SET007 Axioms: SET007+258.ax
%------------------------------------------------------------------------------
% File : SET007+258 : TPTP v8.2.0. Released v3.4.0.
% Domain : Set Theory
% Axioms : Finite Sequence over Ring and Left-, Right-, and Bi-Modules
% Version : [Urb08] axioms.
% English : Finite Sequence over Ring and Left-, Right-, and Bi-Modules over
% a Ring
% 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 : algseq_1 [Urb08]
% Status : Satisfiable
% Syntax : Number of formulae : 49 ( 17 unt; 0 def)
% Number of atoms : 233 ( 38 equ)
% Maximal formula atoms : 15 ( 4 avg)
% Number of connectives : 211 ( 27 ~; 1 |; 96 &)
% ( 11 <=>; 76 =>; 0 <=; 0 <~>)
% Maximal formula depth : 13 ( 6 avg)
% Maximal term depth : 4 ( 1 avg)
% Number of predicates : 16 ( 14 usr; 1 prp; 0-3 aty)
% Number of functors : 26 ( 26 usr; 8 con; 0-3 aty)
% Number of variables : 73 ( 68 !; 5 ?)
% 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(rc1_algseq_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l2_struct_0(A) )
=> ? [B] :
( m1_relset_1(B,k5_numbers,u1_struct_0(A))
& v1_relat_1(B)
& v1_funct_1(B)
& v1_funct_2(B,k5_numbers,u1_struct_0(A))
& v1_algseq_1(B,A) ) ) ).
fof(t1_algseq_1,axiom,
$true ).
fof(t2_algseq_1,axiom,
$true ).
fof(t3_algseq_1,axiom,
$true ).
fof(t4_algseq_1,axiom,
$true ).
fof(t5_algseq_1,axiom,
$true ).
fof(t6_algseq_1,axiom,
$true ).
fof(t7_algseq_1,axiom,
$true ).
fof(t8_algseq_1,axiom,
$true ).
fof(t9_algseq_1,axiom,
$true ).
fof(t10_algseq_1,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> ( r2_hidden(A,k2_algseq_1(B))
<=> ~ r1_xreal_0(B,A) ) ) ) ).
fof(t11_algseq_1,axiom,
( k2_algseq_1(np__0) = k1_xboole_0
& k2_algseq_1(np__1) = k1_tarski(np__0)
& k2_algseq_1(np__2) = k2_tarski(np__0,np__1) ) ).
fof(t12_algseq_1,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> r2_hidden(A,k2_algseq_1(k1_nat_1(A,np__1))) ) ).
fof(t13_algseq_1,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> ( r1_xreal_0(A,B)
<=> r1_tarski(k2_algseq_1(A),k2_algseq_1(B)) ) ) ) ).
fof(t14_algseq_1,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> ( k2_algseq_1(A) = k2_algseq_1(B)
=> A = B ) ) ) ).
fof(t15_algseq_1,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> ( r1_xreal_0(A,B)
=> ( k2_algseq_1(A) = k5_subset_1(k5_numbers,k2_algseq_1(A),k2_algseq_1(B))
& k2_algseq_1(A) = k5_subset_1(k5_numbers,k2_algseq_1(B),k2_algseq_1(A)) ) ) ) ) ).
fof(t16_algseq_1,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> ( ( k2_algseq_1(A) = k5_subset_1(k5_numbers,k2_algseq_1(A),k2_algseq_1(B))
| k2_algseq_1(A) = k5_subset_1(k5_numbers,k2_algseq_1(B),k2_algseq_1(A)) )
=> r1_xreal_0(A,B) ) ) ) ).
fof(t17_algseq_1,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> k2_xboole_0(k2_algseq_1(A),k1_tarski(A)) = k2_algseq_1(k1_nat_1(A,np__1)) ) ).
fof(d2_algseq_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l2_struct_0(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_algseq_1(B,A)
<=> ? [C] :
( m2_subset_1(C,k1_numbers,k5_numbers)
& ! [D] :
( m2_subset_1(D,k1_numbers,k5_numbers)
=> ( r1_xreal_0(C,D)
=> k2_normsp_1(A,B,D) = k1_rlvect_1(A) ) ) ) ) ) ) ).
fof(d3_algseq_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l2_struct_0(A) )
=> ! [B] :
( ( v1_funct_1(B)
& v1_funct_2(B,k5_numbers,u1_struct_0(A))
& v1_algseq_1(B,A)
& m2_relset_1(B,k5_numbers,u1_struct_0(A)) )
=> ! [C] :
( m2_subset_1(C,k1_numbers,k5_numbers)
=> ( r1_algseq_1(A,B,C)
<=> ! [D] :
( m2_subset_1(D,k1_numbers,k5_numbers)
=> ( r1_xreal_0(C,D)
=> k2_normsp_1(A,B,D) = k1_rlvect_1(A) ) ) ) ) ) ) ).
fof(d4_algseq_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l2_struct_0(A) )
=> ! [B] :
( ( v1_funct_1(B)
& v1_funct_2(B,k5_numbers,u1_struct_0(A))
& v1_algseq_1(B,A)
& m2_relset_1(B,k5_numbers,u1_struct_0(A)) )
=> ! [C] :
( m2_subset_1(C,k1_numbers,k5_numbers)
=> ( C = k3_algseq_1(A,B)
<=> ( r1_algseq_1(A,B,C)
& ! [D] :
( m2_subset_1(D,k1_numbers,k5_numbers)
=> ( r1_algseq_1(A,B,D)
=> r1_xreal_0(C,D) ) ) ) ) ) ) ) ).
fof(t18_algseq_1,axiom,
$true ).
fof(t19_algseq_1,axiom,
$true ).
fof(t20_algseq_1,axiom,
$true ).
fof(t21_algseq_1,axiom,
$true ).
fof(t22_algseq_1,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( ( ~ v3_struct_0(B)
& l2_struct_0(B) )
=> ! [C] :
( ( v1_funct_1(C)
& v1_funct_2(C,k5_numbers,u1_struct_0(B))
& v1_algseq_1(C,B)
& m2_relset_1(C,k5_numbers,u1_struct_0(B)) )
=> ( r1_xreal_0(k3_algseq_1(B,C),A)
=> k2_normsp_1(B,C,A) = k1_rlvect_1(B) ) ) ) ) ).
fof(t23_algseq_1,axiom,
$true ).
fof(t24_algseq_1,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( ( ~ v3_struct_0(B)
& l2_struct_0(B) )
=> ! [C] :
( ( v1_funct_1(C)
& v1_funct_2(C,k5_numbers,u1_struct_0(B))
& v1_algseq_1(C,B)
& m2_relset_1(C,k5_numbers,u1_struct_0(B)) )
=> ( ! [D] :
( m2_subset_1(D,k1_numbers,k5_numbers)
=> ~ ( ~ r1_xreal_0(A,D)
& k2_normsp_1(B,C,D) = k1_rlvect_1(B) ) )
=> r1_xreal_0(A,k3_algseq_1(B,C)) ) ) ) ) ).
fof(t25_algseq_1,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( ( ~ v3_struct_0(B)
& l2_struct_0(B) )
=> ! [C] :
( ( v1_funct_1(C)
& v1_funct_2(C,k5_numbers,u1_struct_0(B))
& v1_algseq_1(C,B)
& m2_relset_1(C,k5_numbers,u1_struct_0(B)) )
=> ~ ( k3_algseq_1(B,C) = k1_nat_1(A,np__1)
& k2_normsp_1(B,C,A) = k1_rlvect_1(B) ) ) ) ) ).
fof(d5_algseq_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l2_struct_0(A) )
=> ! [B] :
( ( v1_funct_1(B)
& v1_funct_2(B,k5_numbers,u1_struct_0(A))
& v1_algseq_1(B,A)
& m2_relset_1(B,k5_numbers,u1_struct_0(A)) )
=> k4_algseq_1(A,B) = k2_algseq_1(k3_algseq_1(A,B)) ) ) ).
fof(t26_algseq_1,axiom,
$true ).
fof(t27_algseq_1,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( ( ~ v3_struct_0(B)
& l2_struct_0(B) )
=> ! [C] :
( ( v1_funct_1(C)
& v1_funct_2(C,k5_numbers,u1_struct_0(B))
& v1_algseq_1(C,B)
& m2_relset_1(C,k5_numbers,u1_struct_0(B)) )
=> ( A = k3_algseq_1(B,C)
<=> k2_algseq_1(A) = k4_algseq_1(B,C) ) ) ) ) ).
fof(t28_algseq_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l2_struct_0(A) )
=> ! [B] :
( ( v1_funct_1(B)
& v1_funct_2(B,k5_numbers,u1_struct_0(A))
& v1_algseq_1(B,A)
& m2_relset_1(B,k5_numbers,u1_struct_0(A)) )
=> ! [C] :
( ( v1_funct_1(C)
& v1_funct_2(C,k5_numbers,u1_struct_0(A))
& v1_algseq_1(C,A)
& m2_relset_1(C,k5_numbers,u1_struct_0(A)) )
=> ( ( k3_algseq_1(A,B) = k3_algseq_1(A,C)
& ! [D] :
( m2_subset_1(D,k1_numbers,k5_numbers)
=> ( ~ r1_xreal_0(k3_algseq_1(A,B),D)
=> k2_normsp_1(A,B,D) = k2_normsp_1(A,C,D) ) ) )
=> B = C ) ) ) ) ).
fof(t29_algseq_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l2_struct_0(A) )
=> ( u1_struct_0(A) != k1_struct_0(A,k1_rlvect_1(A))
=> ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> ? [C] :
( v1_funct_1(C)
& v1_funct_2(C,k5_numbers,u1_struct_0(A))
& v1_algseq_1(C,A)
& m2_relset_1(C,k5_numbers,u1_struct_0(A))
& k3_algseq_1(A,C) = B ) ) ) ) ).
fof(d6_algseq_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l2_struct_0(A) )
=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> ! [C] :
( ( v1_funct_1(C)
& v1_funct_2(C,k5_numbers,u1_struct_0(A))
& v1_algseq_1(C,A)
& m2_relset_1(C,k5_numbers,u1_struct_0(A)) )
=> ( C = k5_algseq_1(A,B)
<=> ( r1_xreal_0(k3_algseq_1(A,C),np__1)
& k2_normsp_1(A,C,np__0) = B ) ) ) ) ) ).
fof(t30_algseq_1,axiom,
$true ).
fof(t31_algseq_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l2_struct_0(A) )
=> ! [B] :
( ( v1_funct_1(B)
& v1_funct_2(B,k5_numbers,u1_struct_0(A))
& v1_algseq_1(B,A)
& m2_relset_1(B,k5_numbers,u1_struct_0(A)) )
=> ( B = k5_algseq_1(A,k1_rlvect_1(A))
<=> k3_algseq_1(A,B) = np__0 ) ) ) ).
fof(t32_algseq_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l2_struct_0(A) )
=> ! [B] :
( ( v1_funct_1(B)
& v1_funct_2(B,k5_numbers,u1_struct_0(A))
& v1_algseq_1(B,A)
& m2_relset_1(B,k5_numbers,u1_struct_0(A)) )
=> ( B = k5_algseq_1(A,k1_rlvect_1(A))
<=> k4_algseq_1(A,B) = k1_xboole_0 ) ) ) ).
fof(t33_algseq_1,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( ( ~ v3_struct_0(B)
& l2_struct_0(B) )
=> k2_normsp_1(B,k5_algseq_1(B,k1_rlvect_1(B)),A) = k1_rlvect_1(B) ) ) ).
fof(t34_algseq_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l2_struct_0(A) )
=> ! [B] :
( ( v1_funct_1(B)
& v1_funct_2(B,k5_numbers,u1_struct_0(A))
& v1_algseq_1(B,A)
& m2_relset_1(B,k5_numbers,u1_struct_0(A)) )
=> ( B = k5_algseq_1(A,k1_rlvect_1(A))
<=> k2_relat_1(B) = k1_struct_0(A,k1_rlvect_1(A)) ) ) ) ).
fof(s1_algseq_1,axiom,
? [A] :
( v1_funct_1(A)
& v1_funct_2(A,k5_numbers,u1_struct_0(f1_s1_algseq_1))
& v1_algseq_1(A,f1_s1_algseq_1)
& m2_relset_1(A,k5_numbers,u1_struct_0(f1_s1_algseq_1))
& r1_xreal_0(k3_algseq_1(f1_s1_algseq_1,A),f2_s1_algseq_1)
& ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> ( ~ r1_xreal_0(f2_s1_algseq_1,B)
=> k2_normsp_1(f1_s1_algseq_1,A,B) = f3_s1_algseq_1(B) ) ) ) ).
fof(dt_k1_algseq_1,axiom,
$true ).
fof(dt_k2_algseq_1,axiom,
! [A] :
( m1_subset_1(A,k5_numbers)
=> m1_subset_1(k2_algseq_1(A),k1_zfmisc_1(k5_numbers)) ) ).
fof(redefinition_k2_algseq_1,axiom,
! [A] :
( m1_subset_1(A,k5_numbers)
=> k2_algseq_1(A) = k1_algseq_1(A) ) ).
fof(dt_k3_algseq_1,axiom,
! [A,B] :
( ( ~ v3_struct_0(A)
& l2_struct_0(A)
& v1_funct_1(B)
& v1_funct_2(B,k5_numbers,u1_struct_0(A))
& v1_algseq_1(B,A)
& m1_relset_1(B,k5_numbers,u1_struct_0(A)) )
=> m2_subset_1(k3_algseq_1(A,B),k1_numbers,k5_numbers) ) ).
fof(dt_k4_algseq_1,axiom,
! [A,B] :
( ( ~ v3_struct_0(A)
& l2_struct_0(A)
& v1_funct_1(B)
& v1_funct_2(B,k5_numbers,u1_struct_0(A))
& v1_algseq_1(B,A)
& m1_relset_1(B,k5_numbers,u1_struct_0(A)) )
=> m1_subset_1(k4_algseq_1(A,B),k1_zfmisc_1(k5_numbers)) ) ).
fof(dt_k5_algseq_1,axiom,
! [A,B] :
( ( ~ v3_struct_0(A)
& l2_struct_0(A)
& m1_subset_1(B,u1_struct_0(A)) )
=> ( v1_funct_1(k5_algseq_1(A,B))
& v1_funct_2(k5_algseq_1(A,B),k5_numbers,u1_struct_0(A))
& v1_algseq_1(k5_algseq_1(A,B),A)
& m2_relset_1(k5_algseq_1(A,B),k5_numbers,u1_struct_0(A)) ) ) ).
fof(d1_algseq_1,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> k1_algseq_1(A) = a_1_0_algseq_1(A) ) ).
fof(fraenkel_a_1_0_algseq_1,axiom,
! [A,B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> ( r2_hidden(A,a_1_0_algseq_1(B))
<=> ? [C] :
( m2_subset_1(C,k1_numbers,k5_numbers)
& A = C
& ~ r1_xreal_0(B,C) ) ) ) ).
%------------------------------------------------------------------------------