SET007 Axioms: SET007+822.ax


%------------------------------------------------------------------------------
% File     : SET007+822 : TPTP v7.5.0. Released v3.4.0.
% Domain   : Set Theory
% Axioms   : Banach Space of Absolute Summable Complex 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    : csspace3 [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  |; 105  &)
%                                         (   6 <=>;  46 =>;   0 <=)
%                                         (   0 <~>;   0 ~|;   0 ~&)
%            Number of predicates  :   28 (   1 propositional; 0-3 arity)
%            Number of functors    :   45 (  11 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_csspace3,axiom,(
    ~ v1_xboole_0(k1_csspace3) )).

fof(fc2_csspace3,axiom,
    ( ~ v1_xboole_0(k1_csspace3)
    & v3_clvect_1(k1_csspace3,k7_csspace) )).

fof(fc3_csspace3,axiom,
    ( ~ v3_struct_0(g1_clvect_1(k1_csspace3,k10_csspace(k7_csspace,k1_csspace3),k8_csspace(k7_csspace,k1_csspace3),k9_csspace(k7_csspace,k1_csspace3)))
    & v3_rlvect_1(g1_clvect_1(k1_csspace3,k10_csspace(k7_csspace,k1_csspace3),k8_csspace(k7_csspace,k1_csspace3),k9_csspace(k7_csspace,k1_csspace3)))
    & v4_rlvect_1(g1_clvect_1(k1_csspace3,k10_csspace(k7_csspace,k1_csspace3),k8_csspace(k7_csspace,k1_csspace3),k9_csspace(k7_csspace,k1_csspace3)))
    & v5_rlvect_1(g1_clvect_1(k1_csspace3,k10_csspace(k7_csspace,k1_csspace3),k8_csspace(k7_csspace,k1_csspace3),k9_csspace(k7_csspace,k1_csspace3)))
    & v6_rlvect_1(g1_clvect_1(k1_csspace3,k10_csspace(k7_csspace,k1_csspace3),k8_csspace(k7_csspace,k1_csspace3),k9_csspace(k7_csspace,k1_csspace3)))
    & v1_clvect_1(g1_clvect_1(k1_csspace3,k10_csspace(k7_csspace,k1_csspace3),k8_csspace(k7_csspace,k1_csspace3),k9_csspace(k7_csspace,k1_csspace3)))
    & v2_clvect_1(g1_clvect_1(k1_csspace3,k10_csspace(k7_csspace,k1_csspace3),k8_csspace(k7_csspace,k1_csspace3),k9_csspace(k7_csspace,k1_csspace3))) )).

fof(fc4_csspace3,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(k2_numbers,A),A)
        & m1_relset_1(D,k2_zfmisc_1(k2_numbers,A),A)
        & v1_funct_1(E)
        & v1_funct_2(E,A,k1_numbers)
        & m1_relset_1(E,A,k1_numbers) )
     => ( ~ v3_struct_0(g2_clvect_1(A,B,C,D,E))
        & v4_clvect_1(g2_clvect_1(A,B,C,D,E)) ) ) )).

fof(fc5_csspace3,axiom,
    ( ~ v3_struct_0(k3_csspace3)
    & v3_rlvect_1(k3_csspace3)
    & v4_rlvect_1(k3_csspace3)
    & v5_rlvect_1(k3_csspace3)
    & v6_rlvect_1(k3_csspace3)
    & v2_clvect_1(k3_csspace3)
    & v5_clvect_1(k3_csspace3) )).

fof(d1_csspace3,axiom,(
    ! [A] :
      ( m1_subset_1(A,k1_zfmisc_1(u1_struct_0(k7_csspace)))
     => ( A = k1_csspace3
      <=> ! [B] :
            ( r2_hidden(B,A)
          <=> ( r2_hidden(B,k1_csspace)
              & v2_comseq_3(k2_csspace(B)) ) ) ) ) )).

fof(t1_csspace3,axiom,(
    ! [A] :
      ( m1_subset_1(A,k2_numbers)
     => ! [B] :
          ( ( v1_funct_1(B)
            & v1_funct_2(B,k5_numbers,k2_numbers)
            & m2_relset_1(B,k5_numbers,k2_numbers) )
         => ! [C] :
              ( ( v1_funct_1(C)
                & v1_funct_2(C,k5_numbers,k1_numbers)
                & m2_relset_1(C,k5_numbers,k1_numbers) )
             => ( ( v2_comseq_2(B)
                  & ! [D] :
                      ( m2_subset_1(D,k1_numbers,k5_numbers)
                     => k2_seq_1(k5_numbers,k1_numbers,C,D) = k17_complex1(k11_complex1(k1_comseq_1(B,D),A)) ) )
               => ( v4_seq_2(C)
                  & k2_seq_2(C) = k17_complex1(k11_complex1(k2_comseq_2(B),A)) ) ) ) ) ) )).

fof(d2_csspace3,axiom,(
    ! [A] :
      ( ( v1_funct_1(A)
        & v1_funct_2(A,k1_csspace3,k1_numbers)
        & m2_relset_1(A,k1_csspace3,k1_numbers) )
     => ( A = k2_csspace3
      <=> ! [B] :
            ( r2_hidden(B,k1_csspace3)
           => k2_seq_1(k1_csspace3,k1_numbers,A,B) = k2_series_1(k9_comseq_1(k5_numbers,k2_csspace(B))) ) ) ) )).

fof(t2_csspace3,axiom,(
    $true )).

fof(t3_csspace3,axiom,(
    $true )).

fof(t4_csspace3,axiom,(
    ! [A] :
      ( l2_clvect_1(A)
     => ( ( ~ v3_struct_0(g1_clvect_1(u1_struct_0(A),u2_struct_0(A),u1_rlvect_1(A),u1_clvect_1(A)))
          & v3_rlvect_1(g1_clvect_1(u1_struct_0(A),u2_struct_0(A),u1_rlvect_1(A),u1_clvect_1(A)))
          & v4_rlvect_1(g1_clvect_1(u1_struct_0(A),u2_struct_0(A),u1_rlvect_1(A),u1_clvect_1(A)))
          & v5_rlvect_1(g1_clvect_1(u1_struct_0(A),u2_struct_0(A),u1_rlvect_1(A),u1_clvect_1(A)))
          & v6_rlvect_1(g1_clvect_1(u1_struct_0(A),u2_struct_0(A),u1_rlvect_1(A),u1_clvect_1(A)))
          & v2_clvect_1(g1_clvect_1(u1_struct_0(A),u2_struct_0(A),u1_rlvect_1(A),u1_clvect_1(A)))
          & l1_clvect_1(g1_clvect_1(u1_struct_0(A),u2_struct_0(A),u1_rlvect_1(A),u1_clvect_1(A))) )
       => ( ~ v3_struct_0(A)
          & v3_rlvect_1(A)
          & v4_rlvect_1(A)
          & v5_rlvect_1(A)
          & v6_rlvect_1(A)
          & v2_clvect_1(A)
          & l1_clvect_1(A) ) ) ) )).

fof(t5_csspace3,axiom,(
    ! [A] :
      ( ( v1_funct_1(A)
        & v1_funct_2(A,k5_numbers,k2_numbers)
        & m2_relset_1(A,k5_numbers,k2_numbers) )
     => ( ! [B] :
            ( m2_subset_1(B,k1_numbers,k5_numbers)
           => k1_comseq_1(A,B) = k5_complex1 )
       => ( v2_comseq_3(A)
          & k2_series_1(k9_comseq_1(k5_numbers,A)) = np__0 ) ) ) )).

fof(t6_csspace3,axiom,(
    ! [A] :
      ( ( v1_funct_1(A)
        & v1_funct_2(A,k5_numbers,k2_numbers)
        & m2_relset_1(A,k5_numbers,k2_numbers) )
     => ( ( v2_comseq_3(A)
          & k2_series_1(k9_comseq_1(k5_numbers,A)) = np__0 )
       => ! [B] :
            ( m2_subset_1(B,k1_numbers,k5_numbers)
           => k1_comseq_1(A,B) = k5_complex1 ) ) ) )).

fof(t7_csspace3,axiom,
    ( ~ v3_struct_0(g2_clvect_1(k1_csspace3,k10_csspace(k7_csspace,k1_csspace3),k8_csspace(k7_csspace,k1_csspace3),k9_csspace(k7_csspace,k1_csspace3),k2_csspace3))
    & v3_rlvect_1(g2_clvect_1(k1_csspace3,k10_csspace(k7_csspace,k1_csspace3),k8_csspace(k7_csspace,k1_csspace3),k9_csspace(k7_csspace,k1_csspace3),k2_csspace3))
    & v4_rlvect_1(g2_clvect_1(k1_csspace3,k10_csspace(k7_csspace,k1_csspace3),k8_csspace(k7_csspace,k1_csspace3),k9_csspace(k7_csspace,k1_csspace3),k2_csspace3))
    & v5_rlvect_1(g2_clvect_1(k1_csspace3,k10_csspace(k7_csspace,k1_csspace3),k8_csspace(k7_csspace,k1_csspace3),k9_csspace(k7_csspace,k1_csspace3),k2_csspace3))
    & v6_rlvect_1(g2_clvect_1(k1_csspace3,k10_csspace(k7_csspace,k1_csspace3),k8_csspace(k7_csspace,k1_csspace3),k9_csspace(k7_csspace,k1_csspace3),k2_csspace3))
    & v2_clvect_1(g2_clvect_1(k1_csspace3,k10_csspace(k7_csspace,k1_csspace3),k8_csspace(k7_csspace,k1_csspace3),k9_csspace(k7_csspace,k1_csspace3),k2_csspace3))
    & l1_clvect_1(g2_clvect_1(k1_csspace3,k10_csspace(k7_csspace,k1_csspace3),k8_csspace(k7_csspace,k1_csspace3),k9_csspace(k7_csspace,k1_csspace3),k2_csspace3)) )).

fof(d3_csspace3,axiom,(
    k3_csspace3 = g2_clvect_1(k1_csspace3,k10_csspace(k7_csspace,k1_csspace3),k8_csspace(k7_csspace,k1_csspace3),k9_csspace(k7_csspace,k1_csspace3),k2_csspace3) )).

fof(t8_csspace3,axiom,
    ( u1_struct_0(k3_csspace3) = k1_csspace3
    & ! [A] :
        ( m1_subset_1(A,u1_struct_0(k3_csspace3))
      <=> ( v1_funct_1(A)
          & v1_funct_2(A,k5_numbers,k2_numbers)
          & m2_relset_1(A,k5_numbers,k2_numbers)
          & v2_comseq_3(k2_csspace(A)) ) )
    & k1_rlvect_1(k3_csspace3) = k6_csspace
    & ! [A] :
        ( m1_subset_1(A,u1_struct_0(k3_csspace3))
       => A = k2_csspace(A) )
    & ! [A] :
        ( m1_subset_1(A,u1_struct_0(k3_csspace3))
       => ! [B] :
            ( m1_subset_1(B,u1_struct_0(k3_csspace3))
           => k2_rlvect_1(k3_csspace3,A,B) = k2_comseq_1(k5_numbers,k2_csspace(A),k2_csspace(B)) ) )
    & ! [A] :
        ( m1_subset_1(A,k2_numbers)
       => ! [B] :
            ( m1_subset_1(B,u1_struct_0(k3_csspace3))
           => k1_clvect_1(k3_csspace3,B,A) = k4_comseq_1(k5_numbers,k2_csspace(B),A) ) )
    & ! [A] :
        ( m1_subset_1(A,u1_struct_0(k3_csspace3))
       => ( k5_rlvect_1(k3_csspace3,A) = k5_comseq_1(k5_numbers,k2_csspace(A))
          & k2_csspace(k5_rlvect_1(k3_csspace3,A)) = k5_comseq_1(k5_numbers,k2_csspace(A)) ) )
    & ! [A] :
        ( m1_subset_1(A,u1_struct_0(k3_csspace3))
       => ! [B] :
            ( m1_subset_1(B,u1_struct_0(k3_csspace3))
           => k6_rlvect_1(k3_csspace3,A,B) = k6_comseq_1(k5_numbers,k2_csspace(A),k2_csspace(B)) ) )
    & ! [A] :
        ( m1_subset_1(A,u1_struct_0(k3_csspace3))
       => v2_comseq_3(k2_csspace(A)) )
    & ! [A] :
        ( m1_subset_1(A,u1_struct_0(k3_csspace3))
       => k5_clvect_1(k3_csspace3,A) = k2_series_1(k9_comseq_1(k5_numbers,k2_csspace(A))) ) )).

fof(t9_csspace3,axiom,(
    ! [A] :
      ( m1_subset_1(A,u1_struct_0(k3_csspace3))
     => ! [B] :
          ( m1_subset_1(B,u1_struct_0(k3_csspace3))
         => ! [C] :
              ( m1_subset_1(C,k2_numbers)
             => ( ( k5_clvect_1(k3_csspace3,A) = np__0
                 => A = k1_rlvect_1(k3_csspace3) )
                & ( A = k1_rlvect_1(k3_csspace3)
                 => k5_clvect_1(k3_csspace3,A) = np__0 )
                & r1_xreal_0(np__0,k5_clvect_1(k3_csspace3,A))
                & r1_xreal_0(k5_clvect_1(k3_csspace3,k2_rlvect_1(k3_csspace3,A,B)),k3_real_1(k5_clvect_1(k3_csspace3,A),k5_clvect_1(k3_csspace3,B)))
                & k5_clvect_1(k3_csspace3,k1_clvect_1(k3_csspace3,A,C)) = k4_real_1(k17_complex1(C),k5_clvect_1(k3_csspace3,A)) ) ) ) ) )).

fof(d4_csspace3,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & l2_clvect_1(A) )
     => ! [B] :
          ( m1_subset_1(B,u1_struct_0(A))
         => ! [C] :
              ( m1_subset_1(C,u1_struct_0(A))
             => k4_csspace3(A,B,C) = k5_clvect_1(A,k6_rlvect_1(A,B,C)) ) ) ) )).

fof(d5_csspace3,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & l2_clvect_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_csspace3(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_csspace3(A,k2_normsp_1(A,B,E),k2_normsp_1(A,B,F))) ) ) ) ) ) ) ) ) )).

fof(t10_csspace3,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v3_rlvect_1(A)
        & v4_rlvect_1(A)
        & v5_rlvect_1(A)
        & v6_rlvect_1(A)
        & v2_clvect_1(A)
        & v5_clvect_1(A)
        & l2_clvect_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_csspace3(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,k5_clvect_1(A,k6_rlvect_1(A,k2_normsp_1(A,B,E),k2_normsp_1(A,B,F)))) ) ) ) ) ) ) ) ) )).

fof(t11_csspace3,axiom,(
    ! [A] :
      ( ( v1_funct_1(A)
        & v1_funct_2(A,k5_numbers,u1_struct_0(k3_csspace3))
        & m2_relset_1(A,k5_numbers,u1_struct_0(k3_csspace3)) )
     => ( v1_csspace3(A,k3_csspace3)
       => v6_clvect_1(A,k3_csspace3) ) ) )).

fof(dt_k1_csspace3,axiom,(
    m1_subset_1(k1_csspace3,k1_zfmisc_1(u1_struct_0(k7_csspace))) )).

fof(dt_k2_csspace3,axiom,
    ( v1_funct_1(k2_csspace3)
    & v1_funct_2(k2_csspace3,k1_csspace3,k1_numbers)
    & m2_relset_1(k2_csspace3,k1_csspace3,k1_numbers) )).

fof(dt_k3_csspace3,axiom,
    ( ~ v3_struct_0(k3_csspace3)
    & l2_clvect_1(k3_csspace3) )).

fof(dt_k4_csspace3,axiom,(
    ! [A,B,C] :
      ( ( ~ v3_struct_0(A)
        & l2_clvect_1(A)
        & m1_subset_1(B,u1_struct_0(A))
        & m1_subset_1(C,u1_struct_0(A)) )
     => m1_subset_1(k4_csspace3(A,B,C),k1_numbers) ) )).
%------------------------------------------------------------------------------