SET007 Axioms: SET007+917.ax
%------------------------------------------------------------------------------
% File : SET007+917 : TPTP v9.0.0. Released v3.4.0.
% Domain : Set Theory
% Axioms : The Fashoda Meet Theorem for Continuous Mappings
% 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 : jgraph_8 [Urb08]
% Status : Satisfiable
% Syntax : Number of formulae : 8 ( 0 unt; 0 def)
% Number of atoms : 131 ( 23 equ)
% Maximal formula atoms : 30 ( 16 avg)
% Number of connectives : 131 ( 8 ~; 0 |; 64 &)
% ( 0 <=>; 59 =>; 0 <=; 0 <~>)
% Maximal formula depth : 43 ( 21 avg)
% Maximal term depth : 5 ( 2 avg)
% Number of predicates : 22 ( 21 usr; 0 prp; 1-4 aty)
% Number of functors : 27 ( 27 usr; 8 con; 0-4 aty)
% Number of variables : 60 ( 59 !; 1 ?)
% 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_jgraph_8,axiom,
! [A,B,C,D] :
( ( v1_xreal_0(A)
& v1_xreal_0(B)
& v1_xreal_0(C)
& v1_xreal_0(D) )
=> v1_jordan1(k2_jgraph_6(A,B,C,D),np__2) ) ).
fof(fc2_jgraph_8,axiom,
! [A,B,C,D] :
( ( v1_xreal_0(A)
& v1_xreal_0(B)
& v1_xreal_0(C)
& v1_xreal_0(D) )
=> ( v2_pre_topc(k1_topreala(A,B,C,D))
& v1_topalg_2(k1_topreala(A,B,C,D),np__2) ) ) ).
fof(t1_jgraph_8,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( ( v1_xreal_0(B)
& v2_xreal_0(B) )
=> ! [C] :
( ( v1_funct_1(C)
& v1_funct_2(C,u1_struct_0(k22_borsuk_1),u1_struct_0(k15_euclid(A)))
& v5_pre_topc(C,k22_borsuk_1,k15_euclid(A))
& m2_relset_1(C,u1_struct_0(k22_borsuk_1),u1_struct_0(k15_euclid(A))) )
=> ? [D] :
( m2_finseq_1(D,k1_numbers)
& k1_goboard1(D,np__1) = np__0
& k1_goboard1(D,k3_finseq_1(D)) = np__1
& r1_xreal_0(np__5,k3_finseq_1(D))
& r1_tarski(k5_relset_1(k5_numbers,k1_numbers,D),u1_struct_0(k22_borsuk_1))
& v1_goboard1(D)
& ! [E] :
( m2_subset_1(E,k1_numbers,k5_numbers)
=> ! [F] :
( m1_subset_1(F,k1_zfmisc_1(u1_struct_0(k22_borsuk_1)))
=> ! [G] :
( m1_subset_1(G,k1_zfmisc_1(u1_struct_0(k14_euclid(A))))
=> ~ ( r1_xreal_0(np__1,E)
& ~ r1_xreal_0(k3_finseq_1(D),E)
& F = k1_rcomp_1(k4_finseq_4(k5_numbers,k1_numbers,D,E),k4_finseq_4(k5_numbers,k1_numbers,D,k1_nat_1(E,np__1)))
& G = k4_pre_topc(k22_borsuk_1,k15_euclid(A),C,F)
& r1_xreal_0(B,k2_tbsp_1(k14_euclid(A),G)) ) ) ) ) ) ) ) ) ).
fof(t2_jgraph_8,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] :
( m1_subset_1(D,k1_zfmisc_1(u1_struct_0(k15_euclid(A))))
=> ( ( r1_tarski(D,k3_topreal1(A,B,C))
& r2_hidden(B,D)
& r2_hidden(C,D)
& v2_connsp_1(D,k15_euclid(A)) )
=> D = k3_topreal1(A,B,C) ) ) ) ) ) ).
fof(t3_jgraph_8,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] :
( m1_borsuk_2(D,k15_euclid(A),B,C)
=> ( r1_tarski(k1_pscomp_1(u1_struct_0(k5_topmetr),u1_struct_0(k15_euclid(A)),D),k3_topreal1(A,B,C))
=> k1_pscomp_1(u1_struct_0(k5_topmetr),u1_struct_0(k15_euclid(A)),D) = k3_topreal1(A,B,C) ) ) ) ) ) ).
fof(t4_jgraph_8,axiom,
! [A] :
( ( ~ v1_xboole_0(A)
& m1_subset_1(A,k1_zfmisc_1(u1_struct_0(k15_euclid(np__2)))) )
=> ! [B] :
( ( ~ v1_xboole_0(B)
& m1_subset_1(B,k1_zfmisc_1(u1_struct_0(k15_euclid(np__2)))) )
=> ! [C] :
( m1_subset_1(C,u1_struct_0(k15_euclid(np__2)))
=> ! [D] :
( m1_subset_1(D,u1_struct_0(k15_euclid(np__2)))
=> ! [E] :
( m1_subset_1(E,u1_struct_0(k15_euclid(np__2)))
=> ! [F] :
( m1_subset_1(F,u1_struct_0(k15_euclid(np__2)))
=> ! [G] :
( m1_borsuk_2(G,k15_euclid(np__2),C,D)
=> ! [H] :
( m1_borsuk_2(H,k15_euclid(np__2),E,F)
=> ~ ( k1_pscomp_1(u1_struct_0(k5_topmetr),u1_struct_0(k15_euclid(np__2)),G) = A
& k1_pscomp_1(u1_struct_0(k5_topmetr),u1_struct_0(k15_euclid(np__2)),H) = B
& ! [I] :
( m1_subset_1(I,u1_struct_0(k15_euclid(np__2)))
=> ( r2_hidden(I,A)
=> ( r1_xreal_0(k21_euclid(C),k21_euclid(I))
& r1_xreal_0(k21_euclid(I),k21_euclid(D)) ) ) )
& ! [I] :
( m1_subset_1(I,u1_struct_0(k15_euclid(np__2)))
=> ( r2_hidden(I,B)
=> ( r1_xreal_0(k21_euclid(C),k21_euclid(I))
& r1_xreal_0(k21_euclid(I),k21_euclid(D)) ) ) )
& ! [I] :
( m1_subset_1(I,u1_struct_0(k15_euclid(np__2)))
=> ( r2_hidden(I,A)
=> ( r1_xreal_0(k22_euclid(E),k22_euclid(I))
& r1_xreal_0(k22_euclid(I),k22_euclid(F)) ) ) )
& ! [I] :
( m1_subset_1(I,u1_struct_0(k15_euclid(np__2)))
=> ( r2_hidden(I,B)
=> ( r1_xreal_0(k22_euclid(E),k22_euclid(I))
& r1_xreal_0(k22_euclid(I),k22_euclid(F)) ) ) )
& r2_subset_1(A,B) ) ) ) ) ) ) ) ) ) ).
fof(t5_jgraph_8,axiom,
! [A] :
( v1_xreal_0(A)
=> ! [B] :
( v1_xreal_0(B)
=> ! [C] :
( v1_xreal_0(C)
=> ! [D] :
( v1_xreal_0(D)
=> ! [E] :
( ( v1_funct_1(E)
& v1_funct_2(E,u1_struct_0(k22_borsuk_1),u1_struct_0(k15_euclid(np__2)))
& v5_pre_topc(E,k22_borsuk_1,k15_euclid(np__2))
& m2_relset_1(E,u1_struct_0(k22_borsuk_1),u1_struct_0(k15_euclid(np__2))) )
=> ! [F] :
( ( v1_funct_1(F)
& v1_funct_2(F,u1_struct_0(k22_borsuk_1),u1_struct_0(k15_euclid(np__2)))
& v5_pre_topc(F,k22_borsuk_1,k15_euclid(np__2))
& m2_relset_1(F,u1_struct_0(k22_borsuk_1),u1_struct_0(k15_euclid(np__2))) )
=> ! [G] :
( m1_subset_1(G,u1_struct_0(k22_borsuk_1))
=> ! [H] :
( m1_subset_1(H,u1_struct_0(k22_borsuk_1))
=> ~ ( G = np__0
& H = np__1
& k21_euclid(k8_funct_2(u1_struct_0(k22_borsuk_1),u1_struct_0(k15_euclid(np__2)),E,G)) = A
& k21_euclid(k8_funct_2(u1_struct_0(k22_borsuk_1),u1_struct_0(k15_euclid(np__2)),E,H)) = B
& k22_euclid(k8_funct_2(u1_struct_0(k22_borsuk_1),u1_struct_0(k15_euclid(np__2)),F,G)) = C
& k22_euclid(k8_funct_2(u1_struct_0(k22_borsuk_1),u1_struct_0(k15_euclid(np__2)),F,H)) = D
& ! [I] :
( m1_subset_1(I,u1_struct_0(k22_borsuk_1))
=> ( r1_xreal_0(A,k21_euclid(k8_funct_2(u1_struct_0(k22_borsuk_1),u1_struct_0(k15_euclid(np__2)),E,I)))
& r1_xreal_0(k21_euclid(k8_funct_2(u1_struct_0(k22_borsuk_1),u1_struct_0(k15_euclid(np__2)),E,I)),B)
& r1_xreal_0(A,k21_euclid(k8_funct_2(u1_struct_0(k22_borsuk_1),u1_struct_0(k15_euclid(np__2)),F,I)))
& r1_xreal_0(k21_euclid(k8_funct_2(u1_struct_0(k22_borsuk_1),u1_struct_0(k15_euclid(np__2)),F,I)),B)
& r1_xreal_0(C,k22_euclid(k8_funct_2(u1_struct_0(k22_borsuk_1),u1_struct_0(k15_euclid(np__2)),E,I)))
& r1_xreal_0(k22_euclid(k8_funct_2(u1_struct_0(k22_borsuk_1),u1_struct_0(k15_euclid(np__2)),E,I)),D)
& r1_xreal_0(C,k22_euclid(k8_funct_2(u1_struct_0(k22_borsuk_1),u1_struct_0(k15_euclid(np__2)),F,I)))
& r1_xreal_0(k22_euclid(k8_funct_2(u1_struct_0(k22_borsuk_1),u1_struct_0(k15_euclid(np__2)),F,I)),D) ) )
& r1_xboole_0(k1_pscomp_1(u1_struct_0(k22_borsuk_1),u1_struct_0(k15_euclid(np__2)),E),k1_pscomp_1(u1_struct_0(k22_borsuk_1),u1_struct_0(k15_euclid(np__2)),F)) ) ) ) ) ) ) ) ) ) ).
fof(t6_jgraph_8,axiom,
! [A] :
( v1_xreal_0(A)
=> ! [B] :
( v1_xreal_0(B)
=> ! [C] :
( v1_xreal_0(C)
=> ! [D] :
( v1_xreal_0(D)
=> ! [E] :
( m1_subset_1(E,u1_struct_0(k1_topreala(A,B,C,D)))
=> ! [F] :
( m1_subset_1(F,u1_struct_0(k1_topreala(A,B,C,D)))
=> ! [G] :
( m1_subset_1(G,u1_struct_0(k1_topreala(A,B,C,D)))
=> ! [H] :
( m1_subset_1(H,u1_struct_0(k1_topreala(A,B,C,D)))
=> ! [I] :
( m1_borsuk_2(I,k1_topreala(A,B,C,D),E,F)
=> ! [J] :
( m1_borsuk_2(J,k1_topreala(A,B,C,D),H,G)
=> ! [K] :
( m1_subset_1(K,u1_struct_0(k15_euclid(np__2)))
=> ! [L] :
( m1_subset_1(L,u1_struct_0(k15_euclid(np__2)))
=> ! [M] :
( m1_subset_1(M,u1_struct_0(k15_euclid(np__2)))
=> ! [N] :
( m1_subset_1(N,u1_struct_0(k15_euclid(np__2)))
=> ~ ( k21_euclid(K) = A
& k21_euclid(L) = B
& k22_euclid(M) = C
& k22_euclid(N) = D
& E = K
& F = L
& G = M
& H = N
& ! [O] :
( m1_subset_1(O,u1_struct_0(k22_borsuk_1))
=> ! [P] :
( m1_subset_1(P,u1_struct_0(k22_borsuk_1))
=> k8_funct_2(u1_struct_0(k22_borsuk_1),u1_struct_0(k1_topreala(A,B,C,D)),I,O) != k8_funct_2(u1_struct_0(k22_borsuk_1),u1_struct_0(k1_topreala(A,B,C,D)),J,P) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
%------------------------------------------------------------------------------