SET007 Axioms: SET007+38.ax


%------------------------------------------------------------------------------
% File     : SET007+38 : TPTP v7.5.0. Released v3.4.0.
% Domain   : Set Theory
% Axioms   : Complex Numbers - Basic Definitions
% 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    : xcmplx_0 [Urb08]

% Status   : Satisfiable
% Syntax   : Number of formulae    :   33 (   8 unit)
%            Number of atoms       :  103 (  24 equality)
%            Maximal formula depth :   17 (   5 average)
%            Number of connectives :   82 (  12 ~  ;   0  |;  32  &)
%                                         (   7 <=>;  31 =>;   0 <=)
%                                         (   0 <~>;   0 ~|;   0 ~&)
%            Number of predicates  :    7 (   1 propositional; 0-2 arity)
%            Number of functors    :   16 (   5 constant; 0-5 arity)
%            Number of variables   :   50 (   0 singleton;  40 !;  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.
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fof(fc1_xcmplx_0,axiom,(
    v1_xcmplx_0(k1_xcmplx_0) )).

fof(rc1_xcmplx_0,axiom,(
    ? [A] : v1_xcmplx_0(A) )).

fof(fc2_xcmplx_0,axiom,(
    ! [A,B] :
      ( ( v1_xcmplx_0(A)
        & v1_xcmplx_0(B) )
     => v1_xcmplx_0(k2_xcmplx_0(A,B)) ) )).

fof(fc3_xcmplx_0,axiom,(
    ! [A,B] :
      ( ( v1_xcmplx_0(A)
        & v1_xcmplx_0(B) )
     => v1_xcmplx_0(k3_xcmplx_0(A,B)) ) )).

fof(fc4_xcmplx_0,axiom,(
    ! [A,B] :
      ( ( v1_xcmplx_0(A)
        & v1_xcmplx_0(B) )
     => v1_xcmplx_0(k6_xcmplx_0(A,B)) ) )).

fof(fc5_xcmplx_0,axiom,(
    ! [A,B] :
      ( ( v1_xcmplx_0(A)
        & v1_xcmplx_0(B) )
     => v1_xcmplx_0(k7_xcmplx_0(A,B)) ) )).

fof(rc2_xcmplx_0,axiom,(
    ? [A] :
      ( ~ v1_xboole_0(A)
      & v1_xcmplx_0(A) ) )).

fof(fc6_xcmplx_0,axiom,(
    ! [A] :
      ( ( ~ v1_xboole_0(A)
        & v1_xcmplx_0(A) )
     => ( ~ v1_xboole_0(k4_xcmplx_0(A))
        & v1_xcmplx_0(k4_xcmplx_0(A)) ) ) )).

fof(fc7_xcmplx_0,axiom,(
    ! [A] :
      ( ( ~ v1_xboole_0(A)
        & v1_xcmplx_0(A) )
     => ( ~ v1_xboole_0(k5_xcmplx_0(A))
        & v1_xcmplx_0(k5_xcmplx_0(A)) ) ) )).

fof(fc8_xcmplx_0,axiom,(
    ! [A,B] :
      ( ( ~ v1_xboole_0(A)
        & v1_xcmplx_0(A)
        & ~ v1_xboole_0(B)
        & v1_xcmplx_0(B) )
     => ( ~ v1_xboole_0(k3_xcmplx_0(A,B))
        & v1_xcmplx_0(k3_xcmplx_0(A,B)) ) ) )).

fof(fc9_xcmplx_0,axiom,(
    ! [A,B] :
      ( ( ~ v1_xboole_0(A)
        & v1_xcmplx_0(A)
        & ~ v1_xboole_0(B)
        & v1_xcmplx_0(B) )
     => ( ~ v1_xboole_0(k7_xcmplx_0(A,B))
        & v1_xcmplx_0(k7_xcmplx_0(A,B)) ) ) )).

fof(cc1_xcmplx_0,axiom,(
    ! [A] :
      ( m1_subset_1(A,k1_numbers)
     => v1_xcmplx_0(A) ) )).

fof(cc2_xcmplx_0,axiom,(
    ! [A] :
      ( v4_ordinal2(A)
     => v1_xcmplx_0(A) ) )).

fof(d1_xcmplx_0,axiom,(
    k1_xcmplx_0 = k5_funct_4(k1_numbers,np__0,np__1,np__0,np__1) )).

fof(d2_xcmplx_0,axiom,(
    ! [A] :
      ( v1_xcmplx_0(A)
    <=> r2_hidden(A,k2_numbers) ) )).

fof(d3_xcmplx_0,axiom,(
    ! [A] :
      ( v1_xcmplx_0(A)
     => ( v1_xboole_0(A)
      <=> A = np__0 ) ) )).

fof(d4_xcmplx_0,axiom,(
    ! [A] :
      ( v1_xcmplx_0(A)
     => ! [B] :
          ( v1_xcmplx_0(B)
         => ! [C] :
              ( C = k2_xcmplx_0(A,B)
            <=> ? [D] :
                  ( m1_subset_1(D,k1_numbers)
                  & ? [E] :
                      ( m1_subset_1(E,k1_numbers)
                      & ? [F] :
                          ( m1_subset_1(F,k1_numbers)
                          & ? [G] :
                              ( m1_subset_1(G,k1_numbers)
                              & A = k5_arytm_0(D,E)
                              & B = k5_arytm_0(F,G)
                              & C = k5_arytm_0(k1_arytm_0(D,F),k1_arytm_0(E,G)) ) ) ) ) ) ) ) )).

fof(d5_xcmplx_0,axiom,(
    ! [A] :
      ( v1_xcmplx_0(A)
     => ! [B] :
          ( v1_xcmplx_0(B)
         => ! [C] :
              ( C = k3_xcmplx_0(A,B)
            <=> ? [D] :
                  ( m1_subset_1(D,k1_numbers)
                  & ? [E] :
                      ( m1_subset_1(E,k1_numbers)
                      & ? [F] :
                          ( m1_subset_1(F,k1_numbers)
                          & ? [G] :
                              ( m1_subset_1(G,k1_numbers)
                              & A = k5_arytm_0(D,E)
                              & B = k5_arytm_0(F,G)
                              & C = k5_arytm_0(k1_arytm_0(k2_arytm_0(D,F),k3_arytm_0(k2_arytm_0(E,G))),k1_arytm_0(k2_arytm_0(D,G),k2_arytm_0(E,F))) ) ) ) ) ) ) ) )).

fof(d6_xcmplx_0,axiom,(
    ! [A] :
      ( v1_xcmplx_0(A)
     => ! [B] :
          ( v1_xcmplx_0(B)
         => ( B = k4_xcmplx_0(A)
          <=> k2_xcmplx_0(A,B) = np__0 ) ) ) )).

fof(d7_xcmplx_0,axiom,(
    ! [A] :
      ( v1_xcmplx_0(A)
     => ! [B] :
          ( v1_xcmplx_0(B)
         => ( ( A != np__0
             => ( B = k5_xcmplx_0(A)
              <=> k3_xcmplx_0(A,B) = np__1 ) )
            & ( A = np__0
             => ( B = k5_xcmplx_0(A)
              <=> B = np__0 ) ) ) ) ) )).

fof(d8_xcmplx_0,axiom,(
    ! [A] :
      ( v1_xcmplx_0(A)
     => ! [B] :
          ( v1_xcmplx_0(B)
         => k6_xcmplx_0(A,B) = k2_xcmplx_0(A,k4_xcmplx_0(B)) ) ) )).

fof(d9_xcmplx_0,axiom,(
    ! [A] :
      ( v1_xcmplx_0(A)
     => ! [B] :
          ( v1_xcmplx_0(B)
         => k7_xcmplx_0(A,B) = k3_xcmplx_0(A,k5_xcmplx_0(B)) ) ) )).

fof(dt_k1_xcmplx_0,axiom,(
    $true )).

fof(dt_k2_xcmplx_0,axiom,(
    $true )).

fof(commutativity_k2_xcmplx_0,axiom,(
    ! [A,B] :
      ( ( v1_xcmplx_0(A)
        & v1_xcmplx_0(B) )
     => k2_xcmplx_0(A,B) = k2_xcmplx_0(B,A) ) )).

fof(dt_k3_xcmplx_0,axiom,(
    $true )).

fof(commutativity_k3_xcmplx_0,axiom,(
    ! [A,B] :
      ( ( v1_xcmplx_0(A)
        & v1_xcmplx_0(B) )
     => k3_xcmplx_0(A,B) = k3_xcmplx_0(B,A) ) )).

fof(dt_k4_xcmplx_0,axiom,(
    ! [A] :
      ( v1_xcmplx_0(A)
     => v1_xcmplx_0(k4_xcmplx_0(A)) ) )).

fof(involutiveness_k4_xcmplx_0,axiom,(
    ! [A] :
      ( v1_xcmplx_0(A)
     => k4_xcmplx_0(k4_xcmplx_0(A)) = A ) )).

fof(dt_k5_xcmplx_0,axiom,(
    ! [A] :
      ( v1_xcmplx_0(A)
     => v1_xcmplx_0(k5_xcmplx_0(A)) ) )).

fof(involutiveness_k5_xcmplx_0,axiom,(
    ! [A] :
      ( v1_xcmplx_0(A)
     => k5_xcmplx_0(k5_xcmplx_0(A)) = A ) )).

fof(dt_k6_xcmplx_0,axiom,(
    $true )).

fof(dt_k7_xcmplx_0,axiom,(
    $true )).
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