SET007 Axioms: SET007+773.ax
%------------------------------------------------------------------------------
% File : SET007+773 : TPTP v8.2.0. Released v3.4.0.
% Domain : Set Theory
% Axioms : On Some Properties of Real Hilbert Space. Part II
% 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 : bhsp_7 [Urb08]
% Status : Satisfiable
% Syntax : Number of formulae : 8 ( 0 unt; 0 def)
% Number of atoms : 167 ( 8 equ)
% Maximal formula atoms : 29 ( 20 avg)
% Number of connectives : 181 ( 22 ~; 1 |; 101 &)
% ( 2 <=>; 55 =>; 0 <=; 0 <~>)
% Maximal formula depth : 23 ( 15 avg)
% Maximal term depth : 5 ( 1 avg)
% Number of predicates : 27 ( 26 usr; 0 prp; 1-3 aty)
% Number of functors : 16 ( 16 usr; 4 con; 0-5 aty)
% Number of variables : 35 ( 35 !; 0 ?)
% 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(t1_bhsp_7,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_bhsp_1(A)
& l1_bhsp_1(A) )
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
=> ! [C] :
( ( v1_funct_1(C)
& v1_funct_2(C,u1_struct_0(A),k6_supinf_1)
& m2_relset_1(C,u1_struct_0(A),k6_supinf_1) )
=> ( r1_bhsp_6(A,B,C)
<=> ! [D] :
( m1_subset_1(D,k1_numbers)
=> ~ ( ~ r1_xreal_0(D,np__0)
& ! [E] :
( ( v1_finset_1(E)
& m1_subset_1(E,k1_zfmisc_1(u1_struct_0(A))) )
=> ~ ( ~ v1_xboole_0(E)
& r1_tarski(E,B)
& ! [F] :
( ( v1_finset_1(F)
& m1_subset_1(F,k1_zfmisc_1(u1_struct_0(A))) )
=> ~ ( ~ v1_xboole_0(F)
& r1_tarski(F,B)
& r1_xboole_0(E,F)
& r1_xreal_0(D,k18_complex1(k5_bhsp_5(k1_numbers,u1_struct_0(A),k33_binop_2,F,C))) ) ) ) ) ) ) ) ) ) ) ).
fof(t2_bhsp_7,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_bhsp_1(A)
& l1_bhsp_1(A) )
=> ( ( v1_binop_1(u1_rlvect_1(A),u1_struct_0(A))
& v2_binop_1(u1_rlvect_1(A),u1_struct_0(A))
& v1_setwiseo(u1_rlvect_1(A),u1_struct_0(A)) )
=> ! [B] :
( ( v1_finset_1(B)
& m1_bhsp_5(B,A) )
=> ( ~ v1_xboole_0(B)
=> ! [C] :
( ( v1_funct_1(C)
& v1_funct_2(C,u1_struct_0(A),u1_struct_0(A))
& m2_relset_1(C,u1_struct_0(A),u1_struct_0(A)) )
=> ( ( r1_tarski(B,k1_relat_1(C))
& ! [D] :
( m1_subset_1(D,u1_struct_0(A))
=> ( r2_hidden(D,B)
=> k8_funct_2(u1_struct_0(A),u1_struct_0(A),C,D) = D ) ) )
=> ! [D] :
( ( v1_funct_1(D)
& v1_funct_2(D,u1_struct_0(A),k1_numbers)
& m2_relset_1(D,u1_struct_0(A),k1_numbers) )
=> ( ( r1_tarski(B,k1_relat_1(D))
& ! [E] :
( m1_subset_1(E,u1_struct_0(A))
=> ( r2_hidden(E,B)
=> k8_funct_2(u1_struct_0(A),k1_numbers,D,E) = k2_bhsp_1(A,E,E) ) ) )
=> k2_bhsp_1(A,k5_bhsp_5(u1_struct_0(A),u1_struct_0(A),u1_rlvect_1(A),B,C),k5_bhsp_5(u1_struct_0(A),u1_struct_0(A),u1_rlvect_1(A),B,C)) = k5_bhsp_5(k1_numbers,u1_struct_0(A),k33_binop_2,B,D) ) ) ) ) ) ) ) ) ).
fof(t3_bhsp_7,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_bhsp_1(A)
& l1_bhsp_1(A) )
=> ( ( v1_binop_1(u1_rlvect_1(A),u1_struct_0(A))
& v2_binop_1(u1_rlvect_1(A),u1_struct_0(A))
& v1_setwiseo(u1_rlvect_1(A),u1_struct_0(A)) )
=> ! [B] :
( ( v1_finset_1(B)
& m1_bhsp_5(B,A) )
=> ( ~ v1_xboole_0(B)
=> ! [C] :
( ( v1_funct_1(C)
& v1_funct_2(C,u1_struct_0(A),k6_supinf_1)
& m2_relset_1(C,u1_struct_0(A),k6_supinf_1) )
=> ( ( r1_tarski(B,k1_relat_1(C))
& ! [D] :
( m1_subset_1(D,u1_struct_0(A))
=> ( r2_hidden(D,B)
=> k8_funct_2(u1_struct_0(A),k6_supinf_1,C,D) = k2_bhsp_1(A,D,D) ) ) )
=> k2_bhsp_1(A,k1_bhsp_6(A,B),k1_bhsp_6(A,B)) = k5_bhsp_5(k1_numbers,u1_struct_0(A),k33_binop_2,B,C) ) ) ) ) ) ) ).
fof(t4_bhsp_7,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_bhsp_1(A)
& l1_bhsp_1(A) )
=> ! [B] :
( m1_bhsp_5(B,A)
=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A)))
=> ( m1_subset_1(C,k1_zfmisc_1(B))
=> m1_bhsp_5(C,A) ) ) ) ) ).
fof(t5_bhsp_7,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_bhsp_1(A)
& l1_bhsp_1(A) )
=> ! [B] :
( m2_bhsp_5(B,A)
=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A)))
=> ( m1_subset_1(C,k1_zfmisc_1(B))
=> m2_bhsp_5(C,A) ) ) ) ) ).
fof(t6_bhsp_7,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_bhsp_1(A)
& l1_bhsp_1(A) )
=> ( ( v1_binop_1(u1_rlvect_1(A),u1_struct_0(A))
& v2_binop_1(u1_rlvect_1(A),u1_struct_0(A))
& v1_setwiseo(u1_rlvect_1(A),u1_struct_0(A))
& v4_bhsp_3(A) )
=> ! [B] :
( m2_bhsp_5(B,A)
=> ! [C] :
( ( v1_funct_1(C)
& v1_funct_2(C,u1_struct_0(A),k6_supinf_1)
& m2_relset_1(C,u1_struct_0(A),k6_supinf_1) )
=> ( ( r1_tarski(B,k1_relat_1(C))
& ! [D] :
( m1_subset_1(D,u1_struct_0(A))
=> ( r2_hidden(D,B)
=> k8_funct_2(u1_struct_0(A),k6_supinf_1,C,D) = k2_bhsp_1(A,D,D) ) ) )
=> ( v1_bhsp_6(B,A)
<=> r1_bhsp_6(A,B,C) ) ) ) ) ) ) ).
fof(t7_bhsp_7,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_bhsp_1(A)
& l1_bhsp_1(A) )
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
=> ( v1_bhsp_6(B,A)
=> ( v1_xboole_0(B)
| ! [C] :
( m1_subset_1(C,k1_numbers)
=> ~ ( ~ r1_xreal_0(C,np__0)
& ! [D] :
( ( v1_finset_1(D)
& m1_subset_1(D,k1_zfmisc_1(u1_struct_0(A))) )
=> ~ ( ~ v1_xboole_0(D)
& r1_tarski(D,B)
& ! [E] :
( ( v1_finset_1(E)
& m1_subset_1(E,k1_zfmisc_1(u1_struct_0(A))) )
=> ~ ( r1_tarski(D,E)
& r1_tarski(E,B)
& r1_xreal_0(C,k18_complex1(k5_real_1(k2_bhsp_1(A,k2_bhsp_6(A,B),k2_bhsp_6(A,B)),k2_bhsp_1(A,k1_bhsp_6(A,E),k1_bhsp_6(A,E))))) ) ) ) ) ) ) ) ) ) ) ).
fof(t8_bhsp_7,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_bhsp_1(A)
& l1_bhsp_1(A) )
=> ( ( v1_binop_1(u1_rlvect_1(A),u1_struct_0(A))
& v2_binop_1(u1_rlvect_1(A),u1_struct_0(A))
& v1_setwiseo(u1_rlvect_1(A),u1_struct_0(A))
& v4_bhsp_3(A) )
=> ! [B] :
( m2_bhsp_5(B,A)
=> ( ~ v1_xboole_0(B)
=> ! [C] :
( ( v1_funct_1(C)
& v1_funct_2(C,u1_struct_0(A),k6_supinf_1)
& m2_relset_1(C,u1_struct_0(A),k6_supinf_1) )
=> ( ( r1_tarski(B,k1_relat_1(C))
& ! [D] :
( m1_subset_1(D,u1_struct_0(A))
=> ( r2_hidden(D,B)
=> k8_funct_2(u1_struct_0(A),k6_supinf_1,C,D) = k2_bhsp_1(A,D,D) ) )
& v1_bhsp_6(B,A) )
=> k2_bhsp_1(A,k2_bhsp_6(A,B),k2_bhsp_6(A,B)) = k3_bhsp_6(A,B,C) ) ) ) ) ) ) ).
%------------------------------------------------------------------------------