SET007 Axioms: SET007+909.ax
%------------------------------------------------------------------------------
% File : SET007+909 : TPTP v9.0.0. Released v3.4.0.
% Domain : Set Theory
% Axioms : Homeomorphism between Finite Topological Spaces
% Version : [Urb08] axioms.
% English : Homeomorphism between Finite Topological Spaces Two-Dimensional
% Lattice Spaces and a Fixed Point Theorem
% 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 : fintopo5 [Urb08]
% Status : Satisfiable
% Syntax : Number of formulae : 27 ( 0 unt; 0 def)
% Number of atoms : 196 ( 17 equ)
% Maximal formula atoms : 15 ( 7 avg)
% Number of connectives : 207 ( 38 ~; 0 |; 78 &)
% ( 6 <=>; 85 =>; 0 <=; 0 <~>)
% Maximal formula depth : 17 ( 9 avg)
% Maximal term depth : 5 ( 1 avg)
% Number of predicates : 19 ( 18 usr; 0 prp; 1-4 aty)
% Number of functors : 32 ( 32 usr; 6 con; 0-4 aty)
% Number of variables : 86 ( 84 !; 2 ?)
% 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_fintopo5,axiom,
! [A,B] :
( ( ~ v1_xboole_0(A)
& m1_subset_1(A,k5_numbers)
& ~ v1_xboole_0(B)
& m1_subset_1(B,k5_numbers) )
=> ( ~ v3_struct_0(k2_fintopo5(A,B))
& v1_fin_topo(k2_fintopo5(A,B)) ) ) ).
fof(fc2_fintopo5,axiom,
! [A,B] :
( ( ~ v1_xboole_0(A)
& m1_subset_1(A,k5_numbers)
& ~ v1_xboole_0(B)
& m1_subset_1(B,k5_numbers) )
=> ( ~ v3_struct_0(k4_fintopo5(A,B))
& v1_fin_topo(k4_fintopo5(A,B)) ) ) ).
fof(t1_fintopo5,axiom,
! [A,B] :
( ~ v1_xboole_0(B)
=> ! [C] :
( ( v1_funct_1(C)
& v1_funct_2(C,A,B)
& m2_relset_1(C,A,B) )
=> ! [D] :
( m1_subset_1(D,k1_zfmisc_1(A))
=> ( v2_funct_1(C)
=> k9_relat_1(k2_funct_1(C),k2_funct_2(A,B,C,D)) = D ) ) ) ) ).
fof(t2_fintopo5,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ( ~ ( ~ r1_xreal_0(A,np__0)
& k2_finseq_1(A) = k1_xboole_0 )
& ~ ( k2_finseq_1(A) != k1_xboole_0
& r1_xreal_0(A,np__0) ) ) ) ).
fof(d1_fintopo5,axiom,
! [A] :
( l1_fin_topo(A)
=> ! [B] :
( l1_fin_topo(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)) )
=> ( r1_fintopo5(A,B,C)
<=> ( v2_funct_1(C)
& v2_funct_2(C,u1_struct_0(A),u1_struct_0(B))
& ! [D] :
( m1_subset_1(D,u1_struct_0(A))
=> k2_funct_2(u1_struct_0(A),u1_struct_0(B),C,k1_funct_1(u1_fin_topo(A),D)) = k1_funct_1(u1_fin_topo(B),k1_funct_1(C,D)) ) ) ) ) ) ) ).
fof(t3_fintopo5,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l1_fin_topo(A) )
=> ! [B] :
( ( ~ v3_struct_0(B)
& l1_fin_topo(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)) )
=> ~ ( r1_fintopo5(A,B,C)
& ! [D] :
( ( v1_funct_1(D)
& v1_funct_2(D,u1_struct_0(B),u1_struct_0(A))
& m2_relset_1(D,u1_struct_0(B),u1_struct_0(A)) )
=> ~ ( D = k2_funct_1(C)
& r1_fintopo5(B,A,D) ) ) ) ) ) ) ).
fof(t4_fintopo5,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l1_fin_topo(A) )
=> ! [B] :
( ( ~ v3_struct_0(B)
& l1_fin_topo(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] :
( m2_subset_1(D,k1_numbers,k5_numbers)
=> ! [E] :
( m1_subset_1(E,u1_struct_0(A))
=> ! [F] :
( m1_subset_1(F,u1_struct_0(B))
=> ( ( r1_fintopo5(A,B,C)
& F = k8_funct_2(u1_struct_0(A),u1_struct_0(B),C,E) )
=> ! [G] :
( m1_subset_1(G,u1_struct_0(A))
=> ( r2_hidden(G,k10_fintopo3(A,D,E))
<=> r2_hidden(k8_funct_2(u1_struct_0(A),u1_struct_0(B),C,G),k10_fintopo3(B,D,F)) ) ) ) ) ) ) ) ) ) ).
fof(t5_fintopo5,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l1_fin_topo(A) )
=> ! [B] :
( ( ~ v3_struct_0(B)
& l1_fin_topo(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] :
( m2_subset_1(D,k1_numbers,k5_numbers)
=> ! [E] :
( m1_subset_1(E,u1_struct_0(A))
=> ! [F] :
( m1_subset_1(F,u1_struct_0(B))
=> ( ( r1_fintopo5(A,B,C)
& F = k8_funct_2(u1_struct_0(A),u1_struct_0(B),C,E) )
=> ! [G] :
( m1_subset_1(G,u1_struct_0(B))
=> ( r2_hidden(k1_funct_1(k2_funct_1(C),G),k10_fintopo3(A,D,E))
<=> r2_hidden(G,k10_fintopo3(B,D,F)) ) ) ) ) ) ) ) ) ) ).
fof(t6_fintopo5,axiom,
! [A] :
( ( ~ v1_xboole_0(A)
& m2_subset_1(A,k1_numbers,k5_numbers) )
=> ! [B] :
( ( v1_funct_1(B)
& v1_funct_2(B,u1_struct_0(k2_fintopo4(A)),u1_struct_0(k2_fintopo4(A)))
& m2_relset_1(B,u1_struct_0(k2_fintopo4(A)),u1_struct_0(k2_fintopo4(A))) )
=> ~ ( r2_fintopo4(k2_fintopo4(A),k2_fintopo4(A),B,np__0)
& ! [C] :
( m1_subset_1(C,u1_struct_0(k2_fintopo4(A)))
=> ~ r2_hidden(k8_funct_2(u1_struct_0(k2_fintopo4(A)),u1_struct_0(k2_fintopo4(A)),B,C),k10_fintopo3(k2_fintopo4(A),np__0,C)) ) ) ) ) ).
fof(t7_fintopo5,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l1_fin_topo(A) )
=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> ! [C] :
( m2_subset_1(C,k1_numbers,k5_numbers)
=> ( v2_fin_topo(A)
=> r1_tarski(k10_fintopo3(A,C,B),k10_fintopo3(A,k1_nat_1(C,np__1),B)) ) ) ) ) ).
fof(t8_fintopo5,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l1_fin_topo(A) )
=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> ! [C] :
( m2_subset_1(C,k1_numbers,k5_numbers)
=> ( v2_fin_topo(A)
=> r1_tarski(k10_fintopo3(A,np__0,B),k10_fintopo3(A,C,B)) ) ) ) ) ).
fof(t9_fintopo5,axiom,
! [A] :
( ( ~ v1_xboole_0(A)
& m2_subset_1(A,k1_numbers,k5_numbers) )
=> ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> ! [C] :
( m2_subset_1(C,k1_numbers,k5_numbers)
=> ! [D] :
( m2_subset_1(D,k1_numbers,k5_numbers)
=> ! [E] :
( m1_subset_1(E,u1_struct_0(k2_fintopo4(A)))
=> ( E = B
=> ( r2_hidden(C,k10_fintopo3(k2_fintopo4(A),D,E))
<=> ( r2_hidden(C,k2_finseq_1(A))
& r1_xreal_0(k18_complex1(k6_xcmplx_0(B,C)),k1_nat_1(D,np__1)) ) ) ) ) ) ) ) ) ).
fof(t10_fintopo5,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> ! [C] :
( ( ~ v1_xboole_0(C)
& m2_subset_1(C,k1_numbers,k5_numbers) )
=> ! [D] :
( ( v1_funct_1(D)
& v1_funct_2(D,u1_struct_0(k2_fintopo4(C)),u1_struct_0(k2_fintopo4(C)))
& m2_relset_1(D,u1_struct_0(k2_fintopo4(C)),u1_struct_0(k2_fintopo4(C))) )
=> ~ ( r2_fintopo4(k2_fintopo4(C),k2_fintopo4(C),D,A)
& B = k2_int_1(k7_xcmplx_0(A,np__2))
& ! [E] :
( m1_subset_1(E,u1_struct_0(k2_fintopo4(C)))
=> ~ r2_hidden(k8_funct_2(u1_struct_0(k2_fintopo4(C)),u1_struct_0(k2_fintopo4(C)),D,E),k10_fintopo3(k2_fintopo4(C),B,E)) ) ) ) ) ) ) ).
fof(d2_fintopo5,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> ! [C] :
( ( v1_funct_1(C)
& v1_funct_2(C,k2_zfmisc_1(k2_finseq_1(A),k2_finseq_1(B)),k1_zfmisc_1(k2_zfmisc_1(k2_finseq_1(A),k2_finseq_1(B))))
& m2_relset_1(C,k2_zfmisc_1(k2_finseq_1(A),k2_finseq_1(B)),k1_zfmisc_1(k2_zfmisc_1(k2_finseq_1(A),k2_finseq_1(B)))) )
=> ( C = k1_fintopo5(A,B)
<=> ! [D] :
( r2_hidden(D,k2_zfmisc_1(k2_finseq_1(A),k2_finseq_1(B)))
=> ! [E] :
( m2_subset_1(E,k1_numbers,k5_numbers)
=> ! [F] :
( m2_subset_1(F,k1_numbers,k5_numbers)
=> ( D = k4_tarski(E,F)
=> k1_funct_1(C,D) = k2_zfmisc_1(k1_funct_1(k1_fintopo4(A),E),k1_funct_1(k1_fintopo4(B),F)) ) ) ) ) ) ) ) ) ).
fof(d3_fintopo5,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> k2_fintopo5(A,B) = g1_fin_topo(k2_zfmisc_1(k2_finseq_1(A),k2_finseq_1(B)),k1_fintopo5(A,B)) ) ) ).
fof(t11_fintopo5,axiom,
! [A] :
( ( ~ v1_xboole_0(A)
& m2_subset_1(A,k1_numbers,k5_numbers) )
=> ! [B] :
( ( ~ v1_xboole_0(B)
& m2_subset_1(B,k1_numbers,k5_numbers) )
=> v2_fin_topo(k2_fintopo5(A,B)) ) ) ).
fof(t12_fintopo5,axiom,
! [A] :
( ( ~ v1_xboole_0(A)
& m2_subset_1(A,k1_numbers,k5_numbers) )
=> ! [B] :
( ( ~ v1_xboole_0(B)
& m2_subset_1(B,k1_numbers,k5_numbers) )
=> v3_fin_topo(k2_fintopo5(A,B)) ) ) ).
fof(t13_fintopo5,axiom,
! [A] :
( ( ~ v1_xboole_0(A)
& m2_subset_1(A,k1_numbers,k5_numbers) )
=> ? [B] :
( v1_funct_1(B)
& v1_funct_2(B,u1_struct_0(k2_fintopo5(A,np__1)),u1_struct_0(k2_fintopo4(A)))
& m2_relset_1(B,u1_struct_0(k2_fintopo5(A,np__1)),u1_struct_0(k2_fintopo4(A)))
& r1_fintopo5(k2_fintopo5(A,np__1),k2_fintopo4(A),B) ) ) ).
fof(d4_fintopo5,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> ! [C] :
( ( v1_funct_1(C)
& v1_funct_2(C,k2_zfmisc_1(k2_finseq_1(A),k2_finseq_1(B)),k1_zfmisc_1(k2_zfmisc_1(k2_finseq_1(A),k2_finseq_1(B))))
& m2_relset_1(C,k2_zfmisc_1(k2_finseq_1(A),k2_finseq_1(B)),k1_zfmisc_1(k2_zfmisc_1(k2_finseq_1(A),k2_finseq_1(B)))) )
=> ( C = k3_fintopo5(A,B)
<=> ! [D] :
( r2_hidden(D,k2_zfmisc_1(k2_finseq_1(A),k2_finseq_1(B)))
=> ! [E] :
( m2_subset_1(E,k1_numbers,k5_numbers)
=> ! [F] :
( m2_subset_1(F,k1_numbers,k5_numbers)
=> ( D = k4_tarski(E,F)
=> k1_funct_1(C,D) = k2_xboole_0(k2_zfmisc_1(k1_tarski(E),k1_funct_1(k1_fintopo4(B),F)),k2_zfmisc_1(k1_funct_1(k1_fintopo4(A),E),k1_tarski(F))) ) ) ) ) ) ) ) ) ).
fof(d5_fintopo5,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> k4_fintopo5(A,B) = g1_fin_topo(k2_zfmisc_1(k2_finseq_1(A),k2_finseq_1(B)),k3_fintopo5(A,B)) ) ) ).
fof(t14_fintopo5,axiom,
! [A] :
( ( ~ v1_xboole_0(A)
& m2_subset_1(A,k1_numbers,k5_numbers) )
=> ! [B] :
( ( ~ v1_xboole_0(B)
& m2_subset_1(B,k1_numbers,k5_numbers) )
=> v2_fin_topo(k4_fintopo5(A,B)) ) ) ).
fof(t15_fintopo5,axiom,
! [A] :
( ( ~ v1_xboole_0(A)
& m2_subset_1(A,k1_numbers,k5_numbers) )
=> ! [B] :
( ( ~ v1_xboole_0(B)
& m2_subset_1(B,k1_numbers,k5_numbers) )
=> v3_fin_topo(k4_fintopo5(A,B)) ) ) ).
fof(t16_fintopo5,axiom,
! [A] :
( ( ~ v1_xboole_0(A)
& m2_subset_1(A,k1_numbers,k5_numbers) )
=> ? [B] :
( v1_funct_1(B)
& v1_funct_2(B,u1_struct_0(k4_fintopo5(A,np__1)),u1_struct_0(k2_fintopo4(A)))
& m2_relset_1(B,u1_struct_0(k4_fintopo5(A,np__1)),u1_struct_0(k2_fintopo4(A)))
& r1_fintopo5(k4_fintopo5(A,np__1),k2_fintopo4(A),B) ) ) ).
fof(dt_k1_fintopo5,axiom,
! [A,B] :
( ( m1_subset_1(A,k5_numbers)
& m1_subset_1(B,k5_numbers) )
=> ( v1_funct_1(k1_fintopo5(A,B))
& v1_funct_2(k1_fintopo5(A,B),k2_zfmisc_1(k2_finseq_1(A),k2_finseq_1(B)),k1_zfmisc_1(k2_zfmisc_1(k2_finseq_1(A),k2_finseq_1(B))))
& m2_relset_1(k1_fintopo5(A,B),k2_zfmisc_1(k2_finseq_1(A),k2_finseq_1(B)),k1_zfmisc_1(k2_zfmisc_1(k2_finseq_1(A),k2_finseq_1(B)))) ) ) ).
fof(dt_k2_fintopo5,axiom,
! [A,B] :
( ( m1_subset_1(A,k5_numbers)
& m1_subset_1(B,k5_numbers) )
=> ( v1_fin_topo(k2_fintopo5(A,B))
& l1_fin_topo(k2_fintopo5(A,B)) ) ) ).
fof(dt_k3_fintopo5,axiom,
! [A,B] :
( ( m1_subset_1(A,k5_numbers)
& m1_subset_1(B,k5_numbers) )
=> ( v1_funct_1(k3_fintopo5(A,B))
& v1_funct_2(k3_fintopo5(A,B),k2_zfmisc_1(k2_finseq_1(A),k2_finseq_1(B)),k1_zfmisc_1(k2_zfmisc_1(k2_finseq_1(A),k2_finseq_1(B))))
& m2_relset_1(k3_fintopo5(A,B),k2_zfmisc_1(k2_finseq_1(A),k2_finseq_1(B)),k1_zfmisc_1(k2_zfmisc_1(k2_finseq_1(A),k2_finseq_1(B)))) ) ) ).
fof(dt_k4_fintopo5,axiom,
! [A,B] :
( ( m1_subset_1(A,k5_numbers)
& m1_subset_1(B,k5_numbers) )
=> ( v1_fin_topo(k4_fintopo5(A,B))
& l1_fin_topo(k4_fintopo5(A,B)) ) ) ).
%------------------------------------------------------------------------------