SET007 Axioms: SET007+271.ax


%------------------------------------------------------------------------------
% File     : SET007+271 : TPTP v7.5.0. Released v3.4.0.
% Domain   : Set Theory
% Axioms   : Ordered Rings - Part I
% 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    : o_ring_1 [Urb08]

% Status   : Satisfiable
% Syntax   : Number of formulae    :   23 (   0 unit)
%            Number of atoms       :  189 (  33 equality)
%            Maximal formula depth :   23 (  11 average)
%            Number of connectives :  227 (  61 ~  ;   2  |;  85  &)
%                                         (  13 <=>;  66 =>;   0 <=)
%                                         (   0 <~>;   0 ~|;   0 ~&)
%            Number of predicates  :   22 (   0 propositional; 1-3 arity)
%            Number of functors    :   12 (   4 constant; 0-3 arity)
%            Number of variables   :   69 (   0 singleton;  62 !;   7 ?)
%            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(d1_o_ring_1,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ! [B] :
          ( m2_finseq_1(B,A)
         => ! [C] :
              ( m2_subset_1(C,k1_numbers,k5_numbers)
             => ( r1_xreal_0(C,k3_finseq_1(B))
               => ( np__0 = C
                  | k1_o_ring_1(A,B,C) = k1_funct_1(B,C) ) ) ) ) ) )).

fof(d2_o_ring_1,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & l3_vectsp_1(A) )
     => ! [B] :
          ( m1_subset_1(B,u1_struct_0(A))
         => k2_o_ring_1(A,B) = k1_group_1(A,B,B) ) ) )).

fof(d3_o_ring_1,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & l3_vectsp_1(A) )
     => ! [B] :
          ( m1_subset_1(B,u1_struct_0(A))
         => ( v1_o_ring_1(B,A)
          <=> ? [C] :
                ( m1_subset_1(C,u1_struct_0(A))
                & B = k2_o_ring_1(A,C) ) ) ) ) )).

fof(d4_o_ring_1,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & l3_vectsp_1(A) )
     => ! [B] :
          ( m2_finseq_1(B,u1_struct_0(A))
         => ( v2_o_ring_1(B,A)
          <=> ( k3_finseq_1(B) != np__0
              & v1_o_ring_1(k1_o_ring_1(u1_struct_0(A),B,np__1),A)
              & ! [C] :
                  ( m2_subset_1(C,k1_numbers,k5_numbers)
                 => ~ ( C != np__0
                      & ~ r1_xreal_0(k3_finseq_1(B),C)
                      & ! [D] :
                          ( m1_subset_1(D,u1_struct_0(A))
                         => ~ ( v1_o_ring_1(D,A)
                              & k1_o_ring_1(u1_struct_0(A),B,k1_nat_1(C,np__1)) = k2_rlvect_1(A,k1_o_ring_1(u1_struct_0(A),B,C),D) ) ) ) ) ) ) ) ) )).

fof(d5_o_ring_1,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & l3_vectsp_1(A) )
     => ! [B] :
          ( m1_subset_1(B,u1_struct_0(A))
         => ( v3_o_ring_1(B,A)
          <=> ? [C] :
                ( m2_finseq_1(C,u1_struct_0(A))
                & v2_o_ring_1(C,A)
                & B = k1_o_ring_1(u1_struct_0(A),C,k3_finseq_1(C)) ) ) ) ) )).

fof(d6_o_ring_1,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & l3_vectsp_1(A) )
     => ! [B] :
          ( m2_finseq_1(B,u1_struct_0(A))
         => ( v4_o_ring_1(B,A)
          <=> ( k3_finseq_1(B) != np__0
              & v1_o_ring_1(k1_o_ring_1(u1_struct_0(A),B,np__1),A)
              & ! [C] :
                  ( m2_subset_1(C,k1_numbers,k5_numbers)
                 => ~ ( C != np__0
                      & ~ r1_xreal_0(k3_finseq_1(B),C)
                      & ! [D] :
                          ( m1_subset_1(D,u1_struct_0(A))
                         => ~ ( v1_o_ring_1(D,A)
                              & k1_o_ring_1(u1_struct_0(A),B,k1_nat_1(C,np__1)) = k1_group_1(A,k1_o_ring_1(u1_struct_0(A),B,C),D) ) ) ) ) ) ) ) ) )).

fof(d7_o_ring_1,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & l3_vectsp_1(A) )
     => ! [B] :
          ( m1_subset_1(B,u1_struct_0(A))
         => ( v5_o_ring_1(B,A)
          <=> ? [C] :
                ( m2_finseq_1(C,u1_struct_0(A))
                & v4_o_ring_1(C,A)
                & B = k1_o_ring_1(u1_struct_0(A),C,k3_finseq_1(C)) ) ) ) ) )).

fof(d8_o_ring_1,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & l3_vectsp_1(A) )
     => ! [B] :
          ( m2_finseq_1(B,u1_struct_0(A))
         => ( v6_o_ring_1(B,A)
          <=> ( k3_finseq_1(B) != np__0
              & v5_o_ring_1(k1_o_ring_1(u1_struct_0(A),B,np__1),A)
              & ! [C] :
                  ( m2_subset_1(C,k1_numbers,k5_numbers)
                 => ~ ( C != np__0
                      & ~ r1_xreal_0(k3_finseq_1(B),C)
                      & ! [D] :
                          ( m1_subset_1(D,u1_struct_0(A))
                         => ~ ( v5_o_ring_1(D,A)
                              & k1_o_ring_1(u1_struct_0(A),B,k1_nat_1(C,np__1)) = k2_rlvect_1(A,k1_o_ring_1(u1_struct_0(A),B,C),D) ) ) ) ) ) ) ) ) )).

fof(d9_o_ring_1,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & l3_vectsp_1(A) )
     => ! [B] :
          ( m1_subset_1(B,u1_struct_0(A))
         => ( v7_o_ring_1(B,A)
          <=> ? [C] :
                ( m2_finseq_1(C,u1_struct_0(A))
                & v6_o_ring_1(C,A)
                & B = k1_o_ring_1(u1_struct_0(A),C,k3_finseq_1(C)) ) ) ) ) )).

fof(d10_o_ring_1,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & l3_vectsp_1(A) )
     => ! [B] :
          ( m2_finseq_1(B,u1_struct_0(A))
         => ( v8_o_ring_1(B,A)
          <=> ( k3_finseq_1(B) != np__0
              & ! [C] :
                  ( m2_subset_1(C,k1_numbers,k5_numbers)
                 => ~ ( C != np__0
                      & r1_xreal_0(C,k3_finseq_1(B))
                      & ~ v5_o_ring_1(k1_o_ring_1(u1_struct_0(A),B,C),A)
                      & ! [D] :
                          ( m2_subset_1(D,k1_numbers,k5_numbers)
                         => ! [E] :
                              ( m2_subset_1(E,k1_numbers,k5_numbers)
                             => ~ ( k1_o_ring_1(u1_struct_0(A),B,C) = k1_group_1(A,k1_o_ring_1(u1_struct_0(A),B,D),k1_o_ring_1(u1_struct_0(A),B,E))
                                  & D != np__0
                                  & ~ r1_xreal_0(C,D)
                                  & E != np__0
                                  & ~ r1_xreal_0(C,E) ) ) ) ) ) ) ) ) ) )).

fof(d11_o_ring_1,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & l3_vectsp_1(A) )
     => ! [B] :
          ( m1_subset_1(B,u1_struct_0(A))
         => ( v9_o_ring_1(B,A)
          <=> ? [C] :
                ( m2_finseq_1(C,u1_struct_0(A))
                & v8_o_ring_1(C,A)
                & B = k1_o_ring_1(u1_struct_0(A),C,k3_finseq_1(C)) ) ) ) ) )).

fof(d12_o_ring_1,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & l3_vectsp_1(A) )
     => ! [B] :
          ( m2_finseq_1(B,u1_struct_0(A))
         => ( v10_o_ring_1(B,A)
          <=> ( k3_finseq_1(B) != np__0
              & v9_o_ring_1(k1_o_ring_1(u1_struct_0(A),B,np__1),A)
              & ! [C] :
                  ( m2_subset_1(C,k1_numbers,k5_numbers)
                 => ~ ( C != np__0
                      & ~ r1_xreal_0(k3_finseq_1(B),C)
                      & ! [D] :
                          ( m1_subset_1(D,u1_struct_0(A))
                         => ~ ( v9_o_ring_1(D,A)
                              & k1_o_ring_1(u1_struct_0(A),B,k1_nat_1(C,np__1)) = k2_rlvect_1(A,k1_o_ring_1(u1_struct_0(A),B,C),D) ) ) ) ) ) ) ) ) )).

fof(d13_o_ring_1,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & l3_vectsp_1(A) )
     => ! [B] :
          ( m1_subset_1(B,u1_struct_0(A))
         => ( v11_o_ring_1(B,A)
          <=> ? [C] :
                ( m2_finseq_1(C,u1_struct_0(A))
                & v10_o_ring_1(C,A)
                & B = k1_o_ring_1(u1_struct_0(A),C,k3_finseq_1(C)) ) ) ) ) )).

fof(d14_o_ring_1,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & l3_vectsp_1(A) )
     => ! [B] :
          ( m2_finseq_1(B,u1_struct_0(A))
         => ( v12_o_ring_1(B,A)
          <=> ( k3_finseq_1(B) != np__0
              & ! [C] :
                  ( m2_subset_1(C,k1_numbers,k5_numbers)
                 => ~ ( C != np__0
                      & r1_xreal_0(C,k3_finseq_1(B))
                      & ~ v9_o_ring_1(k1_o_ring_1(u1_struct_0(A),B,C),A)
                      & ! [D] :
                          ( m2_subset_1(D,k1_numbers,k5_numbers)
                         => ! [E] :
                              ( m2_subset_1(E,k1_numbers,k5_numbers)
                             => ~ ( ( k1_o_ring_1(u1_struct_0(A),B,C) = k1_group_1(A,k1_o_ring_1(u1_struct_0(A),B,D),k1_o_ring_1(u1_struct_0(A),B,E))
                                    | k1_o_ring_1(u1_struct_0(A),B,C) = k2_rlvect_1(A,k1_o_ring_1(u1_struct_0(A),B,D),k1_o_ring_1(u1_struct_0(A),B,E)) )
                                  & D != np__0
                                  & ~ r1_xreal_0(C,D)
                                  & E != np__0
                                  & ~ r1_xreal_0(C,E) ) ) ) ) ) ) ) ) ) )).

fof(d15_o_ring_1,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & l3_vectsp_1(A) )
     => ! [B] :
          ( m1_subset_1(B,u1_struct_0(A))
         => ( v13_o_ring_1(B,A)
          <=> ? [C] :
                ( m2_finseq_1(C,u1_struct_0(A))
                & v12_o_ring_1(C,A)
                & B = k1_o_ring_1(u1_struct_0(A),C,k3_finseq_1(C)) ) ) ) ) )).

fof(t1_o_ring_1,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & l3_vectsp_1(A) )
     => ! [B] :
          ( m1_subset_1(B,u1_struct_0(A))
         => ( v1_o_ring_1(B,A)
           => ( v3_o_ring_1(B,A)
              & v5_o_ring_1(B,A)
              & v7_o_ring_1(B,A)
              & v9_o_ring_1(B,A)
              & v11_o_ring_1(B,A)
              & v13_o_ring_1(B,A) ) ) ) ) )).

fof(t2_o_ring_1,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & l3_vectsp_1(A) )
     => ! [B] :
          ( m1_subset_1(B,u1_struct_0(A))
         => ( v3_o_ring_1(B,A)
           => ( v7_o_ring_1(B,A)
              & v11_o_ring_1(B,A)
              & v13_o_ring_1(B,A) ) ) ) ) )).

fof(t3_o_ring_1,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & l3_vectsp_1(A) )
     => ! [B] :
          ( m1_subset_1(B,u1_struct_0(A))
         => ( v5_o_ring_1(B,A)
           => ( v7_o_ring_1(B,A)
              & v9_o_ring_1(B,A)
              & v11_o_ring_1(B,A)
              & v13_o_ring_1(B,A) ) ) ) ) )).

fof(t4_o_ring_1,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & l3_vectsp_1(A) )
     => ! [B] :
          ( m1_subset_1(B,u1_struct_0(A))
         => ( v7_o_ring_1(B,A)
           => ( v11_o_ring_1(B,A)
              & v13_o_ring_1(B,A) ) ) ) ) )).

fof(t5_o_ring_1,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & l3_vectsp_1(A) )
     => ! [B] :
          ( m1_subset_1(B,u1_struct_0(A))
         => ( v9_o_ring_1(B,A)
           => ( v11_o_ring_1(B,A)
              & v13_o_ring_1(B,A) ) ) ) ) )).

fof(t6_o_ring_1,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & l3_vectsp_1(A) )
     => ! [B] :
          ( m1_subset_1(B,u1_struct_0(A))
         => ( v11_o_ring_1(B,A)
           => v13_o_ring_1(B,A) ) ) ) )).

fof(dt_k1_o_ring_1,axiom,(
    ! [A,B,C] :
      ( ( ~ v1_xboole_0(A)
        & m1_finseq_1(B,A)
        & m1_subset_1(C,k5_numbers) )
     => m1_subset_1(k1_o_ring_1(A,B,C),A) ) )).

fof(dt_k2_o_ring_1,axiom,(
    ! [A,B] :
      ( ( ~ v3_struct_0(A)
        & l3_vectsp_1(A)
        & m1_subset_1(B,u1_struct_0(A)) )
     => m1_subset_1(k2_o_ring_1(A,B),u1_struct_0(A)) ) )).
%------------------------------------------------------------------------------