TPTP Problem File: TOP030+2.p
View Solutions
- Solve Problem
%------------------------------------------------------------------------------
% File : TOP030+2 : TPTP v8.2.0. Released v3.4.0.
% Domain : Topology
% Problem : Maximal Kolmogorov Subspaces of a Topological Space T15
% Version : [Urb08] axioms : Especial.
% English :
% Refs : [Kar96] Karno (1996), Maximal Kolmogorov Subspaces of a Topolo
% : [Urb07] Urban (2007), MPTP 0.2: Design, Implementation, and In
% : [Urb08] Urban (2006), Email to G. Sutcliffe
% Source : [Urb08]
% Names : t15_tsp_2 [Urb08]
% Status : Theorem
% Rating : 1.00 v3.4.0
% Syntax : Number of formulae : 3611 ( 764 unt; 0 def)
% Number of atoms : 19088 (2119 equ)
% Maximal formula atoms : 49 ( 5 avg)
% Number of connectives : 18311 (2834 ~; 141 |;8010 &)
% ( 669 <=>;6657 =>; 0 <=; 0 <~>)
% Maximal formula depth : 28 ( 7 avg)
% Maximal term depth : 7 ( 1 avg)
% Number of predicates : 217 ( 215 usr; 1 prp; 0-3 aty)
% Number of functors : 419 ( 419 usr; 132 con; 0-8 aty)
% Number of variables : 8509 (8114 !; 395 ?)
% SPC : FOF_THM_RFO_SEQ
% Comments : Bushy version: includes all articles that contribute axioms to the
% Normal version.
% : Translated by MPTP from the Mizar Mathematical Library 4.48.930.
% : The problem encoding is based on set theory.
%------------------------------------------------------------------------------
include('Axioms/SET007/SET007+0.ax').
include('Axioms/SET007/SET007+1.ax').
include('Axioms/SET007/SET007+2.ax').
include('Axioms/SET007/SET007+3.ax').
include('Axioms/SET007/SET007+4.ax').
include('Axioms/SET007/SET007+6.ax').
include('Axioms/SET007/SET007+7.ax').
include('Axioms/SET007/SET007+9.ax').
include('Axioms/SET007/SET007+10.ax').
include('Axioms/SET007/SET007+11.ax').
include('Axioms/SET007/SET007+13.ax').
include('Axioms/SET007/SET007+14.ax').
include('Axioms/SET007/SET007+16.ax').
include('Axioms/SET007/SET007+17.ax').
include('Axioms/SET007/SET007+20.ax').
include('Axioms/SET007/SET007+23.ax').
include('Axioms/SET007/SET007+24.ax').
include('Axioms/SET007/SET007+26.ax').
include('Axioms/SET007/SET007+35.ax').
include('Axioms/SET007/SET007+40.ax').
include('Axioms/SET007/SET007+51.ax').
include('Axioms/SET007/SET007+60.ax').
include('Axioms/SET007/SET007+200.ax').
include('Axioms/SET007/SET007+206.ax').
include('Axioms/SET007/SET007+207.ax').
include('Axioms/SET007/SET007+209.ax').
include('Axioms/SET007/SET007+227.ax').
include('Axioms/SET007/SET007+256.ax').
include('Axioms/SET007/SET007+301.ax').
include('Axioms/SET007/SET007+309.ax').
include('Axioms/SET007/SET007+327.ax').
include('Axioms/SET007/SET007+370.ax').
include('Axioms/SET007/SET007+387.ax').
include('Axioms/SET007/SET007+388.ax').
include('Axioms/SET007/SET007+399.ax').
include('Axioms/SET007/SET007+405.ax').
include('Axioms/SET007/SET007+406.ax').
%------------------------------------------------------------------------------
fof(dt_k1_tsp_2,axiom,
! [A,B] :
( ( ~ v3_struct_0(A)
& v2_pre_topc(A)
& l1_pre_topc(A)
& ~ v3_struct_0(B)
& v2_tsp_2(B,A)
& m1_pre_topc(B,A) )
=> ( v1_funct_1(k1_tsp_2(A,B))
& v1_funct_2(k1_tsp_2(A,B),u1_struct_0(A),u1_struct_0(B))
& v5_pre_topc(k1_tsp_2(A,B),A,B)
& m2_relset_1(k1_tsp_2(A,B),u1_struct_0(A),u1_struct_0(B)) ) ) ).
fof(dt_k2_tsp_2,axiom,
! [A,B] :
( ( ~ v3_struct_0(A)
& v2_pre_topc(A)
& l1_pre_topc(A)
& ~ v3_struct_0(B)
& v2_tsp_2(B,A)
& m1_pre_topc(B,A) )
=> ( v1_funct_1(k2_tsp_2(A,B))
& v1_funct_2(k2_tsp_2(A,B),u1_struct_0(A),u1_struct_0(B))
& v5_pre_topc(k2_tsp_2(A,B),A,B)
& m2_relset_1(k2_tsp_2(A,B),u1_struct_0(A),u1_struct_0(B)) ) ) ).
fof(redefinition_k2_tsp_2,axiom,
! [A,B] :
( ( ~ v3_struct_0(A)
& v2_pre_topc(A)
& l1_pre_topc(A)
& ~ v3_struct_0(B)
& v2_tsp_2(B,A)
& m1_pre_topc(B,A) )
=> k2_tsp_2(A,B) = k1_tsp_2(A,B) ) ).
fof(dt_k3_tsp_2,axiom,
! [A,B] :
( ( ~ v3_struct_0(A)
& v2_pre_topc(A)
& l1_pre_topc(A)
& ~ v3_struct_0(B)
& v2_tsp_2(B,A)
& m1_pre_topc(B,A) )
=> ( v1_funct_1(k3_tsp_2(A,B))
& v1_funct_2(k3_tsp_2(A,B),u1_struct_0(A),u1_struct_0(B))
& v5_pre_topc(k3_tsp_2(A,B),A,B)
& m2_relset_1(k3_tsp_2(A,B),u1_struct_0(A),u1_struct_0(B)) ) ) ).
fof(redefinition_k3_tsp_2,axiom,
! [A,B] :
( ( ~ v3_struct_0(A)
& v2_pre_topc(A)
& l1_pre_topc(A)
& ~ v3_struct_0(B)
& v2_tsp_2(B,A)
& m1_pre_topc(B,A) )
=> k3_tsp_2(A,B) = k1_tsp_2(A,B) ) ).
fof(dt_k4_tsp_2,axiom,
! [A,B] :
( ( ~ v3_struct_0(A)
& v2_pre_topc(A)
& l1_pre_topc(A)
& ~ v3_struct_0(B)
& v2_tsp_2(B,A)
& m1_pre_topc(B,A) )
=> ( v1_funct_1(k4_tsp_2(A,B))
& v1_funct_2(k4_tsp_2(A,B),u1_struct_0(A),u1_struct_0(B))
& v5_pre_topc(k4_tsp_2(A,B),A,B)
& m2_relset_1(k4_tsp_2(A,B),u1_struct_0(A),u1_struct_0(B)) ) ) ).
fof(redefinition_k4_tsp_2,axiom,
! [A,B] :
( ( ~ v3_struct_0(A)
& v2_pre_topc(A)
& l1_pre_topc(A)
& ~ v3_struct_0(B)
& v2_tsp_2(B,A)
& m1_pre_topc(B,A) )
=> k4_tsp_2(A,B) = k1_tsp_2(A,B) ) ).
fof(d1_tsp_2,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)))
=> ( v1_tsp_1(B,A)
<=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> ! [D] :
( m1_subset_1(D,u1_struct_0(A))
=> ( ( r2_hidden(C,B)
& r2_hidden(D,B) )
=> ( C = D
| r1_subset_1(k4_tex_4(A,C),k4_tex_4(A,D)) ) ) ) ) ) ) ) ).
fof(d2_tsp_2,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)))
=> ( v1_tsp_1(B,A)
<=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> ( r2_hidden(C,B)
=> k5_subset_1(u1_struct_0(A),B,k4_tex_4(A,C)) = k1_struct_0(A,C) ) ) ) ) ) ).
fof(d3_tsp_2,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)))
=> ( v1_tsp_1(B,A)
<=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> ~ ( r2_hidden(C,B)
& ! [D] :
( m1_subset_1(D,k1_zfmisc_1(u1_struct_0(A)))
=> ~ ( v3_tex_4(D,A)
& k5_subset_1(u1_struct_0(A),B,D) = k1_struct_0(A,C) ) ) ) ) ) ) ) ).
fof(d4_tsp_2,axiom,
! [A] :
( l1_pre_topc(A)
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
=> ( v1_tsp_2(B,A)
<=> ( v1_tsp_1(B,A)
& ! [C] :
( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A)))
=> ( ( v1_tsp_1(C,A)
& r1_tarski(B,C) )
=> B = C ) ) ) ) ) ) ).
fof(t1_tsp_2,axiom,
! [A] :
( l1_pre_topc(A)
=> ! [B] :
( l1_pre_topc(B)
=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A)))
=> ! [D] :
( m1_subset_1(D,k1_zfmisc_1(u1_struct_0(B)))
=> ( ( g1_pre_topc(u1_struct_0(A),u1_pre_topc(A)) = g1_pre_topc(u1_struct_0(B),u1_pre_topc(B))
& C = D
& v1_tsp_2(C,A) )
=> v1_tsp_2(D,B) ) ) ) ) ) ).
fof(d5_tsp_2,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)))
=> ( v1_tsp_2(B,A)
<=> ( v1_tsp_1(B,A)
& k3_tex_4(A,B) = u1_struct_0(A) ) ) ) ) ).
fof(t2_tsp_2,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)))
=> ( v1_tsp_2(B,A)
=> v1_tops_1(B,A) ) ) ) ).
fof(t3_tsp_2,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)))
=> ~ ( ( v3_pre_topc(B,A)
| v4_pre_topc(B,A) )
& v1_tsp_2(B,A)
& v1_tex_2(B,k1_zfmisc_1(u1_struct_0(A))) ) ) ) ).
fof(t4_tsp_2,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_pre_topc(A)
& l1_pre_topc(A) )
=> ! [B] :
( ( v1_xboole_0(B)
& m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) )
=> ~ v1_tsp_2(B,A) ) ) ).
fof(t5_tsp_2,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)))
=> ( v1_tsp_2(B,A)
=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A)))
=> ( v4_pre_topc(C,A)
=> C = k3_tex_4(A,k5_subset_1(u1_struct_0(A),B,C)) ) ) ) ) ) ).
fof(t6_tsp_2,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)))
=> ( v1_tsp_2(B,A)
=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A)))
=> ( v3_pre_topc(C,A)
=> C = k3_tex_4(A,k5_subset_1(u1_struct_0(A),B,C)) ) ) ) ) ) ).
fof(t7_tsp_2,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)))
=> ( v1_tsp_2(B,A)
=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A)))
=> k6_pre_topc(A,C) = k3_tex_4(A,k5_subset_1(u1_struct_0(A),B,k6_pre_topc(A,C))) ) ) ) ) ).
fof(t8_tsp_2,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)))
=> ( v1_tsp_2(B,A)
=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A)))
=> k1_tops_1(A,C) = k3_tex_4(A,k5_subset_1(u1_struct_0(A),B,k1_tops_1(A,C))) ) ) ) ) ).
fof(d6_tsp_2,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)))
=> ( v1_tsp_2(B,A)
<=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> ? [D] :
( m1_subset_1(D,u1_struct_0(A))
& r2_hidden(D,B)
& k5_subset_1(u1_struct_0(A),B,k4_tex_4(A,C)) = k1_struct_0(A,D) ) ) ) ) ) ).
fof(t9_tsp_2,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)))
=> ~ ( v1_tsp_1(B,A)
& ! [C] :
( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A)))
=> ~ ( r1_tarski(B,C)
& v1_tsp_2(C,A) ) ) ) ) ) ).
fof(t10_tsp_2,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)))
& v1_tsp_2(B,A) ) ) ).
fof(d7_tsp_2,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l1_pre_topc(A) )
=> ! [B] :
( m2_tsp_1(B,A)
=> ( v2_tsp_2(B,A)
<=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A)))
=> ( C = u1_struct_0(B)
=> v1_tsp_2(C,A) ) ) ) ) ) ).
fof(t11_tsp_2,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l1_pre_topc(A) )
=> ! [B] :
( m2_tsp_1(B,A)
=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A)))
=> ( C = u1_struct_0(B)
=> ( v1_tsp_2(C,A)
<=> v2_tsp_2(B,A) ) ) ) ) ) ).
fof(l19_tsp_2,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l1_pre_topc(A) )
=> ! [B] :
( m2_tsp_1(B,A)
=> m1_subset_1(u1_struct_0(B),k1_zfmisc_1(u1_struct_0(A))) ) ) ).
fof(cc1_tsp_2,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l1_pre_topc(A) )
=> ! [B] :
( m1_pre_topc(B,A)
=> ( ( ~ v3_struct_0(B)
& v2_tsp_2(B,A) )
=> ( ~ v3_struct_0(B)
& v2_t_0topsp(B) ) ) ) ) ).
fof(cc2_tsp_2,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l1_pre_topc(A) )
=> ! [B] :
( m1_pre_topc(B,A)
=> ( ( ~ v3_struct_0(B)
& ~ v2_t_0topsp(B) )
=> ( ~ v3_struct_0(B)
& ~ v2_tsp_2(B,A) ) ) ) ) ).
fof(d8_tsp_2,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_pre_topc(A)
& l1_pre_topc(A) )
=> ! [B] :
( ( ~ v3_struct_0(B)
& m2_tsp_1(B,A) )
=> ( v2_tsp_2(B,A)
<=> ( v2_t_0topsp(B)
& ! [C] :
( ( ~ v3_struct_0(C)
& v2_t_0topsp(C)
& m2_tsp_1(C,A) )
=> ( m2_tsp_1(B,C)
=> g1_pre_topc(u1_struct_0(B),u1_pre_topc(B)) = g1_pre_topc(u1_struct_0(C),u1_pre_topc(C)) ) ) ) ) ) ) ).
fof(t12_tsp_2,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_pre_topc(A)
& l1_pre_topc(A) )
=> ! [B] :
( ( ~ v1_xboole_0(B)
& m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) )
=> ~ ( v1_tsp_2(B,A)
& ! [C] :
( ( ~ v3_struct_0(C)
& v1_pre_topc(C)
& m2_tsp_1(C,A) )
=> ~ ( v2_tsp_2(C,A)
& B = u1_struct_0(C) ) ) ) ) ) ).
fof(cc3_tsp_2,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_pre_topc(A)
& l1_pre_topc(A) )
=> ! [B] :
( m1_pre_topc(B,A)
=> ( v2_tsp_2(B,A)
=> ( v2_pre_topc(B)
& v1_tex_3(B,A) ) ) ) ) ).
fof(cc4_tsp_2,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_pre_topc(A)
& l1_pre_topc(A) )
=> ! [B] :
( m1_pre_topc(B,A)
=> ( ~ v1_tex_3(B,A)
=> ( v2_pre_topc(B)
& ~ v2_tsp_2(B,A) ) ) ) ) ).
fof(cc5_tsp_2,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_pre_topc(A)
& l1_pre_topc(A) )
=> ! [B] :
( m1_pre_topc(B,A)
=> ( ( v1_tsep_1(B,A)
& v2_tsp_2(B,A) )
=> ( v2_pre_topc(B)
& ~ v2_tex_2(B,A) ) ) ) ) ).
fof(cc6_tsp_2,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_pre_topc(A)
& l1_pre_topc(A) )
=> ! [B] :
( m1_pre_topc(B,A)
=> ( ( v1_tsep_1(B,A)
& v2_tex_2(B,A) )
=> ( v2_pre_topc(B)
& ~ v2_tsp_2(B,A) ) ) ) ) ).
fof(cc7_tsp_2,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_pre_topc(A)
& l1_pre_topc(A) )
=> ! [B] :
( m1_pre_topc(B,A)
=> ( ( v2_tex_2(B,A)
& v2_tsp_2(B,A) )
=> ( v2_pre_topc(B)
& ~ v1_tsep_1(B,A) ) ) ) ) ).
fof(cc8_tsp_2,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_pre_topc(A)
& l1_pre_topc(A) )
=> ! [B] :
( m1_pre_topc(B,A)
=> ( ( v1_borsuk_1(B,A)
& v2_tsp_2(B,A) )
=> ( v2_pre_topc(B)
& ~ v2_tex_2(B,A) ) ) ) ) ).
fof(cc9_tsp_2,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_pre_topc(A)
& l1_pre_topc(A) )
=> ! [B] :
( m1_pre_topc(B,A)
=> ( ( v2_tex_2(B,A)
& v1_borsuk_1(B,A) )
=> ( v2_pre_topc(B)
& ~ v2_tsp_2(B,A) ) ) ) ) ).
fof(cc10_tsp_2,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_pre_topc(A)
& l1_pre_topc(A) )
=> ! [B] :
( m1_pre_topc(B,A)
=> ( ( v2_tex_2(B,A)
& v2_tsp_2(B,A) )
=> ( v2_pre_topc(B)
& ~ v1_borsuk_1(B,A) ) ) ) ) ).
fof(t13_tsp_2,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_pre_topc(A)
& l1_pre_topc(A) )
=> ! [B] :
( ( ~ v3_struct_0(B)
& v2_t_0topsp(B)
& m2_tsp_1(B,A) )
=> ? [C] :
( v1_pre_topc(C)
& m2_tsp_1(C,A)
& m2_tsp_1(B,C)
& v2_tsp_2(C,A) ) ) ) ).
fof(rc1_tsp_2,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_pre_topc(A)
& l1_pre_topc(A) )
=> ? [B] :
( m1_pre_topc(B,A)
& ~ v3_struct_0(B)
& v1_pre_topc(B)
& v2_pre_topc(B)
& v1_tex_3(B,A)
& v2_t_0topsp(B)
& v2_tsp_2(B,A) ) ) ).
fof(t14_tsp_2,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_pre_topc(A)
& l1_pre_topc(A) )
=> ! [B] :
( ( v2_tsp_2(B,A)
& m2_tsp_1(B,A) )
=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A)))
=> ! [D] :
( m1_subset_1(D,k1_zfmisc_1(u1_struct_0(B)))
=> ( D = C
=> ( v3_pre_topc(D,B)
<=> ( v3_pre_topc(k3_tex_4(A,C),A)
& D = k3_xboole_0(k3_tex_4(A,C),u1_struct_0(B)) ) ) ) ) ) ) ) ).
fof(t15_tsp_2,conjecture,
! [A] :
( ( ~ v3_struct_0(A)
& v2_pre_topc(A)
& l1_pre_topc(A) )
=> ! [B] :
( ( v2_tsp_2(B,A)
& m2_tsp_1(B,A) )
=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A)))
=> ( v3_pre_topc(C,A)
<=> ( C = k3_tex_4(A,C)
& ? [D] :
( m1_subset_1(D,k1_zfmisc_1(u1_struct_0(B)))
& v3_pre_topc(D,B)
& D = k3_xboole_0(C,u1_struct_0(B)) ) ) ) ) ) ) ).
%------------------------------------------------------------------------------