SET007 Axioms: SET007+835.ax
%------------------------------------------------------------------------------
% File : SET007+835 : TPTP v8.2.0. Released v3.4.0.
% Domain : Set Theory
% Axioms : The Uniform Continuity of Functions on Normed Linear Spaces
% 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 : nfcont_2 [Urb08]
% Status : Satisfiable
% Syntax : Number of formulae : 21 ( 0 unt; 0 def)
% Number of atoms : 406 ( 9 equ)
% Maximal formula atoms : 30 ( 19 avg)
% Number of connectives : 442 ( 57 ~; 0 |; 291 &)
% ( 3 <=>; 91 =>; 0 <=; 0 <~>)
% Maximal formula depth : 27 ( 15 avg)
% Maximal term depth : 5 ( 1 avg)
% Number of predicates : 27 ( 26 usr; 0 prp; 1-4 aty)
% Number of functors : 27 ( 27 usr; 5 con; 0-4 aty)
% Number of variables : 95 ( 92 !; 3 ?)
% 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(d1_nfcont_2,axiom,
! [A,B] :
( ( ~ v3_struct_0(B)
& v3_rlvect_1(B)
& v4_rlvect_1(B)
& v5_rlvect_1(B)
& v6_rlvect_1(B)
& v7_rlvect_1(B)
& v2_normsp_1(B)
& l1_normsp_1(B) )
=> ! [C] :
( ( ~ v3_struct_0(C)
& v3_rlvect_1(C)
& v4_rlvect_1(C)
& v5_rlvect_1(C)
& v6_rlvect_1(C)
& v7_rlvect_1(C)
& v2_normsp_1(C)
& l1_normsp_1(C) )
=> ! [D] :
( ( v1_funct_1(D)
& m2_relset_1(D,u1_struct_0(B),u1_struct_0(C)) )
=> ( r1_nfcont_2(A,B,C,D)
<=> ( r1_tarski(A,k4_relset_1(u1_struct_0(B),u1_struct_0(C),D))
& ! [E] :
( m1_subset_1(E,k1_numbers)
=> ~ ( ~ r1_xreal_0(E,np__0)
& ! [F] :
( m1_subset_1(F,k1_numbers)
=> ~ ( ~ r1_xreal_0(F,np__0)
& ! [G] :
( m1_subset_1(G,u1_struct_0(B))
=> ! [H] :
( m1_subset_1(H,u1_struct_0(B))
=> ~ ( r2_hidden(G,A)
& r2_hidden(H,A)
& ~ r1_xreal_0(F,k1_normsp_1(B,k6_rlvect_1(B,G,H)))
& r1_xreal_0(E,k1_normsp_1(C,k6_rlvect_1(C,k4_finseq_4(u1_struct_0(B),u1_struct_0(C),D,G),k4_finseq_4(u1_struct_0(B),u1_struct_0(C),D,H)))) ) ) ) ) ) ) ) ) ) ) ) ) ).
fof(d2_nfcont_2,axiom,
! [A,B] :
( ( ~ v3_struct_0(B)
& v3_rlvect_1(B)
& v4_rlvect_1(B)
& v5_rlvect_1(B)
& v6_rlvect_1(B)
& v7_rlvect_1(B)
& v2_normsp_1(B)
& l1_normsp_1(B) )
=> ! [C] :
( ( v1_funct_1(C)
& m2_relset_1(C,u1_struct_0(B),k1_numbers) )
=> ( r2_nfcont_2(A,B,C)
<=> ( r1_tarski(A,k4_relset_1(u1_struct_0(B),k1_numbers,C))
& ! [D] :
( m1_subset_1(D,k1_numbers)
=> ~ ( ~ r1_xreal_0(D,np__0)
& ! [E] :
( m1_subset_1(E,k1_numbers)
=> ~ ( ~ r1_xreal_0(E,np__0)
& ! [F] :
( m1_subset_1(F,u1_struct_0(B))
=> ! [G] :
( m1_subset_1(G,u1_struct_0(B))
=> ~ ( r2_hidden(F,A)
& r2_hidden(G,A)
& ~ r1_xreal_0(E,k1_normsp_1(B,k6_rlvect_1(B,F,G)))
& r1_xreal_0(D,k18_complex1(k5_real_1(k4_finseq_4(u1_struct_0(B),k1_numbers,C,F),k4_finseq_4(u1_struct_0(B),k1_numbers,C,G)))) ) ) ) ) ) ) ) ) ) ) ) ).
fof(t1_nfcont_2,axiom,
! [A,B,C] :
( ( ~ v3_struct_0(C)
& v3_rlvect_1(C)
& v4_rlvect_1(C)
& v5_rlvect_1(C)
& v6_rlvect_1(C)
& v7_rlvect_1(C)
& v2_normsp_1(C)
& l1_normsp_1(C) )
=> ! [D] :
( ( ~ v3_struct_0(D)
& v3_rlvect_1(D)
& v4_rlvect_1(D)
& v5_rlvect_1(D)
& v6_rlvect_1(D)
& v7_rlvect_1(D)
& v2_normsp_1(D)
& l1_normsp_1(D) )
=> ! [E] :
( ( v1_funct_1(E)
& m2_relset_1(E,u1_struct_0(C),u1_struct_0(D)) )
=> ( ( r1_nfcont_2(A,C,D,E)
& r1_tarski(B,A) )
=> r1_nfcont_2(B,C,D,E) ) ) ) ) ).
fof(t2_nfcont_2,axiom,
! [A,B,C] :
( ( ~ v3_struct_0(C)
& v3_rlvect_1(C)
& v4_rlvect_1(C)
& v5_rlvect_1(C)
& v6_rlvect_1(C)
& v7_rlvect_1(C)
& v2_normsp_1(C)
& l1_normsp_1(C) )
=> ! [D] :
( ( ~ v3_struct_0(D)
& v3_rlvect_1(D)
& v4_rlvect_1(D)
& v5_rlvect_1(D)
& v6_rlvect_1(D)
& v7_rlvect_1(D)
& v2_normsp_1(D)
& l1_normsp_1(D) )
=> ! [E] :
( ( v1_funct_1(E)
& m2_relset_1(E,u1_struct_0(C),u1_struct_0(D)) )
=> ! [F] :
( ( v1_funct_1(F)
& m2_relset_1(F,u1_struct_0(C),u1_struct_0(D)) )
=> ( ( r1_nfcont_2(A,C,D,E)
& r1_nfcont_2(B,C,D,F) )
=> r1_nfcont_2(k3_xboole_0(A,B),C,D,k1_vfunct_1(u1_struct_0(C),D,E,F)) ) ) ) ) ) ).
fof(t3_nfcont_2,axiom,
! [A,B,C] :
( ( ~ v3_struct_0(C)
& v3_rlvect_1(C)
& v4_rlvect_1(C)
& v5_rlvect_1(C)
& v6_rlvect_1(C)
& v7_rlvect_1(C)
& v2_normsp_1(C)
& l1_normsp_1(C) )
=> ! [D] :
( ( ~ v3_struct_0(D)
& v3_rlvect_1(D)
& v4_rlvect_1(D)
& v5_rlvect_1(D)
& v6_rlvect_1(D)
& v7_rlvect_1(D)
& v2_normsp_1(D)
& l1_normsp_1(D) )
=> ! [E] :
( ( v1_funct_1(E)
& m2_relset_1(E,u1_struct_0(C),u1_struct_0(D)) )
=> ! [F] :
( ( v1_funct_1(F)
& m2_relset_1(F,u1_struct_0(C),u1_struct_0(D)) )
=> ( ( r1_nfcont_2(A,C,D,E)
& r1_nfcont_2(B,C,D,F) )
=> r1_nfcont_2(k3_xboole_0(A,B),C,D,k2_vfunct_1(u1_struct_0(C),D,E,F)) ) ) ) ) ) ).
fof(t4_nfcont_2,axiom,
! [A,B] :
( m1_subset_1(B,k1_numbers)
=> ! [C] :
( ( ~ v3_struct_0(C)
& v3_rlvect_1(C)
& v4_rlvect_1(C)
& v5_rlvect_1(C)
& v6_rlvect_1(C)
& v7_rlvect_1(C)
& v2_normsp_1(C)
& l1_normsp_1(C) )
=> ! [D] :
( ( ~ v3_struct_0(D)
& v3_rlvect_1(D)
& v4_rlvect_1(D)
& v5_rlvect_1(D)
& v6_rlvect_1(D)
& v7_rlvect_1(D)
& v2_normsp_1(D)
& l1_normsp_1(D) )
=> ! [E] :
( ( v1_funct_1(E)
& m2_relset_1(E,u1_struct_0(C),u1_struct_0(D)) )
=> ( r1_nfcont_2(A,C,D,E)
=> r1_nfcont_2(A,C,D,k4_vfunct_1(u1_struct_0(C),D,E,B)) ) ) ) ) ) ).
fof(t5_nfcont_2,axiom,
! [A,B] :
( ( ~ v3_struct_0(B)
& v3_rlvect_1(B)
& v4_rlvect_1(B)
& v5_rlvect_1(B)
& v6_rlvect_1(B)
& v7_rlvect_1(B)
& v2_normsp_1(B)
& l1_normsp_1(B) )
=> ! [C] :
( ( ~ v3_struct_0(C)
& v3_rlvect_1(C)
& v4_rlvect_1(C)
& v5_rlvect_1(C)
& v6_rlvect_1(C)
& v7_rlvect_1(C)
& v2_normsp_1(C)
& l1_normsp_1(C) )
=> ! [D] :
( ( v1_funct_1(D)
& m2_relset_1(D,u1_struct_0(B),u1_struct_0(C)) )
=> ( r1_nfcont_2(A,B,C,D)
=> r1_nfcont_2(A,B,C,k6_vfunct_1(u1_struct_0(B),C,D)) ) ) ) ) ).
fof(t6_nfcont_2,axiom,
! [A,B] :
( ( ~ v3_struct_0(B)
& v3_rlvect_1(B)
& v4_rlvect_1(B)
& v5_rlvect_1(B)
& v6_rlvect_1(B)
& v7_rlvect_1(B)
& v2_normsp_1(B)
& l1_normsp_1(B) )
=> ! [C] :
( ( ~ v3_struct_0(C)
& v3_rlvect_1(C)
& v4_rlvect_1(C)
& v5_rlvect_1(C)
& v6_rlvect_1(C)
& v7_rlvect_1(C)
& v2_normsp_1(C)
& l1_normsp_1(C) )
=> ! [D] :
( ( v1_funct_1(D)
& m2_relset_1(D,u1_struct_0(B),u1_struct_0(C)) )
=> ( r1_nfcont_2(A,B,C,D)
=> r2_nfcont_2(A,B,k2_nfcont_1(B,C,D)) ) ) ) ) ).
fof(t7_nfcont_2,axiom,
! [A,B] :
( ( ~ v3_struct_0(B)
& v3_rlvect_1(B)
& v4_rlvect_1(B)
& v5_rlvect_1(B)
& v6_rlvect_1(B)
& v7_rlvect_1(B)
& v2_normsp_1(B)
& l1_normsp_1(B) )
=> ! [C] :
( ( ~ v3_struct_0(C)
& v3_rlvect_1(C)
& v4_rlvect_1(C)
& v5_rlvect_1(C)
& v6_rlvect_1(C)
& v7_rlvect_1(C)
& v2_normsp_1(C)
& l1_normsp_1(C) )
=> ! [D] :
( ( v1_funct_1(D)
& m2_relset_1(D,u1_struct_0(B),u1_struct_0(C)) )
=> ( r1_nfcont_2(A,B,C,D)
=> r3_nfcont_1(B,C,D,A) ) ) ) ) ).
fof(t8_nfcont_2,axiom,
! [A,B] :
( ( ~ v3_struct_0(B)
& v3_rlvect_1(B)
& v4_rlvect_1(B)
& v5_rlvect_1(B)
& v6_rlvect_1(B)
& v7_rlvect_1(B)
& v2_normsp_1(B)
& l1_normsp_1(B) )
=> ! [C] :
( ( v1_funct_1(C)
& m2_relset_1(C,u1_struct_0(B),k1_numbers) )
=> ( r2_nfcont_2(A,B,C)
=> r4_nfcont_1(B,C,A) ) ) ) ).
fof(t9_nfcont_2,axiom,
! [A,B] :
( ( ~ v3_struct_0(B)
& v3_rlvect_1(B)
& v4_rlvect_1(B)
& v5_rlvect_1(B)
& v6_rlvect_1(B)
& v7_rlvect_1(B)
& v2_normsp_1(B)
& l1_normsp_1(B) )
=> ! [C] :
( ( ~ v3_struct_0(C)
& v3_rlvect_1(C)
& v4_rlvect_1(C)
& v5_rlvect_1(C)
& v6_rlvect_1(C)
& v7_rlvect_1(C)
& v2_normsp_1(C)
& l1_normsp_1(C) )
=> ! [D] :
( ( v1_funct_1(D)
& m2_relset_1(D,u1_struct_0(B),u1_struct_0(C)) )
=> ( r5_nfcont_1(B,C,A,D)
=> r1_nfcont_2(A,B,C,D) ) ) ) ) ).
fof(t10_nfcont_2,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v3_rlvect_1(A)
& v4_rlvect_1(A)
& v5_rlvect_1(A)
& v6_rlvect_1(A)
& v7_rlvect_1(A)
& v2_normsp_1(A)
& l1_normsp_1(A) )
=> ! [B] :
( ( ~ v3_struct_0(B)
& v3_rlvect_1(B)
& v4_rlvect_1(B)
& v5_rlvect_1(B)
& v6_rlvect_1(B)
& v7_rlvect_1(B)
& v2_normsp_1(B)
& l1_normsp_1(B) )
=> ! [C] :
( ( v1_funct_1(C)
& m2_relset_1(C,u1_struct_0(B),u1_struct_0(A)) )
=> ! [D] :
( m1_subset_1(D,k1_zfmisc_1(u1_struct_0(B)))
=> ( ( v1_nfcont_1(D,B)
& r3_nfcont_1(B,A,C,D) )
=> r1_nfcont_2(D,B,A,C) ) ) ) ) ) ).
fof(t11_nfcont_2,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v3_rlvect_1(A)
& v4_rlvect_1(A)
& v5_rlvect_1(A)
& v6_rlvect_1(A)
& v7_rlvect_1(A)
& v2_normsp_1(A)
& l1_normsp_1(A) )
=> ! [B] :
( ( ~ v3_struct_0(B)
& v3_rlvect_1(B)
& v4_rlvect_1(B)
& v5_rlvect_1(B)
& v6_rlvect_1(B)
& v7_rlvect_1(B)
& v2_normsp_1(B)
& l1_normsp_1(B) )
=> ! [C] :
( ( v1_funct_1(C)
& m2_relset_1(C,u1_struct_0(B),u1_struct_0(A)) )
=> ! [D] :
( m1_subset_1(D,k1_zfmisc_1(u1_struct_0(B)))
=> ( ( r1_tarski(D,k4_relset_1(u1_struct_0(B),u1_struct_0(A),C))
& v1_nfcont_1(D,B)
& r1_nfcont_2(D,B,A,C) )
=> v1_nfcont_1(k10_relset_1(u1_struct_0(B),u1_struct_0(A),C,D),A) ) ) ) ) ) ).
fof(t12_nfcont_2,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v3_rlvect_1(A)
& v4_rlvect_1(A)
& v5_rlvect_1(A)
& v6_rlvect_1(A)
& v7_rlvect_1(A)
& v2_normsp_1(A)
& l1_normsp_1(A) )
=> ! [B] :
( ( v1_funct_1(B)
& m2_relset_1(B,u1_struct_0(A),k1_numbers) )
=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A)))
=> ~ ( C != k1_xboole_0
& r1_tarski(C,k4_relset_1(u1_struct_0(A),k1_numbers,B))
& v1_nfcont_1(C,A)
& r2_nfcont_2(C,A,B)
& ! [D] :
( m1_subset_1(D,u1_struct_0(A))
=> ! [E] :
( m1_subset_1(E,u1_struct_0(A))
=> ~ ( r2_hidden(D,C)
& r2_hidden(E,C)
& k4_finseq_4(u1_struct_0(A),k1_numbers,B,D) = k4_seq_4(k10_relset_1(u1_struct_0(A),k1_numbers,B,C))
& k4_finseq_4(u1_struct_0(A),k1_numbers,B,E) = k5_seq_4(k10_relset_1(u1_struct_0(A),k1_numbers,B,C)) ) ) ) ) ) ) ) ).
fof(t13_nfcont_2,axiom,
! [A,B] :
( ( ~ v3_struct_0(B)
& v3_rlvect_1(B)
& v4_rlvect_1(B)
& v5_rlvect_1(B)
& v6_rlvect_1(B)
& v7_rlvect_1(B)
& v2_normsp_1(B)
& l1_normsp_1(B) )
=> ! [C] :
( ( ~ v3_struct_0(C)
& v3_rlvect_1(C)
& v4_rlvect_1(C)
& v5_rlvect_1(C)
& v6_rlvect_1(C)
& v7_rlvect_1(C)
& v2_normsp_1(C)
& l1_normsp_1(C) )
=> ! [D] :
( ( v1_funct_1(D)
& m2_relset_1(D,u1_struct_0(B),u1_struct_0(C)) )
=> ( ( r1_tarski(A,k4_relset_1(u1_struct_0(B),u1_struct_0(C),D))
& r1_partfun2(u1_struct_0(B),u1_struct_0(C),D,A) )
=> r1_nfcont_2(A,B,C,D) ) ) ) ) ).
fof(d3_nfcont_2,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v3_rlvect_1(A)
& v4_rlvect_1(A)
& v5_rlvect_1(A)
& v6_rlvect_1(A)
& v7_rlvect_1(A)
& v2_normsp_1(A)
& v4_lopban_1(A)
& l1_normsp_1(A) )
=> ! [B] :
( ( v1_funct_1(B)
& v1_funct_2(B,u1_struct_0(A),u1_struct_0(A))
& m2_relset_1(B,u1_struct_0(A),u1_struct_0(A)) )
=> ( m1_nfcont_2(B,A)
<=> ? [C] :
( m1_subset_1(C,k1_numbers)
& ~ r1_xreal_0(C,np__0)
& ~ r1_xreal_0(np__1,C)
& ! [D] :
( m1_subset_1(D,u1_struct_0(A))
=> ! [E] :
( m1_subset_1(E,u1_struct_0(A))
=> r1_xreal_0(k1_normsp_1(A,k6_rlvect_1(A,k8_funct_2(u1_struct_0(A),u1_struct_0(A),B,D),k8_funct_2(u1_struct_0(A),u1_struct_0(A),B,E))),k4_real_1(C,k1_normsp_1(A,k6_rlvect_1(A,D,E)))) ) ) ) ) ) ) ).
fof(t14_nfcont_2,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v3_rlvect_1(A)
& v4_rlvect_1(A)
& v5_rlvect_1(A)
& v6_rlvect_1(A)
& v7_rlvect_1(A)
& v2_normsp_1(A)
& v4_lopban_1(A)
& l1_normsp_1(A) )
=> ! [B] :
( ( v1_funct_1(B)
& v1_funct_2(B,u1_struct_0(A),u1_struct_0(A))
& m2_relset_1(B,u1_struct_0(A),u1_struct_0(A)) )
=> ~ ( m1_nfcont_2(B,A)
& ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> ~ ( k8_funct_2(u1_struct_0(A),u1_struct_0(A),B,C) = C
& ! [D] :
( m1_subset_1(D,u1_struct_0(A))
=> ( k8_funct_2(u1_struct_0(A),u1_struct_0(A),B,D) = D
=> C = D ) ) ) ) ) ) ) ).
fof(t15_nfcont_2,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v3_rlvect_1(A)
& v4_rlvect_1(A)
& v5_rlvect_1(A)
& v6_rlvect_1(A)
& v7_rlvect_1(A)
& v2_normsp_1(A)
& v4_lopban_1(A)
& l1_normsp_1(A) )
=> ! [B] :
( ( v1_funct_1(B)
& v1_funct_2(B,u1_struct_0(A),u1_struct_0(A))
& m2_relset_1(B,u1_struct_0(A),u1_struct_0(A)) )
=> ~ ( ? [C] :
( m2_subset_1(C,k1_numbers,k5_numbers)
& m1_nfcont_2(k1_abian(u1_struct_0(A),B,C),A) )
& ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> ~ ( k8_funct_2(u1_struct_0(A),u1_struct_0(A),B,C) = C
& ! [D] :
( m1_subset_1(D,u1_struct_0(A))
=> ( k8_funct_2(u1_struct_0(A),u1_struct_0(A),B,D) = D
=> C = D ) ) ) ) ) ) ) ).
fof(t16_nfcont_2,axiom,
! [A] :
( v1_xreal_0(A)
=> ! [B] :
( v1_xreal_0(B)
=> ! [C] :
( v1_xreal_0(C)
=> ~ ( ~ r1_xreal_0(A,np__0)
& ~ r1_xreal_0(np__1,A)
& ~ r1_xreal_0(C,np__0)
& ! [D] :
( m2_subset_1(D,k1_numbers,k5_numbers)
=> r1_xreal_0(C,k18_complex1(k3_xcmplx_0(B,k3_power(A,D)))) ) ) ) ) ) ).
fof(dt_m1_nfcont_2,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v3_rlvect_1(A)
& v4_rlvect_1(A)
& v5_rlvect_1(A)
& v6_rlvect_1(A)
& v7_rlvect_1(A)
& v2_normsp_1(A)
& v4_lopban_1(A)
& l1_normsp_1(A) )
=> ! [B] :
( m1_nfcont_2(B,A)
=> ( v1_funct_1(B)
& v1_funct_2(B,u1_struct_0(A),u1_struct_0(A))
& m2_relset_1(B,u1_struct_0(A),u1_struct_0(A)) ) ) ) ).
fof(existence_m1_nfcont_2,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v3_rlvect_1(A)
& v4_rlvect_1(A)
& v5_rlvect_1(A)
& v6_rlvect_1(A)
& v7_rlvect_1(A)
& v2_normsp_1(A)
& v4_lopban_1(A)
& l1_normsp_1(A) )
=> ? [B] : m1_nfcont_2(B,A) ) ).
%------------------------------------------------------------------------------