SET007 Axioms: SET007+338.ax


%------------------------------------------------------------------------------
% File     : SET007+338 : TPTP v7.5.0. Released v3.4.0.
% Domain   : Set Theory
% Axioms   : Isomorphisms of Cyclic Groups. Some Properties of Cyclic Groups
% 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    : gr_cy_2 [Urb08]

% Status   : Satisfiable
% Syntax   : Number of formulae    :   36 (   6 unit)
%            Number of atoms       :  390 (  49 equality)
%            Maximal formula depth :   18 (   9 average)
%            Number of connectives :  412 (  58 ~  ;   2  |; 236  &)
%                                         (   4 <=>; 112 =>;   0 <=)
%                                         (   0 <~>;   0 ~|;   0 ~&)
%            Number of predicates  :   26 (   1 propositional; 0-3 arity)
%            Number of functors    :   21 (   6 constant; 0-3 arity)
%            Number of variables   :   95 (   0 singleton;  88 !;   7 ?)
%            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_gr_cy_2,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v3_group_1(A)
        & v4_group_1(A)
        & l1_group_1(A) )
     => ( ~ v3_struct_0(k10_group_5(A))
        & v1_group_1(k10_group_5(A))
        & v2_group_1(k10_group_5(A))
        & v3_group_1(k10_group_5(A))
        & v4_group_1(k10_group_5(A))
        & v1_group_3(k10_group_5(A),A) ) ) )).

fof(t1_gr_cy_2,axiom,(
    $true )).

fof(t2_gr_cy_2,axiom,(
    $true )).

fof(t3_gr_cy_2,axiom,(
    $true )).

fof(t4_gr_cy_2,axiom,(
    $true )).

fof(t5_gr_cy_2,axiom,(
    $true )).

fof(t6_gr_cy_2,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v3_group_1(A)
        & v4_group_1(A)
        & l1_group_1(A) )
     => ! [B] :
          ( m1_subset_1(B,u1_struct_0(A))
         => ! [C] :
              ( m1_subset_1(C,u1_struct_0(A))
             => ! [D] :
                  ( m2_subset_1(D,k1_numbers,k5_numbers)
                 => ~ ( ~ r1_xreal_0(k7_group_1(A,B),np__1)
                      & B = k6_group_1(A,D,C)
                      & D = np__0 ) ) ) ) ) )).

fof(t7_gr_cy_2,axiom,(
    $true )).

fof(t8_gr_cy_2,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v3_group_1(A)
        & v4_group_1(A)
        & l1_group_1(A) )
     => ! [B] :
          ( m1_subset_1(B,u1_struct_0(A))
         => r1_rlvect_1(k5_group_4(A,k1_struct_0(A,B)),B) ) ) )).

fof(t9_gr_cy_2,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v3_group_1(A)
        & v4_group_1(A)
        & l1_group_1(A) )
     => ! [B] :
          ( m1_group_2(B,A)
         => ! [C] :
              ( m1_subset_1(C,u1_struct_0(A))
             => ! [D] :
                  ( m1_subset_1(D,u1_struct_0(B))
                 => ( C = D
                   => k5_group_4(A,k1_struct_0(A,C)) = k5_group_4(B,k1_struct_0(B,D)) ) ) ) ) ) )).

fof(t10_gr_cy_2,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v3_group_1(A)
        & v4_group_1(A)
        & l1_group_1(A) )
     => ! [B] :
          ( m1_subset_1(B,u1_struct_0(A))
         => ( ~ v3_struct_0(k5_group_4(A,k1_struct_0(A,B)))
            & v3_group_1(k5_group_4(A,k1_struct_0(A,B)))
            & v4_group_1(k5_group_4(A,k1_struct_0(A,B)))
            & v1_gr_cy_1(k5_group_4(A,k1_struct_0(A,B)))
            & l1_group_1(k5_group_4(A,k1_struct_0(A,B))) ) ) ) )).

fof(t11_gr_cy_2,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v1_group_1(A)
        & v3_group_1(A)
        & v4_group_1(A)
        & l1_group_1(A) )
     => ! [B] :
          ( m1_subset_1(B,u1_struct_0(A))
         => ( ! [C] :
                ( m1_subset_1(C,u1_struct_0(A))
               => ? [D] :
                    ( v1_int_1(D)
                    & C = k6_group_1(A,D,B) ) )
          <=> A = k5_group_4(A,k1_struct_0(A,B)) ) ) ) )).

fof(t12_gr_cy_2,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v1_group_1(A)
        & v3_group_1(A)
        & v4_group_1(A)
        & l1_group_1(A) )
     => ! [B] :
          ( m1_subset_1(B,u1_struct_0(A))
         => ( v6_group_1(A)
           => ( ! [C] :
                  ( m1_subset_1(C,u1_struct_0(A))
                 => ? [D] :
                      ( m2_subset_1(D,k1_numbers,k5_numbers)
                      & C = k6_group_1(A,D,B) ) )
            <=> A = k5_group_4(A,k1_struct_0(A,B)) ) ) ) ) )).

fof(t13_gr_cy_2,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v1_group_1(A)
        & v3_group_1(A)
        & v4_group_1(A)
        & l1_group_1(A) )
     => ! [B] :
          ( m1_subset_1(B,u1_struct_0(A))
         => ( ( v6_group_1(A)
              & A = k5_group_4(A,k1_struct_0(A,B)) )
           => ! [C] :
                ( ( v1_group_1(C)
                  & m1_group_2(C,A) )
               => ? [D] :
                    ( m2_subset_1(D,k1_numbers,k5_numbers)
                    & r1_group_2(A,C,k5_group_4(A,k1_struct_0(A,k6_group_1(A,D,B)))) ) ) ) ) ) )).

fof(t14_gr_cy_2,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v3_group_1(A)
        & v4_group_1(A)
        & l1_group_1(A) )
     => ! [B] :
          ( m1_subset_1(B,u1_struct_0(A))
         => ! [C] :
              ( m2_subset_1(C,k1_numbers,k5_numbers)
             => ! [D] :
                  ( m2_subset_1(D,k1_numbers,k5_numbers)
                 => ! [E] :
                      ( m2_subset_1(E,k1_numbers,k5_numbers)
                     => ( ( v6_group_1(A)
                          & A = k5_group_4(A,k1_struct_0(A,B))
                          & k9_group_1(A) = C
                          & C = k2_nat_1(D,E) )
                       => k7_group_1(A,k6_group_1(A,D,B)) = E ) ) ) ) ) ) )).

fof(t15_gr_cy_2,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v3_group_1(A)
        & v4_group_1(A)
        & l1_group_1(A) )
     => ! [B] :
          ( m1_subset_1(B,u1_struct_0(A))
         => ! [C] :
              ( m2_subset_1(C,k1_numbers,k5_numbers)
             => ! [D] :
                  ( m2_subset_1(D,k1_numbers,k5_numbers)
                 => ( r1_nat_1(C,D)
                   => r1_rlvect_1(k5_group_4(A,k1_struct_0(A,k6_group_1(A,C,B))),k6_group_1(A,D,B)) ) ) ) ) ) )).

fof(t16_gr_cy_2,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v3_group_1(A)
        & v4_group_1(A)
        & l1_group_1(A) )
     => ! [B] :
          ( m1_subset_1(B,u1_struct_0(A))
         => ! [C] :
              ( m2_subset_1(C,k1_numbers,k5_numbers)
             => ! [D] :
                  ( m2_subset_1(D,k1_numbers,k5_numbers)
                 => ( ( v6_group_1(A)
                      & k9_group_1(k5_group_4(A,k1_struct_0(A,k6_group_1(A,C,B)))) = k9_group_1(k5_group_4(A,k1_struct_0(A,k6_group_1(A,D,B))))
                      & r1_rlvect_1(k5_group_4(A,k1_struct_0(A,k6_group_1(A,C,B))),k6_group_1(A,D,B)) )
                   => r1_group_2(A,k5_group_4(A,k1_struct_0(A,k6_group_1(A,C,B))),k5_group_4(A,k1_struct_0(A,k6_group_1(A,D,B)))) ) ) ) ) ) )).

fof(t17_gr_cy_2,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v3_group_1(A)
        & v4_group_1(A)
        & l1_group_1(A) )
     => ! [B] :
          ( m1_group_2(B,A)
         => ! [C] :
              ( m1_subset_1(C,u1_struct_0(A))
             => ! [D] :
                  ( m2_subset_1(D,k1_numbers,k5_numbers)
                 => ! [E] :
                      ( m2_subset_1(E,k1_numbers,k5_numbers)
                     => ! [F] :
                          ( m2_subset_1(F,k1_numbers,k5_numbers)
                         => ( ( v6_group_1(A)
                              & k9_group_1(A) = D
                              & A = k5_group_4(A,k1_struct_0(A,C))
                              & k9_group_1(B) = E
                              & B = k5_group_4(A,k1_struct_0(A,k6_group_1(A,F,C))) )
                           => r1_nat_1(D,k2_nat_1(F,E)) ) ) ) ) ) ) ) )).

fof(t18_gr_cy_2,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] :
          ( m2_subset_1(B,k1_numbers,k5_numbers)
         => ! [C] :
              ( ( ~ v3_struct_0(C)
                & v1_group_1(C)
                & v3_group_1(C)
                & v4_group_1(C)
                & l1_group_1(C) )
             => ! [D] :
                  ( m1_subset_1(D,u1_struct_0(C))
                 => ( ( v6_group_1(C)
                      & C = k5_group_4(C,k1_struct_0(C,D))
                      & k9_group_1(C) = A )
                   => ( C = k5_group_4(C,k1_struct_0(C,k6_group_1(C,B,D)))
                    <=> k6_nat_1(B,A) = np__1 ) ) ) ) ) ) )).

fof(t19_gr_cy_2,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v3_group_1(A)
        & v4_group_1(A)
        & v1_gr_cy_1(A)
        & l1_group_1(A) )
     => ! [B] :
          ( m1_group_2(B,A)
         => ! [C] :
              ( m1_subset_1(C,u1_struct_0(A))
             => ( ( A = k5_group_4(A,k1_struct_0(A,C))
                  & r1_rlvect_1(B,C) )
               => g1_group_1(u1_struct_0(A),u1_group_1(A)) = g1_group_1(u1_struct_0(B),u1_group_1(B)) ) ) ) ) )).

fof(t20_gr_cy_2,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v3_group_1(A)
        & v4_group_1(A)
        & v1_gr_cy_1(A)
        & l1_group_1(A) )
     => ! [B] :
          ( m1_subset_1(B,u1_struct_0(A))
         => ( A = k5_group_4(A,k1_struct_0(A,B))
           => ( v6_group_1(A)
            <=> ? [C] :
                  ( v1_int_1(C)
                  & ? [D] :
                      ( v1_int_1(D)
                      & C != D
                      & k6_group_1(A,C,B) = k6_group_1(A,D,B) ) ) ) ) ) ) )).

fof(d1_gr_cy_2,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ( ~ r1_xreal_0(A,np__0)
       => ! [B] :
            ( m1_subset_1(B,u1_struct_0(k5_gr_cy_1(A)))
           => k1_gr_cy_2(A,B) = B ) ) ) )).

fof(t21_gr_cy_2,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] :
          ( ( ~ v3_struct_0(B)
            & v1_group_1(B)
            & v3_group_1(B)
            & v4_group_1(B)
            & v1_gr_cy_1(B)
            & l1_group_1(B) )
         => ( ( v6_group_1(B)
              & k9_group_1(B) = A )
           => r2_group_6(k5_gr_cy_1(A),B) ) ) ) )).

fof(t22_gr_cy_2,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v1_group_1(A)
        & v3_group_1(A)
        & v4_group_1(A)
        & v1_gr_cy_1(A)
        & l1_group_1(A) )
     => ( ~ v6_group_1(A)
       => r2_group_6(k3_gr_cy_1,A) ) ) )).

fof(t23_gr_cy_2,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v1_group_1(A)
        & v3_group_1(A)
        & v4_group_1(A)
        & v1_gr_cy_1(A)
        & l1_group_1(A) )
     => ! [B] :
          ( ( ~ v3_struct_0(B)
            & v1_group_1(B)
            & v3_group_1(B)
            & v4_group_1(B)
            & v1_gr_cy_1(B)
            & l1_group_1(B) )
         => ( ( v6_group_1(B)
              & v6_group_1(A)
              & k9_group_1(B) = k9_group_1(A) )
           => r2_group_6(B,A) ) ) ) )).

fof(t24_gr_cy_2,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] :
          ( ( ~ v3_struct_0(B)
            & v1_group_1(B)
            & v3_group_1(B)
            & v4_group_1(B)
            & l1_group_1(B) )
         => ! [C] :
              ( ( ~ v3_struct_0(C)
                & v1_group_1(C)
                & v3_group_1(C)
                & v4_group_1(C)
                & l1_group_1(C) )
             => ( ( v6_group_1(B)
                  & v6_group_1(C)
                  & k9_group_1(B) = A
                  & k9_group_1(C) = A
                  & v1_int_2(A) )
               => r2_group_6(B,C) ) ) ) ) )).

fof(t25_gr_cy_2,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v1_group_1(A)
        & v3_group_1(A)
        & v4_group_1(A)
        & l1_group_1(A) )
     => ! [B] :
          ( ( ~ v3_struct_0(B)
            & v1_group_1(B)
            & v3_group_1(B)
            & v4_group_1(B)
            & l1_group_1(B) )
         => ( ( v6_group_1(A)
              & v6_group_1(B)
              & k9_group_1(A) = np__2
              & k9_group_1(B) = np__2 )
           => r2_group_6(A,B) ) ) ) )).

fof(t26_gr_cy_2,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v1_group_1(A)
        & v3_group_1(A)
        & v4_group_1(A)
        & l1_group_1(A) )
     => ( ( v6_group_1(A)
          & k9_group_1(A) = np__2 )
       => ! [B] :
            ( ( v1_group_1(B)
              & m1_group_2(B,A) )
           => ( r1_group_2(A,B,k5_group_2(A))
              | B = A ) ) ) ) )).

fof(t27_gr_cy_2,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v1_group_1(A)
        & v3_group_1(A)
        & v4_group_1(A)
        & l1_group_1(A) )
     => ( ( v6_group_1(A)
          & k9_group_1(A) = np__2 )
       => ( ~ v3_struct_0(A)
          & v3_group_1(A)
          & v4_group_1(A)
          & v1_gr_cy_1(A)
          & l1_group_1(A) ) ) ) )).

fof(t28_gr_cy_2,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] :
          ( ( ~ v3_struct_0(B)
            & v1_group_1(B)
            & v3_group_1(B)
            & v4_group_1(B)
            & l1_group_1(B) )
         => ( ( v6_group_1(B)
              & ~ v3_struct_0(B)
              & v3_group_1(B)
              & v4_group_1(B)
              & v1_gr_cy_1(B)
              & l1_group_1(B)
              & k9_group_1(B) = A )
           => ! [C] :
                ( m2_subset_1(C,k1_numbers,k5_numbers)
               => ~ ( r1_nat_1(C,A)
                    & ! [D] :
                        ( ( v1_group_1(D)
                          & m1_group_2(D,B) )
                       => ~ ( k9_group_1(D) = C
                            & ! [E] :
                                ( ( v1_group_1(E)
                                  & m1_group_2(E,B) )
                               => ( k9_group_1(E) = C
                                 => r1_group_2(B,E,D) ) ) ) ) ) ) ) ) ) )).

fof(t29_gr_cy_2,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v3_group_1(A)
        & v4_group_1(A)
        & v1_gr_cy_1(A)
        & l1_group_1(A) )
     => ! [B] :
          ( m1_subset_1(B,u1_struct_0(A))
         => ( A = k5_group_4(A,k1_struct_0(A,B))
           => ! [C] :
                ( ( ~ v3_struct_0(C)
                  & v3_group_1(C)
                  & v4_group_1(C)
                  & l1_group_1(C) )
               => ! [D] :
                    ( ( v1_funct_1(D)
                      & v1_funct_2(D,u1_struct_0(C),u1_struct_0(A))
                      & v1_group_6(D,C,A)
                      & m2_relset_1(D,u1_struct_0(C),u1_struct_0(A)) )
                   => ( r1_rlvect_1(k13_group_6(C,A,D),B)
                     => v3_group_6(D,C,A) ) ) ) ) ) ) )).

fof(t30_gr_cy_2,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] :
          ( ( ~ v3_struct_0(B)
            & v1_group_1(B)
            & v3_group_1(B)
            & v4_group_1(B)
            & v1_gr_cy_1(B)
            & l1_group_1(B) )
         => ~ ( v6_group_1(B)
              & k9_group_1(B) = A
              & ? [C] :
                  ( m2_subset_1(C,k1_numbers,k5_numbers)
                  & A = k2_nat_1(np__2,C) )
              & ! [C] :
                  ( m1_subset_1(C,u1_struct_0(B))
                 => ~ ( k7_group_1(B,C) = np__2
                      & ! [D] :
                          ( m1_subset_1(D,u1_struct_0(B))
                         => ( k7_group_1(B,D) = np__2
                           => C = D ) ) ) ) ) ) ) )).

fof(t31_gr_cy_2,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] :
          ( ( ~ v3_struct_0(B)
            & v1_group_1(B)
            & v3_group_1(B)
            & v4_group_1(B)
            & v1_gr_cy_1(B)
            & l1_group_1(B) )
         => ~ ( v6_group_1(B)
              & k9_group_1(B) = A
              & ? [C] :
                  ( m2_subset_1(C,k1_numbers,k5_numbers)
                  & A = k2_nat_1(np__2,C) )
              & ! [C] :
                  ( m1_group_2(C,B)
                 => ~ ( k9_group_1(C) = np__2
                      & ~ v3_struct_0(C)
                      & v3_group_1(C)
                      & v4_group_1(C)
                      & v1_gr_cy_1(C)
                      & l1_group_1(C) ) ) ) ) ) )).

fof(t32_gr_cy_2,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v3_group_1(A)
        & v4_group_1(A)
        & l1_group_1(A) )
     => ! [B] :
          ( ( ~ v3_struct_0(B)
            & v1_group_1(B)
            & v3_group_1(B)
            & v4_group_1(B)
            & l1_group_1(B) )
         => ! [C] :
              ( ( v1_funct_1(C)
                & v1_funct_2(C,u1_struct_0(B),u1_struct_0(A))
                & v1_group_6(C,B,A)
                & m2_relset_1(C,u1_struct_0(B),u1_struct_0(A)) )
             => ( ( ~ v3_struct_0(B)
                  & v3_group_1(B)
                  & v4_group_1(B)
                  & v1_gr_cy_1(B)
                  & l1_group_1(B) )
               => ( ~ v3_struct_0(k13_group_6(B,A,C))
                  & v3_group_1(k13_group_6(B,A,C))
                  & v4_group_1(k13_group_6(B,A,C))
                  & v1_gr_cy_1(k13_group_6(B,A,C))
                  & l1_group_1(k13_group_6(B,A,C)) ) ) ) ) ) )).

fof(t33_gr_cy_2,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v1_group_1(A)
        & v3_group_1(A)
        & v4_group_1(A)
        & l1_group_1(A) )
     => ! [B] :
          ( ( ~ v3_struct_0(B)
            & v1_group_1(B)
            & v3_group_1(B)
            & v4_group_1(B)
            & l1_group_1(B) )
         => ( r2_group_6(A,B)
           => ( ( ~ ( ~ v3_struct_0(A)
                    & v3_group_1(A)
                    & v4_group_1(A)
                    & v1_gr_cy_1(A)
                    & l1_group_1(A) )
                & ~ ( ~ v3_struct_0(B)
                    & v3_group_1(B)
                    & v4_group_1(B)
                    & v1_gr_cy_1(B)
                    & l1_group_1(B) ) )
              | ( ~ v3_struct_0(A)
                & v3_group_1(A)
                & v4_group_1(A)
                & v1_gr_cy_1(A)
                & l1_group_1(A)
                & ~ v3_struct_0(B)
                & v3_group_1(B)
                & v4_group_1(B)
                & v1_gr_cy_1(B)
                & l1_group_1(B) ) ) ) ) ) )).

fof(dt_k1_gr_cy_2,axiom,(
    ! [A,B] :
      ( ( m1_subset_1(A,k5_numbers)
        & m1_subset_1(B,u1_struct_0(k5_gr_cy_1(A))) )
     => m2_subset_1(k1_gr_cy_2(A,B),k1_numbers,k5_numbers) ) )).
%------------------------------------------------------------------------------