SET007 Axioms: SET007+274.ax


%------------------------------------------------------------------------------
% File     : SET007+274 : TPTP v7.5.0. Released v3.4.0.
% Domain   : Set Theory
% Axioms   : Ternary Fields
% 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    : algstr_3 [Urb08]

% Status   : Satisfiable
% Syntax   : Number of formulae    :   37 (  10 unit)
%            Number of atoms       :  207 (  52 equality)
%            Maximal formula depth :   30 (   7 average)
%            Number of connectives :  194 (  24 ~  ;   3  |;  69  &)
%                                         (   2 <=>;  96 =>;   0 <=)
%                                         (   0 <~>;   0 ~|;   0 ~&)
%            Number of predicates  :   12 (   1 propositional; 0-3 arity)
%            Number of functors    :   18 (   5 constant; 0-8 arity)
%            Number of variables   :  110 (   0 singleton; 100 !;  10 ?)
%            Maximal term depth    :    3 (   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_algstr_3,axiom,(
    ? [A] :
      ( l1_algstr_3(A)
      & v1_algstr_3(A) ) )).

fof(rc2_algstr_3,axiom,(
    ? [A] :
      ( l1_algstr_3(A)
      & ~ v3_struct_0(A) ) )).

fof(fc1_algstr_3,axiom,
    ( ~ v3_struct_0(k4_algstr_3)
    & v1_algstr_3(k4_algstr_3) )).

fof(rc3_algstr_3,axiom,(
    ? [A] :
      ( l1_algstr_3(A)
      & ~ v3_struct_0(A)
      & v1_algstr_3(A)
      & v2_algstr_3(A) ) )).

fof(d1_algstr_3,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & l1_algstr_3(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))
                 => k1_algstr_3(A,B,C,D) = k2_multop_1(u1_struct_0(A),u1_struct_0(A),u1_struct_0(A),u1_struct_0(A),u2_algstr_3(A),B,C,D) ) ) ) ) )).

fof(d2_algstr_3,axiom,(
    $true )).

fof(d3_algstr_3,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & l1_algstr_3(A) )
     => k2_algstr_3(A) = u1_algstr_3(A) ) )).

fof(d4_algstr_3,axiom,(
    ! [A] :
      ( ( v1_funct_1(A)
        & v1_funct_2(A,k3_zfmisc_1(k1_numbers,k1_numbers,k1_numbers),k1_numbers)
        & m2_relset_1(A,k3_zfmisc_1(k1_numbers,k1_numbers,k1_numbers),k1_numbers) )
     => ( A = k3_algstr_3
      <=> ! [B] :
            ( m1_subset_1(B,k1_numbers)
           => ! [C] :
                ( m1_subset_1(C,k1_numbers)
               => ! [D] :
                    ( m1_subset_1(D,k1_numbers)
                   => k2_multop_1(k1_numbers,k1_numbers,k1_numbers,k1_numbers,A,B,C,D) = k3_real_1(k4_real_1(B,C),D) ) ) ) ) ) )).

fof(d5_algstr_3,axiom,(
    k4_algstr_3 = g1_algstr_3(k1_numbers,np__0,np__1,k3_algstr_3) )).

fof(d6_algstr_3,axiom,(
    ! [A] :
      ( m1_subset_1(A,u1_struct_0(k4_algstr_3))
     => ! [B] :
          ( m1_subset_1(B,u1_struct_0(k4_algstr_3))
         => ! [C] :
              ( m1_subset_1(C,u1_struct_0(k4_algstr_3))
             => k5_algstr_3(A,B,C) = k2_multop_1(u1_struct_0(k4_algstr_3),u1_struct_0(k4_algstr_3),u1_struct_0(k4_algstr_3),u1_struct_0(k4_algstr_3),u2_algstr_3(k4_algstr_3),A,B,C) ) ) ) )).

fof(t1_algstr_3,axiom,(
    $true )).

fof(t2_algstr_3,axiom,(
    $true )).

fof(t3_algstr_3,axiom,(
    ! [A] :
      ( m1_subset_1(A,k1_numbers)
     => ! [B] :
          ( m1_subset_1(B,k1_numbers)
         => ! [C] :
              ( m1_subset_1(C,k1_numbers)
             => ! [D] :
                  ( m1_subset_1(D,k1_numbers)
                 => ~ ( A != B
                      & ! [E] :
                          ( m1_subset_1(E,k1_numbers)
                         => k3_real_1(k4_real_1(A,E),C) != k3_real_1(k4_real_1(B,E),D) ) ) ) ) ) ) )).

fof(t4_algstr_3,axiom,(
    $true )).

fof(t5_algstr_3,axiom,(
    ! [A] :
      ( m1_subset_1(A,u1_struct_0(k4_algstr_3))
     => ! [B] :
          ( m1_subset_1(B,u1_struct_0(k4_algstr_3))
         => ! [C] :
              ( m1_subset_1(C,u1_struct_0(k4_algstr_3))
             => ! [D] :
                  ( m1_subset_1(D,k1_numbers)
                 => ! [E] :
                      ( m1_subset_1(E,k1_numbers)
                     => ! [F] :
                          ( m1_subset_1(F,k1_numbers)
                         => ( ( A = D
                              & B = E
                              & C = F )
                           => k1_algstr_3(k4_algstr_3,A,B,C) = k3_real_1(k4_real_1(D,E),F) ) ) ) ) ) ) ) )).

fof(t6_algstr_3,axiom,(
    np__0 = k1_rlvect_1(k4_algstr_3) )).

fof(t7_algstr_3,axiom,(
    np__1 = k2_algstr_3(k4_algstr_3) )).

fof(d7_algstr_3,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & l1_algstr_3(A) )
     => ( v2_algstr_3(A)
      <=> ( k1_rlvect_1(A) != k2_algstr_3(A)
          & ! [B] :
              ( m1_subset_1(B,u1_struct_0(A))
             => k1_algstr_3(A,B,k2_algstr_3(A),k1_rlvect_1(A)) = B )
          & ! [B] :
              ( m1_subset_1(B,u1_struct_0(A))
             => k1_algstr_3(A,k2_algstr_3(A),B,k1_rlvect_1(A)) = B )
          & ! [B] :
              ( m1_subset_1(B,u1_struct_0(A))
             => ! [C] :
                  ( m1_subset_1(C,u1_struct_0(A))
                 => k1_algstr_3(A,B,k1_rlvect_1(A),C) = C ) )
          & ! [B] :
              ( m1_subset_1(B,u1_struct_0(A))
             => ! [C] :
                  ( m1_subset_1(C,u1_struct_0(A))
                 => k1_algstr_3(A,k1_rlvect_1(A),B,C) = C ) )
          & ! [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,u1_struct_0(A))
                          & k1_algstr_3(A,B,C,E) = D ) ) ) )
          & ! [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,u1_struct_0(A))
                         => ( k1_algstr_3(A,B,C,D) = k1_algstr_3(A,B,C,E)
                           => D = E ) ) ) ) )
          & ! [B] :
              ( m1_subset_1(B,u1_struct_0(A))
             => ! [C] :
                  ( m1_subset_1(C,u1_struct_0(A))
                 => ( B != C
                   => ! [D] :
                        ( m1_subset_1(D,u1_struct_0(A))
                       => ! [E] :
                            ( m1_subset_1(E,u1_struct_0(A))
                           => ? [F] :
                                ( m1_subset_1(F,u1_struct_0(A))
                                & ? [G] :
                                    ( m1_subset_1(G,u1_struct_0(A))
                                    & k1_algstr_3(A,F,B,G) = D
                                    & k1_algstr_3(A,F,C,G) = E ) ) ) ) ) ) )
          & ! [B] :
              ( m1_subset_1(B,u1_struct_0(A))
             => ! [C] :
                  ( m1_subset_1(C,u1_struct_0(A))
                 => ( B != C
                   => ! [D] :
                        ( m1_subset_1(D,u1_struct_0(A))
                       => ! [E] :
                            ( m1_subset_1(E,u1_struct_0(A))
                           => ? [F] :
                                ( m1_subset_1(F,u1_struct_0(A))
                                & k1_algstr_3(A,B,F,D) = k1_algstr_3(A,C,F,E) ) ) ) ) ) )
          & ! [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,u1_struct_0(A))
                         => ! [F] :
                              ( m1_subset_1(F,u1_struct_0(A))
                             => ! [G] :
                                  ( m1_subset_1(G,u1_struct_0(A))
                                 => ~ ( k1_algstr_3(A,D,B,F) = k1_algstr_3(A,E,B,G)
                                      & k1_algstr_3(A,D,C,F) = k1_algstr_3(A,E,C,G)
                                      & B != C
                                      & D != E ) ) ) ) ) ) ) ) ) ) )).

fof(t8_algstr_3,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v2_algstr_3(A)
        & l1_algstr_3(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,u1_struct_0(A))
                     => ! [F] :
                          ( m1_subset_1(F,u1_struct_0(A))
                         => ! [G] :
                              ( m1_subset_1(G,u1_struct_0(A))
                             => ( ( k1_algstr_3(A,D,B,E) = k1_algstr_3(A,F,B,G)
                                  & k1_algstr_3(A,D,C,E) = k1_algstr_3(A,F,C,G) )
                               => ( B = C
                                  | ( D = F
                                    & E = G ) ) ) ) ) ) ) ) ) ) )).

fof(t9_algstr_3,axiom,(
    $true )).

fof(t10_algstr_3,axiom,(
    $true )).

fof(t11_algstr_3,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v2_algstr_3(A)
        & l1_algstr_3(A) )
     => ! [B] :
          ( m1_subset_1(B,u1_struct_0(A))
         => ( B != k1_rlvect_1(A)
           => ! [C] :
                ( m1_subset_1(C,u1_struct_0(A))
               => ! [D] :
                    ( m1_subset_1(D,u1_struct_0(A))
                   => ? [E] :
                        ( m1_subset_1(E,u1_struct_0(A))
                        & k1_algstr_3(A,B,E,C) = D ) ) ) ) ) ) )).

fof(t12_algstr_3,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v2_algstr_3(A)
        & l1_algstr_3(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,u1_struct_0(A))
                     => ( k1_algstr_3(A,B,C,D) = k1_algstr_3(A,B,E,D)
                       => ( B = k1_rlvect_1(A)
                          | C = E ) ) ) ) ) ) ) )).

fof(t13_algstr_3,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v2_algstr_3(A)
        & l1_algstr_3(A) )
     => ! [B] :
          ( m1_subset_1(B,u1_struct_0(A))
         => ( B != k1_rlvect_1(A)
           => ! [C] :
                ( m1_subset_1(C,u1_struct_0(A))
               => ! [D] :
                    ( m1_subset_1(D,u1_struct_0(A))
                   => ? [E] :
                        ( m1_subset_1(E,u1_struct_0(A))
                        & k1_algstr_3(A,E,B,C) = D ) ) ) ) ) ) )).

fof(t14_algstr_3,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v2_algstr_3(A)
        & l1_algstr_3(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,u1_struct_0(A))
                     => ( k1_algstr_3(A,C,B,D) = k1_algstr_3(A,E,B,D)
                       => ( B = k1_rlvect_1(A)
                          | C = E ) ) ) ) ) ) ) )).

fof(dt_l1_algstr_3,axiom,(
    ! [A] :
      ( l1_algstr_3(A)
     => l2_struct_0(A) ) )).

fof(existence_l1_algstr_3,axiom,(
    ? [A] : l1_algstr_3(A) )).

fof(abstractness_v1_algstr_3,axiom,(
    ! [A] :
      ( l1_algstr_3(A)
     => ( v1_algstr_3(A)
       => A = g1_algstr_3(u1_struct_0(A),u2_struct_0(A),u1_algstr_3(A),u2_algstr_3(A)) ) ) )).

fof(dt_k1_algstr_3,axiom,(
    ! [A,B,C,D] :
      ( ( ~ v3_struct_0(A)
        & l1_algstr_3(A)
        & m1_subset_1(B,u1_struct_0(A))
        & m1_subset_1(C,u1_struct_0(A))
        & m1_subset_1(D,u1_struct_0(A)) )
     => m1_subset_1(k1_algstr_3(A,B,C,D),u1_struct_0(A)) ) )).

fof(dt_k2_algstr_3,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & l1_algstr_3(A) )
     => m1_subset_1(k2_algstr_3(A),u1_struct_0(A)) ) )).

fof(dt_k3_algstr_3,axiom,
    ( v1_funct_1(k3_algstr_3)
    & v1_funct_2(k3_algstr_3,k3_zfmisc_1(k1_numbers,k1_numbers,k1_numbers),k1_numbers)
    & m2_relset_1(k3_algstr_3,k3_zfmisc_1(k1_numbers,k1_numbers,k1_numbers),k1_numbers) )).

fof(dt_k4_algstr_3,axiom,
    ( v1_algstr_3(k4_algstr_3)
    & l1_algstr_3(k4_algstr_3) )).

fof(dt_k5_algstr_3,axiom,(
    ! [A,B,C] :
      ( ( m1_subset_1(A,u1_struct_0(k4_algstr_3))
        & m1_subset_1(B,u1_struct_0(k4_algstr_3))
        & m1_subset_1(C,u1_struct_0(k4_algstr_3)) )
     => m1_subset_1(k5_algstr_3(A,B,C),u1_struct_0(k4_algstr_3)) ) )).

fof(dt_u1_algstr_3,axiom,(
    ! [A] :
      ( l1_algstr_3(A)
     => m1_subset_1(u1_algstr_3(A),u1_struct_0(A)) ) )).

fof(dt_u2_algstr_3,axiom,(
    ! [A] :
      ( l1_algstr_3(A)
     => ( v1_funct_1(u2_algstr_3(A))
        & v1_funct_2(u2_algstr_3(A),k3_zfmisc_1(u1_struct_0(A),u1_struct_0(A),u1_struct_0(A)),u1_struct_0(A))
        & m2_relset_1(u2_algstr_3(A),k3_zfmisc_1(u1_struct_0(A),u1_struct_0(A),u1_struct_0(A)),u1_struct_0(A)) ) ) )).

fof(dt_g1_algstr_3,axiom,(
    ! [A,B,C,D] :
      ( ( m1_subset_1(B,A)
        & m1_subset_1(C,A)
        & v1_funct_1(D)
        & v1_funct_2(D,k3_zfmisc_1(A,A,A),A)
        & m1_relset_1(D,k3_zfmisc_1(A,A,A),A) )
     => ( v1_algstr_3(g1_algstr_3(A,B,C,D))
        & l1_algstr_3(g1_algstr_3(A,B,C,D)) ) ) )).

fof(free_g1_algstr_3,axiom,(
    ! [A,B,C,D] :
      ( ( m1_subset_1(B,A)
        & m1_subset_1(C,A)
        & v1_funct_1(D)
        & v1_funct_2(D,k3_zfmisc_1(A,A,A),A)
        & m1_relset_1(D,k3_zfmisc_1(A,A,A),A) )
     => ! [E,F,G,H] :
          ( g1_algstr_3(A,B,C,D) = g1_algstr_3(E,F,G,H)
         => ( A = E
            & B = F
            & C = G
            & D = H ) ) ) )).
%------------------------------------------------------------------------------