SET007 Axioms: SET007+822.ax
%------------------------------------------------------------------------------
% File : SET007+822 : TPTP v8.2.0. Released v3.4.0.
% Domain : Set Theory
% Axioms : Banach Space of Absolute Summable 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 : csspace3 [Urb08]
% Status : Satisfiable
% Syntax : Number of formulae : 25 ( 5 unt; 0 def)
% Number of atoms : 182 ( 25 equ)
% Maximal formula atoms : 25 ( 7 avg)
% Number of connectives : 175 ( 18 ~; 0 |; 105 &)
% ( 6 <=>; 46 =>; 0 <=; 0 <~>)
% Maximal formula depth : 18 ( 7 avg)
% Maximal term depth : 4 ( 1 avg)
% Number of predicates : 28 ( 26 usr; 1 prp; 0-3 aty)
% Number of functors : 45 ( 45 usr; 11 con; 0-5 aty)
% Number of variables : 51 ( 47 !; 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_csspace3,axiom,
~ v1_xboole_0(k1_csspace3) ).
fof(fc2_csspace3,axiom,
( ~ v1_xboole_0(k1_csspace3)
& v3_clvect_1(k1_csspace3,k7_csspace) ) ).
fof(fc3_csspace3,axiom,
( ~ v3_struct_0(g1_clvect_1(k1_csspace3,k10_csspace(k7_csspace,k1_csspace3),k8_csspace(k7_csspace,k1_csspace3),k9_csspace(k7_csspace,k1_csspace3)))
& v3_rlvect_1(g1_clvect_1(k1_csspace3,k10_csspace(k7_csspace,k1_csspace3),k8_csspace(k7_csspace,k1_csspace3),k9_csspace(k7_csspace,k1_csspace3)))
& v4_rlvect_1(g1_clvect_1(k1_csspace3,k10_csspace(k7_csspace,k1_csspace3),k8_csspace(k7_csspace,k1_csspace3),k9_csspace(k7_csspace,k1_csspace3)))
& v5_rlvect_1(g1_clvect_1(k1_csspace3,k10_csspace(k7_csspace,k1_csspace3),k8_csspace(k7_csspace,k1_csspace3),k9_csspace(k7_csspace,k1_csspace3)))
& v6_rlvect_1(g1_clvect_1(k1_csspace3,k10_csspace(k7_csspace,k1_csspace3),k8_csspace(k7_csspace,k1_csspace3),k9_csspace(k7_csspace,k1_csspace3)))
& v1_clvect_1(g1_clvect_1(k1_csspace3,k10_csspace(k7_csspace,k1_csspace3),k8_csspace(k7_csspace,k1_csspace3),k9_csspace(k7_csspace,k1_csspace3)))
& v2_clvect_1(g1_clvect_1(k1_csspace3,k10_csspace(k7_csspace,k1_csspace3),k8_csspace(k7_csspace,k1_csspace3),k9_csspace(k7_csspace,k1_csspace3))) ) ).
fof(fc4_csspace3,axiom,
! [A,B,C,D,E] :
( ( ~ v1_xboole_0(A)
& m1_subset_1(B,A)
& v1_funct_1(C)
& v1_funct_2(C,k2_zfmisc_1(A,A),A)
& m1_relset_1(C,k2_zfmisc_1(A,A),A)
& v1_funct_1(D)
& v1_funct_2(D,k2_zfmisc_1(k2_numbers,A),A)
& m1_relset_1(D,k2_zfmisc_1(k2_numbers,A),A)
& v1_funct_1(E)
& v1_funct_2(E,A,k1_numbers)
& m1_relset_1(E,A,k1_numbers) )
=> ( ~ v3_struct_0(g2_clvect_1(A,B,C,D,E))
& v4_clvect_1(g2_clvect_1(A,B,C,D,E)) ) ) ).
fof(fc5_csspace3,axiom,
( ~ v3_struct_0(k3_csspace3)
& v3_rlvect_1(k3_csspace3)
& v4_rlvect_1(k3_csspace3)
& v5_rlvect_1(k3_csspace3)
& v6_rlvect_1(k3_csspace3)
& v2_clvect_1(k3_csspace3)
& v5_clvect_1(k3_csspace3) ) ).
fof(d1_csspace3,axiom,
! [A] :
( m1_subset_1(A,k1_zfmisc_1(u1_struct_0(k7_csspace)))
=> ( A = k1_csspace3
<=> ! [B] :
( r2_hidden(B,A)
<=> ( r2_hidden(B,k1_csspace)
& v2_comseq_3(k2_csspace(B)) ) ) ) ) ).
fof(t1_csspace3,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) )
=> ! [C] :
( ( v1_funct_1(C)
& v1_funct_2(C,k5_numbers,k1_numbers)
& m2_relset_1(C,k5_numbers,k1_numbers) )
=> ( ( v2_comseq_2(B)
& ! [D] :
( m2_subset_1(D,k1_numbers,k5_numbers)
=> k2_seq_1(k5_numbers,k1_numbers,C,D) = k17_complex1(k11_complex1(k1_comseq_1(B,D),A)) ) )
=> ( v4_seq_2(C)
& k2_seq_2(C) = k17_complex1(k11_complex1(k2_comseq_2(B),A)) ) ) ) ) ) ).
fof(d2_csspace3,axiom,
! [A] :
( ( v1_funct_1(A)
& v1_funct_2(A,k1_csspace3,k1_numbers)
& m2_relset_1(A,k1_csspace3,k1_numbers) )
=> ( A = k2_csspace3
<=> ! [B] :
( r2_hidden(B,k1_csspace3)
=> k2_seq_1(k1_csspace3,k1_numbers,A,B) = k2_series_1(k9_comseq_1(k5_numbers,k2_csspace(B))) ) ) ) ).
fof(t2_csspace3,axiom,
$true ).
fof(t3_csspace3,axiom,
$true ).
fof(t4_csspace3,axiom,
! [A] :
( l2_clvect_1(A)
=> ( ( ~ v3_struct_0(g1_clvect_1(u1_struct_0(A),u2_struct_0(A),u1_rlvect_1(A),u1_clvect_1(A)))
& v3_rlvect_1(g1_clvect_1(u1_struct_0(A),u2_struct_0(A),u1_rlvect_1(A),u1_clvect_1(A)))
& v4_rlvect_1(g1_clvect_1(u1_struct_0(A),u2_struct_0(A),u1_rlvect_1(A),u1_clvect_1(A)))
& v5_rlvect_1(g1_clvect_1(u1_struct_0(A),u2_struct_0(A),u1_rlvect_1(A),u1_clvect_1(A)))
& v6_rlvect_1(g1_clvect_1(u1_struct_0(A),u2_struct_0(A),u1_rlvect_1(A),u1_clvect_1(A)))
& v2_clvect_1(g1_clvect_1(u1_struct_0(A),u2_struct_0(A),u1_rlvect_1(A),u1_clvect_1(A)))
& l1_clvect_1(g1_clvect_1(u1_struct_0(A),u2_struct_0(A),u1_rlvect_1(A),u1_clvect_1(A))) )
=> ( ~ v3_struct_0(A)
& v3_rlvect_1(A)
& v4_rlvect_1(A)
& v5_rlvect_1(A)
& v6_rlvect_1(A)
& v2_clvect_1(A)
& l1_clvect_1(A) ) ) ) ).
fof(t5_csspace3,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)
=> k1_comseq_1(A,B) = k5_complex1 )
=> ( v2_comseq_3(A)
& k2_series_1(k9_comseq_1(k5_numbers,A)) = np__0 ) ) ) ).
fof(t6_csspace3,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(t7_csspace3,axiom,
( ~ v3_struct_0(g2_clvect_1(k1_csspace3,k10_csspace(k7_csspace,k1_csspace3),k8_csspace(k7_csspace,k1_csspace3),k9_csspace(k7_csspace,k1_csspace3),k2_csspace3))
& v3_rlvect_1(g2_clvect_1(k1_csspace3,k10_csspace(k7_csspace,k1_csspace3),k8_csspace(k7_csspace,k1_csspace3),k9_csspace(k7_csspace,k1_csspace3),k2_csspace3))
& v4_rlvect_1(g2_clvect_1(k1_csspace3,k10_csspace(k7_csspace,k1_csspace3),k8_csspace(k7_csspace,k1_csspace3),k9_csspace(k7_csspace,k1_csspace3),k2_csspace3))
& v5_rlvect_1(g2_clvect_1(k1_csspace3,k10_csspace(k7_csspace,k1_csspace3),k8_csspace(k7_csspace,k1_csspace3),k9_csspace(k7_csspace,k1_csspace3),k2_csspace3))
& v6_rlvect_1(g2_clvect_1(k1_csspace3,k10_csspace(k7_csspace,k1_csspace3),k8_csspace(k7_csspace,k1_csspace3),k9_csspace(k7_csspace,k1_csspace3),k2_csspace3))
& v2_clvect_1(g2_clvect_1(k1_csspace3,k10_csspace(k7_csspace,k1_csspace3),k8_csspace(k7_csspace,k1_csspace3),k9_csspace(k7_csspace,k1_csspace3),k2_csspace3))
& l1_clvect_1(g2_clvect_1(k1_csspace3,k10_csspace(k7_csspace,k1_csspace3),k8_csspace(k7_csspace,k1_csspace3),k9_csspace(k7_csspace,k1_csspace3),k2_csspace3)) ) ).
fof(d3_csspace3,axiom,
k3_csspace3 = g2_clvect_1(k1_csspace3,k10_csspace(k7_csspace,k1_csspace3),k8_csspace(k7_csspace,k1_csspace3),k9_csspace(k7_csspace,k1_csspace3),k2_csspace3) ).
fof(t8_csspace3,axiom,
( u1_struct_0(k3_csspace3) = k1_csspace3
& ! [A] :
( m1_subset_1(A,u1_struct_0(k3_csspace3))
<=> ( v1_funct_1(A)
& v1_funct_2(A,k5_numbers,k2_numbers)
& m2_relset_1(A,k5_numbers,k2_numbers)
& v2_comseq_3(k2_csspace(A)) ) )
& k1_rlvect_1(k3_csspace3) = k6_csspace
& ! [A] :
( m1_subset_1(A,u1_struct_0(k3_csspace3))
=> A = k2_csspace(A) )
& ! [A] :
( m1_subset_1(A,u1_struct_0(k3_csspace3))
=> ! [B] :
( m1_subset_1(B,u1_struct_0(k3_csspace3))
=> k2_rlvect_1(k3_csspace3,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(k3_csspace3))
=> k1_clvect_1(k3_csspace3,B,A) = k4_comseq_1(k5_numbers,k2_csspace(B),A) ) )
& ! [A] :
( m1_subset_1(A,u1_struct_0(k3_csspace3))
=> ( k5_rlvect_1(k3_csspace3,A) = k5_comseq_1(k5_numbers,k2_csspace(A))
& k2_csspace(k5_rlvect_1(k3_csspace3,A)) = k5_comseq_1(k5_numbers,k2_csspace(A)) ) )
& ! [A] :
( m1_subset_1(A,u1_struct_0(k3_csspace3))
=> ! [B] :
( m1_subset_1(B,u1_struct_0(k3_csspace3))
=> k6_rlvect_1(k3_csspace3,A,B) = k6_comseq_1(k5_numbers,k2_csspace(A),k2_csspace(B)) ) )
& ! [A] :
( m1_subset_1(A,u1_struct_0(k3_csspace3))
=> v2_comseq_3(k2_csspace(A)) )
& ! [A] :
( m1_subset_1(A,u1_struct_0(k3_csspace3))
=> k5_clvect_1(k3_csspace3,A) = k2_series_1(k9_comseq_1(k5_numbers,k2_csspace(A))) ) ) ).
fof(t9_csspace3,axiom,
! [A] :
( m1_subset_1(A,u1_struct_0(k3_csspace3))
=> ! [B] :
( m1_subset_1(B,u1_struct_0(k3_csspace3))
=> ! [C] :
( m1_subset_1(C,k2_numbers)
=> ( ( k5_clvect_1(k3_csspace3,A) = np__0
=> A = k1_rlvect_1(k3_csspace3) )
& ( A = k1_rlvect_1(k3_csspace3)
=> k5_clvect_1(k3_csspace3,A) = np__0 )
& r1_xreal_0(np__0,k5_clvect_1(k3_csspace3,A))
& r1_xreal_0(k5_clvect_1(k3_csspace3,k2_rlvect_1(k3_csspace3,A,B)),k3_real_1(k5_clvect_1(k3_csspace3,A),k5_clvect_1(k3_csspace3,B)))
& k5_clvect_1(k3_csspace3,k1_clvect_1(k3_csspace3,A,C)) = k4_real_1(k17_complex1(C),k5_clvect_1(k3_csspace3,A)) ) ) ) ) ).
fof(d4_csspace3,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l2_clvect_1(A) )
=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> k4_csspace3(A,B,C) = k5_clvect_1(A,k6_rlvect_1(A,B,C)) ) ) ) ).
fof(d5_csspace3,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l2_clvect_1(A) )
=> ! [B] :
( ( v1_funct_1(B)
& v1_funct_2(B,k5_numbers,u1_struct_0(A))
& m2_relset_1(B,k5_numbers,u1_struct_0(A)) )
=> ( v1_csspace3(B,A)
<=> ! [C] :
( m1_subset_1(C,k1_numbers)
=> ~ ( ~ r1_xreal_0(C,np__0)
& ! [D] :
( m2_subset_1(D,k1_numbers,k5_numbers)
=> ? [E] :
( m2_subset_1(E,k1_numbers,k5_numbers)
& ? [F] :
( m2_subset_1(F,k1_numbers,k5_numbers)
& r1_xreal_0(D,E)
& r1_xreal_0(D,F)
& r1_xreal_0(C,k4_csspace3(A,k2_normsp_1(A,B,E),k2_normsp_1(A,B,F))) ) ) ) ) ) ) ) ) ).
fof(t10_csspace3,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v3_rlvect_1(A)
& v4_rlvect_1(A)
& v5_rlvect_1(A)
& v6_rlvect_1(A)
& v2_clvect_1(A)
& v5_clvect_1(A)
& l2_clvect_1(A) )
=> ! [B] :
( ( v1_funct_1(B)
& v1_funct_2(B,k5_numbers,u1_struct_0(A))
& m2_relset_1(B,k5_numbers,u1_struct_0(A)) )
=> ( v1_csspace3(B,A)
<=> ! [C] :
( m1_subset_1(C,k1_numbers)
=> ~ ( ~ r1_xreal_0(C,np__0)
& ! [D] :
( m2_subset_1(D,k1_numbers,k5_numbers)
=> ? [E] :
( m2_subset_1(E,k1_numbers,k5_numbers)
& ? [F] :
( m2_subset_1(F,k1_numbers,k5_numbers)
& r1_xreal_0(D,E)
& r1_xreal_0(D,F)
& r1_xreal_0(C,k5_clvect_1(A,k6_rlvect_1(A,k2_normsp_1(A,B,E),k2_normsp_1(A,B,F)))) ) ) ) ) ) ) ) ) ).
fof(t11_csspace3,axiom,
! [A] :
( ( v1_funct_1(A)
& v1_funct_2(A,k5_numbers,u1_struct_0(k3_csspace3))
& m2_relset_1(A,k5_numbers,u1_struct_0(k3_csspace3)) )
=> ( v1_csspace3(A,k3_csspace3)
=> v6_clvect_1(A,k3_csspace3) ) ) ).
fof(dt_k1_csspace3,axiom,
m1_subset_1(k1_csspace3,k1_zfmisc_1(u1_struct_0(k7_csspace))) ).
fof(dt_k2_csspace3,axiom,
( v1_funct_1(k2_csspace3)
& v1_funct_2(k2_csspace3,k1_csspace3,k1_numbers)
& m2_relset_1(k2_csspace3,k1_csspace3,k1_numbers) ) ).
fof(dt_k3_csspace3,axiom,
( ~ v3_struct_0(k3_csspace3)
& l2_clvect_1(k3_csspace3) ) ).
fof(dt_k4_csspace3,axiom,
! [A,B,C] :
( ( ~ v3_struct_0(A)
& l2_clvect_1(A)
& m1_subset_1(B,u1_struct_0(A))
& m1_subset_1(C,u1_struct_0(A)) )
=> m1_subset_1(k4_csspace3(A,B,C),k1_numbers) ) ).
%------------------------------------------------------------------------------