SET007 Axioms: SET007+394.ax
%------------------------------------------------------------------------------
% File : SET007+394 : TPTP v8.2.0. Released v3.4.0.
% Domain : Set Theory
% Axioms : On the Group of Inner Automorphisms
% 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 : autgroup [Urb08]
% Status : Satisfiable
% Syntax : Number of formulae : 44 ( 2 unt; 0 def)
% Number of atoms : 353 ( 34 equ)
% Maximal formula atoms : 19 ( 8 avg)
% Number of connectives : 353 ( 44 ~; 0 |; 204 &)
% ( 9 <=>; 96 =>; 0 <=; 0 <~>)
% Maximal formula depth : 13 ( 8 avg)
% Maximal term depth : 4 ( 1 avg)
% Number of predicates : 24 ( 22 usr; 1 prp; 0-4 aty)
% Number of functors : 22 ( 22 usr; 0 con; 1-6 aty)
% Number of variables : 92 ( 91 !; 1 ?)
% 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(t1_autgroup,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v1_group_1(A)
& v3_group_1(A)
& v4_group_1(A)
& l1_group_1(A) )
=> ! [B] :
( m1_group_2(B,A)
=> ( ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> ! [D] :
( m1_subset_1(D,u1_struct_0(A))
=> ( m1_subset_1(D,u1_struct_0(B))
=> r1_rlvect_1(B,k2_group_3(A,D,C)) ) ) )
<=> v1_group_3(B,A) ) ) ) ).
fof(d1_autgroup,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v1_group_1(A)
& v3_group_1(A)
& v4_group_1(A)
& l1_group_1(A) )
=> ! [B] :
( m1_fraenkel(B,u1_struct_0(A),u1_struct_0(A))
=> ( B = k1_autgroup(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))
& v1_group_6(C,A,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))
& v1_group_6(C,A,A)
& m2_relset_1(C,u1_struct_0(A),u1_struct_0(A)) )
=> ( r2_hidden(C,B)
<=> ( v2_funct_1(C)
& v3_group_6(C,A,A) ) ) ) ) ) ) ) ).
fof(t2_autgroup,axiom,
$true ).
fof(t3_autgroup,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v1_group_1(A)
& v3_group_1(A)
& v4_group_1(A)
& l1_group_1(A) )
=> r1_tarski(k1_autgroup(A),k1_fraenkel(u1_struct_0(A),u1_struct_0(A))) ) ).
fof(t4_autgroup,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v1_group_1(A)
& v3_group_1(A)
& v4_group_1(A)
& l1_group_1(A) )
=> m2_fraenkel(k6_partfun1(u1_struct_0(A)),u1_struct_0(A),u1_struct_0(A),k1_autgroup(A)) ) ).
fof(t5_autgroup,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v1_group_1(A)
& v3_group_1(A)
& v4_group_1(A)
& l1_group_1(A) )
=> ! [B] :
( ( v1_funct_1(B)
& v1_funct_2(B,u1_struct_0(A),u1_struct_0(A))
& v1_group_6(B,A,A)
& m2_relset_1(B,u1_struct_0(A),u1_struct_0(A)) )
=> ( r2_hidden(B,k1_autgroup(A))
<=> v4_group_6(B,A,A) ) ) ) ).
fof(t6_autgroup,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v1_group_1(A)
& v3_group_1(A)
& v4_group_1(A)
& l1_group_1(A) )
=> ! [B] :
( m2_fraenkel(B,u1_struct_0(A),u1_struct_0(A),k1_autgroup(A))
=> ( v1_funct_1(k2_funct_1(B))
& v1_funct_2(k2_funct_1(B),u1_struct_0(A),u1_struct_0(A))
& v1_group_6(k2_funct_1(B),A,A)
& m2_relset_1(k2_funct_1(B),u1_struct_0(A),u1_struct_0(A)) ) ) ) ).
fof(t7_autgroup,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v1_group_1(A)
& v3_group_1(A)
& v4_group_1(A)
& l1_group_1(A) )
=> ! [B] :
( m2_fraenkel(B,u1_struct_0(A),u1_struct_0(A),k1_autgroup(A))
=> m2_fraenkel(k2_funct_1(B),u1_struct_0(A),u1_struct_0(A),k1_autgroup(A)) ) ) ).
fof(t8_autgroup,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v1_group_1(A)
& v3_group_1(A)
& v4_group_1(A)
& l1_group_1(A) )
=> ! [B] :
( m2_fraenkel(B,u1_struct_0(A),u1_struct_0(A),k1_autgroup(A))
=> ! [C] :
( m2_fraenkel(C,u1_struct_0(A),u1_struct_0(A),k1_autgroup(A))
=> m2_fraenkel(k7_funct_2(u1_struct_0(A),u1_struct_0(A),u1_struct_0(A),C,B),u1_struct_0(A),u1_struct_0(A),k1_autgroup(A)) ) ) ) ).
fof(d2_autgroup,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v1_group_1(A)
& v3_group_1(A)
& v4_group_1(A)
& l1_group_1(A) )
=> ! [B] :
( ( v1_funct_1(B)
& v1_funct_2(B,k2_zfmisc_1(k1_autgroup(A),k1_autgroup(A)),k1_autgroup(A))
& m2_relset_1(B,k2_zfmisc_1(k1_autgroup(A),k1_autgroup(A)),k1_autgroup(A)) )
=> ( B = k2_autgroup(A)
<=> ! [C] :
( m2_fraenkel(C,u1_struct_0(A),u1_struct_0(A),k1_autgroup(A))
=> ! [D] :
( m2_fraenkel(D,u1_struct_0(A),u1_struct_0(A),k1_autgroup(A))
=> k2_binop_1(k1_autgroup(A),k1_autgroup(A),k1_autgroup(A),B,C,D) = k7_funct_2(u1_struct_0(A),u1_struct_0(A),u1_struct_0(A),D,C) ) ) ) ) ) ).
fof(d3_autgroup,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v1_group_1(A)
& v3_group_1(A)
& v4_group_1(A)
& l1_group_1(A) )
=> k3_autgroup(A) = g1_group_1(k1_autgroup(A),k2_autgroup(A)) ) ).
fof(t9_autgroup,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v1_group_1(A)
& v3_group_1(A)
& v4_group_1(A)
& l1_group_1(A) )
=> ! [B] :
( m1_subset_1(B,u1_struct_0(k3_autgroup(A)))
=> ! [C] :
( m1_subset_1(C,u1_struct_0(k3_autgroup(A)))
=> ! [D] :
( m2_fraenkel(D,u1_struct_0(A),u1_struct_0(A),k1_autgroup(A))
=> ! [E] :
( m2_fraenkel(E,u1_struct_0(A),u1_struct_0(A),k1_autgroup(A))
=> ( ( B = D
& C = E )
=> k1_group_1(k3_autgroup(A),B,C) = k7_funct_2(u1_struct_0(A),u1_struct_0(A),u1_struct_0(A),E,D) ) ) ) ) ) ) ).
fof(t10_autgroup,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v1_group_1(A)
& v3_group_1(A)
& v4_group_1(A)
& l1_group_1(A) )
=> k6_partfun1(u1_struct_0(A)) = k2_group_1(k3_autgroup(A)) ) ).
fof(t11_autgroup,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v1_group_1(A)
& v3_group_1(A)
& v4_group_1(A)
& l1_group_1(A) )
=> ! [B] :
( m2_fraenkel(B,u1_struct_0(A),u1_struct_0(A),k1_autgroup(A))
=> ! [C] :
( m1_subset_1(C,u1_struct_0(k3_autgroup(A)))
=> ( B = C
=> k2_funct_1(B) = k3_group_1(k3_autgroup(A),C) ) ) ) ) ).
fof(d4_autgroup,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v1_group_1(A)
& v3_group_1(A)
& v4_group_1(A)
& l1_group_1(A) )
=> ! [B] :
( m1_fraenkel(B,u1_struct_0(A),u1_struct_0(A))
=> ( B = k4_autgroup(A)
<=> ! [C] :
( m2_fraenkel(C,u1_struct_0(A),u1_struct_0(A),k1_fraenkel(u1_struct_0(A),u1_struct_0(A)))
=> ( r2_hidden(C,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(A),C,E) = k2_group_3(A,E,D) ) ) ) ) ) ) ) ).
fof(t12_autgroup,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v1_group_1(A)
& v3_group_1(A)
& v4_group_1(A)
& l1_group_1(A) )
=> r1_tarski(k4_autgroup(A),k1_fraenkel(u1_struct_0(A),u1_struct_0(A))) ) ).
fof(t13_autgroup,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v1_group_1(A)
& v3_group_1(A)
& v4_group_1(A)
& l1_group_1(A) )
=> ! [B] :
( m2_fraenkel(B,u1_struct_0(A),u1_struct_0(A),k4_autgroup(A))
=> m2_fraenkel(B,u1_struct_0(A),u1_struct_0(A),k1_autgroup(A)) ) ) ).
fof(t14_autgroup,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v1_group_1(A)
& v3_group_1(A)
& v4_group_1(A)
& l1_group_1(A) )
=> r1_tarski(k4_autgroup(A),k1_autgroup(A)) ) ).
fof(t15_autgroup,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v1_group_1(A)
& v3_group_1(A)
& v4_group_1(A)
& l1_group_1(A) )
=> ! [B] :
( m2_fraenkel(B,u1_struct_0(A),u1_struct_0(A),k4_autgroup(A))
=> ! [C] :
( m2_fraenkel(C,u1_struct_0(A),u1_struct_0(A),k4_autgroup(A))
=> k1_binop_1(k2_autgroup(A),B,C) = k7_funct_2(u1_struct_0(A),u1_struct_0(A),u1_struct_0(A),C,B) ) ) ) ).
fof(t16_autgroup,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v1_group_1(A)
& v3_group_1(A)
& v4_group_1(A)
& l1_group_1(A) )
=> m2_fraenkel(k6_partfun1(u1_struct_0(A)),u1_struct_0(A),u1_struct_0(A),k4_autgroup(A)) ) ).
fof(t17_autgroup,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v1_group_1(A)
& v3_group_1(A)
& v4_group_1(A)
& l1_group_1(A) )
=> ! [B] :
( m2_fraenkel(B,u1_struct_0(A),u1_struct_0(A),k4_autgroup(A))
=> m2_fraenkel(k2_funct_1(B),u1_struct_0(A),u1_struct_0(A),k4_autgroup(A)) ) ) ).
fof(t18_autgroup,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v1_group_1(A)
& v3_group_1(A)
& v4_group_1(A)
& l1_group_1(A) )
=> ! [B] :
( m2_fraenkel(B,u1_struct_0(A),u1_struct_0(A),k4_autgroup(A))
=> ! [C] :
( m2_fraenkel(C,u1_struct_0(A),u1_struct_0(A),k4_autgroup(A))
=> m2_fraenkel(k7_funct_2(u1_struct_0(A),u1_struct_0(A),u1_struct_0(A),C,B),u1_struct_0(A),u1_struct_0(A),k4_autgroup(A)) ) ) ) ).
fof(d5_autgroup,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v1_group_1(A)
& v3_group_1(A)
& v4_group_1(A)
& l1_group_1(A) )
=> ! [B] :
( ( v1_group_1(B)
& v1_group_3(B,k3_autgroup(A))
& m1_group_2(B,k3_autgroup(A)) )
=> ( B = k5_autgroup(A)
<=> u1_struct_0(B) = k4_autgroup(A) ) ) ) ).
fof(t19_autgroup,axiom,
$true ).
fof(t20_autgroup,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v1_group_1(A)
& v3_group_1(A)
& v4_group_1(A)
& l1_group_1(A) )
=> ! [B] :
( m1_subset_1(B,u1_struct_0(k5_autgroup(A)))
=> ! [C] :
( m1_subset_1(C,u1_struct_0(k5_autgroup(A)))
=> ! [D] :
( m2_fraenkel(D,u1_struct_0(A),u1_struct_0(A),k4_autgroup(A))
=> ! [E] :
( m2_fraenkel(E,u1_struct_0(A),u1_struct_0(A),k4_autgroup(A))
=> ( ( B = D
& C = E )
=> k1_group_1(k5_autgroup(A),B,C) = k7_funct_2(u1_struct_0(A),u1_struct_0(A),u1_struct_0(A),E,D) ) ) ) ) ) ) ).
fof(t21_autgroup,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v1_group_1(A)
& v3_group_1(A)
& v4_group_1(A)
& l1_group_1(A) )
=> k6_partfun1(u1_struct_0(A)) = k2_group_1(k5_autgroup(A)) ) ).
fof(t22_autgroup,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v1_group_1(A)
& v3_group_1(A)
& v4_group_1(A)
& l1_group_1(A) )
=> ! [B] :
( m2_fraenkel(B,u1_struct_0(A),u1_struct_0(A),k4_autgroup(A))
=> ! [C] :
( m1_subset_1(C,u1_struct_0(k5_autgroup(A)))
=> ( B = C
=> k2_funct_1(B) = k3_group_1(k5_autgroup(A),C) ) ) ) ) ).
fof(d6_autgroup,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v1_group_1(A)
& v3_group_1(A)
& v4_group_1(A)
& l1_group_1(A) )
=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> ! [C] :
( m2_fraenkel(C,u1_struct_0(A),u1_struct_0(A),k4_autgroup(A))
=> ( C = k6_autgroup(A,B)
<=> ! [D] :
( m1_subset_1(D,u1_struct_0(A))
=> k8_funct_2(u1_struct_0(A),u1_struct_0(A),C,D) = k2_group_3(A,D,B) ) ) ) ) ) ).
fof(t23_autgroup,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v1_group_1(A)
& v3_group_1(A)
& v4_group_1(A)
& l1_group_1(A) )
=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> k6_autgroup(A,k1_group_1(A,B,C)) = k7_funct_2(u1_struct_0(A),u1_struct_0(A),u1_struct_0(A),k6_autgroup(A,B),k6_autgroup(A,C)) ) ) ) ).
fof(t24_autgroup,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v1_group_1(A)
& v3_group_1(A)
& v4_group_1(A)
& l1_group_1(A) )
=> k6_autgroup(A,k2_group_1(A)) = k6_partfun1(u1_struct_0(A)) ) ).
fof(t25_autgroup,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v1_group_1(A)
& v3_group_1(A)
& v4_group_1(A)
& l1_group_1(A) )
=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> k8_funct_2(u1_struct_0(A),u1_struct_0(A),k6_autgroup(A,k2_group_1(A)),B) = B ) ) ).
fof(t26_autgroup,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v1_group_1(A)
& v3_group_1(A)
& v4_group_1(A)
& l1_group_1(A) )
=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> k7_funct_2(u1_struct_0(A),u1_struct_0(A),u1_struct_0(A),k6_autgroup(A,k3_group_1(A,B)),k6_autgroup(A,B)) = k6_autgroup(A,k2_group_1(A)) ) ) ).
fof(t27_autgroup,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v1_group_1(A)
& v3_group_1(A)
& v4_group_1(A)
& l1_group_1(A) )
=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> k7_funct_2(u1_struct_0(A),u1_struct_0(A),u1_struct_0(A),k6_autgroup(A,B),k6_autgroup(A,k3_group_1(A,B))) = k6_autgroup(A,k2_group_1(A)) ) ) ).
fof(t28_autgroup,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v1_group_1(A)
& v3_group_1(A)
& v4_group_1(A)
& l1_group_1(A) )
=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> k6_autgroup(A,k3_group_1(A,B)) = k2_funct_1(k6_autgroup(A,B)) ) ) ).
fof(t29_autgroup,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v1_group_1(A)
& v3_group_1(A)
& v4_group_1(A)
& l1_group_1(A) )
=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> ( k7_funct_2(u1_struct_0(A),u1_struct_0(A),u1_struct_0(A),k6_autgroup(A,k2_group_1(A)),k6_autgroup(A,B)) = k6_autgroup(A,B)
& k7_funct_2(u1_struct_0(A),u1_struct_0(A),u1_struct_0(A),k6_autgroup(A,B),k6_autgroup(A,k2_group_1(A))) = k6_autgroup(A,B) ) ) ) ).
fof(t30_autgroup,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v1_group_1(A)
& v3_group_1(A)
& v4_group_1(A)
& l1_group_1(A) )
=> ! [B] :
( m2_fraenkel(B,u1_struct_0(A),u1_struct_0(A),k4_autgroup(A))
=> ( k7_funct_2(u1_struct_0(A),u1_struct_0(A),u1_struct_0(A),k6_autgroup(A,k2_group_1(A)),B) = B
& k7_funct_2(u1_struct_0(A),u1_struct_0(A),u1_struct_0(A),B,k6_autgroup(A,k2_group_1(A))) = B ) ) ) ).
fof(t31_autgroup,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v1_group_1(A)
& v3_group_1(A)
& v4_group_1(A)
& l1_group_1(A) )
=> r2_group_6(k5_autgroup(A),k6_group_6(A,k10_group_5(A))) ) ).
fof(t32_autgroup,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v1_group_1(A)
& v3_group_1(A)
& v4_group_1(A)
& l1_group_1(A) )
=> ( ( ~ v3_struct_0(A)
& v3_group_1(A)
& v4_group_1(A)
& v7_group_1(A)
& l1_group_1(A) )
=> ! [B] :
( m1_subset_1(B,u1_struct_0(k5_autgroup(A)))
=> B = k2_group_1(k5_autgroup(A)) ) ) ) ).
fof(dt_k1_autgroup,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v1_group_1(A)
& v3_group_1(A)
& v4_group_1(A)
& l1_group_1(A) )
=> m1_fraenkel(k1_autgroup(A),u1_struct_0(A),u1_struct_0(A)) ) ).
fof(dt_k2_autgroup,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v1_group_1(A)
& v3_group_1(A)
& v4_group_1(A)
& l1_group_1(A) )
=> ( v1_funct_1(k2_autgroup(A))
& v1_funct_2(k2_autgroup(A),k2_zfmisc_1(k1_autgroup(A),k1_autgroup(A)),k1_autgroup(A))
& m2_relset_1(k2_autgroup(A),k2_zfmisc_1(k1_autgroup(A),k1_autgroup(A)),k1_autgroup(A)) ) ) ).
fof(dt_k3_autgroup,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v1_group_1(A)
& v3_group_1(A)
& v4_group_1(A)
& l1_group_1(A) )
=> ( ~ v3_struct_0(k3_autgroup(A))
& v1_group_1(k3_autgroup(A))
& v3_group_1(k3_autgroup(A))
& v4_group_1(k3_autgroup(A))
& l1_group_1(k3_autgroup(A)) ) ) ).
fof(dt_k4_autgroup,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v1_group_1(A)
& v3_group_1(A)
& v4_group_1(A)
& l1_group_1(A) )
=> m1_fraenkel(k4_autgroup(A),u1_struct_0(A),u1_struct_0(A)) ) ).
fof(dt_k5_autgroup,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v1_group_1(A)
& v3_group_1(A)
& v4_group_1(A)
& l1_group_1(A) )
=> ( v1_group_1(k5_autgroup(A))
& v1_group_3(k5_autgroup(A),k3_autgroup(A))
& m1_group_2(k5_autgroup(A),k3_autgroup(A)) ) ) ).
fof(dt_k6_autgroup,axiom,
! [A,B] :
( ( ~ v3_struct_0(A)
& v1_group_1(A)
& v3_group_1(A)
& v4_group_1(A)
& l1_group_1(A)
& m1_subset_1(B,u1_struct_0(A)) )
=> m2_fraenkel(k6_autgroup(A,B),u1_struct_0(A),u1_struct_0(A),k4_autgroup(A)) ) ).
%------------------------------------------------------------------------------