SET007 Axioms: SET007+424.ax
%------------------------------------------------------------------------------
% File : SET007+424 : TPTP v8.2.0. Released v3.4.0.
% Domain : Set Theory
% Axioms : Group of Automorphisms of Universal and Many Sorted Algebras
% 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 : autalg_1 [Urb08]
% Status : Satisfiable
% Syntax : Number of formulae : 66 ( 6 unt; 0 def)
% Number of atoms : 470 ( 27 equ)
% Maximal formula atoms : 16 ( 7 avg)
% Number of connectives : 469 ( 65 ~; 0 |; 232 &)
% ( 8 <=>; 164 =>; 0 <=; 0 <~>)
% Maximal formula depth : 16 ( 8 avg)
% Maximal term depth : 4 ( 1 avg)
% Number of predicates : 42 ( 40 usr; 1 prp; 0-4 aty)
% Number of functors : 32 ( 32 usr; 1 con; 0-6 aty)
% Number of variables : 177 ( 174 !; 3 ?)
% 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_autalg_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v6_unialg_1(A)
& v7_unialg_1(A)
& v8_unialg_1(A)
& l1_unialg_1(A) )
=> ( ~ v3_struct_0(k3_autalg_1(A))
& v1_group_1(k3_autalg_1(A))
& v2_group_1(k3_autalg_1(A))
& v3_group_1(k3_autalg_1(A))
& v4_group_1(k3_autalg_1(A)) ) ) ).
fof(fc2_autalg_1,axiom,
! [A,B] :
( m1_pboole(B,A)
=> ( v1_relat_1(k12_pboole(A,B,B))
& v2_relat_1(k12_pboole(A,B,B))
& v1_funct_1(k12_pboole(A,B,B)) ) ) ).
fof(rc1_autalg_1,axiom,
! [A,B] :
( m1_pboole(B,A)
=> ? [C] :
( m1_autalg_1(C,A,B,B)
& ~ v1_xboole_0(C) ) ) ).
fof(fc3_autalg_1,axiom,
! [A,B] :
( ( ~ v3_struct_0(A)
& ~ v2_msualg_1(A)
& l1_msualg_1(A)
& v5_msualg_1(B,A)
& l3_msualg_1(B,A) )
=> ( ~ v3_struct_0(k6_autalg_1(A,B))
& v1_group_1(k6_autalg_1(A,B))
& v2_group_1(k6_autalg_1(A,B))
& v3_group_1(k6_autalg_1(A,B))
& v4_group_1(k6_autalg_1(A,B)) ) ) ).
fof(t1_autalg_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v6_unialg_1(A)
& v7_unialg_1(A)
& v8_unialg_1(A)
& l1_unialg_1(A) )
=> r4_alg_1(A,A,k6_partfun1(u1_struct_0(A))) ) ).
fof(d1_autalg_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v6_unialg_1(A)
& v7_unialg_1(A)
& v8_unialg_1(A)
& l1_unialg_1(A) )
=> ! [B] :
( m1_fraenkel(B,u1_struct_0(A),u1_struct_0(A))
=> ( B = k1_autalg_1(A)
<=> ( ! [C] :
( m2_fraenkel(C,u1_struct_0(A),u1_struct_0(A),B)
=> ( v1_funct_1(C)
& v1_funct_2(C,u1_struct_0(A),u1_struct_0(A))
& m2_relset_1(C,u1_struct_0(A),u1_struct_0(A)) ) )
& ! [C] :
( ( v1_funct_1(C)
& v1_funct_2(C,u1_struct_0(A),u1_struct_0(A))
& m2_relset_1(C,u1_struct_0(A),u1_struct_0(A)) )
=> ( r2_hidden(C,B)
<=> r4_alg_1(A,A,C) ) ) ) ) ) ) ).
fof(t2_autalg_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v6_unialg_1(A)
& v7_unialg_1(A)
& v8_unialg_1(A)
& l1_unialg_1(A) )
=> r1_tarski(k1_autalg_1(A),k1_fraenkel(u1_struct_0(A),u1_struct_0(A))) ) ).
fof(t3_autalg_1,axiom,
$true ).
fof(t4_autalg_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v6_unialg_1(A)
& v7_unialg_1(A)
& v8_unialg_1(A)
& l1_unialg_1(A) )
=> r2_hidden(k6_partfun1(u1_struct_0(A)),k1_autalg_1(A)) ) ).
fof(t5_autalg_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v6_unialg_1(A)
& v7_unialg_1(A)
& v8_unialg_1(A)
& l1_unialg_1(A) )
=> ! [B] :
( ( v1_funct_1(B)
& v1_funct_2(B,u1_struct_0(A),u1_struct_0(A))
& m2_relset_1(B,u1_struct_0(A),u1_struct_0(A)) )
=> ! [C] :
( ( v1_funct_1(C)
& v1_funct_2(C,u1_struct_0(A),u1_struct_0(A))
& m2_relset_1(C,u1_struct_0(A),u1_struct_0(A)) )
=> ( ( m2_fraenkel(B,u1_struct_0(A),u1_struct_0(A),k1_autalg_1(A))
& C = k2_funct_1(B) )
=> r4_alg_1(A,A,C) ) ) ) ) ).
fof(t6_autalg_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v6_unialg_1(A)
& v7_unialg_1(A)
& v8_unialg_1(A)
& l1_unialg_1(A) )
=> ! [B] :
( m2_fraenkel(B,u1_struct_0(A),u1_struct_0(A),k1_autalg_1(A))
=> r2_hidden(k2_funct_1(B),k1_autalg_1(A)) ) ) ).
fof(t7_autalg_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v6_unialg_1(A)
& v7_unialg_1(A)
& v8_unialg_1(A)
& l1_unialg_1(A) )
=> ! [B] :
( m2_fraenkel(B,u1_struct_0(A),u1_struct_0(A),k1_autalg_1(A))
=> ! [C] :
( m2_fraenkel(C,u1_struct_0(A),u1_struct_0(A),k1_autalg_1(A))
=> r2_hidden(k7_funct_2(u1_struct_0(A),u1_struct_0(A),u1_struct_0(A),C,B),k1_autalg_1(A)) ) ) ) ).
fof(d2_autalg_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v6_unialg_1(A)
& v7_unialg_1(A)
& v8_unialg_1(A)
& l1_unialg_1(A) )
=> ! [B] :
( ( v1_funct_1(B)
& v1_funct_2(B,k2_zfmisc_1(k1_autalg_1(A),k1_autalg_1(A)),k1_autalg_1(A))
& m2_relset_1(B,k2_zfmisc_1(k1_autalg_1(A),k1_autalg_1(A)),k1_autalg_1(A)) )
=> ( B = k2_autalg_1(A)
<=> ! [C] :
( m2_fraenkel(C,u1_struct_0(A),u1_struct_0(A),k1_autalg_1(A))
=> ! [D] :
( m2_fraenkel(D,u1_struct_0(A),u1_struct_0(A),k1_autalg_1(A))
=> k2_binop_1(k1_autalg_1(A),k1_autalg_1(A),k1_autalg_1(A),B,C,D) = k7_funct_2(u1_struct_0(A),u1_struct_0(A),u1_struct_0(A),C,D) ) ) ) ) ) ).
fof(d3_autalg_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v6_unialg_1(A)
& v7_unialg_1(A)
& v8_unialg_1(A)
& l1_unialg_1(A) )
=> k3_autalg_1(A) = g1_group_1(k1_autalg_1(A),k2_autalg_1(A)) ) ).
fof(t8_autalg_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v6_unialg_1(A)
& v7_unialg_1(A)
& v8_unialg_1(A)
& l1_unialg_1(A) )
=> ! [B] :
( m1_subset_1(B,u1_struct_0(k3_autalg_1(A)))
=> ! [C] :
( m1_subset_1(C,u1_struct_0(k3_autalg_1(A)))
=> ! [D] :
( m2_fraenkel(D,u1_struct_0(A),u1_struct_0(A),k1_autalg_1(A))
=> ! [E] :
( m2_fraenkel(E,u1_struct_0(A),u1_struct_0(A),k1_autalg_1(A))
=> ( ( B = D
& C = E )
=> k1_group_1(k3_autalg_1(A),B,C) = k7_funct_2(u1_struct_0(A),u1_struct_0(A),u1_struct_0(A),D,E) ) ) ) ) ) ) ).
fof(t9_autalg_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v6_unialg_1(A)
& v7_unialg_1(A)
& v8_unialg_1(A)
& l1_unialg_1(A) )
=> k6_partfun1(u1_struct_0(A)) = k2_group_1(k3_autalg_1(A)) ) ).
fof(t10_autalg_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v6_unialg_1(A)
& v7_unialg_1(A)
& v8_unialg_1(A)
& l1_unialg_1(A) )
=> ! [B] :
( m2_fraenkel(B,u1_struct_0(A),u1_struct_0(A),k1_autalg_1(A))
=> ! [C] :
( m1_subset_1(C,u1_struct_0(k3_autalg_1(A)))
=> ( B = C
=> k2_funct_1(B) = k3_group_1(k3_autalg_1(A),C) ) ) ) ) ).
fof(t11_autalg_1,axiom,
! [A,B] :
( m1_pboole(B,A)
=> ! [C] :
( m1_pboole(C,A)
=> ! [D] :
( m1_pboole(D,A)
=> ( ( r1_pzfmisc1(A,B,C)
& r1_pzfmisc1(A,C,D) )
=> r1_pzfmisc1(A,B,D) ) ) ) ) ).
fof(t12_autalg_1,axiom,
! [A,B] :
( m1_pboole(B,k1_tarski(A))
=> B = k3_cqc_lang(A,k1_funct_1(B,A)) ) ).
fof(t13_autalg_1,axiom,
! [A] :
( ( v1_relat_1(A)
& v1_funct_1(A)
& v1_funcop_1(A) )
=> ! [B] :
( ( v1_relat_1(B)
& v1_funct_1(B)
& v1_funcop_1(B) )
=> ! [C] :
( ( v1_relat_1(C)
& v1_funct_1(C)
& v1_funcop_1(C) )
=> k13_pboole(A,k13_pboole(B,C)) = k13_pboole(k13_pboole(A,B),C) ) ) ) ).
fof(t14_autalg_1,axiom,
! [A,B] :
( ( v2_relat_1(B)
& m1_pboole(B,A) )
=> ! [C] :
( ( v2_relat_1(C)
& m1_pboole(C,A) )
=> ! [D] :
( m3_pboole(D,A,B,C)
=> ( ( v1_msualg_3(D)
& v2_msualg_3(D,A,B,C) )
=> ( v1_msualg_3(k4_msualg_3(A,B,C,D))
& v2_msualg_3(k4_msualg_3(A,B,C,D),A,C,B) ) ) ) ) ) ).
fof(t15_autalg_1,axiom,
! [A,B] :
( ( v2_relat_1(B)
& m1_pboole(B,A) )
=> ! [C] :
( ( v2_relat_1(C)
& m1_pboole(C,A) )
=> ! [D] :
( m3_pboole(D,A,B,C)
=> ( ( v1_msualg_3(D)
& v2_msualg_3(D,A,B,C) )
=> r6_pboole(A,k4_msualg_3(A,C,B,k4_msualg_3(A,B,C,D)),D) ) ) ) ) ).
fof(t16_autalg_1,axiom,
! [A] :
( ( v1_relat_1(A)
& v1_funct_1(A)
& v1_funcop_1(A) )
=> ! [B] :
( ( v1_relat_1(B)
& v1_funct_1(B)
& v1_funcop_1(B) )
=> ( ( v1_msualg_3(A)
& v1_msualg_3(B) )
=> v1_msualg_3(k13_pboole(A,B)) ) ) ) ).
fof(t17_autalg_1,axiom,
! [A,B] :
( m1_pboole(B,A)
=> ! [C] :
( ( v2_relat_1(C)
& m1_pboole(C,A) )
=> ! [D] :
( ( v2_relat_1(D)
& m1_pboole(D,A) )
=> ! [E] :
( m3_pboole(E,A,B,C)
=> ! [F] :
( m3_pboole(F,A,C,D)
=> ( ( v2_msualg_3(E,A,B,C)
& v2_msualg_3(F,A,C,D) )
=> v2_msualg_3(k3_msualg_3(A,B,C,D,E,F),A,B,D) ) ) ) ) ) ) ).
fof(t18_autalg_1,axiom,
! [A,B] :
( ( v2_relat_1(B)
& m1_pboole(B,A) )
=> ! [C] :
( ( v2_relat_1(C)
& m1_pboole(C,A) )
=> ! [D] :
( ( v2_relat_1(D)
& m1_pboole(D,A) )
=> ! [E] :
( m3_pboole(E,A,B,C)
=> ! [F] :
( m3_pboole(F,A,C,D)
=> ( ( v1_msualg_3(E)
& v2_msualg_3(E,A,B,C)
& v1_msualg_3(F)
& v2_msualg_3(F,A,C,D) )
=> r6_pboole(A,k4_msualg_3(A,B,D,k3_msualg_3(A,B,C,D,E,F)),k3_msualg_3(A,D,C,B,k4_msualg_3(A,C,D,F),k4_msualg_3(A,B,C,E))) ) ) ) ) ) ) ).
fof(t19_autalg_1,axiom,
! [A,B] :
( ( v2_relat_1(B)
& m1_pboole(B,A) )
=> ! [C] :
( ( v2_relat_1(C)
& m1_pboole(C,A) )
=> ! [D] :
( m3_pboole(D,A,B,C)
=> ! [E] :
( m3_pboole(E,A,C,B)
=> ( ( v1_msualg_3(D)
& v2_msualg_3(D,A,B,C)
& r6_pboole(A,k3_msualg_3(A,B,C,B,D,E),k2_msualg_3(A,B)) )
=> r6_pboole(A,E,k4_msualg_3(A,B,C,D)) ) ) ) ) ) ).
fof(t20_autalg_1,axiom,
$true ).
fof(t21_autalg_1,axiom,
! [A,B] :
( m1_pboole(B,A)
=> ! [C] :
( m1_pboole(C,A)
=> ( r1_pzfmisc1(A,B,C)
=> ! [D] :
( r2_hidden(D,k4_card_3(k12_pboole(A,B,C)))
=> m3_pboole(D,A,B,C) ) ) ) ) ).
fof(t22_autalg_1,axiom,
! [A,B] :
( m1_pboole(B,A)
=> ! [C] :
( m1_pboole(C,A)
=> ( r1_pzfmisc1(A,B,C)
=> ! [D] :
( m3_pboole(D,A,B,C)
=> r2_hidden(D,k4_card_3(k12_pboole(A,B,C))) ) ) ) ) ).
fof(t23_autalg_1,axiom,
! [A,B] :
( m1_pboole(B,A)
=> ! [C] :
( m1_pboole(C,A)
=> ( r1_pzfmisc1(A,B,C)
=> v2_relat_1(k12_pboole(A,B,C)) ) ) ) ).
fof(d4_autalg_1,axiom,
$true ).
fof(d5_autalg_1,axiom,
$true ).
fof(d6_autalg_1,axiom,
! [A,B] :
( m1_pboole(B,A)
=> ! [C] :
( m1_pboole(C,A)
=> ( r1_pzfmisc1(A,B,C)
=> ! [D] :
( ~ v1_xboole_0(D)
=> ( m1_autalg_1(D,A,B,C)
<=> ! [E] :
( r2_hidden(E,D)
=> m3_pboole(E,A,B,C) ) ) ) ) ) ) ).
fof(t24_autalg_1,axiom,
! [A,B] :
( m1_pboole(B,A)
=> v2_msualg_3(k2_msualg_3(A,B),A,B,B) ) ).
fof(t25_autalg_1,axiom,
! [A,B] :
( m1_pboole(B,A)
=> v1_msualg_3(k2_msualg_3(A,B)) ) ).
fof(t26_autalg_1,axiom,
$true ).
fof(t27_autalg_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& ~ v2_msualg_1(A)
& l1_msualg_1(A) )
=> ! [B] :
( ( v5_msualg_1(B,A)
& l3_msualg_1(B,A) )
=> r2_hidden(k2_msualg_3(u1_struct_0(A),u4_msualg_1(A,B)),k4_card_3(k12_pboole(u1_struct_0(A),u4_msualg_1(A,B),u4_msualg_1(A,B)))) ) ) ).
fof(d7_autalg_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& ~ v2_msualg_1(A)
& l1_msualg_1(A) )
=> ! [B] :
( ( v5_msualg_1(B,A)
& l3_msualg_1(B,A) )
=> ! [C] :
( m1_autalg_1(C,u1_struct_0(A),u4_msualg_1(A,B),u4_msualg_1(A,B))
=> ( C = k4_autalg_1(A,B)
<=> ( ! [D] :
( m2_autalg_1(D,u1_struct_0(A),u4_msualg_1(A,B),C)
=> m3_pboole(D,u1_struct_0(A),u4_msualg_1(A,B),u4_msualg_1(A,B)) )
& ! [D] :
( m3_pboole(D,u1_struct_0(A),u4_msualg_1(A,B),u4_msualg_1(A,B))
=> ( r2_hidden(D,C)
<=> r4_msualg_3(A,B,B,D) ) ) ) ) ) ) ) ).
fof(t28_autalg_1,axiom,
$true ).
fof(t29_autalg_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& ~ v2_msualg_1(A)
& l1_msualg_1(A) )
=> ! [B] :
( ( v5_msualg_1(B,A)
& l3_msualg_1(B,A) )
=> ! [C] :
( m2_autalg_1(C,u1_struct_0(A),u4_msualg_1(A,B),k4_autalg_1(A,B))
=> r2_hidden(C,k4_card_3(k12_pboole(u1_struct_0(A),u4_msualg_1(A,B),u4_msualg_1(A,B)))) ) ) ) ).
fof(t30_autalg_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& ~ v2_msualg_1(A)
& l1_msualg_1(A) )
=> ! [B] :
( ( v5_msualg_1(B,A)
& l3_msualg_1(B,A) )
=> r1_tarski(k4_autalg_1(A,B),k4_card_3(k12_pboole(u1_struct_0(A),u4_msualg_1(A,B),u4_msualg_1(A,B)))) ) ) ).
fof(t31_autalg_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& ~ v2_msualg_1(A)
& l1_msualg_1(A) )
=> ! [B] :
( ( v5_msualg_1(B,A)
& l3_msualg_1(B,A) )
=> r2_hidden(k2_msualg_3(u1_struct_0(A),u4_msualg_1(A,B)),k4_autalg_1(A,B)) ) ) ).
fof(t32_autalg_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& ~ v2_msualg_1(A)
& l1_msualg_1(A) )
=> ! [B] :
( ( v5_msualg_1(B,A)
& l3_msualg_1(B,A) )
=> ! [C] :
( m2_autalg_1(C,u1_struct_0(A),u4_msualg_1(A,B),k4_autalg_1(A,B))
=> r2_hidden(k4_msualg_3(u1_struct_0(A),u4_msualg_1(A,B),u4_msualg_1(A,B),C),k4_autalg_1(A,B)) ) ) ) ).
fof(t33_autalg_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& ~ v2_msualg_1(A)
& l1_msualg_1(A) )
=> ! [B] :
( ( v5_msualg_1(B,A)
& l3_msualg_1(B,A) )
=> ! [C] :
( m2_autalg_1(C,u1_struct_0(A),u4_msualg_1(A,B),k4_autalg_1(A,B))
=> ! [D] :
( m2_autalg_1(D,u1_struct_0(A),u4_msualg_1(A,B),k4_autalg_1(A,B))
=> r2_hidden(k3_msualg_3(u1_struct_0(A),u4_msualg_1(A,B),u4_msualg_1(A,B),u4_msualg_1(A,B),D,C),k4_autalg_1(A,B)) ) ) ) ) ).
fof(t34_autalg_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v6_unialg_1(A)
& v7_unialg_1(A)
& v8_unialg_1(A)
& l1_unialg_1(A) )
=> ! [B] :
( m3_pboole(B,u1_struct_0(k7_msualg_1(A)),u4_msualg_1(k7_msualg_1(A),k10_msualg_1(A)),u4_msualg_1(k7_msualg_1(A),k10_msualg_1(A)))
=> ! [C] :
( m2_fraenkel(C,u1_struct_0(A),u1_struct_0(A),k1_autalg_1(A))
=> ( B = k3_cqc_lang(np__0,C)
=> r2_hidden(B,k4_autalg_1(k7_msualg_1(A),k10_msualg_1(A))) ) ) ) ) ).
fof(d8_autalg_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& ~ v2_msualg_1(A)
& l1_msualg_1(A) )
=> ! [B] :
( ( v5_msualg_1(B,A)
& l3_msualg_1(B,A) )
=> ! [C] :
( ( v1_funct_1(C)
& v1_funct_2(C,k2_zfmisc_1(k4_autalg_1(A,B),k4_autalg_1(A,B)),k4_autalg_1(A,B))
& m2_relset_1(C,k2_zfmisc_1(k4_autalg_1(A,B),k4_autalg_1(A,B)),k4_autalg_1(A,B)) )
=> ( C = k5_autalg_1(A,B)
<=> ! [D] :
( m2_autalg_1(D,u1_struct_0(A),u4_msualg_1(A,B),k4_autalg_1(A,B))
=> ! [E] :
( m2_autalg_1(E,u1_struct_0(A),u4_msualg_1(A,B),k4_autalg_1(A,B))
=> k2_binop_1(k4_autalg_1(A,B),k4_autalg_1(A,B),k4_autalg_1(A,B),C,D,E) = k3_msualg_3(u1_struct_0(A),u4_msualg_1(A,B),u4_msualg_1(A,B),u4_msualg_1(A,B),D,E) ) ) ) ) ) ) ).
fof(d9_autalg_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& ~ v2_msualg_1(A)
& l1_msualg_1(A) )
=> ! [B] :
( ( v5_msualg_1(B,A)
& l3_msualg_1(B,A) )
=> k6_autalg_1(A,B) = g1_group_1(k4_autalg_1(A,B),k5_autalg_1(A,B)) ) ) ).
fof(t35_autalg_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& ~ v2_msualg_1(A)
& l1_msualg_1(A) )
=> ! [B] :
( ( v5_msualg_1(B,A)
& l3_msualg_1(B,A) )
=> ! [C] :
( m1_subset_1(C,u1_struct_0(k6_autalg_1(A,B)))
=> ! [D] :
( m1_subset_1(D,u1_struct_0(k6_autalg_1(A,B)))
=> ! [E] :
( m2_autalg_1(E,u1_struct_0(A),u4_msualg_1(A,B),k4_autalg_1(A,B))
=> ! [F] :
( m2_autalg_1(F,u1_struct_0(A),u4_msualg_1(A,B),k4_autalg_1(A,B))
=> ( ( C = E
& D = F )
=> k1_group_1(k6_autalg_1(A,B),C,D) = k3_msualg_3(u1_struct_0(A),u4_msualg_1(A,B),u4_msualg_1(A,B),u4_msualg_1(A,B),E,F) ) ) ) ) ) ) ) ).
fof(t36_autalg_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& ~ v2_msualg_1(A)
& l1_msualg_1(A) )
=> ! [B] :
( ( v5_msualg_1(B,A)
& l3_msualg_1(B,A) )
=> k2_msualg_3(u1_struct_0(A),u4_msualg_1(A,B)) = k2_group_1(k6_autalg_1(A,B)) ) ) ).
fof(t37_autalg_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& ~ v2_msualg_1(A)
& l1_msualg_1(A) )
=> ! [B] :
( ( v5_msualg_1(B,A)
& l3_msualg_1(B,A) )
=> ! [C] :
( m2_autalg_1(C,u1_struct_0(A),u4_msualg_1(A,B),k4_autalg_1(A,B))
=> ! [D] :
( m1_subset_1(D,u1_struct_0(k6_autalg_1(A,B)))
=> ( C = D
=> k4_msualg_3(u1_struct_0(A),u4_msualg_1(A,B),u4_msualg_1(A,B),C) = k3_group_1(k6_autalg_1(A,B),D) ) ) ) ) ) ).
fof(t38_autalg_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v6_unialg_1(A)
& v7_unialg_1(A)
& v8_unialg_1(A)
& l1_unialg_1(A) )
=> ! [B] :
( ( ~ v3_struct_0(B)
& v6_unialg_1(B)
& v7_unialg_1(B)
& v8_unialg_1(B)
& l1_unialg_1(B) )
=> ( r1_unialg_2(A,B)
=> ! [C] :
( m3_pboole(C,u1_struct_0(k7_msualg_1(A)),u4_msualg_1(k7_msualg_1(A),k10_msualg_1(A)),u4_msualg_1(k7_msualg_1(A),k2_msuhom_1(k7_msualg_1(A),k7_msualg_1(B),k10_msualg_1(B))))
=> ( v1_funct_1(k1_funct_1(C,np__0))
& v1_funct_2(k1_funct_1(C,np__0),u1_struct_0(A),u1_struct_0(B))
& m2_relset_1(k1_funct_1(C,np__0),u1_struct_0(A),u1_struct_0(B)) ) ) ) ) ) ).
fof(t39_autalg_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v6_unialg_1(A)
& v7_unialg_1(A)
& v8_unialg_1(A)
& l1_unialg_1(A) )
=> ! [B] :
( m2_fraenkel(B,u1_struct_0(A),u1_struct_0(A),k1_autalg_1(A))
=> m3_pboole(k3_cqc_lang(np__0,B),u1_struct_0(k7_msualg_1(A)),u4_msualg_1(k7_msualg_1(A),k10_msualg_1(A)),u4_msualg_1(k7_msualg_1(A),k10_msualg_1(A))) ) ) ).
fof(t40_autalg_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v6_unialg_1(A)
& v7_unialg_1(A)
& v8_unialg_1(A)
& l1_unialg_1(A) )
=> ! [B] :
( ( v1_relat_1(B)
& v1_funct_1(B) )
=> ( ( k1_relat_1(B) = k1_autalg_1(A)
& ! [C] :
( r2_hidden(C,k1_autalg_1(A))
=> k1_funct_1(B,C) = k3_cqc_lang(np__0,C) ) )
=> ( v1_funct_1(B)
& v1_funct_2(B,u1_struct_0(k3_autalg_1(A)),u1_struct_0(k6_autalg_1(k7_msualg_1(A),k10_msualg_1(A))))
& v1_group_6(B,k3_autalg_1(A),k6_autalg_1(k7_msualg_1(A),k10_msualg_1(A)))
& m2_relset_1(B,u1_struct_0(k3_autalg_1(A)),u1_struct_0(k6_autalg_1(k7_msualg_1(A),k10_msualg_1(A)))) ) ) ) ) ).
fof(t41_autalg_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v6_unialg_1(A)
& v7_unialg_1(A)
& v8_unialg_1(A)
& l1_unialg_1(A) )
=> ! [B] :
( ( v1_funct_1(B)
& v1_funct_2(B,u1_struct_0(k3_autalg_1(A)),u1_struct_0(k6_autalg_1(k7_msualg_1(A),k10_msualg_1(A))))
& v1_group_6(B,k3_autalg_1(A),k6_autalg_1(k7_msualg_1(A),k10_msualg_1(A)))
& m2_relset_1(B,u1_struct_0(k3_autalg_1(A)),u1_struct_0(k6_autalg_1(k7_msualg_1(A),k10_msualg_1(A)))) )
=> ( ! [C] :
( r2_hidden(C,k1_autalg_1(A))
=> k1_funct_1(B,C) = k3_cqc_lang(np__0,C) )
=> v4_group_6(B,k3_autalg_1(A),k6_autalg_1(k7_msualg_1(A),k10_msualg_1(A))) ) ) ) ).
fof(t42_autalg_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v6_unialg_1(A)
& v7_unialg_1(A)
& v8_unialg_1(A)
& l1_unialg_1(A) )
=> r2_group_6(k3_autalg_1(A),k6_autalg_1(k7_msualg_1(A),k10_msualg_1(A))) ) ).
fof(dt_m1_autalg_1,axiom,
! [A,B,C] :
( ( m1_pboole(B,A)
& m1_pboole(C,A) )
=> ! [D] :
( m1_autalg_1(D,A,B,C)
=> ~ v1_xboole_0(D) ) ) ).
fof(existence_m1_autalg_1,axiom,
! [A,B,C] :
( ( m1_pboole(B,A)
& m1_pboole(C,A) )
=> ? [D] : m1_autalg_1(D,A,B,C) ) ).
fof(dt_m2_autalg_1,axiom,
! [A,B,C] :
( ( m1_pboole(B,A)
& ~ v1_xboole_0(C)
& m1_autalg_1(C,A,B,B) )
=> ! [D] :
( m2_autalg_1(D,A,B,C)
=> m3_pboole(D,A,B,B) ) ) ).
fof(existence_m2_autalg_1,axiom,
! [A,B,C] :
( ( m1_pboole(B,A)
& ~ v1_xboole_0(C)
& m1_autalg_1(C,A,B,B) )
=> ? [D] : m2_autalg_1(D,A,B,C) ) ).
fof(redefinition_m2_autalg_1,axiom,
! [A,B,C] :
( ( m1_pboole(B,A)
& ~ v1_xboole_0(C)
& m1_autalg_1(C,A,B,B) )
=> ! [D] :
( m2_autalg_1(D,A,B,C)
<=> m1_subset_1(D,C) ) ) ).
fof(dt_k1_autalg_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v6_unialg_1(A)
& v7_unialg_1(A)
& v8_unialg_1(A)
& l1_unialg_1(A) )
=> m1_fraenkel(k1_autalg_1(A),u1_struct_0(A),u1_struct_0(A)) ) ).
fof(dt_k2_autalg_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v6_unialg_1(A)
& v7_unialg_1(A)
& v8_unialg_1(A)
& l1_unialg_1(A) )
=> ( v1_funct_1(k2_autalg_1(A))
& v1_funct_2(k2_autalg_1(A),k2_zfmisc_1(k1_autalg_1(A),k1_autalg_1(A)),k1_autalg_1(A))
& m2_relset_1(k2_autalg_1(A),k2_zfmisc_1(k1_autalg_1(A),k1_autalg_1(A)),k1_autalg_1(A)) ) ) ).
fof(dt_k3_autalg_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v6_unialg_1(A)
& v7_unialg_1(A)
& v8_unialg_1(A)
& l1_unialg_1(A) )
=> ( ~ v3_struct_0(k3_autalg_1(A))
& v3_group_1(k3_autalg_1(A))
& v4_group_1(k3_autalg_1(A))
& l1_group_1(k3_autalg_1(A)) ) ) ).
fof(dt_k4_autalg_1,axiom,
! [A,B] :
( ( ~ v3_struct_0(A)
& ~ v2_msualg_1(A)
& l1_msualg_1(A)
& v5_msualg_1(B,A)
& l3_msualg_1(B,A) )
=> m1_autalg_1(k4_autalg_1(A,B),u1_struct_0(A),u4_msualg_1(A,B),u4_msualg_1(A,B)) ) ).
fof(dt_k5_autalg_1,axiom,
! [A,B] :
( ( ~ v3_struct_0(A)
& ~ v2_msualg_1(A)
& l1_msualg_1(A)
& v5_msualg_1(B,A)
& l3_msualg_1(B,A) )
=> ( v1_funct_1(k5_autalg_1(A,B))
& v1_funct_2(k5_autalg_1(A,B),k2_zfmisc_1(k4_autalg_1(A,B),k4_autalg_1(A,B)),k4_autalg_1(A,B))
& m2_relset_1(k5_autalg_1(A,B),k2_zfmisc_1(k4_autalg_1(A,B),k4_autalg_1(A,B)),k4_autalg_1(A,B)) ) ) ).
fof(dt_k6_autalg_1,axiom,
! [A,B] :
( ( ~ v3_struct_0(A)
& ~ v2_msualg_1(A)
& l1_msualg_1(A)
& v5_msualg_1(B,A)
& l3_msualg_1(B,A) )
=> ( ~ v3_struct_0(k6_autalg_1(A,B))
& v3_group_1(k6_autalg_1(A,B))
& v4_group_1(k6_autalg_1(A,B))
& l1_group_1(k6_autalg_1(A,B)) ) ) ).
%------------------------------------------------------------------------------