SET007 Axioms: SET007+279.ax
%------------------------------------------------------------------------------
% File : SET007+279 : TPTP v9.0.0. Released v3.4.0.
% Domain : Set Theory
% Axioms : Operations on Submodules in Right Module over Associative Ring
% Version : [Urb08] axioms.
% English :
% Refs : [Mat90] Matuszewski (1990), Formalized Mathematics
% : [Urb07] Urban (2007), MPTP 0.2: Design, Implementation, and In
% : [Urb08] Urban (2006), Email to G. Sutcliffe
% Source : [Urb08]
% Names : rmod_3 [Urb08]
% Status : Satisfiable
% Syntax : Number of formulae : 82 ( 17 unt; 0 def)
% Number of atoms : 1462 ( 64 equ)
% Maximal formula atoms : 34 ( 17 avg)
% Number of connectives : 1523 ( 143 ~; 3 |;1091 &)
% ( 15 <=>; 271 =>; 0 <=; 0 <~>)
% Maximal formula depth : 29 ( 12 avg)
% Maximal term depth : 4 ( 1 avg)
% Number of predicates : 32 ( 30 usr; 1 prp; 0-4 aty)
% Number of functors : 24 ( 24 usr; 0 con; 1-6 aty)
% Number of variables : 284 ( 271 !; 13 ?)
% SPC :
% Comments : The individual reference can be found in [Mat90] by looking for
% the name provided by [Urb08].
% : Translated by MPTP from the Mizar Mathematical Library 4.48.930.
% : These set theory axioms are used in encodings of problems in
% various domains, including ALG, CAT, GRP, LAT, SET, and TOP.
%------------------------------------------------------------------------------
fof(fc1_rmod_3,axiom,
! [A,B] :
( ( ~ v3_struct_0(A)
& v3_rlvect_1(A)
& v4_rlvect_1(A)
& v5_rlvect_1(A)
& v6_rlvect_1(A)
& v4_group_1(A)
& v6_vectsp_1(A)
& v7_vectsp_1(A)
& v8_vectsp_1(A)
& l3_vectsp_1(A)
& ~ v3_struct_0(B)
& v3_rlvect_1(B)
& v4_rlvect_1(B)
& v5_rlvect_1(B)
& v6_rlvect_1(B)
& v5_vectsp_2(B,A)
& l1_vectsp_2(B,A) )
=> ~ v1_xboole_0(k3_rmod_3(A,B)) ) ).
fof(d2_rmod_3,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v3_rlvect_1(A)
& v4_rlvect_1(A)
& v5_rlvect_1(A)
& v6_rlvect_1(A)
& v4_group_1(A)
& v6_vectsp_1(A)
& v7_vectsp_1(A)
& v8_vectsp_1(A)
& l3_vectsp_1(A) )
=> ! [B] :
( ( ~ v3_struct_0(B)
& v3_rlvect_1(B)
& v4_rlvect_1(B)
& v5_rlvect_1(B)
& v6_rlvect_1(B)
& v5_vectsp_2(B,A)
& l1_vectsp_2(B,A) )
=> ! [C] :
( m1_rmod_2(C,A,B)
=> ! [D] :
( m1_rmod_2(D,A,B)
=> ! [E] :
( ( v3_vectsp_2(E,A)
& m1_rmod_2(E,A,B) )
=> ( E = k2_rmod_3(A,B,C,D)
<=> u1_struct_0(E) = k3_xboole_0(u1_struct_0(C),u1_struct_0(D)) ) ) ) ) ) ) ).
fof(t1_rmod_3,axiom,
$true ).
fof(t2_rmod_3,axiom,
$true ).
fof(t3_rmod_3,axiom,
$true ).
fof(t4_rmod_3,axiom,
$true ).
fof(t5_rmod_3,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v3_rlvect_1(A)
& v4_rlvect_1(A)
& v5_rlvect_1(A)
& v6_rlvect_1(A)
& v4_group_1(A)
& v6_vectsp_1(A)
& v7_vectsp_1(A)
& v8_vectsp_1(A)
& l3_vectsp_1(A) )
=> ! [B] :
( ( ~ v3_struct_0(B)
& v3_rlvect_1(B)
& v4_rlvect_1(B)
& v5_rlvect_1(B)
& v6_rlvect_1(B)
& v5_vectsp_2(B,A)
& l1_vectsp_2(B,A) )
=> ! [C] :
( m1_rmod_2(C,A,B)
=> ! [D] :
( m1_rmod_2(D,A,B)
=> ! [E] :
( r1_rlvect_1(k1_rmod_3(A,B,C,D),E)
<=> ? [F] :
( m1_subset_1(F,u1_struct_0(B))
& ? [G] :
( m1_subset_1(G,u1_struct_0(B))
& r1_rlvect_1(C,F)
& r1_rlvect_1(D,G)
& E = k4_rlvect_1(B,F,G) ) ) ) ) ) ) ) ).
fof(t6_rmod_3,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v3_rlvect_1(A)
& v4_rlvect_1(A)
& v5_rlvect_1(A)
& v6_rlvect_1(A)
& v4_group_1(A)
& v6_vectsp_1(A)
& v7_vectsp_1(A)
& v8_vectsp_1(A)
& l3_vectsp_1(A) )
=> ! [B] :
( ( ~ v3_struct_0(B)
& v3_rlvect_1(B)
& v4_rlvect_1(B)
& v5_rlvect_1(B)
& v6_rlvect_1(B)
& v5_vectsp_2(B,A)
& l1_vectsp_2(B,A) )
=> ! [C] :
( m1_rmod_2(C,A,B)
=> ! [D] :
( m1_rmod_2(D,A,B)
=> ! [E] :
( m1_subset_1(E,u1_struct_0(B))
=> ( ( r1_rlvect_1(C,E)
| r1_rlvect_1(D,E) )
=> r1_rlvect_1(k1_rmod_3(A,B,C,D),E) ) ) ) ) ) ) ).
fof(t7_rmod_3,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v3_rlvect_1(A)
& v4_rlvect_1(A)
& v5_rlvect_1(A)
& v6_rlvect_1(A)
& v4_group_1(A)
& v6_vectsp_1(A)
& v7_vectsp_1(A)
& v8_vectsp_1(A)
& l3_vectsp_1(A) )
=> ! [B] :
( ( ~ v3_struct_0(B)
& v3_rlvect_1(B)
& v4_rlvect_1(B)
& v5_rlvect_1(B)
& v6_rlvect_1(B)
& v5_vectsp_2(B,A)
& l1_vectsp_2(B,A) )
=> ! [C] :
( m1_rmod_2(C,A,B)
=> ! [D] :
( m1_rmod_2(D,A,B)
=> ! [E] :
( r1_rlvect_1(k2_rmod_3(A,B,C,D),E)
<=> ( r1_rlvect_1(C,E)
& r1_rlvect_1(D,E) ) ) ) ) ) ) ).
fof(t8_rmod_3,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v3_rlvect_1(A)
& v4_rlvect_1(A)
& v5_rlvect_1(A)
& v6_rlvect_1(A)
& v4_group_1(A)
& v6_vectsp_1(A)
& v7_vectsp_1(A)
& v8_vectsp_1(A)
& l3_vectsp_1(A) )
=> ! [B] :
( ( ~ v3_struct_0(B)
& v3_rlvect_1(B)
& v4_rlvect_1(B)
& v5_rlvect_1(B)
& v6_rlvect_1(B)
& v5_vectsp_2(B,A)
& l1_vectsp_2(B,A) )
=> ! [C] :
( ( v3_vectsp_2(C,A)
& m1_rmod_2(C,A,B) )
=> k1_rmod_3(A,B,C,C) = C ) ) ) ).
fof(t9_rmod_3,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v3_rlvect_1(A)
& v4_rlvect_1(A)
& v5_rlvect_1(A)
& v6_rlvect_1(A)
& v4_group_1(A)
& v6_vectsp_1(A)
& v7_vectsp_1(A)
& v8_vectsp_1(A)
& l3_vectsp_1(A) )
=> ! [B] :
( ( ~ v3_struct_0(B)
& v3_rlvect_1(B)
& v4_rlvect_1(B)
& v5_rlvect_1(B)
& v6_rlvect_1(B)
& v5_vectsp_2(B,A)
& l1_vectsp_2(B,A) )
=> ! [C] :
( m1_rmod_2(C,A,B)
=> ! [D] :
( m1_rmod_2(D,A,B)
=> k1_rmod_3(A,B,C,D) = k1_rmod_3(A,B,D,C) ) ) ) ) ).
fof(t10_rmod_3,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v3_rlvect_1(A)
& v4_rlvect_1(A)
& v5_rlvect_1(A)
& v6_rlvect_1(A)
& v4_group_1(A)
& v6_vectsp_1(A)
& v7_vectsp_1(A)
& v8_vectsp_1(A)
& l3_vectsp_1(A) )
=> ! [B] :
( ( ~ v3_struct_0(B)
& v3_rlvect_1(B)
& v4_rlvect_1(B)
& v5_rlvect_1(B)
& v6_rlvect_1(B)
& v5_vectsp_2(B,A)
& l1_vectsp_2(B,A) )
=> ! [C] :
( m1_rmod_2(C,A,B)
=> ! [D] :
( m1_rmod_2(D,A,B)
=> ! [E] :
( m1_rmod_2(E,A,B)
=> k1_rmod_3(A,B,C,k1_rmod_3(A,B,D,E)) = k1_rmod_3(A,B,k1_rmod_3(A,B,C,D),E) ) ) ) ) ) ).
fof(t11_rmod_3,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v3_rlvect_1(A)
& v4_rlvect_1(A)
& v5_rlvect_1(A)
& v6_rlvect_1(A)
& v4_group_1(A)
& v6_vectsp_1(A)
& v7_vectsp_1(A)
& v8_vectsp_1(A)
& l3_vectsp_1(A) )
=> ! [B] :
( ( ~ v3_struct_0(B)
& v3_rlvect_1(B)
& v4_rlvect_1(B)
& v5_rlvect_1(B)
& v6_rlvect_1(B)
& v5_vectsp_2(B,A)
& l1_vectsp_2(B,A) )
=> ! [C] :
( m1_rmod_2(C,A,B)
=> ! [D] :
( m1_rmod_2(D,A,B)
=> ( m1_rmod_2(C,A,k1_rmod_3(A,B,C,D))
& m1_rmod_2(D,A,k1_rmod_3(A,B,C,D)) ) ) ) ) ) ).
fof(t12_rmod_3,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v3_rlvect_1(A)
& v4_rlvect_1(A)
& v5_rlvect_1(A)
& v6_rlvect_1(A)
& v4_group_1(A)
& v6_vectsp_1(A)
& v7_vectsp_1(A)
& v8_vectsp_1(A)
& l3_vectsp_1(A) )
=> ! [B] :
( ( ~ v3_struct_0(B)
& v3_rlvect_1(B)
& v4_rlvect_1(B)
& v5_rlvect_1(B)
& v6_rlvect_1(B)
& v5_vectsp_2(B,A)
& l1_vectsp_2(B,A) )
=> ! [C] :
( m1_rmod_2(C,A,B)
=> ! [D] :
( ( v3_vectsp_2(D,A)
& m1_rmod_2(D,A,B) )
=> ( m1_rmod_2(C,A,D)
<=> k1_rmod_3(A,B,C,D) = D ) ) ) ) ) ).
fof(t13_rmod_3,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v3_rlvect_1(A)
& v4_rlvect_1(A)
& v5_rlvect_1(A)
& v6_rlvect_1(A)
& v4_group_1(A)
& v6_vectsp_1(A)
& v7_vectsp_1(A)
& v8_vectsp_1(A)
& l3_vectsp_1(A) )
=> ! [B] :
( ( ~ v3_struct_0(B)
& v3_rlvect_1(B)
& v4_rlvect_1(B)
& v5_rlvect_1(B)
& v6_rlvect_1(B)
& v5_vectsp_2(B,A)
& l1_vectsp_2(B,A) )
=> ! [C] :
( ( v3_vectsp_2(C,A)
& m1_rmod_2(C,A,B) )
=> ( k1_rmod_3(A,B,k1_rmod_2(A,B),C) = C
& k1_rmod_3(A,B,C,k1_rmod_2(A,B)) = C ) ) ) ) ).
fof(t14_rmod_3,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v3_rlvect_1(A)
& v4_rlvect_1(A)
& v5_rlvect_1(A)
& v6_rlvect_1(A)
& v4_group_1(A)
& v6_vectsp_1(A)
& v7_vectsp_1(A)
& v8_vectsp_1(A)
& l3_vectsp_1(A) )
=> ! [B] :
( ( ~ v3_struct_0(B)
& v3_rlvect_1(B)
& v4_rlvect_1(B)
& v5_rlvect_1(B)
& v6_rlvect_1(B)
& v3_vectsp_2(B,A)
& v5_vectsp_2(B,A)
& l1_vectsp_2(B,A) )
=> ( k1_rmod_3(A,B,k1_rmod_2(A,B),k2_rmod_2(A,B)) = B
& k1_rmod_3(A,B,k2_rmod_2(A,B),k1_rmod_2(A,B)) = B ) ) ) ).
fof(t15_rmod_3,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v3_rlvect_1(A)
& v4_rlvect_1(A)
& v5_rlvect_1(A)
& v6_rlvect_1(A)
& v4_group_1(A)
& v6_vectsp_1(A)
& v7_vectsp_1(A)
& v8_vectsp_1(A)
& l3_vectsp_1(A) )
=> ! [B] :
( ( ~ v3_struct_0(B)
& v3_rlvect_1(B)
& v4_rlvect_1(B)
& v5_rlvect_1(B)
& v6_rlvect_1(B)
& v5_vectsp_2(B,A)
& l1_vectsp_2(B,A) )
=> ! [C] :
( m1_rmod_2(C,A,B)
=> ( k1_rmod_3(A,B,k2_rmod_2(A,B),C) = g1_vectsp_2(A,u1_struct_0(B),u1_rlvect_1(B),u2_struct_0(B),u1_vectsp_2(A,B))
& k1_rmod_3(A,B,C,k2_rmod_2(A,B)) = g1_vectsp_2(A,u1_struct_0(B),u1_rlvect_1(B),u2_struct_0(B),u1_vectsp_2(A,B)) ) ) ) ) ).
fof(t16_rmod_3,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v3_rlvect_1(A)
& v4_rlvect_1(A)
& v5_rlvect_1(A)
& v6_rlvect_1(A)
& v4_group_1(A)
& v6_vectsp_1(A)
& v7_vectsp_1(A)
& v8_vectsp_1(A)
& l3_vectsp_1(A) )
=> ! [B] :
( ( ~ v3_struct_0(B)
& v3_rlvect_1(B)
& v4_rlvect_1(B)
& v5_rlvect_1(B)
& v6_rlvect_1(B)
& v3_vectsp_2(B,A)
& v5_vectsp_2(B,A)
& l1_vectsp_2(B,A) )
=> k1_rmod_3(A,B,k2_rmod_2(A,B),k2_rmod_2(A,B)) = B ) ) ).
fof(t17_rmod_3,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v3_rlvect_1(A)
& v4_rlvect_1(A)
& v5_rlvect_1(A)
& v6_rlvect_1(A)
& v4_group_1(A)
& v6_vectsp_1(A)
& v7_vectsp_1(A)
& v8_vectsp_1(A)
& l3_vectsp_1(A) )
=> ! [B] :
( ( ~ v3_struct_0(B)
& v3_rlvect_1(B)
& v4_rlvect_1(B)
& v5_rlvect_1(B)
& v6_rlvect_1(B)
& v5_vectsp_2(B,A)
& l1_vectsp_2(B,A) )
=> ! [C] :
( ( v3_vectsp_2(C,A)
& m1_rmod_2(C,A,B) )
=> k2_rmod_3(A,B,C,C) = C ) ) ) ).
fof(t18_rmod_3,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v3_rlvect_1(A)
& v4_rlvect_1(A)
& v5_rlvect_1(A)
& v6_rlvect_1(A)
& v4_group_1(A)
& v6_vectsp_1(A)
& v7_vectsp_1(A)
& v8_vectsp_1(A)
& l3_vectsp_1(A) )
=> ! [B] :
( ( ~ v3_struct_0(B)
& v3_rlvect_1(B)
& v4_rlvect_1(B)
& v5_rlvect_1(B)
& v6_rlvect_1(B)
& v5_vectsp_2(B,A)
& l1_vectsp_2(B,A) )
=> ! [C] :
( m1_rmod_2(C,A,B)
=> ! [D] :
( m1_rmod_2(D,A,B)
=> k2_rmod_3(A,B,C,D) = k2_rmod_3(A,B,D,C) ) ) ) ) ).
fof(t19_rmod_3,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v3_rlvect_1(A)
& v4_rlvect_1(A)
& v5_rlvect_1(A)
& v6_rlvect_1(A)
& v4_group_1(A)
& v6_vectsp_1(A)
& v7_vectsp_1(A)
& v8_vectsp_1(A)
& l3_vectsp_1(A) )
=> ! [B] :
( ( ~ v3_struct_0(B)
& v3_rlvect_1(B)
& v4_rlvect_1(B)
& v5_rlvect_1(B)
& v6_rlvect_1(B)
& v5_vectsp_2(B,A)
& l1_vectsp_2(B,A) )
=> ! [C] :
( m1_rmod_2(C,A,B)
=> ! [D] :
( m1_rmod_2(D,A,B)
=> ! [E] :
( m1_rmod_2(E,A,B)
=> k2_rmod_3(A,B,C,k2_rmod_3(A,B,D,E)) = k2_rmod_3(A,B,k2_rmod_3(A,B,C,D),E) ) ) ) ) ) ).
fof(t20_rmod_3,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v3_rlvect_1(A)
& v4_rlvect_1(A)
& v5_rlvect_1(A)
& v6_rlvect_1(A)
& v4_group_1(A)
& v6_vectsp_1(A)
& v7_vectsp_1(A)
& v8_vectsp_1(A)
& l3_vectsp_1(A) )
=> ! [B] :
( ( ~ v3_struct_0(B)
& v3_rlvect_1(B)
& v4_rlvect_1(B)
& v5_rlvect_1(B)
& v6_rlvect_1(B)
& v5_vectsp_2(B,A)
& l1_vectsp_2(B,A) )
=> ! [C] :
( m1_rmod_2(C,A,B)
=> ! [D] :
( m1_rmod_2(D,A,B)
=> ( m1_rmod_2(k2_rmod_3(A,B,C,D),A,C)
& m1_rmod_2(k2_rmod_3(A,B,C,D),A,D) ) ) ) ) ) ).
fof(t21_rmod_3,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v3_rlvect_1(A)
& v4_rlvect_1(A)
& v5_rlvect_1(A)
& v6_rlvect_1(A)
& v4_group_1(A)
& v6_vectsp_1(A)
& v7_vectsp_1(A)
& v8_vectsp_1(A)
& l3_vectsp_1(A) )
=> ! [B] :
( ( ~ v3_struct_0(B)
& v3_rlvect_1(B)
& v4_rlvect_1(B)
& v5_rlvect_1(B)
& v6_rlvect_1(B)
& v5_vectsp_2(B,A)
& l1_vectsp_2(B,A) )
=> ! [C] :
( m1_rmod_2(C,A,B)
=> ( ! [D] :
( ( v3_vectsp_2(D,A)
& m1_rmod_2(D,A,B) )
=> ( m1_rmod_2(D,A,C)
=> k2_rmod_3(A,B,D,C) = D ) )
& ! [D] :
( m1_rmod_2(D,A,B)
=> ( k2_rmod_3(A,B,D,C) = D
=> m1_rmod_2(D,A,C) ) ) ) ) ) ) ).
fof(t22_rmod_3,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v3_rlvect_1(A)
& v4_rlvect_1(A)
& v5_rlvect_1(A)
& v6_rlvect_1(A)
& v4_group_1(A)
& v6_vectsp_1(A)
& v7_vectsp_1(A)
& v8_vectsp_1(A)
& l3_vectsp_1(A) )
=> ! [B] :
( ( ~ v3_struct_0(B)
& v3_rlvect_1(B)
& v4_rlvect_1(B)
& v5_rlvect_1(B)
& v6_rlvect_1(B)
& v5_vectsp_2(B,A)
& l1_vectsp_2(B,A) )
=> ! [C] :
( m1_rmod_2(C,A,B)
=> ! [D] :
( m1_rmod_2(D,A,B)
=> ! [E] :
( m1_rmod_2(E,A,B)
=> ( m1_rmod_2(C,A,D)
=> m1_rmod_2(k2_rmod_3(A,B,C,E),A,k2_rmod_3(A,B,D,E)) ) ) ) ) ) ) ).
fof(t23_rmod_3,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v3_rlvect_1(A)
& v4_rlvect_1(A)
& v5_rlvect_1(A)
& v6_rlvect_1(A)
& v4_group_1(A)
& v6_vectsp_1(A)
& v7_vectsp_1(A)
& v8_vectsp_1(A)
& l3_vectsp_1(A) )
=> ! [B] :
( ( ~ v3_struct_0(B)
& v3_rlvect_1(B)
& v4_rlvect_1(B)
& v5_rlvect_1(B)
& v6_rlvect_1(B)
& v5_vectsp_2(B,A)
& l1_vectsp_2(B,A) )
=> ! [C] :
( m1_rmod_2(C,A,B)
=> ! [D] :
( m1_rmod_2(D,A,B)
=> ! [E] :
( m1_rmod_2(E,A,B)
=> ( m1_rmod_2(C,A,D)
=> m1_rmod_2(k2_rmod_3(A,B,C,E),A,D) ) ) ) ) ) ) ).
fof(t24_rmod_3,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v3_rlvect_1(A)
& v4_rlvect_1(A)
& v5_rlvect_1(A)
& v6_rlvect_1(A)
& v4_group_1(A)
& v6_vectsp_1(A)
& v7_vectsp_1(A)
& v8_vectsp_1(A)
& l3_vectsp_1(A) )
=> ! [B] :
( ( ~ v3_struct_0(B)
& v3_rlvect_1(B)
& v4_rlvect_1(B)
& v5_rlvect_1(B)
& v6_rlvect_1(B)
& v5_vectsp_2(B,A)
& l1_vectsp_2(B,A) )
=> ! [C] :
( m1_rmod_2(C,A,B)
=> ! [D] :
( m1_rmod_2(D,A,B)
=> ! [E] :
( m1_rmod_2(E,A,B)
=> ( ( m1_rmod_2(C,A,D)
& m1_rmod_2(C,A,E) )
=> m1_rmod_2(C,A,k2_rmod_3(A,B,D,E)) ) ) ) ) ) ) ).
fof(t25_rmod_3,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v3_rlvect_1(A)
& v4_rlvect_1(A)
& v5_rlvect_1(A)
& v6_rlvect_1(A)
& v4_group_1(A)
& v6_vectsp_1(A)
& v7_vectsp_1(A)
& v8_vectsp_1(A)
& l3_vectsp_1(A) )
=> ! [B] :
( ( ~ v3_struct_0(B)
& v3_rlvect_1(B)
& v4_rlvect_1(B)
& v5_rlvect_1(B)
& v6_rlvect_1(B)
& v5_vectsp_2(B,A)
& l1_vectsp_2(B,A) )
=> ! [C] :
( m1_rmod_2(C,A,B)
=> ( k2_rmod_3(A,B,k1_rmod_2(A,B),C) = k1_rmod_2(A,B)
& k2_rmod_3(A,B,C,k1_rmod_2(A,B)) = k1_rmod_2(A,B) ) ) ) ) ).
fof(t26_rmod_3,axiom,
$true ).
fof(t27_rmod_3,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v3_rlvect_1(A)
& v4_rlvect_1(A)
& v5_rlvect_1(A)
& v6_rlvect_1(A)
& v4_group_1(A)
& v6_vectsp_1(A)
& v7_vectsp_1(A)
& v8_vectsp_1(A)
& l3_vectsp_1(A) )
=> ! [B] :
( ( ~ v3_struct_0(B)
& v3_rlvect_1(B)
& v4_rlvect_1(B)
& v5_rlvect_1(B)
& v6_rlvect_1(B)
& v5_vectsp_2(B,A)
& l1_vectsp_2(B,A) )
=> ! [C] :
( ( v3_vectsp_2(C,A)
& m1_rmod_2(C,A,B) )
=> ( k2_rmod_3(A,B,k2_rmod_2(A,B),C) = C
& k2_rmod_3(A,B,C,k2_rmod_2(A,B)) = C ) ) ) ) ).
fof(t28_rmod_3,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v3_rlvect_1(A)
& v4_rlvect_1(A)
& v5_rlvect_1(A)
& v6_rlvect_1(A)
& v4_group_1(A)
& v6_vectsp_1(A)
& v7_vectsp_1(A)
& v8_vectsp_1(A)
& l3_vectsp_1(A) )
=> ! [B] :
( ( ~ v3_struct_0(B)
& v3_rlvect_1(B)
& v4_rlvect_1(B)
& v5_rlvect_1(B)
& v6_rlvect_1(B)
& v3_vectsp_2(B,A)
& v5_vectsp_2(B,A)
& l1_vectsp_2(B,A) )
=> k2_rmod_3(A,B,k2_rmod_2(A,B),k2_rmod_2(A,B)) = B ) ) ).
fof(t29_rmod_3,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v3_rlvect_1(A)
& v4_rlvect_1(A)
& v5_rlvect_1(A)
& v6_rlvect_1(A)
& v4_group_1(A)
& v6_vectsp_1(A)
& v7_vectsp_1(A)
& v8_vectsp_1(A)
& l3_vectsp_1(A) )
=> ! [B] :
( ( ~ v3_struct_0(B)
& v3_rlvect_1(B)
& v4_rlvect_1(B)
& v5_rlvect_1(B)
& v6_rlvect_1(B)
& v5_vectsp_2(B,A)
& l1_vectsp_2(B,A) )
=> ! [C] :
( m1_rmod_2(C,A,B)
=> ! [D] :
( m1_rmod_2(D,A,B)
=> m1_rmod_2(k2_rmod_3(A,B,C,D),A,k1_rmod_3(A,B,C,D)) ) ) ) ) ).
fof(t30_rmod_3,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v3_rlvect_1(A)
& v4_rlvect_1(A)
& v5_rlvect_1(A)
& v6_rlvect_1(A)
& v4_group_1(A)
& v6_vectsp_1(A)
& v7_vectsp_1(A)
& v8_vectsp_1(A)
& l3_vectsp_1(A) )
=> ! [B] :
( ( ~ v3_struct_0(B)
& v3_rlvect_1(B)
& v4_rlvect_1(B)
& v5_rlvect_1(B)
& v6_rlvect_1(B)
& v5_vectsp_2(B,A)
& l1_vectsp_2(B,A) )
=> ! [C] :
( m1_rmod_2(C,A,B)
=> ! [D] :
( ( v3_vectsp_2(D,A)
& m1_rmod_2(D,A,B) )
=> k1_rmod_3(A,B,k2_rmod_3(A,B,C,D),D) = D ) ) ) ) ).
fof(t31_rmod_3,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v3_rlvect_1(A)
& v4_rlvect_1(A)
& v5_rlvect_1(A)
& v6_rlvect_1(A)
& v4_group_1(A)
& v6_vectsp_1(A)
& v7_vectsp_1(A)
& v8_vectsp_1(A)
& l3_vectsp_1(A) )
=> ! [B] :
( ( ~ v3_struct_0(B)
& v3_rlvect_1(B)
& v4_rlvect_1(B)
& v5_rlvect_1(B)
& v6_rlvect_1(B)
& v5_vectsp_2(B,A)
& l1_vectsp_2(B,A) )
=> ! [C] :
( m1_rmod_2(C,A,B)
=> ! [D] :
( ( v3_vectsp_2(D,A)
& m1_rmod_2(D,A,B) )
=> k2_rmod_3(A,B,D,k1_rmod_3(A,B,D,C)) = D ) ) ) ) ).
fof(t32_rmod_3,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v3_rlvect_1(A)
& v4_rlvect_1(A)
& v5_rlvect_1(A)
& v6_rlvect_1(A)
& v4_group_1(A)
& v6_vectsp_1(A)
& v7_vectsp_1(A)
& v8_vectsp_1(A)
& l3_vectsp_1(A) )
=> ! [B] :
( ( ~ v3_struct_0(B)
& v3_rlvect_1(B)
& v4_rlvect_1(B)
& v5_rlvect_1(B)
& v6_rlvect_1(B)
& v5_vectsp_2(B,A)
& l1_vectsp_2(B,A) )
=> ! [C] :
( m1_rmod_2(C,A,B)
=> ! [D] :
( m1_rmod_2(D,A,B)
=> ! [E] :
( m1_rmod_2(E,A,B)
=> m1_rmod_2(k1_rmod_3(A,B,k2_rmod_3(A,B,C,D),k2_rmod_3(A,B,D,E)),A,k2_rmod_3(A,B,D,k1_rmod_3(A,B,C,E))) ) ) ) ) ) ).
fof(t33_rmod_3,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v3_rlvect_1(A)
& v4_rlvect_1(A)
& v5_rlvect_1(A)
& v6_rlvect_1(A)
& v4_group_1(A)
& v6_vectsp_1(A)
& v7_vectsp_1(A)
& v8_vectsp_1(A)
& l3_vectsp_1(A) )
=> ! [B] :
( ( ~ v3_struct_0(B)
& v3_rlvect_1(B)
& v4_rlvect_1(B)
& v5_rlvect_1(B)
& v6_rlvect_1(B)
& v5_vectsp_2(B,A)
& l1_vectsp_2(B,A) )
=> ! [C] :
( m1_rmod_2(C,A,B)
=> ! [D] :
( m1_rmod_2(D,A,B)
=> ! [E] :
( m1_rmod_2(E,A,B)
=> ( m1_rmod_2(C,A,D)
=> k2_rmod_3(A,B,D,k1_rmod_3(A,B,C,E)) = k1_rmod_3(A,B,k2_rmod_3(A,B,C,D),k2_rmod_3(A,B,D,E)) ) ) ) ) ) ) ).
fof(t34_rmod_3,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v3_rlvect_1(A)
& v4_rlvect_1(A)
& v5_rlvect_1(A)
& v6_rlvect_1(A)
& v4_group_1(A)
& v6_vectsp_1(A)
& v7_vectsp_1(A)
& v8_vectsp_1(A)
& l3_vectsp_1(A) )
=> ! [B] :
( ( ~ v3_struct_0(B)
& v3_rlvect_1(B)
& v4_rlvect_1(B)
& v5_rlvect_1(B)
& v6_rlvect_1(B)
& v5_vectsp_2(B,A)
& l1_vectsp_2(B,A) )
=> ! [C] :
( m1_rmod_2(C,A,B)
=> ! [D] :
( m1_rmod_2(D,A,B)
=> ! [E] :
( m1_rmod_2(E,A,B)
=> m1_rmod_2(k1_rmod_3(A,B,C,k2_rmod_3(A,B,D,E)),A,k2_rmod_3(A,B,k1_rmod_3(A,B,D,C),k1_rmod_3(A,B,C,E))) ) ) ) ) ) ).
fof(t35_rmod_3,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v3_rlvect_1(A)
& v4_rlvect_1(A)
& v5_rlvect_1(A)
& v6_rlvect_1(A)
& v4_group_1(A)
& v6_vectsp_1(A)
& v7_vectsp_1(A)
& v8_vectsp_1(A)
& l3_vectsp_1(A) )
=> ! [B] :
( ( ~ v3_struct_0(B)
& v3_rlvect_1(B)
& v4_rlvect_1(B)
& v5_rlvect_1(B)
& v6_rlvect_1(B)
& v5_vectsp_2(B,A)
& l1_vectsp_2(B,A) )
=> ! [C] :
( m1_rmod_2(C,A,B)
=> ! [D] :
( m1_rmod_2(D,A,B)
=> ! [E] :
( m1_rmod_2(E,A,B)
=> ( m1_rmod_2(C,A,D)
=> k1_rmod_3(A,B,D,k2_rmod_3(A,B,C,E)) = k2_rmod_3(A,B,k1_rmod_3(A,B,C,D),k1_rmod_3(A,B,D,E)) ) ) ) ) ) ) ).
fof(t36_rmod_3,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v3_rlvect_1(A)
& v4_rlvect_1(A)
& v5_rlvect_1(A)
& v6_rlvect_1(A)
& v4_group_1(A)
& v6_vectsp_1(A)
& v7_vectsp_1(A)
& v8_vectsp_1(A)
& l3_vectsp_1(A) )
=> ! [B] :
( ( ~ v3_struct_0(B)
& v3_rlvect_1(B)
& v4_rlvect_1(B)
& v5_rlvect_1(B)
& v6_rlvect_1(B)
& v5_vectsp_2(B,A)
& l1_vectsp_2(B,A) )
=> ! [C] :
( m1_rmod_2(C,A,B)
=> ! [D] :
( m1_rmod_2(D,A,B)
=> ! [E] :
( ( v3_vectsp_2(E,A)
& m1_rmod_2(E,A,B) )
=> ( m1_rmod_2(E,A,C)
=> k1_rmod_3(A,B,E,k2_rmod_3(A,B,D,C)) = k2_rmod_3(A,B,k1_rmod_3(A,B,E,D),C) ) ) ) ) ) ) ).
fof(t37_rmod_3,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v3_rlvect_1(A)
& v4_rlvect_1(A)
& v5_rlvect_1(A)
& v6_rlvect_1(A)
& v4_group_1(A)
& v6_vectsp_1(A)
& v7_vectsp_1(A)
& v8_vectsp_1(A)
& l3_vectsp_1(A) )
=> ! [B] :
( ( ~ v3_struct_0(B)
& v3_rlvect_1(B)
& v4_rlvect_1(B)
& v5_rlvect_1(B)
& v6_rlvect_1(B)
& v5_vectsp_2(B,A)
& l1_vectsp_2(B,A) )
=> ! [C] :
( ( v3_vectsp_2(C,A)
& m1_rmod_2(C,A,B) )
=> ! [D] :
( ( v3_vectsp_2(D,A)
& m1_rmod_2(D,A,B) )
=> ( k1_rmod_3(A,B,C,D) = D
<=> k2_rmod_3(A,B,C,D) = C ) ) ) ) ) ).
fof(t38_rmod_3,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v3_rlvect_1(A)
& v4_rlvect_1(A)
& v5_rlvect_1(A)
& v6_rlvect_1(A)
& v4_group_1(A)
& v6_vectsp_1(A)
& v7_vectsp_1(A)
& v8_vectsp_1(A)
& l3_vectsp_1(A) )
=> ! [B] :
( ( ~ v3_struct_0(B)
& v3_rlvect_1(B)
& v4_rlvect_1(B)
& v5_rlvect_1(B)
& v6_rlvect_1(B)
& v5_vectsp_2(B,A)
& l1_vectsp_2(B,A) )
=> ! [C] :
( m1_rmod_2(C,A,B)
=> ! [D] :
( ( v3_vectsp_2(D,A)
& m1_rmod_2(D,A,B) )
=> ! [E] :
( ( v3_vectsp_2(E,A)
& m1_rmod_2(E,A,B) )
=> ( m1_rmod_2(C,A,D)
=> m1_rmod_2(k1_rmod_3(A,B,C,E),A,k1_rmod_3(A,B,D,E)) ) ) ) ) ) ) ).
fof(t39_rmod_3,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v3_rlvect_1(A)
& v4_rlvect_1(A)
& v5_rlvect_1(A)
& v6_rlvect_1(A)
& v4_group_1(A)
& v6_vectsp_1(A)
& v7_vectsp_1(A)
& v8_vectsp_1(A)
& l3_vectsp_1(A) )
=> ! [B] :
( ( ~ v3_struct_0(B)
& v3_rlvect_1(B)
& v4_rlvect_1(B)
& v5_rlvect_1(B)
& v6_rlvect_1(B)
& v5_vectsp_2(B,A)
& l1_vectsp_2(B,A) )
=> ! [C] :
( m1_rmod_2(C,A,B)
=> ! [D] :
( m1_rmod_2(D,A,B)
=> ! [E] :
( m1_rmod_2(E,A,B)
=> ( m1_rmod_2(C,A,D)
=> m1_rmod_2(C,A,k1_rmod_3(A,B,D,E)) ) ) ) ) ) ) ).
fof(t40_rmod_3,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v3_rlvect_1(A)
& v4_rlvect_1(A)
& v5_rlvect_1(A)
& v6_rlvect_1(A)
& v4_group_1(A)
& v6_vectsp_1(A)
& v7_vectsp_1(A)
& v8_vectsp_1(A)
& l3_vectsp_1(A) )
=> ! [B] :
( ( ~ v3_struct_0(B)
& v3_rlvect_1(B)
& v4_rlvect_1(B)
& v5_rlvect_1(B)
& v6_rlvect_1(B)
& v5_vectsp_2(B,A)
& l1_vectsp_2(B,A) )
=> ! [C] :
( m1_rmod_2(C,A,B)
=> ! [D] :
( m1_rmod_2(D,A,B)
=> ! [E] :
( m1_rmod_2(E,A,B)
=> ( ( m1_rmod_2(C,A,D)
& m1_rmod_2(E,A,D) )
=> m1_rmod_2(k1_rmod_3(A,B,C,E),A,D) ) ) ) ) ) ) ).
fof(t41_rmod_3,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v3_rlvect_1(A)
& v4_rlvect_1(A)
& v5_rlvect_1(A)
& v6_rlvect_1(A)
& v4_group_1(A)
& v6_vectsp_1(A)
& v7_vectsp_1(A)
& v8_vectsp_1(A)
& l3_vectsp_1(A) )
=> ! [B] :
( ( ~ v3_struct_0(B)
& v3_rlvect_1(B)
& v4_rlvect_1(B)
& v5_rlvect_1(B)
& v6_rlvect_1(B)
& v5_vectsp_2(B,A)
& l1_vectsp_2(B,A) )
=> ! [C] :
( m1_rmod_2(C,A,B)
=> ! [D] :
( m1_rmod_2(D,A,B)
=> ( ~ ( ? [E] :
( m1_rmod_2(E,A,B)
& u1_struct_0(E) = k2_xboole_0(u1_struct_0(C),u1_struct_0(D)) )
& ~ m1_rmod_2(C,A,D)
& ~ m1_rmod_2(D,A,C) )
& ~ ( ( m1_rmod_2(C,A,D)
| m1_rmod_2(D,A,C) )
& ! [E] :
( m1_rmod_2(E,A,B)
=> u1_struct_0(E) != k2_xboole_0(u1_struct_0(C),u1_struct_0(D)) ) ) ) ) ) ) ) ).
fof(d3_rmod_3,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v3_rlvect_1(A)
& v4_rlvect_1(A)
& v5_rlvect_1(A)
& v6_rlvect_1(A)
& v4_group_1(A)
& v6_vectsp_1(A)
& v7_vectsp_1(A)
& v8_vectsp_1(A)
& l3_vectsp_1(A) )
=> ! [B] :
( ( ~ v3_struct_0(B)
& v3_rlvect_1(B)
& v4_rlvect_1(B)
& v5_rlvect_1(B)
& v6_rlvect_1(B)
& v5_vectsp_2(B,A)
& l1_vectsp_2(B,A) )
=> ! [C] :
( C = k3_rmod_3(A,B)
<=> ! [D] :
( r2_hidden(D,C)
<=> ? [E] :
( v3_vectsp_2(E,A)
& m1_rmod_2(E,A,B)
& E = D ) ) ) ) ) ).
fof(t42_rmod_3,axiom,
$true ).
fof(t43_rmod_3,axiom,
$true ).
fof(t44_rmod_3,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v3_rlvect_1(A)
& v4_rlvect_1(A)
& v5_rlvect_1(A)
& v6_rlvect_1(A)
& v4_group_1(A)
& v6_vectsp_1(A)
& v7_vectsp_1(A)
& v8_vectsp_1(A)
& l3_vectsp_1(A) )
=> ! [B] :
( ( ~ v3_struct_0(B)
& v3_rlvect_1(B)
& v4_rlvect_1(B)
& v5_rlvect_1(B)
& v6_rlvect_1(B)
& v3_vectsp_2(B,A)
& v5_vectsp_2(B,A)
& l1_vectsp_2(B,A) )
=> r2_hidden(B,k3_rmod_3(A,B)) ) ) ).
fof(d4_rmod_3,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v3_rlvect_1(A)
& v4_rlvect_1(A)
& v5_rlvect_1(A)
& v6_rlvect_1(A)
& v4_group_1(A)
& v6_vectsp_1(A)
& v7_vectsp_1(A)
& v8_vectsp_1(A)
& l3_vectsp_1(A) )
=> ! [B] :
( ( ~ v3_struct_0(B)
& v3_rlvect_1(B)
& v4_rlvect_1(B)
& v5_rlvect_1(B)
& v6_rlvect_1(B)
& v5_vectsp_2(B,A)
& l1_vectsp_2(B,A) )
=> ! [C] :
( m1_rmod_2(C,A,B)
=> ! [D] :
( m1_rmod_2(D,A,B)
=> ( r1_rmod_3(A,B,C,D)
<=> ( g1_vectsp_2(A,u1_struct_0(B),u1_rlvect_1(B),u2_struct_0(B),u1_vectsp_2(A,B)) = k1_rmod_3(A,B,C,D)
& k2_rmod_3(A,B,C,D) = k1_rmod_2(A,B) ) ) ) ) ) ) ).
fof(t45_rmod_3,axiom,
$true ).
fof(t46_rmod_3,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v3_rlvect_1(A)
& v4_rlvect_1(A)
& v5_rlvect_1(A)
& v6_rlvect_1(A)
& v4_group_1(A)
& v6_vectsp_1(A)
& v7_vectsp_1(A)
& v8_vectsp_1(A)
& l3_vectsp_1(A) )
=> ! [B] :
( ( ~ v3_struct_0(B)
& v3_rlvect_1(B)
& v4_rlvect_1(B)
& v5_rlvect_1(B)
& v6_rlvect_1(B)
& v5_vectsp_2(B,A)
& l1_vectsp_2(B,A) )
=> ! [C] :
( m1_rmod_2(C,A,B)
=> ! [D] :
( m1_rmod_2(D,A,B)
=> ( r1_rmod_3(A,B,C,D)
=> r1_rmod_3(A,B,D,C) ) ) ) ) ) ).
fof(t47_rmod_3,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v3_rlvect_1(A)
& v4_rlvect_1(A)
& v5_rlvect_1(A)
& v6_rlvect_1(A)
& v4_group_1(A)
& v6_vectsp_1(A)
& v7_vectsp_1(A)
& v8_vectsp_1(A)
& l3_vectsp_1(A) )
=> ! [B] :
( ( ~ v3_struct_0(B)
& v3_rlvect_1(B)
& v4_rlvect_1(B)
& v5_rlvect_1(B)
& v6_rlvect_1(B)
& v3_vectsp_2(B,A)
& v5_vectsp_2(B,A)
& l1_vectsp_2(B,A) )
=> ( r1_rmod_3(A,B,k1_rmod_2(A,B),k2_rmod_2(A,B))
& r1_rmod_3(A,B,k2_rmod_2(A,B),k1_rmod_2(A,B)) ) ) ) ).
fof(t48_rmod_3,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v3_rlvect_1(A)
& v4_rlvect_1(A)
& v5_rlvect_1(A)
& v6_rlvect_1(A)
& v4_group_1(A)
& v6_vectsp_1(A)
& v7_vectsp_1(A)
& v8_vectsp_1(A)
& l3_vectsp_1(A) )
=> ! [B] :
( ( ~ v3_struct_0(B)
& v3_rlvect_1(B)
& v4_rlvect_1(B)
& v5_rlvect_1(B)
& v6_rlvect_1(B)
& v5_vectsp_2(B,A)
& l1_vectsp_2(B,A) )
=> ! [C] :
( m1_rmod_2(C,A,B)
=> ! [D] :
( m1_rmod_2(D,A,B)
=> ! [E] :
( m2_rmod_2(E,A,B,C)
=> ! [F] :
( m2_rmod_2(F,A,B,D)
=> ( ~ r1_xboole_0(E,F)
=> m2_rmod_2(k5_subset_1(u1_struct_0(B),E,F),A,B,k2_rmod_3(A,B,C,D)) ) ) ) ) ) ) ) ).
fof(t49_rmod_3,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v3_rlvect_1(A)
& v4_rlvect_1(A)
& v5_rlvect_1(A)
& v6_rlvect_1(A)
& v4_group_1(A)
& v6_vectsp_1(A)
& v7_vectsp_1(A)
& v8_vectsp_1(A)
& l3_vectsp_1(A) )
=> ! [B] :
( ( ~ v3_struct_0(B)
& v3_rlvect_1(B)
& v4_rlvect_1(B)
& v5_rlvect_1(B)
& v6_rlvect_1(B)
& v5_vectsp_2(B,A)
& l1_vectsp_2(B,A) )
=> ! [C] :
( m1_rmod_2(C,A,B)
=> ! [D] :
( m1_rmod_2(D,A,B)
=> ( r1_rmod_3(A,B,C,D)
<=> ! [E] :
( m2_rmod_2(E,A,B,C)
=> ! [F] :
( m2_rmod_2(F,A,B,D)
=> ? [G] :
( m1_subset_1(G,u1_struct_0(B))
& k5_subset_1(u1_struct_0(B),E,F) = k6_domain_1(u1_struct_0(B),G) ) ) ) ) ) ) ) ) ).
fof(t50_rmod_3,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v3_rlvect_1(A)
& v4_rlvect_1(A)
& v5_rlvect_1(A)
& v6_rlvect_1(A)
& v4_group_1(A)
& v6_vectsp_1(A)
& v7_vectsp_1(A)
& v8_vectsp_1(A)
& l3_vectsp_1(A) )
=> ! [B] :
( ( ~ v3_struct_0(B)
& v3_rlvect_1(B)
& v4_rlvect_1(B)
& v5_rlvect_1(B)
& v6_rlvect_1(B)
& v3_vectsp_2(B,A)
& v5_vectsp_2(B,A)
& l1_vectsp_2(B,A) )
=> ! [C] :
( m1_rmod_2(C,A,B)
=> ! [D] :
( m1_rmod_2(D,A,B)
=> ( k1_rmod_3(A,B,C,D) = B
<=> ! [E] :
( m1_subset_1(E,u1_struct_0(B))
=> ? [F] :
( m1_subset_1(F,u1_struct_0(B))
& ? [G] :
( m1_subset_1(G,u1_struct_0(B))
& r1_rlvect_1(C,F)
& r1_rlvect_1(D,G)
& E = k4_rlvect_1(B,F,G) ) ) ) ) ) ) ) ) ).
fof(t51_rmod_3,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v3_rlvect_1(A)
& v4_rlvect_1(A)
& v5_rlvect_1(A)
& v6_rlvect_1(A)
& v4_group_1(A)
& v6_vectsp_1(A)
& v7_vectsp_1(A)
& v8_vectsp_1(A)
& l3_vectsp_1(A) )
=> ! [B] :
( ( ~ v3_struct_0(B)
& v3_rlvect_1(B)
& v4_rlvect_1(B)
& v5_rlvect_1(B)
& v6_rlvect_1(B)
& v5_vectsp_2(B,A)
& l1_vectsp_2(B,A) )
=> ! [C] :
( m1_rmod_2(C,A,B)
=> ! [D] :
( m1_rmod_2(D,A,B)
=> ! [E] :
( m1_subset_1(E,u1_struct_0(B))
=> ! [F] :
( m1_subset_1(F,u1_struct_0(B))
=> ! [G] :
( m1_subset_1(G,u1_struct_0(B))
=> ! [H] :
( m1_subset_1(H,u1_struct_0(B))
=> ! [I] :
( m1_subset_1(I,u1_struct_0(B))
=> ( ( r1_rmod_3(A,B,C,D)
& E = k4_rlvect_1(B,F,G)
& E = k4_rlvect_1(B,H,I)
& r1_rlvect_1(C,F)
& r1_rlvect_1(C,H)
& r1_rlvect_1(D,G)
& r1_rlvect_1(D,I) )
=> ( F = H
& G = I ) ) ) ) ) ) ) ) ) ) ) ).
fof(t52_rmod_3,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v3_rlvect_1(A)
& v4_rlvect_1(A)
& v5_rlvect_1(A)
& v6_rlvect_1(A)
& v4_group_1(A)
& v6_vectsp_1(A)
& v7_vectsp_1(A)
& v8_vectsp_1(A)
& l3_vectsp_1(A) )
=> ! [B] :
( ( ~ v3_struct_0(B)
& v3_rlvect_1(B)
& v4_rlvect_1(B)
& v5_rlvect_1(B)
& v6_rlvect_1(B)
& v5_vectsp_2(B,A)
& l1_vectsp_2(B,A) )
=> ! [C] :
( m1_rmod_2(C,A,B)
=> ! [D] :
( m1_rmod_2(D,A,B)
=> ( B = k1_rmod_3(A,B,C,D)
=> ( ! [E] :
( m1_subset_1(E,u1_struct_0(B))
=> ? [F] :
( m1_subset_1(F,u1_struct_0(B))
& ? [G] :
( m1_subset_1(G,u1_struct_0(B))
& ? [H] :
( m1_subset_1(H,u1_struct_0(B))
& ? [I] :
( m1_subset_1(I,u1_struct_0(B))
& E = k4_rlvect_1(B,F,G)
& E = k4_rlvect_1(B,H,I)
& r1_rlvect_1(C,F)
& r1_rlvect_1(C,H)
& r1_rlvect_1(D,G)
& r1_rlvect_1(D,I)
& ~ ( F = H
& G = I ) ) ) ) ) )
| r1_rmod_3(A,B,C,D) ) ) ) ) ) ) ).
fof(d5_rmod_3,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v3_rlvect_1(A)
& v4_rlvect_1(A)
& v5_rlvect_1(A)
& v6_rlvect_1(A)
& v4_group_1(A)
& v6_vectsp_1(A)
& v7_vectsp_1(A)
& v8_vectsp_1(A)
& l3_vectsp_1(A) )
=> ! [B] :
( ( ~ v3_struct_0(B)
& v3_rlvect_1(B)
& v4_rlvect_1(B)
& v5_rlvect_1(B)
& v6_rlvect_1(B)
& v5_vectsp_2(B,A)
& l1_vectsp_2(B,A) )
=> ! [C] :
( m1_subset_1(C,u1_struct_0(B))
=> ! [D] :
( m1_rmod_2(D,A,B)
=> ! [E] :
( m1_rmod_2(E,A,B)
=> ( r1_rmod_3(A,B,D,E)
=> ! [F] :
( m1_subset_1(F,k2_zfmisc_1(u1_struct_0(B),u1_struct_0(B)))
=> ( F = k4_rmod_3(A,B,C,D,E)
<=> ( C = k4_rlvect_1(B,k2_domain_1(u1_struct_0(B),u1_struct_0(B),F),k3_domain_1(u1_struct_0(B),u1_struct_0(B),F))
& r1_rlvect_1(D,k2_domain_1(u1_struct_0(B),u1_struct_0(B),F))
& r1_rlvect_1(E,k3_domain_1(u1_struct_0(B),u1_struct_0(B),F)) ) ) ) ) ) ) ) ) ) ).
fof(t53_rmod_3,axiom,
$true ).
fof(t54_rmod_3,axiom,
$true ).
fof(t55_rmod_3,axiom,
$true ).
fof(t56_rmod_3,axiom,
$true ).
fof(t57_rmod_3,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v3_rlvect_1(A)
& v4_rlvect_1(A)
& v5_rlvect_1(A)
& v6_rlvect_1(A)
& v4_group_1(A)
& v6_vectsp_1(A)
& v7_vectsp_1(A)
& v8_vectsp_1(A)
& l3_vectsp_1(A) )
=> ! [B] :
( ( ~ v3_struct_0(B)
& v3_rlvect_1(B)
& v4_rlvect_1(B)
& v5_rlvect_1(B)
& v6_rlvect_1(B)
& v5_vectsp_2(B,A)
& l1_vectsp_2(B,A) )
=> ! [C] :
( m1_rmod_2(C,A,B)
=> ! [D] :
( m1_rmod_2(D,A,B)
=> ! [E] :
( m1_subset_1(E,u1_struct_0(B))
=> ( r1_rmod_3(A,B,C,D)
=> k2_domain_1(u1_struct_0(B),u1_struct_0(B),k4_rmod_3(A,B,E,C,D)) = k3_domain_1(u1_struct_0(B),u1_struct_0(B),k4_rmod_3(A,B,E,D,C)) ) ) ) ) ) ) ).
fof(t58_rmod_3,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v3_rlvect_1(A)
& v4_rlvect_1(A)
& v5_rlvect_1(A)
& v6_rlvect_1(A)
& v4_group_1(A)
& v6_vectsp_1(A)
& v7_vectsp_1(A)
& v8_vectsp_1(A)
& l3_vectsp_1(A) )
=> ! [B] :
( ( ~ v3_struct_0(B)
& v3_rlvect_1(B)
& v4_rlvect_1(B)
& v5_rlvect_1(B)
& v6_rlvect_1(B)
& v5_vectsp_2(B,A)
& l1_vectsp_2(B,A) )
=> ! [C] :
( m1_rmod_2(C,A,B)
=> ! [D] :
( m1_rmod_2(D,A,B)
=> ! [E] :
( m1_subset_1(E,u1_struct_0(B))
=> ( r1_rmod_3(A,B,C,D)
=> k3_domain_1(u1_struct_0(B),u1_struct_0(B),k4_rmod_3(A,B,E,C,D)) = k2_domain_1(u1_struct_0(B),u1_struct_0(B),k4_rmod_3(A,B,E,D,C)) ) ) ) ) ) ) ).
fof(d6_rmod_3,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v3_rlvect_1(A)
& v4_rlvect_1(A)
& v5_rlvect_1(A)
& v6_rlvect_1(A)
& v4_group_1(A)
& v6_vectsp_1(A)
& v7_vectsp_1(A)
& v8_vectsp_1(A)
& l3_vectsp_1(A) )
=> ! [B] :
( ( ~ v3_struct_0(B)
& v3_rlvect_1(B)
& v4_rlvect_1(B)
& v5_rlvect_1(B)
& v6_rlvect_1(B)
& v5_vectsp_2(B,A)
& l1_vectsp_2(B,A) )
=> ! [C] :
( ( v1_funct_1(C)
& v1_funct_2(C,k2_zfmisc_1(k3_rmod_3(A,B),k3_rmod_3(A,B)),k3_rmod_3(A,B))
& m2_relset_1(C,k2_zfmisc_1(k3_rmod_3(A,B),k3_rmod_3(A,B)),k3_rmod_3(A,B)) )
=> ( C = k5_rmod_3(A,B)
<=> ! [D] :
( m1_subset_1(D,k3_rmod_3(A,B))
=> ! [E] :
( m1_subset_1(E,k3_rmod_3(A,B))
=> ! [F] :
( m1_rmod_2(F,A,B)
=> ! [G] :
( m1_rmod_2(G,A,B)
=> ( ( D = F
& E = G )
=> k2_binop_1(k3_rmod_3(A,B),k3_rmod_3(A,B),k3_rmod_3(A,B),C,D,E) = k1_rmod_3(A,B,F,G) ) ) ) ) ) ) ) ) ) ).
fof(d7_rmod_3,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v3_rlvect_1(A)
& v4_rlvect_1(A)
& v5_rlvect_1(A)
& v6_rlvect_1(A)
& v4_group_1(A)
& v6_vectsp_1(A)
& v7_vectsp_1(A)
& v8_vectsp_1(A)
& l3_vectsp_1(A) )
=> ! [B] :
( ( ~ v3_struct_0(B)
& v3_rlvect_1(B)
& v4_rlvect_1(B)
& v5_rlvect_1(B)
& v6_rlvect_1(B)
& v5_vectsp_2(B,A)
& l1_vectsp_2(B,A) )
=> ! [C] :
( ( v1_funct_1(C)
& v1_funct_2(C,k2_zfmisc_1(k3_rmod_3(A,B),k3_rmod_3(A,B)),k3_rmod_3(A,B))
& m2_relset_1(C,k2_zfmisc_1(k3_rmod_3(A,B),k3_rmod_3(A,B)),k3_rmod_3(A,B)) )
=> ( C = k6_rmod_3(A,B)
<=> ! [D] :
( m1_subset_1(D,k3_rmod_3(A,B))
=> ! [E] :
( m1_subset_1(E,k3_rmod_3(A,B))
=> ! [F] :
( m1_rmod_2(F,A,B)
=> ! [G] :
( m1_rmod_2(G,A,B)
=> ( ( D = F
& E = G )
=> k2_binop_1(k3_rmod_3(A,B),k3_rmod_3(A,B),k3_rmod_3(A,B),C,D,E) = k2_rmod_3(A,B,F,G) ) ) ) ) ) ) ) ) ) ).
fof(t59_rmod_3,axiom,
$true ).
fof(t60_rmod_3,axiom,
$true ).
fof(t61_rmod_3,axiom,
$true ).
fof(t62_rmod_3,axiom,
$true ).
fof(t63_rmod_3,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v3_rlvect_1(A)
& v4_rlvect_1(A)
& v5_rlvect_1(A)
& v6_rlvect_1(A)
& v4_group_1(A)
& v6_vectsp_1(A)
& v7_vectsp_1(A)
& v8_vectsp_1(A)
& l3_vectsp_1(A) )
=> ! [B] :
( ( ~ v3_struct_0(B)
& v3_rlvect_1(B)
& v4_rlvect_1(B)
& v5_rlvect_1(B)
& v6_rlvect_1(B)
& v5_vectsp_2(B,A)
& l1_vectsp_2(B,A) )
=> ( ~ v3_struct_0(g3_lattices(k3_rmod_3(A,B),k5_rmod_3(A,B),k6_rmod_3(A,B)))
& v10_lattices(g3_lattices(k3_rmod_3(A,B),k5_rmod_3(A,B),k6_rmod_3(A,B)))
& l3_lattices(g3_lattices(k3_rmod_3(A,B),k5_rmod_3(A,B),k6_rmod_3(A,B))) ) ) ) ).
fof(t64_rmod_3,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v3_rlvect_1(A)
& v4_rlvect_1(A)
& v5_rlvect_1(A)
& v6_rlvect_1(A)
& v4_group_1(A)
& v6_vectsp_1(A)
& v7_vectsp_1(A)
& v8_vectsp_1(A)
& l3_vectsp_1(A) )
=> ! [B] :
( ( ~ v3_struct_0(B)
& v3_rlvect_1(B)
& v4_rlvect_1(B)
& v5_rlvect_1(B)
& v6_rlvect_1(B)
& v5_vectsp_2(B,A)
& l1_vectsp_2(B,A) )
=> ( ~ v3_struct_0(g3_lattices(k3_rmod_3(A,B),k5_rmod_3(A,B),k6_rmod_3(A,B)))
& v10_lattices(g3_lattices(k3_rmod_3(A,B),k5_rmod_3(A,B),k6_rmod_3(A,B)))
& v13_lattices(g3_lattices(k3_rmod_3(A,B),k5_rmod_3(A,B),k6_rmod_3(A,B)))
& l3_lattices(g3_lattices(k3_rmod_3(A,B),k5_rmod_3(A,B),k6_rmod_3(A,B))) ) ) ) ).
fof(t65_rmod_3,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v3_rlvect_1(A)
& v4_rlvect_1(A)
& v5_rlvect_1(A)
& v6_rlvect_1(A)
& v4_group_1(A)
& v6_vectsp_1(A)
& v7_vectsp_1(A)
& v8_vectsp_1(A)
& l3_vectsp_1(A) )
=> ! [B] :
( ( ~ v3_struct_0(B)
& v3_rlvect_1(B)
& v4_rlvect_1(B)
& v5_rlvect_1(B)
& v6_rlvect_1(B)
& v5_vectsp_2(B,A)
& l1_vectsp_2(B,A) )
=> ( ~ v3_struct_0(g3_lattices(k3_rmod_3(A,B),k5_rmod_3(A,B),k6_rmod_3(A,B)))
& v10_lattices(g3_lattices(k3_rmod_3(A,B),k5_rmod_3(A,B),k6_rmod_3(A,B)))
& v14_lattices(g3_lattices(k3_rmod_3(A,B),k5_rmod_3(A,B),k6_rmod_3(A,B)))
& l3_lattices(g3_lattices(k3_rmod_3(A,B),k5_rmod_3(A,B),k6_rmod_3(A,B))) ) ) ) ).
fof(t66_rmod_3,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v3_rlvect_1(A)
& v4_rlvect_1(A)
& v5_rlvect_1(A)
& v6_rlvect_1(A)
& v4_group_1(A)
& v6_vectsp_1(A)
& v7_vectsp_1(A)
& v8_vectsp_1(A)
& l3_vectsp_1(A) )
=> ! [B] :
( ( ~ v3_struct_0(B)
& v3_rlvect_1(B)
& v4_rlvect_1(B)
& v5_rlvect_1(B)
& v6_rlvect_1(B)
& v5_vectsp_2(B,A)
& l1_vectsp_2(B,A) )
=> ( ~ v3_struct_0(g3_lattices(k3_rmod_3(A,B),k5_rmod_3(A,B),k6_rmod_3(A,B)))
& v10_lattices(g3_lattices(k3_rmod_3(A,B),k5_rmod_3(A,B),k6_rmod_3(A,B)))
& v15_lattices(g3_lattices(k3_rmod_3(A,B),k5_rmod_3(A,B),k6_rmod_3(A,B)))
& l3_lattices(g3_lattices(k3_rmod_3(A,B),k5_rmod_3(A,B),k6_rmod_3(A,B))) ) ) ) ).
fof(t67_rmod_3,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v3_rlvect_1(A)
& v4_rlvect_1(A)
& v5_rlvect_1(A)
& v6_rlvect_1(A)
& v4_group_1(A)
& v6_vectsp_1(A)
& v7_vectsp_1(A)
& v8_vectsp_1(A)
& l3_vectsp_1(A) )
=> ! [B] :
( ( ~ v3_struct_0(B)
& v3_rlvect_1(B)
& v4_rlvect_1(B)
& v5_rlvect_1(B)
& v6_rlvect_1(B)
& v5_vectsp_2(B,A)
& l1_vectsp_2(B,A) )
=> ( ~ v3_struct_0(g3_lattices(k3_rmod_3(A,B),k5_rmod_3(A,B),k6_rmod_3(A,B)))
& v10_lattices(g3_lattices(k3_rmod_3(A,B),k5_rmod_3(A,B),k6_rmod_3(A,B)))
& v12_lattices(g3_lattices(k3_rmod_3(A,B),k5_rmod_3(A,B),k6_rmod_3(A,B)))
& l3_lattices(g3_lattices(k3_rmod_3(A,B),k5_rmod_3(A,B),k6_rmod_3(A,B))) ) ) ) ).
fof(dt_k1_rmod_3,axiom,
! [A,B,C,D] :
( ( ~ v3_struct_0(A)
& v3_rlvect_1(A)
& v4_rlvect_1(A)
& v5_rlvect_1(A)
& v6_rlvect_1(A)
& v4_group_1(A)
& v6_vectsp_1(A)
& v7_vectsp_1(A)
& v8_vectsp_1(A)
& l3_vectsp_1(A)
& ~ v3_struct_0(B)
& v3_rlvect_1(B)
& v4_rlvect_1(B)
& v5_rlvect_1(B)
& v6_rlvect_1(B)
& v5_vectsp_2(B,A)
& l1_vectsp_2(B,A)
& m1_rmod_2(C,A,B)
& m1_rmod_2(D,A,B) )
=> ( v3_vectsp_2(k1_rmod_3(A,B,C,D),A)
& m1_rmod_2(k1_rmod_3(A,B,C,D),A,B) ) ) ).
fof(dt_k2_rmod_3,axiom,
! [A,B,C,D] :
( ( ~ v3_struct_0(A)
& v3_rlvect_1(A)
& v4_rlvect_1(A)
& v5_rlvect_1(A)
& v6_rlvect_1(A)
& v4_group_1(A)
& v6_vectsp_1(A)
& v7_vectsp_1(A)
& v8_vectsp_1(A)
& l3_vectsp_1(A)
& ~ v3_struct_0(B)
& v3_rlvect_1(B)
& v4_rlvect_1(B)
& v5_rlvect_1(B)
& v6_rlvect_1(B)
& v5_vectsp_2(B,A)
& l1_vectsp_2(B,A)
& m1_rmod_2(C,A,B)
& m1_rmod_2(D,A,B) )
=> ( v3_vectsp_2(k2_rmod_3(A,B,C,D),A)
& m1_rmod_2(k2_rmod_3(A,B,C,D),A,B) ) ) ).
fof(dt_k3_rmod_3,axiom,
$true ).
fof(dt_k4_rmod_3,axiom,
! [A,B,C,D,E] :
( ( ~ v3_struct_0(A)
& v3_rlvect_1(A)
& v4_rlvect_1(A)
& v5_rlvect_1(A)
& v6_rlvect_1(A)
& v4_group_1(A)
& v6_vectsp_1(A)
& v7_vectsp_1(A)
& v8_vectsp_1(A)
& l3_vectsp_1(A)
& ~ v3_struct_0(B)
& v3_rlvect_1(B)
& v4_rlvect_1(B)
& v5_rlvect_1(B)
& v6_rlvect_1(B)
& v5_vectsp_2(B,A)
& l1_vectsp_2(B,A)
& m1_subset_1(C,u1_struct_0(B))
& m1_rmod_2(D,A,B)
& m1_rmod_2(E,A,B) )
=> m1_subset_1(k4_rmod_3(A,B,C,D,E),k2_zfmisc_1(u1_struct_0(B),u1_struct_0(B))) ) ).
fof(dt_k5_rmod_3,axiom,
! [A,B] :
( ( ~ v3_struct_0(A)
& v3_rlvect_1(A)
& v4_rlvect_1(A)
& v5_rlvect_1(A)
& v6_rlvect_1(A)
& v4_group_1(A)
& v6_vectsp_1(A)
& v7_vectsp_1(A)
& v8_vectsp_1(A)
& l3_vectsp_1(A)
& ~ v3_struct_0(B)
& v3_rlvect_1(B)
& v4_rlvect_1(B)
& v5_rlvect_1(B)
& v6_rlvect_1(B)
& v5_vectsp_2(B,A)
& l1_vectsp_2(B,A) )
=> ( v1_funct_1(k5_rmod_3(A,B))
& v1_funct_2(k5_rmod_3(A,B),k2_zfmisc_1(k3_rmod_3(A,B),k3_rmod_3(A,B)),k3_rmod_3(A,B))
& m2_relset_1(k5_rmod_3(A,B),k2_zfmisc_1(k3_rmod_3(A,B),k3_rmod_3(A,B)),k3_rmod_3(A,B)) ) ) ).
fof(dt_k6_rmod_3,axiom,
! [A,B] :
( ( ~ v3_struct_0(A)
& v3_rlvect_1(A)
& v4_rlvect_1(A)
& v5_rlvect_1(A)
& v6_rlvect_1(A)
& v4_group_1(A)
& v6_vectsp_1(A)
& v7_vectsp_1(A)
& v8_vectsp_1(A)
& l3_vectsp_1(A)
& ~ v3_struct_0(B)
& v3_rlvect_1(B)
& v4_rlvect_1(B)
& v5_rlvect_1(B)
& v6_rlvect_1(B)
& v5_vectsp_2(B,A)
& l1_vectsp_2(B,A) )
=> ( v1_funct_1(k6_rmod_3(A,B))
& v1_funct_2(k6_rmod_3(A,B),k2_zfmisc_1(k3_rmod_3(A,B),k3_rmod_3(A,B)),k3_rmod_3(A,B))
& m2_relset_1(k6_rmod_3(A,B),k2_zfmisc_1(k3_rmod_3(A,B),k3_rmod_3(A,B)),k3_rmod_3(A,B)) ) ) ).
fof(d1_rmod_3,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v3_rlvect_1(A)
& v4_rlvect_1(A)
& v5_rlvect_1(A)
& v6_rlvect_1(A)
& v4_group_1(A)
& v6_vectsp_1(A)
& v7_vectsp_1(A)
& v8_vectsp_1(A)
& l3_vectsp_1(A) )
=> ! [B] :
( ( ~ v3_struct_0(B)
& v3_rlvect_1(B)
& v4_rlvect_1(B)
& v5_rlvect_1(B)
& v6_rlvect_1(B)
& v5_vectsp_2(B,A)
& l1_vectsp_2(B,A) )
=> ! [C] :
( m1_rmod_2(C,A,B)
=> ! [D] :
( m1_rmod_2(D,A,B)
=> ! [E] :
( ( v3_vectsp_2(E,A)
& m1_rmod_2(E,A,B) )
=> ( E = k1_rmod_3(A,B,C,D)
<=> u1_struct_0(E) = a_4_0_rmod_3(A,B,C,D) ) ) ) ) ) ) ).
fof(fraenkel_a_4_0_rmod_3,axiom,
! [A,B,C,D,E] :
( ( ~ v3_struct_0(B)
& v3_rlvect_1(B)
& v4_rlvect_1(B)
& v5_rlvect_1(B)
& v6_rlvect_1(B)
& v4_group_1(B)
& v6_vectsp_1(B)
& v7_vectsp_1(B)
& v8_vectsp_1(B)
& l3_vectsp_1(B)
& ~ v3_struct_0(C)
& v3_rlvect_1(C)
& v4_rlvect_1(C)
& v5_rlvect_1(C)
& v6_rlvect_1(C)
& v5_vectsp_2(C,B)
& l1_vectsp_2(C,B)
& m1_rmod_2(D,B,C)
& m1_rmod_2(E,B,C) )
=> ( r2_hidden(A,a_4_0_rmod_3(B,C,D,E))
<=> ? [F,G] :
( m1_subset_1(F,u1_struct_0(C))
& m1_subset_1(G,u1_struct_0(C))
& A = k4_rlvect_1(C,F,G)
& r1_rlvect_1(D,F)
& r1_rlvect_1(E,G) ) ) ) ).
%------------------------------------------------------------------------------