SET007 Axioms: SET007+821.ax


%------------------------------------------------------------------------------
% File     : SET007+821 : TPTP v7.5.0. Released v3.4.0.
% Domain   : Set Theory
% Axioms   : Hilbert Space of 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    : csspace2 [Urb08]

% Status   : Satisfiable
% Syntax   : Number of formulae    :   21 (   0 unit)
%            Number of atoms       :  184 (  38 equality)
%            Maximal formula depth :   19 (   8 average)
%            Number of connectives :  165 (   2 ~  ;   0  |;  89  &)
%                                         (   2 <=>;  72 =>;   0 <=)
%                                         (   0 <~>;   0 ~|;   0 ~&)
%            Number of predicates  :   22 (   0 propositional; 1-3 arity)
%            Number of functors    :   45 (   9 constant; 0-4 arity)
%            Number of variables   :   57 (   0 singleton;  57 !;   0 ?)
%            Maximal term depth    :    6 (   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(fc1_csspace2,axiom,
    ( ~ v3_struct_0(k20_csspace)
    & v3_rlvect_1(k20_csspace)
    & v4_rlvect_1(k20_csspace)
    & v5_rlvect_1(k20_csspace)
    & v6_rlvect_1(k20_csspace)
    & v2_clvect_1(k20_csspace)
    & v2_csspace(k20_csspace) )).

fof(fc2_csspace2,axiom,
    ( ~ v3_struct_0(k20_csspace)
    & v3_rlvect_1(k20_csspace)
    & v4_rlvect_1(k20_csspace)
    & v5_rlvect_1(k20_csspace)
    & v6_rlvect_1(k20_csspace)
    & v2_clvect_1(k20_csspace)
    & v2_csspace(k20_csspace)
    & v5_clvect_2(k20_csspace) )).

fof(t1_csspace2,axiom,
    ( u1_struct_0(k20_csspace) = k11_csspace
    & ! [A] :
        ( m1_subset_1(A,u1_struct_0(k20_csspace))
      <=> ( v1_funct_1(A)
          & v1_funct_2(A,k5_numbers,k2_numbers)
          & m2_relset_1(A,k5_numbers,k2_numbers)
          & v1_series_1(k11_seq_1(k9_comseq_1(k5_numbers,k2_csspace(A)),k9_comseq_1(k5_numbers,k2_csspace(A)))) ) )
    & ! [A] :
        ( m1_subset_1(A,u1_struct_0(k20_csspace))
      <=> ( v1_funct_1(A)
          & v1_funct_2(A,k5_numbers,k2_numbers)
          & m2_relset_1(A,k5_numbers,k2_numbers)
          & v2_comseq_3(k3_comseq_1(k5_numbers,k2_csspace(A),k1_comseq_2(k5_numbers,k2_csspace(A)))) ) )
    & k1_rlvect_1(k20_csspace) = k6_csspace
    & ! [A] :
        ( m1_subset_1(A,u1_struct_0(k20_csspace))
       => A = k2_csspace(A) )
    & ! [A] :
        ( m1_subset_1(A,u1_struct_0(k20_csspace))
       => ! [B] :
            ( m1_subset_1(B,u1_struct_0(k20_csspace))
           => k4_rlvect_1(k20_csspace,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(k20_csspace))
           => k1_clvect_1(k20_csspace,B,A) = k4_comseq_1(k5_numbers,k2_csspace(B),A) ) )
    & ! [A] :
        ( m1_subset_1(A,u1_struct_0(k20_csspace))
       => ( k5_rlvect_1(k20_csspace,A) = k5_comseq_1(k5_numbers,k2_csspace(A))
          & k2_csspace(k5_rlvect_1(k20_csspace,A)) = k5_comseq_1(k5_numbers,k2_csspace(A)) ) )
    & ! [A] :
        ( m1_subset_1(A,u1_struct_0(k20_csspace))
       => ! [B] :
            ( m1_subset_1(B,u1_struct_0(k20_csspace))
           => k6_rlvect_1(k20_csspace,A,B) = k6_comseq_1(k5_numbers,k2_csspace(A),k2_csspace(B)) ) )
    & ! [A] :
        ( m1_subset_1(A,u1_struct_0(k20_csspace))
       => ! [B] :
            ( m1_subset_1(B,u1_struct_0(k20_csspace))
           => ( v1_series_1(k11_seq_1(k9_comseq_1(k5_numbers,k2_csspace(A)),k9_comseq_1(k5_numbers,k2_csspace(B))))
              & ! [C] :
                  ( m1_subset_1(C,u1_struct_0(k20_csspace))
                 => ! [D] :
                      ( m1_subset_1(D,u1_struct_0(k20_csspace))
                     => k12_csspace(k20_csspace,C,D) = k8_comseq_3(k3_comseq_1(k5_numbers,k2_csspace(C),k1_comseq_2(k5_numbers,k2_csspace(D)))) ) ) ) ) ) )).

fof(t2_csspace2,axiom,(
    ! [A] :
      ( m1_subset_1(A,u1_struct_0(k20_csspace))
     => ! [B] :
          ( m1_subset_1(B,u1_struct_0(k20_csspace))
         => ! [C] :
              ( m1_subset_1(C,u1_struct_0(k20_csspace))
             => ! [D] :
                  ( m1_subset_1(D,k2_numbers)
                 => ( ( k12_csspace(k20_csspace,A,A) = np__0
                     => A = k1_rlvect_1(k20_csspace) )
                    & ( A = k1_rlvect_1(k20_csspace)
                     => k12_csspace(k20_csspace,A,A) = np__0 )
                    & r1_xreal_0(np__0,k3_complex1(k12_csspace(k20_csspace,A,A)))
                    & k4_complex1(k12_csspace(k20_csspace,A,A)) = np__0
                    & k12_csspace(k20_csspace,A,B) = k15_complex1(k12_csspace(k20_csspace,B,A))
                    & k12_csspace(k20_csspace,k4_rlvect_1(k20_csspace,A,B),C) = k8_complex1(k12_csspace(k20_csspace,A,C),k12_csspace(k20_csspace,B,C))
                    & k12_csspace(k20_csspace,k1_clvect_1(k20_csspace,A,D),B) = k9_complex1(D,k12_csspace(k20_csspace,A,B)) ) ) ) ) ) )).

fof(t3_csspace2,axiom,(
    ! [A] :
      ( ( v1_funct_1(A)
        & v1_funct_2(A,k5_numbers,u1_struct_0(k20_csspace))
        & m2_relset_1(A,k5_numbers,u1_struct_0(k20_csspace)) )
     => ( v2_clvect_2(A,k20_csspace)
       => v1_clvect_2(A,k20_csspace) ) ) )).

fof(t4_csspace2,axiom,(
    ! [A] :
      ( m1_subset_1(A,k2_numbers)
     => ! [B] :
          ( m1_subset_1(B,k2_numbers)
         => ( ( k4_real_1(k3_complex1(A),k4_complex1(B)) = k4_real_1(k3_complex1(B),k4_complex1(A))
              & r1_xreal_0(np__0,k3_real_1(k4_real_1(k3_complex1(A),k3_complex1(B)),k4_real_1(k4_complex1(A),k4_complex1(B)))) )
           => k17_complex1(k8_complex1(A,B)) = k3_real_1(k17_complex1(A),k17_complex1(B)) ) ) ) )).

fof(t5_csspace2,axiom,(
    ! [A] :
      ( m1_subset_1(A,k2_numbers)
     => ! [B] :
          ( m1_subset_1(B,k2_numbers)
         => r1_xreal_0(k4_real_1(np__2,k17_complex1(k9_complex1(A,B))),k3_real_1(k7_square_1(k17_complex1(A)),k7_square_1(k17_complex1(B)))) ) ) )).

fof(t6_csspace2,axiom,(
    ! [A] :
      ( m1_subset_1(A,k2_numbers)
     => ! [B] :
          ( m1_subset_1(B,k2_numbers)
         => ( r1_xreal_0(k4_real_1(k17_complex1(k8_complex1(A,B)),k17_complex1(k8_complex1(A,B))),k3_real_1(k4_real_1(k4_real_1(np__2,k17_complex1(A)),k17_complex1(A)),k4_real_1(k4_real_1(np__2,k17_complex1(B)),k17_complex1(B))))
            & r1_xreal_0(k4_real_1(k17_complex1(A),k17_complex1(A)),k3_real_1(k4_real_1(k4_real_1(np__2,k17_complex1(k11_complex1(A,B))),k17_complex1(k11_complex1(A,B))),k4_real_1(k4_real_1(np__2,k17_complex1(B)),k17_complex1(B)))) ) ) ) )).

fof(t7_csspace2,axiom,(
    ! [A] :
      ( ( v1_funct_1(A)
        & v1_funct_2(A,k5_numbers,k2_numbers)
        & m2_relset_1(A,k5_numbers,k2_numbers) )
     => A = k1_comseq_2(k5_numbers,k1_comseq_2(k5_numbers,A)) ) )).

fof(t8_csspace2,axiom,(
    ! [A] :
      ( ( v1_funct_1(A)
        & v1_funct_2(A,k5_numbers,k2_numbers)
        & m2_relset_1(A,k5_numbers,k2_numbers) )
     => k7_comseq_3(k1_comseq_2(k5_numbers,A)) = k1_comseq_2(k5_numbers,k7_comseq_3(A)) ) )).

fof(t9_csspace2,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)
         => ( ! [C] :
                ( m2_subset_1(C,k1_numbers,k5_numbers)
               => ( r1_xreal_0(np__0,k2_seq_1(k5_numbers,k1_numbers,k3_comseq_3(A),C))
                  & k2_seq_1(k5_numbers,k1_numbers,k4_comseq_3(A),C) = np__0 ) )
           => k2_seq_1(k5_numbers,k1_numbers,k9_comseq_1(k5_numbers,k7_comseq_3(A)),B) = k2_seq_1(k5_numbers,k1_numbers,k1_series_1(k9_comseq_1(k5_numbers,A)),B) ) ) ) )).

fof(t10_csspace2,axiom,(
    ! [A] :
      ( ( v1_funct_1(A)
        & v1_funct_2(A,k5_numbers,k2_numbers)
        & m2_relset_1(A,k5_numbers,k2_numbers) )
     => ( v1_comseq_3(A)
       => k8_comseq_3(k1_comseq_2(k5_numbers,A)) = k15_complex1(k8_comseq_3(A)) ) ) )).

fof(t11_csspace2,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)
       => r1_xreal_0(k17_complex1(k8_comseq_3(A)),k2_series_1(k9_comseq_1(k5_numbers,A))) ) ) )).

fof(t12_csspace2,axiom,(
    ! [A] :
      ( ( v1_funct_1(A)
        & v1_funct_2(A,k5_numbers,k2_numbers)
        & m2_relset_1(A,k5_numbers,k2_numbers) )
     => ( ( v1_comseq_3(A)
          & ! [B] :
              ( m2_subset_1(B,k1_numbers,k5_numbers)
             => ( r1_xreal_0(np__0,k2_seq_1(k5_numbers,k1_numbers,k3_comseq_3(A),B))
                & k2_seq_1(k5_numbers,k1_numbers,k4_comseq_3(A),B) = np__0 ) ) )
       => k17_complex1(k8_comseq_3(A)) = k2_series_1(k9_comseq_1(k5_numbers,A)) ) ) )).

fof(t13_csspace2,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)
         => ( r1_xreal_0(np__0,k2_seq_1(k5_numbers,k1_numbers,k3_comseq_3(k3_comseq_1(k5_numbers,A,k1_comseq_2(k5_numbers,A))),B))
            & k2_seq_1(k5_numbers,k1_numbers,k4_comseq_3(k3_comseq_1(k5_numbers,A,k1_comseq_2(k5_numbers,A))),B) = np__0 ) ) ) )).

fof(t14_csspace2,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(t15_csspace2,axiom,(
    ! [A] :
      ( ( v1_funct_1(A)
        & v1_funct_2(A,k5_numbers,k2_numbers)
        & m2_relset_1(A,k5_numbers,k2_numbers) )
     => k9_comseq_1(k5_numbers,A) = k9_comseq_1(k5_numbers,k1_comseq_2(k5_numbers,A)) ) )).

fof(t16_csspace2,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) )
         => ( v2_comseq_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) = k4_real_1(k17_complex1(k11_complex1(k1_comseq_1(B,D),A)),k17_complex1(k11_complex1(k1_comseq_1(B,D),A))) )
                 => ( v4_seq_2(C)
                    & k2_seq_2(C) = k4_real_1(k17_complex1(k11_complex1(k2_comseq_2(B),A)),k17_complex1(k11_complex1(k2_comseq_2(B),A))) ) ) ) ) ) ) )).

fof(t17_csspace2,axiom,(
    ! [A] :
      ( m1_subset_1(A,k2_numbers)
     => ! [B] :
          ( ( v1_funct_1(B)
            & v1_funct_2(B,k5_numbers,k1_numbers)
            & m2_relset_1(B,k5_numbers,k1_numbers) )
         => ! [C] :
              ( ( v1_funct_1(C)
                & v1_funct_2(C,k5_numbers,k2_numbers)
                & m2_relset_1(C,k5_numbers,k2_numbers) )
             => ( ( v2_comseq_2(C)
                  & v4_seq_2(B) )
               => ! [D] :
                    ( ( v1_funct_1(D)
                      & v1_funct_2(D,k5_numbers,k1_numbers)
                      & m2_relset_1(D,k5_numbers,k1_numbers) )
                   => ( ! [E] :
                          ( m2_subset_1(E,k1_numbers,k5_numbers)
                         => k2_seq_1(k5_numbers,k1_numbers,D,E) = k3_real_1(k4_real_1(k17_complex1(k11_complex1(k1_comseq_1(C,E),A)),k17_complex1(k11_complex1(k1_comseq_1(C,E),A))),k2_seq_1(k5_numbers,k1_numbers,B,E)) )
                     => ( v4_seq_2(D)
                        & k2_seq_2(D) = k3_real_1(k4_real_1(k17_complex1(k11_complex1(k2_comseq_2(C),A)),k17_complex1(k11_complex1(k2_comseq_2(C),A))),k2_seq_2(B)) ) ) ) ) ) ) ) )).

fof(t18_csspace2,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) )
         => ( v2_comseq_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) = k4_real_1(k17_complex1(k11_complex1(k1_comseq_1(B,D),A)),k17_complex1(k11_complex1(k1_comseq_1(B,D),A))) )
                 => ( v4_seq_2(C)
                    & k2_seq_2(C) = k4_real_1(k17_complex1(k11_complex1(k2_comseq_2(B),A)),k17_complex1(k11_complex1(k2_comseq_2(B),A))) ) ) ) ) ) ) )).

fof(t19_csspace2,axiom,(
    ! [A] :
      ( m1_subset_1(A,k2_numbers)
     => ! [B] :
          ( ( v1_funct_1(B)
            & v1_funct_2(B,k5_numbers,k1_numbers)
            & m2_relset_1(B,k5_numbers,k1_numbers) )
         => ! [C] :
              ( ( v1_funct_1(C)
                & v1_funct_2(C,k5_numbers,k2_numbers)
                & m2_relset_1(C,k5_numbers,k2_numbers) )
             => ( ( v2_comseq_2(C)
                  & v4_seq_2(B) )
               => ! [D] :
                    ( ( v1_funct_1(D)
                      & v1_funct_2(D,k5_numbers,k1_numbers)
                      & m2_relset_1(D,k5_numbers,k1_numbers) )
                   => ( ! [E] :
                          ( m2_subset_1(E,k1_numbers,k5_numbers)
                         => k2_seq_1(k5_numbers,k1_numbers,D,E) = k3_real_1(k4_real_1(k17_complex1(k11_complex1(k1_comseq_1(C,E),A)),k17_complex1(k11_complex1(k1_comseq_1(C,E),A))),k2_seq_1(k5_numbers,k1_numbers,B,E)) )
                     => ( v4_seq_2(D)
                        & k2_seq_2(D) = k3_real_1(k4_real_1(k17_complex1(k11_complex1(k2_comseq_2(C),A)),k17_complex1(k11_complex1(k2_comseq_2(C),A))),k2_seq_2(B)) ) ) ) ) ) ) ) )).
%------------------------------------------------------------------------------