## SET007 Axioms: SET007+410.ax

```%------------------------------------------------------------------------------
% File     : SET007+410 : TPTP v7.5.0. Released v3.4.0.
% Domain   : Set Theory
% Axioms   : Solvable Groups
% 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    : grsolv_1 [Urb08]

% Status   : Satisfiable
% Syntax   : Number of formulae    :   23 (   0 unit)
%            Number of atoms       :  400 (  21 equality)
%            Maximal formula depth :   27 (  13 average)
%            Number of connectives :  422 (  45 ~  ;   0  |; 275  &)
%                                         (   1 <=>; 101 =>;   0 <=)
%                                         (   0 <~>;   0 ~|;   0 ~&)
%            Number of predicates  :   25 (   0 propositional; 1-3 arity)
%            Number of functors    :   27 (   4 constant; 0-4 arity)
%            Number of variables   :   97 (   0 singleton;  94 !;   3 ?)
%            Maximal term depth    :    4 (   1 average)
% 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_grsolv_1,axiom,(
? [A] :
( l1_group_1(A)
& ~ v3_struct_0(A)
& v1_group_1(A)
& v2_group_1(A)
& v3_group_1(A)
& v4_group_1(A)
& v1_grsolv_1(A) ) )).

fof(d1_grsolv_1,axiom,(
! [A] :
( ( ~ v3_struct_0(A)
& v3_group_1(A)
& v4_group_1(A)
& l1_group_1(A) )
=> ( v1_grsolv_1(A)
<=> ? [B] :
( m2_finseq_1(B,k1_group_3(A))
& ~ r1_xreal_0(k3_finseq_1(B),np__0)
& k1_funct_1(B,np__1) = k6_group_2(A)
& k1_funct_1(B,k3_finseq_1(B)) = k5_group_2(A)
& ! [C] :
( m2_subset_1(C,k1_numbers,k5_numbers)
=> ( ( r2_hidden(C,k4_finseq_1(B))
& r2_hidden(k2_xcmplx_0(C,np__1),k4_finseq_1(B)) )
=> ! [D] :
( ( v1_group_1(D)
& m1_group_2(D,A) )
=> ! [E] :
( ( v1_group_1(E)
& m1_group_2(E,A) )
=> ( ( D = k1_funct_1(B,C)
& E = k1_funct_1(B,k2_xcmplx_0(C,np__1)) )
=> ( v1_group_1(E)
& v1_group_3(E,D)
& m1_group_6(E,A,D)
& ! [F] :
( ( v1_group_3(F,D)
& m1_group_6(F,A,D) )
=> ( F = E
=> v7_group_1(k6_group_6(D,F)) ) ) ) ) ) ) ) ) ) ) ) )).

fof(t1_grsolv_1,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)
& m1_group_2(B,A) )
=> ! [C] :
( ( v1_group_1(C)
& m1_group_2(C,A) )
=> ! [D] :
( ( v1_group_1(D)
& m1_group_2(D,A) )
=> ( ( v1_group_3(C,D)
& m1_group_6(C,A,D) )
=> ( v1_group_3(k9_group_2(A,C,B),k9_group_2(A,D,B))
& m1_group_6(k9_group_2(A,C,B),A,k9_group_2(A,D,B)) ) ) ) ) ) ) )).

fof(t2_grsolv_1,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)
& m1_group_2(B,A) )
=> ! [C] :
( ( v1_group_1(C)
& v1_group_3(C,B)
& m1_group_6(C,A,B) )
=> ! [D] :
( m1_subset_1(D,u1_struct_0(B))
=> ! [E] :
( m1_subset_1(E,u1_struct_0(B))
=> k2_group_2(B,k12_group_2(B,C,D),k12_group_2(B,C,E)) = k12_group_2(B,C,k1_group_1(B,D,E)) ) ) ) ) ) )).

fof(t3_grsolv_1,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)
& m1_group_2(B,A) )
=> ! [C] :
( ( v1_group_1(C)
& m1_group_2(C,A) )
=> ! [D] :
( ( v1_group_1(D)
& v1_group_3(D,C)
& m1_group_6(D,A,C) )
=> ! [E] :
( ( v1_group_1(E)
& m1_group_2(E,A) )
=> ( r1_group_2(A,E,k9_group_2(A,B,C))
=> ! [F] :
( ( v1_group_3(F,E)
& m1_group_6(F,A,E) )
=> ~ ( F = k2_group_6(A,B,D)
& ! [G] :
( m1_group_2(G,k6_group_6(C,D))
=> ~ r1_group_6(k6_group_6(E,F),G) ) ) ) ) ) ) ) ) ) )).

fof(t4_grsolv_1,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)
& m1_group_2(B,A) )
=> ! [C] :
( ( v1_group_1(C)
& m1_group_2(C,A) )
=> ! [D] :
( ( v1_group_1(D)
& v1_group_3(D,C)
& m1_group_6(D,A,C) )
=> ! [E] :
( ( v1_group_1(E)
& m1_group_2(E,A) )
=> ( r1_group_2(A,E,k9_group_2(A,C,B))
=> ! [F] :
( ( v1_group_3(F,E)
& m1_group_6(F,A,E) )
=> ~ ( F = k9_group_2(A,D,B)
& ! [G] :
( m1_group_2(G,k6_group_6(C,D))
=> ~ r1_group_6(k6_group_6(E,F),G) ) ) ) ) ) ) ) ) ) )).

fof(t5_grsolv_1,axiom,(
! [A] :
( ( ~ v3_struct_0(A)
& v1_group_1(A)
& v3_group_1(A)
& v4_group_1(A)
& v1_grsolv_1(A)
& l1_group_1(A) )
=> ! [B] :
( ( v1_group_1(B)
& m1_group_2(B,A) )
=> v1_grsolv_1(B) ) ) )).

fof(t6_grsolv_1,axiom,(
! [A] :
( ( ~ v3_struct_0(A)
& v1_group_1(A)
& v3_group_1(A)
& v4_group_1(A)
& l1_group_1(A) )
=> ( ? [B] :
( m2_finseq_1(B,k1_group_3(A))
& ~ r1_xreal_0(k3_finseq_1(B),np__0)
& k1_funct_1(B,np__1) = k6_group_2(A)
& k1_funct_1(B,k3_finseq_1(B)) = k5_group_2(A)
& ! [C] :
( m2_subset_1(C,k1_numbers,k5_numbers)
=> ( ( r2_hidden(C,k4_finseq_1(B))
& r2_hidden(k2_xcmplx_0(C,np__1),k4_finseq_1(B)) )
=> ! [D] :
( ( v1_group_1(D)
& m1_group_2(D,A) )
=> ! [E] :
( ( v1_group_1(E)
& m1_group_2(E,A) )
=> ( ( D = k1_funct_1(B,C)
& E = k1_funct_1(B,k2_xcmplx_0(C,np__1)) )
=> ( v1_group_1(E)
& v1_group_3(E,D)
& m1_group_6(E,A,D)
& ! [F] :
( ( v1_group_3(F,D)
& m1_group_6(F,A,D) )
=> ( F = E
=> ( ~ v3_struct_0(k6_group_6(D,F))
& v3_group_1(k6_group_6(D,F))
& v4_group_1(k6_group_6(D,F))
& v1_gr_cy_1(k6_group_6(D,F))
& l1_group_1(k6_group_6(D,F)) ) ) ) ) ) ) ) ) ) )
=> v1_grsolv_1(A) ) ) )).

fof(t7_grsolv_1,axiom,(
! [A] :
( ( ~ v3_struct_0(A)
& v1_group_1(A)
& v3_group_1(A)
& v4_group_1(A)
& v7_group_1(A)
& l1_group_1(A) )
=> ( v1_group_1(A)
& v1_grsolv_1(A) ) ) )).

fof(d2_grsolv_1,axiom,(
! [A] :
( ( ~ v3_struct_0(A)
& v1_group_1(A)
& v3_group_1(A)
& v4_group_1(A)
& l1_group_1(A) )
=> ! [B] :
( ( ~ v3_struct_0(B)
& v1_group_1(B)
& v3_group_1(B)
& v4_group_1(B)
& l1_group_1(B) )
=> ! [C] :
( ( v1_funct_1(C)
& v1_funct_2(C,u1_struct_0(A),u1_struct_0(B))
& v1_group_6(C,A,B)
& m2_relset_1(C,u1_struct_0(A),u1_struct_0(B)) )
=> ! [D] :
( m1_group_2(D,A)
=> k1_grsolv_1(A,B,C,D) = k7_relat_1(C,u1_struct_0(D)) ) ) ) ) )).

fof(d3_grsolv_1,axiom,(
! [A] :
( ( ~ v3_struct_0(A)
& v1_group_1(A)
& v3_group_1(A)
& v4_group_1(A)
& l1_group_1(A) )
=> ! [B] :
( ( ~ v3_struct_0(B)
& v1_group_1(B)
& v3_group_1(B)
& v4_group_1(B)
& l1_group_1(B) )
=> ! [C] :
( ( v1_funct_1(C)
& v1_funct_2(C,u1_struct_0(A),u1_struct_0(B))
& v1_group_6(C,A,B)
& m2_relset_1(C,u1_struct_0(A),u1_struct_0(B)) )
=> ! [D] :
( m1_group_2(D,A)
=> k2_grsolv_1(A,B,C,D) = k13_group_6(D,B,k1_grsolv_1(A,B,C,D)) ) ) ) ) )).

fof(t8_grsolv_1,axiom,(
! [A] :
( ( ~ v3_struct_0(A)
& v1_group_1(A)
& v3_group_1(A)
& v4_group_1(A)
& l1_group_1(A) )
=> ! [B] :
( ( ~ v3_struct_0(B)
& v1_group_1(B)
& v3_group_1(B)
& v4_group_1(B)
& l1_group_1(B) )
=> ! [C] :
( ( v1_funct_1(C)
& v1_funct_2(C,u1_struct_0(A),u1_struct_0(B))
& v1_group_6(C,A,B)
& m2_relset_1(C,u1_struct_0(A),u1_struct_0(B)) )
=> ! [D] :
( m1_group_2(D,A)
=> k9_group_6(D,B,k1_grsolv_1(A,B,C,D)) = k2_funct_2(u1_struct_0(A),u1_struct_0(B),C,u1_struct_0(D)) ) ) ) ) )).

fof(t9_grsolv_1,axiom,(
! [A] :
( ( ~ v3_struct_0(A)
& v1_group_1(A)
& v3_group_1(A)
& v4_group_1(A)
& l1_group_1(A) )
=> ! [B] :
( ( ~ v3_struct_0(B)
& v1_group_1(B)
& v3_group_1(B)
& v4_group_1(B)
& l1_group_1(B) )
=> ! [C] :
( ( v1_funct_1(C)
& v1_funct_2(C,u1_struct_0(A),u1_struct_0(B))
& v1_group_6(C,A,B)
& m2_relset_1(C,u1_struct_0(A),u1_struct_0(B)) )
=> ! [D] :
( ( v1_group_1(D)
& m1_group_2(D,A) )
=> u1_struct_0(k2_grsolv_1(A,B,C,D)) = k2_funct_2(u1_struct_0(A),u1_struct_0(B),C,u1_struct_0(D)) ) ) ) ) )).

fof(t10_grsolv_1,axiom,(
! [A] :
( ( ~ v3_struct_0(A)
& v1_group_1(A)
& v3_group_1(A)
& v4_group_1(A)
& l1_group_1(A) )
=> ! [B] :
( ( ~ v3_struct_0(B)
& v1_group_1(B)
& v3_group_1(B)
& v4_group_1(B)
& l1_group_1(B) )
=> ! [C] :
( ( v1_funct_1(C)
& v1_funct_2(C,u1_struct_0(A),u1_struct_0(B))
& v1_group_6(C,A,B)
& m2_relset_1(C,u1_struct_0(A),u1_struct_0(B)) )
=> ! [D] :
( ( v1_group_1(D)
& m1_group_2(D,A) )
=> ( v1_group_1(k13_group_6(D,B,k1_grsolv_1(A,B,C,D)))
& m1_group_6(k13_group_6(D,B,k1_grsolv_1(A,B,C,D)),B,k13_group_6(A,B,C)) ) ) ) ) ) )).

fof(t11_grsolv_1,axiom,(
! [A] :
( ( ~ v3_struct_0(A)
& v1_group_1(A)
& v3_group_1(A)
& v4_group_1(A)
& l1_group_1(A) )
=> ! [B] :
( ( ~ v3_struct_0(B)
& v1_group_1(B)
& v3_group_1(B)
& v4_group_1(B)
& l1_group_1(B) )
=> ! [C] :
( ( v1_funct_1(C)
& v1_funct_2(C,u1_struct_0(A),u1_struct_0(B))
& v1_group_6(C,A,B)
& m2_relset_1(C,u1_struct_0(A),u1_struct_0(B)) )
=> ! [D] :
( ( v1_group_1(D)
& m1_group_2(D,A) )
=> ( v1_group_1(k2_grsolv_1(A,B,C,D))
& m1_group_6(k2_grsolv_1(A,B,C,D),B,k13_group_6(A,B,C)) ) ) ) ) ) )).

fof(t12_grsolv_1,axiom,(
! [A] :
( ( ~ v3_struct_0(A)
& v1_group_1(A)
& v3_group_1(A)
& v4_group_1(A)
& l1_group_1(A) )
=> ! [B] :
( ( ~ v3_struct_0(B)
& v1_group_1(B)
& v3_group_1(B)
& v4_group_1(B)
& l1_group_1(B) )
=> ! [C] :
( ( v1_funct_1(C)
& v1_funct_2(C,u1_struct_0(A),u1_struct_0(B))
& v1_group_6(C,A,B)
& m2_relset_1(C,u1_struct_0(A),u1_struct_0(B)) )
=> ( r1_group_2(B,k2_grsolv_1(A,B,C,k5_group_2(A)),k5_group_2(B))
& k2_grsolv_1(A,B,C,k6_group_2(A)) = k6_group_2(k13_group_6(A,B,C)) ) ) ) ) )).

fof(t13_grsolv_1,axiom,(
! [A] :
( ( ~ v3_struct_0(A)
& v1_group_1(A)
& v3_group_1(A)
& v4_group_1(A)
& l1_group_1(A) )
=> ! [B] :
( ( ~ v3_struct_0(B)
& v1_group_1(B)
& v3_group_1(B)
& v4_group_1(B)
& l1_group_1(B) )
=> ! [C] :
( ( v1_funct_1(C)
& v1_funct_2(C,u1_struct_0(A),u1_struct_0(B))
& v1_group_6(C,A,B)
& m2_relset_1(C,u1_struct_0(A),u1_struct_0(B)) )
=> ! [D] :
( ( v1_group_1(D)
& m1_group_2(D,A) )
=> ! [E] :
( ( v1_group_1(E)
& m1_group_2(E,A) )
=> ( m1_group_6(D,A,E)
=> m1_group_6(k2_grsolv_1(A,B,C,D),B,k2_grsolv_1(A,B,C,E)) ) ) ) ) ) ) )).

fof(t14_grsolv_1,axiom,(
! [A] :
( ( ~ v3_struct_0(A)
& v1_group_1(A)
& v3_group_1(A)
& v4_group_1(A)
& l1_group_1(A) )
=> ! [B] :
( ( ~ v3_struct_0(B)
& v1_group_1(B)
& v3_group_1(B)
& v4_group_1(B)
& l1_group_1(B) )
=> ! [C] :
( ( v1_funct_1(C)
& v1_funct_2(C,u1_struct_0(A),u1_struct_0(B))
& v1_group_6(C,A,B)
& m2_relset_1(C,u1_struct_0(A),u1_struct_0(B)) )
=> ! [D] :
( ( v1_group_1(D)
& m1_group_2(D,A) )
=> ! [E] :
( m1_subset_1(E,u1_struct_0(A))
=> ( k12_group_2(B,k2_grsolv_1(A,B,C,D),k8_funct_2(u1_struct_0(A),u1_struct_0(B),C,E)) = k2_funct_2(u1_struct_0(A),u1_struct_0(B),C,k12_group_2(A,D,E))
& k13_group_2(B,k2_grsolv_1(A,B,C,D),k8_funct_2(u1_struct_0(A),u1_struct_0(B),C,E)) = k2_funct_2(u1_struct_0(A),u1_struct_0(B),C,k13_group_2(A,D,E)) ) ) ) ) ) ) )).

fof(t15_grsolv_1,axiom,(
! [A] :
( ( ~ v3_struct_0(A)
& v1_group_1(A)
& v3_group_1(A)
& v4_group_1(A)
& l1_group_1(A) )
=> ! [B] :
( ( ~ v3_struct_0(B)
& v1_group_1(B)
& v3_group_1(B)
& v4_group_1(B)
& l1_group_1(B) )
=> ! [C] :
( ( v1_funct_1(C)
& v1_funct_2(C,u1_struct_0(A),u1_struct_0(B))
& v1_group_6(C,A,B)
& m2_relset_1(C,u1_struct_0(A),u1_struct_0(B)) )
=> ! [D] :
( m1_subset_1(D,k1_zfmisc_1(u1_struct_0(A)))
=> ! [E] :
( m1_subset_1(E,k1_zfmisc_1(u1_struct_0(A)))
=> k2_group_2(B,k2_funct_2(u1_struct_0(A),u1_struct_0(B),C,D),k2_funct_2(u1_struct_0(A),u1_struct_0(B),C,E)) = k2_funct_2(u1_struct_0(A),u1_struct_0(B),C,k2_group_2(A,D,E)) ) ) ) ) ) )).

fof(t16_grsolv_1,axiom,(
! [A] :
( ( ~ v3_struct_0(A)
& v1_group_1(A)
& v3_group_1(A)
& v4_group_1(A)
& l1_group_1(A) )
=> ! [B] :
( ( ~ v3_struct_0(B)
& v1_group_1(B)
& v3_group_1(B)
& v4_group_1(B)
& l1_group_1(B) )
=> ! [C] :
( ( v1_funct_1(C)
& v1_funct_2(C,u1_struct_0(A),u1_struct_0(B))
& v1_group_6(C,A,B)
& m2_relset_1(C,u1_struct_0(A),u1_struct_0(B)) )
=> ! [D] :
( ( v1_group_1(D)
& m1_group_2(D,A) )
=> ! [E] :
( ( v1_group_1(E)
& m1_group_2(E,A) )
=> ( ( v1_group_1(D)
& v1_group_3(D,E)
& m1_group_6(D,A,E) )
=> ( v1_group_1(k2_grsolv_1(A,B,C,D))
& v1_group_3(k2_grsolv_1(A,B,C,D),k2_grsolv_1(A,B,C,E))
& m1_group_6(k2_grsolv_1(A,B,C,D),B,k2_grsolv_1(A,B,C,E)) ) ) ) ) ) ) ) )).

fof(t17_grsolv_1,axiom,(
! [A] :
( ( ~ v3_struct_0(A)
& v1_group_1(A)
& v3_group_1(A)
& v4_group_1(A)
& l1_group_1(A) )
=> ! [B] :
( ( ~ v3_struct_0(B)
& v1_group_1(B)
& v3_group_1(B)
& v4_group_1(B)
& l1_group_1(B) )
=> ! [C] :
( ( v1_funct_1(C)
& v1_funct_2(C,u1_struct_0(A),u1_struct_0(B))
& v1_group_6(C,A,B)
& m2_relset_1(C,u1_struct_0(A),u1_struct_0(B)) )
=> ( ( ~ v3_struct_0(A)
& v3_group_1(A)
& v4_group_1(A)
& v1_grsolv_1(A)
& l1_group_1(A) )
=> v1_grsolv_1(k13_group_6(A,B,C)) ) ) ) ) )).

fof(dt_k1_grsolv_1,axiom,(
! [A,B,C,D] :
( ( ~ v3_struct_0(A)
& v1_group_1(A)
& v3_group_1(A)
& v4_group_1(A)
& l1_group_1(A)
& ~ v3_struct_0(B)
& v1_group_1(B)
& v3_group_1(B)
& v4_group_1(B)
& l1_group_1(B)
& v1_funct_1(C)
& v1_funct_2(C,u1_struct_0(A),u1_struct_0(B))
& v1_group_6(C,A,B)
& m1_relset_1(C,u1_struct_0(A),u1_struct_0(B))
& m1_group_2(D,A) )
=> ( v1_funct_1(k1_grsolv_1(A,B,C,D))
& v1_funct_2(k1_grsolv_1(A,B,C,D),u1_struct_0(D),u1_struct_0(B))
& v1_group_6(k1_grsolv_1(A,B,C,D),D,B)
& m2_relset_1(k1_grsolv_1(A,B,C,D),u1_struct_0(D),u1_struct_0(B)) ) ) )).

fof(dt_k2_grsolv_1,axiom,(
! [A,B,C,D] :
( ( ~ v3_struct_0(A)
& v1_group_1(A)
& v3_group_1(A)
& v4_group_1(A)
& l1_group_1(A)
& ~ v3_struct_0(B)
& v1_group_1(B)
& v3_group_1(B)
& v4_group_1(B)
& l1_group_1(B)
& v1_funct_1(C)
& v1_funct_2(C,u1_struct_0(A),u1_struct_0(B))
& v1_group_6(C,A,B)
& m1_relset_1(C,u1_struct_0(A),u1_struct_0(B))
& m1_group_2(D,A) )
=> ( v1_group_1(k2_grsolv_1(A,B,C,D))
& m1_group_2(k2_grsolv_1(A,B,C,D),B) ) ) )).
%------------------------------------------------------------------------------
```