SET007 Axioms: SET007+245.ax


%------------------------------------------------------------------------------
% File     : SET007+245 : TPTP v7.5.0. Released v3.4.0.
% Domain   : Set Theory
% Axioms   : Finite Sums of Vectors in Vector Space
% 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    : vectsp_3 [Urb08]

% Status   : Satisfiable
% Syntax   : Number of formulae    :   35 (  23 unit)
%            Number of atoms       :  211 (  23 equality)
%            Maximal formula depth :   19 (   5 average)
%            Number of connectives :  194 (  18 ~  ;   0  |; 121  &)
%                                         (   0 <=>;  55 =>;   0 <=)
%                                         (   0 <~>;   0 ~|;   0 ~&)
%            Number of predicates  :   18 (   1 propositional; 0-3 arity)
%            Number of functors    :   16 (   2 constant; 0-4 arity)
%            Number of variables   :   49 (   0 singleton;  49 !;   0 ?)
%            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(t1_vectsp_3,axiom,(
    $true )).

fof(t2_vectsp_3,axiom,(
    $true )).

fof(t3_vectsp_3,axiom,(
    $true )).

fof(t4_vectsp_3,axiom,(
    $true )).

fof(t5_vectsp_3,axiom,(
    $true )).

fof(t6_vectsp_3,axiom,(
    $true )).

fof(t7_vectsp_3,axiom,(
    $true )).

fof(t8_vectsp_3,axiom,(
    $true )).

fof(t9_vectsp_3,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)
        & v7_vectsp_1(A)
        & v8_vectsp_1(A)
        & l3_vectsp_1(A) )
     => ! [B] :
          ( m1_subset_1(B,u1_struct_0(A))
         => ! [C] :
              ( ( ~ v3_struct_0(C)
                & v3_rlvect_1(C)
                & v4_rlvect_1(C)
                & v5_rlvect_1(C)
                & v6_rlvect_1(C)
                & v12_vectsp_1(C,A)
                & l4_vectsp_1(C,A) )
             => ! [D] :
                  ( m2_finseq_1(D,u1_struct_0(C))
                 => ! [E] :
                      ( m2_finseq_1(E,u1_struct_0(C))
                     => ( ( k3_finseq_1(D) = k3_finseq_1(E)
                          & ! [F] :
                              ( m2_subset_1(F,k1_numbers,k5_numbers)
                             => ! [G] :
                                  ( m1_subset_1(G,u1_struct_0(C))
                                 => ( ( r2_hidden(F,k1_relat_1(D))
                                      & G = k1_funct_1(E,F) )
                                   => k1_funct_1(D,F) = k6_vectsp_1(A,C,B,G) ) ) ) )
                       => k9_rlvect_1(C,D) = k6_vectsp_1(A,C,B,k9_rlvect_1(C,E)) ) ) ) ) ) ) )).

fof(t10_vectsp_3,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)
        & v7_vectsp_1(A)
        & v8_vectsp_1(A)
        & l3_vectsp_1(A) )
     => ! [B] :
          ( m1_subset_1(B,u1_struct_0(A))
         => ! [C] :
              ( ( ~ v3_struct_0(C)
                & v3_rlvect_1(C)
                & v4_rlvect_1(C)
                & v5_rlvect_1(C)
                & v6_rlvect_1(C)
                & v12_vectsp_1(C,A)
                & l4_vectsp_1(C,A) )
             => ! [D] :
                  ( m2_finseq_1(D,u1_struct_0(C))
                 => ! [E] :
                      ( m2_finseq_1(E,u1_struct_0(C))
                     => ( ( k3_finseq_1(D) = k3_finseq_1(E)
                          & ! [F] :
                              ( m2_subset_1(F,k1_numbers,k5_numbers)
                             => ( r2_hidden(F,k1_relat_1(D))
                               => k1_funct_1(E,F) = k6_vectsp_1(A,C,B,k4_finseq_4(k5_numbers,u1_struct_0(C),D,F)) ) ) )
                       => k9_rlvect_1(C,E) = k6_vectsp_1(A,C,B,k9_rlvect_1(C,D)) ) ) ) ) ) ) )).

fof(t11_vectsp_3,axiom,(
    $true )).

fof(t12_vectsp_3,axiom,(
    $true )).

fof(t13_vectsp_3,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)
        & v7_vectsp_1(A)
        & v8_vectsp_1(A)
        & l3_vectsp_1(A) )
     => ! [B] :
          ( ( ~ v3_struct_0(B)
            & v3_rlvect_1(B)
            & v4_rlvect_1(B)
            & v5_rlvect_1(B)
            & v6_rlvect_1(B)
            & l4_vectsp_1(B,A) )
         => ! [C] :
              ( m2_finseq_1(C,u1_struct_0(B))
             => ! [D] :
                  ( m2_finseq_1(D,u1_struct_0(B))
                 => ! [E] :
                      ( m2_finseq_1(E,u1_struct_0(B))
                     => ( ( k3_finseq_1(C) = k3_finseq_1(D)
                          & k3_finseq_1(C) = k3_finseq_1(E)
                          & ! [F] :
                              ( m2_subset_1(F,k1_numbers,k5_numbers)
                             => ( r2_hidden(F,k1_relat_1(C))
                               => k1_funct_1(E,F) = k6_rlvect_1(B,k4_finseq_4(k5_numbers,u1_struct_0(B),C,F),k4_finseq_4(k5_numbers,u1_struct_0(B),D,F)) ) ) )
                       => k9_rlvect_1(B,E) = k6_rlvect_1(B,k9_rlvect_1(B,C),k9_rlvect_1(B,D)) ) ) ) ) ) ) )).

fof(t14_vectsp_3,axiom,(
    $true )).

fof(t15_vectsp_3,axiom,(
    $true )).

fof(t16_vectsp_3,axiom,(
    $true )).

fof(t17_vectsp_3,axiom,(
    $true )).

fof(t18_vectsp_3,axiom,(
    $true )).

fof(t19_vectsp_3,axiom,(
    $true )).

fof(t20_vectsp_3,axiom,(
    $true )).

fof(t21_vectsp_3,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)
        & v7_vectsp_1(A)
        & v8_vectsp_1(A)
        & l3_vectsp_1(A) )
     => ! [B] :
          ( m1_subset_1(B,u1_struct_0(A))
         => ! [C] :
              ( ( ~ v3_struct_0(C)
                & v3_rlvect_1(C)
                & v4_rlvect_1(C)
                & v5_rlvect_1(C)
                & v6_rlvect_1(C)
                & v12_vectsp_1(C,A)
                & l4_vectsp_1(C,A) )
             => k6_vectsp_1(A,C,B,k9_rlvect_1(C,k6_finseq_1(u1_struct_0(C)))) = k1_rlvect_1(C) ) ) ) )).

fof(t22_vectsp_3,axiom,(
    $true )).

fof(t23_vectsp_3,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)
        & v7_vectsp_1(A)
        & v8_vectsp_1(A)
        & l3_vectsp_1(A) )
     => ! [B] :
          ( m1_subset_1(B,u1_struct_0(A))
         => ! [C] :
              ( ( ~ v3_struct_0(C)
                & v3_rlvect_1(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))
                 => ! [E] :
                      ( m1_subset_1(E,u1_struct_0(C))
                     => k6_vectsp_1(A,C,B,k9_rlvect_1(C,k2_finseq_4(u1_struct_0(C),D,E))) = k4_rlvect_1(C,k6_vectsp_1(A,C,B,D),k6_vectsp_1(A,C,B,E)) ) ) ) ) ) )).

fof(t24_vectsp_3,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)
        & v7_vectsp_1(A)
        & v8_vectsp_1(A)
        & l3_vectsp_1(A) )
     => ! [B] :
          ( m1_subset_1(B,u1_struct_0(A))
         => ! [C] :
              ( ( ~ v3_struct_0(C)
                & v3_rlvect_1(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))
                 => ! [E] :
                      ( m1_subset_1(E,u1_struct_0(C))
                     => ! [F] :
                          ( m1_subset_1(F,u1_struct_0(C))
                         => k6_vectsp_1(A,C,B,k9_rlvect_1(C,k3_finseq_4(u1_struct_0(C),D,E,F))) = k4_rlvect_1(C,k4_rlvect_1(C,k6_vectsp_1(A,C,B,D),k6_vectsp_1(A,C,B,E)),k6_vectsp_1(A,C,B,F)) ) ) ) ) ) ) )).

fof(t25_vectsp_3,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v3_rlvect_1(A)
        & v4_rlvect_1(A)
        & v5_rlvect_1(A)
        & v6_rlvect_1(A)
        & l1_rlvect_1(A) )
     => k5_rlvect_1(A,k9_rlvect_1(A,k6_finseq_1(u1_struct_0(A)))) = k1_rlvect_1(A) ) )).

fof(t26_vectsp_3,axiom,(
    $true )).

fof(t27_vectsp_3,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v3_rlvect_1(A)
        & v4_rlvect_1(A)
        & v5_rlvect_1(A)
        & v6_rlvect_1(A)
        & l1_rlvect_1(A) )
     => ! [B] :
          ( m1_subset_1(B,u1_struct_0(A))
         => ! [C] :
              ( m1_subset_1(C,u1_struct_0(A))
             => k5_rlvect_1(A,k9_rlvect_1(A,k2_finseq_4(u1_struct_0(A),B,C))) = k6_rlvect_1(A,k5_rlvect_1(A,B),C) ) ) ) )).

fof(t28_vectsp_3,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v3_rlvect_1(A)
        & v4_rlvect_1(A)
        & v5_rlvect_1(A)
        & v6_rlvect_1(A)
        & l1_rlvect_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))
                 => k5_rlvect_1(A,k9_rlvect_1(A,k3_finseq_4(u1_struct_0(A),B,C,D))) = k6_rlvect_1(A,k6_rlvect_1(A,k5_rlvect_1(A,B),C),D) ) ) ) ) )).

fof(t29_vectsp_3,axiom,(
    $true )).

fof(t30_vectsp_3,axiom,(
    $true )).

fof(t31_vectsp_3,axiom,(
    $true )).

fof(t32_vectsp_3,axiom,(
    $true )).

fof(t33_vectsp_3,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v3_rlvect_1(A)
        & v4_rlvect_1(A)
        & v5_rlvect_1(A)
        & v6_rlvect_1(A)
        & l1_rlvect_1(A) )
     => ! [B] :
          ( m1_subset_1(B,u1_struct_0(A))
         => ( k9_rlvect_1(A,k2_finseq_4(u1_struct_0(A),B,k5_rlvect_1(A,B))) = k1_rlvect_1(A)
            & k9_rlvect_1(A,k2_finseq_4(u1_struct_0(A),k5_rlvect_1(A,B),B)) = k1_rlvect_1(A) ) ) ) )).

fof(t34_vectsp_3,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v3_rlvect_1(A)
        & v4_rlvect_1(A)
        & v5_rlvect_1(A)
        & v6_rlvect_1(A)
        & l1_rlvect_1(A) )
     => ! [B] :
          ( m1_subset_1(B,u1_struct_0(A))
         => ! [C] :
              ( m1_subset_1(C,u1_struct_0(A))
             => ( k9_rlvect_1(A,k2_finseq_4(u1_struct_0(A),B,k5_rlvect_1(A,C))) = k6_rlvect_1(A,B,C)
                & k9_rlvect_1(A,k2_finseq_4(u1_struct_0(A),k5_rlvect_1(A,C),B)) = k6_rlvect_1(A,B,C) ) ) ) ) )).

fof(t35_vectsp_3,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v3_rlvect_1(A)
        & v4_rlvect_1(A)
        & v5_rlvect_1(A)
        & v6_rlvect_1(A)
        & l1_rlvect_1(A) )
     => ! [B] :
          ( m1_subset_1(B,u1_struct_0(A))
         => ! [C] :
              ( m1_subset_1(C,u1_struct_0(A))
             => ( k9_rlvect_1(A,k2_finseq_4(u1_struct_0(A),k5_rlvect_1(A,B),k5_rlvect_1(A,C))) = k5_rlvect_1(A,k4_rlvect_1(A,B,C))
                & k9_rlvect_1(A,k2_finseq_4(u1_struct_0(A),k5_rlvect_1(A,C),k5_rlvect_1(A,B))) = k5_rlvect_1(A,k4_rlvect_1(A,B,C)) ) ) ) ) )).
%------------------------------------------------------------------------------