SET007 Axioms: SET007+323.ax
%------------------------------------------------------------------------------
% File : SET007+323 : TPTP v8.2.0. Released v3.4.0.
% Domain : Set Theory
% Axioms : Category of Rings
% 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 : ringcat1 [Urb08]
% Status : Satisfiable
% Syntax : Number of formulae : 111 ( 9 unt; 0 def)
% Number of atoms : 1336 ( 93 equ)
% Maximal formula atoms : 48 ( 12 avg)
% Number of connectives : 1368 ( 143 ~; 0 |; 973 &)
% ( 25 <=>; 227 =>; 0 <=; 0 <~>)
% Maximal formula depth : 38 ( 11 avg)
% Maximal term depth : 4 ( 1 avg)
% Number of predicates : 38 ( 36 usr; 1 prp; 0-4 aty)
% Number of functors : 50 ( 50 usr; 1 con; 0-6 aty)
% Number of variables : 272 ( 243 !; 29 ?)
% 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(rc1_ringcat1,axiom,
? [A] :
( l1_ringcat1(A)
& v2_ringcat1(A) ) ).
fof(fc1_ringcat1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l3_vectsp_1(A) )
=> ( v1_relat_1(k7_grcat_1(A))
& v1_funct_1(k7_grcat_1(A))
& v1_funct_2(k7_grcat_1(A),u1_struct_0(A),u1_struct_0(A))
& v1_ringcat1(k7_grcat_1(A),A,A) ) ) ).
fof(rc2_ringcat1,axiom,
? [A] :
( l1_ringcat1(A)
& v2_ringcat1(A)
& v3_ringcat1(A) ) ).
fof(fc2_ringcat1,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) )
=> ( v2_ringcat1(k4_ringcat1(A))
& v3_ringcat1(k4_ringcat1(A)) ) ) ).
fof(rc3_ringcat1,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)
& v4_group_1(B)
& v6_vectsp_1(B)
& v7_vectsp_1(B)
& v8_vectsp_1(B)
& l3_vectsp_1(B) )
=> ? [C] :
( m1_ringcat1(C,A,B)
& v2_ringcat1(C)
& v3_ringcat1(C) ) ) ).
fof(rc4_ringcat1,axiom,
? [A] :
( ~ v1_xboole_0(A)
& v4_ringcat1(A) ) ).
fof(rc5_ringcat1,axiom,
! [A] :
( ( ~ v1_xboole_0(A)
& v4_ringcat1(A) )
=> ? [B] :
( m1_subset_1(B,A)
& ~ v3_struct_0(B)
& v3_rlvect_1(B)
& v4_rlvect_1(B)
& v5_rlvect_1(B)
& v6_rlvect_1(B)
& v4_group_1(B)
& v3_vectsp_1(B)
& v6_vectsp_1(B)
& v7_vectsp_1(B)
& v8_vectsp_1(B) ) ) ).
fof(rc6_ringcat1,axiom,
? [A] :
( ~ v1_xboole_0(A)
& v5_ringcat1(A) ) ).
fof(rc7_ringcat1,axiom,
! [A] :
( ( ~ v1_xboole_0(A)
& v5_ringcat1(A) )
=> ? [B] :
( m1_subset_1(B,A)
& v2_ringcat1(B)
& v3_ringcat1(B) ) ) ).
fof(rc8_ringcat1,axiom,
! [A,B,C] :
( ( ~ 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)
& v4_group_1(B)
& v6_vectsp_1(B)
& v7_vectsp_1(B)
& v8_vectsp_1(B)
& l3_vectsp_1(B)
& m4_ringcat1(C,A,B) )
=> ? [D] :
( m1_subset_1(D,C)
& v2_ringcat1(D)
& v3_ringcat1(D) ) ) ).
fof(fc3_ringcat1,axiom,
! [A] :
( ( ~ v1_xboole_0(A)
& v1_classes2(A) )
=> ~ v1_xboole_0(k9_ringcat1(A)) ) ).
fof(fc4_ringcat1,axiom,
! [A] :
( ( ~ v1_xboole_0(A)
& v1_classes2(A) )
=> ( ~ v1_xboole_0(k9_ringcat1(A))
& v4_ringcat1(k9_ringcat1(A)) ) ) ).
fof(fc5_ringcat1,axiom,
! [A] :
( ( ~ v1_xboole_0(A)
& v1_classes2(A) )
=> v1_cat_1(k18_ringcat1(A)) ) ).
fof(fc6_ringcat1,axiom,
! [A] :
( ( ~ v1_xboole_0(A)
& v1_classes2(A) )
=> ( v1_cat_1(k18_ringcat1(A))
& v2_cat_1(k18_ringcat1(A)) ) ) ).
fof(d1_ringcat1,axiom,
$true ).
fof(d2_ringcat1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l3_vectsp_1(A) )
=> ! [B] :
( ( ~ v3_struct_0(B)
& l3_vectsp_1(B) )
=> ! [C] :
( ( v1_funct_1(C)
& v1_funct_2(C,u1_struct_0(A),u1_struct_0(B))
& m2_relset_1(C,u1_struct_0(A),u1_struct_0(B)) )
=> ( v1_ringcat1(C,A,B)
<=> ( ! [D] :
( m1_subset_1(D,u1_struct_0(A))
=> ! [E] :
( m1_subset_1(E,u1_struct_0(A))
=> k8_funct_2(u1_struct_0(A),u1_struct_0(B),C,k2_rlvect_1(A,D,E)) = k2_rlvect_1(B,k8_funct_2(u1_struct_0(A),u1_struct_0(B),C,D),k8_funct_2(u1_struct_0(A),u1_struct_0(B),C,E)) ) )
& ! [D] :
( m1_subset_1(D,u1_struct_0(A))
=> ! [E] :
( m1_subset_1(E,u1_struct_0(A))
=> k8_funct_2(u1_struct_0(A),u1_struct_0(B),C,k1_group_1(A,D,E)) = k1_group_1(B,k8_funct_2(u1_struct_0(A),u1_struct_0(B),C,D),k8_funct_2(u1_struct_0(A),u1_struct_0(B),C,E)) ) )
& k8_funct_2(u1_struct_0(A),u1_struct_0(B),C,k2_group_1(A)) = k2_group_1(B) ) ) ) ) ) ).
fof(t1_ringcat1,axiom,
$true ).
fof(t2_ringcat1,axiom,
$true ).
fof(t3_ringcat1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l3_vectsp_1(A) )
=> ! [B] :
( ( ~ v3_struct_0(B)
& l3_vectsp_1(B) )
=> ! [C] :
( ( ~ v3_struct_0(C)
& l3_vectsp_1(C) )
=> ! [D] :
( ( v1_funct_1(D)
& v1_funct_2(D,u1_struct_0(A),u1_struct_0(B))
& m2_relset_1(D,u1_struct_0(A),u1_struct_0(B)) )
=> ! [E] :
( ( v1_funct_1(E)
& v1_funct_2(E,u1_struct_0(B),u1_struct_0(C))
& m2_relset_1(E,u1_struct_0(B),u1_struct_0(C)) )
=> ( ( v1_ringcat1(D,A,B)
& v1_ringcat1(E,B,C) )
=> v1_ringcat1(k7_funct_2(u1_struct_0(A),u1_struct_0(B),u1_struct_0(C),D,E),A,C) ) ) ) ) ) ) ).
fof(d3_ringcat1,axiom,
! [A] :
( l1_ringcat1(A)
=> k1_ringcat1(A) = u1_ringcat1(A) ) ).
fof(d4_ringcat1,axiom,
! [A] :
( l1_ringcat1(A)
=> k2_ringcat1(A) = u2_ringcat1(A) ) ).
fof(d5_ringcat1,axiom,
! [A] :
( l1_ringcat1(A)
=> k3_ringcat1(A) = u3_ringcat1(A) ) ).
fof(d6_ringcat1,axiom,
! [A] :
( l1_ringcat1(A)
=> ( v3_ringcat1(A)
<=> v1_ringcat1(k3_ringcat1(A),u1_ringcat1(A),u2_ringcat1(A)) ) ) ).
fof(d7_ringcat1,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) )
=> k4_ringcat1(A) = g1_ringcat1(A,A,k7_grcat_1(A)) ) ).
fof(d8_ringcat1,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)
& v4_group_1(B)
& v6_vectsp_1(B)
& v7_vectsp_1(B)
& v8_vectsp_1(B)
& l3_vectsp_1(B) )
=> ( r1_ringcat1(A,B)
<=> ? [C] :
( v3_ringcat1(C)
& l1_ringcat1(C)
& k1_ringcat1(C) = A
& k2_ringcat1(C) = B ) ) ) ) ).
fof(d9_ringcat1,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)
& v4_group_1(B)
& v6_vectsp_1(B)
& v7_vectsp_1(B)
& v8_vectsp_1(B)
& l3_vectsp_1(B) )
=> ( r1_ringcat1(A,B)
=> ! [C] :
( ( v2_ringcat1(C)
& v3_ringcat1(C)
& l1_ringcat1(C) )
=> ( m1_ringcat1(C,A,B)
<=> ( k1_ringcat1(C) = A
& k2_ringcat1(C) = B ) ) ) ) ) ) ).
fof(t4_ringcat1,axiom,
$true ).
fof(t5_ringcat1,axiom,
! [A] :
( ( v3_ringcat1(A)
& l1_ringcat1(A) )
=> ! [B] :
( ( v3_ringcat1(B)
& l1_ringcat1(B) )
=> ~ ( k1_ringcat1(A) = k2_ringcat1(B)
& ! [C] :
( ( ~ v3_struct_0(C)
& v3_rlvect_1(C)
& v4_rlvect_1(C)
& v5_rlvect_1(C)
& v6_rlvect_1(C)
& v4_group_1(C)
& v6_vectsp_1(C)
& v7_vectsp_1(C)
& v8_vectsp_1(C)
& l3_vectsp_1(C) )
=> ! [D] :
( ( ~ v3_struct_0(D)
& v3_rlvect_1(D)
& v4_rlvect_1(D)
& v5_rlvect_1(D)
& v6_rlvect_1(D)
& v4_group_1(D)
& v6_vectsp_1(D)
& v7_vectsp_1(D)
& v8_vectsp_1(D)
& l3_vectsp_1(D) )
=> ! [E] :
( ( ~ v3_struct_0(E)
& v3_rlvect_1(E)
& v4_rlvect_1(E)
& v5_rlvect_1(E)
& v6_rlvect_1(E)
& v4_group_1(E)
& v6_vectsp_1(E)
& v7_vectsp_1(E)
& v8_vectsp_1(E)
& l3_vectsp_1(E) )
=> ~ ( r1_ringcat1(C,D)
& r1_ringcat1(D,E)
& m1_ringcat1(g1_ringcat1(u1_ringcat1(A),u2_ringcat1(A),u3_ringcat1(A)),D,E)
& m1_ringcat1(g1_ringcat1(u1_ringcat1(B),u2_ringcat1(B),u3_ringcat1(B)),C,D) ) ) ) ) ) ) ) ).
fof(t6_ringcat1,axiom,
! [A] :
( ( v2_ringcat1(A)
& v3_ringcat1(A)
& l1_ringcat1(A) )
=> ( m1_ringcat1(A,k1_ringcat1(A),k2_ringcat1(A))
& r1_ringcat1(k1_ringcat1(A),k2_ringcat1(A)) ) ) ).
fof(t7_ringcat1,axiom,
! [A] :
( ( v2_ringcat1(A)
& v3_ringcat1(A)
& l1_ringcat1(A) )
=> ? [B] :
( ~ 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)
& ? [C] :
( ~ v3_struct_0(C)
& v3_rlvect_1(C)
& v4_rlvect_1(C)
& v5_rlvect_1(C)
& v6_rlvect_1(C)
& v4_group_1(C)
& v6_vectsp_1(C)
& v7_vectsp_1(C)
& v8_vectsp_1(C)
& l3_vectsp_1(C)
& ? [D] :
( v1_funct_1(D)
& v1_funct_2(D,u1_struct_0(B),u1_struct_0(C))
& m2_relset_1(D,u1_struct_0(B),u1_struct_0(C))
& m1_ringcat1(A,B,C)
& A = g1_ringcat1(B,C,D)
& v1_ringcat1(D,B,C) ) ) ) ) ).
fof(d10_ringcat1,axiom,
! [A] :
( ( v3_ringcat1(A)
& l1_ringcat1(A) )
=> ! [B] :
( ( v3_ringcat1(B)
& l1_ringcat1(B) )
=> ( k1_ringcat1(A) = k2_ringcat1(B)
=> ! [C] :
( ( v2_ringcat1(C)
& v3_ringcat1(C)
& l1_ringcat1(C) )
=> ( C = k6_ringcat1(A,B)
<=> ! [D] :
( ( ~ v3_struct_0(D)
& v3_rlvect_1(D)
& v4_rlvect_1(D)
& v5_rlvect_1(D)
& v6_rlvect_1(D)
& v4_group_1(D)
& v6_vectsp_1(D)
& v7_vectsp_1(D)
& v8_vectsp_1(D)
& l3_vectsp_1(D) )
=> ! [E] :
( ( ~ v3_struct_0(E)
& v3_rlvect_1(E)
& v4_rlvect_1(E)
& v5_rlvect_1(E)
& v6_rlvect_1(E)
& v4_group_1(E)
& v6_vectsp_1(E)
& v7_vectsp_1(E)
& v8_vectsp_1(E)
& l3_vectsp_1(E) )
=> ! [F] :
( ( ~ v3_struct_0(F)
& v3_rlvect_1(F)
& v4_rlvect_1(F)
& v5_rlvect_1(F)
& v6_rlvect_1(F)
& v4_group_1(F)
& v6_vectsp_1(F)
& v7_vectsp_1(F)
& v8_vectsp_1(F)
& l3_vectsp_1(F) )
=> ! [G] :
( ( v1_funct_1(G)
& v1_funct_2(G,u1_struct_0(E),u1_struct_0(F))
& m2_relset_1(G,u1_struct_0(E),u1_struct_0(F)) )
=> ! [H] :
( ( v1_funct_1(H)
& v1_funct_2(H,u1_struct_0(D),u1_struct_0(E))
& m2_relset_1(H,u1_struct_0(D),u1_struct_0(E)) )
=> ( ( g1_ringcat1(u1_ringcat1(A),u2_ringcat1(A),u3_ringcat1(A)) = g1_ringcat1(E,F,G)
& g1_ringcat1(u1_ringcat1(B),u2_ringcat1(B),u3_ringcat1(B)) = g1_ringcat1(D,E,H) )
=> C = g1_ringcat1(D,F,k7_funct_2(u1_struct_0(D),u1_struct_0(E),u1_struct_0(F),H,G)) ) ) ) ) ) ) ) ) ) ) ) ).
fof(t8_ringcat1,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)
& v4_group_1(B)
& v6_vectsp_1(B)
& v7_vectsp_1(B)
& v8_vectsp_1(B)
& l3_vectsp_1(B) )
=> ! [C] :
( ( ~ v3_struct_0(C)
& v3_rlvect_1(C)
& v4_rlvect_1(C)
& v5_rlvect_1(C)
& v6_rlvect_1(C)
& v4_group_1(C)
& v6_vectsp_1(C)
& v7_vectsp_1(C)
& v8_vectsp_1(C)
& l3_vectsp_1(C) )
=> ( ( r1_ringcat1(A,B)
& r1_ringcat1(B,C) )
=> r1_ringcat1(A,C) ) ) ) ) ).
fof(t9_ringcat1,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)
& v4_group_1(B)
& v6_vectsp_1(B)
& v7_vectsp_1(B)
& v8_vectsp_1(B)
& l3_vectsp_1(B) )
=> ! [C] :
( ( ~ v3_struct_0(C)
& v3_rlvect_1(C)
& v4_rlvect_1(C)
& v5_rlvect_1(C)
& v6_rlvect_1(C)
& v4_group_1(C)
& v6_vectsp_1(C)
& v7_vectsp_1(C)
& v8_vectsp_1(C)
& l3_vectsp_1(C) )
=> ! [D] :
( m1_ringcat1(D,A,B)
=> ! [E] :
( m1_ringcat1(E,C,A)
=> ( ( r1_ringcat1(C,A)
& r1_ringcat1(A,B) )
=> m1_ringcat1(k6_ringcat1(D,E),C,B) ) ) ) ) ) ) ).
fof(d11_ringcat1,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)
& v4_group_1(B)
& v6_vectsp_1(B)
& v7_vectsp_1(B)
& v8_vectsp_1(B)
& l3_vectsp_1(B) )
=> ! [C] :
( ( ~ v3_struct_0(C)
& v3_rlvect_1(C)
& v4_rlvect_1(C)
& v5_rlvect_1(C)
& v6_rlvect_1(C)
& v4_group_1(C)
& v6_vectsp_1(C)
& v7_vectsp_1(C)
& v8_vectsp_1(C)
& l3_vectsp_1(C) )
=> ! [D] :
( m1_ringcat1(D,B,C)
=> ! [E] :
( m1_ringcat1(E,A,B)
=> ( ( r1_ringcat1(A,B)
& r1_ringcat1(B,C) )
=> k7_ringcat1(A,B,C,D,E) = k6_ringcat1(D,E) ) ) ) ) ) ) ).
fof(t10_ringcat1,axiom,
! [A] :
( ( v2_ringcat1(A)
& v3_ringcat1(A)
& l1_ringcat1(A) )
=> ! [B] :
( ( v2_ringcat1(B)
& v3_ringcat1(B)
& l1_ringcat1(B) )
=> ~ ( k1_ringcat1(B) = k2_ringcat1(A)
& ! [C] :
( ( ~ v3_struct_0(C)
& v3_rlvect_1(C)
& v4_rlvect_1(C)
& v5_rlvect_1(C)
& v6_rlvect_1(C)
& v4_group_1(C)
& v6_vectsp_1(C)
& v7_vectsp_1(C)
& v8_vectsp_1(C)
& l3_vectsp_1(C) )
=> ! [D] :
( ( ~ v3_struct_0(D)
& v3_rlvect_1(D)
& v4_rlvect_1(D)
& v5_rlvect_1(D)
& v6_rlvect_1(D)
& v4_group_1(D)
& v6_vectsp_1(D)
& v7_vectsp_1(D)
& v8_vectsp_1(D)
& l3_vectsp_1(D) )
=> ! [E] :
( ( ~ v3_struct_0(E)
& v3_rlvect_1(E)
& v4_rlvect_1(E)
& v5_rlvect_1(E)
& v6_rlvect_1(E)
& v4_group_1(E)
& v6_vectsp_1(E)
& v7_vectsp_1(E)
& v8_vectsp_1(E)
& l3_vectsp_1(E) )
=> ! [F] :
( ( v1_funct_1(F)
& v1_funct_2(F,u1_struct_0(C),u1_struct_0(D))
& m2_relset_1(F,u1_struct_0(C),u1_struct_0(D)) )
=> ! [G] :
( ( v1_funct_1(G)
& v1_funct_2(G,u1_struct_0(D),u1_struct_0(E))
& m2_relset_1(G,u1_struct_0(D),u1_struct_0(E)) )
=> ~ ( A = g1_ringcat1(C,D,F)
& B = g1_ringcat1(D,E,G)
& k6_ringcat1(B,A) = g1_ringcat1(C,E,k7_funct_2(u1_struct_0(C),u1_struct_0(D),u1_struct_0(E),F,G)) ) ) ) ) ) ) ) ) ) ).
fof(t11_ringcat1,axiom,
! [A] :
( ( v2_ringcat1(A)
& v3_ringcat1(A)
& l1_ringcat1(A) )
=> ! [B] :
( ( v2_ringcat1(B)
& v3_ringcat1(B)
& l1_ringcat1(B) )
=> ( k1_ringcat1(B) = k2_ringcat1(A)
=> ( k1_ringcat1(k6_ringcat1(B,A)) = k1_ringcat1(A)
& k2_ringcat1(k6_ringcat1(B,A)) = k2_ringcat1(B) ) ) ) ) ).
fof(t12_ringcat1,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)
& v4_group_1(B)
& v6_vectsp_1(B)
& v7_vectsp_1(B)
& v8_vectsp_1(B)
& l3_vectsp_1(B) )
=> ! [C] :
( ( ~ v3_struct_0(C)
& v3_rlvect_1(C)
& v4_rlvect_1(C)
& v5_rlvect_1(C)
& v6_rlvect_1(C)
& v4_group_1(C)
& v6_vectsp_1(C)
& v7_vectsp_1(C)
& v8_vectsp_1(C)
& l3_vectsp_1(C) )
=> ! [D] :
( ( ~ v3_struct_0(D)
& v3_rlvect_1(D)
& v4_rlvect_1(D)
& v5_rlvect_1(D)
& v6_rlvect_1(D)
& v4_group_1(D)
& v6_vectsp_1(D)
& v7_vectsp_1(D)
& v8_vectsp_1(D)
& l3_vectsp_1(D) )
=> ! [E] :
( m1_ringcat1(E,A,B)
=> ! [F] :
( m1_ringcat1(F,B,C)
=> ! [G] :
( m1_ringcat1(G,C,D)
=> ( ( r1_ringcat1(A,B)
& r1_ringcat1(B,C)
& r1_ringcat1(C,D) )
=> k6_ringcat1(G,k6_ringcat1(F,E)) = k6_ringcat1(k6_ringcat1(G,F),E) ) ) ) ) ) ) ) ) ).
fof(t13_ringcat1,axiom,
! [A] :
( ( v2_ringcat1(A)
& v3_ringcat1(A)
& l1_ringcat1(A) )
=> ! [B] :
( ( v2_ringcat1(B)
& v3_ringcat1(B)
& l1_ringcat1(B) )
=> ! [C] :
( ( v2_ringcat1(C)
& v3_ringcat1(C)
& l1_ringcat1(C) )
=> ( ( k1_ringcat1(C) = k2_ringcat1(B)
& k1_ringcat1(B) = k2_ringcat1(A) )
=> k6_ringcat1(C,k6_ringcat1(B,A)) = k6_ringcat1(k6_ringcat1(C,B),A) ) ) ) ) ).
fof(t14_ringcat1,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) )
=> ( k1_ringcat1(k5_ringcat1(A)) = A
& k2_ringcat1(k5_ringcat1(A)) = A
& ! [B] :
( ( v2_ringcat1(B)
& v3_ringcat1(B)
& l1_ringcat1(B) )
=> ( k2_ringcat1(B) = A
=> k6_ringcat1(k5_ringcat1(A),B) = B ) )
& ! [B] :
( ( v2_ringcat1(B)
& v3_ringcat1(B)
& l1_ringcat1(B) )
=> ( k1_ringcat1(B) = A
=> k6_ringcat1(B,k5_ringcat1(A)) = B ) ) ) ) ).
fof(d12_ringcat1,axiom,
! [A] :
( v4_ringcat1(A)
<=> ! [B] :
( m1_subset_1(B,A)
=> ( ~ v3_struct_0(B)
& v3_rlvect_1(B)
& v4_rlvect_1(B)
& v5_rlvect_1(B)
& v6_rlvect_1(B)
& v4_group_1(B)
& v3_vectsp_1(B)
& v6_vectsp_1(B)
& v7_vectsp_1(B)
& v8_vectsp_1(B)
& l3_vectsp_1(B) ) ) ) ).
fof(d13_ringcat1,axiom,
! [A] :
( v5_ringcat1(A)
<=> ! [B] :
( r2_hidden(B,A)
=> ( v2_ringcat1(B)
& v3_ringcat1(B)
& l1_ringcat1(B) ) ) ) ).
fof(t15_ringcat1,axiom,
$true ).
fof(t16_ringcat1,axiom,
$true ).
fof(t17_ringcat1,axiom,
! [A] :
( ( v2_ringcat1(A)
& v3_ringcat1(A)
& l1_ringcat1(A) )
=> ( ~ v1_xboole_0(k1_tarski(A))
& v5_ringcat1(k1_tarski(A)) ) ) ).
fof(d14_ringcat1,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)
& v4_group_1(B)
& v6_vectsp_1(B)
& v7_vectsp_1(B)
& v8_vectsp_1(B)
& l3_vectsp_1(B) )
=> ! [C] :
( ( ~ v1_xboole_0(C)
& v5_ringcat1(C) )
=> ( m4_ringcat1(C,A,B)
<=> ! [D] :
( m3_ringcat1(D,C)
=> m1_ringcat1(D,A,B) ) ) ) ) ) ).
fof(t18_ringcat1,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( ( ~ 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) )
=> ! [C] :
( ( ~ v3_struct_0(C)
& v3_rlvect_1(C)
& v4_rlvect_1(C)
& v5_rlvect_1(C)
& v6_rlvect_1(C)
& v4_group_1(C)
& v6_vectsp_1(C)
& v7_vectsp_1(C)
& v8_vectsp_1(C)
& l3_vectsp_1(C) )
=> ( m4_ringcat1(A,B,C)
<=> ! [D] :
( m1_subset_1(D,A)
=> m1_ringcat1(D,B,C) ) ) ) ) ) ).
fof(t19_ringcat1,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)
& v4_group_1(B)
& v6_vectsp_1(B)
& v7_vectsp_1(B)
& v8_vectsp_1(B)
& l3_vectsp_1(B) )
=> ! [C] :
( m1_ringcat1(C,A,B)
=> m4_ringcat1(k1_tarski(C),A,B) ) ) ) ).
fof(d15_ringcat1,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)
& v4_group_1(B)
& v6_vectsp_1(B)
& v7_vectsp_1(B)
& v8_vectsp_1(B)
& l3_vectsp_1(B) )
=> ( r1_ringcat1(A,B)
=> ! [C] :
( m4_ringcat1(C,A,B)
=> ( C = k8_ringcat1(A,B)
<=> ! [D] :
( r2_hidden(D,C)
<=> m1_ringcat1(D,A,B) ) ) ) ) ) ) ).
fof(d16_ringcat1,axiom,
! [A,B] :
( r2_ringcat1(A,B)
<=> ? [C,D,E,F,G,H] :
( A = k3_mcart_1(k4_mcart_1(C,D,E,F),G,H)
& ? [I] :
( ~ v3_struct_0(I)
& v3_rlvect_1(I)
& v4_rlvect_1(I)
& v5_rlvect_1(I)
& v6_rlvect_1(I)
& v4_group_1(I)
& v3_vectsp_1(I)
& v6_vectsp_1(I)
& v7_vectsp_1(I)
& v8_vectsp_1(I)
& l3_vectsp_1(I)
& B = I
& C = u1_struct_0(I)
& D = u1_rlvect_1(I)
& E = k7_vectsp_1(I)
& F = u2_struct_0(I)
& G = u1_group_1(I)
& H = u1_vectsp_1(I) ) ) ) ).
fof(t20_ringcat1,axiom,
! [A,B,C] :
( ( r2_ringcat1(A,B)
& r2_ringcat1(A,C) )
=> B = C ) ).
fof(t21_ringcat1,axiom,
! [A] :
( ( ~ v1_xboole_0(A)
& v1_classes2(A) )
=> ? [B] :
( r2_hidden(B,A)
& r2_ringcat1(B,k18_mod_2) ) ) ).
fof(d17_ringcat1,axiom,
! [A] :
( ( ~ v1_xboole_0(A)
& v1_classes2(A) )
=> ! [B] :
( B = k9_ringcat1(A)
<=> ! [C] :
( r2_hidden(C,B)
<=> ? [D] :
( r2_hidden(D,A)
& r2_ringcat1(D,C) ) ) ) ) ).
fof(t22_ringcat1,axiom,
! [A] :
( ( ~ v1_xboole_0(A)
& v1_classes2(A) )
=> r2_hidden(k18_mod_2,k9_ringcat1(A)) ) ).
fof(t23_ringcat1,axiom,
! [A] :
( ( ~ v1_xboole_0(A)
& v1_classes2(A) )
=> ! [B] :
( m1_subset_1(B,k9_ringcat1(A))
=> ( ~ v3_struct_0(B)
& v3_rlvect_1(B)
& v4_rlvect_1(B)
& v5_rlvect_1(B)
& v6_rlvect_1(B)
& v4_group_1(B)
& v3_vectsp_1(B)
& v6_vectsp_1(B)
& v7_vectsp_1(B)
& v8_vectsp_1(B)
& l3_vectsp_1(B) ) ) ) ).
fof(d18_ringcat1,axiom,
! [A] :
( ( ~ v1_xboole_0(A)
& v4_ringcat1(A) )
=> ! [B] :
( ( ~ v1_xboole_0(B)
& v5_ringcat1(B) )
=> ( B = k10_ringcat1(A)
<=> ! [C] :
( r2_hidden(C,B)
<=> ? [D] :
( m2_ringcat1(D,A)
& ? [E] :
( m2_ringcat1(E,A)
& r1_ringcat1(D,E)
& m1_ringcat1(C,D,E) ) ) ) ) ) ) ).
fof(d19_ringcat1,axiom,
! [A] :
( ( ~ v1_xboole_0(A)
& v4_ringcat1(A) )
=> ! [B] :
( m2_ringcat1(B,A)
=> k13_ringcat1(A,B) = k5_ringcat1(B) ) ) ).
fof(d20_ringcat1,axiom,
! [A] :
( ( ~ v1_xboole_0(A)
& v4_ringcat1(A) )
=> ! [B] :
( ( v1_funct_1(B)
& v1_funct_2(B,k10_ringcat1(A),A)
& m2_relset_1(B,k10_ringcat1(A),A) )
=> ( B = k14_ringcat1(A)
<=> ! [C] :
( m3_ringcat1(C,k10_ringcat1(A))
=> k8_funct_2(k10_ringcat1(A),A,B,C) = k11_ringcat1(A,C) ) ) ) ) ).
fof(d21_ringcat1,axiom,
! [A] :
( ( ~ v1_xboole_0(A)
& v4_ringcat1(A) )
=> ! [B] :
( ( v1_funct_1(B)
& v1_funct_2(B,k10_ringcat1(A),A)
& m2_relset_1(B,k10_ringcat1(A),A) )
=> ( B = k15_ringcat1(A)
<=> ! [C] :
( m3_ringcat1(C,k10_ringcat1(A))
=> k8_funct_2(k10_ringcat1(A),A,B,C) = k12_ringcat1(A,C) ) ) ) ) ).
fof(d22_ringcat1,axiom,
! [A] :
( ( ~ v1_xboole_0(A)
& v4_ringcat1(A) )
=> ! [B] :
( ( v1_funct_1(B)
& v1_funct_2(B,A,k10_ringcat1(A))
& m2_relset_1(B,A,k10_ringcat1(A)) )
=> ( B = k16_ringcat1(A)
<=> ! [C] :
( m2_ringcat1(C,A)
=> k8_funct_2(A,k10_ringcat1(A),B,C) = k13_ringcat1(A,C) ) ) ) ) ).
fof(t24_ringcat1,axiom,
! [A] :
( ( ~ v1_xboole_0(A)
& v4_ringcat1(A) )
=> ! [B] :
( m3_ringcat1(B,k10_ringcat1(A))
=> ! [C] :
( m3_ringcat1(C,k10_ringcat1(A))
=> ~ ( k11_ringcat1(A,B) = k12_ringcat1(A,C)
& ! [D] :
( m2_ringcat1(D,A)
=> ! [E] :
( m2_ringcat1(E,A)
=> ! [F] :
( m2_ringcat1(F,A)
=> ~ ( r1_ringcat1(D,E)
& r1_ringcat1(E,F)
& m1_ringcat1(B,E,F)
& m1_ringcat1(C,D,E) ) ) ) ) ) ) ) ) ).
fof(t25_ringcat1,axiom,
! [A] :
( ( ~ v1_xboole_0(A)
& v4_ringcat1(A) )
=> ! [B] :
( m3_ringcat1(B,k10_ringcat1(A))
=> ! [C] :
( m3_ringcat1(C,k10_ringcat1(A))
=> ( k11_ringcat1(A,B) = k12_ringcat1(A,C)
=> r2_hidden(k6_ringcat1(B,C),k10_ringcat1(A)) ) ) ) ) ).
fof(d23_ringcat1,axiom,
! [A] :
( ( ~ v1_xboole_0(A)
& v4_ringcat1(A) )
=> ! [B] :
( ( v1_funct_1(B)
& m2_relset_1(B,k2_zfmisc_1(k10_ringcat1(A),k10_ringcat1(A)),k10_ringcat1(A)) )
=> ( B = k17_ringcat1(A)
<=> ( ! [C] :
( m3_ringcat1(C,k10_ringcat1(A))
=> ! [D] :
( m3_ringcat1(D,k10_ringcat1(A))
=> ( r2_hidden(k4_tarski(C,D),k1_relat_1(B))
<=> k11_ringcat1(A,C) = k12_ringcat1(A,D) ) ) )
& ! [C] :
( m3_ringcat1(C,k10_ringcat1(A))
=> ! [D] :
( m3_ringcat1(D,k10_ringcat1(A))
=> ( r2_hidden(k4_tarski(C,D),k1_relat_1(B))
=> k1_funct_1(B,k4_tarski(C,D)) = k6_ringcat1(C,D) ) ) ) ) ) ) ) ).
fof(d24_ringcat1,axiom,
! [A] :
( ( ~ v1_xboole_0(A)
& v1_classes2(A) )
=> k18_ringcat1(A) = g1_cat_1(k9_ringcat1(A),k10_ringcat1(k9_ringcat1(A)),k14_ringcat1(k9_ringcat1(A)),k15_ringcat1(k9_ringcat1(A)),k17_ringcat1(k9_ringcat1(A)),k16_ringcat1(k9_ringcat1(A))) ) ).
fof(t26_ringcat1,axiom,
! [A] :
( ( ~ v1_xboole_0(A)
& v1_classes2(A) )
=> ! [B] :
( m1_subset_1(B,u2_cat_1(k18_ringcat1(A)))
=> ! [C] :
( m1_subset_1(C,u2_cat_1(k18_ringcat1(A)))
=> ( r2_hidden(k4_tarski(C,B),k1_relat_1(u5_cat_1(k18_ringcat1(A))))
<=> k2_cat_1(k18_ringcat1(A),C) = k3_cat_1(k18_ringcat1(A),B) ) ) ) ) ).
fof(t27_ringcat1,axiom,
! [A] :
( ( ~ v1_xboole_0(A)
& v1_classes2(A) )
=> ! [B] :
( m1_subset_1(B,u2_cat_1(k18_ringcat1(A)))
=> ! [C] :
( m3_ringcat1(C,k10_ringcat1(k9_ringcat1(A)))
=> ! [D] :
( m1_subset_1(D,u1_cat_1(k18_ringcat1(A)))
=> ! [E] :
( m2_ringcat1(E,k9_ringcat1(A))
=> ( v2_ringcat1(B)
& m3_ringcat1(B,k10_ringcat1(k9_ringcat1(A)))
& m1_subset_1(C,u2_cat_1(k18_ringcat1(A)))
& v3_vectsp_1(D)
& m2_ringcat1(D,k9_ringcat1(A))
& m1_subset_1(E,u1_cat_1(k18_ringcat1(A))) ) ) ) ) ) ) ).
fof(t28_ringcat1,axiom,
! [A] :
( ( ~ v1_xboole_0(A)
& v1_classes2(A) )
=> ! [B] :
( m1_subset_1(B,u1_cat_1(k18_ringcat1(A)))
=> ! [C] :
( m2_ringcat1(C,k9_ringcat1(A))
=> ( B = C
=> k5_cat_1(k18_ringcat1(A),B) = k13_ringcat1(k9_ringcat1(A),C) ) ) ) ) ).
fof(t29_ringcat1,axiom,
! [A] :
( ( ~ v1_xboole_0(A)
& v1_classes2(A) )
=> ! [B] :
( m1_subset_1(B,u2_cat_1(k18_ringcat1(A)))
=> ! [C] :
( m3_ringcat1(C,k10_ringcat1(k9_ringcat1(A)))
=> ( B = C
=> ( k2_cat_1(k18_ringcat1(A),B) = k11_ringcat1(k9_ringcat1(A),C)
& k3_cat_1(k18_ringcat1(A),B) = k12_ringcat1(k9_ringcat1(A),C) ) ) ) ) ) ).
fof(t30_ringcat1,axiom,
! [A] :
( ( ~ v1_xboole_0(A)
& v1_classes2(A) )
=> ! [B] :
( m1_subset_1(B,u2_cat_1(k18_ringcat1(A)))
=> ! [C] :
( m1_subset_1(C,u2_cat_1(k18_ringcat1(A)))
=> ! [D] :
( m3_ringcat1(D,k10_ringcat1(k9_ringcat1(A)))
=> ! [E] :
( m3_ringcat1(E,k10_ringcat1(k9_ringcat1(A)))
=> ( ( B = D
& C = E )
=> ( ( k2_cat_1(k18_ringcat1(A),C) = k3_cat_1(k18_ringcat1(A),B)
=> k11_ringcat1(k9_ringcat1(A),E) = k12_ringcat1(k9_ringcat1(A),D) )
& ( k11_ringcat1(k9_ringcat1(A),E) = k12_ringcat1(k9_ringcat1(A),D)
=> k2_cat_1(k18_ringcat1(A),C) = k3_cat_1(k18_ringcat1(A),B) )
& ( k2_cat_1(k18_ringcat1(A),C) = k3_cat_1(k18_ringcat1(A),B)
=> r2_hidden(k4_tarski(E,D),k1_relat_1(k17_ringcat1(k9_ringcat1(A)))) )
& ( r2_hidden(k4_tarski(E,D),k1_relat_1(k17_ringcat1(k9_ringcat1(A))))
=> k2_cat_1(k18_ringcat1(A),C) = k3_cat_1(k18_ringcat1(A),B) )
& ( k2_cat_1(k18_ringcat1(A),C) = k3_cat_1(k18_ringcat1(A),B)
=> k4_cat_1(k18_ringcat1(A),B,C) = k6_ringcat1(E,D) )
& ( k2_cat_1(k18_ringcat1(A),B) = k2_cat_1(k18_ringcat1(A),C)
=> k11_ringcat1(k9_ringcat1(A),D) = k11_ringcat1(k9_ringcat1(A),E) )
& ( k11_ringcat1(k9_ringcat1(A),D) = k11_ringcat1(k9_ringcat1(A),E)
=> k2_cat_1(k18_ringcat1(A),B) = k2_cat_1(k18_ringcat1(A),C) )
& ( k3_cat_1(k18_ringcat1(A),B) = k3_cat_1(k18_ringcat1(A),C)
=> k12_ringcat1(k9_ringcat1(A),D) = k12_ringcat1(k9_ringcat1(A),E) )
& ( k12_ringcat1(k9_ringcat1(A),D) = k12_ringcat1(k9_ringcat1(A),E)
=> k3_cat_1(k18_ringcat1(A),B) = k3_cat_1(k18_ringcat1(A),C) ) ) ) ) ) ) ) ) ).
fof(dt_m1_ringcat1,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)
& v4_group_1(B)
& v6_vectsp_1(B)
& v7_vectsp_1(B)
& v8_vectsp_1(B)
& l3_vectsp_1(B) )
=> ! [C] :
( m1_ringcat1(C,A,B)
=> ( v2_ringcat1(C)
& v3_ringcat1(C)
& l1_ringcat1(C) ) ) ) ).
fof(existence_m1_ringcat1,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)
& v4_group_1(B)
& v6_vectsp_1(B)
& v7_vectsp_1(B)
& v8_vectsp_1(B)
& l3_vectsp_1(B) )
=> ? [C] : m1_ringcat1(C,A,B) ) ).
fof(dt_m2_ringcat1,axiom,
! [A] :
( ( ~ v1_xboole_0(A)
& v4_ringcat1(A) )
=> ! [B] :
( m2_ringcat1(B,A)
=> ( ~ 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) ) ) ) ).
fof(existence_m2_ringcat1,axiom,
! [A] :
( ( ~ v1_xboole_0(A)
& v4_ringcat1(A) )
=> ? [B] : m2_ringcat1(B,A) ) ).
fof(redefinition_m2_ringcat1,axiom,
! [A] :
( ( ~ v1_xboole_0(A)
& v4_ringcat1(A) )
=> ! [B] :
( m2_ringcat1(B,A)
<=> m1_subset_1(B,A) ) ) ).
fof(dt_m3_ringcat1,axiom,
! [A] :
( ( ~ v1_xboole_0(A)
& v5_ringcat1(A) )
=> ! [B] :
( m3_ringcat1(B,A)
=> ( v3_ringcat1(B)
& l1_ringcat1(B) ) ) ) ).
fof(existence_m3_ringcat1,axiom,
! [A] :
( ( ~ v1_xboole_0(A)
& v5_ringcat1(A) )
=> ? [B] : m3_ringcat1(B,A) ) ).
fof(redefinition_m3_ringcat1,axiom,
! [A] :
( ( ~ v1_xboole_0(A)
& v5_ringcat1(A) )
=> ! [B] :
( m3_ringcat1(B,A)
<=> m1_subset_1(B,A) ) ) ).
fof(dt_m4_ringcat1,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)
& v4_group_1(B)
& v6_vectsp_1(B)
& v7_vectsp_1(B)
& v8_vectsp_1(B)
& l3_vectsp_1(B) )
=> ! [C] :
( m4_ringcat1(C,A,B)
=> ( ~ v1_xboole_0(C)
& v5_ringcat1(C) ) ) ) ).
fof(existence_m4_ringcat1,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)
& v4_group_1(B)
& v6_vectsp_1(B)
& v7_vectsp_1(B)
& v8_vectsp_1(B)
& l3_vectsp_1(B) )
=> ? [C] : m4_ringcat1(C,A,B) ) ).
fof(dt_m5_ringcat1,axiom,
! [A,B,C] :
( ( ~ 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)
& v4_group_1(B)
& v6_vectsp_1(B)
& v7_vectsp_1(B)
& v8_vectsp_1(B)
& l3_vectsp_1(B)
& m4_ringcat1(C,A,B) )
=> ! [D] :
( m5_ringcat1(D,A,B,C)
=> m1_ringcat1(D,A,B) ) ) ).
fof(existence_m5_ringcat1,axiom,
! [A,B,C] :
( ( ~ 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)
& v4_group_1(B)
& v6_vectsp_1(B)
& v7_vectsp_1(B)
& v8_vectsp_1(B)
& l3_vectsp_1(B)
& m4_ringcat1(C,A,B) )
=> ? [D] : m5_ringcat1(D,A,B,C) ) ).
fof(redefinition_m5_ringcat1,axiom,
! [A,B,C] :
( ( ~ 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)
& v4_group_1(B)
& v6_vectsp_1(B)
& v7_vectsp_1(B)
& v8_vectsp_1(B)
& l3_vectsp_1(B)
& m4_ringcat1(C,A,B) )
=> ! [D] :
( m5_ringcat1(D,A,B,C)
<=> m1_subset_1(D,C) ) ) ).
fof(dt_l1_ringcat1,axiom,
$true ).
fof(existence_l1_ringcat1,axiom,
? [A] : l1_ringcat1(A) ).
fof(abstractness_v2_ringcat1,axiom,
! [A] :
( l1_ringcat1(A)
=> ( v2_ringcat1(A)
=> A = g1_ringcat1(u1_ringcat1(A),u2_ringcat1(A),u3_ringcat1(A)) ) ) ).
fof(reflexivity_r1_ringcat1,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)
& v4_group_1(B)
& v6_vectsp_1(B)
& v7_vectsp_1(B)
& v8_vectsp_1(B)
& l3_vectsp_1(B) )
=> r1_ringcat1(A,A) ) ).
fof(dt_k1_ringcat1,axiom,
! [A] :
( l1_ringcat1(A)
=> ( ~ v3_struct_0(k1_ringcat1(A))
& v3_rlvect_1(k1_ringcat1(A))
& v4_rlvect_1(k1_ringcat1(A))
& v5_rlvect_1(k1_ringcat1(A))
& v6_rlvect_1(k1_ringcat1(A))
& v4_group_1(k1_ringcat1(A))
& v6_vectsp_1(k1_ringcat1(A))
& v7_vectsp_1(k1_ringcat1(A))
& v8_vectsp_1(k1_ringcat1(A))
& l3_vectsp_1(k1_ringcat1(A)) ) ) ).
fof(dt_k2_ringcat1,axiom,
! [A] :
( l1_ringcat1(A)
=> ( ~ v3_struct_0(k2_ringcat1(A))
& v3_rlvect_1(k2_ringcat1(A))
& v4_rlvect_1(k2_ringcat1(A))
& v5_rlvect_1(k2_ringcat1(A))
& v6_rlvect_1(k2_ringcat1(A))
& v4_group_1(k2_ringcat1(A))
& v6_vectsp_1(k2_ringcat1(A))
& v7_vectsp_1(k2_ringcat1(A))
& v8_vectsp_1(k2_ringcat1(A))
& l3_vectsp_1(k2_ringcat1(A)) ) ) ).
fof(dt_k3_ringcat1,axiom,
! [A] :
( l1_ringcat1(A)
=> ( v1_funct_1(k3_ringcat1(A))
& v1_funct_2(k3_ringcat1(A),u1_struct_0(u1_ringcat1(A)),u1_struct_0(u2_ringcat1(A)))
& m2_relset_1(k3_ringcat1(A),u1_struct_0(u1_ringcat1(A)),u1_struct_0(u2_ringcat1(A))) ) ) ).
fof(dt_k4_ringcat1,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) )
=> ( v3_ringcat1(k4_ringcat1(A))
& l1_ringcat1(k4_ringcat1(A)) ) ) ).
fof(dt_k5_ringcat1,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) )
=> ( v2_ringcat1(k5_ringcat1(A))
& m1_ringcat1(k5_ringcat1(A),A,A) ) ) ).
fof(redefinition_k5_ringcat1,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) )
=> k5_ringcat1(A) = k4_ringcat1(A) ) ).
fof(dt_k6_ringcat1,axiom,
! [A,B] :
( ( v3_ringcat1(A)
& l1_ringcat1(A)
& v3_ringcat1(B)
& l1_ringcat1(B) )
=> ( v2_ringcat1(k6_ringcat1(A,B))
& v3_ringcat1(k6_ringcat1(A,B))
& l1_ringcat1(k6_ringcat1(A,B)) ) ) ).
fof(dt_k7_ringcat1,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)
& 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)
& v4_group_1(C)
& v6_vectsp_1(C)
& v7_vectsp_1(C)
& v8_vectsp_1(C)
& l3_vectsp_1(C)
& m1_ringcat1(D,B,C)
& m1_ringcat1(E,A,B) )
=> ( v2_ringcat1(k7_ringcat1(A,B,C,D,E))
& m1_ringcat1(k7_ringcat1(A,B,C,D,E),A,C) ) ) ).
fof(dt_k8_ringcat1,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)
& v4_group_1(B)
& v6_vectsp_1(B)
& v7_vectsp_1(B)
& v8_vectsp_1(B)
& l3_vectsp_1(B) )
=> m4_ringcat1(k8_ringcat1(A,B),A,B) ) ).
fof(dt_k9_ringcat1,axiom,
$true ).
fof(dt_k10_ringcat1,axiom,
! [A] :
( ( ~ v1_xboole_0(A)
& v4_ringcat1(A) )
=> ( ~ v1_xboole_0(k10_ringcat1(A))
& v5_ringcat1(k10_ringcat1(A)) ) ) ).
fof(dt_k11_ringcat1,axiom,
! [A,B] :
( ( ~ v1_xboole_0(A)
& v4_ringcat1(A)
& m1_subset_1(B,k10_ringcat1(A)) )
=> m2_ringcat1(k11_ringcat1(A,B),A) ) ).
fof(redefinition_k11_ringcat1,axiom,
! [A,B] :
( ( ~ v1_xboole_0(A)
& v4_ringcat1(A)
& m1_subset_1(B,k10_ringcat1(A)) )
=> k11_ringcat1(A,B) = k1_ringcat1(B) ) ).
fof(dt_k12_ringcat1,axiom,
! [A,B] :
( ( ~ v1_xboole_0(A)
& v4_ringcat1(A)
& m1_subset_1(B,k10_ringcat1(A)) )
=> m2_ringcat1(k12_ringcat1(A,B),A) ) ).
fof(redefinition_k12_ringcat1,axiom,
! [A,B] :
( ( ~ v1_xboole_0(A)
& v4_ringcat1(A)
& m1_subset_1(B,k10_ringcat1(A)) )
=> k12_ringcat1(A,B) = k2_ringcat1(B) ) ).
fof(dt_k13_ringcat1,axiom,
! [A,B] :
( ( ~ v1_xboole_0(A)
& v4_ringcat1(A)
& m1_subset_1(B,A) )
=> ( v2_ringcat1(k13_ringcat1(A,B))
& m3_ringcat1(k13_ringcat1(A,B),k10_ringcat1(A)) ) ) ).
fof(dt_k14_ringcat1,axiom,
! [A] :
( ( ~ v1_xboole_0(A)
& v4_ringcat1(A) )
=> ( v1_funct_1(k14_ringcat1(A))
& v1_funct_2(k14_ringcat1(A),k10_ringcat1(A),A)
& m2_relset_1(k14_ringcat1(A),k10_ringcat1(A),A) ) ) ).
fof(dt_k15_ringcat1,axiom,
! [A] :
( ( ~ v1_xboole_0(A)
& v4_ringcat1(A) )
=> ( v1_funct_1(k15_ringcat1(A))
& v1_funct_2(k15_ringcat1(A),k10_ringcat1(A),A)
& m2_relset_1(k15_ringcat1(A),k10_ringcat1(A),A) ) ) ).
fof(dt_k16_ringcat1,axiom,
! [A] :
( ( ~ v1_xboole_0(A)
& v4_ringcat1(A) )
=> ( v1_funct_1(k16_ringcat1(A))
& v1_funct_2(k16_ringcat1(A),A,k10_ringcat1(A))
& m2_relset_1(k16_ringcat1(A),A,k10_ringcat1(A)) ) ) ).
fof(dt_k17_ringcat1,axiom,
! [A] :
( ( ~ v1_xboole_0(A)
& v4_ringcat1(A) )
=> ( v1_funct_1(k17_ringcat1(A))
& m2_relset_1(k17_ringcat1(A),k2_zfmisc_1(k10_ringcat1(A),k10_ringcat1(A)),k10_ringcat1(A)) ) ) ).
fof(dt_k18_ringcat1,axiom,
! [A] :
( ( ~ v1_xboole_0(A)
& v1_classes2(A) )
=> l1_cat_1(k18_ringcat1(A)) ) ).
fof(dt_u1_ringcat1,axiom,
! [A] :
( l1_ringcat1(A)
=> ( ~ v3_struct_0(u1_ringcat1(A))
& v3_rlvect_1(u1_ringcat1(A))
& v4_rlvect_1(u1_ringcat1(A))
& v5_rlvect_1(u1_ringcat1(A))
& v6_rlvect_1(u1_ringcat1(A))
& v4_group_1(u1_ringcat1(A))
& v6_vectsp_1(u1_ringcat1(A))
& v7_vectsp_1(u1_ringcat1(A))
& v8_vectsp_1(u1_ringcat1(A))
& l3_vectsp_1(u1_ringcat1(A)) ) ) ).
fof(dt_u2_ringcat1,axiom,
! [A] :
( l1_ringcat1(A)
=> ( ~ v3_struct_0(u2_ringcat1(A))
& v3_rlvect_1(u2_ringcat1(A))
& v4_rlvect_1(u2_ringcat1(A))
& v5_rlvect_1(u2_ringcat1(A))
& v6_rlvect_1(u2_ringcat1(A))
& v4_group_1(u2_ringcat1(A))
& v6_vectsp_1(u2_ringcat1(A))
& v7_vectsp_1(u2_ringcat1(A))
& v8_vectsp_1(u2_ringcat1(A))
& l3_vectsp_1(u2_ringcat1(A)) ) ) ).
fof(dt_u3_ringcat1,axiom,
! [A] :
( l1_ringcat1(A)
=> ( v1_funct_1(u3_ringcat1(A))
& v1_funct_2(u3_ringcat1(A),u1_struct_0(u1_ringcat1(A)),u1_struct_0(u2_ringcat1(A)))
& m2_relset_1(u3_ringcat1(A),u1_struct_0(u1_ringcat1(A)),u1_struct_0(u2_ringcat1(A))) ) ) ).
fof(dt_g1_ringcat1,axiom,
! [A,B,C] :
( ( ~ 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)
& v4_group_1(B)
& v6_vectsp_1(B)
& v7_vectsp_1(B)
& v8_vectsp_1(B)
& l3_vectsp_1(B)
& v1_funct_1(C)
& v1_funct_2(C,u1_struct_0(A),u1_struct_0(B))
& m1_relset_1(C,u1_struct_0(A),u1_struct_0(B)) )
=> ( v2_ringcat1(g1_ringcat1(A,B,C))
& l1_ringcat1(g1_ringcat1(A,B,C)) ) ) ).
fof(free_g1_ringcat1,axiom,
! [A,B,C] :
( ( ~ 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)
& v4_group_1(B)
& v6_vectsp_1(B)
& v7_vectsp_1(B)
& v8_vectsp_1(B)
& l3_vectsp_1(B)
& v1_funct_1(C)
& v1_funct_2(C,u1_struct_0(A),u1_struct_0(B))
& m1_relset_1(C,u1_struct_0(A),u1_struct_0(B)) )
=> ! [D,E,F] :
( g1_ringcat1(A,B,C) = g1_ringcat1(D,E,F)
=> ( A = D
& B = E
& C = F ) ) ) ).
%------------------------------------------------------------------------------