SET007 Axioms: SET007+53.ax


%------------------------------------------------------------------------------
% File     : SET007+53 : TPTP v7.5.0. Released v3.4.0.
% Domain   : Set Theory
% Axioms   : Some Properties of Function Modul and Signum
% 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    : absvalue [Urb08]

% Status   : Satisfiable
% Syntax   : Number of formulae    :   48 (  18 unit)
%            Number of atoms       :  119 (  31 equality)
%            Maximal formula depth :   11 (   3 average)
%            Number of connectives :   81 (  10 ~  ;   1  |;  12  &)
%                                         (   2 <=>;  56 =>;   0 <=)
%                                         (   0 <~>;   0 ~|;   0 ~&)
%            Number of predicates  :    6 (   1 propositional; 0-2 arity)
%            Number of functors    :   18 (   3 constant; 0-2 arity)
%            Number of variables   :   43 (   0 singleton;  43 !;   0 ?)
%            Maximal term depth    :    5 (   2 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_absvalue,axiom,(
    ! [A] :
      ( v1_xreal_0(A)
     => ( v1_xcmplx_0(k1_absvalue(A))
        & v1_xreal_0(k1_absvalue(A)) ) ) )).

fof(d1_absvalue,axiom,(
    ! [A] :
      ( v1_xreal_0(A)
     => ( ( r1_xreal_0(np__0,A)
         => k16_complex1(A) = A )
        & ( ~ r1_xreal_0(np__0,A)
         => k16_complex1(A) = k4_xcmplx_0(A) ) ) ) )).

fof(t1_absvalue,axiom,(
    ! [A] :
      ( v1_xreal_0(A)
     => ( k18_complex1(A) = A
        | k18_complex1(A) = k4_xcmplx_0(A) ) ) )).

fof(t2_absvalue,axiom,(
    $true )).

fof(t3_absvalue,axiom,(
    $true )).

fof(t4_absvalue,axiom,(
    $true )).

fof(t5_absvalue,axiom,(
    $true )).

fof(t6_absvalue,axiom,(
    $true )).

fof(t7_absvalue,axiom,(
    ! [A] :
      ( v1_xreal_0(A)
     => ( A = np__0
      <=> k18_complex1(A) = np__0 ) ) )).

fof(t8_absvalue,axiom,(
    $true )).

fof(t9_absvalue,axiom,(
    ! [A] :
      ( v1_xreal_0(A)
     => ~ ( k18_complex1(A) = k4_xcmplx_0(A)
          & A != np__0
          & r1_xreal_0(np__0,A) ) ) )).

fof(t10_absvalue,axiom,(
    $true )).

fof(t11_absvalue,axiom,(
    ! [A] :
      ( v1_xreal_0(A)
     => ( r1_xreal_0(k1_real_1(k18_complex1(A)),A)
        & r1_xreal_0(A,k18_complex1(A)) ) ) )).

fof(t12_absvalue,axiom,(
    ! [A] :
      ( v1_xreal_0(A)
     => ! [B] :
          ( v1_xreal_0(B)
         => ( ( r1_xreal_0(k4_xcmplx_0(A),B)
              & r1_xreal_0(B,A) )
          <=> r1_xreal_0(k18_complex1(B),A) ) ) ) )).

fof(t13_absvalue,axiom,(
    $true )).

fof(t14_absvalue,axiom,(
    ! [A] :
      ( v1_xreal_0(A)
     => ( A != np__0
       => k4_real_1(k18_complex1(A),k18_complex1(k7_xcmplx_0(np__1,A))) = np__1 ) ) )).

fof(t15_absvalue,axiom,(
    ! [A] :
      ( v1_xreal_0(A)
     => k18_complex1(k7_xcmplx_0(np__1,A)) = k6_real_1(np__1,k18_complex1(A)) ) )).

fof(t16_absvalue,axiom,(
    $true )).

fof(t17_absvalue,axiom,(
    $true )).

fof(t18_absvalue,axiom,(
    $true )).

fof(t19_absvalue,axiom,(
    $true )).

fof(t20_absvalue,axiom,(
    ! [A] :
      ( v1_xreal_0(A)
     => ! [B] :
          ( v1_xreal_0(B)
         => ( r1_xreal_0(np__0,k3_xcmplx_0(A,B))
           => k8_square_1(k3_xcmplx_0(A,B)) = k4_real_1(k9_square_1(k18_complex1(A)),k9_square_1(k18_complex1(B))) ) ) ) )).

fof(t21_absvalue,axiom,(
    ! [A] :
      ( v1_xreal_0(A)
     => ! [B] :
          ( v1_xreal_0(B)
         => ! [C] :
              ( v1_xreal_0(C)
             => ! [D] :
                  ( v1_xreal_0(D)
                 => ( ( r1_xreal_0(k18_complex1(A),B)
                      & r1_xreal_0(k18_complex1(C),D) )
                   => r1_xreal_0(k18_complex1(k2_xcmplx_0(A,C)),k2_xcmplx_0(B,D)) ) ) ) ) ) )).

fof(t22_absvalue,axiom,(
    $true )).

fof(t23_absvalue,axiom,(
    ! [A] :
      ( v1_xreal_0(A)
     => ! [B] :
          ( v1_xreal_0(B)
         => ( ~ r1_xreal_0(k7_xcmplx_0(A,B),np__0)
           => k8_square_1(k7_xcmplx_0(A,B)) = k6_real_1(k9_square_1(k18_complex1(A)),k9_square_1(k18_complex1(B))) ) ) ) )).

fof(t24_absvalue,axiom,(
    ! [A] :
      ( v1_xreal_0(A)
     => ! [B] :
          ( v1_xreal_0(B)
         => ( r1_xreal_0(np__0,k3_xcmplx_0(A,B))
           => k18_complex1(k2_xcmplx_0(A,B)) = k3_real_1(k18_complex1(A),k18_complex1(B)) ) ) ) )).

fof(t25_absvalue,axiom,(
    ! [A] :
      ( v1_xreal_0(A)
     => ! [B] :
          ( v1_xreal_0(B)
         => ( k18_complex1(k2_xcmplx_0(A,B)) = k3_real_1(k18_complex1(A),k18_complex1(B))
           => r1_xreal_0(np__0,k3_xcmplx_0(A,B)) ) ) ) )).

fof(t26_absvalue,axiom,(
    ! [A] :
      ( v1_xreal_0(A)
     => ! [B] :
          ( v1_xreal_0(B)
         => r1_xreal_0(k6_real_1(k18_complex1(k2_xcmplx_0(A,B)),k3_real_1(np__1,k18_complex1(k2_xcmplx_0(A,B)))),k3_real_1(k6_real_1(k18_complex1(A),k3_real_1(np__1,k18_complex1(A))),k6_real_1(k18_complex1(B),k3_real_1(np__1,k18_complex1(B))))) ) ) )).

fof(d2_absvalue,axiom,(
    ! [A] :
      ( v1_xreal_0(A)
     => ( ( ~ r1_xreal_0(A,np__0)
         => k1_absvalue(A) = np__1 )
        & ( ~ r1_xreal_0(np__0,A)
         => k1_absvalue(A) = k1_real_1(np__1) )
        & ( ( r1_xreal_0(A,np__0)
            & r1_xreal_0(np__0,A) )
         => k1_absvalue(A) = np__0 ) ) ) )).

fof(t27_absvalue,axiom,(
    $true )).

fof(t28_absvalue,axiom,(
    $true )).

fof(t29_absvalue,axiom,(
    $true )).

fof(t30_absvalue,axiom,(
    $true )).

fof(t31_absvalue,axiom,(
    ! [A] :
      ( v1_xreal_0(A)
     => ~ ( k1_absvalue(A) = np__1
          & r1_xreal_0(A,np__0) ) ) )).

fof(t32_absvalue,axiom,(
    ! [A] :
      ( v1_xreal_0(A)
     => ~ ( k1_absvalue(A) = k1_real_1(np__1)
          & r1_xreal_0(np__0,A) ) ) )).

fof(t33_absvalue,axiom,(
    ! [A] :
      ( v1_xreal_0(A)
     => ( k1_absvalue(A) = np__0
       => A = np__0 ) ) )).

fof(t34_absvalue,axiom,(
    ! [A] :
      ( v1_xreal_0(A)
     => A = k3_xcmplx_0(k18_complex1(A),k1_absvalue(A)) ) )).

fof(t35_absvalue,axiom,(
    ! [A] :
      ( v1_xreal_0(A)
     => ! [B] :
          ( v1_xreal_0(B)
         => k1_absvalue(k3_xcmplx_0(A,B)) = k3_xcmplx_0(k1_absvalue(A),k1_absvalue(B)) ) ) )).

fof(t36_absvalue,axiom,(
    ! [A] :
      ( v1_xreal_0(A)
     => k1_absvalue(k1_absvalue(A)) = k1_absvalue(A) ) )).

fof(t37_absvalue,axiom,(
    ! [A] :
      ( v1_xreal_0(A)
     => ! [B] :
          ( v1_xreal_0(B)
         => r1_xreal_0(k1_absvalue(k2_xcmplx_0(A,B)),k2_xcmplx_0(k2_xcmplx_0(k1_absvalue(A),k1_absvalue(B)),np__1)) ) ) )).

fof(t38_absvalue,axiom,(
    ! [A] :
      ( v1_xreal_0(A)
     => ( A != np__0
       => k3_xcmplx_0(k1_absvalue(A),k1_absvalue(k7_xcmplx_0(np__1,A))) = np__1 ) ) )).

fof(t39_absvalue,axiom,(
    ! [A] :
      ( v1_xreal_0(A)
     => k7_xcmplx_0(np__1,k1_absvalue(A)) = k1_absvalue(k7_xcmplx_0(np__1,A)) ) )).

fof(t40_absvalue,axiom,(
    ! [A] :
      ( v1_xreal_0(A)
     => ! [B] :
          ( v1_xreal_0(B)
         => r1_xreal_0(k6_xcmplx_0(k2_xcmplx_0(k1_absvalue(A),k1_absvalue(B)),np__1),k1_absvalue(k2_xcmplx_0(A,B))) ) ) )).

fof(t41_absvalue,axiom,(
    ! [A] :
      ( v1_xreal_0(A)
     => k1_absvalue(A) = k1_absvalue(k7_xcmplx_0(np__1,A)) ) )).

fof(t42_absvalue,axiom,(
    ! [A] :
      ( v1_xreal_0(A)
     => ! [B] :
          ( v1_xreal_0(B)
         => k1_absvalue(k7_xcmplx_0(A,B)) = k7_xcmplx_0(k1_absvalue(A),k1_absvalue(B)) ) ) )).

fof(dt_k1_absvalue,axiom,(
    $true )).

fof(dt_k2_absvalue,axiom,(
    ! [A] :
      ( m1_subset_1(A,k1_numbers)
     => m1_subset_1(k2_absvalue(A),k1_numbers) ) )).

fof(redefinition_k2_absvalue,axiom,(
    ! [A] :
      ( m1_subset_1(A,k1_numbers)
     => k2_absvalue(A) = k1_absvalue(A) ) )).
%------------------------------------------------------------------------------