SET007 Axioms: SET007+867.ax


%------------------------------------------------------------------------------
% File     : SET007+867 : TPTP v7.5.0. Released v3.4.0.
% Domain   : Set Theory
% Axioms   : Holder's Inequality and Minkowski's Inequality
% 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    : holder_1 [Urb08]

% Status   : Satisfiable
% Syntax   : Number of formulae    :   14 (   0 unit)
%            Number of atoms       :  212 (  30 equality)
%            Maximal formula depth :   23 (  14 average)
%            Number of connectives :  212 (  14 ~  ;   6  |; 104  &)
%                                         (   0 <=>;  88 =>;   0 <=)
%                                         (   0 <~>;   0 ~|;   0 ~&)
%            Number of predicates  :   14 (   0 propositional; 1-3 arity)
%            Number of functors    :   15 (   4 constant; 0-4 arity)
%            Number of variables   :   68 (   0 singleton;  66 !;   2 ?)
%            Maximal term depth    :    5 (   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_holder_1,axiom,(
    ! [A] :
      ( v1_xreal_0(A)
     => ( ~ v1_xboole_0(k3_limfunc1(A))
        & v1_membered(k3_limfunc1(A))
        & v2_membered(k3_limfunc1(A)) ) ) )).

fof(t1_holder_1,axiom,(
    ! [A] :
      ( m1_subset_1(A,k1_numbers)
     => ! [B] :
          ( m1_subset_1(B,k1_numbers)
         => ~ ( ~ r1_xreal_0(A,np__0)
              & ~ r1_xreal_0(B,np__0)
              & ? [C] :
                  ( m1_subset_1(C,k1_numbers)
                  & r1_xreal_0(np__0,C)
                  & k4_real_1(k4_power(C,A),k4_power(C,B)) != k4_power(C,k3_real_1(A,B)) ) ) ) ) )).

fof(t2_holder_1,axiom,(
    ! [A] :
      ( m1_subset_1(A,k1_numbers)
     => ! [B] :
          ( m1_subset_1(B,k1_numbers)
         => ~ ( ~ r1_xreal_0(A,np__0)
              & ~ r1_xreal_0(B,np__0)
              & ? [C] :
                  ( m1_subset_1(C,k1_numbers)
                  & r1_xreal_0(np__0,C)
                  & k4_power(k4_power(C,A),B) != k4_power(C,k4_real_1(A,B)) ) ) ) ) )).

fof(t3_holder_1,axiom,(
    ! [A] :
      ( m1_subset_1(A,k1_numbers)
     => ( ~ r1_xreal_0(A,np__0)
       => ! [B] :
            ( m1_subset_1(B,k1_numbers)
           => ! [C] :
                ( m1_subset_1(C,k1_numbers)
               => ( ( r1_xreal_0(np__0,B)
                    & r1_xreal_0(B,C) )
                 => r1_xreal_0(k4_power(B,A),k4_power(C,A)) ) ) ) ) ) )).

fof(t4_holder_1,axiom,(
    ! [A] :
      ( m1_subset_1(A,k1_numbers)
     => ! [B] :
          ( m1_subset_1(B,k1_numbers)
         => ! [C] :
              ( m1_subset_1(C,k1_numbers)
             => ! [D] :
                  ( m1_subset_1(D,k1_numbers)
                 => ( k3_real_1(k6_real_1(np__1,A),k6_real_1(np__1,B)) = np__1
                   => ( r1_xreal_0(A,np__1)
                      | r1_xreal_0(C,np__0)
                      | r1_xreal_0(D,np__0)
                      | ( r1_xreal_0(k4_real_1(C,D),k3_real_1(k6_real_1(k13_prepower(C,A),A),k6_real_1(k13_prepower(D,B),B)))
                        & ( k4_real_1(C,D) = k3_real_1(k6_real_1(k13_prepower(C,A),A),k6_real_1(k13_prepower(D,B),B))
                         => k13_prepower(C,A) = k13_prepower(D,B) )
                        & ( k13_prepower(C,A) = k13_prepower(D,B)
                         => k4_real_1(C,D) = k3_real_1(k6_real_1(k13_prepower(C,A),A),k6_real_1(k13_prepower(D,B),B)) ) ) ) ) ) ) ) ) )).

fof(t5_holder_1,axiom,(
    ! [A] :
      ( m1_subset_1(A,k1_numbers)
     => ! [B] :
          ( m1_subset_1(B,k1_numbers)
         => ! [C] :
              ( m1_subset_1(C,k1_numbers)
             => ! [D] :
                  ( m1_subset_1(D,k1_numbers)
                 => ( ( k3_real_1(k6_real_1(np__1,A),k6_real_1(np__1,B)) = np__1
                      & r1_xreal_0(np__0,C)
                      & r1_xreal_0(np__0,D) )
                   => ( r1_xreal_0(A,np__1)
                      | ( r1_xreal_0(k4_real_1(C,D),k3_real_1(k6_real_1(k4_power(C,A),A),k6_real_1(k4_power(D,B),B)))
                        & ( k4_real_1(C,D) = k3_real_1(k6_real_1(k4_power(C,A),A),k6_real_1(k4_power(D,B),B))
                         => k4_power(C,A) = k4_power(D,B) )
                        & ( k4_power(C,A) = k4_power(D,B)
                         => k4_real_1(C,D) = k3_real_1(k6_real_1(k4_power(C,A),A),k6_real_1(k4_power(D,B),B)) ) ) ) ) ) ) ) ) )).

fof(t6_holder_1,axiom,(
    ! [A] :
      ( m1_subset_1(A,k1_numbers)
     => ! [B] :
          ( m1_subset_1(B,k1_numbers)
         => ( k3_real_1(k6_real_1(np__1,A),k6_real_1(np__1,B)) = np__1
           => ( r1_xreal_0(A,np__1)
              | ! [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,k1_numbers)
                        & m2_relset_1(D,k5_numbers,k1_numbers) )
                     => ! [E] :
                          ( ( v1_funct_1(E)
                            & v1_funct_2(E,k5_numbers,k1_numbers)
                            & m2_relset_1(E,k5_numbers,k1_numbers) )
                         => ! [F] :
                              ( ( v1_funct_1(F)
                                & v1_funct_2(F,k5_numbers,k1_numbers)
                                & m2_relset_1(F,k5_numbers,k1_numbers) )
                             => ! [G] :
                                  ( ( v1_funct_1(G)
                                    & v1_funct_2(G,k5_numbers,k1_numbers)
                                    & m2_relset_1(G,k5_numbers,k1_numbers) )
                                 => ( ! [H] :
                                        ( m2_subset_1(H,k1_numbers,k5_numbers)
                                       => ( k2_seq_1(k5_numbers,k1_numbers,E,H) = k4_power(k18_complex1(k2_seq_1(k5_numbers,k1_numbers,C,H)),A)
                                          & k2_seq_1(k5_numbers,k1_numbers,F,H) = k4_power(k18_complex1(k2_seq_1(k5_numbers,k1_numbers,D,H)),B)
                                          & k2_seq_1(k5_numbers,k1_numbers,G,H) = k18_complex1(k4_real_1(k2_seq_1(k5_numbers,k1_numbers,C,H),k2_seq_1(k5_numbers,k1_numbers,D,H))) ) )
                                   => ! [H] :
                                        ( m2_subset_1(H,k1_numbers,k5_numbers)
                                       => r1_xreal_0(k2_seq_1(k5_numbers,k1_numbers,k1_series_1(G),H),k4_real_1(k4_power(k2_seq_1(k5_numbers,k1_numbers,k1_series_1(E),H),k6_real_1(np__1,A)),k4_power(k2_seq_1(k5_numbers,k1_numbers,k1_series_1(F),H),k6_real_1(np__1,B)))) ) ) ) ) ) ) ) ) ) ) ) )).

fof(t7_holder_1,axiom,(
    ! [A] :
      ( m1_subset_1(A,k1_numbers)
     => ( ~ r1_xreal_0(A,np__1)
       => ! [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] :
                    ( ( v1_funct_1(D)
                      & v1_funct_2(D,k5_numbers,k1_numbers)
                      & m2_relset_1(D,k5_numbers,k1_numbers) )
                   => ! [E] :
                        ( ( v1_funct_1(E)
                          & v1_funct_2(E,k5_numbers,k1_numbers)
                          & m2_relset_1(E,k5_numbers,k1_numbers) )
                       => ! [F] :
                            ( ( v1_funct_1(F)
                              & v1_funct_2(F,k5_numbers,k1_numbers)
                              & m2_relset_1(F,k5_numbers,k1_numbers) )
                           => ( ! [G] :
                                  ( m2_subset_1(G,k1_numbers,k5_numbers)
                                 => ( k2_seq_1(k5_numbers,k1_numbers,D,G) = k4_power(k18_complex1(k2_seq_1(k5_numbers,k1_numbers,B,G)),A)
                                    & k2_seq_1(k5_numbers,k1_numbers,E,G) = k4_power(k18_complex1(k2_seq_1(k5_numbers,k1_numbers,C,G)),A)
                                    & k2_seq_1(k5_numbers,k1_numbers,F,G) = k4_power(k18_complex1(k3_real_1(k2_seq_1(k5_numbers,k1_numbers,B,G),k2_seq_1(k5_numbers,k1_numbers,C,G))),A) ) )
                             => ! [G] :
                                  ( m2_subset_1(G,k1_numbers,k5_numbers)
                                 => r1_xreal_0(k4_power(k2_seq_1(k5_numbers,k1_numbers,k1_series_1(F),G),k6_real_1(np__1,A)),k3_real_1(k4_power(k2_seq_1(k5_numbers,k1_numbers,k1_series_1(D),G),k6_real_1(np__1,A)),k4_power(k2_seq_1(k5_numbers,k1_numbers,k1_series_1(E),G),k6_real_1(np__1,A)))) ) ) ) ) ) ) ) ) ) )).

fof(t8_holder_1,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] :
                  ( m2_subset_1(C,k1_numbers,k5_numbers)
                 => r1_xreal_0(k2_seq_1(k5_numbers,k1_numbers,A,C),k2_seq_1(k5_numbers,k1_numbers,B,C)) )
              & v4_seq_2(B)
              & v3_seqm_3(A) )
           => ( v4_seq_2(A)
              & r1_xreal_0(k2_seq_2(A),k2_seq_2(B)) ) ) ) ) )).

fof(t9_holder_1,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] :
              ( ( 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(k2_seq_1(k5_numbers,k1_numbers,A,D),k3_real_1(k2_seq_1(k5_numbers,k1_numbers,B,D),k2_seq_1(k5_numbers,k1_numbers,C,D))) )
                  & v4_seq_2(B)
                  & v4_seq_2(C)
                  & v3_seqm_3(A) )
               => ( v4_seq_2(A)
                  & r1_xreal_0(k2_seq_2(A),k3_real_1(k2_seq_2(B),k2_seq_2(C))) ) ) ) ) ) )).

fof(t10_holder_1,axiom,(
    ! [A] :
      ( m1_subset_1(A,k1_numbers)
     => ( ~ r1_xreal_0(A,np__0)
       => ! [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) )
               => ( ( v4_seq_2(B)
                    & ! [D] :
                        ( m2_subset_1(D,k1_numbers,k5_numbers)
                       => r1_xreal_0(np__0,k2_seq_1(k5_numbers,k1_numbers,B,D)) )
                    & ! [D] :
                        ( m2_subset_1(D,k1_numbers,k5_numbers)
                       => k2_seq_1(k5_numbers,k1_numbers,C,D) = k4_power(k2_seq_1(k5_numbers,k1_numbers,B,D),A) ) )
                 => ( v4_seq_2(C)
                    & k2_seq_2(C) = k4_power(k2_seq_2(B),A) ) ) ) ) ) ) )).

fof(t11_holder_1,axiom,(
    ! [A] :
      ( m1_subset_1(A,k1_numbers)
     => ( ~ r1_xreal_0(A,np__0)
       => ! [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) )
               => ( ( v1_series_1(B)
                    & ! [D] :
                        ( m2_subset_1(D,k1_numbers,k5_numbers)
                       => r1_xreal_0(np__0,k2_seq_1(k5_numbers,k1_numbers,B,D)) )
                    & ! [D] :
                        ( m2_subset_1(D,k1_numbers,k5_numbers)
                       => k2_seq_1(k5_numbers,k1_numbers,C,D) = k4_power(k2_seq_1(k5_numbers,k1_numbers,k1_series_1(B),D),A) ) )
                 => ( v4_seq_2(C)
                    & k2_seq_2(C) = k4_power(k2_series_1(B),A)
                    & v3_seqm_3(C)
                    & ! [D] :
                        ( m2_subset_1(D,k1_numbers,k5_numbers)
                       => r1_xreal_0(k2_seq_1(k5_numbers,k1_numbers,C,D),k4_power(k2_series_1(B),A)) ) ) ) ) ) ) ) )).

fof(t12_holder_1,axiom,(
    ! [A] :
      ( m1_subset_1(A,k1_numbers)
     => ! [B] :
          ( m1_subset_1(B,k1_numbers)
         => ( k3_real_1(k6_real_1(np__1,A),k6_real_1(np__1,B)) = np__1
           => ( r1_xreal_0(A,np__1)
              | ! [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,k1_numbers)
                        & m2_relset_1(D,k5_numbers,k1_numbers) )
                     => ! [E] :
                          ( ( v1_funct_1(E)
                            & v1_funct_2(E,k5_numbers,k1_numbers)
                            & m2_relset_1(E,k5_numbers,k1_numbers) )
                         => ! [F] :
                              ( ( v1_funct_1(F)
                                & v1_funct_2(F,k5_numbers,k1_numbers)
                                & m2_relset_1(F,k5_numbers,k1_numbers) )
                             => ! [G] :
                                  ( ( v1_funct_1(G)
                                    & v1_funct_2(G,k5_numbers,k1_numbers)
                                    & m2_relset_1(G,k5_numbers,k1_numbers) )
                                 => ( ( ! [H] :
                                          ( m2_subset_1(H,k1_numbers,k5_numbers)
                                         => ( k2_seq_1(k5_numbers,k1_numbers,E,H) = k4_power(k18_complex1(k2_seq_1(k5_numbers,k1_numbers,C,H)),A)
                                            & k2_seq_1(k5_numbers,k1_numbers,F,H) = k4_power(k18_complex1(k2_seq_1(k5_numbers,k1_numbers,D,H)),B)
                                            & k2_seq_1(k5_numbers,k1_numbers,G,H) = k18_complex1(k4_real_1(k2_seq_1(k5_numbers,k1_numbers,C,H),k2_seq_1(k5_numbers,k1_numbers,D,H))) ) )
                                      & v1_series_1(E)
                                      & v1_series_1(F) )
                                   => ( v1_series_1(G)
                                      & r1_xreal_0(k2_series_1(G),k4_real_1(k4_power(k2_series_1(E),k6_real_1(np__1,A)),k4_power(k2_series_1(F),k6_real_1(np__1,B)))) ) ) ) ) ) ) ) ) ) ) ) )).

fof(t13_holder_1,axiom,(
    ! [A] :
      ( m1_subset_1(A,k1_numbers)
     => ( ~ r1_xreal_0(A,np__1)
       => ! [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] :
                    ( ( v1_funct_1(D)
                      & v1_funct_2(D,k5_numbers,k1_numbers)
                      & m2_relset_1(D,k5_numbers,k1_numbers) )
                   => ! [E] :
                        ( ( v1_funct_1(E)
                          & v1_funct_2(E,k5_numbers,k1_numbers)
                          & m2_relset_1(E,k5_numbers,k1_numbers) )
                       => ! [F] :
                            ( ( v1_funct_1(F)
                              & v1_funct_2(F,k5_numbers,k1_numbers)
                              & m2_relset_1(F,k5_numbers,k1_numbers) )
                           => ( ( ! [G] :
                                    ( m2_subset_1(G,k1_numbers,k5_numbers)
                                   => ( k2_seq_1(k5_numbers,k1_numbers,D,G) = k4_power(k18_complex1(k2_seq_1(k5_numbers,k1_numbers,B,G)),A)
                                      & k2_seq_1(k5_numbers,k1_numbers,E,G) = k4_power(k18_complex1(k2_seq_1(k5_numbers,k1_numbers,C,G)),A)
                                      & k2_seq_1(k5_numbers,k1_numbers,F,G) = k4_power(k18_complex1(k3_real_1(k2_seq_1(k5_numbers,k1_numbers,B,G),k2_seq_1(k5_numbers,k1_numbers,C,G))),A) ) )
                                & v1_series_1(D)
                                & v1_series_1(E) )
                             => ( v1_series_1(F)
                                & r1_xreal_0(k4_power(k2_series_1(F),k6_real_1(np__1,A)),k3_real_1(k4_power(k2_series_1(D),k6_real_1(np__1,A)),k4_power(k2_series_1(E),k6_real_1(np__1,A)))) ) ) ) ) ) ) ) ) ) )).
%------------------------------------------------------------------------------