SET007 Axioms: SET007+868.ax
%------------------------------------------------------------------------------
% File : SET007+868 : TPTP v9.0.0. Released v3.4.0.
% Domain : Set Theory
% Axioms : The Banach Space l^p
% 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 : lp_space [Urb08]
% Status : Satisfiable
% Syntax : Number of formulae : 24 ( 0 unt; 0 def)
% Number of atoms : 213 ( 29 equ)
% Maximal formula atoms : 30 ( 8 avg)
% Number of connectives : 203 ( 14 ~; 0 |; 97 &)
% ( 5 <=>; 87 =>; 0 <=; 0 <~>)
% Maximal formula depth : 19 ( 9 avg)
% Maximal term depth : 6 ( 2 avg)
% Number of predicates : 25 ( 24 usr; 0 prp; 1-3 aty)
% Number of functors : 37 ( 37 usr; 7 con; 0-5 aty)
% Number of variables : 63 ( 63 !; 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(d1_lp_space,axiom,
! [A] :
( ( v1_funct_1(A)
& v1_funct_2(A,k5_numbers,k1_numbers)
& m2_relset_1(A,k5_numbers,k1_numbers) )
=> ! [B] :
( m1_subset_1(B,k1_numbers)
=> ! [C] :
( ( v1_funct_1(C)
& v1_funct_2(C,k5_numbers,k1_numbers)
& m2_relset_1(C,k5_numbers,k1_numbers) )
=> ( C = k1_lp_space(A,B)
<=> ! [D] :
( m2_subset_1(D,k1_numbers,k5_numbers)
=> k2_seq_1(k5_numbers,k1_numbers,C,D) = k4_power(k18_complex1(k2_seq_1(k5_numbers,k1_numbers,A,D)),B) ) ) ) ) ) ).
fof(d2_lp_space,axiom,
! [A] :
( m1_subset_1(A,k1_numbers)
=> ( r1_xreal_0(np__1,A)
=> ! [B] :
( ( ~ v1_xboole_0(B)
& m1_subset_1(B,k1_zfmisc_1(u1_struct_0(k7_rsspace))) )
=> ( B = k2_lp_space(A)
<=> ! [C] :
( r2_hidden(C,B)
<=> ( r2_hidden(C,k1_rsspace)
& v1_series_1(k1_lp_space(k2_rsspace(C),A)) ) ) ) ) ) ) ).
fof(t1_lp_space,axiom,
! [A] :
( m1_subset_1(A,k1_numbers)
=> ! [B] :
( m1_subset_1(B,k1_numbers)
=> ! [C] :
( m1_subset_1(C,k1_numbers)
=> ~ ( r1_xreal_0(np__0,A)
& ~ r1_xreal_0(B,A)
& ~ r1_xreal_0(C,np__0)
& r1_xreal_0(k4_power(B,C),k4_power(A,C)) ) ) ) ) ).
fof(t2_lp_space,axiom,
! [A] :
( m1_subset_1(A,k1_numbers)
=> ( r1_xreal_0(np__1,A)
=> ! [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,k1_numbers)
& m2_relset_1(C,k5_numbers,k1_numbers) )
=> ! [D] :
( m2_subset_1(D,k1_numbers,k5_numbers)
=> r1_xreal_0(k4_power(k2_seq_1(k5_numbers,k1_numbers,k1_series_1(k1_lp_space(k9_seq_1(B,C),A)),D),k6_real_1(np__1,A)),k3_real_1(k4_power(k2_seq_1(k5_numbers,k1_numbers,k1_series_1(k1_lp_space(B,A)),D),k6_real_1(np__1,A)),k4_power(k2_seq_1(k5_numbers,k1_numbers,k1_series_1(k1_lp_space(C,A)),D),k6_real_1(np__1,A)))) ) ) ) ) ) ).
fof(t3_lp_space,axiom,
! [A] :
( ( v1_funct_1(A)
& v1_funct_2(A,k5_numbers,k1_numbers)
& m2_relset_1(A,k5_numbers,k1_numbers) )
=> ! [B] :
( ( v1_funct_1(B)
& v1_funct_2(B,k5_numbers,k1_numbers)
& m2_relset_1(B,k5_numbers,k1_numbers) )
=> ! [C] :
( m1_subset_1(C,k1_numbers)
=> ( ( r1_xreal_0(np__1,C)
& v1_series_1(k1_lp_space(A,C))
& v1_series_1(k1_lp_space(B,C)) )
=> ( v1_series_1(k1_lp_space(k9_seq_1(A,B),C))
& r1_xreal_0(k4_power(k2_series_1(k1_lp_space(k9_seq_1(A,B),C)),k6_real_1(np__1,C)),k3_real_1(k4_power(k2_series_1(k1_lp_space(A,C)),k6_real_1(np__1,C)),k4_power(k2_series_1(k1_lp_space(B,C)),k6_real_1(np__1,C)))) ) ) ) ) ) ).
fof(t4_lp_space,axiom,
! [A] :
( m1_subset_1(A,k1_numbers)
=> ( r1_xreal_0(np__1,A)
=> v1_rlsub_1(k2_lp_space(A),k7_rsspace) ) ) ).
fof(t5_lp_space,axiom,
! [A] :
( m1_subset_1(A,k1_numbers)
=> ( r1_xreal_0(np__1,A)
=> m1_rlsub_1(g2_rlvect_1(k2_lp_space(A),k10_rsspace(k7_rsspace,k2_lp_space(A)),k8_rsspace(k7_rsspace,k2_lp_space(A)),k9_rsspace(k7_rsspace,k2_lp_space(A))),k7_rsspace) ) ) ).
fof(t6_lp_space,axiom,
! [A] :
( m1_subset_1(A,k1_numbers)
=> ( r1_xreal_0(np__1,A)
=> ( v3_rlvect_1(g2_rlvect_1(k2_lp_space(A),k10_rsspace(k7_rsspace,k2_lp_space(A)),k8_rsspace(k7_rsspace,k2_lp_space(A)),k9_rsspace(k7_rsspace,k2_lp_space(A))))
& v4_rlvect_1(g2_rlvect_1(k2_lp_space(A),k10_rsspace(k7_rsspace,k2_lp_space(A)),k8_rsspace(k7_rsspace,k2_lp_space(A)),k9_rsspace(k7_rsspace,k2_lp_space(A))))
& v5_rlvect_1(g2_rlvect_1(k2_lp_space(A),k10_rsspace(k7_rsspace,k2_lp_space(A)),k8_rsspace(k7_rsspace,k2_lp_space(A)),k9_rsspace(k7_rsspace,k2_lp_space(A))))
& v6_rlvect_1(g2_rlvect_1(k2_lp_space(A),k10_rsspace(k7_rsspace,k2_lp_space(A)),k8_rsspace(k7_rsspace,k2_lp_space(A)),k9_rsspace(k7_rsspace,k2_lp_space(A))))
& v7_rlvect_1(g2_rlvect_1(k2_lp_space(A),k10_rsspace(k7_rsspace,k2_lp_space(A)),k8_rsspace(k7_rsspace,k2_lp_space(A)),k9_rsspace(k7_rsspace,k2_lp_space(A)))) ) ) ) ).
fof(t7_lp_space,axiom,
! [A] :
( m1_subset_1(A,k1_numbers)
=> ( r1_xreal_0(np__1,A)
=> ( ~ v3_struct_0(g2_rlvect_1(k2_lp_space(A),k10_rsspace(k7_rsspace,k2_lp_space(A)),k8_rsspace(k7_rsspace,k2_lp_space(A)),k9_rsspace(k7_rsspace,k2_lp_space(A))))
& v3_rlvect_1(g2_rlvect_1(k2_lp_space(A),k10_rsspace(k7_rsspace,k2_lp_space(A)),k8_rsspace(k7_rsspace,k2_lp_space(A)),k9_rsspace(k7_rsspace,k2_lp_space(A))))
& v4_rlvect_1(g2_rlvect_1(k2_lp_space(A),k10_rsspace(k7_rsspace,k2_lp_space(A)),k8_rsspace(k7_rsspace,k2_lp_space(A)),k9_rsspace(k7_rsspace,k2_lp_space(A))))
& v5_rlvect_1(g2_rlvect_1(k2_lp_space(A),k10_rsspace(k7_rsspace,k2_lp_space(A)),k8_rsspace(k7_rsspace,k2_lp_space(A)),k9_rsspace(k7_rsspace,k2_lp_space(A))))
& v6_rlvect_1(g2_rlvect_1(k2_lp_space(A),k10_rsspace(k7_rsspace,k2_lp_space(A)),k8_rsspace(k7_rsspace,k2_lp_space(A)),k9_rsspace(k7_rsspace,k2_lp_space(A))))
& v7_rlvect_1(g2_rlvect_1(k2_lp_space(A),k10_rsspace(k7_rsspace,k2_lp_space(A)),k8_rsspace(k7_rsspace,k2_lp_space(A)),k9_rsspace(k7_rsspace,k2_lp_space(A))))
& l2_rlvect_1(g2_rlvect_1(k2_lp_space(A),k10_rsspace(k7_rsspace,k2_lp_space(A)),k8_rsspace(k7_rsspace,k2_lp_space(A)),k9_rsspace(k7_rsspace,k2_lp_space(A)))) ) ) ) ).
fof(d3_lp_space,axiom,
! [A] :
( m1_subset_1(A,k1_numbers)
=> ! [B] :
( ( v1_funct_1(B)
& v1_funct_2(B,k2_lp_space(A),k1_numbers)
& m2_relset_1(B,k2_lp_space(A),k1_numbers) )
=> ( B = k3_lp_space(A)
<=> ! [C] :
( r2_hidden(C,k2_lp_space(A))
=> k2_seq_1(k2_lp_space(A),k1_numbers,B,C) = k4_power(k2_series_1(k1_lp_space(k2_rsspace(C),A)),k6_real_1(np__1,A)) ) ) ) ) ).
fof(t8_lp_space,axiom,
! [A] :
( m1_subset_1(A,k1_numbers)
=> ( r1_xreal_0(np__1,A)
=> ( ~ v3_struct_0(g1_normsp_1(k2_lp_space(A),k10_rsspace(k7_rsspace,k2_lp_space(A)),k8_rsspace(k7_rsspace,k2_lp_space(A)),k9_rsspace(k7_rsspace,k2_lp_space(A)),k3_lp_space(A)))
& v3_rlvect_1(g1_normsp_1(k2_lp_space(A),k10_rsspace(k7_rsspace,k2_lp_space(A)),k8_rsspace(k7_rsspace,k2_lp_space(A)),k9_rsspace(k7_rsspace,k2_lp_space(A)),k3_lp_space(A)))
& v4_rlvect_1(g1_normsp_1(k2_lp_space(A),k10_rsspace(k7_rsspace,k2_lp_space(A)),k8_rsspace(k7_rsspace,k2_lp_space(A)),k9_rsspace(k7_rsspace,k2_lp_space(A)),k3_lp_space(A)))
& v5_rlvect_1(g1_normsp_1(k2_lp_space(A),k10_rsspace(k7_rsspace,k2_lp_space(A)),k8_rsspace(k7_rsspace,k2_lp_space(A)),k9_rsspace(k7_rsspace,k2_lp_space(A)),k3_lp_space(A)))
& v6_rlvect_1(g1_normsp_1(k2_lp_space(A),k10_rsspace(k7_rsspace,k2_lp_space(A)),k8_rsspace(k7_rsspace,k2_lp_space(A)),k9_rsspace(k7_rsspace,k2_lp_space(A)),k3_lp_space(A)))
& v7_rlvect_1(g1_normsp_1(k2_lp_space(A),k10_rsspace(k7_rsspace,k2_lp_space(A)),k8_rsspace(k7_rsspace,k2_lp_space(A)),k9_rsspace(k7_rsspace,k2_lp_space(A)),k3_lp_space(A)))
& l2_rlvect_1(g1_normsp_1(k2_lp_space(A),k10_rsspace(k7_rsspace,k2_lp_space(A)),k8_rsspace(k7_rsspace,k2_lp_space(A)),k9_rsspace(k7_rsspace,k2_lp_space(A)),k3_lp_space(A))) ) ) ) ).
fof(t9_lp_space,axiom,
! [A] :
( m1_subset_1(A,k1_numbers)
=> ( r1_xreal_0(np__1,A)
=> m1_rlsub_1(g1_normsp_1(k2_lp_space(A),k10_rsspace(k7_rsspace,k2_lp_space(A)),k8_rsspace(k7_rsspace,k2_lp_space(A)),k9_rsspace(k7_rsspace,k2_lp_space(A)),k3_lp_space(A)),k7_rsspace) ) ) ).
fof(t10_lp_space,axiom,
! [A] :
( m1_subset_1(A,k1_numbers)
=> ( r1_xreal_0(np__1,A)
=> ! [B] :
( ( ~ v3_struct_0(B)
& l1_normsp_1(B) )
=> ( B = g1_normsp_1(k2_lp_space(A),k10_rsspace(k7_rsspace,k2_lp_space(A)),k8_rsspace(k7_rsspace,k2_lp_space(A)),k9_rsspace(k7_rsspace,k2_lp_space(A)),k3_lp_space(A))
=> ( u1_struct_0(B) = k2_lp_space(A)
& ! [C] :
( m1_subset_1(C,u1_struct_0(B))
<=> ( v1_funct_1(C)
& v1_funct_2(C,k5_numbers,k1_numbers)
& m2_relset_1(C,k5_numbers,k1_numbers)
& v1_series_1(k1_lp_space(k2_rsspace(C),A)) ) )
& k1_rlvect_1(B) = k6_rsspace
& ! [C] :
( m1_subset_1(C,u1_struct_0(B))
=> C = k2_rsspace(C) )
& ! [C] :
( m1_subset_1(C,u1_struct_0(B))
=> ! [D] :
( m1_subset_1(D,u1_struct_0(B))
=> k2_rlvect_1(B,C,D) = k9_seq_1(k2_rsspace(C),k2_rsspace(D)) ) )
& ! [C] :
( m1_subset_1(C,k1_numbers)
=> ! [D] :
( m1_subset_1(D,u1_struct_0(B))
=> k3_rlvect_1(B,D,C) = k14_seq_1(k2_rsspace(D),C) ) )
& ! [C] :
( m1_subset_1(C,u1_struct_0(B))
=> ( k5_rlvect_1(B,C) = k17_seq_1(k2_rsspace(C))
& k2_rsspace(k5_rlvect_1(B,C)) = k17_seq_1(k2_rsspace(C)) ) )
& ! [C] :
( m1_subset_1(C,u1_struct_0(B))
=> ! [D] :
( m1_subset_1(D,u1_struct_0(B))
=> k6_rlvect_1(B,C,D) = k10_seq_1(k2_rsspace(C),k2_rsspace(D)) ) )
& ! [C] :
( m1_subset_1(C,u1_struct_0(B))
=> v1_series_1(k1_lp_space(k2_rsspace(C),A)) )
& ! [C] :
( m1_subset_1(C,u1_struct_0(B))
=> k1_normsp_1(B,C) = k4_power(k2_series_1(k1_lp_space(k2_rsspace(C),A)),k6_real_1(np__1,A)) ) ) ) ) ) ) ).
fof(t11_lp_space,axiom,
! [A] :
( m1_subset_1(A,k1_numbers)
=> ( r1_xreal_0(np__1,A)
=> ! [B] :
( ( v1_funct_1(B)
& v1_funct_2(B,k5_numbers,k1_numbers)
& m2_relset_1(B,k5_numbers,k1_numbers) )
=> ( ! [C] :
( m2_subset_1(C,k1_numbers,k5_numbers)
=> k2_seq_1(k5_numbers,k1_numbers,B,C) = np__0 )
=> ( v1_series_1(k1_lp_space(B,A))
& k4_power(k2_series_1(k1_lp_space(B,A)),k6_real_1(np__1,A)) = np__0 ) ) ) ) ) ).
fof(t12_lp_space,axiom,
! [A] :
( m1_subset_1(A,k1_numbers)
=> ( r1_xreal_0(np__1,A)
=> ! [B] :
( ( v1_funct_1(B)
& v1_funct_2(B,k5_numbers,k1_numbers)
& m2_relset_1(B,k5_numbers,k1_numbers) )
=> ( ( v1_series_1(k1_lp_space(B,A))
& k4_power(k2_series_1(k1_lp_space(B,A)),k6_real_1(np__1,A)) = np__0 )
=> ! [C] :
( m2_subset_1(C,k1_numbers,k5_numbers)
=> k2_seq_1(k5_numbers,k1_numbers,B,C) = np__0 ) ) ) ) ) ).
fof(t13_lp_space,axiom,
! [A] :
( m1_subset_1(A,k1_numbers)
=> ( r1_xreal_0(np__1,A)
=> ! [B] :
( ( ~ v3_struct_0(B)
& l1_normsp_1(B) )
=> ( B = g1_normsp_1(k2_lp_space(A),k10_rsspace(k7_rsspace,k2_lp_space(A)),k8_rsspace(k7_rsspace,k2_lp_space(A)),k9_rsspace(k7_rsspace,k2_lp_space(A)),k3_lp_space(A))
=> ! [C] :
( m1_subset_1(C,u1_struct_0(B))
=> ! [D] :
( m1_subset_1(D,u1_struct_0(B))
=> ! [E] :
( m1_subset_1(E,k1_numbers)
=> ( ( k1_normsp_1(B,C) = np__0
=> C = k1_rlvect_1(B) )
& ( C = k1_rlvect_1(B)
=> k1_normsp_1(B,C) = np__0 )
& r1_xreal_0(np__0,k1_normsp_1(B,C))
& r1_xreal_0(k1_normsp_1(B,k2_rlvect_1(B,C,D)),k3_real_1(k1_normsp_1(B,C),k1_normsp_1(B,D)))
& k1_normsp_1(B,k3_rlvect_1(B,C,E)) = k4_real_1(k18_complex1(E),k1_normsp_1(B,C)) ) ) ) ) ) ) ) ) ).
fof(t14_lp_space,axiom,
! [A] :
( m1_subset_1(A,k1_numbers)
=> ( r1_xreal_0(np__1,A)
=> ! [B] :
( ( ~ v3_struct_0(B)
& l1_normsp_1(B) )
=> ( B = g1_normsp_1(k2_lp_space(A),k10_rsspace(k7_rsspace,k2_lp_space(A)),k8_rsspace(k7_rsspace,k2_lp_space(A)),k9_rsspace(k7_rsspace,k2_lp_space(A)),k3_lp_space(A))
=> v2_normsp_1(B) ) ) ) ) ).
fof(t15_lp_space,axiom,
! [A] :
( m1_subset_1(A,k1_numbers)
=> ( r1_xreal_0(np__1,A)
=> ! [B] :
( ( ~ v3_struct_0(B)
& l1_normsp_1(B) )
=> ( B = g1_normsp_1(k2_lp_space(A),k10_rsspace(k7_rsspace,k2_lp_space(A)),k8_rsspace(k7_rsspace,k2_lp_space(A)),k9_rsspace(k7_rsspace,k2_lp_space(A)),k3_lp_space(A))
=> ( ~ 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) ) ) ) ) ) ).
fof(t16_lp_space,axiom,
! [A] :
( m1_subset_1(A,k1_numbers)
=> ( r1_xreal_0(np__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) )
=> ( B = g1_normsp_1(k2_lp_space(A),k10_rsspace(k7_rsspace,k2_lp_space(A)),k8_rsspace(k7_rsspace,k2_lp_space(A)),k9_rsspace(k7_rsspace,k2_lp_space(A)),k3_lp_space(A))
=> ! [C] :
( ( v1_funct_1(C)
& v1_funct_2(C,k5_numbers,u1_struct_0(B))
& m2_relset_1(C,k5_numbers,u1_struct_0(B)) )
=> ( v1_rsspace3(C,B)
=> v4_normsp_1(C,B) ) ) ) ) ) ) ).
fof(d4_lp_space,axiom,
! [A] :
( m1_subset_1(A,k1_numbers)
=> ( r1_xreal_0(np__1,A)
=> k4_lp_space(A) = g1_normsp_1(k2_lp_space(A),k10_rsspace(k7_rsspace,k2_lp_space(A)),k8_rsspace(k7_rsspace,k2_lp_space(A)),k9_rsspace(k7_rsspace,k2_lp_space(A)),k3_lp_space(A)) ) ) ).
fof(dt_k1_lp_space,axiom,
! [A,B] :
( ( v1_funct_1(A)
& v1_funct_2(A,k5_numbers,k1_numbers)
& m1_relset_1(A,k5_numbers,k1_numbers)
& m1_subset_1(B,k1_numbers) )
=> ( v1_funct_1(k1_lp_space(A,B))
& v1_funct_2(k1_lp_space(A,B),k5_numbers,k1_numbers)
& m2_relset_1(k1_lp_space(A,B),k5_numbers,k1_numbers) ) ) ).
fof(dt_k2_lp_space,axiom,
! [A] :
( m1_subset_1(A,k1_numbers)
=> ( ~ v1_xboole_0(k2_lp_space(A))
& m1_subset_1(k2_lp_space(A),k1_zfmisc_1(u1_struct_0(k7_rsspace))) ) ) ).
fof(dt_k3_lp_space,axiom,
! [A] :
( m1_subset_1(A,k1_numbers)
=> ( v1_funct_1(k3_lp_space(A))
& v1_funct_2(k3_lp_space(A),k2_lp_space(A),k1_numbers)
& m2_relset_1(k3_lp_space(A),k2_lp_space(A),k1_numbers) ) ) ).
fof(dt_k4_lp_space,axiom,
! [A] :
( m1_subset_1(A,k1_numbers)
=> ( ~ v3_struct_0(k4_lp_space(A))
& v3_rlvect_1(k4_lp_space(A))
& v4_rlvect_1(k4_lp_space(A))
& v5_rlvect_1(k4_lp_space(A))
& v6_rlvect_1(k4_lp_space(A))
& v7_rlvect_1(k4_lp_space(A))
& v2_normsp_1(k4_lp_space(A))
& v4_lopban_1(k4_lp_space(A))
& l1_normsp_1(k4_lp_space(A)) ) ) ).
%------------------------------------------------------------------------------