SET007 Axioms: SET007+273.ax


%------------------------------------------------------------------------------
% File     : SET007+273 : TPTP v8.2.0. Released v3.4.0.
% Domain   : Set Theory
% Axioms   : Ordered Rings - Part III
% 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_3 [Urb08]

% Status   : Satisfiable
% Syntax   : Number of formulae    :   12 (   0 unt;   0 def)
%            Number of atoms       :  158 (   0 equ)
%            Maximal formula atoms :   19 (  13 avg)
%            Number of connectives :  209 (  63   ~;   2   |;  96   &)
%                                         (   0 <=>;  48  =>;   0  <=;   0 <~>)
%            Maximal formula depth :   17 (  13 avg)
%            Maximal term depth    :    2 (   1 avg)
%            Number of predicates  :   10 (  10 usr;   0 prp; 1-2 aty)
%            Number of functors    :    2 (   2 usr;   0 con; 1-3 aty)
%            Number of variables   :   36 (  36   !;   0   ?)
% 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(t1_o_ring_3,axiom,
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & l3_vectsp_1(A) )
     => ! [B] :
          ( m1_subset_1(B,u1_struct_0(A))
         => ! [C] :
              ( m1_subset_1(C,u1_struct_0(A))
             => ( ( v1_o_ring_1(B,A)
                  & v1_o_ring_1(C,A) )
               => v5_o_ring_1(k1_group_1(A,B,C),A) ) ) ) ) ).

fof(t2_o_ring_3,axiom,
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & l3_vectsp_1(A) )
     => ! [B] :
          ( m1_subset_1(B,u1_struct_0(A))
         => ! [C] :
              ( m1_subset_1(C,u1_struct_0(A))
             => ( ( v5_o_ring_1(B,A)
                  & v1_o_ring_1(C,A) )
               => v5_o_ring_1(k1_group_1(A,B,C),A) ) ) ) ) ).

fof(t3_o_ring_3,axiom,
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & l3_vectsp_1(A) )
     => ! [B] :
          ( m1_subset_1(B,u1_struct_0(A))
         => ! [C] :
              ( m1_subset_1(C,u1_struct_0(A))
             => ( ( ( v1_o_ring_1(B,A)
                    & v5_o_ring_1(C,A) )
                  | ( v1_o_ring_1(B,A)
                    & v9_o_ring_1(C,A) ) )
               => v9_o_ring_1(k1_group_1(A,B,C),A) ) ) ) ) ).

fof(t4_o_ring_3,axiom,
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & l3_vectsp_1(A) )
     => ! [B] :
          ( m1_subset_1(B,u1_struct_0(A))
         => ! [C] :
              ( m1_subset_1(C,u1_struct_0(A))
             => ( ( ( v5_o_ring_1(B,A)
                    & v5_o_ring_1(C,A) )
                  | ( v5_o_ring_1(B,A)
                    & v9_o_ring_1(C,A) ) )
               => v9_o_ring_1(k1_group_1(A,B,C),A) ) ) ) ) ).

fof(t5_o_ring_3,axiom,
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & l3_vectsp_1(A) )
     => ! [B] :
          ( m1_subset_1(B,u1_struct_0(A))
         => ! [C] :
              ( m1_subset_1(C,u1_struct_0(A))
             => ( ~ ( ~ ( v9_o_ring_1(B,A)
                        & v1_o_ring_1(C,A) )
                    & ~ ( v9_o_ring_1(B,A)
                        & v5_o_ring_1(C,A) )
                    & ~ ( v9_o_ring_1(B,A)
                        & v9_o_ring_1(C,A) ) )
               => v9_o_ring_1(k1_group_1(A,B,C),A) ) ) ) ) ).

fof(t6_o_ring_3,axiom,
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & l3_vectsp_1(A) )
     => ! [B] :
          ( m1_subset_1(B,u1_struct_0(A))
         => ! [C] :
              ( m1_subset_1(C,u1_struct_0(A))
             => ( ~ ( ~ ( v1_o_ring_1(B,A)
                        & v3_o_ring_1(C,A) )
                    & ~ ( v1_o_ring_1(B,A)
                        & v7_o_ring_1(C,A) )
                    & ~ ( v1_o_ring_1(B,A)
                        & v11_o_ring_1(C,A) )
                    & ~ ( v1_o_ring_1(B,A)
                        & v13_o_ring_1(C,A) ) )
               => v13_o_ring_1(k1_group_1(A,B,C),A) ) ) ) ) ).

fof(t7_o_ring_3,axiom,
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & l3_vectsp_1(A) )
     => ! [B] :
          ( m1_subset_1(B,u1_struct_0(A))
         => ! [C] :
              ( m1_subset_1(C,u1_struct_0(A))
             => ( ~ ( ~ ( v3_o_ring_1(B,A)
                        & v1_o_ring_1(C,A) )
                    & ~ ( v3_o_ring_1(B,A)
                        & v3_o_ring_1(C,A) )
                    & ~ ( v3_o_ring_1(B,A)
                        & v5_o_ring_1(C,A) )
                    & ~ ( v3_o_ring_1(B,A)
                        & v7_o_ring_1(C,A) )
                    & ~ ( v3_o_ring_1(B,A)
                        & v9_o_ring_1(C,A) )
                    & ~ ( v3_o_ring_1(B,A)
                        & v11_o_ring_1(C,A) )
                    & ~ ( v3_o_ring_1(B,A)
                        & v13_o_ring_1(C,A) ) )
               => v13_o_ring_1(k1_group_1(A,B,C),A) ) ) ) ) ).

fof(t8_o_ring_3,axiom,
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & l3_vectsp_1(A) )
     => ! [B] :
          ( m1_subset_1(B,u1_struct_0(A))
         => ! [C] :
              ( m1_subset_1(C,u1_struct_0(A))
             => ( ~ ( ~ ( v5_o_ring_1(B,A)
                        & v3_o_ring_1(C,A) )
                    & ~ ( v5_o_ring_1(B,A)
                        & v7_o_ring_1(C,A) )
                    & ~ ( v5_o_ring_1(B,A)
                        & v11_o_ring_1(C,A) )
                    & ~ ( v5_o_ring_1(B,A)
                        & v13_o_ring_1(C,A) ) )
               => v13_o_ring_1(k1_group_1(A,B,C),A) ) ) ) ) ).

fof(t9_o_ring_3,axiom,
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & l3_vectsp_1(A) )
     => ! [B] :
          ( m1_subset_1(B,u1_struct_0(A))
         => ! [C] :
              ( m1_subset_1(C,u1_struct_0(A))
             => ( ~ ( ~ ( v7_o_ring_1(B,A)
                        & v1_o_ring_1(C,A) )
                    & ~ ( v7_o_ring_1(B,A)
                        & v3_o_ring_1(C,A) )
                    & ~ ( v7_o_ring_1(B,A)
                        & v5_o_ring_1(C,A) )
                    & ~ ( v7_o_ring_1(B,A)
                        & v7_o_ring_1(C,A) )
                    & ~ ( v7_o_ring_1(B,A)
                        & v9_o_ring_1(C,A) )
                    & ~ ( v7_o_ring_1(B,A)
                        & v11_o_ring_1(C,A) )
                    & ~ ( v7_o_ring_1(B,A)
                        & v13_o_ring_1(C,A) ) )
               => v13_o_ring_1(k1_group_1(A,B,C),A) ) ) ) ) ).

fof(t10_o_ring_3,axiom,
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & l3_vectsp_1(A) )
     => ! [B] :
          ( m1_subset_1(B,u1_struct_0(A))
         => ! [C] :
              ( m1_subset_1(C,u1_struct_0(A))
             => ( ~ ( ~ ( v9_o_ring_1(B,A)
                        & v3_o_ring_1(C,A) )
                    & ~ ( v9_o_ring_1(B,A)
                        & v7_o_ring_1(C,A) )
                    & ~ ( v9_o_ring_1(B,A)
                        & v11_o_ring_1(C,A) )
                    & ~ ( v9_o_ring_1(B,A)
                        & v13_o_ring_1(C,A) ) )
               => v13_o_ring_1(k1_group_1(A,B,C),A) ) ) ) ) ).

fof(t11_o_ring_3,axiom,
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & l3_vectsp_1(A) )
     => ! [B] :
          ( m1_subset_1(B,u1_struct_0(A))
         => ! [C] :
              ( m1_subset_1(C,u1_struct_0(A))
             => ( ~ ( ~ ( v11_o_ring_1(B,A)
                        & v1_o_ring_1(C,A) )
                    & ~ ( v11_o_ring_1(B,A)
                        & v3_o_ring_1(C,A) )
                    & ~ ( v11_o_ring_1(B,A)
                        & v5_o_ring_1(C,A) )
                    & ~ ( v11_o_ring_1(B,A)
                        & v7_o_ring_1(C,A) )
                    & ~ ( v11_o_ring_1(B,A)
                        & v9_o_ring_1(C,A) )
                    & ~ ( v11_o_ring_1(B,A)
                        & v11_o_ring_1(C,A) )
                    & ~ ( v11_o_ring_1(B,A)
                        & v13_o_ring_1(C,A) ) )
               => v13_o_ring_1(k1_group_1(A,B,C),A) ) ) ) ) ).

fof(t12_o_ring_3,axiom,
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & l3_vectsp_1(A) )
     => ! [B] :
          ( m1_subset_1(B,u1_struct_0(A))
         => ! [C] :
              ( m1_subset_1(C,u1_struct_0(A))
             => ( ~ ( ~ ( v13_o_ring_1(B,A)
                        & v1_o_ring_1(C,A) )
                    & ~ ( v13_o_ring_1(B,A)
                        & v3_o_ring_1(C,A) )
                    & ~ ( v13_o_ring_1(B,A)
                        & v5_o_ring_1(C,A) )
                    & ~ ( v13_o_ring_1(B,A)
                        & v7_o_ring_1(C,A) )
                    & ~ ( v13_o_ring_1(B,A)
                        & v9_o_ring_1(C,A) )
                    & ~ ( v13_o_ring_1(B,A)
                        & v11_o_ring_1(C,A) )
                    & ~ ( v13_o_ring_1(B,A)
                        & v13_o_ring_1(C,A) ) )
               => v13_o_ring_1(k1_group_1(A,B,C),A) ) ) ) ) ).

%------------------------------------------------------------------------------