SET007 Axioms: SET007+846.ax


%------------------------------------------------------------------------------
% File     : SET007+846 : TPTP v7.5.0. Released v3.4.0.
% Domain   : Set Theory
% Axioms   : Properties of 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    : group_8 [Urb08]

% Status   : Satisfiable
% Syntax   : Number of formulae    :   28 (   0 unit)
%            Number of atoms       :  335 (  38 equality)
%            Maximal formula depth :   23 (  11 average)
%            Number of connectives :  356 (  49 ~  ;   3  |; 181  &)
%                                         (   6 <=>; 117 =>;   0 <=)
%                                         (   0 <~>;   0 ~|;   0 ~&)
%            Number of predicates  :   23 (   0 propositional; 1-3 arity)
%            Number of functors    :   26 (   3 constant; 0-3 arity)
%            Number of variables   :  105 (   0 singleton; 100 !;   5 ?)
%            Maximal term depth    :    5 (   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(fc1_group_8,axiom,(
    ! [A,B] :
      ( ( ~ v3_struct_0(A)
        & v1_group_1(A)
        & v3_group_1(A)
        & v4_group_1(A)
        & l1_group_1(A)
        & v1_group_1(B)
        & m1_group_2(B,A) )
     => ~ v1_xboole_0(k14_group_2(A,B)) ) )).

fof(t1_group_8,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v1_group_1(A)
        & v3_group_1(A)
        & v4_group_1(A)
        & l1_group_1(A) )
     => ! [B] :
          ( m2_subset_1(B,k1_numbers,k5_numbers)
         => ~ ( v1_int_2(B)
              & k9_group_1(A) = B
              & v6_group_1(A)
              & ! [C] :
                  ( m1_subset_1(C,u1_struct_0(A))
                 => k7_group_1(A,C) != B ) ) ) ) )).

fof(t2_group_8,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] :
              ( m1_subset_1(C,u1_struct_0(B))
             => ! [D] :
                  ( m1_subset_1(D,u1_struct_0(B))
                 => ! [E] :
                      ( m1_subset_1(E,u1_struct_0(A))
                     => ! [F] :
                          ( m1_subset_1(F,u1_struct_0(A))
                         => ( ( C = E
                              & D = F )
                           => k1_group_1(B,C,D) = k1_group_1(A,E,F) ) ) ) ) ) ) ) )).

fof(t3_group_8,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] :
              ( m1_subset_1(C,u1_struct_0(B))
             => ! [D] :
                  ( m1_subset_1(D,u1_struct_0(A))
                 => ( C = D
                   => ! [E] :
                        ( m2_subset_1(E,k1_numbers,k5_numbers)
                       => k6_group_1(B,E,C) = k6_group_1(A,E,D) ) ) ) ) ) ) )).

fof(t4_group_8,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] :
              ( m1_subset_1(C,u1_struct_0(B))
             => ! [D] :
                  ( m1_subset_1(D,u1_struct_0(A))
                 => ( C = D
                   => ! [E] :
                        ( v1_int_1(E)
                       => k6_group_1(B,E,C) = k6_group_1(A,E,D) ) ) ) ) ) ) )).

fof(t5_group_8,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] :
              ( m1_subset_1(C,u1_struct_0(B))
             => ! [D] :
                  ( m1_subset_1(D,u1_struct_0(A))
                 => ( ( C = D
                      & v6_group_1(A) )
                   => k7_group_1(B,C) = k7_group_1(A,D) ) ) ) ) ) )).

fof(t6_group_8,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] :
              ( m1_subset_1(C,u1_struct_0(A))
             => ( r1_rlvect_1(B,C)
               => r1_tarski(k13_group_2(A,B,C),u1_struct_0(B)) ) ) ) ) )).

fof(t7_group_8,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))
         => ~ ( B != k2_group_1(A)
              & k5_group_4(A,k1_struct_0(A,B)) = k5_group_2(A) ) ) ) )).

fof(t8_group_8,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v1_group_1(A)
        & v3_group_1(A)
        & v4_group_1(A)
        & l1_group_1(A) )
     => ! [B] :
          ( v1_int_1(B)
         => k6_group_1(A,B,k2_group_1(A)) = k2_group_1(A) ) ) )).

fof(t9_group_8,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] :
              ( v1_int_1(C)
             => k6_group_1(A,k3_xcmplx_0(C,k7_group_1(A,B)),B) = k2_group_1(A) ) ) ) )).

fof(t10_group_8,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))
         => ( ~ v5_group_1(B,A)
           => ! [C] :
                ( v1_int_1(C)
               => k6_group_1(A,C,B) = k6_group_1(A,k6_int_1(C,k7_group_1(A,B)),B) ) ) ) ) )).

fof(t11_group_8,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))
         => ( ~ v5_group_1(B,A)
           => v6_group_1(k5_group_4(A,k1_struct_0(A,B))) ) ) ) )).

fof(t12_group_8,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))
         => ( v5_group_1(B,A)
           => v5_group_1(k3_group_1(A,B),A) ) ) ) )).

fof(t13_group_8,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))
         => ( v5_group_1(B,A)
          <=> ! [C] :
                ( v1_int_1(C)
               => ( k6_group_1(A,C,B) = k2_group_1(A)
                 => C = np__0 ) ) ) ) ) )).

fof(t14_group_8,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))
             => B = k2_group_1(A) )
       => ( ! [B] :
              ( ( v1_group_1(B)
                & m1_group_2(B,A) )
             => ( B = A
                | r1_group_2(A,B,k5_group_2(A)) ) )
        <=> ( ~ v3_struct_0(A)
            & v3_group_1(A)
            & v4_group_1(A)
            & v1_gr_cy_1(A)
            & l1_group_1(A)
            & v6_group_1(A)
            & ? [B] :
                ( m2_subset_1(B,k1_numbers,k5_numbers)
                & k9_group_1(A) = B
                & v1_int_2(B) ) ) ) ) ) )).

fof(t15_group_8,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))
             => ! [D] :
                  ( m1_subset_1(D,u1_struct_0(A))
                 => ! [E] :
                      ( m1_subset_1(E,k1_zfmisc_1(u1_struct_0(A)))
                     => ( r2_hidden(D,k4_group_2(A,C,k3_group_2(A,B,E)))
                      <=> ? [F] :
                            ( m1_subset_1(F,u1_struct_0(A))
                            & D = k1_group_1(A,k1_group_1(A,B,F),C)
                            & r2_hidden(F,E) ) ) ) ) ) ) ) )).

fof(t16_group_8,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v1_group_1(A)
        & v3_group_1(A)
        & v4_group_1(A)
        & l1_group_1(A) )
     => ! [B] :
          ( ( ~ v1_xboole_0(B)
            & m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) )
         => ! [C] :
              ( m1_subset_1(C,u1_struct_0(A))
             => k1_card_1(B) = k1_card_1(k4_group_2(A,C,k3_group_2(A,k3_group_1(A,C),B))) ) ) ) )).

fof(d1_group_8,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] :
                  ( m1_subset_1(D,k1_zfmisc_1(k1_zfmisc_1(u1_struct_0(A))))
                 => ( D = k1_group_8(A,B,C)
                  <=> ! [E] :
                        ( m1_subset_1(E,k1_zfmisc_1(u1_struct_0(A)))
                       => ( r2_hidden(E,D)
                        <=> ? [F] :
                              ( m1_subset_1(F,u1_struct_0(A))
                              & E = k10_group_2(A,C,k13_group_2(A,B,F)) ) ) ) ) ) ) ) ) )).

fof(t17_group_8,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))
             => ! [D] :
                  ( ( v1_group_1(D)
                    & m1_group_2(D,A) )
                 => ! [E] :
                      ( ( v1_group_1(E)
                        & m1_group_2(E,A) )
                     => ( r2_hidden(B,k10_group_2(A,E,k13_group_2(A,D,C)))
                      <=> ? [F] :
                            ( m1_subset_1(F,u1_struct_0(A))
                            & ? [G] :
                                ( m1_subset_1(G,u1_struct_0(A))
                                & B = k1_group_1(A,k1_group_1(A,F,C),G)
                                & r1_rlvect_1(D,F)
                                & r1_rlvect_1(E,G) ) ) ) ) ) ) ) ) )).

fof(t18_group_8,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))
             => ! [D] :
                  ( ( v1_group_1(D)
                    & m1_group_2(D,A) )
                 => ! [E] :
                      ( ( v1_group_1(E)
                        & m1_group_2(E,A) )
                     => ( k10_group_2(A,E,k13_group_2(A,D,B)) = k10_group_2(A,E,k13_group_2(A,D,C))
                        | ! [F] :
                            ( m1_subset_1(F,u1_struct_0(A))
                           => ~ ( r2_hidden(F,k10_group_2(A,E,k13_group_2(A,D,B)))
                                & r2_hidden(F,k10_group_2(A,E,k13_group_2(A,D,C))) ) ) ) ) ) ) ) ) )).

fof(t19_group_8,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_6(D,A,B) )
                 => ( ( A = k8_group_4(A,B,C)
                      & r1_group_2(A,D,k9_group_2(A,B,C))
                      & v6_group_1(A) )
                   => r1_xreal_0(k17_group_2(B,D),k17_group_2(A,C)) ) ) ) ) ) )).

fof(t20_group_8,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) )
         => ~ ( v6_group_1(A)
              & r1_xreal_0(k17_group_2(A,B),np__0) ) ) ) )).

fof(t21_group_8,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v1_group_1(A)
        & v3_group_1(A)
        & v4_group_1(A)
        & l1_group_1(A) )
     => ( v6_group_1(A)
       => ! [B] :
            ( ( v1_group_1(B)
              & m1_group_2(B,A) )
           => ! [C] :
                ( ( v1_group_1(C)
                  & m1_group_6(C,A,B) )
               => ! [D] :
                    ( ( v1_group_1(D)
                      & m1_group_6(D,A,B) )
                   => ( B = k8_group_4(B,C,D)
                     => ! [E] :
                          ( ( v1_group_1(E)
                            & m1_group_6(E,B,C) )
                         => ( r1_group_2(B,E,k9_group_2(B,C,D))
                           => ! [F] :
                                ( ( v1_group_1(F)
                                  & m1_group_6(F,B,D) )
                               => ( r1_group_2(B,F,k9_group_2(B,C,D))
                                 => ! [G] :
                                      ( ( v1_group_1(G)
                                        & m1_group_6(G,A,B) )
                                     => ( ( r1_group_2(B,G,k9_group_2(B,C,D))
                                          & v1_finset_1(k14_group_2(B,D))
                                          & v1_finset_1(k14_group_2(B,C))
                                          & r2_int_2(k17_group_2(B,C),k17_group_2(B,D)) )
                                       => ( k17_group_2(B,D) = k17_group_2(C,E)
                                          & k17_group_2(B,C) = k17_group_2(D,F) ) ) ) ) ) ) ) ) ) ) ) ) ) )).

fof(t22_group_8,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] :
              ( m1_subset_1(C,u1_struct_0(A))
             => ( r1_rlvect_1(B,C)
               => ! [D] :
                    ( v1_int_1(D)
                   => r1_rlvect_1(B,k6_group_1(A,D,C)) ) ) ) ) ) )).

fof(t23_group_8,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v1_group_1(A)
        & v3_group_1(A)
        & v4_group_1(A)
        & l1_group_1(A) )
     => ~ ( A != k5_group_2(A)
          & ! [B] :
              ( m1_subset_1(B,u1_struct_0(A))
             => B = k2_group_1(A) ) ) ) )).

fof(t24_group_8,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))
         => ( A = k5_group_4(A,k1_struct_0(A,B))
           => ( A = k5_group_2(A)
              | ! [C] :
                  ( ( v1_group_1(C)
                    & m1_group_2(C,A) )
                 => ~ ( C != k5_group_2(A)
                      & ! [D] :
                          ( m2_subset_1(D,k1_numbers,k5_numbers)
                         => ~ ( ~ r1_xreal_0(D,np__0)
                              & r1_rlvect_1(C,k6_group_1(A,D,B)) ) ) ) ) ) ) ) ) )).

fof(t25_group_8,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v1_group_1(A)
        & v3_group_1(A)
        & v4_group_1(A)
        & v1_gr_cy_1(A)
        & l1_group_1(A) )
     => ( A != k5_group_2(A)
       => ! [B] :
            ( ( v1_group_1(B)
              & m1_group_2(B,A) )
           => ( B != k5_group_2(A)
             => ( ~ v3_struct_0(B)
                & v3_group_1(B)
                & v4_group_1(B)
                & v1_gr_cy_1(B)
                & l1_group_1(B) ) ) ) ) ) )).

fof(dt_k1_group_8,axiom,(
    ! [A,B,C] :
      ( ( ~ v3_struct_0(A)
        & v1_group_1(A)
        & v3_group_1(A)
        & v4_group_1(A)
        & l1_group_1(A)
        & v1_group_1(B)
        & m1_group_2(B,A)
        & v1_group_1(C)
        & m1_group_2(C,A) )
     => m1_subset_1(k1_group_8(A,B,C),k1_zfmisc_1(k1_zfmisc_1(u1_struct_0(A)))) ) )).
%------------------------------------------------------------------------------