SET007 Axioms: SET007+276.ax


%------------------------------------------------------------------------------
% File     : SET007+276 : TPTP v7.5.0. Released v3.4.0.
% Domain   : Set Theory
% Axioms   : Groups, Rings, Left- and Right-Modules
% 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    : mod_1 [Urb08]

% Status   : Satisfiable
% Syntax   : Number of formulae    :   43 (  33 unit)
%            Number of atoms       :  222 (  19 equality)
%            Maximal formula depth :   15 (   4 average)
%            Number of connectives :  203 (  24 ~  ;   2  |; 135  &)
%                                         (   2 <=>;  40 =>;   0 <=)
%                                         (   0 <~>;   0 ~|;   0 ~&)
%            Number of predicates  :   21 (   1 propositional; 0-2 arity)
%            Number of functors    :   10 (   0 constant; 1-4 arity)
%            Number of variables   :   38 (   0 singleton;  38 !;   0 ?)
%            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(t1_mod_1,axiom,(
    $true )).

fof(t2_mod_1,axiom,(
    $true )).

fof(t3_mod_1,axiom,(
    $true )).

fof(t4_mod_1,axiom,(
    $true )).

fof(t5_mod_1,axiom,(
    $true )).

fof(t6_mod_1,axiom,(
    $true )).

fof(t7_mod_1,axiom,(
    $true )).

fof(t8_mod_1,axiom,(
    $true )).

fof(t9_mod_1,axiom,(
    $true )).

fof(t10_mod_1,axiom,(
    $true )).

fof(t11_mod_1,axiom,(
    $true )).

fof(t12_mod_1,axiom,(
    $true )).

fof(t13_mod_1,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v4_rlvect_1(A)
        & v5_rlvect_1(A)
        & v6_rlvect_1(A)
        & v4_vectsp_1(A)
        & v6_vectsp_1(A)
        & l3_vectsp_1(A) )
     => ! [B] :
          ( m1_subset_1(B,u1_struct_0(A))
         => k1_group_1(A,B,k5_rlvect_1(A,k2_group_1(A))) = k5_rlvect_1(A,B) ) ) )).

fof(t14_mod_1,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v4_rlvect_1(A)
        & v5_rlvect_1(A)
        & v6_rlvect_1(A)
        & v5_vectsp_1(A)
        & v8_vectsp_1(A)
        & l3_vectsp_1(A) )
     => ! [B] :
          ( m1_subset_1(B,u1_struct_0(A))
         => k1_group_1(A,k5_rlvect_1(A,k2_group_1(A)),B) = k5_rlvect_1(A,B) ) ) )).

fof(t15_mod_1,axiom,(
    $true )).

fof(t16_mod_1,axiom,(
    $true )).

fof(t17_mod_1,axiom,(
    $true )).

fof(t18_mod_1,axiom,(
    $true )).

fof(t19_mod_1,axiom,(
    $true )).

fof(t20_mod_1,axiom,(
    $true )).

fof(t21_mod_1,axiom,(
    $true )).

fof(t22_mod_1,axiom,(
    $true )).

fof(t23_mod_1,axiom,(
    $true )).

fof(t24_mod_1,axiom,(
    $true )).

fof(t25_mod_1,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v3_rlvect_1(A)
        & v4_rlvect_1(A)
        & v5_rlvect_1(A)
        & v6_rlvect_1(A)
        & v4_group_1(A)
        & v6_vectsp_1(A)
        & v7_vectsp_1(A)
        & v8_vectsp_1(A)
        & v9_vectsp_1(A)
        & ~ v10_vectsp_1(A)
        & l3_vectsp_1(A) )
     => ! [B] :
          ( m1_subset_1(B,u1_struct_0(A))
         => ! [C] :
              ( ( ~ v3_struct_0(C)
                & v4_rlvect_1(C)
                & v5_rlvect_1(C)
                & v6_rlvect_1(C)
                & v12_vectsp_1(C,A)
                & l4_vectsp_1(C,A) )
             => ! [D] :
                  ( m1_subset_1(D,u1_struct_0(C))
                 => ( k6_vectsp_1(A,C,B,D) = k1_rlvect_1(C)
                  <=> ( B = k1_rlvect_1(A)
                      | D = k1_rlvect_1(C) ) ) ) ) ) ) )).

fof(t26_mod_1,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v3_rlvect_1(A)
        & v4_rlvect_1(A)
        & v5_rlvect_1(A)
        & v6_rlvect_1(A)
        & v4_group_1(A)
        & v6_vectsp_1(A)
        & v7_vectsp_1(A)
        & v8_vectsp_1(A)
        & v9_vectsp_1(A)
        & ~ v10_vectsp_1(A)
        & l3_vectsp_1(A) )
     => ! [B] :
          ( m1_subset_1(B,u1_struct_0(A))
         => ! [C] :
              ( ( ~ v3_struct_0(C)
                & v4_rlvect_1(C)
                & v5_rlvect_1(C)
                & v6_rlvect_1(C)
                & v12_vectsp_1(C,A)
                & l4_vectsp_1(C,A) )
             => ! [D] :
                  ( m1_subset_1(D,u1_struct_0(C))
                 => ( B != k1_rlvect_1(A)
                   => k6_vectsp_1(A,C,k1_vectsp_2(A,B),k6_vectsp_1(A,C,B,D)) = D ) ) ) ) ) )).

fof(t27_mod_1,axiom,(
    $true )).

fof(t28_mod_1,axiom,(
    $true )).

fof(t29_mod_1,axiom,(
    $true )).

fof(t30_mod_1,axiom,(
    $true )).

fof(t31_mod_1,axiom,(
    $true )).

fof(t32_mod_1,axiom,(
    $true )).

fof(t33_mod_1,axiom,(
    $true )).

fof(t34_mod_1,axiom,(
    $true )).

fof(t35_mod_1,axiom,(
    $true )).

fof(t36_mod_1,axiom,(
    $true )).

fof(t37_mod_1,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v3_rlvect_1(A)
        & v4_rlvect_1(A)
        & v5_rlvect_1(A)
        & v6_rlvect_1(A)
        & v4_group_1(A)
        & v6_vectsp_1(A)
        & v7_vectsp_1(A)
        & v8_vectsp_1(A)
        & l3_vectsp_1(A) )
     => ! [B] :
          ( ( ~ v3_struct_0(B)
            & v4_rlvect_1(B)
            & v5_rlvect_1(B)
            & v6_rlvect_1(B)
            & v5_vectsp_2(B,A)
            & l1_vectsp_2(B,A) )
         => ! [C] :
              ( m1_subset_1(C,u1_struct_0(A))
             => ! [D] :
                  ( m1_subset_1(D,u1_struct_0(B))
                 => ( k6_vectsp_2(A,B,k1_rlvect_1(A),D) = k1_rlvect_1(B)
                    & k6_vectsp_2(A,B,k5_rlvect_1(A,k2_group_1(A)),D) = k5_rlvect_1(B,D)
                    & k6_vectsp_2(A,B,C,k1_rlvect_1(B)) = k1_rlvect_1(B) ) ) ) ) ) )).

fof(t38_mod_1,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v3_rlvect_1(A)
        & v4_rlvect_1(A)
        & v5_rlvect_1(A)
        & v6_rlvect_1(A)
        & v4_group_1(A)
        & v6_vectsp_1(A)
        & v7_vectsp_1(A)
        & v8_vectsp_1(A)
        & l3_vectsp_1(A) )
     => ! [B] :
          ( ( ~ v3_struct_0(B)
            & v4_rlvect_1(B)
            & v5_rlvect_1(B)
            & v6_rlvect_1(B)
            & v5_vectsp_2(B,A)
            & l1_vectsp_2(B,A) )
         => ! [C] :
              ( m1_subset_1(C,u1_struct_0(A))
             => ! [D] :
                  ( m1_subset_1(D,u1_struct_0(B))
                 => ! [E] :
                      ( m1_subset_1(E,u1_struct_0(B))
                     => ( k5_rlvect_1(B,k6_vectsp_2(A,B,C,D)) = k6_vectsp_2(A,B,k5_rlvect_1(A,C),D)
                        & k6_rlvect_1(B,E,k6_vectsp_2(A,B,C,D)) = k2_rlvect_1(B,E,k6_vectsp_2(A,B,k5_rlvect_1(A,C),D)) ) ) ) ) ) ) )).

fof(t39_mod_1,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v3_rlvect_1(A)
        & v4_rlvect_1(A)
        & v5_rlvect_1(A)
        & v6_rlvect_1(A)
        & v4_group_1(A)
        & v6_vectsp_1(A)
        & v7_vectsp_1(A)
        & v8_vectsp_1(A)
        & l3_vectsp_1(A) )
     => ! [B] :
          ( ( ~ v3_struct_0(B)
            & v4_rlvect_1(B)
            & v5_rlvect_1(B)
            & v6_rlvect_1(B)
            & v5_vectsp_2(B,A)
            & l1_vectsp_2(B,A) )
         => ! [C] :
              ( m1_subset_1(C,u1_struct_0(A))
             => ! [D] :
                  ( m1_subset_1(D,u1_struct_0(B))
                 => k6_vectsp_2(A,B,C,k5_rlvect_1(B,D)) = k5_rlvect_1(B,k6_vectsp_2(A,B,C,D)) ) ) ) ) )).

fof(t40_mod_1,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v3_rlvect_1(A)
        & v4_rlvect_1(A)
        & v5_rlvect_1(A)
        & v6_rlvect_1(A)
        & v4_group_1(A)
        & v6_vectsp_1(A)
        & v7_vectsp_1(A)
        & v8_vectsp_1(A)
        & l3_vectsp_1(A) )
     => ! [B] :
          ( ( ~ v3_struct_0(B)
            & v4_rlvect_1(B)
            & v5_rlvect_1(B)
            & v6_rlvect_1(B)
            & v5_vectsp_2(B,A)
            & l1_vectsp_2(B,A) )
         => ! [C] :
              ( m1_subset_1(C,u1_struct_0(A))
             => ! [D] :
                  ( m1_subset_1(D,u1_struct_0(B))
                 => ! [E] :
                      ( m1_subset_1(E,u1_struct_0(B))
                     => k6_vectsp_2(A,B,C,k6_rlvect_1(B,D,E)) = k6_rlvect_1(B,k6_vectsp_2(A,B,C,D),k6_vectsp_2(A,B,C,E)) ) ) ) ) ) )).

fof(t41_mod_1,axiom,(
    $true )).

fof(t42_mod_1,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v3_rlvect_1(A)
        & v4_rlvect_1(A)
        & v5_rlvect_1(A)
        & v6_rlvect_1(A)
        & v4_group_1(A)
        & v6_vectsp_1(A)
        & v7_vectsp_1(A)
        & v8_vectsp_1(A)
        & v9_vectsp_1(A)
        & ~ v10_vectsp_1(A)
        & l3_vectsp_1(A) )
     => ! [B] :
          ( m1_subset_1(B,u1_struct_0(A))
         => ! [C] :
              ( ( ~ v3_struct_0(C)
                & v4_rlvect_1(C)
                & v5_rlvect_1(C)
                & v6_rlvect_1(C)
                & v5_vectsp_2(C,A)
                & l1_vectsp_2(C,A) )
             => ! [D] :
                  ( m1_subset_1(D,u1_struct_0(C))
                 => ( k6_vectsp_2(A,C,B,D) = k1_rlvect_1(C)
                  <=> ( B = k1_rlvect_1(A)
                      | D = k1_rlvect_1(C) ) ) ) ) ) ) )).

fof(t43_mod_1,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v3_rlvect_1(A)
        & v4_rlvect_1(A)
        & v5_rlvect_1(A)
        & v6_rlvect_1(A)
        & v4_group_1(A)
        & v6_vectsp_1(A)
        & v7_vectsp_1(A)
        & v8_vectsp_1(A)
        & v9_vectsp_1(A)
        & ~ v10_vectsp_1(A)
        & l3_vectsp_1(A) )
     => ! [B] :
          ( m1_subset_1(B,u1_struct_0(A))
         => ! [C] :
              ( ( ~ v3_struct_0(C)
                & v4_rlvect_1(C)
                & v5_rlvect_1(C)
                & v6_rlvect_1(C)
                & v5_vectsp_2(C,A)
                & l1_vectsp_2(C,A) )
             => ! [D] :
                  ( m1_subset_1(D,u1_struct_0(C))
                 => ( B != k1_rlvect_1(A)
                   => k6_vectsp_2(A,C,k1_vectsp_2(A,B),k6_vectsp_2(A,C,B,D)) = D ) ) ) ) ) )).
%------------------------------------------------------------------------------