SET007 Axioms: SET007+773.ax


%------------------------------------------------------------------------------
% File     : SET007+773 : TPTP v7.5.0. Released v3.4.0.
% Domain   : Set Theory
% Axioms   : On Some Properties of Real Hilbert Space. Part II
% 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    : bhsp_7 [Urb08]

% Status   : Satisfiable
% Syntax   : Number of formulae    :    8 (   0 unit)
%            Number of atoms       :  167 (   8 equality)
%            Maximal formula depth :   23 (  15 average)
%            Number of connectives :  181 (  22 ~  ;   1  |; 101  &)
%                                         (   2 <=>;  55 =>;   0 <=)
%                                         (   0 <~>;   0 ~|;   0 ~&)
%            Number of predicates  :   27 (   0 propositional; 1-3 arity)
%            Number of functors    :   16 (   4 constant; 0-5 arity)
%            Number of variables   :   35 (   0 singleton;  35 !;   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_bhsp_7,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v3_rlvect_1(A)
        & v4_rlvect_1(A)
        & v5_rlvect_1(A)
        & v6_rlvect_1(A)
        & v7_rlvect_1(A)
        & v2_bhsp_1(A)
        & l1_bhsp_1(A) )
     => ! [B] :
          ( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
         => ! [C] :
              ( ( v1_funct_1(C)
                & v1_funct_2(C,u1_struct_0(A),k6_supinf_1)
                & m2_relset_1(C,u1_struct_0(A),k6_supinf_1) )
             => ( r1_bhsp_6(A,B,C)
              <=> ! [D] :
                    ( m1_subset_1(D,k1_numbers)
                   => ~ ( ~ r1_xreal_0(D,np__0)
                        & ! [E] :
                            ( ( v1_finset_1(E)
                              & m1_subset_1(E,k1_zfmisc_1(u1_struct_0(A))) )
                           => ~ ( ~ v1_xboole_0(E)
                                & r1_tarski(E,B)
                                & ! [F] :
                                    ( ( v1_finset_1(F)
                                      & m1_subset_1(F,k1_zfmisc_1(u1_struct_0(A))) )
                                   => ~ ( ~ v1_xboole_0(F)
                                        & r1_tarski(F,B)
                                        & r1_xboole_0(E,F)
                                        & r1_xreal_0(D,k18_complex1(k5_bhsp_5(k1_numbers,u1_struct_0(A),k33_binop_2,F,C))) ) ) ) ) ) ) ) ) ) ) )).

fof(t2_bhsp_7,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v3_rlvect_1(A)
        & v4_rlvect_1(A)
        & v5_rlvect_1(A)
        & v6_rlvect_1(A)
        & v7_rlvect_1(A)
        & v2_bhsp_1(A)
        & l1_bhsp_1(A) )
     => ( ( v1_binop_1(u1_rlvect_1(A),u1_struct_0(A))
          & v2_binop_1(u1_rlvect_1(A),u1_struct_0(A))
          & v1_setwiseo(u1_rlvect_1(A),u1_struct_0(A)) )
       => ! [B] :
            ( ( v1_finset_1(B)
              & m1_bhsp_5(B,A) )
           => ( ~ v1_xboole_0(B)
             => ! [C] :
                  ( ( v1_funct_1(C)
                    & v1_funct_2(C,u1_struct_0(A),u1_struct_0(A))
                    & m2_relset_1(C,u1_struct_0(A),u1_struct_0(A)) )
                 => ( ( r1_tarski(B,k1_relat_1(C))
                      & ! [D] :
                          ( m1_subset_1(D,u1_struct_0(A))
                         => ( r2_hidden(D,B)
                           => k8_funct_2(u1_struct_0(A),u1_struct_0(A),C,D) = D ) ) )
                   => ! [D] :
                        ( ( v1_funct_1(D)
                          & v1_funct_2(D,u1_struct_0(A),k1_numbers)
                          & m2_relset_1(D,u1_struct_0(A),k1_numbers) )
                       => ( ( r1_tarski(B,k1_relat_1(D))
                            & ! [E] :
                                ( m1_subset_1(E,u1_struct_0(A))
                               => ( r2_hidden(E,B)
                                 => k8_funct_2(u1_struct_0(A),k1_numbers,D,E) = k2_bhsp_1(A,E,E) ) ) )
                         => k2_bhsp_1(A,k5_bhsp_5(u1_struct_0(A),u1_struct_0(A),u1_rlvect_1(A),B,C),k5_bhsp_5(u1_struct_0(A),u1_struct_0(A),u1_rlvect_1(A),B,C)) = k5_bhsp_5(k1_numbers,u1_struct_0(A),k33_binop_2,B,D) ) ) ) ) ) ) ) ) )).

fof(t3_bhsp_7,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v3_rlvect_1(A)
        & v4_rlvect_1(A)
        & v5_rlvect_1(A)
        & v6_rlvect_1(A)
        & v7_rlvect_1(A)
        & v2_bhsp_1(A)
        & l1_bhsp_1(A) )
     => ( ( v1_binop_1(u1_rlvect_1(A),u1_struct_0(A))
          & v2_binop_1(u1_rlvect_1(A),u1_struct_0(A))
          & v1_setwiseo(u1_rlvect_1(A),u1_struct_0(A)) )
       => ! [B] :
            ( ( v1_finset_1(B)
              & m1_bhsp_5(B,A) )
           => ( ~ v1_xboole_0(B)
             => ! [C] :
                  ( ( v1_funct_1(C)
                    & v1_funct_2(C,u1_struct_0(A),k6_supinf_1)
                    & m2_relset_1(C,u1_struct_0(A),k6_supinf_1) )
                 => ( ( r1_tarski(B,k1_relat_1(C))
                      & ! [D] :
                          ( m1_subset_1(D,u1_struct_0(A))
                         => ( r2_hidden(D,B)
                           => k8_funct_2(u1_struct_0(A),k6_supinf_1,C,D) = k2_bhsp_1(A,D,D) ) ) )
                   => k2_bhsp_1(A,k1_bhsp_6(A,B),k1_bhsp_6(A,B)) = k5_bhsp_5(k1_numbers,u1_struct_0(A),k33_binop_2,B,C) ) ) ) ) ) ) )).

fof(t4_bhsp_7,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v3_rlvect_1(A)
        & v4_rlvect_1(A)
        & v5_rlvect_1(A)
        & v6_rlvect_1(A)
        & v7_rlvect_1(A)
        & v2_bhsp_1(A)
        & l1_bhsp_1(A) )
     => ! [B] :
          ( m1_bhsp_5(B,A)
         => ! [C] :
              ( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A)))
             => ( m1_subset_1(C,k1_zfmisc_1(B))
               => m1_bhsp_5(C,A) ) ) ) ) )).

fof(t5_bhsp_7,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v3_rlvect_1(A)
        & v4_rlvect_1(A)
        & v5_rlvect_1(A)
        & v6_rlvect_1(A)
        & v7_rlvect_1(A)
        & v2_bhsp_1(A)
        & l1_bhsp_1(A) )
     => ! [B] :
          ( m2_bhsp_5(B,A)
         => ! [C] :
              ( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A)))
             => ( m1_subset_1(C,k1_zfmisc_1(B))
               => m2_bhsp_5(C,A) ) ) ) ) )).

fof(t6_bhsp_7,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v3_rlvect_1(A)
        & v4_rlvect_1(A)
        & v5_rlvect_1(A)
        & v6_rlvect_1(A)
        & v7_rlvect_1(A)
        & v2_bhsp_1(A)
        & l1_bhsp_1(A) )
     => ( ( v1_binop_1(u1_rlvect_1(A),u1_struct_0(A))
          & v2_binop_1(u1_rlvect_1(A),u1_struct_0(A))
          & v1_setwiseo(u1_rlvect_1(A),u1_struct_0(A))
          & v4_bhsp_3(A) )
       => ! [B] :
            ( m2_bhsp_5(B,A)
           => ! [C] :
                ( ( v1_funct_1(C)
                  & v1_funct_2(C,u1_struct_0(A),k6_supinf_1)
                  & m2_relset_1(C,u1_struct_0(A),k6_supinf_1) )
               => ( ( r1_tarski(B,k1_relat_1(C))
                    & ! [D] :
                        ( m1_subset_1(D,u1_struct_0(A))
                       => ( r2_hidden(D,B)
                         => k8_funct_2(u1_struct_0(A),k6_supinf_1,C,D) = k2_bhsp_1(A,D,D) ) ) )
                 => ( v1_bhsp_6(B,A)
                  <=> r1_bhsp_6(A,B,C) ) ) ) ) ) ) )).

fof(t7_bhsp_7,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v3_rlvect_1(A)
        & v4_rlvect_1(A)
        & v5_rlvect_1(A)
        & v6_rlvect_1(A)
        & v7_rlvect_1(A)
        & v2_bhsp_1(A)
        & l1_bhsp_1(A) )
     => ! [B] :
          ( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
         => ( v1_bhsp_6(B,A)
           => ( v1_xboole_0(B)
              | ! [C] :
                  ( m1_subset_1(C,k1_numbers)
                 => ~ ( ~ r1_xreal_0(C,np__0)
                      & ! [D] :
                          ( ( v1_finset_1(D)
                            & m1_subset_1(D,k1_zfmisc_1(u1_struct_0(A))) )
                         => ~ ( ~ v1_xboole_0(D)
                              & r1_tarski(D,B)
                              & ! [E] :
                                  ( ( v1_finset_1(E)
                                    & m1_subset_1(E,k1_zfmisc_1(u1_struct_0(A))) )
                                 => ~ ( r1_tarski(D,E)
                                      & r1_tarski(E,B)
                                      & r1_xreal_0(C,k18_complex1(k5_real_1(k2_bhsp_1(A,k2_bhsp_6(A,B),k2_bhsp_6(A,B)),k2_bhsp_1(A,k1_bhsp_6(A,E),k1_bhsp_6(A,E))))) ) ) ) ) ) ) ) ) ) ) )).

fof(t8_bhsp_7,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v3_rlvect_1(A)
        & v4_rlvect_1(A)
        & v5_rlvect_1(A)
        & v6_rlvect_1(A)
        & v7_rlvect_1(A)
        & v2_bhsp_1(A)
        & l1_bhsp_1(A) )
     => ( ( v1_binop_1(u1_rlvect_1(A),u1_struct_0(A))
          & v2_binop_1(u1_rlvect_1(A),u1_struct_0(A))
          & v1_setwiseo(u1_rlvect_1(A),u1_struct_0(A))
          & v4_bhsp_3(A) )
       => ! [B] :
            ( m2_bhsp_5(B,A)
           => ( ~ v1_xboole_0(B)
             => ! [C] :
                  ( ( v1_funct_1(C)
                    & v1_funct_2(C,u1_struct_0(A),k6_supinf_1)
                    & m2_relset_1(C,u1_struct_0(A),k6_supinf_1) )
                 => ( ( r1_tarski(B,k1_relat_1(C))
                      & ! [D] :
                          ( m1_subset_1(D,u1_struct_0(A))
                         => ( r2_hidden(D,B)
                           => k8_funct_2(u1_struct_0(A),k6_supinf_1,C,D) = k2_bhsp_1(A,D,D) ) )
                      & v1_bhsp_6(B,A) )
                   => k2_bhsp_1(A,k2_bhsp_6(A,B),k2_bhsp_6(A,B)) = k3_bhsp_6(A,B,C) ) ) ) ) ) ) )).
%------------------------------------------------------------------------------