SET007 Axioms: SET007+786.ax
%------------------------------------------------------------------------------
% File : SET007+786 : TPTP v8.2.0. Released v3.4.0.
% Domain : Set Theory
% Axioms : Banach Space of Absolute Summable Real 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 : rsspace3 [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 |; 104 &)
% ( 6 <=>; 47 =>; 0 <=; 0 <~>)
% Maximal formula depth : 18 ( 7 avg)
% Maximal term depth : 4 ( 1 avg)
% Number of predicates : 27 ( 25 usr; 1 prp; 0-3 aty)
% Number of functors : 41 ( 41 usr; 9 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_rsspace3,axiom,
~ v1_xboole_0(k1_rsspace3) ).
fof(fc2_rsspace3,axiom,
( ~ v1_xboole_0(k1_rsspace3)
& v1_rlsub_1(k1_rsspace3,k7_rsspace) ) ).
fof(fc3_rsspace3,axiom,
( ~ v3_struct_0(g2_rlvect_1(k1_rsspace3,k10_rsspace(k7_rsspace,k1_rsspace3),k8_rsspace(k7_rsspace,k1_rsspace3),k9_rsspace(k7_rsspace,k1_rsspace3)))
& v2_rlvect_1(g2_rlvect_1(k1_rsspace3,k10_rsspace(k7_rsspace,k1_rsspace3),k8_rsspace(k7_rsspace,k1_rsspace3),k9_rsspace(k7_rsspace,k1_rsspace3)))
& v3_rlvect_1(g2_rlvect_1(k1_rsspace3,k10_rsspace(k7_rsspace,k1_rsspace3),k8_rsspace(k7_rsspace,k1_rsspace3),k9_rsspace(k7_rsspace,k1_rsspace3)))
& v4_rlvect_1(g2_rlvect_1(k1_rsspace3,k10_rsspace(k7_rsspace,k1_rsspace3),k8_rsspace(k7_rsspace,k1_rsspace3),k9_rsspace(k7_rsspace,k1_rsspace3)))
& v5_rlvect_1(g2_rlvect_1(k1_rsspace3,k10_rsspace(k7_rsspace,k1_rsspace3),k8_rsspace(k7_rsspace,k1_rsspace3),k9_rsspace(k7_rsspace,k1_rsspace3)))
& v6_rlvect_1(g2_rlvect_1(k1_rsspace3,k10_rsspace(k7_rsspace,k1_rsspace3),k8_rsspace(k7_rsspace,k1_rsspace3),k9_rsspace(k7_rsspace,k1_rsspace3)))
& v7_rlvect_1(g2_rlvect_1(k1_rsspace3,k10_rsspace(k7_rsspace,k1_rsspace3),k8_rsspace(k7_rsspace,k1_rsspace3),k9_rsspace(k7_rsspace,k1_rsspace3))) ) ).
fof(fc4_rsspace3,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(k1_numbers,A),A)
& m1_relset_1(D,k2_zfmisc_1(k1_numbers,A),A)
& v1_funct_1(E)
& v1_funct_2(E,A,k1_numbers)
& m1_relset_1(E,A,k1_numbers) )
=> ( ~ v3_struct_0(g1_normsp_1(A,B,C,D,E))
& v1_normsp_1(g1_normsp_1(A,B,C,D,E)) ) ) ).
fof(fc5_rsspace3,axiom,
( ~ v3_struct_0(k3_rsspace3)
& v3_rlvect_1(k3_rsspace3)
& v4_rlvect_1(k3_rsspace3)
& v5_rlvect_1(k3_rsspace3)
& v6_rlvect_1(k3_rsspace3)
& v7_rlvect_1(k3_rsspace3)
& v2_normsp_1(k3_rsspace3) ) ).
fof(d1_rsspace3,axiom,
! [A] :
( m1_subset_1(A,k1_zfmisc_1(u1_struct_0(k7_rsspace)))
=> ( A = k1_rsspace3
<=> ! [B] :
( r2_hidden(B,A)
<=> ( r2_hidden(B,k1_rsspace)
& v2_series_1(k2_rsspace(B)) ) ) ) ) ).
fof(t1_rsspace3,axiom,
! [A] :
( m1_subset_1(A,k1_numbers)
=> ! [B] :
( ( v1_funct_1(B)
& v1_funct_2(B,k5_numbers,k1_numbers)
& m2_relset_1(B,k5_numbers,k1_numbers) )
=> ( v4_seq_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) = k18_complex1(k5_real_1(k2_seq_1(k5_numbers,k1_numbers,B,D),A)) )
=> ( v4_seq_2(C)
& k2_seq_2(C) = k18_complex1(k5_real_1(k2_seq_2(B),A)) ) ) ) ) ) ) ).
fof(d2_rsspace3,axiom,
! [A] :
( ( v1_funct_1(A)
& v1_funct_2(A,k1_rsspace3,k1_numbers)
& m2_relset_1(A,k1_rsspace3,k1_numbers) )
=> ( A = k2_rsspace3
<=> ! [B] :
( r2_hidden(B,k1_rsspace3)
=> k2_seq_1(k1_rsspace3,k1_numbers,A,B) = k2_series_1(k22_seq_1(k2_rsspace(B))) ) ) ) ).
fof(t2_rsspace3,axiom,
$true ).
fof(t3_rsspace3,axiom,
$true ).
fof(t4_rsspace3,axiom,
! [A] :
( l1_normsp_1(A)
=> ( ( ~ v3_struct_0(g2_rlvect_1(u1_struct_0(A),u2_struct_0(A),u1_rlvect_1(A),u2_rlvect_1(A)))
& v3_rlvect_1(g2_rlvect_1(u1_struct_0(A),u2_struct_0(A),u1_rlvect_1(A),u2_rlvect_1(A)))
& v4_rlvect_1(g2_rlvect_1(u1_struct_0(A),u2_struct_0(A),u1_rlvect_1(A),u2_rlvect_1(A)))
& v5_rlvect_1(g2_rlvect_1(u1_struct_0(A),u2_struct_0(A),u1_rlvect_1(A),u2_rlvect_1(A)))
& v6_rlvect_1(g2_rlvect_1(u1_struct_0(A),u2_struct_0(A),u1_rlvect_1(A),u2_rlvect_1(A)))
& v7_rlvect_1(g2_rlvect_1(u1_struct_0(A),u2_struct_0(A),u1_rlvect_1(A),u2_rlvect_1(A)))
& l2_rlvect_1(g2_rlvect_1(u1_struct_0(A),u2_struct_0(A),u1_rlvect_1(A),u2_rlvect_1(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)
& l2_rlvect_1(A) ) ) ) ).
fof(t5_rsspace3,axiom,
! [A] :
( ( v1_funct_1(A)
& v1_funct_2(A,k5_numbers,k1_numbers)
& m2_relset_1(A,k5_numbers,k1_numbers) )
=> ( ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> k2_seq_1(k5_numbers,k1_numbers,A,B) = np__0 )
=> ( v2_series_1(A)
& k2_series_1(k22_seq_1(A)) = np__0 ) ) ) ).
fof(t6_rsspace3,axiom,
! [A] :
( ( v1_funct_1(A)
& v1_funct_2(A,k5_numbers,k1_numbers)
& m2_relset_1(A,k5_numbers,k1_numbers) )
=> ( ( v2_series_1(A)
& k2_series_1(k22_seq_1(A)) = np__0 )
=> ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> k2_seq_1(k5_numbers,k1_numbers,A,B) = np__0 ) ) ) ).
fof(t7_rsspace3,axiom,
( ~ v3_struct_0(g1_normsp_1(k1_rsspace3,k10_rsspace(k7_rsspace,k1_rsspace3),k8_rsspace(k7_rsspace,k1_rsspace3),k9_rsspace(k7_rsspace,k1_rsspace3),k2_rsspace3))
& v3_rlvect_1(g1_normsp_1(k1_rsspace3,k10_rsspace(k7_rsspace,k1_rsspace3),k8_rsspace(k7_rsspace,k1_rsspace3),k9_rsspace(k7_rsspace,k1_rsspace3),k2_rsspace3))
& v4_rlvect_1(g1_normsp_1(k1_rsspace3,k10_rsspace(k7_rsspace,k1_rsspace3),k8_rsspace(k7_rsspace,k1_rsspace3),k9_rsspace(k7_rsspace,k1_rsspace3),k2_rsspace3))
& v5_rlvect_1(g1_normsp_1(k1_rsspace3,k10_rsspace(k7_rsspace,k1_rsspace3),k8_rsspace(k7_rsspace,k1_rsspace3),k9_rsspace(k7_rsspace,k1_rsspace3),k2_rsspace3))
& v6_rlvect_1(g1_normsp_1(k1_rsspace3,k10_rsspace(k7_rsspace,k1_rsspace3),k8_rsspace(k7_rsspace,k1_rsspace3),k9_rsspace(k7_rsspace,k1_rsspace3),k2_rsspace3))
& v7_rlvect_1(g1_normsp_1(k1_rsspace3,k10_rsspace(k7_rsspace,k1_rsspace3),k8_rsspace(k7_rsspace,k1_rsspace3),k9_rsspace(k7_rsspace,k1_rsspace3),k2_rsspace3))
& l2_rlvect_1(g1_normsp_1(k1_rsspace3,k10_rsspace(k7_rsspace,k1_rsspace3),k8_rsspace(k7_rsspace,k1_rsspace3),k9_rsspace(k7_rsspace,k1_rsspace3),k2_rsspace3)) ) ).
fof(d3_rsspace3,axiom,
k3_rsspace3 = g1_normsp_1(k1_rsspace3,k10_rsspace(k7_rsspace,k1_rsspace3),k8_rsspace(k7_rsspace,k1_rsspace3),k9_rsspace(k7_rsspace,k1_rsspace3),k2_rsspace3) ).
fof(t8_rsspace3,axiom,
( u1_struct_0(k3_rsspace3) = k1_rsspace3
& ! [A] :
( m1_subset_1(A,u1_struct_0(k3_rsspace3))
<=> ( v1_funct_1(A)
& v1_funct_2(A,k5_numbers,k1_numbers)
& m2_relset_1(A,k5_numbers,k1_numbers)
& v2_series_1(k2_rsspace(A)) ) )
& k1_rlvect_1(k3_rsspace3) = k6_rsspace
& ! [A] :
( m1_subset_1(A,u1_struct_0(k3_rsspace3))
=> A = k2_rsspace(A) )
& ! [A] :
( m1_subset_1(A,u1_struct_0(k3_rsspace3))
=> ! [B] :
( m1_subset_1(B,u1_struct_0(k3_rsspace3))
=> k2_rlvect_1(k3_rsspace3,A,B) = k9_seq_1(k2_rsspace(A),k2_rsspace(B)) ) )
& ! [A] :
( m1_subset_1(A,k1_numbers)
=> ! [B] :
( m1_subset_1(B,u1_struct_0(k3_rsspace3))
=> k3_rlvect_1(k3_rsspace3,B,A) = k14_seq_1(k2_rsspace(B),A) ) )
& ! [A] :
( m1_subset_1(A,u1_struct_0(k3_rsspace3))
=> ( k5_rlvect_1(k3_rsspace3,A) = k17_seq_1(k2_rsspace(A))
& k2_rsspace(k5_rlvect_1(k3_rsspace3,A)) = k17_seq_1(k2_rsspace(A)) ) )
& ! [A] :
( m1_subset_1(A,u1_struct_0(k3_rsspace3))
=> ! [B] :
( m1_subset_1(B,u1_struct_0(k3_rsspace3))
=> k6_rlvect_1(k3_rsspace3,A,B) = k10_seq_1(k2_rsspace(A),k2_rsspace(B)) ) )
& ! [A] :
( m1_subset_1(A,u1_struct_0(k3_rsspace3))
=> v2_series_1(k2_rsspace(A)) )
& ! [A] :
( m1_subset_1(A,u1_struct_0(k3_rsspace3))
=> k1_normsp_1(k3_rsspace3,A) = k2_series_1(k22_seq_1(k2_rsspace(A))) ) ) ).
fof(t9_rsspace3,axiom,
! [A] :
( m1_subset_1(A,u1_struct_0(k3_rsspace3))
=> ! [B] :
( m1_subset_1(B,u1_struct_0(k3_rsspace3))
=> ! [C] :
( m1_subset_1(C,k1_numbers)
=> ( ( k1_normsp_1(k3_rsspace3,A) = np__0
=> A = k1_rlvect_1(k3_rsspace3) )
& ( A = k1_rlvect_1(k3_rsspace3)
=> k1_normsp_1(k3_rsspace3,A) = np__0 )
& r1_xreal_0(np__0,k1_normsp_1(k3_rsspace3,A))
& r1_xreal_0(k1_normsp_1(k3_rsspace3,k2_rlvect_1(k3_rsspace3,A,B)),k3_real_1(k1_normsp_1(k3_rsspace3,A),k1_normsp_1(k3_rsspace3,B)))
& k1_normsp_1(k3_rsspace3,k3_rlvect_1(k3_rsspace3,A,C)) = k4_real_1(k18_complex1(C),k1_normsp_1(k3_rsspace3,A)) ) ) ) ) ).
fof(d4_rsspace3,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l1_normsp_1(A) )
=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> k4_rsspace3(A,B,C) = k1_normsp_1(A,k6_rlvect_1(A,B,C)) ) ) ) ).
fof(d5_rsspace3,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l1_normsp_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_rsspace3(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_rsspace3(A,k2_normsp_1(A,B,E),k2_normsp_1(A,B,F))) ) ) ) ) ) ) ) ) ).
fof(t10_rsspace3,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)
& v1_funct_2(B,k5_numbers,u1_struct_0(A))
& m2_relset_1(B,k5_numbers,u1_struct_0(A)) )
=> ( v1_rsspace3(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,k1_normsp_1(A,k6_rlvect_1(A,k2_normsp_1(A,B,E),k2_normsp_1(A,B,F)))) ) ) ) ) ) ) ) ) ).
fof(t11_rsspace3,axiom,
! [A] :
( ( v1_funct_1(A)
& v1_funct_2(A,k5_numbers,u1_struct_0(k3_rsspace3))
& m2_relset_1(A,k5_numbers,u1_struct_0(k3_rsspace3)) )
=> ( v1_rsspace3(A,k3_rsspace3)
=> v4_normsp_1(A,k3_rsspace3) ) ) ).
fof(dt_k1_rsspace3,axiom,
m1_subset_1(k1_rsspace3,k1_zfmisc_1(u1_struct_0(k7_rsspace))) ).
fof(dt_k2_rsspace3,axiom,
( v1_funct_1(k2_rsspace3)
& v1_funct_2(k2_rsspace3,k1_rsspace3,k1_numbers)
& m2_relset_1(k2_rsspace3,k1_rsspace3,k1_numbers) ) ).
fof(dt_k3_rsspace3,axiom,
( ~ v3_struct_0(k3_rsspace3)
& l1_normsp_1(k3_rsspace3) ) ).
fof(dt_k4_rsspace3,axiom,
! [A,B,C] :
( ( ~ v3_struct_0(A)
& l1_normsp_1(A)
& m1_subset_1(B,u1_struct_0(A))
& m1_subset_1(C,u1_struct_0(A)) )
=> m1_subset_1(k4_rsspace3(A,B,C),k1_numbers) ) ).
%------------------------------------------------------------------------------