SET007 Axioms: SET007+258.ax


%------------------------------------------------------------------------------
% File     : SET007+258 : TPTP v7.5.0. Released v3.4.0.
% Domain   : Set Theory
% Axioms   : Finite Sequence over Ring and Left-, Right-, and Bi-Modules
% Version  : [Urb08] axioms.
% English  : Finite Sequence over Ring and Left-, Right-, and Bi-Modules over
%            a Ring

% 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    : algseq_1 [Urb08]

% Status   : Satisfiable
% Syntax   : Number of formulae    :   49 (  17 unit)
%            Number of atoms       :  233 (  38 equality)
%            Maximal formula depth :   13 (   6 average)
%            Number of connectives :  211 (  27 ~  ;   1  |;  96  &)
%                                         (  11 <=>;  76 =>;   0 <=)
%                                         (   0 <~>;   0 ~|;   0 ~&)
%            Number of predicates  :   16 (   1 propositional; 0-3 arity)
%            Number of functors    :   26 (   8 constant; 0-3 arity)
%            Number of variables   :   73 (   0 singleton;  68 !;   5 ?)
%            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(rc1_algseq_1,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & l2_struct_0(A) )
     => ? [B] :
          ( m1_relset_1(B,k5_numbers,u1_struct_0(A))
          & v1_relat_1(B)
          & v1_funct_1(B)
          & v1_funct_2(B,k5_numbers,u1_struct_0(A))
          & v1_algseq_1(B,A) ) ) )).

fof(t1_algseq_1,axiom,(
    $true )).

fof(t2_algseq_1,axiom,(
    $true )).

fof(t3_algseq_1,axiom,(
    $true )).

fof(t4_algseq_1,axiom,(
    $true )).

fof(t5_algseq_1,axiom,(
    $true )).

fof(t6_algseq_1,axiom,(
    $true )).

fof(t7_algseq_1,axiom,(
    $true )).

fof(t8_algseq_1,axiom,(
    $true )).

fof(t9_algseq_1,axiom,(
    $true )).

fof(t10_algseq_1,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] :
          ( m2_subset_1(B,k1_numbers,k5_numbers)
         => ( r2_hidden(A,k2_algseq_1(B))
          <=> ~ r1_xreal_0(B,A) ) ) ) )).

fof(t11_algseq_1,axiom,
    ( k2_algseq_1(np__0) = k1_xboole_0
    & k2_algseq_1(np__1) = k1_tarski(np__0)
    & k2_algseq_1(np__2) = k2_tarski(np__0,np__1) )).

fof(t12_algseq_1,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => r2_hidden(A,k2_algseq_1(k1_nat_1(A,np__1))) ) )).

fof(t13_algseq_1,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] :
          ( m2_subset_1(B,k1_numbers,k5_numbers)
         => ( r1_xreal_0(A,B)
          <=> r1_tarski(k2_algseq_1(A),k2_algseq_1(B)) ) ) ) )).

fof(t14_algseq_1,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] :
          ( m2_subset_1(B,k1_numbers,k5_numbers)
         => ( k2_algseq_1(A) = k2_algseq_1(B)
           => A = B ) ) ) )).

fof(t15_algseq_1,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] :
          ( m2_subset_1(B,k1_numbers,k5_numbers)
         => ( r1_xreal_0(A,B)
           => ( k2_algseq_1(A) = k5_subset_1(k5_numbers,k2_algseq_1(A),k2_algseq_1(B))
              & k2_algseq_1(A) = k5_subset_1(k5_numbers,k2_algseq_1(B),k2_algseq_1(A)) ) ) ) ) )).

fof(t16_algseq_1,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] :
          ( m2_subset_1(B,k1_numbers,k5_numbers)
         => ( ( k2_algseq_1(A) = k5_subset_1(k5_numbers,k2_algseq_1(A),k2_algseq_1(B))
              | k2_algseq_1(A) = k5_subset_1(k5_numbers,k2_algseq_1(B),k2_algseq_1(A)) )
           => r1_xreal_0(A,B) ) ) ) )).

fof(t17_algseq_1,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => k2_xboole_0(k2_algseq_1(A),k1_tarski(A)) = k2_algseq_1(k1_nat_1(A,np__1)) ) )).

fof(d2_algseq_1,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & l2_struct_0(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_algseq_1(B,A)
          <=> ? [C] :
                ( m2_subset_1(C,k1_numbers,k5_numbers)
                & ! [D] :
                    ( m2_subset_1(D,k1_numbers,k5_numbers)
                   => ( r1_xreal_0(C,D)
                     => k2_normsp_1(A,B,D) = k1_rlvect_1(A) ) ) ) ) ) ) )).

fof(d3_algseq_1,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & l2_struct_0(A) )
     => ! [B] :
          ( ( v1_funct_1(B)
            & v1_funct_2(B,k5_numbers,u1_struct_0(A))
            & v1_algseq_1(B,A)
            & m2_relset_1(B,k5_numbers,u1_struct_0(A)) )
         => ! [C] :
              ( m2_subset_1(C,k1_numbers,k5_numbers)
             => ( r1_algseq_1(A,B,C)
              <=> ! [D] :
                    ( m2_subset_1(D,k1_numbers,k5_numbers)
                   => ( r1_xreal_0(C,D)
                     => k2_normsp_1(A,B,D) = k1_rlvect_1(A) ) ) ) ) ) ) )).

fof(d4_algseq_1,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & l2_struct_0(A) )
     => ! [B] :
          ( ( v1_funct_1(B)
            & v1_funct_2(B,k5_numbers,u1_struct_0(A))
            & v1_algseq_1(B,A)
            & m2_relset_1(B,k5_numbers,u1_struct_0(A)) )
         => ! [C] :
              ( m2_subset_1(C,k1_numbers,k5_numbers)
             => ( C = k3_algseq_1(A,B)
              <=> ( r1_algseq_1(A,B,C)
                  & ! [D] :
                      ( m2_subset_1(D,k1_numbers,k5_numbers)
                     => ( r1_algseq_1(A,B,D)
                       => r1_xreal_0(C,D) ) ) ) ) ) ) ) )).

fof(t18_algseq_1,axiom,(
    $true )).

fof(t19_algseq_1,axiom,(
    $true )).

fof(t20_algseq_1,axiom,(
    $true )).

fof(t21_algseq_1,axiom,(
    $true )).

fof(t22_algseq_1,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] :
          ( ( ~ v3_struct_0(B)
            & l2_struct_0(B) )
         => ! [C] :
              ( ( v1_funct_1(C)
                & v1_funct_2(C,k5_numbers,u1_struct_0(B))
                & v1_algseq_1(C,B)
                & m2_relset_1(C,k5_numbers,u1_struct_0(B)) )
             => ( r1_xreal_0(k3_algseq_1(B,C),A)
               => k2_normsp_1(B,C,A) = k1_rlvect_1(B) ) ) ) ) )).

fof(t23_algseq_1,axiom,(
    $true )).

fof(t24_algseq_1,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] :
          ( ( ~ v3_struct_0(B)
            & l2_struct_0(B) )
         => ! [C] :
              ( ( v1_funct_1(C)
                & v1_funct_2(C,k5_numbers,u1_struct_0(B))
                & v1_algseq_1(C,B)
                & m2_relset_1(C,k5_numbers,u1_struct_0(B)) )
             => ( ! [D] :
                    ( m2_subset_1(D,k1_numbers,k5_numbers)
                   => ~ ( ~ r1_xreal_0(A,D)
                        & k2_normsp_1(B,C,D) = k1_rlvect_1(B) ) )
               => r1_xreal_0(A,k3_algseq_1(B,C)) ) ) ) ) )).

fof(t25_algseq_1,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] :
          ( ( ~ v3_struct_0(B)
            & l2_struct_0(B) )
         => ! [C] :
              ( ( v1_funct_1(C)
                & v1_funct_2(C,k5_numbers,u1_struct_0(B))
                & v1_algseq_1(C,B)
                & m2_relset_1(C,k5_numbers,u1_struct_0(B)) )
             => ~ ( k3_algseq_1(B,C) = k1_nat_1(A,np__1)
                  & k2_normsp_1(B,C,A) = k1_rlvect_1(B) ) ) ) ) )).

fof(d5_algseq_1,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & l2_struct_0(A) )
     => ! [B] :
          ( ( v1_funct_1(B)
            & v1_funct_2(B,k5_numbers,u1_struct_0(A))
            & v1_algseq_1(B,A)
            & m2_relset_1(B,k5_numbers,u1_struct_0(A)) )
         => k4_algseq_1(A,B) = k2_algseq_1(k3_algseq_1(A,B)) ) ) )).

fof(t26_algseq_1,axiom,(
    $true )).

fof(t27_algseq_1,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] :
          ( ( ~ v3_struct_0(B)
            & l2_struct_0(B) )
         => ! [C] :
              ( ( v1_funct_1(C)
                & v1_funct_2(C,k5_numbers,u1_struct_0(B))
                & v1_algseq_1(C,B)
                & m2_relset_1(C,k5_numbers,u1_struct_0(B)) )
             => ( A = k3_algseq_1(B,C)
              <=> k2_algseq_1(A) = k4_algseq_1(B,C) ) ) ) ) )).

fof(t28_algseq_1,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & l2_struct_0(A) )
     => ! [B] :
          ( ( v1_funct_1(B)
            & v1_funct_2(B,k5_numbers,u1_struct_0(A))
            & v1_algseq_1(B,A)
            & m2_relset_1(B,k5_numbers,u1_struct_0(A)) )
         => ! [C] :
              ( ( v1_funct_1(C)
                & v1_funct_2(C,k5_numbers,u1_struct_0(A))
                & v1_algseq_1(C,A)
                & m2_relset_1(C,k5_numbers,u1_struct_0(A)) )
             => ( ( k3_algseq_1(A,B) = k3_algseq_1(A,C)
                  & ! [D] :
                      ( m2_subset_1(D,k1_numbers,k5_numbers)
                     => ( ~ r1_xreal_0(k3_algseq_1(A,B),D)
                       => k2_normsp_1(A,B,D) = k2_normsp_1(A,C,D) ) ) )
               => B = C ) ) ) ) )).

fof(t29_algseq_1,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & l2_struct_0(A) )
     => ( u1_struct_0(A) != k1_struct_0(A,k1_rlvect_1(A))
       => ! [B] :
            ( m2_subset_1(B,k1_numbers,k5_numbers)
           => ? [C] :
                ( v1_funct_1(C)
                & v1_funct_2(C,k5_numbers,u1_struct_0(A))
                & v1_algseq_1(C,A)
                & m2_relset_1(C,k5_numbers,u1_struct_0(A))
                & k3_algseq_1(A,C) = B ) ) ) ) )).

fof(d6_algseq_1,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & l2_struct_0(A) )
     => ! [B] :
          ( m1_subset_1(B,u1_struct_0(A))
         => ! [C] :
              ( ( v1_funct_1(C)
                & v1_funct_2(C,k5_numbers,u1_struct_0(A))
                & v1_algseq_1(C,A)
                & m2_relset_1(C,k5_numbers,u1_struct_0(A)) )
             => ( C = k5_algseq_1(A,B)
              <=> ( r1_xreal_0(k3_algseq_1(A,C),np__1)
                  & k2_normsp_1(A,C,np__0) = B ) ) ) ) ) )).

fof(t30_algseq_1,axiom,(
    $true )).

fof(t31_algseq_1,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & l2_struct_0(A) )
     => ! [B] :
          ( ( v1_funct_1(B)
            & v1_funct_2(B,k5_numbers,u1_struct_0(A))
            & v1_algseq_1(B,A)
            & m2_relset_1(B,k5_numbers,u1_struct_0(A)) )
         => ( B = k5_algseq_1(A,k1_rlvect_1(A))
          <=> k3_algseq_1(A,B) = np__0 ) ) ) )).

fof(t32_algseq_1,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & l2_struct_0(A) )
     => ! [B] :
          ( ( v1_funct_1(B)
            & v1_funct_2(B,k5_numbers,u1_struct_0(A))
            & v1_algseq_1(B,A)
            & m2_relset_1(B,k5_numbers,u1_struct_0(A)) )
         => ( B = k5_algseq_1(A,k1_rlvect_1(A))
          <=> k4_algseq_1(A,B) = k1_xboole_0 ) ) ) )).

fof(t33_algseq_1,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] :
          ( ( ~ v3_struct_0(B)
            & l2_struct_0(B) )
         => k2_normsp_1(B,k5_algseq_1(B,k1_rlvect_1(B)),A) = k1_rlvect_1(B) ) ) )).

fof(t34_algseq_1,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & l2_struct_0(A) )
     => ! [B] :
          ( ( v1_funct_1(B)
            & v1_funct_2(B,k5_numbers,u1_struct_0(A))
            & v1_algseq_1(B,A)
            & m2_relset_1(B,k5_numbers,u1_struct_0(A)) )
         => ( B = k5_algseq_1(A,k1_rlvect_1(A))
          <=> k2_relat_1(B) = k1_struct_0(A,k1_rlvect_1(A)) ) ) ) )).

fof(s1_algseq_1,axiom,(
    ? [A] :
      ( v1_funct_1(A)
      & v1_funct_2(A,k5_numbers,u1_struct_0(f1_s1_algseq_1))
      & v1_algseq_1(A,f1_s1_algseq_1)
      & m2_relset_1(A,k5_numbers,u1_struct_0(f1_s1_algseq_1))
      & r1_xreal_0(k3_algseq_1(f1_s1_algseq_1,A),f2_s1_algseq_1)
      & ! [B] :
          ( m2_subset_1(B,k1_numbers,k5_numbers)
         => ( ~ r1_xreal_0(f2_s1_algseq_1,B)
           => k2_normsp_1(f1_s1_algseq_1,A,B) = f3_s1_algseq_1(B) ) ) ) )).

fof(dt_k1_algseq_1,axiom,(
    $true )).

fof(dt_k2_algseq_1,axiom,(
    ! [A] :
      ( m1_subset_1(A,k5_numbers)
     => m1_subset_1(k2_algseq_1(A),k1_zfmisc_1(k5_numbers)) ) )).

fof(redefinition_k2_algseq_1,axiom,(
    ! [A] :
      ( m1_subset_1(A,k5_numbers)
     => k2_algseq_1(A) = k1_algseq_1(A) ) )).

fof(dt_k3_algseq_1,axiom,(
    ! [A,B] :
      ( ( ~ v3_struct_0(A)
        & l2_struct_0(A)
        & v1_funct_1(B)
        & v1_funct_2(B,k5_numbers,u1_struct_0(A))
        & v1_algseq_1(B,A)
        & m1_relset_1(B,k5_numbers,u1_struct_0(A)) )
     => m2_subset_1(k3_algseq_1(A,B),k1_numbers,k5_numbers) ) )).

fof(dt_k4_algseq_1,axiom,(
    ! [A,B] :
      ( ( ~ v3_struct_0(A)
        & l2_struct_0(A)
        & v1_funct_1(B)
        & v1_funct_2(B,k5_numbers,u1_struct_0(A))
        & v1_algseq_1(B,A)
        & m1_relset_1(B,k5_numbers,u1_struct_0(A)) )
     => m1_subset_1(k4_algseq_1(A,B),k1_zfmisc_1(k5_numbers)) ) )).

fof(dt_k5_algseq_1,axiom,(
    ! [A,B] :
      ( ( ~ v3_struct_0(A)
        & l2_struct_0(A)
        & m1_subset_1(B,u1_struct_0(A)) )
     => ( v1_funct_1(k5_algseq_1(A,B))
        & v1_funct_2(k5_algseq_1(A,B),k5_numbers,u1_struct_0(A))
        & v1_algseq_1(k5_algseq_1(A,B),A)
        & m2_relset_1(k5_algseq_1(A,B),k5_numbers,u1_struct_0(A)) ) ) )).

fof(d1_algseq_1,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => k1_algseq_1(A) = a_1_0_algseq_1(A) ) )).

fof(fraenkel_a_1_0_algseq_1,axiom,(
    ! [A,B] :
      ( m2_subset_1(B,k1_numbers,k5_numbers)
     => ( r2_hidden(A,a_1_0_algseq_1(B))
      <=> ? [C] :
            ( m2_subset_1(C,k1_numbers,k5_numbers)
            & A = C
            & ~ r1_xreal_0(B,C) ) ) ) )).
%------------------------------------------------------------------------------