SET007 Axioms: SET007+245.ax
%------------------------------------------------------------------------------
% File : SET007+245 : TPTP v8.2.0. Released v3.4.0.
% Domain : Set Theory
% Axioms : Finite Sums of Vectors in Vector Space
% 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 : vectsp_3 [Urb08]
% Status : Satisfiable
% Syntax : Number of formulae : 35 ( 23 unt; 0 def)
% Number of atoms : 211 ( 23 equ)
% Maximal formula atoms : 26 ( 6 avg)
% Number of connectives : 194 ( 18 ~; 0 |; 121 &)
% ( 0 <=>; 55 =>; 0 <=; 0 <~>)
% Maximal formula depth : 19 ( 5 avg)
% Maximal term depth : 5 ( 1 avg)
% Number of predicates : 18 ( 16 usr; 1 prp; 0-3 aty)
% Number of functors : 16 ( 16 usr; 2 con; 0-4 aty)
% Number of variables : 49 ( 49 !; 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(t1_vectsp_3,axiom,
$true ).
fof(t2_vectsp_3,axiom,
$true ).
fof(t3_vectsp_3,axiom,
$true ).
fof(t4_vectsp_3,axiom,
$true ).
fof(t5_vectsp_3,axiom,
$true ).
fof(t6_vectsp_3,axiom,
$true ).
fof(t7_vectsp_3,axiom,
$true ).
fof(t8_vectsp_3,axiom,
$true ).
fof(t9_vectsp_3,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v3_rlvect_1(A)
& v4_rlvect_1(A)
& v5_rlvect_1(A)
& v6_rlvect_1(A)
& v4_group_1(A)
& v7_vectsp_1(A)
& v8_vectsp_1(A)
& l3_vectsp_1(A) )
=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> ! [C] :
( ( ~ v3_struct_0(C)
& v3_rlvect_1(C)
& v4_rlvect_1(C)
& v5_rlvect_1(C)
& v6_rlvect_1(C)
& v12_vectsp_1(C,A)
& l4_vectsp_1(C,A) )
=> ! [D] :
( m2_finseq_1(D,u1_struct_0(C))
=> ! [E] :
( m2_finseq_1(E,u1_struct_0(C))
=> ( ( k3_finseq_1(D) = k3_finseq_1(E)
& ! [F] :
( m2_subset_1(F,k1_numbers,k5_numbers)
=> ! [G] :
( m1_subset_1(G,u1_struct_0(C))
=> ( ( r2_hidden(F,k1_relat_1(D))
& G = k1_funct_1(E,F) )
=> k1_funct_1(D,F) = k6_vectsp_1(A,C,B,G) ) ) ) )
=> k9_rlvect_1(C,D) = k6_vectsp_1(A,C,B,k9_rlvect_1(C,E)) ) ) ) ) ) ) ).
fof(t10_vectsp_3,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v3_rlvect_1(A)
& v4_rlvect_1(A)
& v5_rlvect_1(A)
& v6_rlvect_1(A)
& v4_group_1(A)
& v7_vectsp_1(A)
& v8_vectsp_1(A)
& l3_vectsp_1(A) )
=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> ! [C] :
( ( ~ v3_struct_0(C)
& v3_rlvect_1(C)
& v4_rlvect_1(C)
& v5_rlvect_1(C)
& v6_rlvect_1(C)
& v12_vectsp_1(C,A)
& l4_vectsp_1(C,A) )
=> ! [D] :
( m2_finseq_1(D,u1_struct_0(C))
=> ! [E] :
( m2_finseq_1(E,u1_struct_0(C))
=> ( ( k3_finseq_1(D) = k3_finseq_1(E)
& ! [F] :
( m2_subset_1(F,k1_numbers,k5_numbers)
=> ( r2_hidden(F,k1_relat_1(D))
=> k1_funct_1(E,F) = k6_vectsp_1(A,C,B,k4_finseq_4(k5_numbers,u1_struct_0(C),D,F)) ) ) )
=> k9_rlvect_1(C,E) = k6_vectsp_1(A,C,B,k9_rlvect_1(C,D)) ) ) ) ) ) ) ).
fof(t11_vectsp_3,axiom,
$true ).
fof(t12_vectsp_3,axiom,
$true ).
fof(t13_vectsp_3,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v3_rlvect_1(A)
& v4_rlvect_1(A)
& v5_rlvect_1(A)
& v6_rlvect_1(A)
& v4_group_1(A)
& v7_vectsp_1(A)
& v8_vectsp_1(A)
& l3_vectsp_1(A) )
=> ! [B] :
( ( ~ v3_struct_0(B)
& v3_rlvect_1(B)
& v4_rlvect_1(B)
& v5_rlvect_1(B)
& v6_rlvect_1(B)
& l4_vectsp_1(B,A) )
=> ! [C] :
( m2_finseq_1(C,u1_struct_0(B))
=> ! [D] :
( m2_finseq_1(D,u1_struct_0(B))
=> ! [E] :
( m2_finseq_1(E,u1_struct_0(B))
=> ( ( k3_finseq_1(C) = k3_finseq_1(D)
& k3_finseq_1(C) = k3_finseq_1(E)
& ! [F] :
( m2_subset_1(F,k1_numbers,k5_numbers)
=> ( r2_hidden(F,k1_relat_1(C))
=> k1_funct_1(E,F) = k6_rlvect_1(B,k4_finseq_4(k5_numbers,u1_struct_0(B),C,F),k4_finseq_4(k5_numbers,u1_struct_0(B),D,F)) ) ) )
=> k9_rlvect_1(B,E) = k6_rlvect_1(B,k9_rlvect_1(B,C),k9_rlvect_1(B,D)) ) ) ) ) ) ) ).
fof(t14_vectsp_3,axiom,
$true ).
fof(t15_vectsp_3,axiom,
$true ).
fof(t16_vectsp_3,axiom,
$true ).
fof(t17_vectsp_3,axiom,
$true ).
fof(t18_vectsp_3,axiom,
$true ).
fof(t19_vectsp_3,axiom,
$true ).
fof(t20_vectsp_3,axiom,
$true ).
fof(t21_vectsp_3,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v3_rlvect_1(A)
& v4_rlvect_1(A)
& v5_rlvect_1(A)
& v6_rlvect_1(A)
& v4_group_1(A)
& v7_vectsp_1(A)
& v8_vectsp_1(A)
& l3_vectsp_1(A) )
=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> ! [C] :
( ( ~ v3_struct_0(C)
& v3_rlvect_1(C)
& v4_rlvect_1(C)
& v5_rlvect_1(C)
& v6_rlvect_1(C)
& v12_vectsp_1(C,A)
& l4_vectsp_1(C,A) )
=> k6_vectsp_1(A,C,B,k9_rlvect_1(C,k6_finseq_1(u1_struct_0(C)))) = k1_rlvect_1(C) ) ) ) ).
fof(t22_vectsp_3,axiom,
$true ).
fof(t23_vectsp_3,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v3_rlvect_1(A)
& v4_rlvect_1(A)
& v5_rlvect_1(A)
& v6_rlvect_1(A)
& v4_group_1(A)
& v7_vectsp_1(A)
& v8_vectsp_1(A)
& l3_vectsp_1(A) )
=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> ! [C] :
( ( ~ v3_struct_0(C)
& v3_rlvect_1(C)
& v4_rlvect_1(C)
& v5_rlvect_1(C)
& v6_rlvect_1(C)
& v12_vectsp_1(C,A)
& l4_vectsp_1(C,A) )
=> ! [D] :
( m1_subset_1(D,u1_struct_0(C))
=> ! [E] :
( m1_subset_1(E,u1_struct_0(C))
=> k6_vectsp_1(A,C,B,k9_rlvect_1(C,k2_finseq_4(u1_struct_0(C),D,E))) = k4_rlvect_1(C,k6_vectsp_1(A,C,B,D),k6_vectsp_1(A,C,B,E)) ) ) ) ) ) ).
fof(t24_vectsp_3,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v3_rlvect_1(A)
& v4_rlvect_1(A)
& v5_rlvect_1(A)
& v6_rlvect_1(A)
& v4_group_1(A)
& v7_vectsp_1(A)
& v8_vectsp_1(A)
& l3_vectsp_1(A) )
=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> ! [C] :
( ( ~ v3_struct_0(C)
& v3_rlvect_1(C)
& v4_rlvect_1(C)
& v5_rlvect_1(C)
& v6_rlvect_1(C)
& v12_vectsp_1(C,A)
& l4_vectsp_1(C,A) )
=> ! [D] :
( m1_subset_1(D,u1_struct_0(C))
=> ! [E] :
( m1_subset_1(E,u1_struct_0(C))
=> ! [F] :
( m1_subset_1(F,u1_struct_0(C))
=> k6_vectsp_1(A,C,B,k9_rlvect_1(C,k3_finseq_4(u1_struct_0(C),D,E,F))) = k4_rlvect_1(C,k4_rlvect_1(C,k6_vectsp_1(A,C,B,D),k6_vectsp_1(A,C,B,E)),k6_vectsp_1(A,C,B,F)) ) ) ) ) ) ) ).
fof(t25_vectsp_3,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v3_rlvect_1(A)
& v4_rlvect_1(A)
& v5_rlvect_1(A)
& v6_rlvect_1(A)
& l1_rlvect_1(A) )
=> k5_rlvect_1(A,k9_rlvect_1(A,k6_finseq_1(u1_struct_0(A)))) = k1_rlvect_1(A) ) ).
fof(t26_vectsp_3,axiom,
$true ).
fof(t27_vectsp_3,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v3_rlvect_1(A)
& v4_rlvect_1(A)
& v5_rlvect_1(A)
& v6_rlvect_1(A)
& l1_rlvect_1(A) )
=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> k5_rlvect_1(A,k9_rlvect_1(A,k2_finseq_4(u1_struct_0(A),B,C))) = k6_rlvect_1(A,k5_rlvect_1(A,B),C) ) ) ) ).
fof(t28_vectsp_3,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v3_rlvect_1(A)
& v4_rlvect_1(A)
& v5_rlvect_1(A)
& v6_rlvect_1(A)
& l1_rlvect_1(A) )
=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> ! [D] :
( m1_subset_1(D,u1_struct_0(A))
=> k5_rlvect_1(A,k9_rlvect_1(A,k3_finseq_4(u1_struct_0(A),B,C,D))) = k6_rlvect_1(A,k6_rlvect_1(A,k5_rlvect_1(A,B),C),D) ) ) ) ) ).
fof(t29_vectsp_3,axiom,
$true ).
fof(t30_vectsp_3,axiom,
$true ).
fof(t31_vectsp_3,axiom,
$true ).
fof(t32_vectsp_3,axiom,
$true ).
fof(t33_vectsp_3,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v3_rlvect_1(A)
& v4_rlvect_1(A)
& v5_rlvect_1(A)
& v6_rlvect_1(A)
& l1_rlvect_1(A) )
=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> ( k9_rlvect_1(A,k2_finseq_4(u1_struct_0(A),B,k5_rlvect_1(A,B))) = k1_rlvect_1(A)
& k9_rlvect_1(A,k2_finseq_4(u1_struct_0(A),k5_rlvect_1(A,B),B)) = k1_rlvect_1(A) ) ) ) ).
fof(t34_vectsp_3,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v3_rlvect_1(A)
& v4_rlvect_1(A)
& v5_rlvect_1(A)
& v6_rlvect_1(A)
& l1_rlvect_1(A) )
=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> ( k9_rlvect_1(A,k2_finseq_4(u1_struct_0(A),B,k5_rlvect_1(A,C))) = k6_rlvect_1(A,B,C)
& k9_rlvect_1(A,k2_finseq_4(u1_struct_0(A),k5_rlvect_1(A,C),B)) = k6_rlvect_1(A,B,C) ) ) ) ) ).
fof(t35_vectsp_3,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v3_rlvect_1(A)
& v4_rlvect_1(A)
& v5_rlvect_1(A)
& v6_rlvect_1(A)
& l1_rlvect_1(A) )
=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> ( k9_rlvect_1(A,k2_finseq_4(u1_struct_0(A),k5_rlvect_1(A,B),k5_rlvect_1(A,C))) = k5_rlvect_1(A,k4_rlvect_1(A,B,C))
& k9_rlvect_1(A,k2_finseq_4(u1_struct_0(A),k5_rlvect_1(A,C),k5_rlvect_1(A,B))) = k5_rlvect_1(A,k4_rlvect_1(A,B,C)) ) ) ) ) ).
%------------------------------------------------------------------------------