SET007 Axioms: SET007+317.ax
%------------------------------------------------------------------------------
% File : SET007+317 : TPTP v8.2.0. Released v3.4.0.
% Domain : Set Theory
% Axioms : Metric Spaces as Topological Spaces - Fundamental Concepts
% 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 : topmetr [Urb08]
% Status : Satisfiable
% Syntax : Number of formulae : 49 ( 7 unt; 0 def)
% Number of atoms : 353 ( 33 equ)
% Maximal formula atoms : 16 ( 7 avg)
% Number of connectives : 341 ( 37 ~; 0 |; 166 &)
% ( 9 <=>; 129 =>; 0 <=; 0 <~>)
% Maximal formula depth : 16 ( 8 avg)
% Maximal term depth : 4 ( 1 avg)
% Number of predicates : 36 ( 34 usr; 1 prp; 0-3 aty)
% Number of functors : 35 ( 35 usr; 8 con; 0-5 aty)
% Number of variables : 119 ( 114 !; 5 ?)
% 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_topmetr,axiom,
! [A] :
( ( v6_metric_1(A)
& v7_metric_1(A)
& v8_metric_1(A)
& v9_metric_1(A)
& l1_metric_1(A) )
=> ? [B] :
( m1_topmetr(B,A)
& v1_metric_1(B)
& v6_metric_1(B)
& v7_metric_1(B)
& v8_metric_1(B)
& v9_metric_1(B) ) ) ).
fof(rc2_topmetr,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v6_metric_1(A)
& v7_metric_1(A)
& v8_metric_1(A)
& v9_metric_1(A)
& l1_metric_1(A) )
=> ? [B] :
( m1_topmetr(B,A)
& ~ v3_struct_0(B)
& v1_metric_1(B)
& v6_metric_1(B)
& v7_metric_1(B)
& v8_metric_1(B)
& v9_metric_1(B) ) ) ).
fof(fc1_topmetr,axiom,
! [A,B] :
( ( ~ v3_struct_0(A)
& v6_metric_1(A)
& v7_metric_1(A)
& v8_metric_1(A)
& v9_metric_1(A)
& l1_metric_1(A)
& ~ v1_xboole_0(B)
& m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) )
=> ( ~ v3_struct_0(k1_topmetr(A,B))
& v1_metric_1(k1_topmetr(A,B))
& v6_metric_1(k1_topmetr(A,B))
& v7_metric_1(k1_topmetr(A,B))
& v8_metric_1(k1_topmetr(A,B))
& v9_metric_1(k1_topmetr(A,B)) ) ) ).
fof(fc2_topmetr,axiom,
( ~ v3_struct_0(k3_topmetr)
& v1_pre_topc(k3_topmetr)
& v2_pre_topc(k3_topmetr) ) ).
fof(fc3_topmetr,axiom,
! [A,B,C] :
( ( v1_funct_1(B)
& m1_relset_1(B,A,u1_struct_0(k3_topmetr)) )
=> ( v1_xcmplx_0(k1_funct_1(B,C))
& v1_xreal_0(k1_funct_1(B,C)) ) ) ).
fof(t1_topmetr,axiom,
! [A] :
( l1_pre_topc(A)
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(k1_zfmisc_1(u1_struct_0(A))))
=> ( r1_pre_topc(A,B)
<=> r1_tarski(u1_struct_0(A),k5_setfam_1(u1_struct_0(A),B)) ) ) ) ).
fof(t2_topmetr,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_pre_topc(A)
& l1_pre_topc(A) )
=> ! [B] :
( ( ~ v3_struct_0(B)
& m1_pre_topc(B,A) )
=> ! [C] :
( m1_subset_1(C,u1_struct_0(B))
=> m1_subset_1(C,u1_struct_0(A)) ) ) ) ).
fof(t3_topmetr,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_pre_topc(A)
& l1_pre_topc(A) )
=> ! [B] :
( ( ~ v3_struct_0(B)
& m1_pre_topc(B,A) )
=> ( v3_compts_1(A)
=> v3_compts_1(B) ) ) ) ).
fof(t4_topmetr,axiom,
! [A] :
( ( v2_pre_topc(A)
& l1_pre_topc(A) )
=> ! [B] :
( m1_pre_topc(B,A)
=> ! [C] :
( m1_pre_topc(C,A)
=> ( r1_tarski(u1_struct_0(B),u1_struct_0(C))
=> m1_pre_topc(B,C) ) ) ) ) ).
fof(t5_topmetr,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_pre_topc(A)
& l1_pre_topc(A) )
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A)))
=> ( m1_pre_topc(k3_pre_topc(A,B),k3_pre_topc(A,k4_subset_1(u1_struct_0(A),B,C)))
& m1_pre_topc(k3_pre_topc(A,C),k3_pre_topc(A,k4_subset_1(u1_struct_0(A),B,C))) ) ) ) ) ).
fof(t6_topmetr,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_pre_topc(A)
& l1_pre_topc(A) )
=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> ! [C] :
( ( ~ v1_xboole_0(C)
& m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A))) )
=> ( r2_hidden(B,C)
=> ! [D] :
( m1_connsp_2(D,A,B)
=> ! [E] :
( m1_subset_1(E,u1_struct_0(k3_pre_topc(A,C)))
=> ! [F] :
( m1_subset_1(F,k1_zfmisc_1(u1_struct_0(k3_pre_topc(A,C))))
=> ( ( F = k5_subset_1(u1_struct_0(A),D,C)
& E = B )
=> m1_connsp_2(F,k3_pre_topc(A,C),E) ) ) ) ) ) ) ) ) ).
fof(t7_topmetr,axiom,
! [A] :
( ( v2_pre_topc(A)
& l1_pre_topc(A) )
=> ! [B] :
( ( v2_pre_topc(B)
& l1_pre_topc(B) )
=> ! [C] :
( ( v2_pre_topc(C)
& l1_pre_topc(C) )
=> ! [D] :
( ( v1_funct_1(D)
& v1_funct_2(D,u1_struct_0(A),u1_struct_0(C))
& m2_relset_1(D,u1_struct_0(A),u1_struct_0(C)) )
=> ( ( v5_pre_topc(D,A,C)
& m1_pre_topc(C,B) )
=> ! [E] :
( ( v1_funct_1(E)
& v1_funct_2(E,u1_struct_0(A),u1_struct_0(B))
& m2_relset_1(E,u1_struct_0(A),u1_struct_0(B)) )
=> ( E = D
=> v5_pre_topc(E,A,B) ) ) ) ) ) ) ) ).
fof(t8_topmetr,axiom,
! [A] :
( ( v2_pre_topc(A)
& l1_pre_topc(A) )
=> ! [B] :
( ( ~ v3_struct_0(B)
& v2_pre_topc(B)
& l1_pre_topc(B) )
=> ! [C] :
( ( v1_funct_1(C)
& v1_funct_2(C,u1_struct_0(A),u1_struct_0(B))
& m2_relset_1(C,u1_struct_0(A),u1_struct_0(B)) )
=> ! [D] :
( m1_pre_topc(D,B)
=> ( v5_pre_topc(C,A,B)
=> ! [E] :
( ( v1_funct_1(E)
& v1_funct_2(E,u1_struct_0(A),u1_struct_0(D))
& m2_relset_1(E,u1_struct_0(A),u1_struct_0(D)) )
=> ( E = C
=> v5_pre_topc(E,A,D) ) ) ) ) ) ) ) ).
fof(t9_topmetr,axiom,
! [A] :
( ( v2_pre_topc(A)
& l1_pre_topc(A) )
=> ! [B] :
( ( v2_pre_topc(B)
& l1_pre_topc(B) )
=> ! [C] :
( ( v1_funct_1(C)
& v1_funct_2(C,u1_struct_0(A),u1_struct_0(B))
& m2_relset_1(C,u1_struct_0(A),u1_struct_0(B)) )
=> ! [D] :
( m1_subset_1(D,k1_zfmisc_1(u1_struct_0(B)))
=> ( v5_pre_topc(C,A,B)
=> ! [E] :
( ( v1_funct_1(E)
& v1_funct_2(E,u1_struct_0(A),u1_struct_0(k3_pre_topc(B,D)))
& m2_relset_1(E,u1_struct_0(A),u1_struct_0(k3_pre_topc(B,D))) )
=> ( E = C
=> v5_pre_topc(E,A,k3_pre_topc(B,D)) ) ) ) ) ) ) ) ).
fof(t10_topmetr,axiom,
! [A] :
( l1_pre_topc(A)
=> ! [B] :
( ( ~ v3_struct_0(B)
& l1_pre_topc(B) )
=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A)))
=> ! [D] :
( ( v1_funct_1(D)
& v1_funct_2(D,u1_struct_0(A),u1_struct_0(B))
& m2_relset_1(D,u1_struct_0(A),u1_struct_0(B)) )
=> ! [E] :
( ( v1_funct_1(E)
& v1_funct_2(E,u1_struct_0(k3_pre_topc(A,C)),u1_struct_0(B))
& m2_relset_1(E,u1_struct_0(k3_pre_topc(A,C)),u1_struct_0(B)) )
=> ( ( v5_pre_topc(D,A,B)
& E = k2_partfun1(u1_struct_0(A),u1_struct_0(B),D,C) )
=> v5_pre_topc(E,k3_pre_topc(A,C),B) ) ) ) ) ) ) ).
fof(d1_topmetr,axiom,
! [A] :
( ( v6_metric_1(A)
& v7_metric_1(A)
& v8_metric_1(A)
& v9_metric_1(A)
& l1_metric_1(A) )
=> ! [B] :
( ( v6_metric_1(B)
& v7_metric_1(B)
& v8_metric_1(B)
& v9_metric_1(B)
& l1_metric_1(B) )
=> ( m1_topmetr(B,A)
<=> ( r1_tarski(u1_struct_0(B),u1_struct_0(A))
& ! [C] :
( m1_subset_1(C,u1_struct_0(B))
=> ! [D] :
( m1_subset_1(D,u1_struct_0(B))
=> k1_metric_1(u1_struct_0(B),u1_struct_0(B),u1_metric_1(B),C,D) = k1_binop_1(u1_metric_1(A),C,D) ) ) ) ) ) ) ).
fof(t11_topmetr,axiom,
$true ).
fof(t12_topmetr,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v6_metric_1(A)
& v7_metric_1(A)
& v8_metric_1(A)
& v9_metric_1(A)
& l1_metric_1(A) )
=> ! [B] :
( ( ~ v3_struct_0(B)
& m1_topmetr(B,A) )
=> ! [C] :
( m1_subset_1(C,u1_struct_0(B))
=> m1_subset_1(C,u1_struct_0(A)) ) ) ) ).
fof(t13_topmetr,axiom,
! [A] :
( v1_xreal_0(A)
=> ! [B] :
( ( v6_metric_1(B)
& v7_metric_1(B)
& v8_metric_1(B)
& v9_metric_1(B)
& l1_metric_1(B) )
=> ! [C] :
( m1_topmetr(C,B)
=> ! [D] :
( m1_subset_1(D,u1_struct_0(B))
=> ! [E] :
( m1_subset_1(E,u1_struct_0(C))
=> ( D = E
=> k9_metric_1(C,E,A) = k3_xboole_0(k9_metric_1(B,D,A),u1_struct_0(C)) ) ) ) ) ) ) ).
fof(d2_topmetr,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v6_metric_1(A)
& v7_metric_1(A)
& v8_metric_1(A)
& v9_metric_1(A)
& l1_metric_1(A) )
=> ! [B] :
( ( ~ v1_xboole_0(B)
& m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) )
=> ! [C] :
( ( v1_metric_1(C)
& m1_topmetr(C,A) )
=> ( C = k1_topmetr(A,B)
<=> u1_struct_0(C) = B ) ) ) ) ).
fof(d3_topmetr,axiom,
! [A] :
( v1_xreal_0(A)
=> ! [B] :
( v1_xreal_0(B)
=> ( r1_xreal_0(A,B)
=> ! [C] :
( ( ~ v3_struct_0(C)
& v1_metric_1(C)
& m1_topmetr(C,k8_metric_1) )
=> ( C = k2_topmetr(A,B)
<=> ! [D] :
( ( ~ v1_xboole_0(D)
& m1_subset_1(D,k1_zfmisc_1(u1_struct_0(k8_metric_1))) )
=> ( D = k1_rcomp_1(A,B)
=> C = k1_topmetr(k8_metric_1,D) ) ) ) ) ) ) ) ).
fof(t14_topmetr,axiom,
! [A] :
( v1_xreal_0(A)
=> ! [B] :
( v1_xreal_0(B)
=> ( r1_xreal_0(A,B)
=> u1_struct_0(k2_topmetr(A,B)) = k1_rcomp_1(A,B) ) ) ) ).
fof(d4_topmetr,axiom,
! [A] :
( l1_metric_1(A)
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(k1_zfmisc_1(u1_struct_0(A))))
=> ( v1_topmetr(B,A)
<=> ! [C] :
~ ( r2_hidden(C,B)
& ! [D] :
( m1_subset_1(D,u1_struct_0(A))
=> ! [E] :
( m1_subset_1(E,k1_numbers)
=> C != k9_metric_1(A,D,E) ) ) ) ) ) ) ).
fof(d5_topmetr,axiom,
! [A] :
( l1_metric_1(A)
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(k1_zfmisc_1(u1_struct_0(A))))
=> ( r1_topmetr(A,B)
<=> r1_tarski(u1_struct_0(A),k5_setfam_1(u1_struct_0(A),B)) ) ) ) ).
fof(t15_topmetr,axiom,
! [A] :
( m1_subset_1(A,u1_struct_0(k8_metric_1))
=> ! [B] :
( m1_subset_1(B,u1_struct_0(k8_metric_1))
=> ! [C] :
( v1_xreal_0(C)
=> ! [D] :
( v1_xreal_0(D)
=> ( ( C = A
& D = B )
=> k4_metric_1(k8_metric_1,A,B) = k18_complex1(k6_xcmplx_0(C,D)) ) ) ) ) ) ).
fof(t16_topmetr,axiom,
! [A] :
( l1_metric_1(A)
=> ( u1_struct_0(A) = u1_struct_0(k5_pcomps_1(A))
& u1_pre_topc(k5_pcomps_1(A)) = k4_pcomps_1(A) ) ) ).
fof(t17_topmetr,axiom,
$true ).
fof(t18_topmetr,axiom,
$true ).
fof(t19_topmetr,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v6_metric_1(A)
& v7_metric_1(A)
& v8_metric_1(A)
& v9_metric_1(A)
& l1_metric_1(A) )
=> ! [B] :
( ( ~ v3_struct_0(B)
& m1_topmetr(B,A) )
=> m1_pre_topc(k5_pcomps_1(B),k5_pcomps_1(A)) ) ) ).
fof(t20_topmetr,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(k15_euclid(A))))
=> ! [C] :
( ( ~ v1_xboole_0(C)
& m1_subset_1(C,k1_zfmisc_1(u1_struct_0(k14_euclid(A)))) )
=> ( B = C
=> k3_pre_topc(k15_euclid(A),B) = k5_pcomps_1(k1_topmetr(k14_euclid(A),C)) ) ) ) ) ).
fof(t21_topmetr,axiom,
! [A] :
( v1_xreal_0(A)
=> ! [B] :
( ( v9_metric_1(B)
& l1_metric_1(B) )
=> ! [C] :
( m1_subset_1(C,u1_struct_0(B))
=> ! [D] :
( m1_subset_1(D,k1_zfmisc_1(u1_struct_0(k5_pcomps_1(B))))
=> ( D = k9_metric_1(B,C,A)
=> v3_pre_topc(D,k5_pcomps_1(B)) ) ) ) ) ) ).
fof(t22_topmetr,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v6_metric_1(A)
& v7_metric_1(A)
& v8_metric_1(A)
& v9_metric_1(A)
& l1_metric_1(A) )
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(k5_pcomps_1(A))))
=> ( v3_pre_topc(B,k5_pcomps_1(A))
<=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> ~ ( r2_hidden(C,B)
& ! [D] :
( v1_xreal_0(D)
=> ~ ( ~ r1_xreal_0(D,np__0)
& r1_tarski(k9_metric_1(A,C,D),B) ) ) ) ) ) ) ) ).
fof(d6_topmetr,axiom,
! [A] :
( l1_metric_1(A)
=> ( v2_topmetr(A)
<=> v2_compts_1(k5_pcomps_1(A)) ) ) ).
fof(t23_topmetr,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v6_metric_1(A)
& v7_metric_1(A)
& v8_metric_1(A)
& v9_metric_1(A)
& l1_metric_1(A) )
=> ( v2_topmetr(A)
<=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(k1_zfmisc_1(u1_struct_0(A))))
=> ~ ( v1_topmetr(B,A)
& r1_topmetr(A,B)
& ! [C] :
( m1_subset_1(C,k1_zfmisc_1(k1_zfmisc_1(u1_struct_0(A))))
=> ~ ( r1_tarski(C,B)
& r1_topmetr(A,C)
& v1_finset_1(C) ) ) ) ) ) ) ).
fof(d7_topmetr,axiom,
k3_topmetr = k5_pcomps_1(k8_metric_1) ).
fof(t24_topmetr,axiom,
u1_struct_0(k3_topmetr) = k1_numbers ).
fof(d8_topmetr,axiom,
! [A] :
( v1_xreal_0(A)
=> ! [B] :
( v1_xreal_0(B)
=> k4_topmetr(A,B) = k5_pcomps_1(k2_topmetr(A,B)) ) ) ).
fof(t25_topmetr,axiom,
! [A] :
( v1_xreal_0(A)
=> ! [B] :
( v1_xreal_0(B)
=> ( r1_xreal_0(A,B)
=> u1_struct_0(k4_topmetr(A,B)) = k1_rcomp_1(A,B) ) ) ) ).
fof(t26_topmetr,axiom,
! [A] :
( v1_xreal_0(A)
=> ! [B] :
( v1_xreal_0(B)
=> ( r1_xreal_0(A,B)
=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(k3_topmetr)))
=> ( C = k1_rcomp_1(A,B)
=> k4_topmetr(A,B) = k3_pre_topc(k3_topmetr,C) ) ) ) ) ) ).
fof(t27_topmetr,axiom,
k4_topmetr(np__0,np__1) = k22_borsuk_1 ).
fof(t28_topmetr,axiom,
! [A] :
( ( v1_funct_1(A)
& v1_funct_2(A,u1_struct_0(k3_topmetr),u1_struct_0(k3_topmetr))
& m2_relset_1(A,u1_struct_0(k3_topmetr),u1_struct_0(k3_topmetr)) )
=> ( ? [B] :
( m1_subset_1(B,k1_numbers)
& ? [C] :
( m1_subset_1(C,k1_numbers)
& ! [D] :
( m1_subset_1(D,k1_numbers)
=> k1_funct_1(A,D) = k3_real_1(k4_real_1(B,D),C) ) ) )
=> v5_pre_topc(A,k3_topmetr,k3_topmetr) ) ) ).
fof(dt_m1_topmetr,axiom,
! [A] :
( ( v6_metric_1(A)
& v7_metric_1(A)
& v8_metric_1(A)
& v9_metric_1(A)
& l1_metric_1(A) )
=> ! [B] :
( m1_topmetr(B,A)
=> ( v6_metric_1(B)
& v7_metric_1(B)
& v8_metric_1(B)
& v9_metric_1(B)
& l1_metric_1(B) ) ) ) ).
fof(existence_m1_topmetr,axiom,
! [A] :
( ( v6_metric_1(A)
& v7_metric_1(A)
& v8_metric_1(A)
& v9_metric_1(A)
& l1_metric_1(A) )
=> ? [B] : m1_topmetr(B,A) ) ).
fof(dt_k1_topmetr,axiom,
! [A,B] :
( ( ~ v3_struct_0(A)
& v6_metric_1(A)
& v7_metric_1(A)
& v8_metric_1(A)
& v9_metric_1(A)
& l1_metric_1(A)
& ~ v1_xboole_0(B)
& m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) )
=> ( v1_metric_1(k1_topmetr(A,B))
& m1_topmetr(k1_topmetr(A,B),A) ) ) ).
fof(dt_k2_topmetr,axiom,
! [A,B] :
( ( v1_xreal_0(A)
& v1_xreal_0(B) )
=> ( ~ v3_struct_0(k2_topmetr(A,B))
& v1_metric_1(k2_topmetr(A,B))
& m1_topmetr(k2_topmetr(A,B),k8_metric_1) ) ) ).
fof(dt_k3_topmetr,axiom,
( v1_pre_topc(k3_topmetr)
& v2_pre_topc(k3_topmetr)
& l1_pre_topc(k3_topmetr) ) ).
fof(dt_k4_topmetr,axiom,
! [A,B] :
( ( v1_xreal_0(A)
& v1_xreal_0(B) )
=> ( ~ v3_struct_0(k4_topmetr(A,B))
& v1_pre_topc(k4_topmetr(A,B))
& m1_pre_topc(k4_topmetr(A,B),k3_topmetr) ) ) ).
fof(dt_k5_topmetr,axiom,
( v1_pre_topc(k5_topmetr)
& m1_pre_topc(k5_topmetr,k3_topmetr) ) ).
fof(redefinition_k5_topmetr,axiom,
k5_topmetr = k22_borsuk_1 ).
%------------------------------------------------------------------------------