SET007 Axioms: SET007+585.ax
%------------------------------------------------------------------------------
% File : SET007+585 : TPTP v9.0.0. Released v3.4.0.
% Domain : Set Theory
% Axioms : The Ring of Integers, Euclidean Rings and Modulo Integers
% 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 : int_3 [Urb08]
% Status : Satisfiable
% Syntax : Number of formulae : 62 ( 6 unt; 0 def)
% Number of atoms : 495 ( 65 equ)
% Maximal formula atoms : 29 ( 7 avg)
% Number of connectives : 501 ( 68 ~; 7 |; 287 &)
% ( 13 <=>; 126 =>; 0 <=; 0 <~>)
% Maximal formula depth : 23 ( 9 avg)
% Maximal term depth : 4 ( 1 avg)
% Number of predicates : 35 ( 33 usr; 1 prp; 0-3 aty)
% Number of functors : 47 ( 47 usr; 14 con; 0-6 aty)
% Number of variables : 112 ( 101 !; 11 ?)
% 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_int_3,axiom,
( ~ v3_struct_0(k1_int_3)
& v3_vectsp_1(k1_int_3) ) ).
fof(fc2_int_3,axiom,
( ~ v3_struct_0(k1_int_3)
& v2_group_1(k1_int_3)
& v3_vectsp_1(k1_int_3)
& v6_vectsp_1(k1_int_3)
& v8_vectsp_1(k1_int_3) ) ).
fof(fc3_int_3,axiom,
( ~ v3_struct_0(k1_int_3)
& v2_group_1(k1_int_3)
& v4_group_1(k1_int_3)
& v7_group_1(k1_int_3)
& v3_rlvect_1(k1_int_3)
& v4_rlvect_1(k1_int_3)
& v5_rlvect_1(k1_int_3)
& v6_rlvect_1(k1_int_3)
& v3_vectsp_1(k1_int_3)
& v4_vectsp_1(k1_int_3)
& v5_vectsp_1(k1_int_3)
& v6_vectsp_1(k1_int_3)
& v7_vectsp_1(k1_int_3)
& v8_vectsp_1(k1_int_3)
& ~ v10_vectsp_1(k1_int_3)
& v2_vectsp_2(k1_int_3) ) ).
fof(fc4_int_3,axiom,
( ~ v3_struct_0(k1_int_3)
& v2_group_1(k1_int_3)
& v4_group_1(k1_int_3)
& v7_group_1(k1_int_3)
& v3_rlvect_1(k1_int_3)
& v4_rlvect_1(k1_int_3)
& v5_rlvect_1(k1_int_3)
& v6_rlvect_1(k1_int_3)
& v3_vectsp_1(k1_int_3)
& v4_vectsp_1(k1_int_3)
& v5_vectsp_1(k1_int_3)
& v6_vectsp_1(k1_int_3)
& v7_vectsp_1(k1_int_3)
& v8_vectsp_1(k1_int_3)
& ~ v10_vectsp_1(k1_int_3)
& v2_vectsp_2(k1_int_3)
& v1_int_3(k1_int_3) ) ).
fof(rc1_int_3,axiom,
? [A] :
( l3_vectsp_1(A)
& ~ v3_struct_0(A)
& v2_group_1(A)
& v4_group_1(A)
& v7_group_1(A)
& v3_rlvect_1(A)
& v4_rlvect_1(A)
& v5_rlvect_1(A)
& v6_rlvect_1(A)
& v3_vectsp_1(A)
& v4_vectsp_1(A)
& v5_vectsp_1(A)
& v6_vectsp_1(A)
& v7_vectsp_1(A)
& v8_vectsp_1(A)
& ~ v10_vectsp_1(A)
& v2_vectsp_2(A)
& v1_int_3(A) ) ).
fof(rc2_int_3,axiom,
? [A] :
( l3_vectsp_1(A)
& ~ v3_struct_0(A)
& v2_group_1(A)
& v4_group_1(A)
& v7_group_1(A)
& v3_rlvect_1(A)
& v4_rlvect_1(A)
& v5_rlvect_1(A)
& v6_rlvect_1(A)
& v3_vectsp_1(A)
& v4_vectsp_1(A)
& v5_vectsp_1(A)
& v6_vectsp_1(A)
& v7_vectsp_1(A)
& v8_vectsp_1(A)
& ~ v10_vectsp_1(A)
& v2_vectsp_2(A)
& v1_int_3(A) ) ).
fof(cc1_int_3,axiom,
! [A] :
( l3_vectsp_1(A)
=> ( ( ~ v3_struct_0(A)
& v4_group_1(A)
& v7_group_1(A)
& v3_rlvect_1(A)
& v4_rlvect_1(A)
& v5_rlvect_1(A)
& v6_rlvect_1(A)
& v4_vectsp_1(A)
& v6_vectsp_1(A)
& ~ v10_vectsp_1(A)
& v2_vectsp_2(A)
& v1_int_3(A) )
=> ( ~ v3_struct_0(A)
& v2_group_1(A)
& v4_group_1(A)
& v7_group_1(A)
& v3_rlvect_1(A)
& v4_rlvect_1(A)
& v5_rlvect_1(A)
& v6_rlvect_1(A)
& v4_vectsp_1(A)
& v5_vectsp_1(A)
& v6_vectsp_1(A)
& v7_vectsp_1(A)
& v8_vectsp_1(A)
& ~ v10_vectsp_1(A)
& v2_vectsp_2(A)
& v3_gcd_1(A) ) ) ) ).
fof(cc2_int_3,axiom,
! [A] :
( l3_vectsp_1(A)
=> ( ( ~ v3_struct_0(A)
& v4_group_1(A)
& v7_group_1(A)
& v5_rlvect_1(A)
& v8_vectsp_1(A)
& v9_vectsp_1(A) )
=> ( ~ v3_struct_0(A)
& v1_int_3(A) ) ) ) ).
fof(fc5_int_3,axiom,
! [A] :
( v4_ordinal2(A)
=> ( ~ v3_struct_0(k9_int_3(A))
& v3_vectsp_1(k9_int_3(A)) ) ) ).
fof(rc3_int_3,axiom,
? [A] :
( l3_vectsp_1(A)
& ~ v3_struct_0(A)
& v2_group_1(A)
& v4_group_1(A)
& v7_group_1(A)
& v3_rlvect_1(A)
& v4_rlvect_1(A)
& v5_rlvect_1(A)
& v6_rlvect_1(A)
& v3_vectsp_1(A)
& v4_vectsp_1(A)
& v5_vectsp_1(A)
& v6_vectsp_1(A)
& v7_vectsp_1(A)
& v8_vectsp_1(A)
& v9_vectsp_1(A)
& v10_vectsp_1(A)
& v3_gcd_1(A)
& v1_int_3(A) ) ).
fof(fc6_int_3,axiom,
! [A] :
( ( v1_int_2(A)
& m1_subset_1(A,k5_numbers) )
=> ( ~ v3_struct_0(k9_int_3(A))
& v2_group_1(k9_int_3(A))
& v4_group_1(k9_int_3(A))
& v7_group_1(k9_int_3(A))
& v3_rlvect_1(k9_int_3(A))
& v4_rlvect_1(k9_int_3(A))
& v5_rlvect_1(k9_int_3(A))
& v6_rlvect_1(k9_int_3(A))
& v3_vectsp_1(k9_int_3(A))
& v4_vectsp_1(k9_int_3(A))
& v5_vectsp_1(k9_int_3(A))
& v6_vectsp_1(k9_int_3(A))
& v7_vectsp_1(k9_int_3(A))
& v8_vectsp_1(k9_int_3(A))
& v9_vectsp_1(k9_int_3(A))
& ~ v10_vectsp_1(k9_int_3(A))
& v3_gcd_1(k9_int_3(A))
& v1_int_3(k9_int_3(A)) ) ) ).
fof(d1_int_3,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 = k46_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(k35_binop_2,B,C) ) ) ) ) ).
fof(d2_int_3,axiom,
! [A] :
( ( v1_funct_1(A)
& v1_funct_2(A,k4_numbers,k4_numbers)
& m2_relset_1(A,k4_numbers,k4_numbers) )
=> ( A = k43_binop_2
<=> ! [B] :
( m1_subset_1(B,k4_numbers)
=> k8_funct_2(k4_numbers,k4_numbers,A,B) = k1_funct_1(k31_binop_2,B) ) ) ) ).
fof(d3_int_3,axiom,
k1_int_3 = g3_vectsp_1(k4_numbers,k44_binop_2,k46_binop_2,k1_funct_7(np__1,k4_numbers),k1_funct_7(np__0,k4_numbers)) ).
fof(d4_int_3,axiom,
! [A] :
( m1_subset_1(A,u1_struct_0(k1_int_3))
=> ! [B] :
( m1_subset_1(B,u1_struct_0(k1_int_3))
=> ( r1_int_3(A,B)
<=> ? [C] :
( v1_int_1(C)
& ? [D] :
( v1_int_1(D)
& C = A
& D = B
& r1_xreal_0(C,D) ) ) ) ) ) ).
fof(d5_int_3,axiom,
! [A] :
( m1_subset_1(A,u1_struct_0(k1_int_3))
=> ( ( r1_int_3(k1_rlvect_1(k1_int_3),A)
=> k2_int_3(A) = A )
& ( ~ r1_int_3(k1_rlvect_1(k1_int_3),A)
=> k2_int_3(A) = k5_rlvect_1(k1_int_3,A) ) ) ) ).
fof(d6_int_3,axiom,
! [A] :
( ( v1_funct_1(A)
& v1_funct_2(A,u1_struct_0(k1_int_3),k5_numbers)
& m2_relset_1(A,u1_struct_0(k1_int_3),k5_numbers) )
=> ( A = k3_int_3
<=> ! [B] :
( m1_subset_1(B,u1_struct_0(k1_int_3))
=> k8_funct_2(u1_struct_0(k1_int_3),k5_numbers,A,B) = k1_funct_1(k2_euclid,B) ) ) ) ).
fof(t1_int_3,axiom,
! [A] :
( m1_subset_1(A,u1_struct_0(k1_int_3))
=> k8_funct_2(u1_struct_0(k1_int_3),k5_numbers,k3_int_3,A) = k2_int_3(A) ) ).
fof(t2_int_3,axiom,
! [A] :
( m1_subset_1(A,u1_struct_0(k1_int_3))
=> ! [B] :
( m1_subset_1(B,u1_struct_0(k1_int_3))
=> ! [C] :
( m1_subset_1(C,u1_struct_0(k1_int_3))
=> ! [D] :
( m1_subset_1(D,u1_struct_0(k1_int_3))
=> ! [E] :
( m1_subset_1(E,u1_struct_0(k1_int_3))
=> ! [F] :
( m1_subset_1(F,u1_struct_0(k1_int_3))
=> ( ( A = k4_rlvect_1(k1_int_3,k10_group_1(k1_int_3,C,B),E)
& r1_int_3(k1_rlvect_1(k1_int_3),E)
& A = k4_rlvect_1(k1_int_3,k10_group_1(k1_int_3,D,B),F)
& r1_int_3(k1_rlvect_1(k1_int_3),F) )
=> ( B = k1_rlvect_1(k1_int_3)
| r1_int_3(k2_int_3(B),E)
| r1_int_3(k2_int_3(B),F)
| ( C = D
& E = F ) ) ) ) ) ) ) ) ) ).
fof(d7_int_3,axiom,
! [A] :
( m1_subset_1(A,u1_struct_0(k1_int_3))
=> ! [B] :
( m1_subset_1(B,u1_struct_0(k1_int_3))
=> ( B != k1_rlvect_1(k1_int_3)
=> ! [C] :
( m1_subset_1(C,u1_struct_0(k1_int_3))
=> ( C = k4_int_3(A,B)
<=> ? [D] :
( m1_subset_1(D,u1_struct_0(k1_int_3))
& A = k4_rlvect_1(k1_int_3,k10_group_1(k1_int_3,C,B),D)
& r1_int_3(k1_rlvect_1(k1_int_3),D)
& ~ r1_int_3(k2_int_3(B),D) ) ) ) ) ) ) ).
fof(d8_int_3,axiom,
! [A] :
( m1_subset_1(A,u1_struct_0(k1_int_3))
=> ! [B] :
( m1_subset_1(B,u1_struct_0(k1_int_3))
=> ( B != k1_rlvect_1(k1_int_3)
=> ! [C] :
( m1_subset_1(C,u1_struct_0(k1_int_3))
=> ( C = k5_int_3(A,B)
<=> ? [D] :
( m1_subset_1(D,u1_struct_0(k1_int_3))
& A = k4_rlvect_1(k1_int_3,k10_group_1(k1_int_3,D,B),C)
& r1_int_3(k1_rlvect_1(k1_int_3),C)
& ~ r1_int_3(k2_int_3(B),C) ) ) ) ) ) ) ).
fof(t3_int_3,axiom,
! [A] :
( m1_subset_1(A,u1_struct_0(k1_int_3))
=> ! [B] :
( m1_subset_1(B,u1_struct_0(k1_int_3))
=> ( B != k1_rlvect_1(k1_int_3)
=> A = k4_rlvect_1(k1_int_3,k10_group_1(k1_int_3,k4_int_3(A,B),B),k5_int_3(A,B)) ) ) ) ).
fof(d9_int_3,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l3_vectsp_1(A) )
=> ( v1_int_3(A)
<=> ? [B] :
( v1_funct_1(B)
& v1_funct_2(B,u1_struct_0(A),k5_numbers)
& m2_relset_1(B,u1_struct_0(A),k5_numbers)
& ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> ! [D] :
( m1_subset_1(D,u1_struct_0(A))
=> ~ ( D != k1_rlvect_1(A)
& ! [E] :
( m1_subset_1(E,u1_struct_0(A))
=> ! [F] :
( m1_subset_1(F,u1_struct_0(A))
=> ~ ( C = k2_rlvect_1(A,k1_group_1(A,E,D),F)
& ~ ( F != k1_rlvect_1(A)
& r1_xreal_0(k8_funct_2(u1_struct_0(A),k5_numbers,B,D),k8_funct_2(u1_struct_0(A),k5_numbers,B,F)) ) ) ) ) ) ) ) ) ) ) ).
fof(d10_int_3,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v1_int_3(A)
& l3_vectsp_1(A) )
=> ! [B] :
( ( v1_funct_1(B)
& v1_funct_2(B,u1_struct_0(A),k5_numbers)
& m2_relset_1(B,u1_struct_0(A),k5_numbers) )
=> ( m1_int_3(B,A)
<=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> ! [D] :
( m1_subset_1(D,u1_struct_0(A))
=> ~ ( D != k1_rlvect_1(A)
& ! [E] :
( m1_subset_1(E,u1_struct_0(A))
=> ! [F] :
( m1_subset_1(F,u1_struct_0(A))
=> ~ ( C = k2_rlvect_1(A,k1_group_1(A,E,D),F)
& ~ ( F != k1_rlvect_1(A)
& r1_xreal_0(k8_funct_2(u1_struct_0(A),k5_numbers,B,D),k8_funct_2(u1_struct_0(A),k5_numbers,B,F)) ) ) ) ) ) ) ) ) ) ) ).
fof(t4_int_3,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v4_group_1(A)
& v7_group_1(A)
& v3_rlvect_1(A)
& v4_rlvect_1(A)
& v5_rlvect_1(A)
& v6_rlvect_1(A)
& v6_vectsp_1(A)
& v7_vectsp_1(A)
& v8_vectsp_1(A)
& ~ v10_vectsp_1(A)
& v2_vectsp_2(A)
& v1_int_3(A)
& l3_vectsp_1(A) )
=> ( ~ v3_struct_0(A)
& v4_group_1(A)
& v7_group_1(A)
& v3_rlvect_1(A)
& v4_rlvect_1(A)
& v5_rlvect_1(A)
& v6_rlvect_1(A)
& v6_vectsp_1(A)
& v7_vectsp_1(A)
& v8_vectsp_1(A)
& ~ v10_vectsp_1(A)
& v2_vectsp_2(A)
& v3_gcd_1(A)
& l3_vectsp_1(A) ) ) ).
fof(t5_int_3,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v4_group_1(A)
& v7_group_1(A)
& v5_rlvect_1(A)
& v8_vectsp_1(A)
& v9_vectsp_1(A)
& l3_vectsp_1(A) )
=> v1_int_3(A) ) ).
fof(t6_int_3,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v4_group_1(A)
& v7_group_1(A)
& v5_rlvect_1(A)
& v8_vectsp_1(A)
& v9_vectsp_1(A)
& l3_vectsp_1(A) )
=> ! [B] :
( ( v1_funct_1(B)
& v1_funct_2(B,u1_struct_0(A),k5_numbers)
& m2_relset_1(B,u1_struct_0(A),k5_numbers) )
=> m1_int_3(B,A) ) ) ).
fof(t7_int_3,axiom,
$true ).
fof(t8_int_3,axiom,
! [A] :
( v1_int_1(A)
=> ! [B] :
( v1_int_1(B)
=> ! [C] :
( v1_int_1(C)
=> ( ( A != np__0
=> k5_int_1(k2_xcmplx_0(B,k3_xcmplx_0(A,C)),A) = k2_xcmplx_0(k5_int_1(B,A),C) )
& k6_int_1(k2_xcmplx_0(B,k3_xcmplx_0(A,C)),A) = k6_int_1(B,A) ) ) ) ) ).
fof(t9_int_3,axiom,
! [A] :
( v4_ordinal2(A)
=> ( ~ r1_xreal_0(A,np__0)
=> ! [B] :
( v1_int_1(B)
=> ( r1_xreal_0(np__0,k6_int_1(B,A))
& ~ r1_xreal_0(A,k6_int_1(B,A)) ) ) ) ) ).
fof(t10_int_3,axiom,
! [A] :
( v1_int_1(A)
=> ! [B] :
( v1_int_1(B)
=> ( ( r1_xreal_0(np__0,B)
=> ( r1_xreal_0(A,B)
| k6_int_1(B,A) = B ) )
& ( r1_xreal_0(k4_xcmplx_0(A),B)
=> ( r1_xreal_0(np__0,B)
| k6_int_1(B,A) = k2_xcmplx_0(A,B) ) ) ) ) ) ).
fof(t11_int_3,axiom,
! [A] :
( v4_ordinal2(A)
=> ( ~ r1_xreal_0(A,np__0)
=> ! [B] :
( v1_int_1(B)
=> ( k6_int_1(B,A) = np__0
<=> r2_int_1(A,B) ) ) ) ) ).
fof(t12_int_3,axiom,
! [A] :
( v1_int_1(A)
=> ! [B] :
( v1_int_1(B)
=> ! [C] :
( v1_int_1(C)
=> ( ( k6_int_1(B,A) = k6_int_1(C,A)
=> ( A = np__0
| r1_int_1(B,C,A) ) )
& ( r1_int_1(B,C,A)
=> k6_int_1(B,A) = k6_int_1(C,A) ) ) ) ) ) ).
fof(t13_int_3,axiom,
! [A] :
( v4_ordinal2(A)
=> ! [B] :
( v1_int_1(B)
=> k6_int_1(k6_int_1(B,A),A) = k6_int_1(B,A) ) ) ).
fof(t14_int_3,axiom,
! [A] :
( v1_int_1(A)
=> ! [B] :
( v1_int_1(B)
=> ! [C] :
( v1_int_1(C)
=> k6_int_1(k2_xcmplx_0(B,C),A) = k6_int_1(k2_xcmplx_0(k6_int_1(B,A),k6_int_1(C,A)),A) ) ) ) ).
fof(t15_int_3,axiom,
! [A] :
( v1_int_1(A)
=> ! [B] :
( v1_int_1(B)
=> ! [C] :
( v1_int_1(C)
=> k6_int_1(k3_xcmplx_0(B,C),A) = k6_int_1(k3_xcmplx_0(k6_int_1(B,A),k6_int_1(C,A)),A) ) ) ) ).
fof(t16_int_3,axiom,
! [A] :
( v1_int_1(A)
=> ! [B] :
( v1_int_1(B)
=> ? [C] :
( v1_int_1(C)
& ? [D] :
( v1_int_1(D)
& k3_int_2(A,B) = k2_xcmplx_0(k3_xcmplx_0(C,A),k3_xcmplx_0(D,B)) ) ) ) ) ).
fof(d11_int_3,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 = k7_int_3(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(k2_nat_1(C,D),A) ) ) ) ) ) ) ).
fof(d12_int_3,axiom,
! [A] :
( v4_ordinal2(A)
=> ( ~ r1_xreal_0(A,np__0)
=> ! [B] :
( ( v1_funct_1(B)
& v1_funct_2(B,k1_gr_cy_1(A),k1_gr_cy_1(A))
& m2_relset_1(B,k1_gr_cy_1(A),k1_gr_cy_1(A)) )
=> ( B = k8_int_3(A)
<=> ! [C] :
( m2_subset_1(C,k5_numbers,k1_gr_cy_1(A))
=> k8_funct_2(k1_gr_cy_1(A),k1_gr_cy_1(A),B,C) = k6_int_1(k6_xcmplx_0(A,C),A) ) ) ) ) ) ).
fof(t17_int_3,axiom,
! [A] :
( v4_ordinal2(A)
=> ( ~ r1_xreal_0(A,np__0)
=> ! [B] :
( m2_subset_1(B,k5_numbers,k1_gr_cy_1(A))
=> ! [C] :
( m2_subset_1(C,k5_numbers,k1_gr_cy_1(A))
=> ( ( ~ r1_xreal_0(A,k1_nat_1(B,C))
=> k2_binop_1(k1_gr_cy_1(A),k1_gr_cy_1(A),k1_gr_cy_1(A),k4_gr_cy_1(A),B,C) = k1_nat_1(B,C) )
& ~ ( k2_binop_1(k1_gr_cy_1(A),k1_gr_cy_1(A),k1_gr_cy_1(A),k4_gr_cy_1(A),B,C) = k1_nat_1(B,C)
& r1_xreal_0(A,k1_nat_1(B,C)) )
& ( r1_xreal_0(A,k1_nat_1(B,C))
=> k2_binop_1(k1_gr_cy_1(A),k1_gr_cy_1(A),k1_gr_cy_1(A),k4_gr_cy_1(A),B,C) = k6_xcmplx_0(k1_nat_1(B,C),A) )
& ( k2_binop_1(k1_gr_cy_1(A),k1_gr_cy_1(A),k1_gr_cy_1(A),k4_gr_cy_1(A),B,C) = k6_xcmplx_0(k1_nat_1(B,C),A)
=> r1_xreal_0(A,k1_nat_1(B,C)) ) ) ) ) ) ) ).
fof(t18_int_3,axiom,
! [A] :
( v4_ordinal2(A)
=> ( ~ r1_xreal_0(A,np__0)
=> ! [B] :
( m2_subset_1(B,k5_numbers,k1_gr_cy_1(A))
=> ! [C] :
( m2_subset_1(C,k5_numbers,k1_gr_cy_1(A))
=> ! [D] :
( v4_ordinal2(D)
=> ( ( r1_xreal_0(k3_xcmplx_0(D,A),k2_nat_1(B,C))
& ~ r1_xreal_0(k3_xcmplx_0(k2_xcmplx_0(D,np__1),A),k2_nat_1(B,C)) )
<=> k2_binop_1(k1_gr_cy_1(A),k1_gr_cy_1(A),k1_gr_cy_1(A),k7_int_3(A),B,C) = k6_xcmplx_0(k2_nat_1(B,C),k3_xcmplx_0(D,A)) ) ) ) ) ) ) ).
fof(t19_int_3,axiom,
! [A] :
( v4_ordinal2(A)
=> ( ~ r1_xreal_0(A,np__0)
=> ! [B] :
( m2_subset_1(B,k5_numbers,k1_gr_cy_1(A))
=> ( ( B = np__0
=> k8_funct_2(k1_gr_cy_1(A),k1_gr_cy_1(A),k8_int_3(A),B) = np__0 )
& ( k8_funct_2(k1_gr_cy_1(A),k1_gr_cy_1(A),k8_int_3(A),B) = np__0
=> B = np__0 )
& ( B != np__0
=> k8_funct_2(k1_gr_cy_1(A),k1_gr_cy_1(A),k8_int_3(A),B) = k6_xcmplx_0(A,B) )
& ~ ( k8_funct_2(k1_gr_cy_1(A),k1_gr_cy_1(A),k8_int_3(A),B) = k6_xcmplx_0(A,B)
& B = np__0 ) ) ) ) ) ).
fof(d13_int_3,axiom,
! [A] :
( v4_ordinal2(A)
=> k9_int_3(A) = g3_vectsp_1(k1_gr_cy_1(A),k4_gr_cy_1(A),k7_int_3(A),k1_funct_7(np__1,k1_gr_cy_1(A)),k1_funct_7(np__0,k1_gr_cy_1(A))) ) ).
fof(t20_int_3,axiom,
( v10_vectsp_1(k9_int_3(np__1))
& ~ v3_struct_0(k9_int_3(np__1))
& v4_group_1(k9_int_3(np__1))
& v3_rlvect_1(k9_int_3(np__1))
& v4_rlvect_1(k9_int_3(np__1))
& v5_rlvect_1(k9_int_3(np__1))
& v6_rlvect_1(k9_int_3(np__1))
& v6_vectsp_1(k9_int_3(np__1))
& v7_vectsp_1(k9_int_3(np__1))
& v8_vectsp_1(k9_int_3(np__1))
& l3_vectsp_1(k9_int_3(np__1))
& v9_vectsp_1(k9_int_3(np__1))
& v2_group_1(k9_int_3(np__1))
& v7_vectsp_1(k9_int_3(np__1))
& v7_group_1(k9_int_3(np__1)) ) ).
fof(t21_int_3,axiom,
! [A] :
( v4_ordinal2(A)
=> ( ~ r1_xreal_0(A,np__1)
=> ( ~ v10_vectsp_1(k9_int_3(A))
& ~ v3_struct_0(k9_int_3(A))
& v2_group_1(k9_int_3(A))
& v4_group_1(k9_int_3(A))
& v7_group_1(k9_int_3(A))
& v3_rlvect_1(k9_int_3(A))
& v4_rlvect_1(k9_int_3(A))
& v5_rlvect_1(k9_int_3(A))
& v6_rlvect_1(k9_int_3(A))
& v6_vectsp_1(k9_int_3(A))
& v7_vectsp_1(k9_int_3(A))
& v8_vectsp_1(k9_int_3(A))
& l3_vectsp_1(k9_int_3(A)) ) ) ) ).
fof(t22_int_3,axiom,
! [A] :
( v4_ordinal2(A)
=> ( ~ r1_xreal_0(A,np__1)
=> ( ( ~ v3_struct_0(k9_int_3(A))
& v4_group_1(k9_int_3(A))
& v7_group_1(k9_int_3(A))
& v3_rlvect_1(k9_int_3(A))
& v4_rlvect_1(k9_int_3(A))
& v5_rlvect_1(k9_int_3(A))
& v6_rlvect_1(k9_int_3(A))
& v7_vectsp_1(k9_int_3(A))
& v8_vectsp_1(k9_int_3(A))
& v9_vectsp_1(k9_int_3(A))
& ~ v10_vectsp_1(k9_int_3(A))
& l3_vectsp_1(k9_int_3(A)) )
<=> ( v1_int_2(A)
& m2_subset_1(A,k1_numbers,k5_numbers) ) ) ) ) ).
fof(t23_int_3,axiom,
k2_group_1(k1_int_3) = np__1 ).
fof(t24_int_3,axiom,
! [A] :
( v4_ordinal2(A)
=> ( ~ r1_xreal_0(A,np__1)
=> k2_group_1(k9_int_3(A)) = np__1 ) ) ).
fof(dt_m1_int_3,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v1_int_3(A)
& l3_vectsp_1(A) )
=> ! [B] :
( m1_int_3(B,A)
=> ( v1_funct_1(B)
& v1_funct_2(B,u1_struct_0(A),k5_numbers)
& m2_relset_1(B,u1_struct_0(A),k5_numbers) ) ) ) ).
fof(existence_m1_int_3,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v1_int_3(A)
& l3_vectsp_1(A) )
=> ? [B] : m1_int_3(B,A) ) ).
fof(reflexivity_r1_int_3,axiom,
! [A,B] :
( ( m1_subset_1(A,u1_struct_0(k1_int_3))
& m1_subset_1(B,u1_struct_0(k1_int_3)) )
=> r1_int_3(A,A) ) ).
fof(connectedness_r1_int_3,axiom,
! [A,B] :
( ( m1_subset_1(A,u1_struct_0(k1_int_3))
& m1_subset_1(B,u1_struct_0(k1_int_3)) )
=> ( r1_int_3(A,B)
| r1_int_3(B,A) ) ) ).
fof(dt_k1_int_3,axiom,
l3_vectsp_1(k1_int_3) ).
fof(dt_k2_int_3,axiom,
! [A] :
( m1_subset_1(A,u1_struct_0(k1_int_3))
=> m1_subset_1(k2_int_3(A),u1_struct_0(k1_int_3)) ) ).
fof(dt_k3_int_3,axiom,
( v1_funct_1(k3_int_3)
& v1_funct_2(k3_int_3,u1_struct_0(k1_int_3),k5_numbers)
& m2_relset_1(k3_int_3,u1_struct_0(k1_int_3),k5_numbers) ) ).
fof(dt_k4_int_3,axiom,
! [A,B] :
( ( m1_subset_1(A,u1_struct_0(k1_int_3))
& m1_subset_1(B,u1_struct_0(k1_int_3)) )
=> m1_subset_1(k4_int_3(A,B),u1_struct_0(k1_int_3)) ) ).
fof(dt_k5_int_3,axiom,
! [A,B] :
( ( m1_subset_1(A,u1_struct_0(k1_int_3))
& m1_subset_1(B,u1_struct_0(k1_int_3)) )
=> m1_subset_1(k5_int_3(A,B),u1_struct_0(k1_int_3)) ) ).
fof(dt_k6_int_3,axiom,
m1_int_3(k6_int_3,k1_int_3) ).
fof(redefinition_k6_int_3,axiom,
k6_int_3 = k3_int_3 ).
fof(dt_k7_int_3,axiom,
! [A] :
( v4_ordinal2(A)
=> ( v1_funct_1(k7_int_3(A))
& v1_funct_2(k7_int_3(A),k2_zfmisc_1(k1_gr_cy_1(A),k1_gr_cy_1(A)),k1_gr_cy_1(A))
& m2_relset_1(k7_int_3(A),k2_zfmisc_1(k1_gr_cy_1(A),k1_gr_cy_1(A)),k1_gr_cy_1(A)) ) ) ).
fof(dt_k8_int_3,axiom,
! [A] :
( v4_ordinal2(A)
=> ( v1_funct_1(k8_int_3(A))
& v1_funct_2(k8_int_3(A),k1_gr_cy_1(A),k1_gr_cy_1(A))
& m2_relset_1(k8_int_3(A),k1_gr_cy_1(A),k1_gr_cy_1(A)) ) ) ).
fof(dt_k9_int_3,axiom,
! [A] :
( v4_ordinal2(A)
=> l3_vectsp_1(k9_int_3(A)) ) ).
%------------------------------------------------------------------------------