SET007 Axioms: SET007+731.ax
%------------------------------------------------------------------------------
% File : SET007+731 : TPTP v8.2.0. Released v3.4.0.
% Domain : Set Theory
% Axioms : Metric Spaces and an Abstract Intermediate Value Theorem
% 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 : topmetr3 [Urb08]
% Status : Satisfiable
% Syntax : Number of formulae : 19 ( 3 unt; 0 def)
% Number of atoms : 223 ( 15 equ)
% Maximal formula atoms : 26 ( 11 avg)
% Number of connectives : 226 ( 22 ~; 0 |; 128 &)
% ( 0 <=>; 76 =>; 0 <=; 0 <~>)
% Maximal formula depth : 28 ( 11 avg)
% Maximal term depth : 4 ( 1 avg)
% Number of predicates : 29 ( 27 usr; 1 prp; 0-4 aty)
% Number of functors : 27 ( 27 usr; 7 con; 0-3 aty)
% Number of variables : 58 ( 58 !; 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(t1_topmetr3,axiom,
! [A] :
( ( ~ v1_xboole_0(A)
& m1_subset_1(A,k1_zfmisc_1(k1_numbers)) )
=> ! [B] :
( v1_xreal_0(B)
=> ( ! [C] :
( v1_xreal_0(C)
=> ( r2_hidden(C,A)
=> r1_xreal_0(C,B) ) )
=> r1_xreal_0(k4_seq_4(A),B) ) ) ) ).
fof(t2_topmetr3,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_funct_1(B)
& v1_funct_2(B,k5_numbers,u1_struct_0(A))
& m2_relset_1(B,k5_numbers,u1_struct_0(A)) )
=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(k5_pcomps_1(A))))
=> ( ( v2_tbsp_1(B,A)
& ! [D] :
( m2_subset_1(D,k1_numbers,k5_numbers)
=> r2_hidden(k2_normsp_1(A,B,D),C) )
& v4_pre_topc(C,k5_pcomps_1(A)) )
=> r2_hidden(k1_tbsp_1(A,B),C) ) ) ) ) ).
fof(t3_topmetr3,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)
& v6_metric_1(B)
& v7_metric_1(B)
& v8_metric_1(B)
& v9_metric_1(B)
& l1_metric_1(B) )
=> ! [C] :
( ( v1_funct_1(C)
& v1_funct_2(C,u1_struct_0(k5_pcomps_1(A)),u1_struct_0(k5_pcomps_1(B)))
& m2_relset_1(C,u1_struct_0(k5_pcomps_1(A)),u1_struct_0(k5_pcomps_1(B))) )
=> ! [D] :
( ( v1_funct_1(D)
& v1_funct_2(D,k5_numbers,u1_struct_0(A))
& m2_relset_1(D,k5_numbers,u1_struct_0(A)) )
=> ( v1_funct_1(k5_relat_1(D,C))
& v1_funct_2(k5_relat_1(D,C),k5_numbers,u1_struct_0(B))
& m2_relset_1(k5_relat_1(D,C),k5_numbers,u1_struct_0(B)) ) ) ) ) ) ).
fof(t4_topmetr3,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)
& v6_metric_1(B)
& v7_metric_1(B)
& v8_metric_1(B)
& v9_metric_1(B)
& l1_metric_1(B) )
=> ! [C] :
( ( v1_funct_1(C)
& v1_funct_2(C,u1_struct_0(k5_pcomps_1(A)),u1_struct_0(k5_pcomps_1(B)))
& m2_relset_1(C,u1_struct_0(k5_pcomps_1(A)),u1_struct_0(k5_pcomps_1(B))) )
=> ! [D] :
( ( v1_funct_1(D)
& v1_funct_2(D,k5_numbers,u1_struct_0(A))
& m2_relset_1(D,k5_numbers,u1_struct_0(A)) )
=> ! [E] :
( ( v1_funct_1(E)
& v1_funct_2(E,k5_numbers,u1_struct_0(B))
& m2_relset_1(E,k5_numbers,u1_struct_0(B)) )
=> ( ( v2_tbsp_1(D,A)
& E = k5_relat_1(D,C)
& v5_pre_topc(C,k5_pcomps_1(A),k5_pcomps_1(B)) )
=> v2_tbsp_1(E,B) ) ) ) ) ) ) ).
fof(t5_topmetr3,axiom,
$true ).
fof(t6_topmetr3,axiom,
! [A] :
( ( v1_funct_1(A)
& v1_funct_2(A,k5_numbers,k1_numbers)
& m2_relset_1(A,k5_numbers,k1_numbers) )
=> ! [B] :
( ( v1_funct_1(B)
& v1_funct_2(B,k5_numbers,u1_struct_0(k8_metric_1))
& m2_relset_1(B,k5_numbers,u1_struct_0(k8_metric_1)) )
=> ( A = B
=> ( ( v4_seq_2(A)
=> v2_tbsp_1(B,k8_metric_1) )
& ( v2_tbsp_1(B,k8_metric_1)
=> v4_seq_2(A) )
& ( v4_seq_2(A)
=> k2_seq_2(A) = k1_tbsp_1(k8_metric_1,B) ) ) ) ) ) ).
fof(t7_topmetr3,axiom,
! [A] :
( v1_xreal_0(A)
=> ! [B] :
( v1_xreal_0(B)
=> ! [C] :
( ( v1_funct_1(C)
& v1_funct_2(C,k5_numbers,k1_numbers)
& m2_relset_1(C,k5_numbers,k1_numbers) )
=> ( r1_tarski(k1_pscomp_1(k5_numbers,k1_numbers,C),k1_rcomp_1(A,B))
=> ( v1_funct_1(C)
& v1_funct_2(C,k5_numbers,u1_struct_0(k2_topmetr(A,B)))
& m2_relset_1(C,k5_numbers,u1_struct_0(k2_topmetr(A,B))) ) ) ) ) ) ).
fof(t8_topmetr3,axiom,
! [A] :
( v1_xreal_0(A)
=> ! [B] :
( v1_xreal_0(B)
=> ! [C] :
( ( v1_funct_1(C)
& v1_funct_2(C,k5_numbers,u1_struct_0(k2_topmetr(A,B)))
& m2_relset_1(C,k5_numbers,u1_struct_0(k2_topmetr(A,B))) )
=> ( r1_xreal_0(A,B)
=> ( v1_funct_1(C)
& v1_funct_2(C,k5_numbers,u1_struct_0(k8_metric_1))
& m2_relset_1(C,k5_numbers,u1_struct_0(k8_metric_1)) ) ) ) ) ) ).
fof(t9_topmetr3,axiom,
! [A] :
( v1_xreal_0(A)
=> ! [B] :
( v1_xreal_0(B)
=> ! [C] :
( ( v1_funct_1(C)
& v1_funct_2(C,k5_numbers,u1_struct_0(k2_topmetr(A,B)))
& m2_relset_1(C,k5_numbers,u1_struct_0(k2_topmetr(A,B))) )
=> ! [D] :
( ( v1_funct_1(D)
& v1_funct_2(D,k5_numbers,u1_struct_0(k8_metric_1))
& m2_relset_1(D,k5_numbers,u1_struct_0(k8_metric_1)) )
=> ( ( D = C
& r1_xreal_0(A,B) )
=> ( ( v2_tbsp_1(D,k8_metric_1)
=> v2_tbsp_1(C,k2_topmetr(A,B)) )
& ( v2_tbsp_1(C,k2_topmetr(A,B))
=> v2_tbsp_1(D,k8_metric_1) )
& ( v2_tbsp_1(D,k8_metric_1)
=> k1_tbsp_1(k8_metric_1,D) = k1_tbsp_1(k2_topmetr(A,B),C) ) ) ) ) ) ) ) ).
fof(t10_topmetr3,axiom,
! [A] :
( v1_xreal_0(A)
=> ! [B] :
( v1_xreal_0(B)
=> ! [C] :
( ( v1_funct_1(C)
& v1_funct_2(C,k5_numbers,k1_numbers)
& m2_relset_1(C,k5_numbers,k1_numbers) )
=> ! [D] :
( ( v1_funct_1(D)
& v1_funct_2(D,k5_numbers,u1_struct_0(k2_topmetr(A,B)))
& m2_relset_1(D,k5_numbers,u1_struct_0(k2_topmetr(A,B))) )
=> ( ( D = C
& r1_xreal_0(A,B)
& v4_seq_2(C) )
=> ( v2_tbsp_1(D,k2_topmetr(A,B))
& k2_seq_2(C) = k1_tbsp_1(k2_topmetr(A,B),D) ) ) ) ) ) ) ).
fof(t11_topmetr3,axiom,
! [A] :
( v1_xreal_0(A)
=> ! [B] :
( v1_xreal_0(B)
=> ! [C] :
( ( v1_funct_1(C)
& v1_funct_2(C,k5_numbers,k1_numbers)
& m2_relset_1(C,k5_numbers,k1_numbers) )
=> ! [D] :
( ( v1_funct_1(D)
& v1_funct_2(D,k5_numbers,u1_struct_0(k2_topmetr(A,B)))
& m2_relset_1(D,k5_numbers,u1_struct_0(k2_topmetr(A,B))) )
=> ( ( D = C
& r1_xreal_0(A,B)
& v3_seqm_3(C) )
=> v2_tbsp_1(D,k2_topmetr(A,B)) ) ) ) ) ) ).
fof(t12_topmetr3,axiom,
! [A] :
( v1_xreal_0(A)
=> ! [B] :
( v1_xreal_0(B)
=> ! [C] :
( ( v1_funct_1(C)
& v1_funct_2(C,k5_numbers,k1_numbers)
& m2_relset_1(C,k5_numbers,k1_numbers) )
=> ! [D] :
( ( v1_funct_1(D)
& v1_funct_2(D,k5_numbers,u1_struct_0(k2_topmetr(A,B)))
& m2_relset_1(D,k5_numbers,u1_struct_0(k2_topmetr(A,B))) )
=> ( ( D = C
& r1_xreal_0(A,B)
& v4_seqm_3(C) )
=> v2_tbsp_1(D,k2_topmetr(A,B)) ) ) ) ) ) ).
fof(t13_topmetr3,axiom,
$true ).
fof(t14_topmetr3,axiom,
$true ).
fof(t15_topmetr3,axiom,
! [A] :
( ( ~ v1_xboole_0(A)
& m1_subset_1(A,k1_zfmisc_1(k1_numbers)) )
=> ~ ( v1_seq_4(A)
& ! [B] :
( ( v1_funct_1(B)
& v1_funct_2(B,k5_numbers,k1_numbers)
& m2_relset_1(B,k5_numbers,k1_numbers) )
=> ~ ( v3_seqm_3(B)
& v4_seq_2(B)
& r1_tarski(k1_pscomp_1(k5_numbers,k1_numbers,B),A)
& k2_seq_2(B) = k4_seq_4(A) ) ) ) ) ).
fof(t16_topmetr3,axiom,
! [A] :
( ( ~ v1_xboole_0(A)
& m1_subset_1(A,k1_zfmisc_1(k1_numbers)) )
=> ~ ( v2_seq_4(A)
& ! [B] :
( ( v1_funct_1(B)
& v1_funct_2(B,k5_numbers,k1_numbers)
& m2_relset_1(B,k5_numbers,k1_numbers) )
=> ~ ( v4_seqm_3(B)
& v4_seq_2(B)
& r1_tarski(k1_pscomp_1(k5_numbers,k1_numbers,B),A)
& k2_seq_2(B) = k5_seq_4(A) ) ) ) ) ).
fof(t17_topmetr3,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_funct_1(B)
& v1_funct_2(B,u1_struct_0(k5_topmetr),u1_struct_0(k5_pcomps_1(A)))
& m2_relset_1(B,u1_struct_0(k5_topmetr),u1_struct_0(k5_pcomps_1(A))) )
=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(k5_pcomps_1(A))))
=> ! [D] :
( m1_subset_1(D,k1_zfmisc_1(u1_struct_0(k5_pcomps_1(A))))
=> ! [E] :
( m1_subset_1(E,k1_numbers)
=> ! [F] :
( m1_subset_1(F,k1_numbers)
=> ~ ( r1_xreal_0(np__0,E)
& r1_xreal_0(F,np__1)
& r1_xreal_0(E,F)
& r2_hidden(k1_funct_1(B,E),C)
& r2_hidden(k1_funct_1(B,F),D)
& v4_pre_topc(C,k5_pcomps_1(A))
& v4_pre_topc(D,k5_pcomps_1(A))
& v5_pre_topc(B,k5_topmetr,k5_pcomps_1(A))
& k4_subset_1(u1_struct_0(k5_pcomps_1(A)),C,D) = u1_struct_0(A)
& ! [G] :
( m1_subset_1(G,k1_numbers)
=> ~ ( r1_xreal_0(E,G)
& r1_xreal_0(G,F)
& r2_hidden(k1_funct_1(B,G),k5_subset_1(u1_struct_0(k5_pcomps_1(A)),C,D)) ) ) ) ) ) ) ) ) ) ).
fof(t18_topmetr3,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m1_subset_1(B,u1_struct_0(k15_euclid(A)))
=> ! [C] :
( m1_subset_1(C,u1_struct_0(k15_euclid(A)))
=> ! [D] :
( ( ~ v1_xboole_0(D)
& m1_subset_1(D,k1_zfmisc_1(u1_struct_0(k15_euclid(A)))) )
=> ! [E] :
( ( ~ v1_xboole_0(E)
& m1_subset_1(E,k1_zfmisc_1(u1_struct_0(k15_euclid(A)))) )
=> ( ( r1_topreal1(k15_euclid(A),B,C,D)
& r1_topreal1(k15_euclid(A),C,B,E)
& r1_tarski(E,D) )
=> E = D ) ) ) ) ) ) ).
fof(t19_topmetr3,axiom,
! [A] :
( ( ~ v1_xboole_0(A)
& v6_compts_1(A,k15_euclid(np__2))
& m1_subset_1(A,k1_zfmisc_1(u1_struct_0(k15_euclid(np__2)))) )
=> ! [B] :
( ( ~ v1_xboole_0(B)
& v6_compts_1(B,k15_euclid(np__2))
& m1_subset_1(B,k1_zfmisc_1(u1_struct_0(k15_euclid(np__2)))) )
=> ~ ( v1_topreal2(A)
& r1_topreal1(k15_euclid(np__2),k30_pscomp_1(A),k34_pscomp_1(A),B)
& r1_tarski(B,A)
& B != k8_jordan6(A)
& B != k9_jordan6(A) ) ) ) ).
%------------------------------------------------------------------------------