SET007 Axioms: SET007+731.ax


%------------------------------------------------------------------------------
% File     : SET007+731 : TPTP v7.5.0. Released v3.4.0.
% Domain   : Set Theory
% Axioms   : Metric Spaces and an Abstract Intermediate Value Theorem
% 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    : topmetr3 [Urb08]

% Status   : Satisfiable
% Syntax   : Number of formulae    :   19 (   3 unit)
%            Number of atoms       :  223 (  15 equality)
%            Maximal formula depth :   28 (  11 average)
%            Number of connectives :  226 (  22 ~  ;   0  |; 128  &)
%                                         (   0 <=>;  76 =>;   0 <=)
%                                         (   0 <~>;   0 ~|;   0 ~&)
%            Number of predicates  :   29 (   1 propositional; 0-4 arity)
%            Number of functors    :   27 (   7 constant; 0-3 arity)
%            Number of variables   :   58 (   0 singleton;  58 !;   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_topmetr3,axiom,(
    ! [A] :
      ( ( ~ v1_xboole_0(A)
        & m1_subset_1(A,k1_zfmisc_1(k1_numbers)) )
     => ! [B] :
          ( v1_xreal_0(B)
         => ( ! [C] :
                ( v1_xreal_0(C)
               => ( r2_hidden(C,A)
                 => r1_xreal_0(C,B) ) )
           => r1_xreal_0(k4_seq_4(A),B) ) ) ) )).

fof(t2_topmetr3,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v6_metric_1(A)
        & v7_metric_1(A)
        & v8_metric_1(A)
        & v9_metric_1(A)
        & l1_metric_1(A) )
     => ! [B] :
          ( ( v1_funct_1(B)
            & v1_funct_2(B,k5_numbers,u1_struct_0(A))
            & m2_relset_1(B,k5_numbers,u1_struct_0(A)) )
         => ! [C] :
              ( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(k5_pcomps_1(A))))
             => ( ( v2_tbsp_1(B,A)
                  & ! [D] :
                      ( m2_subset_1(D,k1_numbers,k5_numbers)
                     => r2_hidden(k2_normsp_1(A,B,D),C) )
                  & v4_pre_topc(C,k5_pcomps_1(A)) )
               => r2_hidden(k1_tbsp_1(A,B),C) ) ) ) ) )).

fof(t3_topmetr3,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v6_metric_1(A)
        & v7_metric_1(A)
        & v8_metric_1(A)
        & v9_metric_1(A)
        & l1_metric_1(A) )
     => ! [B] :
          ( ( ~ v3_struct_0(B)
            & v6_metric_1(B)
            & v7_metric_1(B)
            & v8_metric_1(B)
            & v9_metric_1(B)
            & l1_metric_1(B) )
         => ! [C] :
              ( ( v1_funct_1(C)
                & v1_funct_2(C,u1_struct_0(k5_pcomps_1(A)),u1_struct_0(k5_pcomps_1(B)))
                & m2_relset_1(C,u1_struct_0(k5_pcomps_1(A)),u1_struct_0(k5_pcomps_1(B))) )
             => ! [D] :
                  ( ( v1_funct_1(D)
                    & v1_funct_2(D,k5_numbers,u1_struct_0(A))
                    & m2_relset_1(D,k5_numbers,u1_struct_0(A)) )
                 => ( v1_funct_1(k5_relat_1(D,C))
                    & v1_funct_2(k5_relat_1(D,C),k5_numbers,u1_struct_0(B))
                    & m2_relset_1(k5_relat_1(D,C),k5_numbers,u1_struct_0(B)) ) ) ) ) ) )).

fof(t4_topmetr3,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v6_metric_1(A)
        & v7_metric_1(A)
        & v8_metric_1(A)
        & v9_metric_1(A)
        & l1_metric_1(A) )
     => ! [B] :
          ( ( ~ v3_struct_0(B)
            & v6_metric_1(B)
            & v7_metric_1(B)
            & v8_metric_1(B)
            & v9_metric_1(B)
            & l1_metric_1(B) )
         => ! [C] :
              ( ( v1_funct_1(C)
                & v1_funct_2(C,u1_struct_0(k5_pcomps_1(A)),u1_struct_0(k5_pcomps_1(B)))
                & m2_relset_1(C,u1_struct_0(k5_pcomps_1(A)),u1_struct_0(k5_pcomps_1(B))) )
             => ! [D] :
                  ( ( v1_funct_1(D)
                    & v1_funct_2(D,k5_numbers,u1_struct_0(A))
                    & m2_relset_1(D,k5_numbers,u1_struct_0(A)) )
                 => ! [E] :
                      ( ( v1_funct_1(E)
                        & v1_funct_2(E,k5_numbers,u1_struct_0(B))
                        & m2_relset_1(E,k5_numbers,u1_struct_0(B)) )
                     => ( ( v2_tbsp_1(D,A)
                          & E = k5_relat_1(D,C)
                          & v5_pre_topc(C,k5_pcomps_1(A),k5_pcomps_1(B)) )
                       => v2_tbsp_1(E,B) ) ) ) ) ) ) )).

fof(t5_topmetr3,axiom,(
    $true )).

fof(t6_topmetr3,axiom,(
    ! [A] :
      ( ( v1_funct_1(A)
        & v1_funct_2(A,k5_numbers,k1_numbers)
        & m2_relset_1(A,k5_numbers,k1_numbers) )
     => ! [B] :
          ( ( v1_funct_1(B)
            & v1_funct_2(B,k5_numbers,u1_struct_0(k8_metric_1))
            & m2_relset_1(B,k5_numbers,u1_struct_0(k8_metric_1)) )
         => ( A = B
           => ( ( v4_seq_2(A)
               => v2_tbsp_1(B,k8_metric_1) )
              & ( v2_tbsp_1(B,k8_metric_1)
               => v4_seq_2(A) )
              & ( v4_seq_2(A)
               => k2_seq_2(A) = k1_tbsp_1(k8_metric_1,B) ) ) ) ) ) )).

fof(t7_topmetr3,axiom,(
    ! [A] :
      ( v1_xreal_0(A)
     => ! [B] :
          ( v1_xreal_0(B)
         => ! [C] :
              ( ( v1_funct_1(C)
                & v1_funct_2(C,k5_numbers,k1_numbers)
                & m2_relset_1(C,k5_numbers,k1_numbers) )
             => ( r1_tarski(k1_pscomp_1(k5_numbers,k1_numbers,C),k1_rcomp_1(A,B))
               => ( v1_funct_1(C)
                  & v1_funct_2(C,k5_numbers,u1_struct_0(k2_topmetr(A,B)))
                  & m2_relset_1(C,k5_numbers,u1_struct_0(k2_topmetr(A,B))) ) ) ) ) ) )).

fof(t8_topmetr3,axiom,(
    ! [A] :
      ( v1_xreal_0(A)
     => ! [B] :
          ( v1_xreal_0(B)
         => ! [C] :
              ( ( v1_funct_1(C)
                & v1_funct_2(C,k5_numbers,u1_struct_0(k2_topmetr(A,B)))
                & m2_relset_1(C,k5_numbers,u1_struct_0(k2_topmetr(A,B))) )
             => ( r1_xreal_0(A,B)
               => ( v1_funct_1(C)
                  & v1_funct_2(C,k5_numbers,u1_struct_0(k8_metric_1))
                  & m2_relset_1(C,k5_numbers,u1_struct_0(k8_metric_1)) ) ) ) ) ) )).

fof(t9_topmetr3,axiom,(
    ! [A] :
      ( v1_xreal_0(A)
     => ! [B] :
          ( v1_xreal_0(B)
         => ! [C] :
              ( ( v1_funct_1(C)
                & v1_funct_2(C,k5_numbers,u1_struct_0(k2_topmetr(A,B)))
                & m2_relset_1(C,k5_numbers,u1_struct_0(k2_topmetr(A,B))) )
             => ! [D] :
                  ( ( v1_funct_1(D)
                    & v1_funct_2(D,k5_numbers,u1_struct_0(k8_metric_1))
                    & m2_relset_1(D,k5_numbers,u1_struct_0(k8_metric_1)) )
                 => ( ( D = C
                      & r1_xreal_0(A,B) )
                   => ( ( v2_tbsp_1(D,k8_metric_1)
                       => v2_tbsp_1(C,k2_topmetr(A,B)) )
                      & ( v2_tbsp_1(C,k2_topmetr(A,B))
                       => v2_tbsp_1(D,k8_metric_1) )
                      & ( v2_tbsp_1(D,k8_metric_1)
                       => k1_tbsp_1(k8_metric_1,D) = k1_tbsp_1(k2_topmetr(A,B),C) ) ) ) ) ) ) ) )).

fof(t10_topmetr3,axiom,(
    ! [A] :
      ( v1_xreal_0(A)
     => ! [B] :
          ( v1_xreal_0(B)
         => ! [C] :
              ( ( v1_funct_1(C)
                & v1_funct_2(C,k5_numbers,k1_numbers)
                & m2_relset_1(C,k5_numbers,k1_numbers) )
             => ! [D] :
                  ( ( v1_funct_1(D)
                    & v1_funct_2(D,k5_numbers,u1_struct_0(k2_topmetr(A,B)))
                    & m2_relset_1(D,k5_numbers,u1_struct_0(k2_topmetr(A,B))) )
                 => ( ( D = C
                      & r1_xreal_0(A,B)
                      & v4_seq_2(C) )
                   => ( v2_tbsp_1(D,k2_topmetr(A,B))
                      & k2_seq_2(C) = k1_tbsp_1(k2_topmetr(A,B),D) ) ) ) ) ) ) )).

fof(t11_topmetr3,axiom,(
    ! [A] :
      ( v1_xreal_0(A)
     => ! [B] :
          ( v1_xreal_0(B)
         => ! [C] :
              ( ( v1_funct_1(C)
                & v1_funct_2(C,k5_numbers,k1_numbers)
                & m2_relset_1(C,k5_numbers,k1_numbers) )
             => ! [D] :
                  ( ( v1_funct_1(D)
                    & v1_funct_2(D,k5_numbers,u1_struct_0(k2_topmetr(A,B)))
                    & m2_relset_1(D,k5_numbers,u1_struct_0(k2_topmetr(A,B))) )
                 => ( ( D = C
                      & r1_xreal_0(A,B)
                      & v3_seqm_3(C) )
                   => v2_tbsp_1(D,k2_topmetr(A,B)) ) ) ) ) ) )).

fof(t12_topmetr3,axiom,(
    ! [A] :
      ( v1_xreal_0(A)
     => ! [B] :
          ( v1_xreal_0(B)
         => ! [C] :
              ( ( v1_funct_1(C)
                & v1_funct_2(C,k5_numbers,k1_numbers)
                & m2_relset_1(C,k5_numbers,k1_numbers) )
             => ! [D] :
                  ( ( v1_funct_1(D)
                    & v1_funct_2(D,k5_numbers,u1_struct_0(k2_topmetr(A,B)))
                    & m2_relset_1(D,k5_numbers,u1_struct_0(k2_topmetr(A,B))) )
                 => ( ( D = C
                      & r1_xreal_0(A,B)
                      & v4_seqm_3(C) )
                   => v2_tbsp_1(D,k2_topmetr(A,B)) ) ) ) ) ) )).

fof(t13_topmetr3,axiom,(
    $true )).

fof(t14_topmetr3,axiom,(
    $true )).

fof(t15_topmetr3,axiom,(
    ! [A] :
      ( ( ~ v1_xboole_0(A)
        & m1_subset_1(A,k1_zfmisc_1(k1_numbers)) )
     => ~ ( v1_seq_4(A)
          & ! [B] :
              ( ( v1_funct_1(B)
                & v1_funct_2(B,k5_numbers,k1_numbers)
                & m2_relset_1(B,k5_numbers,k1_numbers) )
             => ~ ( v3_seqm_3(B)
                  & v4_seq_2(B)
                  & r1_tarski(k1_pscomp_1(k5_numbers,k1_numbers,B),A)
                  & k2_seq_2(B) = k4_seq_4(A) ) ) ) ) )).

fof(t16_topmetr3,axiom,(
    ! [A] :
      ( ( ~ v1_xboole_0(A)
        & m1_subset_1(A,k1_zfmisc_1(k1_numbers)) )
     => ~ ( v2_seq_4(A)
          & ! [B] :
              ( ( v1_funct_1(B)
                & v1_funct_2(B,k5_numbers,k1_numbers)
                & m2_relset_1(B,k5_numbers,k1_numbers) )
             => ~ ( v4_seqm_3(B)
                  & v4_seq_2(B)
                  & r1_tarski(k1_pscomp_1(k5_numbers,k1_numbers,B),A)
                  & k2_seq_2(B) = k5_seq_4(A) ) ) ) ) )).

fof(t17_topmetr3,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v6_metric_1(A)
        & v7_metric_1(A)
        & v8_metric_1(A)
        & v9_metric_1(A)
        & l1_metric_1(A) )
     => ! [B] :
          ( ( v1_funct_1(B)
            & v1_funct_2(B,u1_struct_0(k5_topmetr),u1_struct_0(k5_pcomps_1(A)))
            & m2_relset_1(B,u1_struct_0(k5_topmetr),u1_struct_0(k5_pcomps_1(A))) )
         => ! [C] :
              ( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(k5_pcomps_1(A))))
             => ! [D] :
                  ( m1_subset_1(D,k1_zfmisc_1(u1_struct_0(k5_pcomps_1(A))))
                 => ! [E] :
                      ( m1_subset_1(E,k1_numbers)
                     => ! [F] :
                          ( m1_subset_1(F,k1_numbers)
                         => ~ ( r1_xreal_0(np__0,E)
                              & r1_xreal_0(F,np__1)
                              & r1_xreal_0(E,F)
                              & r2_hidden(k1_funct_1(B,E),C)
                              & r2_hidden(k1_funct_1(B,F),D)
                              & v4_pre_topc(C,k5_pcomps_1(A))
                              & v4_pre_topc(D,k5_pcomps_1(A))
                              & v5_pre_topc(B,k5_topmetr,k5_pcomps_1(A))
                              & k4_subset_1(u1_struct_0(k5_pcomps_1(A)),C,D) = u1_struct_0(A)
                              & ! [G] :
                                  ( m1_subset_1(G,k1_numbers)
                                 => ~ ( r1_xreal_0(E,G)
                                      & r1_xreal_0(G,F)
                                      & r2_hidden(k1_funct_1(B,G),k5_subset_1(u1_struct_0(k5_pcomps_1(A)),C,D)) ) ) ) ) ) ) ) ) ) )).

fof(t18_topmetr3,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] :
          ( m1_subset_1(B,u1_struct_0(k15_euclid(A)))
         => ! [C] :
              ( m1_subset_1(C,u1_struct_0(k15_euclid(A)))
             => ! [D] :
                  ( ( ~ v1_xboole_0(D)
                    & m1_subset_1(D,k1_zfmisc_1(u1_struct_0(k15_euclid(A)))) )
                 => ! [E] :
                      ( ( ~ v1_xboole_0(E)
                        & m1_subset_1(E,k1_zfmisc_1(u1_struct_0(k15_euclid(A)))) )
                     => ( ( r1_topreal1(k15_euclid(A),B,C,D)
                          & r1_topreal1(k15_euclid(A),C,B,E)
                          & r1_tarski(E,D) )
                       => E = D ) ) ) ) ) ) )).

fof(t19_topmetr3,axiom,(
    ! [A] :
      ( ( ~ v1_xboole_0(A)
        & v6_compts_1(A,k15_euclid(np__2))
        & m1_subset_1(A,k1_zfmisc_1(u1_struct_0(k15_euclid(np__2)))) )
     => ! [B] :
          ( ( ~ v1_xboole_0(B)
            & v6_compts_1(B,k15_euclid(np__2))
            & m1_subset_1(B,k1_zfmisc_1(u1_struct_0(k15_euclid(np__2)))) )
         => ~ ( v1_topreal2(A)
              & r1_topreal1(k15_euclid(np__2),k30_pscomp_1(A),k34_pscomp_1(A),B)
              & r1_tarski(B,A)
              & B != k8_jordan6(A)
              & B != k9_jordan6(A) ) ) ) )).
%------------------------------------------------------------------------------