TPTP Problem File: ITP027^1.p
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%------------------------------------------------------------------------------
% File : ITP027^1 : 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 : 0.20 v8.2.0, 0.23 v8.1.0, 0.27 v7.5.0
% Syntax : Number of formulae : 587 ( 39 unt; 229 typ; 0 def)
% Number of atoms : 1402 ( 259 equ; 0 cnn)
% Maximal formula atoms : 14 ( 3 avg)
% Number of connectives : 6019 ( 29 ~; 1 |; 8 &;5008 @)
% ( 0 <=>; 973 =>; 0 <=; 0 <~>)
% Maximal formula depth : 17 ( 10 avg)
% Number of types : 55 ( 54 usr)
% Number of type conns : 1891 (1891 >; 0 *; 0 +; 0 <<)
% Number of symbols : 176 ( 175 usr; 2 con; 0-6 aty)
% Number of variables : 1543 ( 170 ^;1369 !; 4 ?;1543 :)
% SPC : TH0_THM_EQU_NAR
% Comments : This file was generated by Sledgehammer 2021-02-23 15:48:27.663
%------------------------------------------------------------------------------
% Could-be-implicit typings (54)
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% Explicit typings (175)
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thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_Itf__a_J,type,
ord_less_eq_set_a: set_a > set_a > $o ).
thf(sy_c_Set_OCollect_001_062_I_062_Itf__c_Mtf__d_J_M_062_Itf__c_Mtf__d_J_J,type,
collect_c_d_c_d: ( ( ( c > d ) > c > d ) > $o ) > set_c_d_c_d ).
thf(sy_c_Set_OCollect_001_062_Itf__a_M_062_Itf__a_M_062_Itf__a_Mtf__a_J_J_J,type,
collect_a_a_a_a: ( ( a > a > a > a ) > $o ) > set_a_a_a_a ).
thf(sy_c_Set_OCollect_001_062_Itf__a_M_062_Itf__a_M_062_Itf__c_Mtf__d_J_J_J,type,
collect_a_a_c_d: ( ( a > a > c > d ) > $o ) > set_a_a_c_d ).
thf(sy_c_Set_OCollect_001_062_Itf__a_M_062_Itf__a_Mtf__a_J_J,type,
collect_a_a_a: ( ( a > a > a ) > $o ) > set_a_a_a ).
thf(sy_c_Set_OCollect_001_062_Itf__a_Mtf__a_J,type,
collect_a_a: ( ( a > a ) > $o ) > set_a_a ).
thf(sy_c_Set_OCollect_001_062_Itf__c_Mtf__d_J,type,
collect_c_d: ( ( c > d ) > $o ) > set_c_d ).
thf(sy_c_Set_OCollect_001t__Nat__Onat,type,
collect_nat: ( nat > $o ) > set_nat ).
thf(sy_c_Set_OCollect_001tf__a,type,
collect_a: ( a > $o ) > set_a ).
thf(sy_c_Set_OCollect_001tf__c,type,
collect_c: ( c > $o ) > set_c ).
thf(sy_c_Set_OCollect_001tf__d,type,
collect_d: ( d > $o ) > set_d ).
thf(sy_c_member_001_062_I_062_I_062_Itf__c_Mtf__d_J_M_062_Itf__c_Mtf__d_J_J_Mtf__a_J,type,
member_c_d_c_d_a: ( ( ( c > d ) > c > d ) > a ) > set_c_d_c_d_a2 > $o ).
thf(sy_c_member_001_062_I_062_I_062_Itf__c_Mtf__d_J_M_062_Itf__c_Mtf__d_J_J_Mtf__d_J,type,
member_c_d_c_d_d: ( ( ( c > d ) > c > d ) > d ) > set_c_d_c_d_d > $o ).
thf(sy_c_member_001_062_I_062_Itf__a_M_062_Itf__a_M_062_Itf__a_M_062_Itf__a_Mtf__a_J_J_J_J_Mtf__d_J,type,
member_a_a_a_a_a_d: ( ( a > a > a > a > a ) > d ) > set_a_a_a_a_a_d > $o ).
thf(sy_c_member_001_062_I_062_Itf__a_M_062_Itf__a_M_062_Itf__a_Mtf__a_J_J_J_Mtf__a_J,type,
member_a_a_a_a_a: ( ( a > a > a > a ) > a ) > set_a_a_a_a_a3 > $o ).
thf(sy_c_member_001_062_I_062_Itf__a_M_062_Itf__a_M_062_Itf__a_Mtf__a_J_J_J_Mtf__d_J,type,
member_a_a_a_a_d: ( ( a > a > a > a ) > d ) > set_a_a_a_a_d > $o ).
thf(sy_c_member_001_062_I_062_Itf__a_M_062_Itf__a_M_062_Itf__c_Mtf__d_J_J_J_Mtf__a_J,type,
member_a_a_c_d_a: ( ( a > a > c > d ) > a ) > set_a_a_c_d_a > $o ).
thf(sy_c_member_001_062_I_062_Itf__a_M_062_Itf__a_M_062_Itf__c_Mtf__d_J_J_J_Mtf__d_J,type,
member_a_a_c_d_d: ( ( a > a > c > d ) > d ) > set_a_a_c_d_d > $o ).
thf(sy_c_member_001_062_I_062_Itf__a_M_062_Itf__a_Mtf__a_J_J_M_062_Itf__a_Mtf__a_J_J,type,
member_a_a_a_a_a2: ( ( a > a > a ) > a > a ) > set_a_a_a_a_a2 > $o ).
thf(sy_c_member_001_062_I_062_Itf__a_M_062_Itf__a_Mtf__a_J_J_M_062_Itf__c_Mtf__d_J_J,type,
member_a_a_a_c_d: ( ( a > a > a ) > c > d ) > set_a_a_a_c_d > $o ).
thf(sy_c_member_001_062_I_062_Itf__a_M_062_Itf__a_Mtf__a_J_J_Mtf__a_J,type,
member_a_a_a_a: ( ( a > a > a ) > a ) > set_a_a_a_a4 > $o ).
thf(sy_c_member_001_062_I_062_Itf__a_M_062_Itf__a_Mtf__a_J_J_Mtf__d_J,type,
member_a_a_a_d: ( ( a > a > a ) > d ) > set_a_a_a_d > $o ).
thf(sy_c_member_001_062_I_062_Itf__a_Mtf__a_J_M_062_Itf__a_Mtf__a_J_J,type,
member_a_a_a_a2: ( ( a > a ) > a > a ) > set_a_a_a_a3 > $o ).
thf(sy_c_member_001_062_I_062_Itf__a_Mtf__a_J_M_062_Itf__c_Mtf__d_J_J,type,
member_a_a_c_d: ( ( a > a ) > c > d ) > set_a_a_c_d2 > $o ).
thf(sy_c_member_001_062_I_062_Itf__a_Mtf__a_J_Mtf__a_J,type,
member_a_a_a: ( ( a > a ) > a ) > set_a_a_a2 > $o ).
thf(sy_c_member_001_062_I_062_Itf__a_Mtf__a_J_Mtf__d_J,type,
member_a_a_d: ( ( a > a ) > d ) > set_a_a_d > $o ).
thf(sy_c_member_001_062_I_062_Itf__c_Mtf__d_J_M_062_I_062_Itf__a_Mtf__a_J_Mtf__a_J_J,type,
member_c_d_a_a_a: ( ( c > d ) > ( a > a ) > a ) > set_c_d_a_a_a2 > $o ).
thf(sy_c_member_001_062_I_062_Itf__c_Mtf__d_J_M_062_I_062_Itf__c_Mtf__d_J_M_062_Itf__c_Mtf__d_J_J_J,type,
member_c_d_c_d_c_d: ( ( c > d ) > ( c > d ) > c > d ) > set_c_d_c_d_c_d > $o ).
thf(sy_c_member_001_062_I_062_Itf__c_Mtf__d_J_M_062_I_062_Itf__c_Mtf__d_J_Mtf__a_J_J,type,
member_c_d_c_d_a2: ( ( c > d ) > ( c > d ) > a ) > set_c_d_c_d_a > $o ).
thf(sy_c_member_001_062_I_062_Itf__c_Mtf__d_J_M_062_Itf__a_M_062_Itf__a_Mtf__a_J_J_J,type,
member_c_d_a_a_a2: ( ( c > d ) > a > a > a ) > set_c_d_a_a_a > $o ).
thf(sy_c_member_001_062_I_062_Itf__c_Mtf__d_J_M_062_Itf__a_Mtf__a_J_J,type,
member_c_d_a_a: ( ( c > d ) > a > a ) > set_c_d_a_a > $o ).
thf(sy_c_member_001_062_I_062_Itf__c_Mtf__d_J_M_062_Itf__c_Mtf__d_J_J,type,
member_c_d_c_d: ( ( c > d ) > c > d ) > set_c_d_c_d > $o ).
thf(sy_c_member_001_062_I_062_Itf__c_Mtf__d_J_Mtf__a_J,type,
member_c_d_a: ( ( c > d ) > a ) > set_c_d_a > $o ).
thf(sy_c_member_001_062_I_062_Itf__c_Mtf__d_J_Mtf__d_J,type,
member_c_d_d: ( ( c > d ) > d ) > set_c_d_d > $o ).
thf(sy_c_member_001_062_Itf__a_M_062_I_062_Itf__a_Mtf__a_J_Mtf__a_J_J,type,
member_a_a_a_a3: ( a > ( a > a ) > a ) > set_a_a_a_a2 > $o ).
thf(sy_c_member_001_062_Itf__a_M_062_I_062_Itf__c_Mtf__d_J_M_062_Itf__c_Mtf__d_J_J_J,type,
member_a_c_d_c_d: ( a > ( c > d ) > c > d ) > set_a_c_d_c_d > $o ).
thf(sy_c_member_001_062_Itf__a_M_062_I_062_Itf__c_Mtf__d_J_Mtf__a_J_J,type,
member_a_c_d_a: ( a > ( c > d ) > a ) > set_a_c_d_a > $o ).
thf(sy_c_member_001_062_Itf__a_M_062_Itf__a_M_062_I_062_Itf__c_Mtf__d_J_M_062_Itf__c_Mtf__d_J_J_J_J,type,
member_a_a_c_d_c_d: ( a > a > ( c > d ) > c > d ) > set_a_a_c_d_c_d > $o ).
thf(sy_c_member_001_062_Itf__a_M_062_Itf__a_M_062_Itf__a_M_062_Itf__a_Mtf__a_J_J_J_J,type,
member_a_a_a_a_a3: ( a > a > a > a > a ) > set_a_a_a_a_a > $o ).
thf(sy_c_member_001_062_Itf__a_M_062_Itf__a_M_062_Itf__a_Mtf__a_J_J_J,type,
member_a_a_a_a4: ( a > a > a > a ) > set_a_a_a_a > $o ).
thf(sy_c_member_001_062_Itf__a_M_062_Itf__a_M_062_Itf__c_Mtf__d_J_J_J,type,
member_a_a_c_d2: ( a > a > c > d ) > set_a_a_c_d > $o ).
thf(sy_c_member_001_062_Itf__a_M_062_Itf__a_Mtf__a_J_J,type,
member_a_a_a2: ( a > a > a ) > set_a_a_a > $o ).
thf(sy_c_member_001_062_Itf__a_M_062_Itf__c_Mtf__d_J_J,type,
member_a_c_d: ( a > c > d ) > set_a_c_d > $o ).
thf(sy_c_member_001_062_Itf__a_Mtf__a_J,type,
member_a_a: ( a > a ) > set_a_a > $o ).
thf(sy_c_member_001_062_Itf__a_Mtf__c_J,type,
member_a_c: ( a > c ) > set_a_c > $o ).
thf(sy_c_member_001_062_Itf__a_Mtf__d_J,type,
member_a_d: ( a > d ) > set_a_d > $o ).
thf(sy_c_member_001_062_Itf__c_M_062_Itf__c_Mtf__d_J_J,type,
member_c_c_d: ( c > c > d ) > set_c_c_d > $o ).
thf(sy_c_member_001_062_Itf__c_Mtf__a_J,type,
member_c_a: ( c > a ) > set_c_a > $o ).
thf(sy_c_member_001_062_Itf__c_Mtf__c_J,type,
member_c_c: ( c > c ) > set_c_c > $o ).
thf(sy_c_member_001_062_Itf__c_Mtf__d_J,type,
member_c_d: ( c > d ) > set_c_d > $o ).
thf(sy_c_member_001_062_Itf__d_M_062_Itf__c_Mtf__d_J_J,type,
member_d_c_d: ( d > c > d ) > set_d_c_d > $o ).
thf(sy_c_member_001_062_Itf__d_Mtf__a_J,type,
member_d_a: ( d > a ) > set_d_a > $o ).
thf(sy_c_member_001_062_Itf__d_Mtf__c_J,type,
member_d_c: ( d > c ) > set_d_c > $o ).
thf(sy_c_member_001_062_Itf__d_Mtf__d_J,type,
member_d_d: ( d > d ) > set_d_d > $o ).
thf(sy_c_member_001t__Nat__Onat,type,
member_nat: nat > set_nat > $o ).
thf(sy_c_member_001tf__a,type,
member_a: a > set_a > $o ).
thf(sy_c_member_001tf__c,type,
member_c: c > set_c > $o ).
thf(sy_c_member_001tf__d,type,
member_d: d > set_d > $o ).
thf(sy_v_I,type,
i: set_c ).
thf(sy_v_M,type,
m: c > carrie1948989069_d_a_e ).
thf(sy_v_R,type,
r: carrie1105631105xt_a_b ).
% Relevant facts (355)
thf(fact_0_Ring__axioms,axiom,
ring_a_b @ r ).
% Ring_axioms
thf(fact_1_bivar__fun,axiom,
! [F: a > c > d,A: set_a,B: set_c,C: set_d,A2: a] :
( ( member_a_c_d @ F
@ ( pi_a_c_d @ A
@ ^ [Uu: a] :
( pi_c_d @ B
@ ^ [Uv: c] : C ) ) )
=> ( ( member_a @ A2 @ A )
=> ( member_c_d @ ( F @ A2 )
@ ( pi_c_d @ B
@ ^ [Uu: c] : C ) ) ) ) ).
% bivar_fun
thf(fact_2_bivar__fun,axiom,
! [F: a > a > a,A: set_a,B: set_a,C: set_a,A2: a] :
( ( member_a_a_a2 @ F
@ ( pi_a_a_a2 @ A
@ ^ [Uu: a] :
( pi_a_a @ B
@ ^ [Uv: a] : C ) ) )
=> ( ( member_a @ A2 @ A )
=> ( member_a_a @ ( F @ A2 )
@ ( pi_a_a @ B
@ ^ [Uu: a] : C ) ) ) ) ).
% bivar_fun
thf(fact_3_bivar__fun,axiom,
! [F: ( c > d ) > a > a,A: set_c_d,B: set_a,C: set_a,A2: c > d] :
( ( member_c_d_a_a @ F
@ ( pi_c_d_a_a @ A
@ ^ [Uu: c > d] :
( pi_a_a @ B
@ ^ [Uv: a] : C ) ) )
=> ( ( member_c_d @ A2 @ A )
=> ( member_a_a @ ( F @ A2 )
@ ( pi_a_a @ B
@ ^ [Uu: a] : C ) ) ) ) ).
% bivar_fun
thf(fact_4_bivar__fun,axiom,
! [F: ( a > a ) > c > d,A: set_a_a,B: set_c,C: set_d,A2: a > a] :
( ( member_a_a_c_d @ F
@ ( pi_a_a_c_d @ A
@ ^ [Uu: a > a] :
( pi_c_d @ B
@ ^ [Uv: c] : C ) ) )
=> ( ( member_a_a @ A2 @ A )
=> ( member_c_d @ ( F @ A2 )
@ ( pi_c_d @ B
@ ^ [Uu: c] : C ) ) ) ) ).
% bivar_fun
thf(fact_5_bivar__fun,axiom,
! [F: ( a > a ) > a > a,A: set_a_a,B: set_a,C: set_a,A2: a > a] :
( ( member_a_a_a_a2 @ F
@ ( pi_a_a_a_a2 @ A
@ ^ [Uu: a > a] :
( pi_a_a @ B
@ ^ [Uv: a] : C ) ) )
=> ( ( member_a_a @ A2 @ A )
=> ( member_a_a @ ( F @ A2 )
@ ( pi_a_a @ B
@ ^ [Uu: a] : C ) ) ) ) ).
% bivar_fun
thf(fact_6_bivar__fun,axiom,
! [F: a > a > a > a,A: set_a,B: set_a,C: set_a_a,A2: a] :
( ( member_a_a_a_a4 @ F
@ ( pi_a_a_a_a4 @ A
@ ^ [Uu: a] :
( pi_a_a_a2 @ B
@ ^ [Uv: a] : C ) ) )
=> ( ( member_a @ A2 @ A )
=> ( member_a_a_a2 @ ( F @ A2 )
@ ( pi_a_a_a2 @ B
@ ^ [Uu: a] : C ) ) ) ) ).
% bivar_fun
thf(fact_7_bivar__fun,axiom,
! [F: ( c > d ) > c > d,A: set_c_d,B: set_c,C: set_d,A2: c > d] :
( ( member_c_d_c_d @ F
@ ( pi_c_d_c_d @ A
@ ^ [Uu: c > d] :
( pi_c_d @ B
@ ^ [Uv: c] : C ) ) )
=> ( ( member_c_d @ A2 @ A )
=> ( member_c_d @ ( F @ A2 )
@ ( pi_c_d @ B
@ ^ [Uu: c] : C ) ) ) ) ).
% bivar_fun
thf(fact_8_bivar__fun,axiom,
! [F: ( c > d ) > a > a > a,A: set_c_d,B: set_a,C: set_a_a,A2: c > d] :
( ( member_c_d_a_a_a2 @ F
@ ( pi_c_d_a_a_a2 @ A
@ ^ [Uu: c > d] :
( pi_a_a_a2 @ B
@ ^ [Uv: a] : C ) ) )
=> ( ( member_c_d @ A2 @ A )
=> ( member_a_a_a2 @ ( F @ A2 )
@ ( pi_a_a_a2 @ B
@ ^ [Uu: a] : C ) ) ) ) ).
% bivar_fun
thf(fact_9_bivar__fun,axiom,
! [F: ( a > a > a ) > c > d,A: set_a_a_a,B: set_c,C: set_d,A2: a > a > a] :
( ( member_a_a_a_c_d @ F
@ ( pi_a_a_a_c_d @ A
@ ^ [Uu: a > a > a] :
( pi_c_d @ B
@ ^ [Uv: c] : C ) ) )
=> ( ( member_a_a_a2 @ A2 @ A )
=> ( member_c_d @ ( F @ A2 )
@ ( pi_c_d @ B
@ ^ [Uu: c] : C ) ) ) ) ).
% bivar_fun
thf(fact_10_bivar__fun,axiom,
! [F: ( a > a > a ) > a > a,A: set_a_a_a,B: set_a,C: set_a,A2: a > a > a] :
( ( member_a_a_a_a_a2 @ F
@ ( pi_a_a_a_a_a @ A
@ ^ [Uu: a > a > a] :
( pi_a_a @ B
@ ^ [Uv: a] : C ) ) )
=> ( ( member_a_a_a2 @ A2 @ A )
=> ( member_a_a @ ( F @ A2 )
@ ( pi_a_a @ B
@ ^ [Uu: a] : C ) ) ) ) ).
% bivar_fun
thf(fact_11_bivar__fun__mem,axiom,
! [F: a > a > a,A: set_a,B: set_a,C: set_a,A2: a,B2: a] :
( ( member_a_a_a2 @ F
@ ( pi_a_a_a2 @ A
@ ^ [Uu: a] :
( pi_a_a @ B
@ ^ [Uv: a] : C ) ) )
=> ( ( member_a @ A2 @ A )
=> ( ( member_a @ B2 @ B )
=> ( member_a @ ( F @ A2 @ B2 ) @ C ) ) ) ) ).
% bivar_fun_mem
thf(fact_12_bivar__fun__mem,axiom,
! [F: ( c > d ) > a > a,A: set_c_d,B: set_a,C: set_a,A2: c > d,B2: a] :
( ( member_c_d_a_a @ F
@ ( pi_c_d_a_a @ A
@ ^ [Uu: c > d] :
( pi_a_a @ B
@ ^ [Uv: a] : C ) ) )
=> ( ( member_c_d @ A2 @ A )
=> ( ( member_a @ B2 @ B )
=> ( member_a @ ( F @ A2 @ B2 ) @ C ) ) ) ) ).
% bivar_fun_mem
thf(fact_13_bivar__fun__mem,axiom,
! [F: ( a > a ) > a > a,A: set_a_a,B: set_a,C: set_a,A2: a > a,B2: a] :
( ( member_a_a_a_a2 @ F
@ ( pi_a_a_a_a2 @ A
@ ^ [Uu: a > a] :
( pi_a_a @ B
@ ^ [Uv: a] : C ) ) )
=> ( ( member_a_a @ A2 @ A )
=> ( ( member_a @ B2 @ B )
=> ( member_a @ ( F @ A2 @ B2 ) @ C ) ) ) ) ).
% bivar_fun_mem
thf(fact_14_bivar__fun__mem,axiom,
! [F: a > ( c > d ) > a,A: set_a,B: set_c_d,C: set_a,A2: a,B2: c > d] :
( ( member_a_c_d_a @ F
@ ( pi_a_c_d_a @ A
@ ^ [Uu: a] :
( pi_c_d_a @ B
@ ^ [Uv: c > d] : C ) ) )
=> ( ( member_a @ A2 @ A )
=> ( ( member_c_d @ B2 @ B )
=> ( member_a @ ( F @ A2 @ B2 ) @ C ) ) ) ) ).
% bivar_fun_mem
thf(fact_15_bivar__fun__mem,axiom,
! [F: a > ( a > a ) > a,A: set_a,B: set_a_a,C: set_a,A2: a,B2: a > a] :
( ( member_a_a_a_a3 @ F
@ ( pi_a_a_a_a3 @ A
@ ^ [Uu: a] :
( pi_a_a_a @ B
@ ^ [Uv: a > a] : C ) ) )
=> ( ( member_a @ A2 @ A )
=> ( ( member_a_a @ B2 @ B )
=> ( member_a @ ( F @ A2 @ B2 ) @ C ) ) ) ) ).
% bivar_fun_mem
thf(fact_16_bivar__fun__mem,axiom,
! [F: a > a > c > d,A: set_a,B: set_a,C: set_c_d,A2: a,B2: a] :
( ( member_a_a_c_d2 @ F
@ ( pi_a_a_c_d2 @ A
@ ^ [Uu: a] :
( pi_a_c_d @ B
@ ^ [Uv: a] : C ) ) )
=> ( ( member_a @ A2 @ A )
=> ( ( member_a @ B2 @ B )
=> ( member_c_d @ ( F @ A2 @ B2 ) @ C ) ) ) ) ).
% bivar_fun_mem
thf(fact_17_bivar__fun__mem,axiom,
! [F: a > a > a > a,A: set_a,B: set_a,C: set_a_a,A2: a,B2: a] :
( ( member_a_a_a_a4 @ F
@ ( pi_a_a_a_a4 @ A
@ ^ [Uu: a] :
( pi_a_a_a2 @ B
@ ^ [Uv: a] : C ) ) )
=> ( ( member_a @ A2 @ A )
=> ( ( member_a @ B2 @ B )
=> ( member_a_a @ ( F @ A2 @ B2 ) @ C ) ) ) ) ).
% bivar_fun_mem
thf(fact_18_bivar__fun__mem,axiom,
! [F: ( c > d ) > c > d,A: set_c_d,B: set_c,C: set_d,A2: c > d,B2: c] :
( ( member_c_d_c_d @ F
@ ( pi_c_d_c_d @ A
@ ^ [Uu: c > d] :
( pi_c_d @ B
@ ^ [Uv: c] : C ) ) )
=> ( ( member_c_d @ A2 @ A )
=> ( ( member_c @ B2 @ B )
=> ( member_d @ ( F @ A2 @ B2 ) @ C ) ) ) ) ).
% bivar_fun_mem
thf(fact_19_bivar__fun__mem,axiom,
! [F: ( c > d ) > ( c > d ) > a,A: set_c_d,B: set_c_d,C: set_a,A2: c > d,B2: c > d] :
( ( member_c_d_c_d_a2 @ F
@ ( pi_c_d_c_d_a @ A
@ ^ [Uu: c > d] :
( pi_c_d_a @ B
@ ^ [Uv: c > d] : C ) ) )
=> ( ( member_c_d @ A2 @ A )
=> ( ( member_c_d @ B2 @ B )
=> ( member_a @ ( F @ A2 @ B2 ) @ C ) ) ) ) ).
% bivar_fun_mem
thf(fact_20_bivar__fun__mem,axiom,
! [F: ( c > d ) > ( a > a ) > a,A: set_c_d,B: set_a_a,C: set_a,A2: c > d,B2: a > a] :
( ( member_c_d_a_a_a @ F
@ ( pi_c_d_a_a_a @ A
@ ^ [Uu: c > d] :
( pi_a_a_a @ B
@ ^ [Uv: a > a] : C ) ) )
=> ( ( member_c_d @ A2 @ A )
=> ( ( member_a_a @ B2 @ B )
=> ( member_a @ ( F @ A2 @ B2 ) @ C ) ) ) ) ).
% bivar_fun_mem
thf(fact_21_Pi__I,axiom,
! [A: set_c,F: c > d,B: c > set_d] :
( ! [X: c] :
( ( member_c @ X @ A )
=> ( member_d @ ( F @ X ) @ ( B @ X ) ) )
=> ( member_c_d @ F @ ( pi_c_d @ A @ B ) ) ) ).
% Pi_I
thf(fact_22_Pi__I,axiom,
! [A: set_a,F: a > a,B: a > set_a] :
( ! [X: a] :
( ( member_a @ X @ A )
=> ( member_a @ ( F @ X ) @ ( B @ X ) ) )
=> ( member_a_a @ F @ ( pi_a_a @ A @ B ) ) ) ).
% Pi_I
thf(fact_23_Pi__I,axiom,
! [A: set_c_d,F: ( c > d ) > a,B: ( c > d ) > set_a] :
( ! [X: c > d] :
( ( member_c_d @ X @ A )
=> ( member_a @ ( F @ X ) @ ( B @ X ) ) )
=> ( member_c_d_a @ F @ ( pi_c_d_a @ A @ B ) ) ) ).
% Pi_I
thf(fact_24_Pi__I,axiom,
! [A: set_a_a,F: ( a > a ) > a,B: ( a > a ) > set_a] :
( ! [X: a > a] :
( ( member_a_a @ X @ A )
=> ( member_a @ ( F @ X ) @ ( B @ X ) ) )
=> ( member_a_a_a @ F @ ( pi_a_a_a @ A @ B ) ) ) ).
% Pi_I
thf(fact_25_Pi__I,axiom,
! [A: set_a,F: a > c > d,B: a > set_c_d] :
( ! [X: a] :
( ( member_a @ X @ A )
=> ( member_c_d @ ( F @ X ) @ ( B @ X ) ) )
=> ( member_a_c_d @ F @ ( pi_a_c_d @ A @ B ) ) ) ).
% Pi_I
thf(fact_26_Pi__I,axiom,
! [A: set_a,F: a > a > a,B: a > set_a_a] :
( ! [X: a] :
( ( member_a @ X @ A )
=> ( member_a_a @ ( F @ X ) @ ( B @ X ) ) )
=> ( member_a_a_a2 @ F @ ( pi_a_a_a2 @ A @ B ) ) ) ).
% Pi_I
thf(fact_27_Pi__I,axiom,
! [A: set_c_d,F: ( c > d ) > c > d,B: ( c > d ) > set_c_d] :
( ! [X: c > d] :
( ( member_c_d @ X @ A )
=> ( member_c_d @ ( F @ X ) @ ( B @ X ) ) )
=> ( member_c_d_c_d @ F @ ( pi_c_d_c_d @ A @ B ) ) ) ).
% Pi_I
thf(fact_28_Pi__I,axiom,
! [A: set_c_d,F: ( c > d ) > a > a,B: ( c > d ) > set_a_a] :
( ! [X: c > d] :
( ( member_c_d @ X @ A )
=> ( member_a_a @ ( F @ X ) @ ( B @ X ) ) )
=> ( member_c_d_a_a @ F @ ( pi_c_d_a_a @ A @ B ) ) ) ).
% Pi_I
thf(fact_29_Pi__I,axiom,
! [A: set_a_a_a,F: ( a > a > a ) > a,B: ( a > a > a ) > set_a] :
( ! [X: a > a > a] :
( ( member_a_a_a2 @ X @ A )
=> ( member_a @ ( F @ X ) @ ( B @ X ) ) )
=> ( member_a_a_a_a @ F @ ( pi_a_a_a_a @ A @ B ) ) ) ).
% Pi_I
thf(fact_30_Pi__I,axiom,
! [A: set_a_a,F: ( a > a ) > c > d,B: ( a > a ) > set_c_d] :
( ! [X: a > a] :
( ( member_a_a @ X @ A )
=> ( member_c_d @ ( F @ X ) @ ( B @ X ) ) )
=> ( member_a_a_c_d @ F @ ( pi_a_a_c_d @ A @ B ) ) ) ).
% Pi_I
thf(fact_31_prodM__sprod__mem,axiom,
! [I: set_c,M: c > carrie1948989069_d_a_e,A2: a,M2: c > d] :
( ! [X: c] :
( ( member_c @ X @ I )
=> ( module_d_a_e_b @ ( M @ X ) @ r ) )
=> ( ( member_a @ A2 @ ( carrie867757212xt_a_b @ r ) )
=> ( ( member_c_d @ M2 @ ( carr_p1636662547_d_a_e @ I @ M ) )
=> ( member_c_d @ ( algebr1697692348_c_d_e @ r @ I @ M @ A2 @ M2 ) @ ( carr_p1636662547_d_a_e @ I @ M ) ) ) ) ) ).
% prodM_sprod_mem
thf(fact_32_prodM__carr,axiom,
! [I: set_c,M: c > carrie1948989069_d_a_e] :
( ! [X: c] :
( ( member_c @ X @ I )
=> ( module_d_a_e_b @ ( M @ X ) @ r ) )
=> ( ( carrie20288686t_unit @ ( algebr1968153176_c_d_e @ r @ I @ M ) )
= ( carr_p1636662547_d_a_e @ I @ M ) ) ) ).
% prodM_carr
thf(fact_33_eq__fun,axiom,
! [F: a > c > d,A: set_a,B: set_c_d,G: a > c > d] :
( ( member_a_c_d @ F
@ ( pi_a_c_d @ A
@ ^ [Uu: a] : B ) )
=> ( ( F = G )
=> ( member_a_c_d @ G
@ ( pi_a_c_d @ A
@ ^ [Uu: a] : B ) ) ) ) ).
% eq_fun
thf(fact_34_eq__fun,axiom,
! [F: a > a > ( c > d ) > c > d,A: set_a,B: set_a_c_d_c_d,G: a > a > ( c > d ) > c > d] :
( ( member_a_a_c_d_c_d @ F
@ ( pi_a_a_c_d_c_d @ A
@ ^ [Uu: a] : B ) )
=> ( ( F = G )
=> ( member_a_a_c_d_c_d @ G
@ ( pi_a_a_c_d_c_d @ A
@ ^ [Uu: a] : B ) ) ) ) ).
% eq_fun
thf(fact_35_eq__fun,axiom,
! [F: a > a > c > d,A: set_a,B: set_a_c_d,G: a > a > c > d] :
( ( member_a_a_c_d2 @ F
@ ( pi_a_a_c_d2 @ A
@ ^ [Uu: a] : B ) )
=> ( ( F = G )
=> ( member_a_a_c_d2 @ G
@ ( pi_a_a_c_d2 @ A
@ ^ [Uu: a] : B ) ) ) ) ).
% eq_fun
thf(fact_36_eq__fun,axiom,
! [F: a > a > a > a > a,A: set_a,B: set_a_a_a_a,G: a > a > a > a > a] :
( ( member_a_a_a_a_a3 @ F
@ ( pi_a_a_a_a_a2 @ A
@ ^ [Uu: a] : B ) )
=> ( ( F = G )
=> ( member_a_a_a_a_a3 @ G
@ ( pi_a_a_a_a_a2 @ A
@ ^ [Uu: a] : B ) ) ) ) ).
% eq_fun
thf(fact_37_eq__fun,axiom,
! [F: a > a > a > a,A: set_a,B: set_a_a_a,G: a > a > a > a] :
( ( member_a_a_a_a4 @ F
@ ( pi_a_a_a_a4 @ A
@ ^ [Uu: a] : B ) )
=> ( ( F = G )
=> ( member_a_a_a_a4 @ G
@ ( pi_a_a_a_a4 @ A
@ ^ [Uu: a] : B ) ) ) ) ).
% eq_fun
thf(fact_38_eq__fun,axiom,
! [F: ( c > d ) > ( c > d ) > c > d,A: set_c_d,B: set_c_d_c_d,G: ( c > d ) > ( c > d ) > c > d] :
( ( member_c_d_c_d_c_d @ F
@ ( pi_c_d_c_d_c_d @ A
@ ^ [Uu: c > d] : B ) )
=> ( ( F = G )
=> ( member_c_d_c_d_c_d @ G
@ ( pi_c_d_c_d_c_d @ A
@ ^ [Uu: c > d] : B ) ) ) ) ).
% eq_fun
thf(fact_39_eq__fun,axiom,
! [F: ( c > d ) > c > d,A: set_c_d,B: set_c_d,G: ( c > d ) > c > d] :
( ( member_c_d_c_d @ F
@ ( pi_c_d_c_d @ A
@ ^ [Uu: c > d] : B ) )
=> ( ( F = G )
=> ( member_c_d_c_d @ G
@ ( pi_c_d_c_d @ A
@ ^ [Uu: c > d] : B ) ) ) ) ).
% eq_fun
thf(fact_40_eq__fun,axiom,
! [F: c > d,A: set_c,B: set_d,G: c > d] :
( ( member_c_d @ F
@ ( pi_c_d @ A
@ ^ [Uu: c] : B ) )
=> ( ( F = G )
=> ( member_c_d @ G
@ ( pi_c_d @ A
@ ^ [Uu: c] : B ) ) ) ) ).
% eq_fun
thf(fact_41_eq__fun,axiom,
! [F: a > a > a,A: set_a,B: set_a_a,G: a > a > a] :
( ( member_a_a_a2 @ F
@ ( pi_a_a_a2 @ A
@ ^ [Uu: a] : B ) )
=> ( ( F = G )
=> ( member_a_a_a2 @ G
@ ( pi_a_a_a2 @ A
@ ^ [Uu: a] : B ) ) ) ) ).
% eq_fun
thf(fact_42_eq__fun,axiom,
! [F: a > a,A: set_a,B: set_a,G: a > a] :
( ( member_a_a @ F
@ ( pi_a_a @ A
@ ^ [Uu: a] : B ) )
=> ( ( F = G )
=> ( member_a_a @ G
@ ( pi_a_a @ A
@ ^ [Uu: a] : B ) ) ) ) ).
% eq_fun
thf(fact_43_funcTr,axiom,
! [F: a > c > d,A: set_a,B: set_c_d,G: a > c > d,A2: a] :
( ( member_a_c_d @ F
@ ( pi_a_c_d @ A
@ ^ [Uu: a] : B ) )
=> ( ( member_a_c_d @ G
@ ( pi_a_c_d @ A
@ ^ [Uu: a] : B ) )
=> ( ( F = G )
=> ( ( member_a @ A2 @ A )
=> ( ( F @ A2 )
= ( G @ A2 ) ) ) ) ) ) ).
% funcTr
thf(fact_44_funcTr,axiom,
! [F: a > a > ( c > d ) > c > d,A: set_a,B: set_a_c_d_c_d,G: a > a > ( c > d ) > c > d,A2: a] :
( ( member_a_a_c_d_c_d @ F
@ ( pi_a_a_c_d_c_d @ A
@ ^ [Uu: a] : B ) )
=> ( ( member_a_a_c_d_c_d @ G
@ ( pi_a_a_c_d_c_d @ A
@ ^ [Uu: a] : B ) )
=> ( ( F = G )
=> ( ( member_a @ A2 @ A )
=> ( ( F @ A2 )
= ( G @ A2 ) ) ) ) ) ) ).
% funcTr
thf(fact_45_funcTr,axiom,
! [F: a > a > c > d,A: set_a,B: set_a_c_d,G: a > a > c > d,A2: a] :
( ( member_a_a_c_d2 @ F
@ ( pi_a_a_c_d2 @ A
@ ^ [Uu: a] : B ) )
=> ( ( member_a_a_c_d2 @ G
@ ( pi_a_a_c_d2 @ A
@ ^ [Uu: a] : B ) )
=> ( ( F = G )
=> ( ( member_a @ A2 @ A )
=> ( ( F @ A2 )
= ( G @ A2 ) ) ) ) ) ) ).
% funcTr
thf(fact_46_funcTr,axiom,
! [F: a > a > a > a > a,A: set_a,B: set_a_a_a_a,G: a > a > a > a > a,A2: a] :
( ( member_a_a_a_a_a3 @ F
@ ( pi_a_a_a_a_a2 @ A
@ ^ [Uu: a] : B ) )
=> ( ( member_a_a_a_a_a3 @ G
@ ( pi_a_a_a_a_a2 @ A
@ ^ [Uu: a] : B ) )
=> ( ( F = G )
=> ( ( member_a @ A2 @ A )
=> ( ( F @ A2 )
= ( G @ A2 ) ) ) ) ) ) ).
% funcTr
thf(fact_47_funcTr,axiom,
! [F: a > a > a > a,A: set_a,B: set_a_a_a,G: a > a > a > a,A2: a] :
( ( member_a_a_a_a4 @ F
@ ( pi_a_a_a_a4 @ A
@ ^ [Uu: a] : B ) )
=> ( ( member_a_a_a_a4 @ G
@ ( pi_a_a_a_a4 @ A
@ ^ [Uu: a] : B ) )
=> ( ( F = G )
=> ( ( member_a @ A2 @ A )
=> ( ( F @ A2 )
= ( G @ A2 ) ) ) ) ) ) ).
% funcTr
thf(fact_48_funcTr,axiom,
! [F: ( c > d ) > ( c > d ) > c > d,A: set_c_d,B: set_c_d_c_d,G: ( c > d ) > ( c > d ) > c > d,A2: c > d] :
( ( member_c_d_c_d_c_d @ F
@ ( pi_c_d_c_d_c_d @ A
@ ^ [Uu: c > d] : B ) )
=> ( ( member_c_d_c_d_c_d @ G
@ ( pi_c_d_c_d_c_d @ A
@ ^ [Uu: c > d] : B ) )
=> ( ( F = G )
=> ( ( member_c_d @ A2 @ A )
=> ( ( F @ A2 )
= ( G @ A2 ) ) ) ) ) ) ).
% funcTr
thf(fact_49_funcTr,axiom,
! [F: ( c > d ) > c > d,A: set_c_d,B: set_c_d,G: ( c > d ) > c > d,A2: c > d] :
( ( member_c_d_c_d @ F
@ ( pi_c_d_c_d @ A
@ ^ [Uu: c > d] : B ) )
=> ( ( member_c_d_c_d @ G
@ ( pi_c_d_c_d @ A
@ ^ [Uu: c > d] : B ) )
=> ( ( F = G )
=> ( ( member_c_d @ A2 @ A )
=> ( ( F @ A2 )
= ( G @ A2 ) ) ) ) ) ) ).
% funcTr
thf(fact_50_funcTr,axiom,
! [F: c > d,A: set_c,B: set_d,G: c > d,A2: c] :
( ( member_c_d @ F
@ ( pi_c_d @ A
@ ^ [Uu: c] : B ) )
=> ( ( member_c_d @ G
@ ( pi_c_d @ A
@ ^ [Uu: c] : B ) )
=> ( ( F = G )
=> ( ( member_c @ A2 @ A )
=> ( ( F @ A2 )
= ( G @ A2 ) ) ) ) ) ) ).
% funcTr
thf(fact_51_funcTr,axiom,
! [F: a > a > a,A: set_a,B: set_a_a,G: a > a > a,A2: a] :
( ( member_a_a_a2 @ F
@ ( pi_a_a_a2 @ A
@ ^ [Uu: a] : B ) )
=> ( ( member_a_a_a2 @ G
@ ( pi_a_a_a2 @ A
@ ^ [Uu: a] : B ) )
=> ( ( F = G )
=> ( ( member_a @ A2 @ A )
=> ( ( F @ A2 )
= ( G @ A2 ) ) ) ) ) ) ).
% funcTr
thf(fact_52_funcTr,axiom,
! [F: a > a,A: set_a,B: set_a,G: a > a,A2: a] :
( ( member_a_a @ F
@ ( pi_a_a @ A
@ ^ [Uu: a] : B ) )
=> ( ( member_a_a @ G
@ ( pi_a_a @ A
@ ^ [Uu: a] : B ) )
=> ( ( F = G )
=> ( ( member_a @ A2 @ A )
=> ( ( F @ A2 )
= ( G @ A2 ) ) ) ) ) ) ).
% funcTr
thf(fact_53_funcsetI,axiom,
! [A: set_c,F: c > d,B: set_d] :
( ! [X: c] :
( ( member_c @ X @ A )
=> ( member_d @ ( F @ X ) @ B ) )
=> ( member_c_d @ F
@ ( pi_c_d @ A
@ ^ [Uu: c] : B ) ) ) ).
% funcsetI
thf(fact_54_funcsetI,axiom,
! [A: set_a,F: a > a,B: set_a] :
( ! [X: a] :
( ( member_a @ X @ A )
=> ( member_a @ ( F @ X ) @ B ) )
=> ( member_a_a @ F
@ ( pi_a_a @ A
@ ^ [Uu: a] : B ) ) ) ).
% funcsetI
thf(fact_55_funcsetI,axiom,
! [A: set_c_d,F: ( c > d ) > a,B: set_a] :
( ! [X: c > d] :
( ( member_c_d @ X @ A )
=> ( member_a @ ( F @ X ) @ B ) )
=> ( member_c_d_a @ F
@ ( pi_c_d_a @ A
@ ^ [Uu: c > d] : B ) ) ) ).
% funcsetI
thf(fact_56_funcsetI,axiom,
! [A: set_a_a,F: ( a > a ) > a,B: set_a] :
( ! [X: a > a] :
( ( member_a_a @ X @ A )
=> ( member_a @ ( F @ X ) @ B ) )
=> ( member_a_a_a @ F
@ ( pi_a_a_a @ A
@ ^ [Uu: a > a] : B ) ) ) ).
% funcsetI
thf(fact_57_funcsetI,axiom,
! [A: set_a,F: a > c > d,B: set_c_d] :
( ! [X: a] :
( ( member_a @ X @ A )
=> ( member_c_d @ ( F @ X ) @ B ) )
=> ( member_a_c_d @ F
@ ( pi_a_c_d @ A
@ ^ [Uu: a] : B ) ) ) ).
% funcsetI
thf(fact_58_funcsetI,axiom,
! [A: set_a,F: a > a > a,B: set_a_a] :
( ! [X: a] :
( ( member_a @ X @ A )
=> ( member_a_a @ ( F @ X ) @ B ) )
=> ( member_a_a_a2 @ F
@ ( pi_a_a_a2 @ A
@ ^ [Uu: a] : B ) ) ) ).
% funcsetI
thf(fact_59_funcsetI,axiom,
! [A: set_c_d,F: ( c > d ) > c > d,B: set_c_d] :
( ! [X: c > d] :
( ( member_c_d @ X @ A )
=> ( member_c_d @ ( F @ X ) @ B ) )
=> ( member_c_d_c_d @ F
@ ( pi_c_d_c_d @ A
@ ^ [Uu: c > d] : B ) ) ) ).
% funcsetI
thf(fact_60_funcsetI,axiom,
! [A: set_c_d,F: ( c > d ) > a > a,B: set_a_a] :
( ! [X: c > d] :
( ( member_c_d @ X @ A )
=> ( member_a_a @ ( F @ X ) @ B ) )
=> ( member_c_d_a_a @ F
@ ( pi_c_d_a_a @ A
@ ^ [Uu: c > d] : B ) ) ) ).
% funcsetI
thf(fact_61_funcsetI,axiom,
! [A: set_a_a_a,F: ( a > a > a ) > a,B: set_a] :
( ! [X: a > a > a] :
( ( member_a_a_a2 @ X @ A )
=> ( member_a @ ( F @ X ) @ B ) )
=> ( member_a_a_a_a @ F
@ ( pi_a_a_a_a @ A
@ ^ [Uu: a > a > a] : B ) ) ) ).
% funcsetI
thf(fact_62_funcsetI,axiom,
! [A: set_a_a,F: ( a > a ) > c > d,B: set_c_d] :
( ! [X: a > a] :
( ( member_a_a @ X @ A )
=> ( member_c_d @ ( F @ X ) @ B ) )
=> ( member_a_a_c_d @ F
@ ( pi_a_a_c_d @ A
@ ^ [Uu: a > a] : B ) ) ) ).
% funcsetI
thf(fact_63_eq__funcs,axiom,
! [F: a > c > d,A: set_a,B: set_c_d,G: a > c > d,X2: a] :
( ( member_a_c_d @ F
@ ( pi_a_c_d @ A
@ ^ [Uu: a] : B ) )
=> ( ( member_a_c_d @ G
@ ( pi_a_c_d @ A
@ ^ [Uu: a] : B ) )
=> ( ( F = G )
=> ( ( member_a @ X2 @ A )
=> ( ( F @ X2 )
= ( G @ X2 ) ) ) ) ) ) ).
% eq_funcs
thf(fact_64_eq__funcs,axiom,
! [F: a > a > ( c > d ) > c > d,A: set_a,B: set_a_c_d_c_d,G: a > a > ( c > d ) > c > d,X2: a] :
( ( member_a_a_c_d_c_d @ F
@ ( pi_a_a_c_d_c_d @ A
@ ^ [Uu: a] : B ) )
=> ( ( member_a_a_c_d_c_d @ G
@ ( pi_a_a_c_d_c_d @ A
@ ^ [Uu: a] : B ) )
=> ( ( F = G )
=> ( ( member_a @ X2 @ A )
=> ( ( F @ X2 )
= ( G @ X2 ) ) ) ) ) ) ).
% eq_funcs
thf(fact_65_eq__funcs,axiom,
! [F: a > a > c > d,A: set_a,B: set_a_c_d,G: a > a > c > d,X2: a] :
( ( member_a_a_c_d2 @ F
@ ( pi_a_a_c_d2 @ A
@ ^ [Uu: a] : B ) )
=> ( ( member_a_a_c_d2 @ G
@ ( pi_a_a_c_d2 @ A
@ ^ [Uu: a] : B ) )
=> ( ( F = G )
=> ( ( member_a @ X2 @ A )
=> ( ( F @ X2 )
= ( G @ X2 ) ) ) ) ) ) ).
% eq_funcs
thf(fact_66_eq__funcs,axiom,
! [F: a > a > a > a > a,A: set_a,B: set_a_a_a_a,G: a > a > a > a > a,X2: a] :
( ( member_a_a_a_a_a3 @ F
@ ( pi_a_a_a_a_a2 @ A
@ ^ [Uu: a] : B ) )
=> ( ( member_a_a_a_a_a3 @ G
@ ( pi_a_a_a_a_a2 @ A
@ ^ [Uu: a] : B ) )
=> ( ( F = G )
=> ( ( member_a @ X2 @ A )
=> ( ( F @ X2 )
= ( G @ X2 ) ) ) ) ) ) ).
% eq_funcs
thf(fact_67_eq__funcs,axiom,
! [F: a > a > a > a,A: set_a,B: set_a_a_a,G: a > a > a > a,X2: a] :
( ( member_a_a_a_a4 @ F
@ ( pi_a_a_a_a4 @ A
@ ^ [Uu: a] : B ) )
=> ( ( member_a_a_a_a4 @ G
@ ( pi_a_a_a_a4 @ A
@ ^ [Uu: a] : B ) )
=> ( ( F = G )
=> ( ( member_a @ X2 @ A )
=> ( ( F @ X2 )
= ( G @ X2 ) ) ) ) ) ) ).
% eq_funcs
thf(fact_68_eq__funcs,axiom,
! [F: ( c > d ) > ( c > d ) > c > d,A: set_c_d,B: set_c_d_c_d,G: ( c > d ) > ( c > d ) > c > d,X2: c > d] :
( ( member_c_d_c_d_c_d @ F
@ ( pi_c_d_c_d_c_d @ A
@ ^ [Uu: c > d] : B ) )
=> ( ( member_c_d_c_d_c_d @ G
@ ( pi_c_d_c_d_c_d @ A
@ ^ [Uu: c > d] : B ) )
=> ( ( F = G )
=> ( ( member_c_d @ X2 @ A )
=> ( ( F @ X2 )
= ( G @ X2 ) ) ) ) ) ) ).
% eq_funcs
thf(fact_69_eq__funcs,axiom,
! [F: ( c > d ) > c > d,A: set_c_d,B: set_c_d,G: ( c > d ) > c > d,X2: c > d] :
( ( member_c_d_c_d @ F
@ ( pi_c_d_c_d @ A
@ ^ [Uu: c > d] : B ) )
=> ( ( member_c_d_c_d @ G
@ ( pi_c_d_c_d @ A
@ ^ [Uu: c > d] : B ) )
=> ( ( F = G )
=> ( ( member_c_d @ X2 @ A )
=> ( ( F @ X2 )
= ( G @ X2 ) ) ) ) ) ) ).
% eq_funcs
thf(fact_70_eq__funcs,axiom,
! [F: c > d,A: set_c,B: set_d,G: c > d,X2: c] :
( ( member_c_d @ F
@ ( pi_c_d @ A
@ ^ [Uu: c] : B ) )
=> ( ( member_c_d @ G
@ ( pi_c_d @ A
@ ^ [Uu: c] : B ) )
=> ( ( F = G )
=> ( ( member_c @ X2 @ A )
=> ( ( F @ X2 )
= ( G @ X2 ) ) ) ) ) ) ).
% eq_funcs
thf(fact_71_eq__funcs,axiom,
! [F: a > a > a,A: set_a,B: set_a_a,G: a > a > a,X2: a] :
( ( member_a_a_a2 @ F
@ ( pi_a_a_a2 @ A
@ ^ [Uu: a] : B ) )
=> ( ( member_a_a_a2 @ G
@ ( pi_a_a_a2 @ A
@ ^ [Uu: a] : B ) )
=> ( ( F = G )
=> ( ( member_a @ X2 @ A )
=> ( ( F @ X2 )
= ( G @ X2 ) ) ) ) ) ) ).
% eq_funcs
thf(fact_72_eq__funcs,axiom,
! [F: a > a,A: set_a,B: set_a,G: a > a,X2: a] :
( ( member_a_a @ F
@ ( pi_a_a @ A
@ ^ [Uu: a] : B ) )
=> ( ( member_a_a @ G
@ ( pi_a_a @ A
@ ^ [Uu: a] : B ) )
=> ( ( F = G )
=> ( ( member_a @ X2 @ A )
=> ( ( F @ X2 )
= ( G @ X2 ) ) ) ) ) ) ).
% eq_funcs
thf(fact_73_funcset__id,axiom,
! [A: set_c_d] :
( member_c_d_c_d
@ ^ [X3: c > d] : X3
@ ( pi_c_d_c_d @ A
@ ^ [Uu: c > d] : A ) ) ).
% funcset_id
thf(fact_74_funcset__id,axiom,
! [A: set_a] :
( member_a_a
@ ^ [X3: a] : X3
@ ( pi_a_a @ A
@ ^ [Uu: a] : A ) ) ).
% funcset_id
thf(fact_75_Algebra1_Ofuncset__mem,axiom,
! [F: a > d,A: set_a,B: set_d,X2: a] :
( ( member_a_d @ F
@ ( pi_a_d @ A
@ ^ [Uu: a] : B ) )
=> ( ( member_a @ X2 @ A )
=> ( member_d @ ( F @ X2 ) @ B ) ) ) ).
% Algebra1.funcset_mem
thf(fact_76_Algebra1_Ofuncset__mem,axiom,
! [F: a > c,A: set_a,B: set_c,X2: a] :
( ( member_a_c @ F
@ ( pi_a_c @ A
@ ^ [Uu: a] : B ) )
=> ( ( member_a @ X2 @ A )
=> ( member_c @ ( F @ X2 ) @ B ) ) ) ).
% Algebra1.funcset_mem
thf(fact_77_Algebra1_Ofuncset__mem,axiom,
! [F: d > a,A: set_d,B: set_a,X2: d] :
( ( member_d_a @ F
@ ( pi_d_a @ A
@ ^ [Uu: d] : B ) )
=> ( ( member_d @ X2 @ A )
=> ( member_a @ ( F @ X2 ) @ B ) ) ) ).
% Algebra1.funcset_mem
thf(fact_78_Algebra1_Ofuncset__mem,axiom,
! [F: d > d,A: set_d,B: set_d,X2: d] :
( ( member_d_d @ F
@ ( pi_d_d @ A
@ ^ [Uu: d] : B ) )
=> ( ( member_d @ X2 @ A )
=> ( member_d @ ( F @ X2 ) @ B ) ) ) ).
% Algebra1.funcset_mem
thf(fact_79_Algebra1_Ofuncset__mem,axiom,
! [F: d > c,A: set_d,B: set_c,X2: d] :
( ( member_d_c @ F
@ ( pi_d_c @ A
@ ^ [Uu: d] : B ) )
=> ( ( member_d @ X2 @ A )
=> ( member_c @ ( F @ X2 ) @ B ) ) ) ).
% Algebra1.funcset_mem
thf(fact_80_Algebra1_Ofuncset__mem,axiom,
! [F: c > a,A: set_c,B: set_a,X2: c] :
( ( member_c_a @ F
@ ( pi_c_a @ A
@ ^ [Uu: c] : B ) )
=> ( ( member_c @ X2 @ A )
=> ( member_a @ ( F @ X2 ) @ B ) ) ) ).
% Algebra1.funcset_mem
thf(fact_81_Algebra1_Ofuncset__mem,axiom,
! [F: c > c,A: set_c,B: set_c,X2: c] :
( ( member_c_c @ F
@ ( pi_c_c @ A
@ ^ [Uu: c] : B ) )
=> ( ( member_c @ X2 @ A )
=> ( member_c @ ( F @ X2 ) @ B ) ) ) ).
% Algebra1.funcset_mem
thf(fact_82_Algebra1_Ofuncset__mem,axiom,
! [F: c > d,A: set_c,B: set_d,X2: c] :
( ( member_c_d @ F
@ ( pi_c_d @ A
@ ^ [Uu: c] : B ) )
=> ( ( member_c @ X2 @ A )
=> ( member_d @ ( F @ X2 ) @ B ) ) ) ).
% Algebra1.funcset_mem
thf(fact_83_Algebra1_Ofuncset__mem,axiom,
! [F: a > a,A: set_a,B: set_a,X2: a] :
( ( member_a_a @ F
@ ( pi_a_a @ A
@ ^ [Uu: a] : B ) )
=> ( ( member_a @ X2 @ A )
=> ( member_a @ ( F @ X2 ) @ B ) ) ) ).
% Algebra1.funcset_mem
thf(fact_84_Algebra1_Ofuncset__mem,axiom,
! [F: ( c > d ) > a,A: set_c_d,B: set_a,X2: c > d] :
( ( member_c_d_a @ F
@ ( pi_c_d_a @ A
@ ^ [Uu: c > d] : B ) )
=> ( ( member_c_d @ X2 @ A )
=> ( member_a @ ( F @ X2 ) @ B ) ) ) ).
% Algebra1.funcset_mem
thf(fact_85_prod__pOp__func,axiom,
! [I: set_c,A: c > carrie1948989069_d_a_e] :
( ! [X: c] :
( ( member_c @ X @ I )
=> ( aGroup1940521469_d_a_e @ ( A @ X ) ) )
=> ( member_c_d_c_d_c_d @ ( prod_p131511076_d_a_e @ I @ A )
@ ( pi_c_d_c_d_c_d @ ( carr_p1636662547_d_a_e @ I @ A )
@ ^ [Uu: c > d] :
( pi_c_d_c_d @ ( carr_p1636662547_d_a_e @ I @ A )
@ ^ [Uv: c > d] : ( carr_p1636662547_d_a_e @ I @ A ) ) ) ) ) ).
% prod_pOp_func
thf(fact_86_ring__is__ag,axiom,
aGroup2097840802xt_a_b @ r ).
% ring_is_ag
thf(fact_87_prodM__mem__eq,axiom,
! [I: set_c,M: c > carrie1948989069_d_a_e,M1: c > d,M22: c > d] :
( ! [X: c] :
( ( member_c @ X @ I )
=> ( module_d_a_e_b @ ( M @ X ) @ r ) )
=> ( ( member_c_d @ M1 @ ( carrie20288686t_unit @ ( algebr1968153176_c_d_e @ r @ I @ M ) ) )
=> ( ( member_c_d @ M22 @ ( carrie20288686t_unit @ ( algebr1968153176_c_d_e @ r @ I @ M ) ) )
=> ( ! [X: c] :
( ( member_c @ X @ I )
=> ( ( M1 @ X )
= ( M22 @ X ) ) )
=> ( M1 = M22 ) ) ) ) ) ).
% prodM_mem_eq
thf(fact_88_J__rad__mem,axiom,
! [X2: a] :
( ( member_a @ X2 @ ( j_rad_a_b @ r ) )
=> ( member_a @ X2 @ ( carrie867757212xt_a_b @ r ) ) ) ).
% J_rad_mem
thf(fact_89_Ring_ORing,axiom,
! [R: carrie1105631105xt_a_b] :
( ( ring_a_b @ R )
=> ( ring_a_b @ R ) ) ).
% Ring.Ring
thf(fact_90_Ring_Oring__is__ag,axiom,
! [R: carrie1105631105xt_a_b] :
( ( ring_a_b @ R )
=> ( aGroup2097840802xt_a_b @ R ) ) ).
% Ring.ring_is_ag
thf(fact_91_Ring_OprodM__mem__eq,axiom,
! [R: carrie1105631105xt_a_b,I: set_c,M: c > carrie1948989069_d_a_e,M1: c > d,M22: c > d] :
( ( ring_a_b @ R )
=> ( ! [X: c] :
( ( member_c @ X @ I )
=> ( module_d_a_e_b @ ( M @ X ) @ R ) )
=> ( ( member_c_d @ M1 @ ( carrie20288686t_unit @ ( algebr1968153176_c_d_e @ R @ I @ M ) ) )
=> ( ( member_c_d @ M22 @ ( carrie20288686t_unit @ ( algebr1968153176_c_d_e @ R @ I @ M ) ) )
=> ( ! [X: c] :
( ( member_c @ X @ I )
=> ( ( M1 @ X )
= ( M22 @ X ) ) )
=> ( M1 = M22 ) ) ) ) ) ) ).
% Ring.prodM_mem_eq
thf(fact_92_carr__prodag__mem__eq,axiom,
! [I: set_a,A: a > carrie1105631105xt_a_b,X4: a > a,Y: a > a] :
( ! [X: a] :
( ( member_a @ X @ I )
=> ( aGroup2097840802xt_a_b @ ( A @ X ) ) )
=> ( ( member_a_a @ X4 @ ( carr_p290614414xt_a_b @ I @ A ) )
=> ( ( member_a_a @ Y @ ( carr_p290614414xt_a_b @ I @ A ) )
=> ( ! [X: a] :
( ( member_a @ X @ I )
=> ( ( X4 @ X )
= ( Y @ X ) ) )
=> ( X4 = Y ) ) ) ) ) ).
% carr_prodag_mem_eq
thf(fact_93_carr__prodag__mem__eq,axiom,
! [I: set_c,A: c > carrie1948989069_d_a_e,X4: c > d,Y: c > d] :
( ! [X: c] :
( ( member_c @ X @ I )
=> ( aGroup1940521469_d_a_e @ ( A @ X ) ) )
=> ( ( member_c_d @ X4 @ ( carr_p1636662547_d_a_e @ I @ A ) )
=> ( ( member_c_d @ Y @ ( carr_p1636662547_d_a_e @ I @ A ) )
=> ( ! [X: c] :
( ( member_c @ X @ I )
=> ( ( X4 @ X )
= ( Y @ X ) ) )
=> ( X4 = Y ) ) ) ) ) ).
% carr_prodag_mem_eq
thf(fact_94_prodag__sameTr1,axiom,
! [I: set_c,A: c > carrie1948989069_d_a_e,B: c > carrie1948989069_d_a_e] :
( ! [X: c] :
( ( member_c @ X @ I )
=> ( aGroup1940521469_d_a_e @ ( A @ X ) ) )
=> ( ! [X: c] :
( ( member_c @ X @ I )
=> ( ( A @ X )
= ( B @ X ) ) )
=> ( ( carr_p1636662547_d_a_e @ I @ A )
= ( carr_p1636662547_d_a_e @ I @ B ) ) ) ) ).
% prodag_sameTr1
thf(fact_95_prodag__comp__i,axiom,
! [A2: d > a,I: set_d,A: d > carrie1105631105xt_a_b,I2: d] :
( ( member_d_a @ A2 @ ( carr_p297433675xt_a_b @ I @ A ) )
=> ( ( member_d @ I2 @ I )
=> ( member_a @ ( A2 @ I2 ) @ ( carrie867757212xt_a_b @ ( A @ I2 ) ) ) ) ) ).
% prodag_comp_i
thf(fact_96_prodag__comp__i,axiom,
! [A2: c > a,I: set_c,A: c > carrie1105631105xt_a_b,I2: c] :
( ( member_c_a @ A2 @ ( carr_p295160588xt_a_b @ I @ A ) )
=> ( ( member_c @ I2 @ I )
=> ( member_a @ ( A2 @ I2 ) @ ( carrie867757212xt_a_b @ ( A @ I2 ) ) ) ) ) ).
% prodag_comp_i
thf(fact_97_prodag__comp__i,axiom,
! [A2: a > a,I: set_a,A: a > carrie1105631105xt_a_b,I2: a] :
( ( member_a_a @ A2 @ ( carr_p290614414xt_a_b @ I @ A ) )
=> ( ( member_a @ I2 @ I )
=> ( member_a @ ( A2 @ I2 ) @ ( carrie867757212xt_a_b @ ( A @ I2 ) ) ) ) ) ).
% prodag_comp_i
thf(fact_98_prodag__comp__i,axiom,
! [A2: c > d,I: set_c,A: c > carrie1948989069_d_a_e,I2: c] :
( ( member_c_d @ A2 @ ( carr_p1636662547_d_a_e @ I @ A ) )
=> ( ( member_c @ I2 @ I )
=> ( member_d @ ( A2 @ I2 ) @ ( carrie11369756_d_a_e @ ( A @ I2 ) ) ) ) ) ).
% prodag_comp_i
thf(fact_99_prodag__comp__i,axiom,
! [A2: ( c > d ) > a,I: set_c_d,A: ( c > d ) > carrie1105631105xt_a_b,I2: c > d] :
( ( member_c_d_a @ A2 @ ( carr_p996491646xt_a_b @ I @ A ) )
=> ( ( member_c_d @ I2 @ I )
=> ( member_a @ ( A2 @ I2 ) @ ( carrie867757212xt_a_b @ ( A @ I2 ) ) ) ) ) ).
% prodag_comp_i
thf(fact_100_prodag__comp__i,axiom,
! [A2: ( a > a ) > a,I: set_a_a,A: ( a > a ) > carrie1105631105xt_a_b,I2: a > a] :
( ( member_a_a_a @ A2 @ ( carr_p740328767xt_a_b @ I @ A ) )
=> ( ( member_a_a @ I2 @ I )
=> ( member_a @ ( A2 @ I2 ) @ ( carrie867757212xt_a_b @ ( A @ I2 ) ) ) ) ) ).
% prodag_comp_i
thf(fact_101_prodag__comp__i,axiom,
! [A2: ( a > a > a ) > a,I: set_a_a_a,A: ( a > a > a ) > carrie1105631105xt_a_b,I2: a > a > a] :
( ( member_a_a_a_a @ A2 @ ( carr_p1481099064xt_a_b @ I @ A ) )
=> ( ( member_a_a_a2 @ I2 @ I )
=> ( member_a @ ( A2 @ I2 ) @ ( carrie867757212xt_a_b @ ( A @ I2 ) ) ) ) ) ).
% prodag_comp_i
thf(fact_102_prodag__comp__i,axiom,
! [A2: a > c > d,I: set_a,A: a > carrie1954260683t_unit,I2: a] :
( ( member_a_c_d @ A2 @ ( carr_p923087418t_unit @ I @ A ) )
=> ( ( member_a @ I2 @ I )
=> ( member_c_d @ ( A2 @ I2 ) @ ( carrie20288686t_unit @ ( A @ I2 ) ) ) ) ) ).
% prodag_comp_i
thf(fact_103_prodag__comp__i,axiom,
! [A2: d > c > d,I: set_d,A: d > carrie1954260683t_unit,I2: d] :
( ( member_d_c_d @ A2 @ ( carr_p1159644861t_unit @ I @ A ) )
=> ( ( member_d @ I2 @ I )
=> ( member_c_d @ ( A2 @ I2 ) @ ( carrie20288686t_unit @ ( A @ I2 ) ) ) ) ) ).
% prodag_comp_i
thf(fact_104_prodag__comp__i,axiom,
! [A2: c > c > d,I: set_c,A: c > carrie1954260683t_unit,I2: c] :
( ( member_c_c_d @ A2 @ ( carr_p1080792380t_unit @ I @ A ) )
=> ( ( member_c @ I2 @ I )
=> ( member_c_d @ ( A2 @ I2 ) @ ( carrie20288686t_unit @ ( A @ I2 ) ) ) ) ) ).
% prodag_comp_i
thf(fact_105_prodag__sameTr2,axiom,
! [I: set_c,A: c > carrie1948989069_d_a_e,B: c > carrie1948989069_d_a_e] :
( ! [X: c] :
( ( member_c @ X @ I )
=> ( aGroup1940521469_d_a_e @ ( A @ X ) ) )
=> ( ! [X: c] :
( ( member_c @ X @ I )
=> ( ( A @ X )
= ( B @ X ) ) )
=> ( ( prod_p131511076_d_a_e @ I @ A )
= ( prod_p131511076_d_a_e @ I @ B ) ) ) ) ).
% prodag_sameTr2
thf(fact_106_prodag__sameTr3,axiom,
! [I: set_c,A: c > carrie1948989069_d_a_e,B: c > carrie1948989069_d_a_e] :
( ! [X: c] :
( ( member_c @ X @ I )
=> ( aGroup1940521469_d_a_e @ ( A @ X ) ) )
=> ( ! [X: c] :
( ( member_c @ X @ I )
=> ( ( A @ X )
= ( B @ X ) ) )
=> ( ( prod_m854557927_d_a_e @ I @ A )
= ( prod_m854557927_d_a_e @ I @ B ) ) ) ) ).
% prodag_sameTr3
thf(fact_107_prodag__sameTr4,axiom,
! [I: set_c,A: c > carrie1948989069_d_a_e,B: c > carrie1948989069_d_a_e] :
( ! [X: c] :
( ( member_c @ X @ I )
=> ( aGroup1940521469_d_a_e @ ( A @ X ) ) )
=> ( ! [X: c] :
( ( member_c @ X @ I )
=> ( ( A @ X )
= ( B @ X ) ) )
=> ( ( prod_z1214987823_d_a_e @ I @ A )
= ( prod_z1214987823_d_a_e @ I @ B ) ) ) ) ).
% prodag_sameTr4
thf(fact_108_Ring_OprodM__carr,axiom,
! [R: carrie1105631105xt_a_b,I: set_c,M: c > carrie1948989069_d_a_e] :
( ( ring_a_b @ R )
=> ( ! [X: c] :
( ( member_c @ X @ I )
=> ( module_d_a_e_b @ ( M @ X ) @ R ) )
=> ( ( carrie20288686t_unit @ ( algebr1968153176_c_d_e @ R @ I @ M ) )
= ( carr_p1636662547_d_a_e @ I @ M ) ) ) ) ).
% Ring.prodM_carr
thf(fact_109_Ring_OprodM__sprod__mem,axiom,
! [R: carrie1105631105xt_a_b,I: set_c,M: c > carrie1948989069_d_a_e,A2: a,M2: c > d] :
( ( ring_a_b @ R )
=> ( ! [X: c] :
( ( member_c @ X @ I )
=> ( module_d_a_e_b @ ( M @ X ) @ R ) )
=> ( ( member_a @ A2 @ ( carrie867757212xt_a_b @ R ) )
=> ( ( member_c_d @ M2 @ ( carr_p1636662547_d_a_e @ I @ M ) )
=> ( member_c_d @ ( algebr1697692348_c_d_e @ R @ I @ M @ A2 @ M2 ) @ ( carr_p1636662547_d_a_e @ I @ M ) ) ) ) ) ) ).
% Ring.prodM_sprod_mem
thf(fact_110_prod__pOp__commute,axiom,
! [I: set_a,A: a > carrie1105631105xt_a_b,A2: a > a,B2: a > a] :
( ! [X: a] :
( ( member_a @ X @ I )
=> ( aGroup2097840802xt_a_b @ ( A @ X ) ) )
=> ( ( member_a_a @ A2 @ ( carr_p290614414xt_a_b @ I @ A ) )
=> ( ( member_a_a @ B2 @ ( carr_p290614414xt_a_b @ I @ A ) )
=> ( ( prod_p704341949xt_a_b @ I @ A @ A2 @ B2 )
= ( prod_p704341949xt_a_b @ I @ A @ B2 @ A2 ) ) ) ) ) ).
% prod_pOp_commute
thf(fact_111_prod__pOp__commute,axiom,
! [I: set_c,A: c > carrie1948989069_d_a_e,A2: c > d,B2: c > d] :
( ! [X: c] :
( ( member_c @ X @ I )
=> ( aGroup1940521469_d_a_e @ ( A @ X ) ) )
=> ( ( member_c_d @ A2 @ ( carr_p1636662547_d_a_e @ I @ A ) )
=> ( ( member_c_d @ B2 @ ( carr_p1636662547_d_a_e @ I @ A ) )
=> ( ( prod_p131511076_d_a_e @ I @ A @ A2 @ B2 )
= ( prod_p131511076_d_a_e @ I @ A @ B2 @ A2 ) ) ) ) ) ).
% prod_pOp_commute
thf(fact_112_prod__pOp__assoc,axiom,
! [I: set_a,A: a > carrie1105631105xt_a_b,A2: a > a,B2: a > a,C2: a > a] :
( ! [X: a] :
( ( member_a @ X @ I )
=> ( aGroup2097840802xt_a_b @ ( A @ X ) ) )
=> ( ( member_a_a @ A2 @ ( carr_p290614414xt_a_b @ I @ A ) )
=> ( ( member_a_a @ B2 @ ( carr_p290614414xt_a_b @ I @ A ) )
=> ( ( member_a_a @ C2 @ ( carr_p290614414xt_a_b @ I @ A ) )
=> ( ( prod_p704341949xt_a_b @ I @ A @ ( prod_p704341949xt_a_b @ I @ A @ A2 @ B2 ) @ C2 )
= ( prod_p704341949xt_a_b @ I @ A @ A2 @ ( prod_p704341949xt_a_b @ I @ A @ B2 @ C2 ) ) ) ) ) ) ) ).
% prod_pOp_assoc
thf(fact_113_prod__pOp__assoc,axiom,
! [I: set_c,A: c > carrie1948989069_d_a_e,A2: c > d,B2: c > d,C2: c > d] :
( ! [X: c] :
( ( member_c @ X @ I )
=> ( aGroup1940521469_d_a_e @ ( A @ X ) ) )
=> ( ( member_c_d @ A2 @ ( carr_p1636662547_d_a_e @ I @ A ) )
=> ( ( member_c_d @ B2 @ ( carr_p1636662547_d_a_e @ I @ A ) )
=> ( ( member_c_d @ C2 @ ( carr_p1636662547_d_a_e @ I @ A ) )
=> ( ( prod_p131511076_d_a_e @ I @ A @ ( prod_p131511076_d_a_e @ I @ A @ A2 @ B2 ) @ C2 )
= ( prod_p131511076_d_a_e @ I @ A @ A2 @ ( prod_p131511076_d_a_e @ I @ A @ B2 @ C2 ) ) ) ) ) ) ) ).
% prod_pOp_assoc
thf(fact_114_prod__pOp__mem,axiom,
! [I: set_a,A: a > carrie1105631105xt_a_b,X4: a > a,Y: a > a] :
( ! [X: a] :
( ( member_a @ X @ I )
=> ( aGroup2097840802xt_a_b @ ( A @ X ) ) )
=> ( ( member_a_a @ X4 @ ( carr_p290614414xt_a_b @ I @ A ) )
=> ( ( member_a_a @ Y @ ( carr_p290614414xt_a_b @ I @ A ) )
=> ( member_a_a @ ( prod_p704341949xt_a_b @ I @ A @ X4 @ Y ) @ ( carr_p290614414xt_a_b @ I @ A ) ) ) ) ) ).
% prod_pOp_mem
thf(fact_115_prod__pOp__mem,axiom,
! [I: set_c,A: c > carrie1948989069_d_a_e,X4: c > d,Y: c > d] :
( ! [X: c] :
( ( member_c @ X @ I )
=> ( aGroup1940521469_d_a_e @ ( A @ X ) ) )
=> ( ( member_c_d @ X4 @ ( carr_p1636662547_d_a_e @ I @ A ) )
=> ( ( member_c_d @ Y @ ( carr_p1636662547_d_a_e @ I @ A ) )
=> ( member_c_d @ ( prod_p131511076_d_a_e @ I @ A @ X4 @ Y ) @ ( carr_p1636662547_d_a_e @ I @ A ) ) ) ) ) ).
% prod_pOp_mem
thf(fact_116_prod__mOp__mem,axiom,
! [I: set_a,A: a > carrie1105631105xt_a_b,X4: a > a] :
( ! [X: a] :
( ( member_a @ X @ I )
=> ( aGroup2097840802xt_a_b @ ( A @ X ) ) )
=> ( ( member_a_a @ X4 @ ( carr_p290614414xt_a_b @ I @ A ) )
=> ( member_a_a @ ( prod_m2022989242xt_a_b @ I @ A @ X4 ) @ ( carr_p290614414xt_a_b @ I @ A ) ) ) ) ).
% prod_mOp_mem
thf(fact_117_prod__mOp__mem,axiom,
! [I: set_c,A: c > carrie1948989069_d_a_e,X4: c > d] :
( ! [X: c] :
( ( member_c @ X @ I )
=> ( aGroup1940521469_d_a_e @ ( A @ X ) ) )
=> ( ( member_c_d @ X4 @ ( carr_p1636662547_d_a_e @ I @ A ) )
=> ( member_c_d @ ( prod_m854557927_d_a_e @ I @ A @ X4 ) @ ( carr_p1636662547_d_a_e @ I @ A ) ) ) ) ).
% prod_mOp_mem
thf(fact_118_prod__zero__func,axiom,
! [I: set_a,A: a > carrie1105631105xt_a_b] :
( ! [X: a] :
( ( member_a @ X @ I )
=> ( aGroup2097840802xt_a_b @ ( A @ X ) ) )
=> ( member_a_a @ ( prod_z851058290xt_a_b @ I @ A ) @ ( carr_p290614414xt_a_b @ I @ A ) ) ) ).
% prod_zero_func
thf(fact_119_prod__zero__func,axiom,
! [I: set_c,A: c > carrie1948989069_d_a_e] :
( ! [X: c] :
( ( member_c @ X @ I )
=> ( aGroup1940521469_d_a_e @ ( A @ X ) ) )
=> ( member_c_d @ ( prod_z1214987823_d_a_e @ I @ A ) @ ( carr_p1636662547_d_a_e @ I @ A ) ) ) ).
% prod_zero_func
thf(fact_120_prod__mOp__func,axiom,
! [I: set_c,A: c > carrie1948989069_d_a_e] :
( ! [X: c] :
( ( member_c @ X @ I )
=> ( aGroup1940521469_d_a_e @ ( A @ X ) ) )
=> ( member_c_d_c_d @ ( prod_m854557927_d_a_e @ I @ A )
@ ( pi_c_d_c_d @ ( carr_p1636662547_d_a_e @ I @ A )
@ ^ [Uu: c > d] : ( carr_p1636662547_d_a_e @ I @ A ) ) ) ) ).
% prod_mOp_func
thf(fact_121_Pi__cong,axiom,
! [A: set_c_d,F: ( c > d ) > ( c > d ) > c > d,G: ( c > d ) > ( c > d ) > c > d,B: ( c > d ) > set_c_d_c_d] :
( ! [W: c > d] :
( ( member_c_d @ W @ A )
=> ( ( F @ W )
= ( G @ W ) ) )
=> ( ( member_c_d_c_d_c_d @ F @ ( pi_c_d_c_d_c_d @ A @ B ) )
= ( member_c_d_c_d_c_d @ G @ ( pi_c_d_c_d_c_d @ A @ B ) ) ) ) ).
% Pi_cong
thf(fact_122_Pi__cong,axiom,
! [A: set_c_d,F: ( c > d ) > c > d,G: ( c > d ) > c > d,B: ( c > d ) > set_c_d] :
( ! [W: c > d] :
( ( member_c_d @ W @ A )
=> ( ( F @ W )
= ( G @ W ) ) )
=> ( ( member_c_d_c_d @ F @ ( pi_c_d_c_d @ A @ B ) )
= ( member_c_d_c_d @ G @ ( pi_c_d_c_d @ A @ B ) ) ) ) ).
% Pi_cong
thf(fact_123_Pi__cong,axiom,
! [A: set_a,F: a > c > d,G: a > c > d,B: a > set_c_d] :
( ! [W: a] :
( ( member_a @ W @ A )
=> ( ( F @ W )
= ( G @ W ) ) )
=> ( ( member_a_c_d @ F @ ( pi_a_c_d @ A @ B ) )
= ( member_a_c_d @ G @ ( pi_a_c_d @ A @ B ) ) ) ) ).
% Pi_cong
thf(fact_124_Pi__cong,axiom,
! [A: set_a,F: a > a > ( c > d ) > c > d,G: a > a > ( c > d ) > c > d,B: a > set_a_c_d_c_d] :
( ! [W: a] :
( ( member_a @ W @ A )
=> ( ( F @ W )
= ( G @ W ) ) )
=> ( ( member_a_a_c_d_c_d @ F @ ( pi_a_a_c_d_c_d @ A @ B ) )
= ( member_a_a_c_d_c_d @ G @ ( pi_a_a_c_d_c_d @ A @ B ) ) ) ) ).
% Pi_cong
thf(fact_125_Pi__cong,axiom,
! [A: set_a,F: a > a > a,G: a > a > a,B: a > set_a_a] :
( ! [W: a] :
( ( member_a @ W @ A )
=> ( ( F @ W )
= ( G @ W ) ) )
=> ( ( member_a_a_a2 @ F @ ( pi_a_a_a2 @ A @ B ) )
= ( member_a_a_a2 @ G @ ( pi_a_a_a2 @ A @ B ) ) ) ) ).
% Pi_cong
thf(fact_126_Pi__cong,axiom,
! [A: set_a,F: a > a,G: a > a,B: a > set_a] :
( ! [W: a] :
( ( member_a @ W @ A )
=> ( ( F @ W )
= ( G @ W ) ) )
=> ( ( member_a_a @ F @ ( pi_a_a @ A @ B ) )
= ( member_a_a @ G @ ( pi_a_a @ A @ B ) ) ) ) ).
% Pi_cong
thf(fact_127_Pi__cong,axiom,
! [A: set_a,F: a > a > c > d,G: a > a > c > d,B: a > set_a_c_d] :
( ! [W: a] :
( ( member_a @ W @ A )
=> ( ( F @ W )
= ( G @ W ) ) )
=> ( ( member_a_a_c_d2 @ F @ ( pi_a_a_c_d2 @ A @ B ) )
= ( member_a_a_c_d2 @ G @ ( pi_a_a_c_d2 @ A @ B ) ) ) ) ).
% Pi_cong
thf(fact_128_Pi__cong,axiom,
! [A: set_a,F: a > a > a > a > a,G: a > a > a > a > a,B: a > set_a_a_a_a] :
( ! [W: a] :
( ( member_a @ W @ A )
=> ( ( F @ W )
= ( G @ W ) ) )
=> ( ( member_a_a_a_a_a3 @ F @ ( pi_a_a_a_a_a2 @ A @ B ) )
= ( member_a_a_a_a_a3 @ G @ ( pi_a_a_a_a_a2 @ A @ B ) ) ) ) ).
% Pi_cong
thf(fact_129_Pi__cong,axiom,
! [A: set_a,F: a > a > a > a,G: a > a > a > a,B: a > set_a_a_a] :
( ! [W: a] :
( ( member_a @ W @ A )
=> ( ( F @ W )
= ( G @ W ) ) )
=> ( ( member_a_a_a_a4 @ F @ ( pi_a_a_a_a4 @ A @ B ) )
= ( member_a_a_a_a4 @ G @ ( pi_a_a_a_a4 @ A @ B ) ) ) ) ).
% Pi_cong
thf(fact_130_Pi__cong,axiom,
! [A: set_c,F: c > d,G: c > d,B: c > set_d] :
( ! [W: c] :
( ( member_c @ W @ A )
=> ( ( F @ W )
= ( G @ W ) ) )
=> ( ( member_c_d @ F @ ( pi_c_d @ A @ B ) )
= ( member_c_d @ G @ ( pi_c_d @ A @ B ) ) ) ) ).
% Pi_cong
thf(fact_131_Pi__mem,axiom,
! [F: a > d,A: set_a,B: a > set_d,X2: a] :
( ( member_a_d @ F @ ( pi_a_d @ A @ B ) )
=> ( ( member_a @ X2 @ A )
=> ( member_d @ ( F @ X2 ) @ ( B @ X2 ) ) ) ) ).
% Pi_mem
thf(fact_132_Pi__mem,axiom,
! [F: a > c,A: set_a,B: a > set_c,X2: a] :
( ( member_a_c @ F @ ( pi_a_c @ A @ B ) )
=> ( ( member_a @ X2 @ A )
=> ( member_c @ ( F @ X2 ) @ ( B @ X2 ) ) ) ) ).
% Pi_mem
thf(fact_133_Pi__mem,axiom,
! [F: d > a,A: set_d,B: d > set_a,X2: d] :
( ( member_d_a @ F @ ( pi_d_a @ A @ B ) )
=> ( ( member_d @ X2 @ A )
=> ( member_a @ ( F @ X2 ) @ ( B @ X2 ) ) ) ) ).
% Pi_mem
thf(fact_134_Pi__mem,axiom,
! [F: d > d,A: set_d,B: d > set_d,X2: d] :
( ( member_d_d @ F @ ( pi_d_d @ A @ B ) )
=> ( ( member_d @ X2 @ A )
=> ( member_d @ ( F @ X2 ) @ ( B @ X2 ) ) ) ) ).
% Pi_mem
thf(fact_135_Pi__mem,axiom,
! [F: d > c,A: set_d,B: d > set_c,X2: d] :
( ( member_d_c @ F @ ( pi_d_c @ A @ B ) )
=> ( ( member_d @ X2 @ A )
=> ( member_c @ ( F @ X2 ) @ ( B @ X2 ) ) ) ) ).
% Pi_mem
thf(fact_136_Pi__mem,axiom,
! [F: c > a,A: set_c,B: c > set_a,X2: c] :
( ( member_c_a @ F @ ( pi_c_a @ A @ B ) )
=> ( ( member_c @ X2 @ A )
=> ( member_a @ ( F @ X2 ) @ ( B @ X2 ) ) ) ) ).
% Pi_mem
thf(fact_137_Pi__mem,axiom,
! [F: c > c,A: set_c,B: c > set_c,X2: c] :
( ( member_c_c @ F @ ( pi_c_c @ A @ B ) )
=> ( ( member_c @ X2 @ A )
=> ( member_c @ ( F @ X2 ) @ ( B @ X2 ) ) ) ) ).
% Pi_mem
thf(fact_138_Pi__mem,axiom,
! [F: c > d,A: set_c,B: c > set_d,X2: c] :
( ( member_c_d @ F @ ( pi_c_d @ A @ B ) )
=> ( ( member_c @ X2 @ A )
=> ( member_d @ ( F @ X2 ) @ ( B @ X2 ) ) ) ) ).
% Pi_mem
thf(fact_139_Pi__mem,axiom,
! [F: a > a,A: set_a,B: a > set_a,X2: a] :
( ( member_a_a @ F @ ( pi_a_a @ A @ B ) )
=> ( ( member_a @ X2 @ A )
=> ( member_a @ ( F @ X2 ) @ ( B @ X2 ) ) ) ) ).
% Pi_mem
thf(fact_140_Pi__mem,axiom,
! [F: ( c > d ) > a,A: set_c_d,B: ( c > d ) > set_a,X2: c > d] :
( ( member_c_d_a @ F @ ( pi_c_d_a @ A @ B ) )
=> ( ( member_c_d @ X2 @ A )
=> ( member_a @ ( F @ X2 ) @ ( B @ X2 ) ) ) ) ).
% Pi_mem
thf(fact_141_Pi__iff,axiom,
! [F: a > a > ( c > d ) > c > d,I: set_a,X4: a > set_a_c_d_c_d] :
( ( member_a_a_c_d_c_d @ F @ ( pi_a_a_c_d_c_d @ I @ X4 ) )
= ( ! [X3: a] :
( ( member_a @ X3 @ I )
=> ( member_a_c_d_c_d @ ( F @ X3 ) @ ( X4 @ X3 ) ) ) ) ) ).
% Pi_iff
thf(fact_142_Pi__iff,axiom,
! [F: ( c > d ) > ( c > d ) > c > d,I: set_c_d,X4: ( c > d ) > set_c_d_c_d] :
( ( member_c_d_c_d_c_d @ F @ ( pi_c_d_c_d_c_d @ I @ X4 ) )
= ( ! [X3: c > d] :
( ( member_c_d @ X3 @ I )
=> ( member_c_d_c_d @ ( F @ X3 ) @ ( X4 @ X3 ) ) ) ) ) ).
% Pi_iff
thf(fact_143_Pi__iff,axiom,
! [F: a > c > d,I: set_a,X4: a > set_c_d] :
( ( member_a_c_d @ F @ ( pi_a_c_d @ I @ X4 ) )
= ( ! [X3: a] :
( ( member_a @ X3 @ I )
=> ( member_c_d @ ( F @ X3 ) @ ( X4 @ X3 ) ) ) ) ) ).
% Pi_iff
thf(fact_144_Pi__iff,axiom,
! [F: ( c > d ) > c > d,I: set_c_d,X4: ( c > d ) > set_c_d] :
( ( member_c_d_c_d @ F @ ( pi_c_d_c_d @ I @ X4 ) )
= ( ! [X3: c > d] :
( ( member_c_d @ X3 @ I )
=> ( member_c_d @ ( F @ X3 ) @ ( X4 @ X3 ) ) ) ) ) ).
% Pi_iff
thf(fact_145_Pi__iff,axiom,
! [F: c > d,I: set_c,X4: c > set_d] :
( ( member_c_d @ F @ ( pi_c_d @ I @ X4 ) )
= ( ! [X3: c] :
( ( member_c @ X3 @ I )
=> ( member_d @ ( F @ X3 ) @ ( X4 @ X3 ) ) ) ) ) ).
% Pi_iff
thf(fact_146_Pi__iff,axiom,
! [F: a > a > a,I: set_a,X4: a > set_a_a] :
( ( member_a_a_a2 @ F @ ( pi_a_a_a2 @ I @ X4 ) )
= ( ! [X3: a] :
( ( member_a @ X3 @ I )
=> ( member_a_a @ ( F @ X3 ) @ ( X4 @ X3 ) ) ) ) ) ).
% Pi_iff
thf(fact_147_Pi__iff,axiom,
! [F: a > a,I: set_a,X4: a > set_a] :
( ( member_a_a @ F @ ( pi_a_a @ I @ X4 ) )
= ( ! [X3: a] :
( ( member_a @ X3 @ I )
=> ( member_a @ ( F @ X3 ) @ ( X4 @ X3 ) ) ) ) ) ).
% Pi_iff
thf(fact_148_Pi__iff,axiom,
! [F: a > a > c > d,I: set_a,X4: a > set_a_c_d] :
( ( member_a_a_c_d2 @ F @ ( pi_a_a_c_d2 @ I @ X4 ) )
= ( ! [X3: a] :
( ( member_a @ X3 @ I )
=> ( member_a_c_d @ ( F @ X3 ) @ ( X4 @ X3 ) ) ) ) ) ).
% Pi_iff
thf(fact_149_Pi__iff,axiom,
! [F: a > a > a > a > a,I: set_a,X4: a > set_a_a_a_a] :
( ( member_a_a_a_a_a3 @ F @ ( pi_a_a_a_a_a2 @ I @ X4 ) )
= ( ! [X3: a] :
( ( member_a @ X3 @ I )
=> ( member_a_a_a_a4 @ ( F @ X3 ) @ ( X4 @ X3 ) ) ) ) ) ).
% Pi_iff
thf(fact_150_Pi__iff,axiom,
! [F: a > a > a > a,I: set_a,X4: a > set_a_a_a] :
( ( member_a_a_a_a4 @ F @ ( pi_a_a_a_a4 @ I @ X4 ) )
= ( ! [X3: a] :
( ( member_a @ X3 @ I )
=> ( member_a_a_a2 @ ( F @ X3 ) @ ( X4 @ X3 ) ) ) ) ) ).
% Pi_iff
thf(fact_151_Pi__I_H,axiom,
! [A: set_a,F: a > a,B: a > set_a] :
( ! [X: a] :
( ( member_a @ X @ A )
=> ( member_a @ ( F @ X ) @ ( B @ X ) ) )
=> ( member_a_a @ F @ ( pi_a_a @ A @ B ) ) ) ).
% Pi_I'
thf(fact_152_Pi__I_H,axiom,
! [A: set_a,F: a > d,B: a > set_d] :
( ! [X: a] :
( ( member_a @ X @ A )
=> ( member_d @ ( F @ X ) @ ( B @ X ) ) )
=> ( member_a_d @ F @ ( pi_a_d @ A @ B ) ) ) ).
% Pi_I'
thf(fact_153_Pi__I_H,axiom,
! [A: set_a,F: a > c,B: a > set_c] :
( ! [X: a] :
( ( member_a @ X @ A )
=> ( member_c @ ( F @ X ) @ ( B @ X ) ) )
=> ( member_a_c @ F @ ( pi_a_c @ A @ B ) ) ) ).
% Pi_I'
thf(fact_154_Pi__I_H,axiom,
! [A: set_d,F: d > a,B: d > set_a] :
( ! [X: d] :
( ( member_d @ X @ A )
=> ( member_a @ ( F @ X ) @ ( B @ X ) ) )
=> ( member_d_a @ F @ ( pi_d_a @ A @ B ) ) ) ).
% Pi_I'
thf(fact_155_Pi__I_H,axiom,
! [A: set_d,F: d > d,B: d > set_d] :
( ! [X: d] :
( ( member_d @ X @ A )
=> ( member_d @ ( F @ X ) @ ( B @ X ) ) )
=> ( member_d_d @ F @ ( pi_d_d @ A @ B ) ) ) ).
% Pi_I'
thf(fact_156_Pi__I_H,axiom,
! [A: set_d,F: d > c,B: d > set_c] :
( ! [X: d] :
( ( member_d @ X @ A )
=> ( member_c @ ( F @ X ) @ ( B @ X ) ) )
=> ( member_d_c @ F @ ( pi_d_c @ A @ B ) ) ) ).
% Pi_I'
thf(fact_157_Pi__I_H,axiom,
! [A: set_c,F: c > a,B: c > set_a] :
( ! [X: c] :
( ( member_c @ X @ A )
=> ( member_a @ ( F @ X ) @ ( B @ X ) ) )
=> ( member_c_a @ F @ ( pi_c_a @ A @ B ) ) ) ).
% Pi_I'
thf(fact_158_Pi__I_H,axiom,
! [A: set_c,F: c > d,B: c > set_d] :
( ! [X: c] :
( ( member_c @ X @ A )
=> ( member_d @ ( F @ X ) @ ( B @ X ) ) )
=> ( member_c_d @ F @ ( pi_c_d @ A @ B ) ) ) ).
% Pi_I'
thf(fact_159_Pi__I_H,axiom,
! [A: set_c,F: c > c,B: c > set_c] :
( ! [X: c] :
( ( member_c @ X @ A )
=> ( member_c @ ( F @ X ) @ ( B @ X ) ) )
=> ( member_c_c @ F @ ( pi_c_c @ A @ B ) ) ) ).
% Pi_I'
thf(fact_160_Pi__I_H,axiom,
! [A: set_c_d,F: ( c > d ) > a,B: ( c > d ) > set_a] :
( ! [X: c > d] :
( ( member_c_d @ X @ A )
=> ( member_a @ ( F @ X ) @ ( B @ X ) ) )
=> ( member_c_d_a @ F @ ( pi_c_d_a @ A @ B ) ) ) ).
% Pi_I'
thf(fact_161_PiE,axiom,
! [F: d > a,A: set_d,B: d > set_a,X2: d] :
( ( member_d_a @ F @ ( pi_d_a @ A @ B ) )
=> ( ~ ( member_a @ ( F @ X2 ) @ ( B @ X2 ) )
=> ~ ( member_d @ X2 @ A ) ) ) ).
% PiE
thf(fact_162_PiE,axiom,
! [F: c > a,A: set_c,B: c > set_a,X2: c] :
( ( member_c_a @ F @ ( pi_c_a @ A @ B ) )
=> ( ~ ( member_a @ ( F @ X2 ) @ ( B @ X2 ) )
=> ~ ( member_c @ X2 @ A ) ) ) ).
% PiE
thf(fact_163_PiE,axiom,
! [F: a > d,A: set_a,B: a > set_d,X2: a] :
( ( member_a_d @ F @ ( pi_a_d @ A @ B ) )
=> ( ~ ( member_d @ ( F @ X2 ) @ ( B @ X2 ) )
=> ~ ( member_a @ X2 @ A ) ) ) ).
% PiE
thf(fact_164_PiE,axiom,
! [F: d > d,A: set_d,B: d > set_d,X2: d] :
( ( member_d_d @ F @ ( pi_d_d @ A @ B ) )
=> ( ~ ( member_d @ ( F @ X2 ) @ ( B @ X2 ) )
=> ~ ( member_d @ X2 @ A ) ) ) ).
% PiE
thf(fact_165_PiE,axiom,
! [F: a > c,A: set_a,B: a > set_c,X2: a] :
( ( member_a_c @ F @ ( pi_a_c @ A @ B ) )
=> ( ~ ( member_c @ ( F @ X2 ) @ ( B @ X2 ) )
=> ~ ( member_a @ X2 @ A ) ) ) ).
% PiE
thf(fact_166_PiE,axiom,
! [F: d > c,A: set_d,B: d > set_c,X2: d] :
( ( member_d_c @ F @ ( pi_d_c @ A @ B ) )
=> ( ~ ( member_c @ ( F @ X2 ) @ ( B @ X2 ) )
=> ~ ( member_d @ X2 @ A ) ) ) ).
% PiE
thf(fact_167_PiE,axiom,
! [F: c > c,A: set_c,B: c > set_c,X2: c] :
( ( member_c_c @ F @ ( pi_c_c @ A @ B ) )
=> ( ~ ( member_c @ ( F @ X2 ) @ ( B @ X2 ) )
=> ~ ( member_c @ X2 @ A ) ) ) ).
% PiE
thf(fact_168_PiE,axiom,
! [F: c > d,A: set_c,B: c > set_d,X2: c] :
( ( member_c_d @ F @ ( pi_c_d @ A @ B ) )
=> ( ~ ( member_d @ ( F @ X2 ) @ ( B @ X2 ) )
=> ~ ( member_c @ X2 @ A ) ) ) ).
% PiE
thf(fact_169_PiE,axiom,
! [F: a > a,A: set_a,B: a > set_a,X2: a] :
( ( member_a_a @ F @ ( pi_a_a @ A @ B ) )
=> ( ~ ( member_a @ ( F @ X2 ) @ ( B @ X2 ) )
=> ~ ( member_a @ X2 @ A ) ) ) ).
% PiE
thf(fact_170_PiE,axiom,
! [F: a > c > d,A: set_a,B: a > set_c_d,X2: a] :
( ( member_a_c_d @ F @ ( pi_a_c_d @ A @ B ) )
=> ( ~ ( member_c_d @ ( F @ X2 ) @ ( B @ X2 ) )
=> ~ ( member_a @ X2 @ A ) ) ) ).
% PiE
thf(fact_171_mem__subring__mem__ring,axiom,
! [S: carrie1105631105xt_a_b,X2: a] :
( ( subring_a_b_b @ r @ S )
=> ( ( member_a @ X2 @ ( carrie867757212xt_a_b @ S ) )
=> ( member_a @ X2 @ ( carrie867757212xt_a_b @ r ) ) ) ) ).
% mem_subring_mem_ring
thf(fact_172_subring__Ring,axiom,
! [S: carrie1105631105xt_a_b] :
( ( subring_a_b_b @ r @ S )
=> ( ring_a_b @ S ) ) ).
% subring_Ring
thf(fact_173_prodM__sprod__val,axiom,
! [I: set_c_d_c_d,M: ( ( c > d ) > c > d ) > carrie1948989069_d_a_e,A2: a,M2: ( ( c > d ) > c > d ) > d,J: ( c > d ) > c > d] :
( ! [X: ( c > d ) > c > d] :
( ( member_c_d_c_d @ X @ I )
=> ( module_d_a_e_b @ ( M @ X ) @ r ) )
=> ( ( member_a @ A2 @ ( carrie867757212xt_a_b @ r ) )
=> ( ( member_c_d_c_d_d @ M2 @ ( carr_p428609062_d_a_e @ I @ M ) )
=> ( ( member_c_d_c_d @ J @ I )
=> ( ( algebr828196337_d_d_e @ r @ I @ M @ A2 @ M2 @ J )
= ( sprod_d_a_e @ ( M @ J ) @ A2 @ ( M2 @ J ) ) ) ) ) ) ) ).
% prodM_sprod_val
thf(fact_174_prodM__sprod__val,axiom,
! [I: set_c_d,M: ( c > d ) > carrie1948989069_d_a_e,A2: a,M2: ( c > d ) > d,J: c > d] :
( ! [X: c > d] :
( ( member_c_d @ X @ I )
=> ( module_d_a_e_b @ ( M @ X ) @ r ) )
=> ( ( member_a @ A2 @ ( carrie867757212xt_a_b @ r ) )
=> ( ( member_c_d_d @ M2 @ ( carr_p124510241_d_a_e @ I @ M ) )
=> ( ( member_c_d @ J @ I )
=> ( ( algebr924632044_d_d_e @ r @ I @ M @ A2 @ M2 @ J )
= ( sprod_d_a_e @ ( M @ J ) @ A2 @ ( M2 @ J ) ) ) ) ) ) ) ).
% prodM_sprod_val
thf(fact_175_prodM__sprod__val,axiom,
! [I: set_a_a_a,M: ( a > a > a ) > carrie1948989069_d_a_e,A2: a,M2: ( a > a > a ) > d,J: a > a > a] :
( ! [X: a > a > a] :
( ( member_a_a_a2 @ X @ I )
=> ( module_d_a_e_b @ ( M @ X ) @ r ) )
=> ( ( member_a @ A2 @ ( carrie867757212xt_a_b @ r ) )
=> ( ( member_a_a_a_d @ M2 @ ( carr_p1364179431_d_a_e @ I @ M ) )
=> ( ( member_a_a_a2 @ J @ I )
=> ( ( algebr1655014672_a_d_e @ r @ I @ M @ A2 @ M2 @ J )
= ( sprod_d_a_e @ ( M @ J ) @ A2 @ ( M2 @ J ) ) ) ) ) ) ) ).
% prodM_sprod_val
thf(fact_176_prodM__sprod__val,axiom,
! [I: set_a_a,M: ( a > a ) > carrie1948989069_d_a_e,A2: a,M2: ( a > a ) > d,J: a > a] :
( ! [X: a > a] :
( ( member_a_a @ X @ I )
=> ( module_d_a_e_b @ ( M @ X ) @ r ) )
=> ( ( member_a @ A2 @ ( carrie867757212xt_a_b @ r ) )
=> ( ( member_a_a_d @ M2 @ ( carr_p1646427936_d_a_e @ I @ M ) )
=> ( ( member_a_a @ J @ I )
=> ( ( algebr137561451_a_d_e @ r @ I @ M @ A2 @ M2 @ J )
= ( sprod_d_a_e @ ( M @ J ) @ A2 @ ( M2 @ J ) ) ) ) ) ) ) ).
% prodM_sprod_val
thf(fact_177_prodM__sprod__val,axiom,
! [I: set_a,M: a > carrie1948989069_d_a_e,A2: a,M2: a > d,J: a] :
( ! [X: a] :
( ( member_a @ X @ I )
=> ( module_d_a_e_b @ ( M @ X ) @ r ) )
=> ( ( member_a @ A2 @ ( carrie867757212xt_a_b @ r ) )
=> ( ( member_a_d @ M2 @ ( carr_p1084104977_d_a_e @ I @ M ) )
=> ( ( member_a @ J @ I )
=> ( ( algebr2109842362_a_d_e @ r @ I @ M @ A2 @ M2 @ J )
= ( sprod_d_a_e @ ( M @ J ) @ A2 @ ( M2 @ J ) ) ) ) ) ) ) ).
% prodM_sprod_val
thf(fact_178_prodM__sprod__val,axiom,
! [I: set_a_a_c_d,M: ( a > a > c > d ) > carrie1948989069_d_a_e,A2: a,M2: ( a > a > c > d ) > d,J: a > a > c > d] :
( ! [X: a > a > c > d] :
( ( member_a_a_c_d2 @ X @ I )
=> ( module_d_a_e_b @ ( M @ X ) @ r ) )
=> ( ( member_a @ A2 @ ( carrie867757212xt_a_b @ r ) )
=> ( ( member_a_a_c_d_d @ M2 @ ( carr_p577380087_d_a_e @ I @ M ) )
=> ( ( member_a_a_c_d2 @ J @ I )
=> ( ( algebr1620716610_d_d_e @ r @ I @ M @ A2 @ M2 @ J )
= ( sprod_d_a_e @ ( M @ J ) @ A2 @ ( M2 @ J ) ) ) ) ) ) ) ).
% prodM_sprod_val
thf(fact_179_prodM__sprod__val,axiom,
! [I: set_a_a_a_a_a,M: ( a > a > a > a > a ) > carrie1948989069_d_a_e,A2: a,M2: ( a > a > a > a > a ) > d,J: a > a > a > a > a] :
( ! [X: a > a > a > a > a] :
( ( member_a_a_a_a_a3 @ X @ I )
=> ( module_d_a_e_b @ ( M @ X ) @ r ) )
=> ( ( member_a @ A2 @ ( carrie867757212xt_a_b @ r ) )
=> ( ( member_a_a_a_a_a_d @ M2 @ ( carr_p1980447933_d_a_e @ I @ M ) )
=> ( ( member_a_a_a_a_a3 @ J @ I )
=> ( ( algebr2122715238_a_d_e @ r @ I @ M @ A2 @ M2 @ J )
= ( sprod_d_a_e @ ( M @ J ) @ A2 @ ( M2 @ J ) ) ) ) ) ) ) ).
% prodM_sprod_val
thf(fact_180_prodM__sprod__val,axiom,
! [I: set_a_a_a_a,M: ( a > a > a > a ) > carrie1948989069_d_a_e,A2: a,M2: ( a > a > a > a ) > d,J: a > a > a > a] :
( ! [X: a > a > a > a] :
( ( member_a_a_a_a4 @ X @ I )
=> ( module_d_a_e_b @ ( M @ X ) @ r ) )
=> ( ( member_a @ A2 @ ( carrie867757212xt_a_b @ r ) )
=> ( ( member_a_a_a_a_d @ M2 @ ( carr_p305487606_d_a_e @ I @ M ) )
=> ( ( member_a_a_a_a4 @ J @ I )
=> ( ( algebr303206081_a_d_e @ r @ I @ M @ A2 @ M2 @ J )
= ( sprod_d_a_e @ ( M @ J ) @ A2 @ ( M2 @ J ) ) ) ) ) ) ) ).
% prodM_sprod_val
thf(fact_181_prodM__sprod__val,axiom,
! [I: set_d,M: d > carrie1948989069_d_a_e,A2: a,M2: d > d,J: d] :
( ! [X: d] :
( ( member_d @ X @ I )
=> ( module_d_a_e_b @ ( M @ X ) @ r ) )
=> ( ( member_a @ A2 @ ( carrie867757212xt_a_b @ r ) )
=> ( ( member_d_d @ M2 @ ( carr_p1912941332_d_a_e @ I @ M ) )
=> ( ( member_d @ J @ I )
=> ( ( algebr417875517_d_d_e @ r @ I @ M @ A2 @ M2 @ J )
= ( sprod_d_a_e @ ( M @ J ) @ A2 @ ( M2 @ J ) ) ) ) ) ) ) ).
% prodM_sprod_val
thf(fact_182_prodM__sprod__val,axiom,
! [I: set_c,M: c > carrie1948989069_d_a_e,A2: a,M2: c > d,J: c] :
( ! [X: c] :
( ( member_c @ X @ I )
=> ( module_d_a_e_b @ ( M @ X ) @ r ) )
=> ( ( member_a @ A2 @ ( carrie867757212xt_a_b @ r ) )
=> ( ( member_c_d @ M2 @ ( carr_p1636662547_d_a_e @ I @ M ) )
=> ( ( member_c @ J @ I )
=> ( ( algebr1697692348_c_d_e @ r @ I @ M @ A2 @ M2 @ J )
= ( sprod_d_a_e @ ( M @ J ) @ A2 @ ( M2 @ J ) ) ) ) ) ) ) ).
% prodM_sprod_val
thf(fact_183_Sr__ring,axiom,
! [S: set_a] :
( ( sr_a_b2 @ r @ S )
=> ( ring_a_b @ ( sr_a_b @ r @ S ) ) ) ).
% Sr_ring
thf(fact_184_mHom__func,axiom,
! [F: ( c > d ) > c > d,M: carrie1954260683t_unit,N: carrie1954260683t_unit] :
( ( member_c_d_c_d @ F @ ( mHom_a1526663829t_unit @ r @ M @ N ) )
=> ( member_c_d_c_d @ F
@ ( pi_c_d_c_d @ ( carrie20288686t_unit @ M )
@ ^ [Uu: c > d] : ( carrie20288686t_unit @ N ) ) ) ) ).
% mHom_func
thf(fact_185_dsum__pOp__func,axiom,
! [I: set_c,A: c > carrie1948989069_d_a_e] :
( ! [X: c] :
( ( member_c @ X @ I )
=> ( aGroup1940521469_d_a_e @ ( A @ X ) ) )
=> ( member_c_d_c_d_c_d @ ( prod_p131511076_d_a_e @ I @ A )
@ ( pi_c_d_c_d_c_d @ ( carr_d56066307_d_a_e @ I @ A )
@ ^ [Uu: c > d] :
( pi_c_d_c_d @ ( carr_d56066307_d_a_e @ I @ A )
@ ^ [Uv: c > d] : ( carr_d56066307_d_a_e @ I @ A ) ) ) ) ) ).
% dsum_pOp_func
thf(fact_186_mem__Collect__eq,axiom,
! [A2: a,P: a > $o] :
( ( member_a @ A2 @ ( collect_a @ P ) )
= ( P @ A2 ) ) ).
% mem_Collect_eq
thf(fact_187_mem__Collect__eq,axiom,
! [A2: d,P: d > $o] :
( ( member_d @ A2 @ ( collect_d @ P ) )
= ( P @ A2 ) ) ).
% mem_Collect_eq
thf(fact_188_mem__Collect__eq,axiom,
! [A2: c,P: c > $o] :
( ( member_c @ A2 @ ( collect_c @ P ) )
= ( P @ A2 ) ) ).
% mem_Collect_eq
thf(fact_189_mem__Collect__eq,axiom,
! [A2: nat,P: nat > $o] :
( ( member_nat @ A2 @ ( collect_nat @ P ) )
= ( P @ A2 ) ) ).
% mem_Collect_eq
thf(fact_190_mem__Collect__eq,axiom,
! [A2: c > d,P: ( c > d ) > $o] :
( ( member_c_d @ A2 @ ( collect_c_d @ P ) )
= ( P @ A2 ) ) ).
% mem_Collect_eq
thf(fact_191_mem__Collect__eq,axiom,
! [A2: a > a,P: ( a > a ) > $o] :
( ( member_a_a @ A2 @ ( collect_a_a @ P ) )
= ( P @ A2 ) ) ).
% mem_Collect_eq
thf(fact_192_mem__Collect__eq,axiom,
! [A2: a > a > a,P: ( a > a > a ) > $o] :
( ( member_a_a_a2 @ A2 @ ( collect_a_a_a @ P ) )
= ( P @ A2 ) ) ).
% mem_Collect_eq
thf(fact_193_mem__Collect__eq,axiom,
! [A2: ( c > d ) > c > d,P: ( ( c > d ) > c > d ) > $o] :
( ( member_c_d_c_d @ A2 @ ( collect_c_d_c_d @ P ) )
= ( P @ A2 ) ) ).
% mem_Collect_eq
thf(fact_194_mem__Collect__eq,axiom,
! [A2: a > a > c > d,P: ( a > a > c > d ) > $o] :
( ( member_a_a_c_d2 @ A2 @ ( collect_a_a_c_d @ P ) )
= ( P @ A2 ) ) ).
% mem_Collect_eq
thf(fact_195_mem__Collect__eq,axiom,
! [A2: a > a > a > a,P: ( a > a > a > a ) > $o] :
( ( member_a_a_a_a4 @ A2 @ ( collect_a_a_a_a @ P ) )
= ( P @ A2 ) ) ).
% mem_Collect_eq
thf(fact_196_Collect__mem__eq,axiom,
! [A: set_a] :
( ( collect_a
@ ^ [X3: a] : ( member_a @ X3 @ A ) )
= A ) ).
% Collect_mem_eq
thf(fact_197_Collect__mem__eq,axiom,
! [A: set_d] :
( ( collect_d
@ ^ [X3: d] : ( member_d @ X3 @ A ) )
= A ) ).
% Collect_mem_eq
thf(fact_198_Collect__mem__eq,axiom,
! [A: set_c] :
( ( collect_c
@ ^ [X3: c] : ( member_c @ X3 @ A ) )
= A ) ).
% Collect_mem_eq
thf(fact_199_Collect__mem__eq,axiom,
! [A: set_nat] :
( ( collect_nat
@ ^ [X3: nat] : ( member_nat @ X3 @ A ) )
= A ) ).
% Collect_mem_eq
thf(fact_200_Collect__mem__eq,axiom,
! [A: set_c_d] :
( ( collect_c_d
@ ^ [X3: c > d] : ( member_c_d @ X3 @ A ) )
= A ) ).
% Collect_mem_eq
thf(fact_201_Collect__mem__eq,axiom,
! [A: set_a_a] :
( ( collect_a_a
@ ^ [X3: a > a] : ( member_a_a @ X3 @ A ) )
= A ) ).
% Collect_mem_eq
thf(fact_202_Collect__mem__eq,axiom,
! [A: set_a_a_a] :
( ( collect_a_a_a
@ ^ [X3: a > a > a] : ( member_a_a_a2 @ X3 @ A ) )
= A ) ).
% Collect_mem_eq
thf(fact_203_Collect__mem__eq,axiom,
! [A: set_c_d_c_d] :
( ( collect_c_d_c_d
@ ^ [X3: ( c > d ) > c > d] : ( member_c_d_c_d @ X3 @ A ) )
= A ) ).
% Collect_mem_eq
thf(fact_204_Collect__mem__eq,axiom,
! [A: set_a_a_c_d] :
( ( collect_a_a_c_d
@ ^ [X3: a > a > c > d] : ( member_a_a_c_d2 @ X3 @ A ) )
= A ) ).
% Collect_mem_eq
thf(fact_205_Collect__mem__eq,axiom,
! [A: set_a_a_a_a] :
( ( collect_a_a_a_a
@ ^ [X3: a > a > a > a] : ( member_a_a_a_a4 @ X3 @ A ) )
= A ) ).
% Collect_mem_eq
thf(fact_206_Collect__cong,axiom,
! [P: nat > $o,Q: nat > $o] :
( ! [X: nat] :
( ( P @ X )
= ( Q @ X ) )
=> ( ( collect_nat @ P )
= ( collect_nat @ Q ) ) ) ).
% Collect_cong
thf(fact_207_nsEqElm,axiom,
! [X2: a,Y2: a,N2: nat] :
( ( member_a @ X2 @ ( carrie867757212xt_a_b @ r ) )
=> ( ( member_a @ Y2 @ ( carrie867757212xt_a_b @ r ) )
=> ( ( X2 = Y2 )
=> ( ( nscal_a_b @ r @ X2 @ N2 )
= ( nscal_a_b @ r @ Y2 @ N2 ) ) ) ) ) ).
% nsEqElm
thf(fact_208_nsClose,axiom,
! [X2: a,N2: nat] :
( ( member_a @ X2 @ ( carrie867757212xt_a_b @ r ) )
=> ( member_a @ ( nscal_a_b @ r @ X2 @ N2 ) @ ( carrie867757212xt_a_b @ r ) ) ) ).
% nsClose
thf(fact_209_tp__closed,axiom,
( member_a_a_a2 @ ( tp_a_b @ r )
@ ( pi_a_a_a2 @ ( carrie867757212xt_a_b @ r )
@ ^ [Uu: a] :
( pi_a_a @ ( carrie867757212xt_a_b @ r )
@ ^ [Uv: a] : ( carrie867757212xt_a_b @ r ) ) ) ) ).
% tp_closed
thf(fact_210_pop__closed,axiom,
( member_a_a_a2 @ ( pop_a_Ring_ext_a_b @ r )
@ ( pi_a_a_a2 @ ( carrie867757212xt_a_b @ r )
@ ^ [Uu: a] :
( pi_a_a @ ( carrie867757212xt_a_b @ r )
@ ^ [Uv: a] : ( carrie867757212xt_a_b @ r ) ) ) ) ).
% pop_closed
thf(fact_211_rEQMulR,axiom,
! [X2: a,Y2: a,Z: a] :
( ( member_a @ X2 @ ( carrie867757212xt_a_b @ r ) )
=> ( ( member_a @ Y2 @ ( carrie867757212xt_a_b @ r ) )
=> ( ( member_a @ Z @ ( carrie867757212xt_a_b @ r ) )
=> ( ( X2 = Y2 )
=> ( ( tp_a_b @ r @ X2 @ Z )
= ( tp_a_b @ r @ Y2 @ Z ) ) ) ) ) ) ).
% rEQMulR
thf(fact_212_rMulLC,axiom,
! [X2: a,Y2: a,Z: a] :
( ( member_a @ X2 @ ( carrie867757212xt_a_b @ r ) )
=> ( ( member_a @ Y2 @ ( carrie867757212xt_a_b @ r ) )
=> ( ( member_a @ Z @ ( carrie867757212xt_a_b @ r ) )
=> ( ( tp_a_b @ r @ X2 @ ( tp_a_b @ r @ Y2 @ Z ) )
= ( tp_a_b @ r @ Y2 @ ( tp_a_b @ r @ X2 @ Z ) ) ) ) ) ) ).
% rMulLC
thf(fact_213_ring__tOp__assoc,axiom,
! [X2: a,Y2: a,Z: a] :
( ( member_a @ X2 @ ( carrie867757212xt_a_b @ r ) )
=> ( ( member_a @ Y2 @ ( carrie867757212xt_a_b @ r ) )
=> ( ( member_a @ Z @ ( carrie867757212xt_a_b @ r ) )
=> ( ( tp_a_b @ r @ ( tp_a_b @ r @ X2 @ Y2 ) @ Z )
= ( tp_a_b @ r @ X2 @ ( tp_a_b @ r @ Y2 @ Z ) ) ) ) ) ) ).
% ring_tOp_assoc
thf(fact_214_ring__tOp__closed,axiom,
! [X2: a,Y2: a] :
( ( member_a @ X2 @ ( carrie867757212xt_a_b @ r ) )
=> ( ( member_a @ Y2 @ ( carrie867757212xt_a_b @ r ) )
=> ( member_a @ ( tp_a_b @ r @ X2 @ Y2 ) @ ( carrie867757212xt_a_b @ r ) ) ) ) ).
% ring_tOp_closed
thf(fact_215_ring__tOp__commute,axiom,
! [X2: a,Y2: a] :
( ( member_a @ X2 @ ( carrie867757212xt_a_b @ r ) )
=> ( ( member_a @ Y2 @ ( carrie867757212xt_a_b @ r ) )
=> ( ( tp_a_b @ r @ X2 @ Y2 )
= ( tp_a_b @ r @ Y2 @ X2 ) ) ) ) ).
% ring_tOp_commute
thf(fact_216_ring__tOp__rel,axiom,
! [X2: a,Xa: a,Y2: a,Ya: a] :
( ( member_a @ X2 @ ( carrie867757212xt_a_b @ r ) )
=> ( ( member_a @ Xa @ ( carrie867757212xt_a_b @ r ) )
=> ( ( member_a @ Y2 @ ( carrie867757212xt_a_b @ r ) )
=> ( ( member_a @ Ya @ ( carrie867757212xt_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_217_tp__assoc,axiom,
! [A2: a,B2: a,C2: a] :
( ( member_a @ A2 @ ( carrie867757212xt_a_b @ r ) )
=> ( ( member_a @ B2 @ ( carrie867757212xt_a_b @ r ) )
=> ( ( member_a @ C2 @ ( carrie867757212xt_a_b @ r ) )
=> ( ( tp_a_b @ r @ ( tp_a_b @ r @ A2 @ B2 ) @ C2 )
= ( tp_a_b @ r @ A2 @ ( tp_a_b @ r @ B2 @ C2 ) ) ) ) ) ) ).
% tp_assoc
thf(fact_218_tp__commute,axiom,
! [A2: a,B2: a] :
( ( member_a @ A2 @ ( carrie867757212xt_a_b @ r ) )
=> ( ( member_a @ B2 @ ( carrie867757212xt_a_b @ r ) )
=> ( ( tp_a_b @ r @ A2 @ B2 )
= ( tp_a_b @ r @ B2 @ A2 ) ) ) ) ).
% tp_commute
thf(fact_219_pop__aassoc,axiom,
! [A2: a,B2: a,C2: a] :
( ( member_a @ A2 @ ( carrie867757212xt_a_b @ r ) )
=> ( ( member_a @ B2 @ ( carrie867757212xt_a_b @ r ) )
=> ( ( member_a @ C2 @ ( carrie867757212xt_a_b @ r ) )
=> ( ( pop_a_Ring_ext_a_b @ r @ ( pop_a_Ring_ext_a_b @ r @ A2 @ B2 ) @ C2 )
= ( pop_a_Ring_ext_a_b @ r @ A2 @ ( pop_a_Ring_ext_a_b @ r @ B2 @ C2 ) ) ) ) ) ) ).
% pop_aassoc
thf(fact_220_pop__commute,axiom,
! [A2: a,B2: a] :
( ( member_a @ A2 @ ( carrie867757212xt_a_b @ r ) )
=> ( ( member_a @ B2 @ ( carrie867757212xt_a_b @ r ) )
=> ( ( pop_a_Ring_ext_a_b @ r @ A2 @ B2 )
= ( pop_a_Ring_ext_a_b @ r @ B2 @ A2 ) ) ) ) ).
% pop_commute
thf(fact_221_Sr__tOp__closed,axiom,
! [S: set_a,X2: a,Y2: a] :
( ( sr_a_b2 @ r @ S )
=> ( ( member_a @ X2 @ S )
=> ( ( member_a @ Y2 @ S )
=> ( member_a @ ( tp_a_b @ r @ X2 @ Y2 ) @ S ) ) ) ) ).
% Sr_tOp_closed
thf(fact_222_Sr__pOp__closed,axiom,
! [S: set_a,X2: a,Y2: a] :
( ( sr_a_b2 @ 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_223_rg__distrib,axiom,
! [A2: a,B2: a,C2: a] :
( ( member_a @ A2 @ ( carrie867757212xt_a_b @ r ) )
=> ( ( member_a @ B2 @ ( carrie867757212xt_a_b @ r ) )
=> ( ( member_a @ C2 @ ( carrie867757212xt_a_b @ r ) )
=> ( ( tp_a_b @ r @ A2 @ ( pop_a_Ring_ext_a_b @ r @ B2 @ C2 ) )
= ( pop_a_Ring_ext_a_b @ r @ ( tp_a_b @ r @ A2 @ B2 ) @ ( tp_a_b @ r @ A2 @ C2 ) ) ) ) ) ) ).
% rg_distrib
thf(fact_224_ring__distrib1,axiom,
! [X2: a,Y2: a,Z: a] :
( ( member_a @ X2 @ ( carrie867757212xt_a_b @ r ) )
=> ( ( member_a @ Y2 @ ( carrie867757212xt_a_b @ r ) )
=> ( ( member_a @ Z @ ( carrie867757212xt_a_b @ r ) )
=> ( ( tp_a_b @ r @ X2 @ ( pop_a_Ring_ext_a_b @ r @ Y2 @ Z ) )
= ( pop_a_Ring_ext_a_b @ r @ ( tp_a_b @ r @ X2 @ Y2 ) @ ( tp_a_b @ r @ X2 @ Z ) ) ) ) ) ) ).
% ring_distrib1
thf(fact_225_ring__distrib2,axiom,
! [X2: a,Y2: a,Z: a] :
( ( member_a @ X2 @ ( carrie867757212xt_a_b @ r ) )
=> ( ( member_a @ Y2 @ ( carrie867757212xt_a_b @ r ) )
=> ( ( member_a @ Z @ ( carrie867757212xt_a_b @ r ) )
=> ( ( tp_a_b @ r @ ( pop_a_Ring_ext_a_b @ r @ Y2 @ Z ) @ X2 )
= ( pop_a_Ring_ext_a_b @ r @ ( tp_a_b @ r @ Y2 @ X2 ) @ ( tp_a_b @ r @ Z @ X2 ) ) ) ) ) ) ).
% ring_distrib2
thf(fact_226_ring__distrib3,axiom,
! [A2: a,B2: a,X2: a,Y2: a] :
( ( member_a @ A2 @ ( carrie867757212xt_a_b @ r ) )
=> ( ( member_a @ B2 @ ( carrie867757212xt_a_b @ r ) )
=> ( ( member_a @ X2 @ ( carrie867757212xt_a_b @ r ) )
=> ( ( member_a @ Y2 @ ( carrie867757212xt_a_b @ r ) )
=> ( ( tp_a_b @ r @ ( pop_a_Ring_ext_a_b @ r @ A2 @ B2 ) @ ( 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 @ A2 @ X2 ) @ ( tp_a_b @ r @ A2 @ Y2 ) ) @ ( tp_a_b @ r @ B2 @ X2 ) ) @ ( tp_a_b @ r @ B2 @ Y2 ) ) ) ) ) ) ) ).
% ring_distrib3
thf(fact_227_nsMulDistrL,axiom,
! [X2: a,Y2: a,N2: nat] :
( ( member_a @ X2 @ ( carrie867757212xt_a_b @ r ) )
=> ( ( member_a @ Y2 @ ( carrie867757212xt_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_228_nsMulDistrR,axiom,
! [X2: a,Y2: a,N2: nat] :
( ( member_a @ X2 @ ( carrie867757212xt_a_b @ r ) )
=> ( ( member_a @ Y2 @ ( carrie867757212xt_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_229_nsDistrL,axiom,
! [X2: a,Y2: a,N2: nat] :
( ( member_a @ X2 @ ( carrie867757212xt_a_b @ r ) )
=> ( ( member_a @ Y2 @ ( carrie867757212xt_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_230_Subring__tOp__ring__tOp,axiom,
! [S: carrie1105631105xt_a_b,A2: a,B2: a] :
( ( subring_a_b_b @ r @ S )
=> ( ( member_a @ A2 @ ( carrie867757212xt_a_b @ S ) )
=> ( ( member_a @ B2 @ ( carrie867757212xt_a_b @ S ) )
=> ( ( tp_a_b @ S @ A2 @ B2 )
= ( tp_a_b @ r @ A2 @ B2 ) ) ) ) ) ).
% Subring_tOp_ring_tOp
thf(fact_231_Subring__pOp__ring__pOp,axiom,
! [S: carrie1105631105xt_a_b,A2: a,B2: a] :
( ( subring_a_b_b @ r @ S )
=> ( ( member_a @ A2 @ ( carrie867757212xt_a_b @ S ) )
=> ( ( member_a @ B2 @ ( carrie867757212xt_a_b @ S ) )
=> ( ( pop_a_Ring_ext_a_b @ S @ A2 @ B2 )
= ( pop_a_Ring_ext_a_b @ r @ A2 @ B2 ) ) ) ) ) ).
% Subring_pOp_ring_pOp
thf(fact_232_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_233_Ring_OSr__tOp__closed,axiom,
! [R: carrie1105631105xt_a_b,S: set_a,X2: a,Y2: a] :
( ( ring_a_b @ R )
=> ( ( sr_a_b2 @ R @ S )
=> ( ( member_a @ X2 @ S )
=> ( ( member_a @ Y2 @ S )
=> ( member_a @ ( tp_a_b @ R @ X2 @ Y2 ) @ S ) ) ) ) ) ).
% Ring.Sr_tOp_closed
thf(fact_234_Ring_OSr__pOp__closed,axiom,
! [R: carrie1105631105xt_a_b,S: set_a,X2: a,Y2: a] :
( ( ring_a_b @ R )
=> ( ( sr_a_b2 @ R @ S )
=> ( ( member_a @ X2 @ S )
=> ( ( member_a @ Y2 @ S )
=> ( member_a @ ( pop_a_Ring_ext_a_b @ R @ X2 @ Y2 ) @ S ) ) ) ) ) ).
% Ring.Sr_pOp_closed
thf(fact_235_Ring_Oring__distrib3,axiom,
! [R: carrie1105631105xt_a_b,A2: a,B2: a,X2: a,Y2: a] :
( ( ring_a_b @ R )
=> ( ( member_a @ A2 @ ( carrie867757212xt_a_b @ R ) )
=> ( ( member_a @ B2 @ ( carrie867757212xt_a_b @ R ) )
=> ( ( member_a @ X2 @ ( carrie867757212xt_a_b @ R ) )
=> ( ( member_a @ Y2 @ ( carrie867757212xt_a_b @ R ) )
=> ( ( tp_a_b @ R @ ( pop_a_Ring_ext_a_b @ R @ A2 @ B2 ) @ ( 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 @ A2 @ X2 ) @ ( tp_a_b @ R @ A2 @ Y2 ) ) @ ( tp_a_b @ R @ B2 @ X2 ) ) @ ( tp_a_b @ R @ B2 @ Y2 ) ) ) ) ) ) ) ) ).
% Ring.ring_distrib3
thf(fact_236_Ring_Oring__distrib2,axiom,
! [R: carrie1105631105xt_a_b,X2: a,Y2: a,Z: a] :
( ( ring_a_b @ R )
=> ( ( member_a @ X2 @ ( carrie867757212xt_a_b @ R ) )
=> ( ( member_a @ Y2 @ ( carrie867757212xt_a_b @ R ) )
=> ( ( member_a @ Z @ ( carrie867757212xt_a_b @ R ) )
=> ( ( tp_a_b @ R @ ( pop_a_Ring_ext_a_b @ R @ Y2 @ Z ) @ X2 )
= ( pop_a_Ring_ext_a_b @ R @ ( tp_a_b @ R @ Y2 @ X2 ) @ ( tp_a_b @ R @ Z @ X2 ) ) ) ) ) ) ) ).
% Ring.ring_distrib2
thf(fact_237_Ring_Oring__distrib1,axiom,
! [R: carrie1105631105xt_a_b,X2: a,Y2: a,Z: a] :
( ( ring_a_b @ R )
=> ( ( member_a @ X2 @ ( carrie867757212xt_a_b @ R ) )
=> ( ( member_a @ Y2 @ ( carrie867757212xt_a_b @ R ) )
=> ( ( member_a @ Z @ ( carrie867757212xt_a_b @ R ) )
=> ( ( tp_a_b @ R @ X2 @ ( pop_a_Ring_ext_a_b @ R @ Y2 @ Z ) )
= ( pop_a_Ring_ext_a_b @ R @ ( tp_a_b @ R @ X2 @ Y2 ) @ ( tp_a_b @ R @ X2 @ Z ) ) ) ) ) ) ) ).
% Ring.ring_distrib1
thf(fact_238_Ring_Org__distrib,axiom,
! [R: carrie1105631105xt_a_b,A2: a,B2: a,C2: a] :
( ( ring_a_b @ R )
=> ( ( member_a @ A2 @ ( carrie867757212xt_a_b @ R ) )
=> ( ( member_a @ B2 @ ( carrie867757212xt_a_b @ R ) )
=> ( ( member_a @ C2 @ ( carrie867757212xt_a_b @ R ) )
=> ( ( tp_a_b @ R @ A2 @ ( pop_a_Ring_ext_a_b @ R @ B2 @ C2 ) )
= ( pop_a_Ring_ext_a_b @ R @ ( tp_a_b @ R @ A2 @ B2 ) @ ( tp_a_b @ R @ A2 @ C2 ) ) ) ) ) ) ) ).
% Ring.rg_distrib
thf(fact_239_Ring_OnsMulDistrR,axiom,
! [R: carrie1105631105xt_a_b,X2: a,Y2: a,N2: nat] :
( ( ring_a_b @ R )
=> ( ( member_a @ X2 @ ( carrie867757212xt_a_b @ R ) )
=> ( ( member_a @ Y2 @ ( carrie867757212xt_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 ) ) ) ) ) ).
% Ring.nsMulDistrR
thf(fact_240_Ring_OnsMulDistrL,axiom,
! [R: carrie1105631105xt_a_b,X2: a,Y2: a,N2: nat] :
( ( ring_a_b @ R )
=> ( ( member_a @ X2 @ ( carrie867757212xt_a_b @ R ) )
=> ( ( member_a @ Y2 @ ( carrie867757212xt_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 ) ) ) ) ) ).
% Ring.nsMulDistrL
thf(fact_241_Ring_OnsDistrL,axiom,
! [R: carrie1105631105xt_a_b,X2: a,Y2: a,N2: nat] :
( ( ring_a_b @ R )
=> ( ( member_a @ X2 @ ( carrie867757212xt_a_b @ R ) )
=> ( ( member_a @ Y2 @ ( carrie867757212xt_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 ) ) ) ) ) ).
% Ring.nsDistrL
thf(fact_242_Ring_OSr__ring,axiom,
! [R: carrie1105631105xt_a_b,S: set_a] :
( ( ring_a_b @ R )
=> ( ( sr_a_b2 @ R @ S )
=> ( ring_a_b @ ( sr_a_b @ R @ S ) ) ) ) ).
% Ring.Sr_ring
thf(fact_243_aGroup_Oag__pOp__commute,axiom,
! [A: carrie1105631105xt_a_b,X2: a,Y2: a] :
( ( aGroup2097840802xt_a_b @ A )
=> ( ( member_a @ X2 @ ( carrie867757212xt_a_b @ A ) )
=> ( ( member_a @ Y2 @ ( carrie867757212xt_a_b @ A ) )
=> ( ( pop_a_Ring_ext_a_b @ A @ X2 @ Y2 )
= ( pop_a_Ring_ext_a_b @ A @ Y2 @ X2 ) ) ) ) ) ).
% aGroup.ag_pOp_commute
thf(fact_244_aGroup_Oag__pOp__commute,axiom,
! [A: carrie1954260683t_unit,X2: c > d,Y2: c > d] :
( ( aGroup2131905830t_unit @ A )
=> ( ( member_c_d @ X2 @ ( carrie20288686t_unit @ A ) )
=> ( ( member_c_d @ Y2 @ ( carrie20288686t_unit @ A ) )
=> ( ( pop_c_449750139t_unit @ A @ X2 @ Y2 )
= ( pop_c_449750139t_unit @ A @ Y2 @ X2 ) ) ) ) ) ).
% aGroup.ag_pOp_commute
thf(fact_245_aGroup_Oag__add__commute,axiom,
! [A: carrie1105631105xt_a_b,A2: a,B2: a] :
( ( aGroup2097840802xt_a_b @ A )
=> ( ( member_a @ A2 @ ( carrie867757212xt_a_b @ A ) )
=> ( ( member_a @ B2 @ ( carrie867757212xt_a_b @ A ) )
=> ( ( pop_a_Ring_ext_a_b @ A @ A2 @ B2 )
= ( pop_a_Ring_ext_a_b @ A @ B2 @ A2 ) ) ) ) ) ).
% aGroup.ag_add_commute
thf(fact_246_aGroup_Oag__add__commute,axiom,
! [A: carrie1954260683t_unit,A2: c > d,B2: c > d] :
( ( aGroup2131905830t_unit @ A )
=> ( ( member_c_d @ A2 @ ( carrie20288686t_unit @ A ) )
=> ( ( member_c_d @ B2 @ ( carrie20288686t_unit @ A ) )
=> ( ( pop_c_449750139t_unit @ A @ A2 @ B2 )
= ( pop_c_449750139t_unit @ A @ B2 @ A2 ) ) ) ) ) ).
% aGroup.ag_add_commute
thf(fact_247_aGroup_OpOp__assocTr43,axiom,
! [A: carrie1105631105xt_a_b,A2: a,B2: a,C2: a,D: a] :
( ( aGroup2097840802xt_a_b @ A )
=> ( ( member_a @ A2 @ ( carrie867757212xt_a_b @ A ) )
=> ( ( member_a @ B2 @ ( carrie867757212xt_a_b @ A ) )
=> ( ( member_a @ C2 @ ( carrie867757212xt_a_b @ A ) )
=> ( ( member_a @ D @ ( carrie867757212xt_a_b @ A ) )
=> ( ( pop_a_Ring_ext_a_b @ A @ ( pop_a_Ring_ext_a_b @ A @ A2 @ B2 ) @ ( pop_a_Ring_ext_a_b @ A @ C2 @ D ) )
= ( pop_a_Ring_ext_a_b @ A @ ( pop_a_Ring_ext_a_b @ A @ A2 @ ( pop_a_Ring_ext_a_b @ A @ B2 @ C2 ) ) @ D ) ) ) ) ) ) ) ).
% aGroup.pOp_assocTr43
thf(fact_248_aGroup_OpOp__assocTr43,axiom,
! [A: carrie1954260683t_unit,A2: c > d,B2: c > d,C2: c > d,D: c > d] :
( ( aGroup2131905830t_unit @ A )
=> ( ( member_c_d @ A2 @ ( carrie20288686t_unit @ A ) )
=> ( ( member_c_d @ B2 @ ( carrie20288686t_unit @ A ) )
=> ( ( member_c_d @ C2 @ ( carrie20288686t_unit @ A ) )
=> ( ( member_c_d @ D @ ( carrie20288686t_unit @ A ) )
=> ( ( pop_c_449750139t_unit @ A @ ( pop_c_449750139t_unit @ A @ A2 @ B2 ) @ ( pop_c_449750139t_unit @ A @ C2 @ D ) )
= ( pop_c_449750139t_unit @ A @ ( pop_c_449750139t_unit @ A @ A2 @ ( pop_c_449750139t_unit @ A @ B2 @ C2 ) ) @ D ) ) ) ) ) ) ) ).
% aGroup.pOp_assocTr43
thf(fact_249_aGroup_OpOp__assocTr42,axiom,
! [A: carrie1105631105xt_a_b,A2: a,B2: a,C2: a,D: a] :
( ( aGroup2097840802xt_a_b @ A )
=> ( ( member_a @ A2 @ ( carrie867757212xt_a_b @ A ) )
=> ( ( member_a @ B2 @ ( carrie867757212xt_a_b @ A ) )
=> ( ( member_a @ C2 @ ( carrie867757212xt_a_b @ A ) )
=> ( ( member_a @ D @ ( carrie867757212xt_a_b @ A ) )
=> ( ( pop_a_Ring_ext_a_b @ A @ ( pop_a_Ring_ext_a_b @ A @ ( pop_a_Ring_ext_a_b @ A @ A2 @ B2 ) @ C2 ) @ D )
= ( pop_a_Ring_ext_a_b @ A @ ( pop_a_Ring_ext_a_b @ A @ A2 @ ( pop_a_Ring_ext_a_b @ A @ B2 @ C2 ) ) @ D ) ) ) ) ) ) ) ).
% aGroup.pOp_assocTr42
thf(fact_250_aGroup_OpOp__assocTr42,axiom,
! [A: carrie1954260683t_unit,A2: c > d,B2: c > d,C2: c > d,D: c > d] :
( ( aGroup2131905830t_unit @ A )
=> ( ( member_c_d @ A2 @ ( carrie20288686t_unit @ A ) )
=> ( ( member_c_d @ B2 @ ( carrie20288686t_unit @ A ) )
=> ( ( member_c_d @ C2 @ ( carrie20288686t_unit @ A ) )
=> ( ( member_c_d @ D @ ( carrie20288686t_unit @ A ) )
=> ( ( pop_c_449750139t_unit @ A @ ( pop_c_449750139t_unit @ A @ ( pop_c_449750139t_unit @ A @ A2 @ B2 ) @ C2 ) @ D )
= ( pop_c_449750139t_unit @ A @ ( pop_c_449750139t_unit @ A @ A2 @ ( pop_c_449750139t_unit @ A @ B2 @ C2 ) ) @ D ) ) ) ) ) ) ) ).
% aGroup.pOp_assocTr42
thf(fact_251_aGroup_OpOp__assocTr41,axiom,
! [A: carrie1105631105xt_a_b,A2: a,B2: a,C2: a,D: a] :
( ( aGroup2097840802xt_a_b @ A )
=> ( ( member_a @ A2 @ ( carrie867757212xt_a_b @ A ) )
=> ( ( member_a @ B2 @ ( carrie867757212xt_a_b @ A ) )
=> ( ( member_a @ C2 @ ( carrie867757212xt_a_b @ A ) )
=> ( ( member_a @ D @ ( carrie867757212xt_a_b @ A ) )
=> ( ( pop_a_Ring_ext_a_b @ A @ ( pop_a_Ring_ext_a_b @ A @ ( pop_a_Ring_ext_a_b @ A @ A2 @ B2 ) @ C2 ) @ D )
= ( pop_a_Ring_ext_a_b @ A @ ( pop_a_Ring_ext_a_b @ A @ A2 @ B2 ) @ ( pop_a_Ring_ext_a_b @ A @ C2 @ D ) ) ) ) ) ) ) ) ).
% aGroup.pOp_assocTr41
thf(fact_252_aGroup_OpOp__assocTr41,axiom,
! [A: carrie1954260683t_unit,A2: c > d,B2: c > d,C2: c > d,D: c > d] :
( ( aGroup2131905830t_unit @ A )
=> ( ( member_c_d @ A2 @ ( carrie20288686t_unit @ A ) )
=> ( ( member_c_d @ B2 @ ( carrie20288686t_unit @ A ) )
=> ( ( member_c_d @ C2 @ ( carrie20288686t_unit @ A ) )
=> ( ( member_c_d @ D @ ( carrie20288686t_unit @ A ) )
=> ( ( pop_c_449750139t_unit @ A @ ( pop_c_449750139t_unit @ A @ ( pop_c_449750139t_unit @ A @ A2 @ B2 ) @ C2 ) @ D )
= ( pop_c_449750139t_unit @ A @ ( pop_c_449750139t_unit @ A @ A2 @ B2 ) @ ( pop_c_449750139t_unit @ A @ C2 @ D ) ) ) ) ) ) ) ) ).
% aGroup.pOp_assocTr41
thf(fact_253_aGroup_Oag__pOp__closed,axiom,
! [A: carrie1105631105xt_a_b,X2: a,Y2: a] :
( ( aGroup2097840802xt_a_b @ A )
=> ( ( member_a @ X2 @ ( carrie867757212xt_a_b @ A ) )
=> ( ( member_a @ Y2 @ ( carrie867757212xt_a_b @ A ) )
=> ( member_a @ ( pop_a_Ring_ext_a_b @ A @ X2 @ Y2 ) @ ( carrie867757212xt_a_b @ A ) ) ) ) ) ).
% aGroup.ag_pOp_closed
thf(fact_254_aGroup_Oag__pOp__closed,axiom,
! [A: carrie1954260683t_unit,X2: c > d,Y2: c > d] :
( ( aGroup2131905830t_unit @ A )
=> ( ( member_c_d @ X2 @ ( carrie20288686t_unit @ A ) )
=> ( ( member_c_d @ Y2 @ ( carrie20288686t_unit @ A ) )
=> ( member_c_d @ ( pop_c_449750139t_unit @ A @ X2 @ Y2 ) @ ( carrie20288686t_unit @ A ) ) ) ) ) ).
% aGroup.ag_pOp_closed
thf(fact_255_aGroup_OpOp__cancel__r,axiom,
! [A: carrie1105631105xt_a_b,A2: a,B2: a,C2: a] :
( ( aGroup2097840802xt_a_b @ A )
=> ( ( member_a @ A2 @ ( carrie867757212xt_a_b @ A ) )
=> ( ( member_a @ B2 @ ( carrie867757212xt_a_b @ A ) )
=> ( ( member_a @ C2 @ ( carrie867757212xt_a_b @ A ) )
=> ( ( ( pop_a_Ring_ext_a_b @ A @ A2 @ C2 )
= ( pop_a_Ring_ext_a_b @ A @ B2 @ C2 ) )
=> ( A2 = B2 ) ) ) ) ) ) ).
% aGroup.pOp_cancel_r
thf(fact_256_aGroup_OpOp__cancel__r,axiom,
! [A: carrie1954260683t_unit,A2: c > d,B2: c > d,C2: c > d] :
( ( aGroup2131905830t_unit @ A )
=> ( ( member_c_d @ A2 @ ( carrie20288686t_unit @ A ) )
=> ( ( member_c_d @ B2 @ ( carrie20288686t_unit @ A ) )
=> ( ( member_c_d @ C2 @ ( carrie20288686t_unit @ A ) )
=> ( ( ( pop_c_449750139t_unit @ A @ A2 @ C2 )
= ( pop_c_449750139t_unit @ A @ B2 @ C2 ) )
=> ( A2 = B2 ) ) ) ) ) ) ).
% aGroup.pOp_cancel_r
thf(fact_257_aGroup_OpOp__cancel__l,axiom,
! [A: carrie1105631105xt_a_b,A2: a,B2: a,C2: a] :
( ( aGroup2097840802xt_a_b @ A )
=> ( ( member_a @ A2 @ ( carrie867757212xt_a_b @ A ) )
=> ( ( member_a @ B2 @ ( carrie867757212xt_a_b @ A ) )
=> ( ( member_a @ C2 @ ( carrie867757212xt_a_b @ A ) )
=> ( ( ( pop_a_Ring_ext_a_b @ A @ C2 @ A2 )
= ( pop_a_Ring_ext_a_b @ A @ C2 @ B2 ) )
=> ( A2 = B2 ) ) ) ) ) ) ).
% aGroup.pOp_cancel_l
thf(fact_258_aGroup_OpOp__cancel__l,axiom,
! [A: carrie1954260683t_unit,A2: c > d,B2: c > d,C2: c > d] :
( ( aGroup2131905830t_unit @ A )
=> ( ( member_c_d @ A2 @ ( carrie20288686t_unit @ A ) )
=> ( ( member_c_d @ B2 @ ( carrie20288686t_unit @ A ) )
=> ( ( member_c_d @ C2 @ ( carrie20288686t_unit @ A ) )
=> ( ( ( pop_c_449750139t_unit @ A @ C2 @ A2 )
= ( pop_c_449750139t_unit @ A @ C2 @ B2 ) )
=> ( A2 = B2 ) ) ) ) ) ) ).
% aGroup.pOp_cancel_l
thf(fact_259_aGroup_Oag__pOp__assoc,axiom,
! [A: carrie1105631105xt_a_b,X2: a,Y2: a,Z: a] :
( ( aGroup2097840802xt_a_b @ A )
=> ( ( member_a @ X2 @ ( carrie867757212xt_a_b @ A ) )
=> ( ( member_a @ Y2 @ ( carrie867757212xt_a_b @ A ) )
=> ( ( member_a @ Z @ ( carrie867757212xt_a_b @ A ) )
=> ( ( pop_a_Ring_ext_a_b @ A @ ( pop_a_Ring_ext_a_b @ A @ X2 @ Y2 ) @ Z )
= ( pop_a_Ring_ext_a_b @ A @ X2 @ ( pop_a_Ring_ext_a_b @ A @ Y2 @ Z ) ) ) ) ) ) ) ).
% aGroup.ag_pOp_assoc
thf(fact_260_aGroup_Oag__pOp__assoc,axiom,
! [A: carrie1954260683t_unit,X2: c > d,Y2: c > d,Z: c > d] :
( ( aGroup2131905830t_unit @ A )
=> ( ( member_c_d @ X2 @ ( carrie20288686t_unit @ A ) )
=> ( ( member_c_d @ Y2 @ ( carrie20288686t_unit @ A ) )
=> ( ( member_c_d @ Z @ ( carrie20288686t_unit @ A ) )
=> ( ( pop_c_449750139t_unit @ A @ ( pop_c_449750139t_unit @ A @ X2 @ Y2 ) @ Z )
= ( pop_c_449750139t_unit @ A @ X2 @ ( pop_c_449750139t_unit @ A @ Y2 @ Z ) ) ) ) ) ) ) ).
% aGroup.ag_pOp_assoc
thf(fact_261_aGroup_Oag__pOp__add__r,axiom,
! [A: carrie1105631105xt_a_b,A2: a,B2: a,C2: a] :
( ( aGroup2097840802xt_a_b @ A )
=> ( ( member_a @ A2 @ ( carrie867757212xt_a_b @ A ) )
=> ( ( member_a @ B2 @ ( carrie867757212xt_a_b @ A ) )
=> ( ( member_a @ C2 @ ( carrie867757212xt_a_b @ A ) )
=> ( ( A2 = B2 )
=> ( ( pop_a_Ring_ext_a_b @ A @ A2 @ C2 )
= ( pop_a_Ring_ext_a_b @ A @ B2 @ C2 ) ) ) ) ) ) ) ).
% aGroup.ag_pOp_add_r
thf(fact_262_aGroup_Oag__pOp__add__r,axiom,
! [A: carrie1954260683t_unit,A2: c > d,B2: c > d,C2: c > d] :
( ( aGroup2131905830t_unit @ A )
=> ( ( member_c_d @ A2 @ ( carrie20288686t_unit @ A ) )
=> ( ( member_c_d @ B2 @ ( carrie20288686t_unit @ A ) )
=> ( ( member_c_d @ C2 @ ( carrie20288686t_unit @ A ) )
=> ( ( A2 = B2 )
=> ( ( pop_c_449750139t_unit @ A @ A2 @ C2 )
= ( pop_c_449750139t_unit @ A @ B2 @ C2 ) ) ) ) ) ) ) ).
% aGroup.ag_pOp_add_r
thf(fact_263_aGroup_Oag__pOp__add__l,axiom,
! [A: carrie1105631105xt_a_b,A2: a,B2: a,C2: a] :
( ( aGroup2097840802xt_a_b @ A )
=> ( ( member_a @ A2 @ ( carrie867757212xt_a_b @ A ) )
=> ( ( member_a @ B2 @ ( carrie867757212xt_a_b @ A ) )
=> ( ( member_a @ C2 @ ( carrie867757212xt_a_b @ A ) )
=> ( ( A2 = B2 )
=> ( ( pop_a_Ring_ext_a_b @ A @ C2 @ A2 )
= ( pop_a_Ring_ext_a_b @ A @ C2 @ B2 ) ) ) ) ) ) ) ).
% aGroup.ag_pOp_add_l
thf(fact_264_aGroup_Oag__pOp__add__l,axiom,
! [A: carrie1954260683t_unit,A2: c > d,B2: c > d,C2: c > d] :
( ( aGroup2131905830t_unit @ A )
=> ( ( member_c_d @ A2 @ ( carrie20288686t_unit @ A ) )
=> ( ( member_c_d @ B2 @ ( carrie20288686t_unit @ A ) )
=> ( ( member_c_d @ C2 @ ( carrie20288686t_unit @ A ) )
=> ( ( A2 = B2 )
=> ( ( pop_c_449750139t_unit @ A @ C2 @ A2 )
= ( pop_c_449750139t_unit @ A @ C2 @ B2 ) ) ) ) ) ) ) ).
% aGroup.ag_pOp_add_l
thf(fact_265_aGroup_OgEQAddcross,axiom,
! [A: carrie1105631105xt_a_b,L1: a,L2: a,R1: a,R2: a] :
( ( aGroup2097840802xt_a_b @ A )
=> ( ( member_a @ L1 @ ( carrie867757212xt_a_b @ A ) )
=> ( ( member_a @ L2 @ ( carrie867757212xt_a_b @ A ) )
=> ( ( member_a @ R1 @ ( carrie867757212xt_a_b @ A ) )
=> ( ( member_a @ R1 @ ( carrie867757212xt_a_b @ A ) )
=> ( ( L1 = R2 )
=> ( ( L2 = R1 )
=> ( ( pop_a_Ring_ext_a_b @ A @ L1 @ L2 )
= ( pop_a_Ring_ext_a_b @ A @ R1 @ R2 ) ) ) ) ) ) ) ) ) ).
% aGroup.gEQAddcross
thf(fact_266_aGroup_OgEQAddcross,axiom,
! [A: carrie1954260683t_unit,L1: c > d,L2: c > d,R1: c > d,R2: c > d] :
( ( aGroup2131905830t_unit @ A )
=> ( ( member_c_d @ L1 @ ( carrie20288686t_unit @ A ) )
=> ( ( member_c_d @ L2 @ ( carrie20288686t_unit @ A ) )
=> ( ( member_c_d @ R1 @ ( carrie20288686t_unit @ A ) )
=> ( ( member_c_d @ R1 @ ( carrie20288686t_unit @ A ) )
=> ( ( L1 = R2 )
=> ( ( L2 = R1 )
=> ( ( pop_c_449750139t_unit @ A @ L1 @ L2 )
= ( pop_c_449750139t_unit @ A @ R1 @ R2 ) ) ) ) ) ) ) ) ) ).
% aGroup.gEQAddcross
thf(fact_267_aGroup_Oag__add4__rel,axiom,
! [A: carrie1105631105xt_a_b,A2: a,B2: a,C2: a,D: a] :
( ( aGroup2097840802xt_a_b @ A )
=> ( ( member_a @ A2 @ ( carrie867757212xt_a_b @ A ) )
=> ( ( member_a @ B2 @ ( carrie867757212xt_a_b @ A ) )
=> ( ( member_a @ C2 @ ( carrie867757212xt_a_b @ A ) )
=> ( ( member_a @ D @ ( carrie867757212xt_a_b @ A ) )
=> ( ( pop_a_Ring_ext_a_b @ A @ ( pop_a_Ring_ext_a_b @ A @ A2 @ B2 ) @ ( pop_a_Ring_ext_a_b @ A @ C2 @ D ) )
= ( pop_a_Ring_ext_a_b @ A @ ( pop_a_Ring_ext_a_b @ A @ A2 @ C2 ) @ ( pop_a_Ring_ext_a_b @ A @ B2 @ D ) ) ) ) ) ) ) ) ).
% aGroup.ag_add4_rel
thf(fact_268_aGroup_Oag__add4__rel,axiom,
! [A: carrie1954260683t_unit,A2: c > d,B2: c > d,C2: c > d,D: c > d] :
( ( aGroup2131905830t_unit @ A )
=> ( ( member_c_d @ A2 @ ( carrie20288686t_unit @ A ) )
=> ( ( member_c_d @ B2 @ ( carrie20288686t_unit @ A ) )
=> ( ( member_c_d @ C2 @ ( carrie20288686t_unit @ A ) )
=> ( ( member_c_d @ D @ ( carrie20288686t_unit @ A ) )
=> ( ( pop_c_449750139t_unit @ A @ ( pop_c_449750139t_unit @ A @ A2 @ B2 ) @ ( pop_c_449750139t_unit @ A @ C2 @ D ) )
= ( pop_c_449750139t_unit @ A @ ( pop_c_449750139t_unit @ A @ A2 @ C2 ) @ ( pop_c_449750139t_unit @ A @ B2 @ D ) ) ) ) ) ) ) ) ).
% aGroup.ag_add4_rel
thf(fact_269_aGroup_Oaassoc,axiom,
! [A: carrie1105631105xt_a_b,A2: a,B2: a,C2: a] :
( ( aGroup2097840802xt_a_b @ A )
=> ( ( member_a @ A2 @ ( carrie867757212xt_a_b @ A ) )
=> ( ( member_a @ B2 @ ( carrie867757212xt_a_b @ A ) )
=> ( ( member_a @ C2 @ ( carrie867757212xt_a_b @ A ) )
=> ( ( pop_a_Ring_ext_a_b @ A @ ( pop_a_Ring_ext_a_b @ A @ A2 @ B2 ) @ C2 )
= ( pop_a_Ring_ext_a_b @ A @ A2 @ ( pop_a_Ring_ext_a_b @ A @ B2 @ C2 ) ) ) ) ) ) ) ).
% aGroup.aassoc
thf(fact_270_aGroup_Oaassoc,axiom,
! [A: carrie1954260683t_unit,A2: c > d,B2: c > d,C2: c > d] :
( ( aGroup2131905830t_unit @ A )
=> ( ( member_c_d @ A2 @ ( carrie20288686t_unit @ A ) )
=> ( ( member_c_d @ B2 @ ( carrie20288686t_unit @ A ) )
=> ( ( member_c_d @ C2 @ ( carrie20288686t_unit @ A ) )
=> ( ( pop_c_449750139t_unit @ A @ ( pop_c_449750139t_unit @ A @ A2 @ B2 ) @ C2 )
= ( pop_c_449750139t_unit @ A @ A2 @ ( pop_c_449750139t_unit @ A @ B2 @ C2 ) ) ) ) ) ) ) ).
% aGroup.aassoc
thf(fact_271_Ring_Oring__tOp__commute,axiom,
! [R: carrie1105631105xt_a_b,X2: a,Y2: a] :
( ( ring_a_b @ R )
=> ( ( member_a @ X2 @ ( carrie867757212xt_a_b @ R ) )
=> ( ( member_a @ Y2 @ ( carrie867757212xt_a_b @ R ) )
=> ( ( tp_a_b @ R @ X2 @ Y2 )
= ( tp_a_b @ R @ Y2 @ X2 ) ) ) ) ) ).
% Ring.ring_tOp_commute
thf(fact_272_Ring_Oring__tOp__closed,axiom,
! [R: carrie1105631105xt_a_b,X2: a,Y2: a] :
( ( ring_a_b @ R )
=> ( ( member_a @ X2 @ ( carrie867757212xt_a_b @ R ) )
=> ( ( member_a @ Y2 @ ( carrie867757212xt_a_b @ R ) )
=> ( member_a @ ( tp_a_b @ R @ X2 @ Y2 ) @ ( carrie867757212xt_a_b @ R ) ) ) ) ) ).
% Ring.ring_tOp_closed
thf(fact_273_Ring_Oring__tOp__assoc,axiom,
! [R: carrie1105631105xt_a_b,X2: a,Y2: a,Z: a] :
( ( ring_a_b @ R )
=> ( ( member_a @ X2 @ ( carrie867757212xt_a_b @ R ) )
=> ( ( member_a @ Y2 @ ( carrie867757212xt_a_b @ R ) )
=> ( ( member_a @ Z @ ( carrie867757212xt_a_b @ R ) )
=> ( ( tp_a_b @ R @ ( tp_a_b @ R @ X2 @ Y2 ) @ Z )
= ( tp_a_b @ R @ X2 @ ( tp_a_b @ R @ Y2 @ Z ) ) ) ) ) ) ) ).
% Ring.ring_tOp_assoc
thf(fact_274_Ring_Oring__tOp__rel,axiom,
! [R: carrie1105631105xt_a_b,X2: a,Xa: a,Y2: a,Ya: a] :
( ( ring_a_b @ R )
=> ( ( member_a @ X2 @ ( carrie867757212xt_a_b @ R ) )
=> ( ( member_a @ Xa @ ( carrie867757212xt_a_b @ R ) )
=> ( ( member_a @ Y2 @ ( carrie867757212xt_a_b @ R ) )
=> ( ( member_a @ Ya @ ( carrie867757212xt_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.ring_tOp_rel
thf(fact_275_Ring_Otp__commute,axiom,
! [R: carrie1105631105xt_a_b,A2: a,B2: a] :
( ( ring_a_b @ R )
=> ( ( member_a @ A2 @ ( carrie867757212xt_a_b @ R ) )
=> ( ( member_a @ B2 @ ( carrie867757212xt_a_b @ R ) )
=> ( ( tp_a_b @ R @ A2 @ B2 )
= ( tp_a_b @ R @ B2 @ A2 ) ) ) ) ) ).
% Ring.tp_commute
thf(fact_276_Ring_Otp__assoc,axiom,
! [R: carrie1105631105xt_a_b,A2: a,B2: a,C2: a] :
( ( ring_a_b @ R )
=> ( ( member_a @ A2 @ ( carrie867757212xt_a_b @ R ) )
=> ( ( member_a @ B2 @ ( carrie867757212xt_a_b @ R ) )
=> ( ( member_a @ C2 @ ( carrie867757212xt_a_b @ R ) )
=> ( ( tp_a_b @ R @ ( tp_a_b @ R @ A2 @ B2 ) @ C2 )
= ( tp_a_b @ R @ A2 @ ( tp_a_b @ R @ B2 @ C2 ) ) ) ) ) ) ) ).
% Ring.tp_assoc
thf(fact_277_Ring_OrEQMulR,axiom,
! [R: carrie1105631105xt_a_b,X2: a,Y2: a,Z: a] :
( ( ring_a_b @ R )
=> ( ( member_a @ X2 @ ( carrie867757212xt_a_b @ R ) )
=> ( ( member_a @ Y2 @ ( carrie867757212xt_a_b @ R ) )
=> ( ( member_a @ Z @ ( carrie867757212xt_a_b @ R ) )
=> ( ( X2 = Y2 )
=> ( ( tp_a_b @ R @ X2 @ Z )
= ( tp_a_b @ R @ Y2 @ Z ) ) ) ) ) ) ) ).
% Ring.rEQMulR
thf(fact_278_Ring_OrMulLC,axiom,
! [R: carrie1105631105xt_a_b,X2: a,Y2: a,Z: a] :
( ( ring_a_b @ R )
=> ( ( member_a @ X2 @ ( carrie867757212xt_a_b @ R ) )
=> ( ( member_a @ Y2 @ ( carrie867757212xt_a_b @ R ) )
=> ( ( member_a @ Z @ ( carrie867757212xt_a_b @ R ) )
=> ( ( tp_a_b @ R @ X2 @ ( tp_a_b @ R @ Y2 @ Z ) )
= ( tp_a_b @ R @ Y2 @ ( tp_a_b @ R @ X2 @ Z ) ) ) ) ) ) ) ).
% Ring.rMulLC
thf(fact_279_Ring_Opop__commute,axiom,
! [R: carrie1105631105xt_a_b,A2: a,B2: a] :
( ( ring_a_b @ R )
=> ( ( member_a @ A2 @ ( carrie867757212xt_a_b @ R ) )
=> ( ( member_a @ B2 @ ( carrie867757212xt_a_b @ R ) )
=> ( ( pop_a_Ring_ext_a_b @ R @ A2 @ B2 )
= ( pop_a_Ring_ext_a_b @ R @ B2 @ A2 ) ) ) ) ) ).
% Ring.pop_commute
thf(fact_280_Ring_Opop__aassoc,axiom,
! [R: carrie1105631105xt_a_b,A2: a,B2: a,C2: a] :
( ( ring_a_b @ R )
=> ( ( member_a @ A2 @ ( carrie867757212xt_a_b @ R ) )
=> ( ( member_a @ B2 @ ( carrie867757212xt_a_b @ R ) )
=> ( ( member_a @ C2 @ ( carrie867757212xt_a_b @ R ) )
=> ( ( pop_a_Ring_ext_a_b @ R @ ( pop_a_Ring_ext_a_b @ R @ A2 @ B2 ) @ C2 )
= ( pop_a_Ring_ext_a_b @ R @ A2 @ ( pop_a_Ring_ext_a_b @ R @ B2 @ C2 ) ) ) ) ) ) ) ).
% Ring.pop_aassoc
thf(fact_281_Ring_OnsEqElm,axiom,
! [R: carrie1105631105xt_a_b,X2: a,Y2: a,N2: nat] :
( ( ring_a_b @ R )
=> ( ( member_a @ X2 @ ( carrie867757212xt_a_b @ R ) )
=> ( ( member_a @ Y2 @ ( carrie867757212xt_a_b @ R ) )
=> ( ( X2 = Y2 )
=> ( ( nscal_a_b @ R @ X2 @ N2 )
= ( nscal_a_b @ R @ Y2 @ N2 ) ) ) ) ) ) ).
% Ring.nsEqElm
thf(fact_282_Ring_OnsClose,axiom,
! [R: carrie1105631105xt_a_b,X2: a,N2: nat] :
( ( ring_a_b @ R )
=> ( ( member_a @ X2 @ ( carrie867757212xt_a_b @ R ) )
=> ( member_a @ ( nscal_a_b @ R @ X2 @ N2 ) @ ( carrie867757212xt_a_b @ R ) ) ) ) ).
% Ring.nsClose
thf(fact_283_prod__pOp__mem__i,axiom,
! [I: set_d,A: d > carrie1105631105xt_a_b,X4: d > a,Y: d > a,I2: d] :
( ! [X: d] :
( ( member_d @ X @ I )
=> ( aGroup2097840802xt_a_b @ ( A @ X ) ) )
=> ( ( member_d_a @ X4 @ ( carr_p297433675xt_a_b @ I @ A ) )
=> ( ( member_d_a @ Y @ ( carr_p297433675xt_a_b @ I @ A ) )
=> ( ( member_d @ I2 @ I )
=> ( ( prod_p711161210xt_a_b @ I @ A @ X4 @ Y @ I2 )
= ( pop_a_Ring_ext_a_b @ ( A @ I2 ) @ ( X4 @ I2 ) @ ( Y @ I2 ) ) ) ) ) ) ) ).
% prod_pOp_mem_i
thf(fact_284_prod__pOp__mem__i,axiom,
! [I: set_c,A: c > carrie1105631105xt_a_b,X4: c > a,Y: c > a,I2: c] :
( ! [X: c] :
( ( member_c @ X @ I )
=> ( aGroup2097840802xt_a_b @ ( A @ X ) ) )
=> ( ( member_c_a @ X4 @ ( carr_p295160588xt_a_b @ I @ A ) )
=> ( ( member_c_a @ Y @ ( carr_p295160588xt_a_b @ I @ A ) )
=> ( ( member_c @ I2 @ I )
=> ( ( prod_p708888123xt_a_b @ I @ A @ X4 @ Y @ I2 )
= ( pop_a_Ring_ext_a_b @ ( A @ I2 ) @ ( X4 @ I2 ) @ ( Y @ I2 ) ) ) ) ) ) ) ).
% prod_pOp_mem_i
thf(fact_285_prod__pOp__mem__i,axiom,
! [I: set_a,A: a > carrie1105631105xt_a_b,X4: a > a,Y: a > a,I2: a] :
( ! [X: a] :
( ( member_a @ X @ I )
=> ( aGroup2097840802xt_a_b @ ( A @ X ) ) )
=> ( ( member_a_a @ X4 @ ( carr_p290614414xt_a_b @ I @ A ) )
=> ( ( member_a_a @ Y @ ( carr_p290614414xt_a_b @ I @ A ) )
=> ( ( member_a @ I2 @ I )
=> ( ( prod_p704341949xt_a_b @ I @ A @ X4 @ Y @ I2 )
= ( pop_a_Ring_ext_a_b @ ( A @ I2 ) @ ( X4 @ I2 ) @ ( Y @ I2 ) ) ) ) ) ) ) ).
% prod_pOp_mem_i
thf(fact_286_prod__pOp__mem__i,axiom,
! [I: set_c,A: c > carrie1948989069_d_a_e,X4: c > d,Y: c > d,I2: c] :
( ! [X: c] :
( ( member_c @ X @ I )
=> ( aGroup1940521469_d_a_e @ ( A @ X ) ) )
=> ( ( member_c_d @ X4 @ ( carr_p1636662547_d_a_e @ I @ A ) )
=> ( ( member_c_d @ Y @ ( carr_p1636662547_d_a_e @ I @ A ) )
=> ( ( member_c @ I2 @ I )
=> ( ( prod_p131511076_d_a_e @ I @ A @ X4 @ Y @ I2 )
= ( pop_d_261713874_d_a_e @ ( A @ I2 ) @ ( X4 @ I2 ) @ ( Y @ I2 ) ) ) ) ) ) ) ).
% prod_pOp_mem_i
thf(fact_287_prod__pOp__mem__i,axiom,
! [I: set_c_d,A: ( c > d ) > carrie1105631105xt_a_b,X4: ( c > d ) > a,Y: ( c > d ) > a,I2: c > d] :
( ! [X: c > d] :
( ( member_c_d @ X @ I )
=> ( aGroup2097840802xt_a_b @ ( A @ X ) ) )
=> ( ( member_c_d_a @ X4 @ ( carr_p996491646xt_a_b @ I @ A ) )
=> ( ( member_c_d_a @ Y @ ( carr_p996491646xt_a_b @ I @ A ) )
=> ( ( member_c_d @ I2 @ I )
=> ( ( prod_p1466615439xt_a_b @ I @ A @ X4 @ Y @ I2 )
= ( pop_a_Ring_ext_a_b @ ( A @ I2 ) @ ( X4 @ I2 ) @ ( Y @ I2 ) ) ) ) ) ) ) ).
% prod_pOp_mem_i
thf(fact_288_prod__pOp__mem__i,axiom,
! [I: set_a_a,A: ( a > a ) > carrie1105631105xt_a_b,X4: ( a > a ) > a,Y: ( a > a ) > a,I2: a > a] :
( ! [X: a > a] :
( ( member_a_a @ X @ I )
=> ( aGroup2097840802xt_a_b @ ( A @ X ) ) )
=> ( ( member_a_a_a @ X4 @ ( carr_p740328767xt_a_b @ I @ A ) )
=> ( ( member_a_a_a @ Y @ ( carr_p740328767xt_a_b @ I @ A ) )
=> ( ( member_a_a @ I2 @ I )
=> ( ( prod_p1210452560xt_a_b @ I @ A @ X4 @ Y @ I2 )
= ( pop_a_Ring_ext_a_b @ ( A @ I2 ) @ ( X4 @ I2 ) @ ( Y @ I2 ) ) ) ) ) ) ) ).
% prod_pOp_mem_i
thf(fact_289_prod__pOp__mem__i,axiom,
! [I: set_a_a_a,A: ( a > a > a ) > carrie1105631105xt_a_b,X4: ( a > a > a ) > a,Y: ( a > a > a ) > a,I2: a > a > a] :
( ! [X: a > a > a] :
( ( member_a_a_a2 @ X @ I )
=> ( aGroup2097840802xt_a_b @ ( A @ X ) ) )
=> ( ( member_a_a_a_a @ X4 @ ( carr_p1481099064xt_a_b @ I @ A ) )
=> ( ( member_a_a_a_a @ Y @ ( carr_p1481099064xt_a_b @ I @ A ) )
=> ( ( member_a_a_a2 @ I2 @ I )
=> ( ( prod_p1460711911xt_a_b @ I @ A @ X4 @ Y @ I2 )
= ( pop_a_Ring_ext_a_b @ ( A @ I2 ) @ ( X4 @ I2 ) @ ( Y @ I2 ) ) ) ) ) ) ) ).
% prod_pOp_mem_i
thf(fact_290_prod__pOp__mem__i,axiom,
! [I: set_c_d_c_d,A: ( ( c > d ) > c > d ) > carrie1105631105xt_a_b,X4: ( ( c > d ) > c > d ) > a,Y: ( ( c > d ) > c > d ) > a,I2: ( c > d ) > c > d] :
( ! [X: ( c > d ) > c > d] :
( ( member_c_d_c_d @ X @ I )
=> ( aGroup2097840802xt_a_b @ ( A @ X ) ) )
=> ( ( member_c_d_c_d_a @ X4 @ ( carr_p1617094073xt_a_b @ I @ A ) )
=> ( ( member_c_d_c_d_a @ Y @ ( carr_p1617094073xt_a_b @ I @ A ) )
=> ( ( member_c_d_c_d @ I2 @ I )
=> ( ( prod_p1959959370xt_a_b @ I @ A @ X4 @ Y @ I2 )
= ( pop_a_Ring_ext_a_b @ ( A @ I2 ) @ ( X4 @ I2 ) @ ( Y @ I2 ) ) ) ) ) ) ) ).
% prod_pOp_mem_i
thf(fact_291_prod__pOp__mem__i,axiom,
! [I: set_a_a_c_d,A: ( a > a > c > d ) > carrie1105631105xt_a_b,X4: ( a > a > c > d ) > a,Y: ( a > a > c > d ) > a,I2: a > a > c > d] :
( ! [X: a > a > c > d] :
( ( member_a_a_c_d2 @ X @ I )
=> ( aGroup2097840802xt_a_b @ ( A @ X ) ) )
=> ( ( member_a_a_c_d_a @ X4 @ ( carr_p109409832xt_a_b @ I @ A ) )
=> ( ( member_a_a_c_d_a @ Y @ ( carr_p109409832xt_a_b @ I @ A ) )
=> ( ( member_a_a_c_d2 @ I2 @ I )
=> ( ( prod_p452275129xt_a_b @ I @ A @ X4 @ Y @ I2 )
= ( pop_a_Ring_ext_a_b @ ( A @ I2 ) @ ( X4 @ I2 ) @ ( Y @ I2 ) ) ) ) ) ) ) ).
% prod_pOp_mem_i
thf(fact_292_prod__pOp__mem__i,axiom,
! [I: set_a_a_a_a,A: ( a > a > a > a ) > carrie1105631105xt_a_b,X4: ( a > a > a > a ) > a,Y: ( a > a > a > a ) > a,I2: a > a > a > a] :
( ! [X: a > a > a > a] :
( ( member_a_a_a_a4 @ X @ I )
=> ( aGroup2097840802xt_a_b @ ( A @ X ) ) )
=> ( ( member_a_a_a_a_a @ X4 @ ( carr_p1850824937xt_a_b @ I @ A ) )
=> ( ( member_a_a_a_a_a @ Y @ ( carr_p1850824937xt_a_b @ I @ A ) )
=> ( ( member_a_a_a_a4 @ I2 @ I )
=> ( ( prod_p46206586xt_a_b @ I @ A @ X4 @ Y @ I2 )
= ( pop_a_Ring_ext_a_b @ ( A @ I2 ) @ ( X4 @ I2 ) @ ( Y @ I2 ) ) ) ) ) ) ) ).
% prod_pOp_mem_i
thf(fact_293_dsum__pOp__mem,axiom,
! [I: set_a,A: a > carrie1105631105xt_a_b,X4: a > a,Y: a > a] :
( ! [X: a] :
( ( member_a @ X @ I )
=> ( aGroup2097840802xt_a_b @ ( A @ X ) ) )
=> ( ( member_a_a @ X4 @ ( carr_d267359390xt_a_b @ I @ A ) )
=> ( ( member_a_a @ Y @ ( carr_d267359390xt_a_b @ I @ A ) )
=> ( member_a_a @ ( prod_p704341949xt_a_b @ I @ A @ X4 @ Y ) @ ( carr_d267359390xt_a_b @ I @ A ) ) ) ) ) ).
% dsum_pOp_mem
thf(fact_294_dsum__pOp__mem,axiom,
! [I: set_c,A: c > carrie1948989069_d_a_e,X4: c > d,Y: c > d] :
( ! [X: c] :
( ( member_c @ X @ I )
=> ( aGroup1940521469_d_a_e @ ( A @ X ) ) )
=> ( ( member_c_d @ X4 @ ( carr_d56066307_d_a_e @ I @ A ) )
=> ( ( member_c_d @ Y @ ( carr_d56066307_d_a_e @ I @ A ) )
=> ( member_c_d @ ( prod_p131511076_d_a_e @ I @ A @ X4 @ Y ) @ ( carr_d56066307_d_a_e @ I @ A ) ) ) ) ) ).
% dsum_pOp_mem
thf(fact_295_dsum__iOp__mem,axiom,
! [I: set_a,A: a > carrie1105631105xt_a_b,X4: a > a] :
( ! [X: a] :
( ( member_a @ X @ I )
=> ( aGroup2097840802xt_a_b @ ( A @ X ) ) )
=> ( ( member_a_a @ X4 @ ( carr_d267359390xt_a_b @ I @ A ) )
=> ( member_a_a @ ( prod_m2022989242xt_a_b @ I @ A @ X4 ) @ ( carr_d267359390xt_a_b @ I @ A ) ) ) ) ).
% dsum_iOp_mem
thf(fact_296_dsum__iOp__mem,axiom,
! [I: set_c,A: c > carrie1948989069_d_a_e,X4: c > d] :
( ! [X: c] :
( ( member_c @ X @ I )
=> ( aGroup1940521469_d_a_e @ ( A @ X ) ) )
=> ( ( member_c_d @ X4 @ ( carr_d56066307_d_a_e @ I @ A ) )
=> ( member_c_d @ ( prod_m854557927_d_a_e @ I @ A @ X4 ) @ ( carr_d56066307_d_a_e @ I @ A ) ) ) ) ).
% dsum_iOp_mem
thf(fact_297_dsum__zero__func,axiom,
! [I: set_a,A: a > carrie1105631105xt_a_b] :
( ! [X: a] :
( ( member_a @ X @ I )
=> ( aGroup2097840802xt_a_b @ ( A @ X ) ) )
=> ( member_a_a @ ( prod_z851058290xt_a_b @ I @ A ) @ ( carr_d267359390xt_a_b @ I @ A ) ) ) ).
% dsum_zero_func
thf(fact_298_dsum__zero__func,axiom,
! [I: set_c,A: c > carrie1948989069_d_a_e] :
( ! [X: c] :
( ( member_c @ X @ I )
=> ( aGroup1940521469_d_a_e @ ( A @ X ) ) )
=> ( member_c_d @ ( prod_z1214987823_d_a_e @ I @ A ) @ ( carr_d56066307_d_a_e @ I @ A ) ) ) ).
% dsum_zero_func
thf(fact_299_aGroup_Opop__closed,axiom,
! [A: carrie1105631105xt_a_b] :
( ( aGroup2097840802xt_a_b @ A )
=> ( member_a_a_a2 @ ( pop_a_Ring_ext_a_b @ A )
@ ( pi_a_a_a2 @ ( carrie867757212xt_a_b @ A )
@ ^ [Uu: a] :
( pi_a_a @ ( carrie867757212xt_a_b @ A )
@ ^ [Uv: a] : ( carrie867757212xt_a_b @ A ) ) ) ) ) ).
% aGroup.pop_closed
thf(fact_300_aGroup_Opop__closed,axiom,
! [A: carrie1954260683t_unit] :
( ( aGroup2131905830t_unit @ A )
=> ( member_c_d_c_d_c_d @ ( pop_c_449750139t_unit @ A )
@ ( pi_c_d_c_d_c_d @ ( carrie20288686t_unit @ A )
@ ^ [Uu: c > d] :
( pi_c_d_c_d @ ( carrie20288686t_unit @ A )
@ ^ [Uv: c > d] : ( carrie20288686t_unit @ A ) ) ) ) ) ).
% aGroup.pop_closed
thf(fact_301_Ring_OJ__rad__mem,axiom,
! [R: carrie1105631105xt_a_b,X2: a] :
( ( ring_a_b @ R )
=> ( ( member_a @ X2 @ ( j_rad_a_b @ R ) )
=> ( member_a @ X2 @ ( carrie867757212xt_a_b @ R ) ) ) ) ).
% Ring.J_rad_mem
thf(fact_302_Ring_Otp__closed,axiom,
! [R: carrie1105631105xt_a_b] :
( ( ring_a_b @ R )
=> ( member_a_a_a2 @ ( tp_a_b @ R )
@ ( pi_a_a_a2 @ ( carrie867757212xt_a_b @ R )
@ ^ [Uu: a] :
( pi_a_a @ ( carrie867757212xt_a_b @ R )
@ ^ [Uv: a] : ( carrie867757212xt_a_b @ R ) ) ) ) ) ).
% Ring.tp_closed
thf(fact_303_Ring_Opop__closed,axiom,
! [R: carrie1105631105xt_a_b] :
( ( ring_a_b @ R )
=> ( member_a_a_a2 @ ( pop_a_Ring_ext_a_b @ R )
@ ( pi_a_a_a2 @ ( carrie867757212xt_a_b @ R )
@ ^ [Uu: a] :
( pi_a_a @ ( carrie867757212xt_a_b @ R )
@ ^ [Uv: a] : ( carrie867757212xt_a_b @ R ) ) ) ) ) ).
% Ring.pop_closed
thf(fact_304_Ring_OprodM__sprod__val,axiom,
! [R: carrie1105631105xt_a_b,I: set_c_d_c_d,M: ( ( c > d ) > c > d ) > carrie1948989069_d_a_e,A2: a,M2: ( ( c > d ) > c > d ) > d,J: ( c > d ) > c > d] :
( ( ring_a_b @ R )
=> ( ! [X: ( c > d ) > c > d] :
( ( member_c_d_c_d @ X @ I )
=> ( module_d_a_e_b @ ( M @ X ) @ R ) )
=> ( ( member_a @ A2 @ ( carrie867757212xt_a_b @ R ) )
=> ( ( member_c_d_c_d_d @ M2 @ ( carr_p428609062_d_a_e @ I @ M ) )
=> ( ( member_c_d_c_d @ J @ I )
=> ( ( algebr828196337_d_d_e @ R @ I @ M @ A2 @ M2 @ J )
= ( sprod_d_a_e @ ( M @ J ) @ A2 @ ( M2 @ J ) ) ) ) ) ) ) ) ).
% Ring.prodM_sprod_val
thf(fact_305_Ring_OprodM__sprod__val,axiom,
! [R: carrie1105631105xt_a_b,I: set_c_d,M: ( c > d ) > carrie1948989069_d_a_e,A2: a,M2: ( c > d ) > d,J: c > d] :
( ( ring_a_b @ R )
=> ( ! [X: c > d] :
( ( member_c_d @ X @ I )
=> ( module_d_a_e_b @ ( M @ X ) @ R ) )
=> ( ( member_a @ A2 @ ( carrie867757212xt_a_b @ R ) )
=> ( ( member_c_d_d @ M2 @ ( carr_p124510241_d_a_e @ I @ M ) )
=> ( ( member_c_d @ J @ I )
=> ( ( algebr924632044_d_d_e @ R @ I @ M @ A2 @ M2 @ J )
= ( sprod_d_a_e @ ( M @ J ) @ A2 @ ( M2 @ J ) ) ) ) ) ) ) ) ).
% Ring.prodM_sprod_val
thf(fact_306_Ring_OprodM__sprod__val,axiom,
! [R: carrie1105631105xt_a_b,I: set_a_a_a,M: ( a > a > a ) > carrie1948989069_d_a_e,A2: a,M2: ( a > a > a ) > d,J: a > a > a] :
( ( ring_a_b @ R )
=> ( ! [X: a > a > a] :
( ( member_a_a_a2 @ X @ I )
=> ( module_d_a_e_b @ ( M @ X ) @ R ) )
=> ( ( member_a @ A2 @ ( carrie867757212xt_a_b @ R ) )
=> ( ( member_a_a_a_d @ M2 @ ( carr_p1364179431_d_a_e @ I @ M ) )
=> ( ( member_a_a_a2 @ J @ I )
=> ( ( algebr1655014672_a_d_e @ R @ I @ M @ A2 @ M2 @ J )
= ( sprod_d_a_e @ ( M @ J ) @ A2 @ ( M2 @ J ) ) ) ) ) ) ) ) ).
% Ring.prodM_sprod_val
thf(fact_307_Ring_OprodM__sprod__val,axiom,
! [R: carrie1105631105xt_a_b,I: set_a_a,M: ( a > a ) > carrie1948989069_d_a_e,A2: a,M2: ( a > a ) > d,J: a > a] :
( ( ring_a_b @ R )
=> ( ! [X: a > a] :
( ( member_a_a @ X @ I )
=> ( module_d_a_e_b @ ( M @ X ) @ R ) )
=> ( ( member_a @ A2 @ ( carrie867757212xt_a_b @ R ) )
=> ( ( member_a_a_d @ M2 @ ( carr_p1646427936_d_a_e @ I @ M ) )
=> ( ( member_a_a @ J @ I )
=> ( ( algebr137561451_a_d_e @ R @ I @ M @ A2 @ M2 @ J )
= ( sprod_d_a_e @ ( M @ J ) @ A2 @ ( M2 @ J ) ) ) ) ) ) ) ) ).
% Ring.prodM_sprod_val
thf(fact_308_Ring_OprodM__sprod__val,axiom,
! [R: carrie1105631105xt_a_b,I: set_a,M: a > carrie1948989069_d_a_e,A2: a,M2: a > d,J: a] :
( ( ring_a_b @ R )
=> ( ! [X: a] :
( ( member_a @ X @ I )
=> ( module_d_a_e_b @ ( M @ X ) @ R ) )
=> ( ( member_a @ A2 @ ( carrie867757212xt_a_b @ R ) )
=> ( ( member_a_d @ M2 @ ( carr_p1084104977_d_a_e @ I @ M ) )
=> ( ( member_a @ J @ I )
=> ( ( algebr2109842362_a_d_e @ R @ I @ M @ A2 @ M2 @ J )
= ( sprod_d_a_e @ ( M @ J ) @ A2 @ ( M2 @ J ) ) ) ) ) ) ) ) ).
% Ring.prodM_sprod_val
thf(fact_309_Ring_OprodM__sprod__val,axiom,
! [R: carrie1105631105xt_a_b,I: set_a_a_c_d,M: ( a > a > c > d ) > carrie1948989069_d_a_e,A2: a,M2: ( a > a > c > d ) > d,J: a > a > c > d] :
( ( ring_a_b @ R )
=> ( ! [X: a > a > c > d] :
( ( member_a_a_c_d2 @ X @ I )
=> ( module_d_a_e_b @ ( M @ X ) @ R ) )
=> ( ( member_a @ A2 @ ( carrie867757212xt_a_b @ R ) )
=> ( ( member_a_a_c_d_d @ M2 @ ( carr_p577380087_d_a_e @ I @ M ) )
=> ( ( member_a_a_c_d2 @ J @ I )
=> ( ( algebr1620716610_d_d_e @ R @ I @ M @ A2 @ M2 @ J )
= ( sprod_d_a_e @ ( M @ J ) @ A2 @ ( M2 @ J ) ) ) ) ) ) ) ) ).
% Ring.prodM_sprod_val
thf(fact_310_Ring_OprodM__sprod__val,axiom,
! [R: carrie1105631105xt_a_b,I: set_a_a_a_a_a,M: ( a > a > a > a > a ) > carrie1948989069_d_a_e,A2: a,M2: ( a > a > a > a > a ) > d,J: a > a > a > a > a] :
( ( ring_a_b @ R )
=> ( ! [X: a > a > a > a > a] :
( ( member_a_a_a_a_a3 @ X @ I )
=> ( module_d_a_e_b @ ( M @ X ) @ R ) )
=> ( ( member_a @ A2 @ ( carrie867757212xt_a_b @ R ) )
=> ( ( member_a_a_a_a_a_d @ M2 @ ( carr_p1980447933_d_a_e @ I @ M ) )
=> ( ( member_a_a_a_a_a3 @ J @ I )
=> ( ( algebr2122715238_a_d_e @ R @ I @ M @ A2 @ M2 @ J )
= ( sprod_d_a_e @ ( M @ J ) @ A2 @ ( M2 @ J ) ) ) ) ) ) ) ) ).
% Ring.prodM_sprod_val
thf(fact_311_Ring_OprodM__sprod__val,axiom,
! [R: carrie1105631105xt_a_b,I: set_a_a_a_a,M: ( a > a > a > a ) > carrie1948989069_d_a_e,A2: a,M2: ( a > a > a > a ) > d,J: a > a > a > a] :
( ( ring_a_b @ R )
=> ( ! [X: a > a > a > a] :
( ( member_a_a_a_a4 @ X @ I )
=> ( module_d_a_e_b @ ( M @ X ) @ R ) )
=> ( ( member_a @ A2 @ ( carrie867757212xt_a_b @ R ) )
=> ( ( member_a_a_a_a_d @ M2 @ ( carr_p305487606_d_a_e @ I @ M ) )
=> ( ( member_a_a_a_a4 @ J @ I )
=> ( ( algebr303206081_a_d_e @ R @ I @ M @ A2 @ M2 @ J )
= ( sprod_d_a_e @ ( M @ J ) @ A2 @ ( M2 @ J ) ) ) ) ) ) ) ) ).
% Ring.prodM_sprod_val
thf(fact_312_Ring_OprodM__sprod__val,axiom,
! [R: carrie1105631105xt_a_b,I: set_d,M: d > carrie1948989069_d_a_e,A2: a,M2: d > d,J: d] :
( ( ring_a_b @ R )
=> ( ! [X: d] :
( ( member_d @ X @ I )
=> ( module_d_a_e_b @ ( M @ X ) @ R ) )
=> ( ( member_a @ A2 @ ( carrie867757212xt_a_b @ R ) )
=> ( ( member_d_d @ M2 @ ( carr_p1912941332_d_a_e @ I @ M ) )
=> ( ( member_d @ J @ I )
=> ( ( algebr417875517_d_d_e @ R @ I @ M @ A2 @ M2 @ J )
= ( sprod_d_a_e @ ( M @ J ) @ A2 @ ( M2 @ J ) ) ) ) ) ) ) ) ).
% Ring.prodM_sprod_val
thf(fact_313_Ring_OprodM__sprod__val,axiom,
! [R: carrie1105631105xt_a_b,I: set_c,M: c > carrie1948989069_d_a_e,A2: a,M2: c > d,J: c] :
( ( ring_a_b @ R )
=> ( ! [X: c] :
( ( member_c @ X @ I )
=> ( module_d_a_e_b @ ( M @ X ) @ R ) )
=> ( ( member_a @ A2 @ ( carrie867757212xt_a_b @ R ) )
=> ( ( member_c_d @ M2 @ ( carr_p1636662547_d_a_e @ I @ M ) )
=> ( ( member_c @ J @ I )
=> ( ( algebr1697692348_c_d_e @ R @ I @ M @ A2 @ M2 @ J )
= ( sprod_d_a_e @ ( M @ J ) @ A2 @ ( M2 @ J ) ) ) ) ) ) ) ) ).
% Ring.prodM_sprod_val
thf(fact_314_dsum__iOp__func,axiom,
! [I: set_c,A: c > carrie1948989069_d_a_e] :
( ! [X: c] :
( ( member_c @ X @ I )
=> ( aGroup1940521469_d_a_e @ ( A @ X ) ) )
=> ( member_c_d_c_d @ ( prod_m854557927_d_a_e @ I @ A )
@ ( pi_c_d_c_d @ ( carr_d56066307_d_a_e @ I @ A )
@ ^ [Uu: c > d] : ( carr_d56066307_d_a_e @ I @ A ) ) ) ) ).
% dsum_iOp_func
thf(fact_315_zeroring__J__rad__empty,axiom,
( ( zeroring_a_b @ r )
=> ( ( j_rad_a_b @ r )
= ( carrie867757212xt_a_b @ r ) ) ) ).
% zeroring_J_rad_empty
thf(fact_316_Module_OmHom__lin,axiom,
! [M: carrie1948989069_d_a_e,R: carrie1105631105xt_a_b,N: carrie1948989069_d_a_e,M2: d,F: d > d,A2: a] :
( ( module_d_a_e_b @ M @ R )
=> ( ( module_d_a_e_b @ N @ R )
=> ( ( member_d @ M2 @ ( carrie11369756_d_a_e @ M ) )
=> ( ( member_d_d @ F @ ( mHom_a_b_d_e_d_e @ R @ M @ N ) )
=> ( ( member_a @ A2 @ ( carrie867757212xt_a_b @ R ) )
=> ( ( F @ ( sprod_d_a_e @ M @ A2 @ M2 ) )
= ( sprod_d_a_e @ N @ A2 @ ( F @ M2 ) ) ) ) ) ) ) ) ).
% Module.mHom_lin
thf(fact_317_Module_OmHom__lin,axiom,
! [M: carrie1954260683t_unit,R: carrie1105631105xt_a_b,N: carrie1948989069_d_a_e,M2: c > d,F: ( c > d ) > d,A2: a] :
( ( module1440397025unit_b @ M @ R )
=> ( ( module_d_a_e_b @ N @ R )
=> ( ( member_c_d @ M2 @ ( carrie20288686t_unit @ M ) )
=> ( ( member_c_d_d @ F @ ( mHom_a764575953it_d_e @ R @ M @ N ) )
=> ( ( member_a @ A2 @ ( carrie867757212xt_a_b @ R ) )
=> ( ( F @ ( sprod_1929599593t_unit @ M @ A2 @ M2 ) )
= ( sprod_d_a_e @ N @ A2 @ ( F @ M2 ) ) ) ) ) ) ) ) ).
% Module.mHom_lin
thf(fact_318_Module_Osprod__assoc,axiom,
! [M: carrie1948989069_d_a_e,R: carrie1105631105xt_a_b,A2: a,B2: a,M2: d] :
( ( module_d_a_e_b @ M @ R )
=> ( ( member_a @ A2 @ ( carrie867757212xt_a_b @ R ) )
=> ( ( member_a @ B2 @ ( carrie867757212xt_a_b @ R ) )
=> ( ( member_d @ M2 @ ( carrie11369756_d_a_e @ M ) )
=> ( ( sprod_d_a_e @ M @ ( tp_a_b @ R @ A2 @ B2 ) @ M2 )
= ( sprod_d_a_e @ M @ A2 @ ( sprod_d_a_e @ M @ B2 @ M2 ) ) ) ) ) ) ) ).
% Module.sprod_assoc
thf(fact_319_Module_Osprod__assoc,axiom,
! [M: carrie1954260683t_unit,R: carrie1105631105xt_a_b,A2: a,B2: a,M2: c > d] :
( ( module1440397025unit_b @ M @ R )
=> ( ( member_a @ A2 @ ( carrie867757212xt_a_b @ R ) )
=> ( ( member_a @ B2 @ ( carrie867757212xt_a_b @ R ) )
=> ( ( member_c_d @ M2 @ ( carrie20288686t_unit @ M ) )
=> ( ( sprod_1929599593t_unit @ M @ ( tp_a_b @ R @ A2 @ B2 ) @ M2 )
= ( sprod_1929599593t_unit @ M @ A2 @ ( sprod_1929599593t_unit @ M @ B2 @ M2 ) ) ) ) ) ) ) ).
% Module.sprod_assoc
thf(fact_320_Ring_OmHom__func,axiom,
! [R: carrie1105631105xt_a_b,F: ( c > d ) > c > d,M: carrie1954260683t_unit,N: carrie1954260683t_unit] :
( ( ring_a_b @ R )
=> ( ( member_c_d_c_d @ F @ ( mHom_a1526663829t_unit @ R @ M @ N ) )
=> ( member_c_d_c_d @ F
@ ( pi_c_d_c_d @ ( carrie20288686t_unit @ M )
@ ^ [Uu: c > d] : ( carrie20288686t_unit @ N ) ) ) ) ) ).
% Ring.mHom_func
thf(fact_321_Ring_OSubring__pOp__ring__pOp,axiom,
! [R: carrie1105631105xt_a_b,S: carrie1105631105xt_a_b,A2: a,B2: a] :
( ( ring_a_b @ R )
=> ( ( subring_a_b_b @ R @ S )
=> ( ( member_a @ A2 @ ( carrie867757212xt_a_b @ S ) )
=> ( ( member_a @ B2 @ ( carrie867757212xt_a_b @ S ) )
=> ( ( pop_a_Ring_ext_a_b @ S @ A2 @ B2 )
= ( pop_a_Ring_ext_a_b @ R @ A2 @ B2 ) ) ) ) ) ) ).
% Ring.Subring_pOp_ring_pOp
thf(fact_322_nsDistr,axiom,
! [X2: a,N2: nat,M2: nat] :
( ( member_a @ X2 @ ( carrie867757212xt_a_b @ r ) )
=> ( ( pop_a_Ring_ext_a_b @ r @ ( nscal_a_b @ r @ X2 @ N2 ) @ ( nscal_a_b @ r @ X2 @ M2 ) )
= ( nscal_a_b @ r @ X2 @ ( plus_plus_nat @ N2 @ M2 ) ) ) ) ).
% nsDistr
thf(fact_323_add__both,axiom,
! [A2: nat,B2: nat,C2: nat] :
( ( A2 = B2 )
=> ( ( plus_plus_nat @ A2 @ C2 )
= ( plus_plus_nat @ B2 @ C2 ) ) ) ).
% add_both
thf(fact_324_mul__closed__set__sub,axiom,
! [S: set_a] :
( ( mul_closed_set_a_b @ r @ S )
=> ( ord_less_eq_set_a @ S @ ( carrie867757212xt_a_b @ r ) ) ) ).
% mul_closed_set_sub
thf(fact_325_zeroring__no__maximal,axiom,
( ( zeroring_a_b @ r )
=> ~ ? [X_1: set_a] : ( maximal_ideal_a_b @ r @ X_1 ) ) ).
% zeroring_no_maximal
thf(fact_326_id__maximal__Exist,axiom,
( ~ ( zeroring_a_b @ r )
=> ? [X_12: set_a] : ( maximal_ideal_a_b @ r @ X_12 ) ) ).
% id_maximal_Exist
thf(fact_327_mop__closed,axiom,
( member_a_a @ ( mop_a_Ring_ext_a_b @ r )
@ ( pi_a_a @ ( carrie867757212xt_a_b @ r )
@ ^ [Uu: a] : ( carrie867757212xt_a_b @ r ) ) ) ).
% mop_closed
thf(fact_328_Sr__mOp__closed,axiom,
! [S: set_a,X2: a] :
( ( sr_a_b2 @ r @ S )
=> ( ( member_a @ X2 @ S )
=> ( member_a @ ( mop_a_Ring_ext_a_b @ r @ X2 ) @ S ) ) ) ).
% Sr_mOp_closed
thf(fact_329_ring__inv1,axiom,
! [A2: a,B2: a] :
( ( member_a @ A2 @ ( carrie867757212xt_a_b @ r ) )
=> ( ( member_a @ B2 @ ( carrie867757212xt_a_b @ r ) )
=> ( ( ( mop_a_Ring_ext_a_b @ r @ ( tp_a_b @ r @ A2 @ B2 ) )
= ( tp_a_b @ r @ ( mop_a_Ring_ext_a_b @ r @ A2 ) @ B2 ) )
& ( ( mop_a_Ring_ext_a_b @ r @ ( tp_a_b @ r @ A2 @ B2 ) )
= ( tp_a_b @ r @ A2 @ ( mop_a_Ring_ext_a_b @ r @ B2 ) ) ) ) ) ) ).
% ring_inv1
thf(fact_330_ring__inv1__1,axiom,
! [A2: a,B2: a] :
( ( member_a @ A2 @ ( carrie867757212xt_a_b @ r ) )
=> ( ( member_a @ B2 @ ( carrie867757212xt_a_b @ r ) )
=> ( ( mop_a_Ring_ext_a_b @ r @ ( tp_a_b @ r @ A2 @ B2 ) )
= ( tp_a_b @ r @ ( mop_a_Ring_ext_a_b @ r @ A2 ) @ B2 ) ) ) ) ).
% ring_inv1_1
thf(fact_331_ring__inv1__2,axiom,
! [A2: a,B2: a] :
( ( member_a @ A2 @ ( carrie867757212xt_a_b @ r ) )
=> ( ( member_a @ B2 @ ( carrie867757212xt_a_b @ r ) )
=> ( ( mop_a_Ring_ext_a_b @ r @ ( tp_a_b @ r @ A2 @ B2 ) )
= ( tp_a_b @ r @ A2 @ ( mop_a_Ring_ext_a_b @ r @ B2 ) ) ) ) ) ).
% ring_inv1_2
thf(fact_332_ring__inv1__3,axiom,
! [A2: a,B2: a] :
( ( member_a @ A2 @ ( carrie867757212xt_a_b @ r ) )
=> ( ( member_a @ B2 @ ( carrie867757212xt_a_b @ r ) )
=> ( ( tp_a_b @ r @ A2 @ B2 )
= ( tp_a_b @ r @ ( mop_a_Ring_ext_a_b @ r @ A2 ) @ ( mop_a_Ring_ext_a_b @ r @ B2 ) ) ) ) ) ).
% ring_inv1_3
thf(fact_333_ring__distrib4,axiom,
! [A2: a,B2: a,X2: a,Y2: a] :
( ( member_a @ A2 @ ( carrie867757212xt_a_b @ r ) )
=> ( ( member_a @ B2 @ ( carrie867757212xt_a_b @ r ) )
=> ( ( member_a @ X2 @ ( carrie867757212xt_a_b @ r ) )
=> ( ( member_a @ Y2 @ ( carrie867757212xt_a_b @ r ) )
=> ( ( pop_a_Ring_ext_a_b @ r @ ( tp_a_b @ r @ A2 @ B2 ) @ ( mop_a_Ring_ext_a_b @ r @ ( tp_a_b @ r @ X2 @ Y2 ) ) )
= ( pop_a_Ring_ext_a_b @ r @ ( tp_a_b @ r @ A2 @ ( pop_a_Ring_ext_a_b @ r @ B2 @ ( mop_a_Ring_ext_a_b @ r @ Y2 ) ) ) @ ( tp_a_b @ r @ ( pop_a_Ring_ext_a_b @ r @ A2 @ ( mop_a_Ring_ext_a_b @ r @ X2 ) ) @ Y2 ) ) ) ) ) ) ) ).
% ring_distrib4
thf(fact_334_nonunit__contained__maxid,axiom,
! [A2: a] :
( ~ ( zeroring_a_b @ r )
=> ( ( member_a @ A2 @ ( carrie867757212xt_a_b @ r ) )
=> ( ~ ( unit_a_b @ r @ A2 )
=> ? [Mx: set_a] :
( ( maximal_ideal_a_b @ r @ Mx )
& ( member_a @ A2 @ Mx ) ) ) ) ) ).
% nonunit_contained_maxid
thf(fact_335_set__sum__mem,axiom,
! [A2: a,I: set_a,B2: a,J2: set_a] :
( ( member_a @ A2 @ I )
=> ( ( member_a @ B2 @ J2 )
=> ( ( ord_less_eq_set_a @ I @ ( carrie867757212xt_a_b @ r ) )
=> ( ( ord_less_eq_set_a @ J2 @ ( carrie867757212xt_a_b @ r ) )
=> ( member_a @ ( pop_a_Ring_ext_a_b @ r @ A2 @ B2 ) @ ( aset_s1743553894xt_a_b @ r @ I @ J2 ) ) ) ) ) ) ).
% set_sum_mem
thf(fact_336_sum__mult__pOp__closed,axiom,
! [A: set_a,B: set_a,A2: a,B2: a] :
( ( ord_less_eq_set_a @ A @ ( carrie867757212xt_a_b @ r ) )
=> ( ( ord_less_eq_set_a @ B @ ( carrie867757212xt_a_b @ r ) )
=> ( ( member_a @ A2 @ ( sum_mult_a_b @ r @ A @ B ) )
=> ( ( member_a @ B2 @ ( sum_mult_a_b @ r @ A @ B ) )
=> ( member_a @ ( pop_a_Ring_ext_a_b @ r @ A2 @ B2 ) @ ( sum_mult_a_b @ r @ A @ B ) ) ) ) ) ) ).
% sum_mult_pOp_closed
thf(fact_337_sum__mult__subR,axiom,
! [A: set_a,B: set_a] :
( ( ord_less_eq_set_a @ A @ ( carrie867757212xt_a_b @ r ) )
=> ( ( ord_less_eq_set_a @ B @ ( carrie867757212xt_a_b @ r ) )
=> ( ord_less_eq_set_a @ ( sum_mult_a_b @ r @ A @ B ) @ ( carrie867757212xt_a_b @ r ) ) ) ) ).
% sum_mult_subR
thf(fact_338_times__mem__sum__mult,axiom,
! [A: set_a,B: set_a,A2: a,B2: a] :
( ( ord_less_eq_set_a @ A @ ( carrie867757212xt_a_b @ r ) )
=> ( ( ord_less_eq_set_a @ B @ ( carrie867757212xt_a_b @ r ) )
=> ( ( member_a @ A2 @ A )
=> ( ( member_a @ B2 @ B )
=> ( member_a @ ( tp_a_b @ r @ A2 @ B2 ) @ ( sum_mult_a_b @ r @ A @ B ) ) ) ) ) ) ).
% times_mem_sum_mult
thf(fact_339_le__imp__add__int,axiom,
! [I2: nat,J: nat] :
( ( ord_less_eq_nat @ I2 @ J )
=> ? [K: nat] :
( J
= ( plus_plus_nat @ I2 @ K ) ) ) ).
% le_imp_add_int
thf(fact_340_J__rad__unit,axiom,
! [X2: a] :
( ~ ( zeroring_a_b @ r )
=> ( ( member_a @ X2 @ ( j_rad_a_b @ r ) )
=> ! [Y3: a] :
( ( member_a @ Y3 @ ( carrie867757212xt_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 ) @ Y3 ) ) ) ) ) ) ).
% J_rad_unit
thf(fact_341_un__closed,axiom,
member_a @ ( un_a_b @ r ) @ ( carrie867757212xt_a_b @ r ) ).
% un_closed
thf(fact_342_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_343_Sr__one,axiom,
! [S: set_a] :
( ( sr_a_b2 @ r @ S )
=> ( member_a @ ( un_a_b @ r ) @ S ) ) ).
% Sr_one
thf(fact_344_rg__l__unit,axiom,
! [A2: a] :
( ( member_a @ A2 @ ( carrie867757212xt_a_b @ r ) )
=> ( ( tp_a_b @ r @ ( un_a_b @ r ) @ A2 )
= A2 ) ) ).
% rg_l_unit
thf(fact_345_ring__l__one,axiom,
! [X2: a] :
( ( member_a @ X2 @ ( carrie867757212xt_a_b @ r ) )
=> ( ( tp_a_b @ r @ ( un_a_b @ r ) @ X2 )
= X2 ) ) ).
% ring_l_one
thf(fact_346_ring__r__one,axiom,
! [X2: a] :
( ( member_a @ X2 @ ( carrie867757212xt_a_b @ r ) )
=> ( ( tp_a_b @ r @ X2 @ ( un_a_b @ r ) )
= X2 ) ) ).
% ring_r_one
thf(fact_347_ring__times__minusr,axiom,
! [A2: a] :
( ( member_a @ A2 @ ( carrie867757212xt_a_b @ r ) )
=> ( ( mop_a_Ring_ext_a_b @ r @ A2 )
= ( tp_a_b @ r @ A2 @ ( mop_a_Ring_ext_a_b @ r @ ( un_a_b @ r ) ) ) ) ) ).
% ring_times_minusr
thf(fact_348_ring__times__minusl,axiom,
! [A2: a] :
( ( member_a @ A2 @ ( carrie867757212xt_a_b @ r ) )
=> ( ( mop_a_Ring_ext_a_b @ r @ A2 )
= ( tp_a_b @ r @ ( mop_a_Ring_ext_a_b @ r @ ( un_a_b @ r ) ) @ A2 ) ) ) ).
% ring_times_minusl
thf(fact_349_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_350_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_351_self__le,axiom,
! [N2: nat] : ( ord_less_eq_nat @ N2 @ N2 ) ).
% self_le
thf(fact_352_nat__eq__le,axiom,
! [M2: nat,N2: nat] :
( ( M2 = N2 )
=> ( ord_less_eq_nat @ M2 @ N2 ) ) ).
% nat_eq_le
thf(fact_353_n__in__Nsetn,axiom,
! [N2: nat] :
( member_nat @ N2
@ ( collect_nat
@ ^ [I3: nat] : ( ord_less_eq_nat @ I3 @ N2 ) ) ) ).
% n_in_Nsetn
thf(fact_354_mem__of__Nset,axiom,
! [X2: nat,N2: nat] :
( ( ord_less_eq_nat @ X2 @ N2 )
=> ( member_nat @ X2
@ ( collect_nat
@ ^ [I3: nat] : ( ord_less_eq_nat @ I3 @ N2 ) ) ) ) ).
% mem_of_Nset
% 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_p131511076_d_a_e @ i @ m )
@ ( pi_c_d_c_d_c_d @ ( carr_p1636662547_d_a_e @ i @ m )
@ ^ [Uu: c > d] :
( pi_c_d_c_d @ ( carr_p1636662547_d_a_e @ i @ m )
@ ^ [Uv: c > d] : ( carr_p1636662547_d_a_e @ i @ m ) ) ) )
& ! [A3: c > d] :
( ( member_c_d @ A3 @ ( carr_p1636662547_d_a_e @ i @ m ) )
=> ! [B3: c > d] :
( ( member_c_d @ B3 @ ( carr_p1636662547_d_a_e @ i @ m ) )
=> ! [C3: c > d] :
( ( member_c_d @ C3 @ ( carr_p1636662547_d_a_e @ i @ m ) )
=> ( ( prod_p131511076_d_a_e @ i @ m @ ( prod_p131511076_d_a_e @ i @ m @ A3 @ B3 ) @ C3 )
= ( prod_p131511076_d_a_e @ i @ m @ A3 @ ( prod_p131511076_d_a_e @ i @ m @ B3 @ C3 ) ) ) ) ) )
& ! [A3: c > d] :
( ( member_c_d @ A3 @ ( carr_p1636662547_d_a_e @ i @ m ) )
=> ! [B3: c > d] :
( ( member_c_d @ B3 @ ( carr_p1636662547_d_a_e @ i @ m ) )
=> ( ( prod_p131511076_d_a_e @ i @ m @ A3 @ B3 )
= ( prod_p131511076_d_a_e @ i @ m @ B3 @ A3 ) ) ) )
& ( member_c_d_c_d @ ( prod_m854557927_d_a_e @ i @ m )
@ ( pi_c_d_c_d @ ( carr_p1636662547_d_a_e @ i @ m )
@ ^ [Uu: c > d] : ( carr_p1636662547_d_a_e @ i @ m ) ) )
& ! [A3: c > d] :
( ( member_c_d @ A3 @ ( carr_p1636662547_d_a_e @ i @ m ) )
=> ( ( prod_p131511076_d_a_e @ i @ m @ ( prod_m854557927_d_a_e @ i @ m @ A3 ) @ A3 )
= ( prod_z1214987823_d_a_e @ i @ m ) ) )
& ( member_c_d @ ( prod_z1214987823_d_a_e @ i @ m ) @ ( carr_p1636662547_d_a_e @ i @ m ) )
& ! [A3: c > d] :
( ( member_c_d @ A3 @ ( carr_p1636662547_d_a_e @ i @ m ) )
=> ( ( prod_p131511076_d_a_e @ i @ m @ ( prod_z1214987823_d_a_e @ i @ m ) @ A3 )
= A3 ) ) ) ).
thf(conj_2,conjecture,
! [A4: c > d] :
( ~ ( member_c_d @ A4 @ ( carr_p1636662547_d_a_e @ i @ m ) )
| ( ( prod_p131511076_d_a_e @ i @ m @ A4 @ ( prod_z1214987823_d_a_e @ i @ m ) )
= A4 ) ) ).
%------------------------------------------------------------------------------