SET007 Axioms: SET007+321.ax


%------------------------------------------------------------------------------
% File     : SET007+321 : TPTP v7.5.0. Released v3.4.0.
% Domain   : Set Theory
% Axioms   : Cyclic Groups and Some of Their Properties - 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    : gr_cy_1 [Urb08]

% Status   : Satisfiable
% Syntax   : Number of formulae    :   70 (  21 unit)
%            Number of atoms       :  312 (  38 equality)
%            Maximal formula depth :   12 (   5 average)
%            Number of connectives :  288 (  46 ~  ;   2  |; 134  &)
%                                         (   9 <=>;  97 =>;   0 <=)
%                                         (   0 <~>;   0 ~|;   0 ~&)
%            Number of predicates  :   31 (   1 propositional; 0-3 arity)
%            Number of functors    :   45 (   8 constant; 0-6 arity)
%            Number of variables   :   86 (   0 singleton;  76 !;  10 ?)
%            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(rc1_gr_cy_1,axiom,(
    ? [A] :
      ( l1_group_1(A)
      & ~ v3_struct_0(A)
      & v1_group_1(A)
      & v2_group_1(A)
      & v3_group_1(A)
      & v4_group_1(A)
      & v1_gr_cy_1(A) ) )).

fof(cc1_gr_cy_1,axiom,(
    ! [A] :
      ( l1_group_1(A)
     => ( ( ~ v3_struct_0(A)
          & v3_group_1(A)
          & v4_group_1(A)
          & v1_gr_cy_1(A) )
       => ( ~ v3_struct_0(A)
          & v2_group_1(A)
          & v3_group_1(A)
          & v4_group_1(A)
          & v7_group_1(A) ) ) ) )).

fof(t1_gr_cy_1,axiom,(
    $true )).

fof(t2_gr_cy_1,axiom,(
    $true )).

fof(t3_gr_cy_1,axiom,(
    $true )).

fof(t4_gr_cy_1,axiom,(
    $true )).

fof(t5_gr_cy_1,axiom,(
    $true )).

fof(t6_gr_cy_1,axiom,(
    $true )).

fof(t7_gr_cy_1,axiom,(
    $true )).

fof(t8_gr_cy_1,axiom,(
    $true )).

fof(t9_gr_cy_1,axiom,(
    $true )).

fof(t10_gr_cy_1,axiom,(
    ! [A] :
      ( v4_ordinal2(A)
     => ! [B] :
          ( v4_ordinal2(B)
         => ( ~ r1_xreal_0(A,np__0)
           => ( r2_hidden(B,k1_gr_cy_1(A))
            <=> ~ r1_xreal_0(A,B) ) ) ) ) )).

fof(t11_gr_cy_1,axiom,(
    $true )).

fof(t12_gr_cy_1,axiom,(
    ! [A] :
      ( v4_ordinal2(A)
     => ( ~ r1_xreal_0(A,np__0)
       => r2_hidden(np__0,k1_gr_cy_1(A)) ) ) )).

fof(t13_gr_cy_1,axiom,(
    k1_gr_cy_1(np__1) = k18_group_2(k5_numbers,np__0) )).

fof(d2_gr_cy_1,axiom,(
    ! [A] :
      ( ( v1_funct_1(A)
        & v1_funct_2(A,k2_zfmisc_1(k4_numbers,k4_numbers),k4_numbers)
        & m2_relset_1(A,k2_zfmisc_1(k4_numbers,k4_numbers),k4_numbers) )
     => ( A = k44_binop_2
      <=> ! [B] :
            ( m1_subset_1(B,k4_numbers)
           => ! [C] :
                ( m1_subset_1(C,k4_numbers)
               => k2_binop_1(k4_numbers,k4_numbers,k4_numbers,A,B,C) = k1_binop_1(k33_binop_2,B,C) ) ) ) ) )).

fof(t14_gr_cy_1,axiom,(
    ! [A] :
      ( v1_int_1(A)
     => ! [B] :
          ( v1_int_1(B)
         => k1_binop_1(k44_binop_2,A,B) = k2_xcmplx_0(A,B) ) ) )).

fof(t15_gr_cy_1,axiom,(
    ! [A] :
      ( m1_subset_1(A,k4_numbers)
     => ( A = np__0
       => r3_binop_1(k4_numbers,A,k44_binop_2) ) ) )).

fof(d3_gr_cy_1,axiom,(
    ! [A] :
      ( m2_finseq_1(A,k4_numbers)
     => k2_gr_cy_1(A) = k2_finsop_1(k4_numbers,A,k44_binop_2) ) )).

fof(t16_gr_cy_1,axiom,(
    $true )).

fof(t17_gr_cy_1,axiom,(
    $true )).

fof(t18_gr_cy_1,axiom,(
    $true )).

fof(t19_gr_cy_1,axiom,(
    $true )).

fof(t20_gr_cy_1,axiom,(
    ! [A] :
      ( m1_subset_1(A,k4_numbers)
     => ! [B] :
          ( m2_finseq_1(B,k4_numbers)
         => k2_gr_cy_1(k8_finseq_1(k4_numbers,B,k12_finseq_1(k4_numbers,A))) = k2_xcmplx_0(k2_gr_cy_1(B),k2_group_4(A)) ) ) )).

fof(t21_gr_cy_1,axiom,(
    ! [A] :
      ( m1_subset_1(A,k4_numbers)
     => k2_gr_cy_1(k12_finseq_1(k4_numbers,A)) = A ) )).

fof(t22_gr_cy_1,axiom,(
    k2_gr_cy_1(k6_finseq_1(k4_numbers)) = np__0 )).

fof(t23_gr_cy_1,axiom,(
    $true )).

fof(t24_gr_cy_1,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_finseq_1(C,k4_numbers)
             => k3_group_4(A,k4_group_4(A,C,k1_finsop_1(u1_struct_0(A),k3_finseq_1(C),B))) = k6_group_1(A,k2_gr_cy_1(C),B) ) ) ) )).

fof(t25_gr_cy_1,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))
             => ( r1_rlvect_1(k5_group_4(A,k1_struct_0(A,C)),B)
              <=> ? [D] :
                    ( v1_int_1(D)
                    & B = k6_group_1(A,D,C) ) ) ) ) ) )).

fof(t26_gr_cy_1,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))
         => ~ ( v6_group_1(A)
              & v5_group_1(B,A) ) ) ) )).

fof(t27_gr_cy_1,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))
         => ( v6_group_1(A)
           => k7_group_1(A,B) = k9_group_1(k5_group_4(A,k1_struct_0(A,B))) ) ) ) )).

fof(t28_gr_cy_1,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))
         => ( v6_group_1(A)
           => r1_nat_1(k7_group_1(A,B),k9_group_1(A)) ) ) ) )).

fof(t29_gr_cy_1,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))
         => ( v6_group_1(A)
           => k6_group_1(A,k9_group_1(A),B) = k2_group_1(A) ) ) ) )).

fof(t30_gr_cy_1,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] :
          ( ( ~ v3_struct_0(B)
            & v3_group_1(B)
            & v4_group_1(B)
            & l1_group_1(B) )
         => ! [C] :
              ( m1_subset_1(C,u1_struct_0(B))
             => ( v6_group_1(B)
               => k3_group_1(B,k6_group_1(B,A,C)) = k6_group_1(B,k10_binop_2(k9_group_1(B),k4_nat_1(A,k9_group_1(B))),C) ) ) ) ) )).

fof(t31_gr_cy_1,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v1_group_1(A)
        & v3_group_1(A)
        & v4_group_1(A)
        & l1_group_1(A) )
     => ~ ( ~ r1_xreal_0(k9_group_1(A),np__1)
          & ! [B] :
              ( m1_subset_1(B,u1_struct_0(A))
             => B = k2_group_1(A) ) ) ) )).

fof(t32_gr_cy_1,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)
              & k9_group_1(B) = A
              & v1_int_2(A) )
           => ! [C] :
                ( ( v1_group_1(C)
                  & m1_group_2(C,B) )
               => ( r1_group_2(B,C,k5_group_2(B))
                  | C = B ) ) ) ) ) )).

fof(t33_gr_cy_1,axiom,
    ( v4_group_1(g1_group_1(k4_numbers,k44_binop_2))
    & v3_group_1(g1_group_1(k4_numbers,k44_binop_2)) )).

fof(d4_gr_cy_1,axiom,(
    k3_gr_cy_1 = g1_group_1(k4_numbers,k44_binop_2) )).

fof(d5_gr_cy_1,axiom,(
    ! [A] :
      ( v4_ordinal2(A)
     => ( ~ r1_xreal_0(A,np__0)
       => ! [B] :
            ( ( v1_funct_1(B)
              & v1_funct_2(B,k2_zfmisc_1(k1_gr_cy_1(A),k1_gr_cy_1(A)),k1_gr_cy_1(A))
              & m2_relset_1(B,k2_zfmisc_1(k1_gr_cy_1(A),k1_gr_cy_1(A)),k1_gr_cy_1(A)) )
           => ( B = k4_gr_cy_1(A)
            <=> ! [C] :
                  ( m2_subset_1(C,k5_numbers,k1_gr_cy_1(A))
                 => ! [D] :
                      ( m2_subset_1(D,k5_numbers,k1_gr_cy_1(A))
                     => k2_binop_1(k1_gr_cy_1(A),k1_gr_cy_1(A),k1_gr_cy_1(A),B,C,D) = k4_nat_1(k1_nat_1(C,D),A) ) ) ) ) ) ) )).

fof(t34_gr_cy_1,axiom,(
    ! [A] :
      ( v4_ordinal2(A)
     => ( ~ r1_xreal_0(A,np__0)
       => ( v4_group_1(g1_group_1(k1_gr_cy_1(A),k4_gr_cy_1(A)))
          & v3_group_1(g1_group_1(k1_gr_cy_1(A),k4_gr_cy_1(A))) ) ) ) )).

fof(d6_gr_cy_1,axiom,(
    ! [A] :
      ( v4_ordinal2(A)
     => ( ~ r1_xreal_0(A,np__0)
       => k5_gr_cy_1(A) = g1_group_1(k1_gr_cy_1(A),k4_gr_cy_1(A)) ) ) )).

fof(t35_gr_cy_1,axiom,(
    k2_group_1(k3_gr_cy_1) = np__0 )).

fof(t36_gr_cy_1,axiom,(
    ! [A] :
      ( v4_ordinal2(A)
     => ( ~ r1_xreal_0(A,np__0)
       => k2_group_1(k5_gr_cy_1(A)) = np__0 ) ) )).

fof(d7_gr_cy_1,axiom,(
    ! [A] :
      ( m1_subset_1(A,u1_struct_0(k3_gr_cy_1))
     => k6_gr_cy_1(A) = A ) )).

fof(d8_gr_cy_1,axiom,(
    ! [A] :
      ( v1_int_1(A)
     => k7_gr_cy_1(A) = A ) )).

fof(t37_gr_cy_1,axiom,(
    ! [A] :
      ( m1_subset_1(A,u1_struct_0(k3_gr_cy_1))
     => k3_group_1(k3_gr_cy_1,A) = k4_xcmplx_0(k6_gr_cy_1(A)) ) )).

fof(t38_gr_cy_1,axiom,(
    ! [A] :
      ( m1_subset_1(A,u1_struct_0(k3_gr_cy_1))
     => ( A = np__1
       => ! [B] :
            ( m2_subset_1(B,k1_numbers,k5_numbers)
           => k6_group_1(k3_gr_cy_1,B,A) = B ) ) ) )).

fof(t39_gr_cy_1,axiom,(
    ! [A] :
      ( m1_subset_1(A,u1_struct_0(k3_gr_cy_1))
     => ! [B] :
          ( v1_int_1(B)
         => ( A = np__1
           => B = k6_group_1(k3_gr_cy_1,B,A) ) ) ) )).

fof(d9_gr_cy_1,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v3_group_1(A)
        & v4_group_1(A)
        & l1_group_1(A) )
     => ( v1_gr_cy_1(A)
      <=> ? [B] :
            ( m1_subset_1(B,u1_struct_0(A))
            & g1_group_1(u1_struct_0(A),u1_group_1(A)) = k5_group_4(A,k1_struct_0(A,B)) ) ) ) )).

fof(t40_gr_cy_1,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v3_group_1(A)
        & v4_group_1(A)
        & l1_group_1(A) )
     => v1_gr_cy_1(k5_group_2(A)) ) )).

fof(t41_gr_cy_1,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v3_group_1(A)
        & v4_group_1(A)
        & l1_group_1(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))
            & ! [C] :
                ( m1_subset_1(C,u1_struct_0(A))
               => ? [D] :
                    ( v1_int_1(D)
                    & C = k6_group_1(A,D,B) ) ) ) ) ) )).

fof(t42_gr_cy_1,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v3_group_1(A)
        & v4_group_1(A)
        & l1_group_1(A) )
     => ( v6_group_1(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))
              & ! [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) ) ) ) ) ) ) )).

fof(t43_gr_cy_1,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)
       => ( ( ~ 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))
              & k7_group_1(A,B) = k9_group_1(A) ) ) ) ) )).

fof(t44_gr_cy_1,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v3_group_1(A)
        & v4_group_1(A)
        & l1_group_1(A) )
     => ! [B] :
          ( ( v1_group_1(B)
            & m1_group_2(B,A) )
         => ( ( v6_group_1(A)
              & ~ 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(t45_gr_cy_1,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)
              & k9_group_1(B) = A
              & v1_int_2(A) )
           => ( ~ v3_struct_0(B)
              & v3_group_1(B)
              & v4_group_1(B)
              & v1_gr_cy_1(B)
              & l1_group_1(B) ) ) ) ) )).

fof(t46_gr_cy_1,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)))
             => ? [C] :
                  ( m1_subset_1(C,u1_struct_0(k5_gr_cy_1(A)))
                  & ! [D] :
                      ( v1_int_1(D)
                     => C != k6_group_1(k5_gr_cy_1(A),D,B) ) ) ) ) ) )).

fof(t47_gr_cy_1,axiom,(
    $true )).

fof(t48_gr_cy_1,axiom,(
    v1_gr_cy_1(k3_gr_cy_1) )).

fof(t49_gr_cy_1,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ( ~ r1_xreal_0(A,np__0)
       => ( ~ v3_struct_0(k5_gr_cy_1(A))
          & v3_group_1(k5_gr_cy_1(A))
          & v4_group_1(k5_gr_cy_1(A))
          & v1_gr_cy_1(k5_gr_cy_1(A))
          & l1_group_1(k5_gr_cy_1(A)) ) ) ) )).

fof(t50_gr_cy_1,axiom,
    ( ~ v3_struct_0(k3_gr_cy_1)
    & v3_group_1(k3_gr_cy_1)
    & v4_group_1(k3_gr_cy_1)
    & v7_group_1(k3_gr_cy_1)
    & l1_group_1(k3_gr_cy_1) )).

fof(t51_gr_cy_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_xreal_0(A,np__0)
              | r1_tarski(k1_gr_cy_1(A),k1_gr_cy_1(B)) ) ) ) ) )).

fof(dt_k1_gr_cy_1,axiom,(
    ! [A] :
      ( v4_ordinal2(A)
     => ( ~ v1_xboole_0(k1_gr_cy_1(A))
        & m1_subset_1(k1_gr_cy_1(A),k1_zfmisc_1(k5_numbers)) ) ) )).

fof(dt_k2_gr_cy_1,axiom,(
    ! [A] :
      ( m1_finseq_1(A,k4_numbers)
     => v1_int_1(k2_gr_cy_1(A)) ) )).

fof(dt_k3_gr_cy_1,axiom,
    ( ~ v3_struct_0(k3_gr_cy_1)
    & v1_group_1(k3_gr_cy_1)
    & v3_group_1(k3_gr_cy_1)
    & v4_group_1(k3_gr_cy_1)
    & l1_group_1(k3_gr_cy_1) )).

fof(dt_k4_gr_cy_1,axiom,(
    ! [A] :
      ( v4_ordinal2(A)
     => ( v1_funct_1(k4_gr_cy_1(A))
        & v1_funct_2(k4_gr_cy_1(A),k2_zfmisc_1(k1_gr_cy_1(A),k1_gr_cy_1(A)),k1_gr_cy_1(A))
        & m2_relset_1(k4_gr_cy_1(A),k2_zfmisc_1(k1_gr_cy_1(A),k1_gr_cy_1(A)),k1_gr_cy_1(A)) ) ) )).

fof(dt_k5_gr_cy_1,axiom,(
    ! [A] :
      ( v4_ordinal2(A)
     => ( ~ v3_struct_0(k5_gr_cy_1(A))
        & v1_group_1(k5_gr_cy_1(A))
        & v3_group_1(k5_gr_cy_1(A))
        & v4_group_1(k5_gr_cy_1(A))
        & l1_group_1(k5_gr_cy_1(A)) ) ) )).

fof(dt_k6_gr_cy_1,axiom,(
    ! [A] :
      ( m1_subset_1(A,u1_struct_0(k3_gr_cy_1))
     => v1_int_1(k6_gr_cy_1(A)) ) )).

fof(dt_k7_gr_cy_1,axiom,(
    ! [A] :
      ( v1_int_1(A)
     => m1_subset_1(k7_gr_cy_1(A),u1_struct_0(k3_gr_cy_1)) ) )).

fof(d1_gr_cy_1,axiom,(
    ! [A] :
      ( v4_ordinal2(A)
     => ( ~ r1_xreal_0(A,np__0)
       => k1_gr_cy_1(A) = a_1_0_gr_cy_1(A) ) ) )).

fof(fraenkel_a_1_0_gr_cy_1,axiom,(
    ! [A,B] :
      ( v4_ordinal2(B)
     => ( r2_hidden(A,a_1_0_gr_cy_1(B))
      <=> ? [C] :
            ( m2_subset_1(C,k1_numbers,k5_numbers)
            & A = C
            & ~ r1_xreal_0(B,C) ) ) ) )).
%------------------------------------------------------------------------------