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) ) ) )).
%------------------------------------------------------------------------------