SET007 Axioms: SET007+868.ax


%------------------------------------------------------------------------------
% File     : SET007+868 : TPTP v7.5.0. Released v3.4.0.
% Domain   : Set Theory
% Axioms   : The Banach Space l^p
% 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    : lp_space [Urb08]

% Status   : Satisfiable
% Syntax   : Number of formulae    :   24 (   0 unit)
%            Number of atoms       :  213 (  29 equality)
%            Maximal formula depth :   19 (   9 average)
%            Number of connectives :  203 (  14 ~  ;   0  |;  97  &)
%                                         (   5 <=>;  87 =>;   0 <=)
%                                         (   0 <~>;   0 ~|;   0 ~&)
%            Number of predicates  :   25 (   0 propositional; 1-3 arity)
%            Number of functors    :   37 (   7 constant; 0-5 arity)
%            Number of variables   :   63 (   0 singleton;  63 !;   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(d1_lp_space,axiom,(
    ! [A] :
      ( ( v1_funct_1(A)
        & v1_funct_2(A,k5_numbers,k1_numbers)
        & m2_relset_1(A,k5_numbers,k1_numbers) )
     => ! [B] :
          ( m1_subset_1(B,k1_numbers)
         => ! [C] :
              ( ( v1_funct_1(C)
                & v1_funct_2(C,k5_numbers,k1_numbers)
                & m2_relset_1(C,k5_numbers,k1_numbers) )
             => ( C = k1_lp_space(A,B)
              <=> ! [D] :
                    ( m2_subset_1(D,k1_numbers,k5_numbers)
                   => k2_seq_1(k5_numbers,k1_numbers,C,D) = k4_power(k18_complex1(k2_seq_1(k5_numbers,k1_numbers,A,D)),B) ) ) ) ) ) )).

fof(d2_lp_space,axiom,(
    ! [A] :
      ( m1_subset_1(A,k1_numbers)
     => ( r1_xreal_0(np__1,A)
       => ! [B] :
            ( ( ~ v1_xboole_0(B)
              & m1_subset_1(B,k1_zfmisc_1(u1_struct_0(k7_rsspace))) )
           => ( B = k2_lp_space(A)
            <=> ! [C] :
                  ( r2_hidden(C,B)
                <=> ( r2_hidden(C,k1_rsspace)
                    & v1_series_1(k1_lp_space(k2_rsspace(C),A)) ) ) ) ) ) ) )).

fof(t1_lp_space,axiom,(
    ! [A] :
      ( m1_subset_1(A,k1_numbers)
     => ! [B] :
          ( m1_subset_1(B,k1_numbers)
         => ! [C] :
              ( m1_subset_1(C,k1_numbers)
             => ~ ( r1_xreal_0(np__0,A)
                  & ~ r1_xreal_0(B,A)
                  & ~ r1_xreal_0(C,np__0)
                  & r1_xreal_0(k4_power(B,C),k4_power(A,C)) ) ) ) ) )).

fof(t2_lp_space,axiom,(
    ! [A] :
      ( m1_subset_1(A,k1_numbers)
     => ( r1_xreal_0(np__1,A)
       => ! [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,k1_numbers)
                  & m2_relset_1(C,k5_numbers,k1_numbers) )
               => ! [D] :
                    ( m2_subset_1(D,k1_numbers,k5_numbers)
                   => r1_xreal_0(k4_power(k2_seq_1(k5_numbers,k1_numbers,k1_series_1(k1_lp_space(k9_seq_1(B,C),A)),D),k6_real_1(np__1,A)),k3_real_1(k4_power(k2_seq_1(k5_numbers,k1_numbers,k1_series_1(k1_lp_space(B,A)),D),k6_real_1(np__1,A)),k4_power(k2_seq_1(k5_numbers,k1_numbers,k1_series_1(k1_lp_space(C,A)),D),k6_real_1(np__1,A)))) ) ) ) ) ) )).

fof(t3_lp_space,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,k1_numbers)
            & m2_relset_1(B,k5_numbers,k1_numbers) )
         => ! [C] :
              ( m1_subset_1(C,k1_numbers)
             => ( ( r1_xreal_0(np__1,C)
                  & v1_series_1(k1_lp_space(A,C))
                  & v1_series_1(k1_lp_space(B,C)) )
               => ( v1_series_1(k1_lp_space(k9_seq_1(A,B),C))
                  & r1_xreal_0(k4_power(k2_series_1(k1_lp_space(k9_seq_1(A,B),C)),k6_real_1(np__1,C)),k3_real_1(k4_power(k2_series_1(k1_lp_space(A,C)),k6_real_1(np__1,C)),k4_power(k2_series_1(k1_lp_space(B,C)),k6_real_1(np__1,C)))) ) ) ) ) ) )).

fof(t4_lp_space,axiom,(
    ! [A] :
      ( m1_subset_1(A,k1_numbers)
     => ( r1_xreal_0(np__1,A)
       => v1_rlsub_1(k2_lp_space(A),k7_rsspace) ) ) )).

fof(t5_lp_space,axiom,(
    ! [A] :
      ( m1_subset_1(A,k1_numbers)
     => ( r1_xreal_0(np__1,A)
       => m1_rlsub_1(g2_rlvect_1(k2_lp_space(A),k10_rsspace(k7_rsspace,k2_lp_space(A)),k8_rsspace(k7_rsspace,k2_lp_space(A)),k9_rsspace(k7_rsspace,k2_lp_space(A))),k7_rsspace) ) ) )).

fof(t6_lp_space,axiom,(
    ! [A] :
      ( m1_subset_1(A,k1_numbers)
     => ( r1_xreal_0(np__1,A)
       => ( v3_rlvect_1(g2_rlvect_1(k2_lp_space(A),k10_rsspace(k7_rsspace,k2_lp_space(A)),k8_rsspace(k7_rsspace,k2_lp_space(A)),k9_rsspace(k7_rsspace,k2_lp_space(A))))
          & v4_rlvect_1(g2_rlvect_1(k2_lp_space(A),k10_rsspace(k7_rsspace,k2_lp_space(A)),k8_rsspace(k7_rsspace,k2_lp_space(A)),k9_rsspace(k7_rsspace,k2_lp_space(A))))
          & v5_rlvect_1(g2_rlvect_1(k2_lp_space(A),k10_rsspace(k7_rsspace,k2_lp_space(A)),k8_rsspace(k7_rsspace,k2_lp_space(A)),k9_rsspace(k7_rsspace,k2_lp_space(A))))
          & v6_rlvect_1(g2_rlvect_1(k2_lp_space(A),k10_rsspace(k7_rsspace,k2_lp_space(A)),k8_rsspace(k7_rsspace,k2_lp_space(A)),k9_rsspace(k7_rsspace,k2_lp_space(A))))
          & v7_rlvect_1(g2_rlvect_1(k2_lp_space(A),k10_rsspace(k7_rsspace,k2_lp_space(A)),k8_rsspace(k7_rsspace,k2_lp_space(A)),k9_rsspace(k7_rsspace,k2_lp_space(A)))) ) ) ) )).

fof(t7_lp_space,axiom,(
    ! [A] :
      ( m1_subset_1(A,k1_numbers)
     => ( r1_xreal_0(np__1,A)
       => ( ~ v3_struct_0(g2_rlvect_1(k2_lp_space(A),k10_rsspace(k7_rsspace,k2_lp_space(A)),k8_rsspace(k7_rsspace,k2_lp_space(A)),k9_rsspace(k7_rsspace,k2_lp_space(A))))
          & v3_rlvect_1(g2_rlvect_1(k2_lp_space(A),k10_rsspace(k7_rsspace,k2_lp_space(A)),k8_rsspace(k7_rsspace,k2_lp_space(A)),k9_rsspace(k7_rsspace,k2_lp_space(A))))
          & v4_rlvect_1(g2_rlvect_1(k2_lp_space(A),k10_rsspace(k7_rsspace,k2_lp_space(A)),k8_rsspace(k7_rsspace,k2_lp_space(A)),k9_rsspace(k7_rsspace,k2_lp_space(A))))
          & v5_rlvect_1(g2_rlvect_1(k2_lp_space(A),k10_rsspace(k7_rsspace,k2_lp_space(A)),k8_rsspace(k7_rsspace,k2_lp_space(A)),k9_rsspace(k7_rsspace,k2_lp_space(A))))
          & v6_rlvect_1(g2_rlvect_1(k2_lp_space(A),k10_rsspace(k7_rsspace,k2_lp_space(A)),k8_rsspace(k7_rsspace,k2_lp_space(A)),k9_rsspace(k7_rsspace,k2_lp_space(A))))
          & v7_rlvect_1(g2_rlvect_1(k2_lp_space(A),k10_rsspace(k7_rsspace,k2_lp_space(A)),k8_rsspace(k7_rsspace,k2_lp_space(A)),k9_rsspace(k7_rsspace,k2_lp_space(A))))
          & l2_rlvect_1(g2_rlvect_1(k2_lp_space(A),k10_rsspace(k7_rsspace,k2_lp_space(A)),k8_rsspace(k7_rsspace,k2_lp_space(A)),k9_rsspace(k7_rsspace,k2_lp_space(A)))) ) ) ) )).

fof(d3_lp_space,axiom,(
    ! [A] :
      ( m1_subset_1(A,k1_numbers)
     => ! [B] :
          ( ( v1_funct_1(B)
            & v1_funct_2(B,k2_lp_space(A),k1_numbers)
            & m2_relset_1(B,k2_lp_space(A),k1_numbers) )
         => ( B = k3_lp_space(A)
          <=> ! [C] :
                ( r2_hidden(C,k2_lp_space(A))
               => k2_seq_1(k2_lp_space(A),k1_numbers,B,C) = k4_power(k2_series_1(k1_lp_space(k2_rsspace(C),A)),k6_real_1(np__1,A)) ) ) ) ) )).

fof(t8_lp_space,axiom,(
    ! [A] :
      ( m1_subset_1(A,k1_numbers)
     => ( r1_xreal_0(np__1,A)
       => ( ~ v3_struct_0(g1_normsp_1(k2_lp_space(A),k10_rsspace(k7_rsspace,k2_lp_space(A)),k8_rsspace(k7_rsspace,k2_lp_space(A)),k9_rsspace(k7_rsspace,k2_lp_space(A)),k3_lp_space(A)))
          & v3_rlvect_1(g1_normsp_1(k2_lp_space(A),k10_rsspace(k7_rsspace,k2_lp_space(A)),k8_rsspace(k7_rsspace,k2_lp_space(A)),k9_rsspace(k7_rsspace,k2_lp_space(A)),k3_lp_space(A)))
          & v4_rlvect_1(g1_normsp_1(k2_lp_space(A),k10_rsspace(k7_rsspace,k2_lp_space(A)),k8_rsspace(k7_rsspace,k2_lp_space(A)),k9_rsspace(k7_rsspace,k2_lp_space(A)),k3_lp_space(A)))
          & v5_rlvect_1(g1_normsp_1(k2_lp_space(A),k10_rsspace(k7_rsspace,k2_lp_space(A)),k8_rsspace(k7_rsspace,k2_lp_space(A)),k9_rsspace(k7_rsspace,k2_lp_space(A)),k3_lp_space(A)))
          & v6_rlvect_1(g1_normsp_1(k2_lp_space(A),k10_rsspace(k7_rsspace,k2_lp_space(A)),k8_rsspace(k7_rsspace,k2_lp_space(A)),k9_rsspace(k7_rsspace,k2_lp_space(A)),k3_lp_space(A)))
          & v7_rlvect_1(g1_normsp_1(k2_lp_space(A),k10_rsspace(k7_rsspace,k2_lp_space(A)),k8_rsspace(k7_rsspace,k2_lp_space(A)),k9_rsspace(k7_rsspace,k2_lp_space(A)),k3_lp_space(A)))
          & l2_rlvect_1(g1_normsp_1(k2_lp_space(A),k10_rsspace(k7_rsspace,k2_lp_space(A)),k8_rsspace(k7_rsspace,k2_lp_space(A)),k9_rsspace(k7_rsspace,k2_lp_space(A)),k3_lp_space(A))) ) ) ) )).

fof(t9_lp_space,axiom,(
    ! [A] :
      ( m1_subset_1(A,k1_numbers)
     => ( r1_xreal_0(np__1,A)
       => m1_rlsub_1(g1_normsp_1(k2_lp_space(A),k10_rsspace(k7_rsspace,k2_lp_space(A)),k8_rsspace(k7_rsspace,k2_lp_space(A)),k9_rsspace(k7_rsspace,k2_lp_space(A)),k3_lp_space(A)),k7_rsspace) ) ) )).

fof(t10_lp_space,axiom,(
    ! [A] :
      ( m1_subset_1(A,k1_numbers)
     => ( r1_xreal_0(np__1,A)
       => ! [B] :
            ( ( ~ v3_struct_0(B)
              & l1_normsp_1(B) )
           => ( B = g1_normsp_1(k2_lp_space(A),k10_rsspace(k7_rsspace,k2_lp_space(A)),k8_rsspace(k7_rsspace,k2_lp_space(A)),k9_rsspace(k7_rsspace,k2_lp_space(A)),k3_lp_space(A))
             => ( u1_struct_0(B) = k2_lp_space(A)
                & ! [C] :
                    ( m1_subset_1(C,u1_struct_0(B))
                  <=> ( v1_funct_1(C)
                      & v1_funct_2(C,k5_numbers,k1_numbers)
                      & m2_relset_1(C,k5_numbers,k1_numbers)
                      & v1_series_1(k1_lp_space(k2_rsspace(C),A)) ) )
                & k1_rlvect_1(B) = k6_rsspace
                & ! [C] :
                    ( m1_subset_1(C,u1_struct_0(B))
                   => C = k2_rsspace(C) )
                & ! [C] :
                    ( m1_subset_1(C,u1_struct_0(B))
                   => ! [D] :
                        ( m1_subset_1(D,u1_struct_0(B))
                       => k2_rlvect_1(B,C,D) = k9_seq_1(k2_rsspace(C),k2_rsspace(D)) ) )
                & ! [C] :
                    ( m1_subset_1(C,k1_numbers)
                   => ! [D] :
                        ( m1_subset_1(D,u1_struct_0(B))
                       => k3_rlvect_1(B,D,C) = k14_seq_1(k2_rsspace(D),C) ) )
                & ! [C] :
                    ( m1_subset_1(C,u1_struct_0(B))
                   => ( k5_rlvect_1(B,C) = k17_seq_1(k2_rsspace(C))
                      & k2_rsspace(k5_rlvect_1(B,C)) = k17_seq_1(k2_rsspace(C)) ) )
                & ! [C] :
                    ( m1_subset_1(C,u1_struct_0(B))
                   => ! [D] :
                        ( m1_subset_1(D,u1_struct_0(B))
                       => k6_rlvect_1(B,C,D) = k10_seq_1(k2_rsspace(C),k2_rsspace(D)) ) )
                & ! [C] :
                    ( m1_subset_1(C,u1_struct_0(B))
                   => v1_series_1(k1_lp_space(k2_rsspace(C),A)) )
                & ! [C] :
                    ( m1_subset_1(C,u1_struct_0(B))
                   => k1_normsp_1(B,C) = k4_power(k2_series_1(k1_lp_space(k2_rsspace(C),A)),k6_real_1(np__1,A)) ) ) ) ) ) ) )).

fof(t11_lp_space,axiom,(
    ! [A] :
      ( m1_subset_1(A,k1_numbers)
     => ( r1_xreal_0(np__1,A)
       => ! [B] :
            ( ( v1_funct_1(B)
              & v1_funct_2(B,k5_numbers,k1_numbers)
              & m2_relset_1(B,k5_numbers,k1_numbers) )
           => ( ! [C] :
                  ( m2_subset_1(C,k1_numbers,k5_numbers)
                 => k2_seq_1(k5_numbers,k1_numbers,B,C) = np__0 )
             => ( v1_series_1(k1_lp_space(B,A))
                & k4_power(k2_series_1(k1_lp_space(B,A)),k6_real_1(np__1,A)) = np__0 ) ) ) ) ) )).

fof(t12_lp_space,axiom,(
    ! [A] :
      ( m1_subset_1(A,k1_numbers)
     => ( r1_xreal_0(np__1,A)
       => ! [B] :
            ( ( v1_funct_1(B)
              & v1_funct_2(B,k5_numbers,k1_numbers)
              & m2_relset_1(B,k5_numbers,k1_numbers) )
           => ( ( v1_series_1(k1_lp_space(B,A))
                & k4_power(k2_series_1(k1_lp_space(B,A)),k6_real_1(np__1,A)) = np__0 )
             => ! [C] :
                  ( m2_subset_1(C,k1_numbers,k5_numbers)
                 => k2_seq_1(k5_numbers,k1_numbers,B,C) = np__0 ) ) ) ) ) )).

fof(t13_lp_space,axiom,(
    ! [A] :
      ( m1_subset_1(A,k1_numbers)
     => ( r1_xreal_0(np__1,A)
       => ! [B] :
            ( ( ~ v3_struct_0(B)
              & l1_normsp_1(B) )
           => ( B = g1_normsp_1(k2_lp_space(A),k10_rsspace(k7_rsspace,k2_lp_space(A)),k8_rsspace(k7_rsspace,k2_lp_space(A)),k9_rsspace(k7_rsspace,k2_lp_space(A)),k3_lp_space(A))
             => ! [C] :
                  ( m1_subset_1(C,u1_struct_0(B))
                 => ! [D] :
                      ( m1_subset_1(D,u1_struct_0(B))
                     => ! [E] :
                          ( m1_subset_1(E,k1_numbers)
                         => ( ( k1_normsp_1(B,C) = np__0
                             => C = k1_rlvect_1(B) )
                            & ( C = k1_rlvect_1(B)
                             => k1_normsp_1(B,C) = np__0 )
                            & r1_xreal_0(np__0,k1_normsp_1(B,C))
                            & r1_xreal_0(k1_normsp_1(B,k2_rlvect_1(B,C,D)),k3_real_1(k1_normsp_1(B,C),k1_normsp_1(B,D)))
                            & k1_normsp_1(B,k3_rlvect_1(B,C,E)) = k4_real_1(k18_complex1(E),k1_normsp_1(B,C)) ) ) ) ) ) ) ) ) )).

fof(t14_lp_space,axiom,(
    ! [A] :
      ( m1_subset_1(A,k1_numbers)
     => ( r1_xreal_0(np__1,A)
       => ! [B] :
            ( ( ~ v3_struct_0(B)
              & l1_normsp_1(B) )
           => ( B = g1_normsp_1(k2_lp_space(A),k10_rsspace(k7_rsspace,k2_lp_space(A)),k8_rsspace(k7_rsspace,k2_lp_space(A)),k9_rsspace(k7_rsspace,k2_lp_space(A)),k3_lp_space(A))
             => v2_normsp_1(B) ) ) ) ) )).

fof(t15_lp_space,axiom,(
    ! [A] :
      ( m1_subset_1(A,k1_numbers)
     => ( r1_xreal_0(np__1,A)
       => ! [B] :
            ( ( ~ v3_struct_0(B)
              & l1_normsp_1(B) )
           => ( B = g1_normsp_1(k2_lp_space(A),k10_rsspace(k7_rsspace,k2_lp_space(A)),k8_rsspace(k7_rsspace,k2_lp_space(A)),k9_rsspace(k7_rsspace,k2_lp_space(A)),k3_lp_space(A))
             => ( ~ v3_struct_0(B)
                & v3_rlvect_1(B)
                & v4_rlvect_1(B)
                & v5_rlvect_1(B)
                & v6_rlvect_1(B)
                & v7_rlvect_1(B)
                & v2_normsp_1(B)
                & l1_normsp_1(B) ) ) ) ) ) )).

fof(t16_lp_space,axiom,(
    ! [A] :
      ( m1_subset_1(A,k1_numbers)
     => ( r1_xreal_0(np__1,A)
       => ! [B] :
            ( ( ~ v3_struct_0(B)
              & v3_rlvect_1(B)
              & v4_rlvect_1(B)
              & v5_rlvect_1(B)
              & v6_rlvect_1(B)
              & v7_rlvect_1(B)
              & v2_normsp_1(B)
              & l1_normsp_1(B) )
           => ( B = g1_normsp_1(k2_lp_space(A),k10_rsspace(k7_rsspace,k2_lp_space(A)),k8_rsspace(k7_rsspace,k2_lp_space(A)),k9_rsspace(k7_rsspace,k2_lp_space(A)),k3_lp_space(A))
             => ! [C] :
                  ( ( v1_funct_1(C)
                    & v1_funct_2(C,k5_numbers,u1_struct_0(B))
                    & m2_relset_1(C,k5_numbers,u1_struct_0(B)) )
                 => ( v1_rsspace3(C,B)
                   => v4_normsp_1(C,B) ) ) ) ) ) ) )).

fof(d4_lp_space,axiom,(
    ! [A] :
      ( m1_subset_1(A,k1_numbers)
     => ( r1_xreal_0(np__1,A)
       => k4_lp_space(A) = g1_normsp_1(k2_lp_space(A),k10_rsspace(k7_rsspace,k2_lp_space(A)),k8_rsspace(k7_rsspace,k2_lp_space(A)),k9_rsspace(k7_rsspace,k2_lp_space(A)),k3_lp_space(A)) ) ) )).

fof(dt_k1_lp_space,axiom,(
    ! [A,B] :
      ( ( v1_funct_1(A)
        & v1_funct_2(A,k5_numbers,k1_numbers)
        & m1_relset_1(A,k5_numbers,k1_numbers)
        & m1_subset_1(B,k1_numbers) )
     => ( v1_funct_1(k1_lp_space(A,B))
        & v1_funct_2(k1_lp_space(A,B),k5_numbers,k1_numbers)
        & m2_relset_1(k1_lp_space(A,B),k5_numbers,k1_numbers) ) ) )).

fof(dt_k2_lp_space,axiom,(
    ! [A] :
      ( m1_subset_1(A,k1_numbers)
     => ( ~ v1_xboole_0(k2_lp_space(A))
        & m1_subset_1(k2_lp_space(A),k1_zfmisc_1(u1_struct_0(k7_rsspace))) ) ) )).

fof(dt_k3_lp_space,axiom,(
    ! [A] :
      ( m1_subset_1(A,k1_numbers)
     => ( v1_funct_1(k3_lp_space(A))
        & v1_funct_2(k3_lp_space(A),k2_lp_space(A),k1_numbers)
        & m2_relset_1(k3_lp_space(A),k2_lp_space(A),k1_numbers) ) ) )).

fof(dt_k4_lp_space,axiom,(
    ! [A] :
      ( m1_subset_1(A,k1_numbers)
     => ( ~ v3_struct_0(k4_lp_space(A))
        & v3_rlvect_1(k4_lp_space(A))
        & v4_rlvect_1(k4_lp_space(A))
        & v5_rlvect_1(k4_lp_space(A))
        & v6_rlvect_1(k4_lp_space(A))
        & v7_rlvect_1(k4_lp_space(A))
        & v2_normsp_1(k4_lp_space(A))
        & v4_lopban_1(k4_lp_space(A))
        & l1_normsp_1(k4_lp_space(A)) ) ) )).
%------------------------------------------------------------------------------