SET007 Axioms: SET007+53.ax
%------------------------------------------------------------------------------
% File : SET007+53 : TPTP v8.2.0. 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 unt; 0 def)
% Number of atoms : 119 ( 31 equ)
% Maximal formula atoms : 8 ( 2 avg)
% Number of connectives : 81 ( 10 ~; 1 |; 12 &)
% ( 2 <=>; 56 =>; 0 <=; 0 <~>)
% Maximal formula depth : 11 ( 3 avg)
% Maximal term depth : 5 ( 2 avg)
% Number of predicates : 6 ( 4 usr; 1 prp; 0-2 aty)
% Number of functors : 18 ( 18 usr; 3 con; 0-2 aty)
% Number of variables : 43 ( 43 !; 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(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) ) ).
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