SET007 Axioms: SET007+589.ax
%------------------------------------------------------------------------------
% File : SET007+589 : TPTP v8.2.0. Released v3.4.0.
% Domain : Set Theory
% Axioms : Homeomorphism between E^i_T, E^j_T and E^i+j_T
% 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 : topreal7 [Urb08]
% Status : Satisfiable
% Syntax : Number of formulae : 40 ( 0 unt; 0 def)
% Number of atoms : 313 ( 50 equ)
% Maximal formula atoms : 30 ( 7 avg)
% Number of connectives : 326 ( 53 ~; 3 |; 155 &)
% ( 1 <=>; 114 =>; 0 <=; 0 <~>)
% Maximal formula depth : 23 ( 9 avg)
% Maximal term depth : 4 ( 1 avg)
% Number of predicates : 18 ( 17 usr; 0 prp; 1-3 aty)
% Number of functors : 33 ( 33 usr; 3 con; 0-5 aty)
% Number of variables : 124 ( 120 !; 4 ?)
% 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_topreal7,axiom,
! [A,B] :
( ( ~ v3_struct_0(A)
& l1_metric_1(A)
& ~ v3_struct_0(B)
& l1_metric_1(B) )
=> ( ~ v3_struct_0(k1_topreal7(A,B))
& v1_metric_1(k1_topreal7(A,B)) ) ) ).
fof(fc2_topreal7,axiom,
! [A,B] :
( ( ~ v3_struct_0(A)
& v6_metric_1(A)
& l1_metric_1(A)
& ~ v3_struct_0(B)
& v6_metric_1(B)
& l1_metric_1(B) )
=> ( ~ v3_struct_0(k1_topreal7(A,B))
& v1_metric_1(k1_topreal7(A,B))
& v6_metric_1(k1_topreal7(A,B)) ) ) ).
fof(fc3_topreal7,axiom,
! [A,B] :
( ( ~ v3_struct_0(A)
& v8_metric_1(A)
& l1_metric_1(A)
& ~ v3_struct_0(B)
& v8_metric_1(B)
& l1_metric_1(B) )
=> ( ~ v3_struct_0(k1_topreal7(A,B))
& v1_metric_1(k1_topreal7(A,B))
& v8_metric_1(k1_topreal7(A,B)) ) ) ).
fof(fc4_topreal7,axiom,
! [A,B] :
( ( ~ v3_struct_0(A)
& v9_metric_1(A)
& l1_metric_1(A)
& ~ v3_struct_0(B)
& v9_metric_1(B)
& l1_metric_1(B) )
=> ( ~ v3_struct_0(k1_topreal7(A,B))
& v1_metric_1(k1_topreal7(A,B))
& v9_metric_1(k1_topreal7(A,B)) ) ) ).
fof(fc5_topreal7,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)
& ~ v3_struct_0(B)
& v6_metric_1(B)
& v7_metric_1(B)
& v8_metric_1(B)
& v9_metric_1(B)
& l1_metric_1(B) )
=> ( ~ v3_struct_0(k1_topreal7(A,B))
& v1_metric_1(k1_topreal7(A,B))
& v6_metric_1(k1_topreal7(A,B))
& v7_metric_1(k1_topreal7(A,B))
& v8_metric_1(k1_topreal7(A,B))
& v9_metric_1(k1_topreal7(A,B)) ) ) ).
fof(t1_topreal7,axiom,
! [A] :
( m1_subset_1(A,k1_numbers)
=> ! [B] :
( m1_subset_1(B,k1_numbers)
=> ( r1_xreal_0(k4_square_1(A,B),A)
=> k4_square_1(A,B) = A ) ) ) ).
fof(t2_topreal7,axiom,
! [A] :
( m1_subset_1(A,k1_numbers)
=> ! [B] :
( m1_subset_1(B,k1_numbers)
=> ! [C] :
( m1_subset_1(C,k1_numbers)
=> ! [D] :
( m1_subset_1(D,k1_numbers)
=> r1_xreal_0(k4_square_1(k3_real_1(A,C),k3_real_1(B,D)),k3_real_1(k4_square_1(A,B),k4_square_1(C,D))) ) ) ) ) ).
fof(t3_topreal7,axiom,
! [A] :
( m1_subset_1(A,k1_numbers)
=> ! [B] :
( m1_subset_1(B,k1_numbers)
=> ! [C] :
( m1_subset_1(C,k1_numbers)
=> ! [D] :
( m1_subset_1(D,k1_numbers)
=> ! [E] :
( m1_subset_1(E,k1_numbers)
=> ! [F] :
( m1_subset_1(F,k1_numbers)
=> ( ( r1_xreal_0(A,k3_real_1(B,C))
& r1_xreal_0(D,k3_real_1(E,F)) )
=> r1_xreal_0(k4_square_1(A,D),k3_real_1(k4_square_1(B,E),k4_square_1(C,F))) ) ) ) ) ) ) ) ).
fof(t4_topreal7,axiom,
! [A] :
( ( v1_relat_1(A)
& v1_funct_1(A)
& v1_finseq_1(A) )
=> ! [B] :
( ( v1_relat_1(B)
& v1_funct_1(B)
& v1_finseq_1(B) )
=> r1_tarski(k4_finseq_1(B),k4_finseq_1(k7_finseq_1(A,B))) ) ) ).
fof(t5_topreal7,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( ( v1_relat_1(B)
& v1_funct_1(B)
& v1_finseq_1(B) )
=> ! [C] :
( ( v1_relat_1(C)
& v1_funct_1(C)
& v1_finseq_1(C) )
=> ( r1_xreal_0(A,k1_nat_1(k3_finseq_1(B),k3_finseq_1(C)))
=> ( r1_xreal_0(A,k3_finseq_1(B))
| r2_hidden(k5_real_1(A,k3_finseq_1(B)),k4_finseq_1(C)) ) ) ) ) ) ).
fof(t6_topreal7,axiom,
! [A] :
( ( v1_relat_1(A)
& v1_funct_1(A)
& v1_finseq_1(A) )
=> ! [B] :
( ( v1_relat_1(B)
& v1_funct_1(B)
& v1_finseq_1(B) )
=> ! [C] :
( ( v1_relat_1(C)
& v1_funct_1(C)
& v1_finseq_1(C) )
=> ! [D] :
( ( v1_relat_1(D)
& v1_funct_1(D)
& v1_finseq_1(D) )
=> ( ( k7_finseq_1(A,B) = k7_finseq_1(C,D)
& k3_finseq_1(A) = k3_finseq_1(C)
& k3_finseq_1(B) = k3_finseq_1(D) )
=> ( A = C
& B = D ) ) ) ) ) ) ).
fof(t7_topreal7,axiom,
! [A] :
( m2_finseq_1(A,k1_numbers)
=> ! [B] :
( m2_finseq_1(B,k1_numbers)
=> ( ( k3_finseq_1(A) = k3_finseq_1(B)
| k4_finseq_1(A) = k4_finseq_1(B) )
=> ( k3_finseq_1(k3_rvsum_1(A,B)) = k3_finseq_1(A)
& k4_finseq_1(k3_rvsum_1(A,B)) = k4_finseq_1(A) ) ) ) ) ).
fof(t8_topreal7,axiom,
! [A] :
( m2_finseq_1(A,k1_numbers)
=> ! [B] :
( m2_finseq_1(B,k1_numbers)
=> ( ( k3_finseq_1(A) = k3_finseq_1(B)
| k4_finseq_1(A) = k4_finseq_1(B) )
=> ( k3_finseq_1(k7_rvsum_1(A,B)) = k3_finseq_1(A)
& k4_finseq_1(k7_rvsum_1(A,B)) = k4_finseq_1(A) ) ) ) ) ).
fof(t9_topreal7,axiom,
! [A] :
( m2_finseq_1(A,k1_numbers)
=> ( k3_finseq_1(A) = k3_finseq_1(k11_rvsum_1(A))
& k4_finseq_1(A) = k4_finseq_1(k11_rvsum_1(A)) ) ) ).
fof(t10_topreal7,axiom,
! [A] :
( m2_finseq_1(A,k1_numbers)
=> ( k3_finseq_1(A) = k3_finseq_1(k3_euclid(A))
& k4_finseq_1(A) = k4_finseq_1(k3_euclid(A)) ) ) ).
fof(t11_topreal7,axiom,
! [A] :
( m2_finseq_1(A,k1_numbers)
=> ! [B] :
( m2_finseq_1(B,k1_numbers)
=> k11_rvsum_1(k8_finseq_1(k1_numbers,A,B)) = k8_finseq_1(k1_numbers,k11_rvsum_1(A),k11_rvsum_1(B)) ) ) ).
fof(t12_topreal7,axiom,
! [A] :
( m2_finseq_1(A,k1_numbers)
=> ! [B] :
( m2_finseq_1(B,k1_numbers)
=> k3_euclid(k8_finseq_1(k1_numbers,A,B)) = k8_finseq_1(k1_numbers,k3_euclid(A),k3_euclid(B)) ) ) ).
fof(t13_topreal7,axiom,
! [A] :
( m2_finseq_1(A,k1_numbers)
=> ! [B] :
( m2_finseq_1(B,k1_numbers)
=> ! [C] :
( m2_finseq_1(C,k1_numbers)
=> ! [D] :
( m2_finseq_1(D,k1_numbers)
=> ( ( k3_finseq_1(A) = k3_finseq_1(B)
& k3_finseq_1(C) = k3_finseq_1(D) )
=> k11_rvsum_1(k3_rvsum_1(k8_finseq_1(k1_numbers,A,C),k8_finseq_1(k1_numbers,B,D))) = k8_finseq_1(k1_numbers,k11_rvsum_1(k3_rvsum_1(A,B)),k11_rvsum_1(k3_rvsum_1(C,D))) ) ) ) ) ) ).
fof(t14_topreal7,axiom,
! [A] :
( m2_finseq_1(A,k1_numbers)
=> ! [B] :
( m2_finseq_1(B,k1_numbers)
=> ! [C] :
( m2_finseq_1(C,k1_numbers)
=> ! [D] :
( m2_finseq_1(D,k1_numbers)
=> ( ( k3_finseq_1(A) = k3_finseq_1(B)
& k3_finseq_1(C) = k3_finseq_1(D) )
=> k3_euclid(k3_rvsum_1(k8_finseq_1(k1_numbers,A,C),k8_finseq_1(k1_numbers,B,D))) = k8_finseq_1(k1_numbers,k3_euclid(k3_rvsum_1(A,B)),k3_euclid(k3_rvsum_1(C,D))) ) ) ) ) ) ).
fof(t15_topreal7,axiom,
! [A] :
( m2_finseq_1(A,k1_numbers)
=> ! [B] :
( m2_finseq_1(B,k1_numbers)
=> ! [C] :
( m2_finseq_1(C,k1_numbers)
=> ! [D] :
( m2_finseq_1(D,k1_numbers)
=> ( ( k3_finseq_1(A) = k3_finseq_1(B)
& k3_finseq_1(C) = k3_finseq_1(D) )
=> k11_rvsum_1(k7_rvsum_1(k8_finseq_1(k1_numbers,A,C),k8_finseq_1(k1_numbers,B,D))) = k8_finseq_1(k1_numbers,k11_rvsum_1(k7_rvsum_1(A,B)),k11_rvsum_1(k7_rvsum_1(C,D))) ) ) ) ) ) ).
fof(t16_topreal7,axiom,
! [A] :
( m2_finseq_1(A,k1_numbers)
=> ! [B] :
( m2_finseq_1(B,k1_numbers)
=> ! [C] :
( m2_finseq_1(C,k1_numbers)
=> ! [D] :
( m2_finseq_1(D,k1_numbers)
=> ( ( k3_finseq_1(A) = k3_finseq_1(B)
& k3_finseq_1(C) = k3_finseq_1(D) )
=> k3_euclid(k7_rvsum_1(k8_finseq_1(k1_numbers,A,C),k8_finseq_1(k1_numbers,B,D))) = k8_finseq_1(k1_numbers,k3_euclid(k7_rvsum_1(A,B)),k3_euclid(k7_rvsum_1(C,D))) ) ) ) ) ) ).
fof(t17_topreal7,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m2_finseq_1(B,k1_numbers)
=> ( k3_finseq_1(B) = A
=> r2_hidden(B,u1_struct_0(k14_euclid(A))) ) ) ) ).
fof(t18_topreal7,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m2_finseq_1(B,k1_numbers)
=> ( k3_finseq_1(B) = A
=> r2_hidden(B,u1_struct_0(k15_euclid(A))) ) ) ) ).
fof(t19_topreal7,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( ( v1_relat_1(B)
& v1_funct_1(B)
& v1_finseq_1(B) )
=> ( r2_hidden(B,u1_struct_0(k14_euclid(A)))
=> k3_finseq_1(B) = A ) ) ) ).
fof(d1_topreal7,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l1_metric_1(A) )
=> ! [B] :
( ( ~ v3_struct_0(B)
& l1_metric_1(B) )
=> ! [C] :
( ( v1_metric_1(C)
& l1_metric_1(C) )
=> ( C = k1_topreal7(A,B)
<=> ( u1_struct_0(C) = k2_zfmisc_1(u1_struct_0(A),u1_struct_0(B))
& ! [D] :
( m1_subset_1(D,u1_struct_0(C))
=> ! [E] :
( m1_subset_1(E,u1_struct_0(C))
=> ? [F] :
( m1_subset_1(F,u1_struct_0(A))
& ? [G] :
( m1_subset_1(G,u1_struct_0(A))
& ? [H] :
( m1_subset_1(H,u1_struct_0(B))
& ? [I] :
( m1_subset_1(I,u1_struct_0(B))
& D = k1_domain_1(u1_struct_0(A),u1_struct_0(B),F,H)
& E = k1_domain_1(u1_struct_0(A),u1_struct_0(B),G,I)
& k1_metric_1(u1_struct_0(C),u1_struct_0(C),u1_metric_1(C),D,E) = k4_square_1(k1_metric_1(u1_struct_0(A),u1_struct_0(A),u1_metric_1(A),F,G),k1_metric_1(u1_struct_0(B),u1_struct_0(B),u1_metric_1(B),H,I)) ) ) ) ) ) ) ) ) ) ) ) ).
fof(t20_topreal7,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l1_metric_1(A) )
=> ! [B] :
( ( ~ v3_struct_0(B)
& l1_metric_1(B) )
=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> ! [D] :
( m1_subset_1(D,u1_struct_0(A))
=> ! [E] :
( m1_subset_1(E,u1_struct_0(B))
=> ! [F] :
( m1_subset_1(F,u1_struct_0(B))
=> k2_metric_1(k1_topreal7(A,B),k2_topreal7(A,B,C,E),k2_topreal7(A,B,D,F)) = k4_square_1(k2_metric_1(A,C,D),k2_metric_1(B,E,F)) ) ) ) ) ) ) ).
fof(t21_topreal7,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l1_metric_1(A) )
=> ! [B] :
( ( ~ v3_struct_0(B)
& l1_metric_1(B) )
=> ! [C] :
( m1_subset_1(C,u1_struct_0(k1_topreal7(A,B)))
=> ! [D] :
( m1_subset_1(D,u1_struct_0(k1_topreal7(A,B)))
=> k2_metric_1(k1_topreal7(A,B),C,D) = k4_square_1(k2_metric_1(A,k3_topreal7(A,B,C),k3_topreal7(A,B,D)),k2_metric_1(B,k4_topreal7(A,B,C),k4_topreal7(A,B,D))) ) ) ) ) ).
fof(t22_topreal7,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v6_metric_1(A)
& l1_metric_1(A) )
=> ! [B] :
( ( ~ v3_struct_0(B)
& v6_metric_1(B)
& l1_metric_1(B) )
=> v6_metric_1(k1_topreal7(A,B)) ) ) ).
fof(t23_topreal7,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v8_metric_1(A)
& l1_metric_1(A) )
=> ! [B] :
( ( ~ v3_struct_0(B)
& v8_metric_1(B)
& l1_metric_1(B) )
=> v8_metric_1(k1_topreal7(A,B)) ) ) ).
fof(t24_topreal7,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v9_metric_1(A)
& l1_metric_1(A) )
=> ! [B] :
( ( ~ v3_struct_0(B)
& v9_metric_1(B)
& l1_metric_1(B) )
=> v9_metric_1(k1_topreal7(A,B)) ) ) ).
fof(t25_topreal7,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) )
=> k6_borsuk_1(k5_pcomps_1(A),k5_pcomps_1(B)) = k5_pcomps_1(k1_topreal7(A,B)) ) ) ).
fof(t26_topreal7,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) )
=> ( ( u1_struct_0(A) = u1_struct_0(B)
& ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> ! [D] :
( m1_subset_1(D,u1_struct_0(B))
=> ! [E] :
( m1_subset_1(E,k1_numbers)
=> ~ ( ~ r1_xreal_0(E,np__0)
& C = D
& ! [F] :
( m1_subset_1(F,k1_numbers)
=> ~ ( ~ r1_xreal_0(F,np__0)
& r1_tarski(k9_metric_1(B,D,F),k9_metric_1(A,C,E)) ) ) ) ) ) )
& ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> ! [D] :
( m1_subset_1(D,u1_struct_0(B))
=> ! [E] :
( m1_subset_1(E,k1_numbers)
=> ~ ( ~ r1_xreal_0(E,np__0)
& C = D
& ! [F] :
( m1_subset_1(F,k1_numbers)
=> ~ ( ~ r1_xreal_0(F,np__0)
& r1_tarski(k9_metric_1(A,C,F),k9_metric_1(B,D,E)) ) ) ) ) ) ) )
=> k5_pcomps_1(A) = k5_pcomps_1(B) ) ) ) ).
fof(t27_topreal7,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> r1_t_0topsp(k6_borsuk_1(k15_euclid(A),k15_euclid(B)),k15_euclid(k1_nat_1(A,B))) ) ) ).
fof(dt_k1_topreal7,axiom,
! [A,B] :
( ( ~ v3_struct_0(A)
& l1_metric_1(A)
& ~ v3_struct_0(B)
& l1_metric_1(B) )
=> ( v1_metric_1(k1_topreal7(A,B))
& l1_metric_1(k1_topreal7(A,B)) ) ) ).
fof(dt_k2_topreal7,axiom,
! [A,B,C,D] :
( ( ~ v3_struct_0(A)
& l1_metric_1(A)
& ~ v3_struct_0(B)
& l1_metric_1(B)
& m1_subset_1(C,u1_struct_0(A))
& m1_subset_1(D,u1_struct_0(B)) )
=> m1_subset_1(k2_topreal7(A,B,C,D),u1_struct_0(k1_topreal7(A,B))) ) ).
fof(redefinition_k2_topreal7,axiom,
! [A,B,C,D] :
( ( ~ v3_struct_0(A)
& l1_metric_1(A)
& ~ v3_struct_0(B)
& l1_metric_1(B)
& m1_subset_1(C,u1_struct_0(A))
& m1_subset_1(D,u1_struct_0(B)) )
=> k2_topreal7(A,B,C,D) = k4_tarski(C,D) ) ).
fof(dt_k3_topreal7,axiom,
! [A,B,C] :
( ( ~ v3_struct_0(A)
& l1_metric_1(A)
& ~ v3_struct_0(B)
& l1_metric_1(B)
& m1_subset_1(C,u1_struct_0(k1_topreal7(A,B))) )
=> m1_subset_1(k3_topreal7(A,B,C),u1_struct_0(A)) ) ).
fof(redefinition_k3_topreal7,axiom,
! [A,B,C] :
( ( ~ v3_struct_0(A)
& l1_metric_1(A)
& ~ v3_struct_0(B)
& l1_metric_1(B)
& m1_subset_1(C,u1_struct_0(k1_topreal7(A,B))) )
=> k3_topreal7(A,B,C) = k1_mcart_1(C) ) ).
fof(dt_k4_topreal7,axiom,
! [A,B,C] :
( ( ~ v3_struct_0(A)
& l1_metric_1(A)
& ~ v3_struct_0(B)
& l1_metric_1(B)
& m1_subset_1(C,u1_struct_0(k1_topreal7(A,B))) )
=> m1_subset_1(k4_topreal7(A,B,C),u1_struct_0(B)) ) ).
fof(redefinition_k4_topreal7,axiom,
! [A,B,C] :
( ( ~ v3_struct_0(A)
& l1_metric_1(A)
& ~ v3_struct_0(B)
& l1_metric_1(B)
& m1_subset_1(C,u1_struct_0(k1_topreal7(A,B))) )
=> k4_topreal7(A,B,C) = k2_mcart_1(C) ) ).
%------------------------------------------------------------------------------