SET007 Axioms: SET007+786.ax


%------------------------------------------------------------------------------
% File     : SET007+786 : TPTP v7.5.0. Released v3.4.0.
% Domain   : Set Theory
% Axioms   : Banach Space of Absolute Summable Real Sequences
% 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    : rsspace3 [Urb08]

% Status   : Satisfiable
% Syntax   : Number of formulae    :   25 (   5 unit)
%            Number of atoms       :  182 (  25 equality)
%            Maximal formula depth :   18 (   7 average)
%            Number of connectives :  175 (  18 ~  ;   0  |; 104  &)
%                                         (   6 <=>;  47 =>;   0 <=)
%                                         (   0 <~>;   0 ~|;   0 ~&)
%            Number of predicates  :   27 (   1 propositional; 0-3 arity)
%            Number of functors    :   41 (   9 constant; 0-5 arity)
%            Number of variables   :   51 (   0 singleton;  47 !;   4 ?)
%            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(fc1_rsspace3,axiom,(
    ~ v1_xboole_0(k1_rsspace3) )).

fof(fc2_rsspace3,axiom,
    ( ~ v1_xboole_0(k1_rsspace3)
    & v1_rlsub_1(k1_rsspace3,k7_rsspace) )).

fof(fc3_rsspace3,axiom,
    ( ~ v3_struct_0(g2_rlvect_1(k1_rsspace3,k10_rsspace(k7_rsspace,k1_rsspace3),k8_rsspace(k7_rsspace,k1_rsspace3),k9_rsspace(k7_rsspace,k1_rsspace3)))
    & v2_rlvect_1(g2_rlvect_1(k1_rsspace3,k10_rsspace(k7_rsspace,k1_rsspace3),k8_rsspace(k7_rsspace,k1_rsspace3),k9_rsspace(k7_rsspace,k1_rsspace3)))
    & v3_rlvect_1(g2_rlvect_1(k1_rsspace3,k10_rsspace(k7_rsspace,k1_rsspace3),k8_rsspace(k7_rsspace,k1_rsspace3),k9_rsspace(k7_rsspace,k1_rsspace3)))
    & v4_rlvect_1(g2_rlvect_1(k1_rsspace3,k10_rsspace(k7_rsspace,k1_rsspace3),k8_rsspace(k7_rsspace,k1_rsspace3),k9_rsspace(k7_rsspace,k1_rsspace3)))
    & v5_rlvect_1(g2_rlvect_1(k1_rsspace3,k10_rsspace(k7_rsspace,k1_rsspace3),k8_rsspace(k7_rsspace,k1_rsspace3),k9_rsspace(k7_rsspace,k1_rsspace3)))
    & v6_rlvect_1(g2_rlvect_1(k1_rsspace3,k10_rsspace(k7_rsspace,k1_rsspace3),k8_rsspace(k7_rsspace,k1_rsspace3),k9_rsspace(k7_rsspace,k1_rsspace3)))
    & v7_rlvect_1(g2_rlvect_1(k1_rsspace3,k10_rsspace(k7_rsspace,k1_rsspace3),k8_rsspace(k7_rsspace,k1_rsspace3),k9_rsspace(k7_rsspace,k1_rsspace3))) )).

fof(fc4_rsspace3,axiom,(
    ! [A,B,C,D,E] :
      ( ( ~ v1_xboole_0(A)
        & m1_subset_1(B,A)
        & v1_funct_1(C)
        & v1_funct_2(C,k2_zfmisc_1(A,A),A)
        & m1_relset_1(C,k2_zfmisc_1(A,A),A)
        & v1_funct_1(D)
        & v1_funct_2(D,k2_zfmisc_1(k1_numbers,A),A)
        & m1_relset_1(D,k2_zfmisc_1(k1_numbers,A),A)
        & v1_funct_1(E)
        & v1_funct_2(E,A,k1_numbers)
        & m1_relset_1(E,A,k1_numbers) )
     => ( ~ v3_struct_0(g1_normsp_1(A,B,C,D,E))
        & v1_normsp_1(g1_normsp_1(A,B,C,D,E)) ) ) )).

fof(fc5_rsspace3,axiom,
    ( ~ v3_struct_0(k3_rsspace3)
    & v3_rlvect_1(k3_rsspace3)
    & v4_rlvect_1(k3_rsspace3)
    & v5_rlvect_1(k3_rsspace3)
    & v6_rlvect_1(k3_rsspace3)
    & v7_rlvect_1(k3_rsspace3)
    & v2_normsp_1(k3_rsspace3) )).

fof(d1_rsspace3,axiom,(
    ! [A] :
      ( m1_subset_1(A,k1_zfmisc_1(u1_struct_0(k7_rsspace)))
     => ( A = k1_rsspace3
      <=> ! [B] :
            ( r2_hidden(B,A)
          <=> ( r2_hidden(B,k1_rsspace)
              & v2_series_1(k2_rsspace(B)) ) ) ) ) )).

fof(t1_rsspace3,axiom,(
    ! [A] :
      ( m1_subset_1(A,k1_numbers)
     => ! [B] :
          ( ( v1_funct_1(B)
            & v1_funct_2(B,k5_numbers,k1_numbers)
            & m2_relset_1(B,k5_numbers,k1_numbers) )
         => ( v4_seq_2(B)
           => ! [C] :
                ( ( v1_funct_1(C)
                  & v1_funct_2(C,k5_numbers,k1_numbers)
                  & m2_relset_1(C,k5_numbers,k1_numbers) )
               => ( ! [D] :
                      ( m2_subset_1(D,k1_numbers,k5_numbers)
                     => k2_seq_1(k5_numbers,k1_numbers,C,D) = k18_complex1(k5_real_1(k2_seq_1(k5_numbers,k1_numbers,B,D),A)) )
                 => ( v4_seq_2(C)
                    & k2_seq_2(C) = k18_complex1(k5_real_1(k2_seq_2(B),A)) ) ) ) ) ) ) )).

fof(d2_rsspace3,axiom,(
    ! [A] :
      ( ( v1_funct_1(A)
        & v1_funct_2(A,k1_rsspace3,k1_numbers)
        & m2_relset_1(A,k1_rsspace3,k1_numbers) )
     => ( A = k2_rsspace3
      <=> ! [B] :
            ( r2_hidden(B,k1_rsspace3)
           => k2_seq_1(k1_rsspace3,k1_numbers,A,B) = k2_series_1(k22_seq_1(k2_rsspace(B))) ) ) ) )).

fof(t2_rsspace3,axiom,(
    $true )).

fof(t3_rsspace3,axiom,(
    $true )).

fof(t4_rsspace3,axiom,(
    ! [A] :
      ( l1_normsp_1(A)
     => ( ( ~ v3_struct_0(g2_rlvect_1(u1_struct_0(A),u2_struct_0(A),u1_rlvect_1(A),u2_rlvect_1(A)))
          & v3_rlvect_1(g2_rlvect_1(u1_struct_0(A),u2_struct_0(A),u1_rlvect_1(A),u2_rlvect_1(A)))
          & v4_rlvect_1(g2_rlvect_1(u1_struct_0(A),u2_struct_0(A),u1_rlvect_1(A),u2_rlvect_1(A)))
          & v5_rlvect_1(g2_rlvect_1(u1_struct_0(A),u2_struct_0(A),u1_rlvect_1(A),u2_rlvect_1(A)))
          & v6_rlvect_1(g2_rlvect_1(u1_struct_0(A),u2_struct_0(A),u1_rlvect_1(A),u2_rlvect_1(A)))
          & v7_rlvect_1(g2_rlvect_1(u1_struct_0(A),u2_struct_0(A),u1_rlvect_1(A),u2_rlvect_1(A)))
          & l2_rlvect_1(g2_rlvect_1(u1_struct_0(A),u2_struct_0(A),u1_rlvect_1(A),u2_rlvect_1(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)
          & l2_rlvect_1(A) ) ) ) )).

fof(t5_rsspace3,axiom,(
    ! [A] :
      ( ( v1_funct_1(A)
        & v1_funct_2(A,k5_numbers,k1_numbers)
        & m2_relset_1(A,k5_numbers,k1_numbers) )
     => ( ! [B] :
            ( m2_subset_1(B,k1_numbers,k5_numbers)
           => k2_seq_1(k5_numbers,k1_numbers,A,B) = np__0 )
       => ( v2_series_1(A)
          & k2_series_1(k22_seq_1(A)) = np__0 ) ) ) )).

fof(t6_rsspace3,axiom,(
    ! [A] :
      ( ( v1_funct_1(A)
        & v1_funct_2(A,k5_numbers,k1_numbers)
        & m2_relset_1(A,k5_numbers,k1_numbers) )
     => ( ( v2_series_1(A)
          & k2_series_1(k22_seq_1(A)) = np__0 )
       => ! [B] :
            ( m2_subset_1(B,k1_numbers,k5_numbers)
           => k2_seq_1(k5_numbers,k1_numbers,A,B) = np__0 ) ) ) )).

fof(t7_rsspace3,axiom,
    ( ~ v3_struct_0(g1_normsp_1(k1_rsspace3,k10_rsspace(k7_rsspace,k1_rsspace3),k8_rsspace(k7_rsspace,k1_rsspace3),k9_rsspace(k7_rsspace,k1_rsspace3),k2_rsspace3))
    & v3_rlvect_1(g1_normsp_1(k1_rsspace3,k10_rsspace(k7_rsspace,k1_rsspace3),k8_rsspace(k7_rsspace,k1_rsspace3),k9_rsspace(k7_rsspace,k1_rsspace3),k2_rsspace3))
    & v4_rlvect_1(g1_normsp_1(k1_rsspace3,k10_rsspace(k7_rsspace,k1_rsspace3),k8_rsspace(k7_rsspace,k1_rsspace3),k9_rsspace(k7_rsspace,k1_rsspace3),k2_rsspace3))
    & v5_rlvect_1(g1_normsp_1(k1_rsspace3,k10_rsspace(k7_rsspace,k1_rsspace3),k8_rsspace(k7_rsspace,k1_rsspace3),k9_rsspace(k7_rsspace,k1_rsspace3),k2_rsspace3))
    & v6_rlvect_1(g1_normsp_1(k1_rsspace3,k10_rsspace(k7_rsspace,k1_rsspace3),k8_rsspace(k7_rsspace,k1_rsspace3),k9_rsspace(k7_rsspace,k1_rsspace3),k2_rsspace3))
    & v7_rlvect_1(g1_normsp_1(k1_rsspace3,k10_rsspace(k7_rsspace,k1_rsspace3),k8_rsspace(k7_rsspace,k1_rsspace3),k9_rsspace(k7_rsspace,k1_rsspace3),k2_rsspace3))
    & l2_rlvect_1(g1_normsp_1(k1_rsspace3,k10_rsspace(k7_rsspace,k1_rsspace3),k8_rsspace(k7_rsspace,k1_rsspace3),k9_rsspace(k7_rsspace,k1_rsspace3),k2_rsspace3)) )).

fof(d3_rsspace3,axiom,(
    k3_rsspace3 = g1_normsp_1(k1_rsspace3,k10_rsspace(k7_rsspace,k1_rsspace3),k8_rsspace(k7_rsspace,k1_rsspace3),k9_rsspace(k7_rsspace,k1_rsspace3),k2_rsspace3) )).

fof(t8_rsspace3,axiom,
    ( u1_struct_0(k3_rsspace3) = k1_rsspace3
    & ! [A] :
        ( m1_subset_1(A,u1_struct_0(k3_rsspace3))
      <=> ( v1_funct_1(A)
          & v1_funct_2(A,k5_numbers,k1_numbers)
          & m2_relset_1(A,k5_numbers,k1_numbers)
          & v2_series_1(k2_rsspace(A)) ) )
    & k1_rlvect_1(k3_rsspace3) = k6_rsspace
    & ! [A] :
        ( m1_subset_1(A,u1_struct_0(k3_rsspace3))
       => A = k2_rsspace(A) )
    & ! [A] :
        ( m1_subset_1(A,u1_struct_0(k3_rsspace3))
       => ! [B] :
            ( m1_subset_1(B,u1_struct_0(k3_rsspace3))
           => k2_rlvect_1(k3_rsspace3,A,B) = k9_seq_1(k2_rsspace(A),k2_rsspace(B)) ) )
    & ! [A] :
        ( m1_subset_1(A,k1_numbers)
       => ! [B] :
            ( m1_subset_1(B,u1_struct_0(k3_rsspace3))
           => k3_rlvect_1(k3_rsspace3,B,A) = k14_seq_1(k2_rsspace(B),A) ) )
    & ! [A] :
        ( m1_subset_1(A,u1_struct_0(k3_rsspace3))
       => ( k5_rlvect_1(k3_rsspace3,A) = k17_seq_1(k2_rsspace(A))
          & k2_rsspace(k5_rlvect_1(k3_rsspace3,A)) = k17_seq_1(k2_rsspace(A)) ) )
    & ! [A] :
        ( m1_subset_1(A,u1_struct_0(k3_rsspace3))
       => ! [B] :
            ( m1_subset_1(B,u1_struct_0(k3_rsspace3))
           => k6_rlvect_1(k3_rsspace3,A,B) = k10_seq_1(k2_rsspace(A),k2_rsspace(B)) ) )
    & ! [A] :
        ( m1_subset_1(A,u1_struct_0(k3_rsspace3))
       => v2_series_1(k2_rsspace(A)) )
    & ! [A] :
        ( m1_subset_1(A,u1_struct_0(k3_rsspace3))
       => k1_normsp_1(k3_rsspace3,A) = k2_series_1(k22_seq_1(k2_rsspace(A))) ) )).

fof(t9_rsspace3,axiom,(
    ! [A] :
      ( m1_subset_1(A,u1_struct_0(k3_rsspace3))
     => ! [B] :
          ( m1_subset_1(B,u1_struct_0(k3_rsspace3))
         => ! [C] :
              ( m1_subset_1(C,k1_numbers)
             => ( ( k1_normsp_1(k3_rsspace3,A) = np__0
                 => A = k1_rlvect_1(k3_rsspace3) )
                & ( A = k1_rlvect_1(k3_rsspace3)
                 => k1_normsp_1(k3_rsspace3,A) = np__0 )
                & r1_xreal_0(np__0,k1_normsp_1(k3_rsspace3,A))
                & r1_xreal_0(k1_normsp_1(k3_rsspace3,k2_rlvect_1(k3_rsspace3,A,B)),k3_real_1(k1_normsp_1(k3_rsspace3,A),k1_normsp_1(k3_rsspace3,B)))
                & k1_normsp_1(k3_rsspace3,k3_rlvect_1(k3_rsspace3,A,C)) = k4_real_1(k18_complex1(C),k1_normsp_1(k3_rsspace3,A)) ) ) ) ) )).

fof(d4_rsspace3,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & l1_normsp_1(A) )
     => ! [B] :
          ( m1_subset_1(B,u1_struct_0(A))
         => ! [C] :
              ( m1_subset_1(C,u1_struct_0(A))
             => k4_rsspace3(A,B,C) = k1_normsp_1(A,k6_rlvect_1(A,B,C)) ) ) ) )).

fof(d5_rsspace3,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & l1_normsp_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)) )
         => ( v1_rsspace3(B,A)
          <=> ! [C] :
                ( m1_subset_1(C,k1_numbers)
               => ~ ( ~ r1_xreal_0(C,np__0)
                    & ! [D] :
                        ( m2_subset_1(D,k1_numbers,k5_numbers)
                       => ? [E] :
                            ( m2_subset_1(E,k1_numbers,k5_numbers)
                            & ? [F] :
                                ( m2_subset_1(F,k1_numbers,k5_numbers)
                                & r1_xreal_0(D,E)
                                & r1_xreal_0(D,F)
                                & r1_xreal_0(C,k4_rsspace3(A,k2_normsp_1(A,B,E),k2_normsp_1(A,B,F))) ) ) ) ) ) ) ) ) )).

fof(t10_rsspace3,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_normsp_1(A)
        & l1_normsp_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)) )
         => ( v1_rsspace3(B,A)
          <=> ! [C] :
                ( m1_subset_1(C,k1_numbers)
               => ~ ( ~ r1_xreal_0(C,np__0)
                    & ! [D] :
                        ( m2_subset_1(D,k1_numbers,k5_numbers)
                       => ? [E] :
                            ( m2_subset_1(E,k1_numbers,k5_numbers)
                            & ? [F] :
                                ( m2_subset_1(F,k1_numbers,k5_numbers)
                                & r1_xreal_0(D,E)
                                & r1_xreal_0(D,F)
                                & r1_xreal_0(C,k1_normsp_1(A,k6_rlvect_1(A,k2_normsp_1(A,B,E),k2_normsp_1(A,B,F)))) ) ) ) ) ) ) ) ) )).

fof(t11_rsspace3,axiom,(
    ! [A] :
      ( ( v1_funct_1(A)
        & v1_funct_2(A,k5_numbers,u1_struct_0(k3_rsspace3))
        & m2_relset_1(A,k5_numbers,u1_struct_0(k3_rsspace3)) )
     => ( v1_rsspace3(A,k3_rsspace3)
       => v4_normsp_1(A,k3_rsspace3) ) ) )).

fof(dt_k1_rsspace3,axiom,(
    m1_subset_1(k1_rsspace3,k1_zfmisc_1(u1_struct_0(k7_rsspace))) )).

fof(dt_k2_rsspace3,axiom,
    ( v1_funct_1(k2_rsspace3)
    & v1_funct_2(k2_rsspace3,k1_rsspace3,k1_numbers)
    & m2_relset_1(k2_rsspace3,k1_rsspace3,k1_numbers) )).

fof(dt_k3_rsspace3,axiom,
    ( ~ v3_struct_0(k3_rsspace3)
    & l1_normsp_1(k3_rsspace3) )).

fof(dt_k4_rsspace3,axiom,(
    ! [A,B,C] :
      ( ( ~ v3_struct_0(A)
        & l1_normsp_1(A)
        & m1_subset_1(B,u1_struct_0(A))
        & m1_subset_1(C,u1_struct_0(A)) )
     => m1_subset_1(k4_rsspace3(A,B,C),k1_numbers) ) )).
%------------------------------------------------------------------------------