SET007 Axioms: SET007+821.ax
%------------------------------------------------------------------------------
% File : SET007+821 : TPTP v8.2.0. Released v3.4.0.
% Domain : Set Theory
% Axioms : Hilbert Space of Complex Sequences
% 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 : csspace2 [Urb08]
% Status : Satisfiable
% Syntax : Number of formulae : 21 ( 0 unt; 0 def)
% Number of atoms : 184 ( 38 equ)
% Maximal formula atoms : 32 ( 8 avg)
% Number of connectives : 165 ( 2 ~; 0 |; 89 &)
% ( 2 <=>; 72 =>; 0 <=; 0 <~>)
% Maximal formula depth : 19 ( 8 avg)
% Maximal term depth : 6 ( 2 avg)
% Number of predicates : 22 ( 21 usr; 0 prp; 1-3 aty)
% Number of functors : 45 ( 45 usr; 9 con; 0-4 aty)
% Number of variables : 57 ( 57 !; 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(fc1_csspace2,axiom,
( ~ v3_struct_0(k20_csspace)
& v3_rlvect_1(k20_csspace)
& v4_rlvect_1(k20_csspace)
& v5_rlvect_1(k20_csspace)
& v6_rlvect_1(k20_csspace)
& v2_clvect_1(k20_csspace)
& v2_csspace(k20_csspace) ) ).
fof(fc2_csspace2,axiom,
( ~ v3_struct_0(k20_csspace)
& v3_rlvect_1(k20_csspace)
& v4_rlvect_1(k20_csspace)
& v5_rlvect_1(k20_csspace)
& v6_rlvect_1(k20_csspace)
& v2_clvect_1(k20_csspace)
& v2_csspace(k20_csspace)
& v5_clvect_2(k20_csspace) ) ).
fof(t1_csspace2,axiom,
( u1_struct_0(k20_csspace) = k11_csspace
& ! [A] :
( m1_subset_1(A,u1_struct_0(k20_csspace))
<=> ( v1_funct_1(A)
& v1_funct_2(A,k5_numbers,k2_numbers)
& m2_relset_1(A,k5_numbers,k2_numbers)
& v1_series_1(k11_seq_1(k9_comseq_1(k5_numbers,k2_csspace(A)),k9_comseq_1(k5_numbers,k2_csspace(A)))) ) )
& ! [A] :
( m1_subset_1(A,u1_struct_0(k20_csspace))
<=> ( v1_funct_1(A)
& v1_funct_2(A,k5_numbers,k2_numbers)
& m2_relset_1(A,k5_numbers,k2_numbers)
& v2_comseq_3(k3_comseq_1(k5_numbers,k2_csspace(A),k1_comseq_2(k5_numbers,k2_csspace(A)))) ) )
& k1_rlvect_1(k20_csspace) = k6_csspace
& ! [A] :
( m1_subset_1(A,u1_struct_0(k20_csspace))
=> A = k2_csspace(A) )
& ! [A] :
( m1_subset_1(A,u1_struct_0(k20_csspace))
=> ! [B] :
( m1_subset_1(B,u1_struct_0(k20_csspace))
=> k4_rlvect_1(k20_csspace,A,B) = k2_comseq_1(k5_numbers,k2_csspace(A),k2_csspace(B)) ) )
& ! [A] :
( m1_subset_1(A,k2_numbers)
=> ! [B] :
( m1_subset_1(B,u1_struct_0(k20_csspace))
=> k1_clvect_1(k20_csspace,B,A) = k4_comseq_1(k5_numbers,k2_csspace(B),A) ) )
& ! [A] :
( m1_subset_1(A,u1_struct_0(k20_csspace))
=> ( k5_rlvect_1(k20_csspace,A) = k5_comseq_1(k5_numbers,k2_csspace(A))
& k2_csspace(k5_rlvect_1(k20_csspace,A)) = k5_comseq_1(k5_numbers,k2_csspace(A)) ) )
& ! [A] :
( m1_subset_1(A,u1_struct_0(k20_csspace))
=> ! [B] :
( m1_subset_1(B,u1_struct_0(k20_csspace))
=> k6_rlvect_1(k20_csspace,A,B) = k6_comseq_1(k5_numbers,k2_csspace(A),k2_csspace(B)) ) )
& ! [A] :
( m1_subset_1(A,u1_struct_0(k20_csspace))
=> ! [B] :
( m1_subset_1(B,u1_struct_0(k20_csspace))
=> ( v1_series_1(k11_seq_1(k9_comseq_1(k5_numbers,k2_csspace(A)),k9_comseq_1(k5_numbers,k2_csspace(B))))
& ! [C] :
( m1_subset_1(C,u1_struct_0(k20_csspace))
=> ! [D] :
( m1_subset_1(D,u1_struct_0(k20_csspace))
=> k12_csspace(k20_csspace,C,D) = k8_comseq_3(k3_comseq_1(k5_numbers,k2_csspace(C),k1_comseq_2(k5_numbers,k2_csspace(D)))) ) ) ) ) ) ) ).
fof(t2_csspace2,axiom,
! [A] :
( m1_subset_1(A,u1_struct_0(k20_csspace))
=> ! [B] :
( m1_subset_1(B,u1_struct_0(k20_csspace))
=> ! [C] :
( m1_subset_1(C,u1_struct_0(k20_csspace))
=> ! [D] :
( m1_subset_1(D,k2_numbers)
=> ( ( k12_csspace(k20_csspace,A,A) = np__0
=> A = k1_rlvect_1(k20_csspace) )
& ( A = k1_rlvect_1(k20_csspace)
=> k12_csspace(k20_csspace,A,A) = np__0 )
& r1_xreal_0(np__0,k3_complex1(k12_csspace(k20_csspace,A,A)))
& k4_complex1(k12_csspace(k20_csspace,A,A)) = np__0
& k12_csspace(k20_csspace,A,B) = k15_complex1(k12_csspace(k20_csspace,B,A))
& k12_csspace(k20_csspace,k4_rlvect_1(k20_csspace,A,B),C) = k8_complex1(k12_csspace(k20_csspace,A,C),k12_csspace(k20_csspace,B,C))
& k12_csspace(k20_csspace,k1_clvect_1(k20_csspace,A,D),B) = k9_complex1(D,k12_csspace(k20_csspace,A,B)) ) ) ) ) ) ).
fof(t3_csspace2,axiom,
! [A] :
( ( v1_funct_1(A)
& v1_funct_2(A,k5_numbers,u1_struct_0(k20_csspace))
& m2_relset_1(A,k5_numbers,u1_struct_0(k20_csspace)) )
=> ( v2_clvect_2(A,k20_csspace)
=> v1_clvect_2(A,k20_csspace) ) ) ).
fof(t4_csspace2,axiom,
! [A] :
( m1_subset_1(A,k2_numbers)
=> ! [B] :
( m1_subset_1(B,k2_numbers)
=> ( ( k4_real_1(k3_complex1(A),k4_complex1(B)) = k4_real_1(k3_complex1(B),k4_complex1(A))
& r1_xreal_0(np__0,k3_real_1(k4_real_1(k3_complex1(A),k3_complex1(B)),k4_real_1(k4_complex1(A),k4_complex1(B)))) )
=> k17_complex1(k8_complex1(A,B)) = k3_real_1(k17_complex1(A),k17_complex1(B)) ) ) ) ).
fof(t5_csspace2,axiom,
! [A] :
( m1_subset_1(A,k2_numbers)
=> ! [B] :
( m1_subset_1(B,k2_numbers)
=> r1_xreal_0(k4_real_1(np__2,k17_complex1(k9_complex1(A,B))),k3_real_1(k7_square_1(k17_complex1(A)),k7_square_1(k17_complex1(B)))) ) ) ).
fof(t6_csspace2,axiom,
! [A] :
( m1_subset_1(A,k2_numbers)
=> ! [B] :
( m1_subset_1(B,k2_numbers)
=> ( r1_xreal_0(k4_real_1(k17_complex1(k8_complex1(A,B)),k17_complex1(k8_complex1(A,B))),k3_real_1(k4_real_1(k4_real_1(np__2,k17_complex1(A)),k17_complex1(A)),k4_real_1(k4_real_1(np__2,k17_complex1(B)),k17_complex1(B))))
& r1_xreal_0(k4_real_1(k17_complex1(A),k17_complex1(A)),k3_real_1(k4_real_1(k4_real_1(np__2,k17_complex1(k11_complex1(A,B))),k17_complex1(k11_complex1(A,B))),k4_real_1(k4_real_1(np__2,k17_complex1(B)),k17_complex1(B)))) ) ) ) ).
fof(t7_csspace2,axiom,
! [A] :
( ( v1_funct_1(A)
& v1_funct_2(A,k5_numbers,k2_numbers)
& m2_relset_1(A,k5_numbers,k2_numbers) )
=> A = k1_comseq_2(k5_numbers,k1_comseq_2(k5_numbers,A)) ) ).
fof(t8_csspace2,axiom,
! [A] :
( ( v1_funct_1(A)
& v1_funct_2(A,k5_numbers,k2_numbers)
& m2_relset_1(A,k5_numbers,k2_numbers) )
=> k7_comseq_3(k1_comseq_2(k5_numbers,A)) = k1_comseq_2(k5_numbers,k7_comseq_3(A)) ) ).
fof(t9_csspace2,axiom,
! [A] :
( ( v1_funct_1(A)
& v1_funct_2(A,k5_numbers,k2_numbers)
& m2_relset_1(A,k5_numbers,k2_numbers) )
=> ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> ( ! [C] :
( m2_subset_1(C,k1_numbers,k5_numbers)
=> ( r1_xreal_0(np__0,k2_seq_1(k5_numbers,k1_numbers,k3_comseq_3(A),C))
& k2_seq_1(k5_numbers,k1_numbers,k4_comseq_3(A),C) = np__0 ) )
=> k2_seq_1(k5_numbers,k1_numbers,k9_comseq_1(k5_numbers,k7_comseq_3(A)),B) = k2_seq_1(k5_numbers,k1_numbers,k1_series_1(k9_comseq_1(k5_numbers,A)),B) ) ) ) ).
fof(t10_csspace2,axiom,
! [A] :
( ( v1_funct_1(A)
& v1_funct_2(A,k5_numbers,k2_numbers)
& m2_relset_1(A,k5_numbers,k2_numbers) )
=> ( v1_comseq_3(A)
=> k8_comseq_3(k1_comseq_2(k5_numbers,A)) = k15_complex1(k8_comseq_3(A)) ) ) ).
fof(t11_csspace2,axiom,
! [A] :
( ( v1_funct_1(A)
& v1_funct_2(A,k5_numbers,k2_numbers)
& m2_relset_1(A,k5_numbers,k2_numbers) )
=> ( v2_comseq_3(A)
=> r1_xreal_0(k17_complex1(k8_comseq_3(A)),k2_series_1(k9_comseq_1(k5_numbers,A))) ) ) ).
fof(t12_csspace2,axiom,
! [A] :
( ( v1_funct_1(A)
& v1_funct_2(A,k5_numbers,k2_numbers)
& m2_relset_1(A,k5_numbers,k2_numbers) )
=> ( ( v1_comseq_3(A)
& ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> ( r1_xreal_0(np__0,k2_seq_1(k5_numbers,k1_numbers,k3_comseq_3(A),B))
& k2_seq_1(k5_numbers,k1_numbers,k4_comseq_3(A),B) = np__0 ) ) )
=> k17_complex1(k8_comseq_3(A)) = k2_series_1(k9_comseq_1(k5_numbers,A)) ) ) ).
fof(t13_csspace2,axiom,
! [A] :
( ( v1_funct_1(A)
& v1_funct_2(A,k5_numbers,k2_numbers)
& m2_relset_1(A,k5_numbers,k2_numbers) )
=> ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> ( r1_xreal_0(np__0,k2_seq_1(k5_numbers,k1_numbers,k3_comseq_3(k3_comseq_1(k5_numbers,A,k1_comseq_2(k5_numbers,A))),B))
& k2_seq_1(k5_numbers,k1_numbers,k4_comseq_3(k3_comseq_1(k5_numbers,A,k1_comseq_2(k5_numbers,A))),B) = np__0 ) ) ) ).
fof(t14_csspace2,axiom,
! [A] :
( ( v1_funct_1(A)
& v1_funct_2(A,k5_numbers,k2_numbers)
& m2_relset_1(A,k5_numbers,k2_numbers) )
=> ( ( v2_comseq_3(A)
& k2_series_1(k9_comseq_1(k5_numbers,A)) = np__0 )
=> ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> k1_comseq_1(A,B) = k5_complex1 ) ) ) ).
fof(t15_csspace2,axiom,
! [A] :
( ( v1_funct_1(A)
& v1_funct_2(A,k5_numbers,k2_numbers)
& m2_relset_1(A,k5_numbers,k2_numbers) )
=> k9_comseq_1(k5_numbers,A) = k9_comseq_1(k5_numbers,k1_comseq_2(k5_numbers,A)) ) ).
fof(t16_csspace2,axiom,
! [A] :
( m1_subset_1(A,k2_numbers)
=> ! [B] :
( ( v1_funct_1(B)
& v1_funct_2(B,k5_numbers,k2_numbers)
& m2_relset_1(B,k5_numbers,k2_numbers) )
=> ( v2_comseq_2(B)
=> ! [C] :
( ( v1_funct_1(C)
& v1_funct_2(C,k5_numbers,k1_numbers)
& m2_relset_1(C,k5_numbers,k1_numbers) )
=> ( ! [D] :
( m2_subset_1(D,k1_numbers,k5_numbers)
=> k2_seq_1(k5_numbers,k1_numbers,C,D) = k4_real_1(k17_complex1(k11_complex1(k1_comseq_1(B,D),A)),k17_complex1(k11_complex1(k1_comseq_1(B,D),A))) )
=> ( v4_seq_2(C)
& k2_seq_2(C) = k4_real_1(k17_complex1(k11_complex1(k2_comseq_2(B),A)),k17_complex1(k11_complex1(k2_comseq_2(B),A))) ) ) ) ) ) ) ).
fof(t17_csspace2,axiom,
! [A] :
( m1_subset_1(A,k2_numbers)
=> ! [B] :
( ( v1_funct_1(B)
& v1_funct_2(B,k5_numbers,k1_numbers)
& m2_relset_1(B,k5_numbers,k1_numbers) )
=> ! [C] :
( ( v1_funct_1(C)
& v1_funct_2(C,k5_numbers,k2_numbers)
& m2_relset_1(C,k5_numbers,k2_numbers) )
=> ( ( v2_comseq_2(C)
& v4_seq_2(B) )
=> ! [D] :
( ( v1_funct_1(D)
& v1_funct_2(D,k5_numbers,k1_numbers)
& m2_relset_1(D,k5_numbers,k1_numbers) )
=> ( ! [E] :
( m2_subset_1(E,k1_numbers,k5_numbers)
=> k2_seq_1(k5_numbers,k1_numbers,D,E) = k3_real_1(k4_real_1(k17_complex1(k11_complex1(k1_comseq_1(C,E),A)),k17_complex1(k11_complex1(k1_comseq_1(C,E),A))),k2_seq_1(k5_numbers,k1_numbers,B,E)) )
=> ( v4_seq_2(D)
& k2_seq_2(D) = k3_real_1(k4_real_1(k17_complex1(k11_complex1(k2_comseq_2(C),A)),k17_complex1(k11_complex1(k2_comseq_2(C),A))),k2_seq_2(B)) ) ) ) ) ) ) ) ).
fof(t18_csspace2,axiom,
! [A] :
( m1_subset_1(A,k2_numbers)
=> ! [B] :
( ( v1_funct_1(B)
& v1_funct_2(B,k5_numbers,k2_numbers)
& m2_relset_1(B,k5_numbers,k2_numbers) )
=> ( v2_comseq_2(B)
=> ! [C] :
( ( v1_funct_1(C)
& v1_funct_2(C,k5_numbers,k1_numbers)
& m2_relset_1(C,k5_numbers,k1_numbers) )
=> ( ! [D] :
( m2_subset_1(D,k1_numbers,k5_numbers)
=> k2_seq_1(k5_numbers,k1_numbers,C,D) = k4_real_1(k17_complex1(k11_complex1(k1_comseq_1(B,D),A)),k17_complex1(k11_complex1(k1_comseq_1(B,D),A))) )
=> ( v4_seq_2(C)
& k2_seq_2(C) = k4_real_1(k17_complex1(k11_complex1(k2_comseq_2(B),A)),k17_complex1(k11_complex1(k2_comseq_2(B),A))) ) ) ) ) ) ) ).
fof(t19_csspace2,axiom,
! [A] :
( m1_subset_1(A,k2_numbers)
=> ! [B] :
( ( v1_funct_1(B)
& v1_funct_2(B,k5_numbers,k1_numbers)
& m2_relset_1(B,k5_numbers,k1_numbers) )
=> ! [C] :
( ( v1_funct_1(C)
& v1_funct_2(C,k5_numbers,k2_numbers)
& m2_relset_1(C,k5_numbers,k2_numbers) )
=> ( ( v2_comseq_2(C)
& v4_seq_2(B) )
=> ! [D] :
( ( v1_funct_1(D)
& v1_funct_2(D,k5_numbers,k1_numbers)
& m2_relset_1(D,k5_numbers,k1_numbers) )
=> ( ! [E] :
( m2_subset_1(E,k1_numbers,k5_numbers)
=> k2_seq_1(k5_numbers,k1_numbers,D,E) = k3_real_1(k4_real_1(k17_complex1(k11_complex1(k1_comseq_1(C,E),A)),k17_complex1(k11_complex1(k1_comseq_1(C,E),A))),k2_seq_1(k5_numbers,k1_numbers,B,E)) )
=> ( v4_seq_2(D)
& k2_seq_2(D) = k3_real_1(k4_real_1(k17_complex1(k11_complex1(k2_comseq_2(C),A)),k17_complex1(k11_complex1(k2_comseq_2(C),A))),k2_seq_2(B)) ) ) ) ) ) ) ) ).
%------------------------------------------------------------------------------