minusl,axiom,
! [B: $tType,A: $tType,R2: carrier_ext @ A @ ( aGroup_ext @ A @ ( ring_ext @ A @ B ) ),A3: A] :
( ( ring @ A @ B @ R2 )
=> ( ( member @ A @ A3 @ ( carrier @ A @ ( aGroup_ext @ A @ ( ring_ext @ A @ B ) ) @ R2 ) )
=> ( ( mop @ A @ ( ring_ext @ A @ B ) @ R2 @ A3 )
= ( tp @ A @ B @ R2 @ ( mop @ A @ ( ring_ext @ A @ B ) @ R2 @ ( un @ A @ B @ R2 ) ) @ A3 ) ) ) ) ).
% Ring.ring_times_minusl
thf(fact_249_Ring_Oring__times__minusr,axiom,
! [B: $tType,A: $tType,R2: carrier_ext @ A @ ( aGroup_ext @ A @ ( ring_ext @ A @ B ) ),A3: A] :
( ( ring @ A @ B @ R2 )
=> ( ( member @ A @ A3 @ ( carrier @ A @ ( aGroup_ext @ A @ ( ring_ext @ A @ B ) ) @ R2 ) )
=> ( ( mop @ A @ ( ring_ext @ A @ B ) @ R2 @ A3 )
= ( tp @ A @ B @ R2 @ A3 @ ( mop @ A @ ( ring_ext @ A @ B ) @ R2 @ ( un @ A @ B @ R2 ) ) ) ) ) ) ).
% Ring.ring_times_minusr
thf(fact_250_sr__def,axiom,
! [B: $tType,A: $tType] :
( ( sr2 @ A @ B )
= ( ^ [R3: carrier_ext @ A @ ( aGroup_ext @ A @ ( ring_ext @ A @ B ) ),S3: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ S3 @ ( carrier @ A @ ( aGroup_ext @ A @ ( ring_ext @ A @ B ) ) @ R3 ) )
& ( member @ A @ ( un @ A @ B @ R3 ) @ S3 )
& ! [X3: A] :
( ( member @ A @ X3 @ S3 )
=> ! [Y3: A] :
( ( member @ A @ Y3 @ S3 )
=> ( ( member @ A @ ( pop @ A @ ( ring_ext @ A @ B ) @ R3 @ X3 @ ( mop @ A @ ( ring_ext @ A @ B ) @ R3 @ Y3 ) ) @ S3 )
& ( member @ A @ ( tp @ A @ B @ R3 @ X3 @ Y3 ) @ S3 ) ) ) ) ) ) ) ).
% sr_def
thf(fact_251_add__mono__thms__linordered__semiring_I3_J,axiom,
! [A: $tType] :
( ( ordere779506340up_add @ A )
=> ! [I3: A,J: A,K4: A,L: A] :
( ( ( ord_less_eq @ A @ I3 @ J )
& ( K4 = L ) )
=> ( ord_less_eq @ A @ ( plus_plus @ A @ I3 @ K4 ) @ ( plus_plus @ A @ J @ L ) ) ) ) ).
% add_mono_thms_linordered_semiring(3)
thf(fact_252_add__mono__thms__linordered__semiring_I2_J,axiom,
! [A: $tType] :
( ( ordere779506340up_add @ A )
=> ! [I3: A,J: A,K4: A,L: A] :
( ( ( I3 = J )
& ( ord_less_eq @ A @ K4 @ L ) )
=> ( ord_less_eq @ A @ ( plus_plus @ A @ I3 @ K4 ) @ ( plus_plus @ A @ J @ L ) ) ) ) ).
% add_mono_thms_linordered_semiring(2)
thf(fact_253_add__mono__thms__linordered__semiring_I1_J,axiom,
! [A: $tType] :
( ( ordere779506340up_add @ A )
=> ! [I3: A,J: A,K4: A,L: A] :
( ( ( ord_less_eq @ A @ I3 @ J )
& ( ord_less_eq @ A @ K4 @ L ) )
=> ( ord_less_eq @ A @ ( plus_plus @ A @ I3 @ K4 ) @ ( plus_plus @ A @ J @ L ) ) ) ) ).
% add_mono_thms_linordered_semiring(1)
thf(fact_254_add__mono,axiom,
! [A: $tType] :
( ( ordere779506340up_add @ A )
=> ! [A3: A,B3: A,C3: A,D2: A] :
( ( ord_less_eq @ A @ A3 @ B3 )
=> ( ( ord_less_eq @ A @ C3 @ D2 )
=> ( ord_less_eq @ A @ ( plus_plus @ A @ A3 @ C3 ) @ ( plus_plus @ A @ B3 @ D2 ) ) ) ) ) ).
% add_mono
thf(fact_255_add__left__mono,axiom,
! [A: $tType] :
( ( ordere779506340up_add @ A )
=> ! [A3: A,B3: A,C3: A] :
( ( ord_less_eq @ A @ A3 @ B3 )
=> ( ord_less_eq @ A @ ( plus_plus @ A @ C3 @ A3 ) @ ( plus_plus @ A @ C3 @ B3 ) ) ) ) ).
% add_left_mono
% Type constructors (12)
thf(tcon_fun___Orderings_Oord,axiom,
! [A8: $tType,A9: $tType] :
( ( ord @ A9 )
=> ( ord @ ( A8 > A9 ) ) ) ).
thf(tcon_Nat_Onat___Groups_Oordered__ab__semigroup__add__imp__le,axiom,
ordere236663937imp_le @ nat ).
thf(tcon_Nat_Onat___Groups_Oordered__ab__semigroup__add,axiom,
ordere779506340up_add @ nat ).
thf(tcon_Nat_Onat___Groups_Ocancel__semigroup__add,axiom,
cancel_semigroup_add @ nat ).
thf(tcon_Nat_Onat___Groups_Oab__semigroup__add,axiom,
ab_semigroup_add @ nat ).
thf(tcon_Nat_Onat___Groups_Ocomm__monoid__add,axiom,
comm_monoid_add @ nat ).
thf(tcon_Nat_Onat___Groups_Osemigroup__add,axiom,
semigroup_add @ nat ).
thf(tcon_Nat_Onat___Orderings_Oord_1,axiom,
ord @ nat ).
thf(tcon_Nat_Onat___Groups_Oplus,axiom,
plus @ nat ).
thf(tcon_Set_Oset___Orderings_Oord_2,axiom,
! [A8: $tType] : ( ord @ ( set @ A8 ) ) ).
thf(tcon_HOL_Obool___Orderings_Oord_3,axiom,
ord @ $o ).
thf(tcon_Product__Type_Ounit___Orderings_Oord_4,axiom,
ord @ product_unit ).
% Conjectures (3)
thf(conj_0,hypothesis,
! [X5: c] :
( ( member @ c @ X5 @ i )
=> ( module @ d @ a @ e @ b @ ( m @ X5 ) @ r ) ) ).
thf(conj_1,hypothesis,
( ( member @ ( ( c > d ) > ( c > d ) > c > d ) @ ( prod_pOp @ c @ d @ ( module_ext @ d @ a @ e ) @ i @ m )
@ ( pi @ ( c > d ) @ ( ( c > d ) > c > d ) @ ( carr_prodag @ c @ d @ ( module_ext @ d @ a @ e ) @ i @ m )
@ ^ [Uu: c > d] :
( pi @ ( c > d ) @ ( c > d ) @ ( carr_prodag @ c @ d @ ( module_ext @ d @ a @ e ) @ i @ m )
@ ^ [Uv: c > d] : ( carr_prodag @ c @ d @ ( module_ext @ d @ a @ e ) @ i @ m ) ) ) )
& ! [A10: c > d] :
( ( member @ ( c > d ) @ A10 @ ( carr_prodag @ c @ d @ ( module_ext @ d @ a @ e ) @ i @ m ) )
=> ! [B5: c > d] :
( ( member @ ( c > d ) @ B5 @ ( carr_prodag @ c @ d @ ( module_ext @ d @ a @ e ) @ i @ m ) )
=> ! [C4: c > d] :
( ( member @ ( c > d ) @ C4 @ ( carr_prodag @ c @ d @ ( module_ext @ d @ a @ e ) @ i @ m ) )
=> ( ( prod_pOp @ c @ d @ ( module_ext @ d @ a @ e ) @ i @ m @ ( prod_pOp @ c @ d @ ( module_ext @ d @ a @ e ) @ i @ m @ A10 @ B5 ) @ C4 )
= ( prod_pOp @ c @ d @ ( module_ext @ d @ a @ e ) @ i @ m @ A10 @ ( prod_pOp @ c @ d @ ( module_ext @ d @ a @ e ) @ i @ m @ B5 @ C4 ) ) ) ) ) )
& ! [A10: c > d] :
( ( member @ ( c > d ) @ A10 @ ( carr_prodag @ c @ d @ ( module_ext @ d @ a @ e ) @ i @ m ) )
=> ! [B5: c > d] :
( ( member @ ( c > d ) @ B5 @ ( carr_prodag @ c @ d @ ( module_ext @ d @ a @ e ) @ i @ m ) )
=> ( ( prod_pOp @ c @ d @ ( module_ext @ d @ a @ e ) @ i @ m @ A10 @ B5 )
= ( prod_pOp @ c @ d @ ( module_ext @ d @ a @ e ) @ i @ m @ B5 @ A10 ) ) ) )
& ( member @ ( ( c > d ) > c > d ) @ ( prod_mOp @ c @ d @ ( module_ext @ d @ a @ e ) @ i @ m )
@ ( pi @ ( c > d ) @ ( c > d ) @ ( carr_prodag @ c @ d @ ( module_ext @ d @ a @ e ) @ i @ m )
@ ^ [Uu: c > d] : ( carr_prodag @ c @ d @ ( module_ext @ d @ a @ e ) @ i @ m ) ) )
& ! [A10: c > d] :
( ( member @ ( c > d ) @ A10 @ ( carr_prodag @ c @ d @ ( module_ext @ d @ a @ e ) @ i @ m ) )
=> ( ( prod_pOp @ c @ d @ ( module_ext @ d @ a @ e ) @ i @ m @ ( prod_mOp @ c @ d @ ( module_ext @ d @ a @ e ) @ i @ m @ A10 ) @ A10 )
= ( prod_zero @ c @ d @ ( module_ext @ d @ a @ e ) @ i @ m ) ) )
& ( member @ ( c > d ) @ ( prod_zero @ c @ d @ ( module_ext @ d @ a @ e ) @ i @ m ) @ ( carr_prodag @ c @ d @ ( module_ext @ d @ a @ e ) @ i @ m ) )
& ! [A10: c > d] :
( ( member @ ( c > d ) @ A10 @ ( carr_prodag @ c @ d @ ( module_ext @ d @ a @ e ) @ i @ m ) )
=> ( ( prod_pOp @ c @ d @ ( module_ext @ d @ a @ e ) @ i @ m @ ( prod_zero @ c @ d @ ( module_ext @ d @ a @ e ) @ i @ m ) @ A10 )
= A10 ) ) ) ).
thf(conj_2,conjecture,
! [A6: c > d] :
( ~ ( member @ ( c > d ) @ A6 @ ( carr_prodag @ c @ d @ ( module_ext @ d @ a @ e ) @ i @ m ) )
| ( ( prod_pOp @ c @ d @ ( module_ext @ d @ a @ e ) @ i @ m @ A6 @ ( prod_zero @ c @ d @ ( module_ext @ d @ a @ e ) @ i @ m ) )
= A6 ) ) ).
%------------------------------------------------------------------------------