SET007 Axioms: SET007+787.ax


%------------------------------------------------------------------------------
% File     : SET007+787 : TPTP v7.5.0. Released v3.4.0.
% Domain   : Set Theory
% Axioms   : Cross Products and Tripple Vector Products in 3-dimensiona
% 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    : euclid_5 [Urb08]

% Status   : Satisfiable
% Syntax   : Number of formulae    :   47 (   1 unit)
%            Number of atoms       :  196 (  58 equality)
%            Maximal formula depth :   13 (   7 average)
%            Number of connectives :  149 (   0 ~  ;   0  |;  23  &)
%                                         (   3 <=>; 123 =>;   0 <=)
%                                         (   0 <~>;   0 ~|;   0 ~&)
%            Number of predicates  :    7 (   0 propositional; 1-2 arity)
%            Number of functors    :   29 (   6 constant; 0-4 arity)
%            Number of variables   :  126 (   0 singleton; 123 !;   3 ?)
%            Maximal term depth    :    5 (   2 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_euclid_5,axiom,(
    ! [A] :
      ( m1_subset_1(A,u1_struct_0(k15_euclid(np__3)))
     => ? [B] :
          ( m1_subset_1(B,k1_numbers)
          & ? [C] :
              ( m1_subset_1(C,k1_numbers)
              & ? [D] :
                  ( m1_subset_1(D,k1_numbers)
                  & A = k3_finseq_4(k1_numbers,B,C,D) ) ) ) ) )).

fof(d1_euclid_5,axiom,(
    ! [A] :
      ( m1_subset_1(A,u1_struct_0(k15_euclid(np__3)))
     => ! [B] :
          ( m1_subset_1(B,k1_numbers)
         => ( B = k1_euclid_5(A)
          <=> ! [C] :
                ( ( v1_relat_1(C)
                  & v1_funct_1(C)
                  & v1_finseq_1(C) )
               => ( A = C
                 => B = k1_funct_1(C,np__1) ) ) ) ) ) )).

fof(d2_euclid_5,axiom,(
    ! [A] :
      ( m1_subset_1(A,u1_struct_0(k15_euclid(np__3)))
     => ! [B] :
          ( m1_subset_1(B,k1_numbers)
         => ( B = k2_euclid_5(A)
          <=> ! [C] :
                ( ( v1_relat_1(C)
                  & v1_funct_1(C)
                  & v1_finseq_1(C) )
               => ( A = C
                 => B = k1_funct_1(C,np__2) ) ) ) ) ) )).

fof(d3_euclid_5,axiom,(
    ! [A] :
      ( m1_subset_1(A,u1_struct_0(k15_euclid(np__3)))
     => ! [B] :
          ( m1_subset_1(B,k1_numbers)
         => ( B = k3_euclid_5(A)
          <=> ! [C] :
                ( ( v1_relat_1(C)
                  & v1_funct_1(C)
                  & v1_finseq_1(C) )
               => ( A = C
                 => B = k1_funct_1(C,np__3) ) ) ) ) ) )).

fof(d4_euclid_5,axiom,(
    ! [A] :
      ( m1_subset_1(A,k1_numbers)
     => ! [B] :
          ( m1_subset_1(B,k1_numbers)
         => ! [C] :
              ( m1_subset_1(C,k1_numbers)
             => k4_euclid_5(A,B,C) = k3_finseq_4(k1_numbers,A,B,C) ) ) ) )).

fof(t2_euclid_5,axiom,(
    ! [A] :
      ( m1_subset_1(A,k1_numbers)
     => ! [B] :
          ( m1_subset_1(B,k1_numbers)
         => ! [C] :
              ( m1_subset_1(C,k1_numbers)
             => ( k1_euclid_5(k4_euclid_5(A,B,C)) = A
                & k2_euclid_5(k4_euclid_5(A,B,C)) = B
                & k3_euclid_5(k4_euclid_5(A,B,C)) = C ) ) ) ) )).

fof(t3_euclid_5,axiom,(
    ! [A] :
      ( m1_subset_1(A,u1_struct_0(k15_euclid(np__3)))
     => A = k4_euclid_5(k1_euclid_5(A),k2_euclid_5(A),k3_euclid_5(A)) ) )).

fof(t4_euclid_5,axiom,(
    k16_euclid(np__3) = k4_euclid_5(np__0,np__0,np__0) )).

fof(t5_euclid_5,axiom,(
    ! [A] :
      ( m1_subset_1(A,u1_struct_0(k15_euclid(np__3)))
     => ! [B] :
          ( m1_subset_1(B,u1_struct_0(k15_euclid(np__3)))
         => k17_euclid(np__3,A,B) = k4_euclid_5(k9_binop_2(k1_euclid_5(A),k1_euclid_5(B)),k9_binop_2(k2_euclid_5(A),k2_euclid_5(B)),k9_binop_2(k3_euclid_5(A),k3_euclid_5(B))) ) ) )).

fof(t6_euclid_5,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)
                 => ! [E] :
                      ( m1_subset_1(E,k1_numbers)
                     => ! [F] :
                          ( m1_subset_1(F,k1_numbers)
                         => k17_euclid(np__3,k4_euclid_5(A,B,C),k4_euclid_5(D,E,F)) = k4_euclid_5(k9_binop_2(A,D),k9_binop_2(B,E),k9_binop_2(C,F)) ) ) ) ) ) ) )).

fof(t7_euclid_5,axiom,(
    ! [A] :
      ( m1_subset_1(A,k1_numbers)
     => ! [B] :
          ( m1_subset_1(B,u1_struct_0(k15_euclid(np__3)))
         => k18_euclid(A,np__3,B) = k4_euclid_5(k11_binop_2(A,k1_euclid_5(B)),k11_binop_2(A,k2_euclid_5(B)),k11_binop_2(A,k3_euclid_5(B))) ) ) )).

fof(t8_euclid_5,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)
                 => k18_euclid(A,np__3,k4_euclid_5(B,C,D)) = k4_euclid_5(k11_binop_2(A,B),k11_binop_2(A,C),k11_binop_2(A,D)) ) ) ) ) )).

fof(t9_euclid_5,axiom,(
    ! [A] :
      ( m1_subset_1(A,k1_numbers)
     => ! [B] :
          ( m1_subset_1(B,u1_struct_0(k15_euclid(np__3)))
         => ( k1_euclid_5(k18_euclid(A,np__3,B)) = k11_binop_2(A,k1_euclid_5(B))
            & k2_euclid_5(k18_euclid(A,np__3,B)) = k11_binop_2(A,k2_euclid_5(B))
            & k3_euclid_5(k18_euclid(A,np__3,B)) = k11_binop_2(A,k3_euclid_5(B)) ) ) ) )).

fof(t10_euclid_5,axiom,(
    ! [A] :
      ( m1_subset_1(A,u1_struct_0(k15_euclid(np__3)))
     => k19_euclid(np__3,A) = k4_euclid_5(k7_binop_2(k1_euclid_5(A)),k7_binop_2(k2_euclid_5(A)),k7_binop_2(k3_euclid_5(A))) ) )).

fof(t11_euclid_5,axiom,(
    ! [A] :
      ( m1_subset_1(A,k1_numbers)
     => ! [B] :
          ( m1_subset_1(B,k1_numbers)
         => ! [C] :
              ( m1_subset_1(C,k1_numbers)
             => k19_euclid(np__3,k4_euclid_5(A,B,C)) = k4_euclid_5(k7_binop_2(A),k7_binop_2(B),k7_binop_2(C)) ) ) ) )).

fof(t12_euclid_5,axiom,(
    ! [A] :
      ( m1_subset_1(A,u1_struct_0(k15_euclid(np__3)))
     => ! [B] :
          ( m1_subset_1(B,u1_struct_0(k15_euclid(np__3)))
         => k20_euclid(np__3,A,B) = k4_euclid_5(k10_binop_2(k1_euclid_5(A),k1_euclid_5(B)),k10_binop_2(k2_euclid_5(A),k2_euclid_5(B)),k10_binop_2(k3_euclid_5(A),k3_euclid_5(B))) ) ) )).

fof(t13_euclid_5,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)
                 => ! [E] :
                      ( m1_subset_1(E,k1_numbers)
                     => ! [F] :
                          ( m1_subset_1(F,k1_numbers)
                         => k20_euclid(np__3,k4_euclid_5(A,B,C),k4_euclid_5(D,E,F)) = k4_euclid_5(k10_binop_2(A,D),k10_binop_2(B,E),k10_binop_2(C,F)) ) ) ) ) ) ) )).

fof(d5_euclid_5,axiom,(
    ! [A] :
      ( m1_subset_1(A,u1_struct_0(k15_euclid(np__3)))
     => ! [B] :
          ( m1_subset_1(B,u1_struct_0(k15_euclid(np__3)))
         => k5_euclid_5(A,B) = k4_euclid_5(k10_binop_2(k11_binop_2(k2_euclid_5(A),k3_euclid_5(B)),k11_binop_2(k3_euclid_5(A),k2_euclid_5(B))),k10_binop_2(k11_binop_2(k3_euclid_5(A),k1_euclid_5(B)),k11_binop_2(k1_euclid_5(A),k3_euclid_5(B))),k10_binop_2(k11_binop_2(k1_euclid_5(A),k2_euclid_5(B)),k11_binop_2(k2_euclid_5(A),k1_euclid_5(B)))) ) ) )).

fof(t14_euclid_5,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,u1_struct_0(k15_euclid(np__3)))
                 => ( D = k4_euclid_5(A,B,C)
                   => ( k1_euclid_5(D) = A
                      & k2_euclid_5(D) = B
                      & k3_euclid_5(D) = C ) ) ) ) ) ) )).

fof(t15_euclid_5,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)
                 => ! [E] :
                      ( m1_subset_1(E,k1_numbers)
                     => ! [F] :
                          ( m1_subset_1(F,k1_numbers)
                         => k5_euclid_5(k4_euclid_5(A,B,C),k4_euclid_5(D,E,F)) = k4_euclid_5(k10_binop_2(k11_binop_2(B,F),k11_binop_2(C,E)),k10_binop_2(k11_binop_2(C,D),k11_binop_2(A,F)),k10_binop_2(k11_binop_2(A,E),k11_binop_2(B,D))) ) ) ) ) ) ) )).

fof(t16_euclid_5,axiom,(
    ! [A] :
      ( m1_subset_1(A,k1_numbers)
     => ! [B] :
          ( m1_subset_1(B,u1_struct_0(k15_euclid(np__3)))
         => ! [C] :
              ( m1_subset_1(C,u1_struct_0(k15_euclid(np__3)))
             => ( k5_euclid_5(k18_euclid(A,np__3,B),C) = k18_euclid(A,np__3,k5_euclid_5(B,C))
                & k5_euclid_5(k18_euclid(A,np__3,B),C) = k5_euclid_5(B,k18_euclid(A,np__3,C)) ) ) ) ) )).

fof(t17_euclid_5,axiom,(
    ! [A] :
      ( m1_subset_1(A,u1_struct_0(k15_euclid(np__3)))
     => ! [B] :
          ( m1_subset_1(B,u1_struct_0(k15_euclid(np__3)))
         => k5_euclid_5(A,B) = k19_euclid(np__3,k5_euclid_5(B,A)) ) ) )).

fof(t18_euclid_5,axiom,(
    ! [A] :
      ( m1_subset_1(A,u1_struct_0(k15_euclid(np__3)))
     => ! [B] :
          ( m1_subset_1(B,u1_struct_0(k15_euclid(np__3)))
         => k5_euclid_5(k19_euclid(np__3,A),B) = k5_euclid_5(A,k19_euclid(np__3,B)) ) ) )).

fof(t19_euclid_5,axiom,(
    ! [A] :
      ( m1_subset_1(A,k1_numbers)
     => ! [B] :
          ( m1_subset_1(B,k1_numbers)
         => ! [C] :
              ( m1_subset_1(C,k1_numbers)
             => k5_euclid_5(k4_euclid_5(np__0,np__0,np__0),k4_euclid_5(A,B,C)) = k16_euclid(np__3) ) ) ) )).

fof(t20_euclid_5,axiom,(
    ! [A] :
      ( m1_subset_1(A,k1_numbers)
     => ! [B] :
          ( m1_subset_1(B,k1_numbers)
         => k5_euclid_5(k4_euclid_5(A,np__0,np__0),k4_euclid_5(B,np__0,np__0)) = k16_euclid(np__3) ) ) )).

fof(t21_euclid_5,axiom,(
    ! [A] :
      ( m1_subset_1(A,k1_numbers)
     => ! [B] :
          ( m1_subset_1(B,k1_numbers)
         => k5_euclid_5(k4_euclid_5(np__0,A,np__0),k4_euclid_5(np__0,B,np__0)) = k16_euclid(np__3) ) ) )).

fof(t22_euclid_5,axiom,(
    ! [A] :
      ( m1_subset_1(A,k1_numbers)
     => ! [B] :
          ( m1_subset_1(B,k1_numbers)
         => k5_euclid_5(k4_euclid_5(np__0,np__0,A),k4_euclid_5(np__0,np__0,B)) = k16_euclid(np__3) ) ) )).

fof(t23_euclid_5,axiom,(
    ! [A] :
      ( m1_subset_1(A,u1_struct_0(k15_euclid(np__3)))
     => ! [B] :
          ( m1_subset_1(B,u1_struct_0(k15_euclid(np__3)))
         => ! [C] :
              ( m1_subset_1(C,u1_struct_0(k15_euclid(np__3)))
             => k5_euclid_5(A,k17_euclid(np__3,B,C)) = k17_euclid(np__3,k5_euclid_5(A,B),k5_euclid_5(A,C)) ) ) ) )).

fof(t24_euclid_5,axiom,(
    ! [A] :
      ( m1_subset_1(A,u1_struct_0(k15_euclid(np__3)))
     => ! [B] :
          ( m1_subset_1(B,u1_struct_0(k15_euclid(np__3)))
         => ! [C] :
              ( m1_subset_1(C,u1_struct_0(k15_euclid(np__3)))
             => k5_euclid_5(k17_euclid(np__3,A,B),C) = k17_euclid(np__3,k5_euclid_5(A,C),k5_euclid_5(B,C)) ) ) ) )).

fof(t25_euclid_5,axiom,(
    ! [A] :
      ( m1_subset_1(A,u1_struct_0(k15_euclid(np__3)))
     => k5_euclid_5(A,A) = k16_euclid(np__3) ) )).

fof(t26_euclid_5,axiom,(
    ! [A] :
      ( m1_subset_1(A,u1_struct_0(k15_euclid(np__3)))
     => ! [B] :
          ( m1_subset_1(B,u1_struct_0(k15_euclid(np__3)))
         => ! [C] :
              ( m1_subset_1(C,u1_struct_0(k15_euclid(np__3)))
             => ! [D] :
                  ( m1_subset_1(D,u1_struct_0(k15_euclid(np__3)))
                 => k5_euclid_5(k17_euclid(np__3,A,B),k17_euclid(np__3,C,D)) = k17_euclid(np__3,k17_euclid(np__3,k17_euclid(np__3,k5_euclid_5(A,C),k5_euclid_5(A,D)),k5_euclid_5(B,C)),k5_euclid_5(B,D)) ) ) ) ) )).

fof(t27_euclid_5,axiom,(
    ! [A] :
      ( m1_subset_1(A,u1_struct_0(k15_euclid(np__3)))
     => A = k3_finseq_4(k1_numbers,k1_euclid_5(A),k2_euclid_5(A),k3_euclid_5(A)) ) )).

fof(t28_euclid_5,axiom,(
    ! [A] :
      ( m2_finseq_1(A,k1_numbers)
     => ! [B] :
          ( m2_finseq_1(B,k1_numbers)
         => ( ( k3_finseq_1(A) = np__3
              & k3_finseq_1(B) = np__3 )
           => k13_rvsum_1(A,B) = k3_finseq_4(k1_numbers,k11_binop_2(k2_seq_1(k5_numbers,k1_numbers,A,np__1),k2_seq_1(k5_numbers,k1_numbers,B,np__1)),k11_binop_2(k2_seq_1(k5_numbers,k1_numbers,A,np__2),k2_seq_1(k5_numbers,k1_numbers,B,np__2)),k11_binop_2(k2_seq_1(k5_numbers,k1_numbers,A,np__3),k2_seq_1(k5_numbers,k1_numbers,B,np__3))) ) ) ) )).

fof(t29_euclid_5,axiom,(
    ! [A] :
      ( m1_subset_1(A,u1_struct_0(k15_euclid(np__3)))
     => ! [B] :
          ( m1_subset_1(B,u1_struct_0(k15_euclid(np__3)))
         => k2_euclid_2(np__3,A,B) = k9_binop_2(k9_binop_2(k11_binop_2(k1_euclid_5(A),k1_euclid_5(B)),k11_binop_2(k2_euclid_5(A),k2_euclid_5(B))),k11_binop_2(k3_euclid_5(A),k3_euclid_5(B))) ) ) )).

fof(t30_euclid_5,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)
                 => ! [E] :
                      ( m1_subset_1(E,k1_numbers)
                     => ! [F] :
                          ( m1_subset_1(F,k1_numbers)
                         => k2_euclid_2(np__3,k4_euclid_5(C,D,A),k4_euclid_5(E,F,B)) = k9_binop_2(k9_binop_2(k11_binop_2(C,E),k11_binop_2(D,F)),k11_binop_2(A,B)) ) ) ) ) ) ) )).

fof(d6_euclid_5,axiom,(
    ! [A] :
      ( m1_subset_1(A,u1_struct_0(k15_euclid(np__3)))
     => ! [B] :
          ( m1_subset_1(B,u1_struct_0(k15_euclid(np__3)))
         => ! [C] :
              ( m1_subset_1(C,u1_struct_0(k15_euclid(np__3)))
             => k6_euclid_5(A,B,C) = k2_euclid_2(np__3,A,k5_euclid_5(B,C)) ) ) ) )).

fof(t31_euclid_5,axiom,(
    ! [A] :
      ( m1_subset_1(A,u1_struct_0(k15_euclid(np__3)))
     => ! [B] :
          ( m1_subset_1(B,u1_struct_0(k15_euclid(np__3)))
         => ( k6_euclid_5(A,A,B) = np__0
            & k6_euclid_5(B,A,B) = np__0 ) ) ) )).

fof(t32_euclid_5,axiom,(
    ! [A] :
      ( m1_subset_1(A,u1_struct_0(k15_euclid(np__3)))
     => ! [B] :
          ( m1_subset_1(B,u1_struct_0(k15_euclid(np__3)))
         => ! [C] :
              ( m1_subset_1(C,u1_struct_0(k15_euclid(np__3)))
             => k5_euclid_5(A,k5_euclid_5(B,C)) = k20_euclid(np__3,k18_euclid(k2_euclid_2(np__3,A,C),np__3,B),k18_euclid(k2_euclid_2(np__3,A,B),np__3,C)) ) ) ) )).

fof(t33_euclid_5,axiom,(
    ! [A] :
      ( m1_subset_1(A,u1_struct_0(k15_euclid(np__3)))
     => ! [B] :
          ( m1_subset_1(B,u1_struct_0(k15_euclid(np__3)))
         => ! [C] :
              ( m1_subset_1(C,u1_struct_0(k15_euclid(np__3)))
             => k6_euclid_5(A,B,C) = k6_euclid_5(B,C,A) ) ) ) )).

fof(t34_euclid_5,axiom,(
    ! [A] :
      ( m1_subset_1(A,u1_struct_0(k15_euclid(np__3)))
     => ! [B] :
          ( m1_subset_1(B,u1_struct_0(k15_euclid(np__3)))
         => ! [C] :
              ( m1_subset_1(C,u1_struct_0(k15_euclid(np__3)))
             => k6_euclid_5(A,B,C) = k6_euclid_5(C,A,B) ) ) ) )).

fof(t35_euclid_5,axiom,(
    ! [A] :
      ( m1_subset_1(A,u1_struct_0(k15_euclid(np__3)))
     => ! [B] :
          ( m1_subset_1(B,u1_struct_0(k15_euclid(np__3)))
         => ! [C] :
              ( m1_subset_1(C,u1_struct_0(k15_euclid(np__3)))
             => k6_euclid_5(A,B,C) = k2_euclid_2(np__3,k5_euclid_5(A,B),C) ) ) ) )).

fof(dt_k1_euclid_5,axiom,(
    ! [A] :
      ( m1_subset_1(A,u1_struct_0(k15_euclid(np__3)))
     => m1_subset_1(k1_euclid_5(A),k1_numbers) ) )).

fof(dt_k2_euclid_5,axiom,(
    ! [A] :
      ( m1_subset_1(A,u1_struct_0(k15_euclid(np__3)))
     => m1_subset_1(k2_euclid_5(A),k1_numbers) ) )).

fof(dt_k3_euclid_5,axiom,(
    ! [A] :
      ( m1_subset_1(A,u1_struct_0(k15_euclid(np__3)))
     => m1_subset_1(k3_euclid_5(A),k1_numbers) ) )).

fof(dt_k4_euclid_5,axiom,(
    ! [A,B,C] :
      ( ( m1_subset_1(A,k1_numbers)
        & m1_subset_1(B,k1_numbers)
        & m1_subset_1(C,k1_numbers) )
     => m1_subset_1(k4_euclid_5(A,B,C),u1_struct_0(k15_euclid(np__3))) ) )).

fof(dt_k5_euclid_5,axiom,(
    ! [A,B] :
      ( ( m1_subset_1(A,u1_struct_0(k15_euclid(np__3)))
        & m1_subset_1(B,u1_struct_0(k15_euclid(np__3))) )
     => m1_subset_1(k5_euclid_5(A,B),u1_struct_0(k15_euclid(np__3))) ) )).

fof(dt_k6_euclid_5,axiom,(
    ! [A,B,C] :
      ( ( m1_subset_1(A,u1_struct_0(k15_euclid(np__3)))
        & m1_subset_1(B,u1_struct_0(k15_euclid(np__3)))
        & m1_subset_1(C,u1_struct_0(k15_euclid(np__3))) )
     => v1_xreal_0(k6_euclid_5(A,B,C)) ) )).
%------------------------------------------------------------------------------