SET007 Axioms: SET007+432.ax
%------------------------------------------------------------------------------
% File : SET007+432 : TPTP v8.2.0. Released v3.4.0.
% Domain : Set Theory
% Axioms : On the Lattice of Subspaces of a 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_8 [Urb08]
% Status : Satisfiable
% Syntax : Number of formulae : 41 ( 1 unt; 0 def)
% Number of atoms : 892 ( 27 equ)
% Maximal formula atoms : 40 ( 21 avg)
% Number of connectives : 977 ( 126 ~; 1 |; 720 &)
% ( 11 <=>; 119 =>; 0 <=; 0 <~>)
% Maximal formula depth : 36 ( 17 avg)
% Maximal term depth : 5 ( 1 avg)
% Number of predicates : 43 ( 41 usr; 1 prp; 0-4 aty)
% Number of functors : 25 ( 25 usr; 0 con; 1-4 aty)
% Number of variables : 142 ( 135 !; 7 ?)
% 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_vectsp_8,axiom,
! [A,B] :
( ( ~ 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_group_1(A)
& v7_vectsp_1(A)
& v8_vectsp_1(A)
& v9_vectsp_1(A)
& ~ v10_vectsp_1(A)
& l3_vectsp_1(A)
& ~ v3_struct_0(B)
& v3_rlvect_1(B)
& v4_rlvect_1(B)
& v5_rlvect_1(B)
& v6_rlvect_1(B)
& v11_vectsp_1(B,A)
& v12_vectsp_1(B,A)
& l4_vectsp_1(B,A) )
=> ? [C] :
( m1_vectsp_8(C,A,B)
& ~ v1_xboole_0(C) ) ) ).
fof(d1_vectsp_8,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_group_1(A)
& v7_vectsp_1(A)
& v8_vectsp_1(A)
& v9_vectsp_1(A)
& ~ v10_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)
& v11_vectsp_1(B,A)
& v12_vectsp_1(B,A)
& l4_vectsp_1(B,A) )
=> k1_vectsp_8(A,B) = g3_lattices(k3_vectsp_5(A,B),k5_vectsp_5(A,B),k6_vectsp_5(A,B)) ) ) ).
fof(d2_vectsp_8,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_group_1(A)
& v7_vectsp_1(A)
& v8_vectsp_1(A)
& v9_vectsp_1(A)
& ~ v10_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)
& v11_vectsp_1(B,A)
& v12_vectsp_1(B,A)
& l4_vectsp_1(B,A) )
=> ! [C] :
( m1_vectsp_8(C,A,B)
<=> ! [D] :
( r2_hidden(D,C)
=> m1_vectsp_4(D,A,B) ) ) ) ) ).
fof(d3_vectsp_8,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_group_1(A)
& v7_vectsp_1(A)
& v8_vectsp_1(A)
& v9_vectsp_1(A)
& ~ v10_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)
& v11_vectsp_1(B,A)
& v12_vectsp_1(B,A)
& l4_vectsp_1(B,A) )
=> ! [C] :
( m2_vectsp_8(C,A,B,k2_vectsp_8(A,B))
=> ! [D] :
( m1_subset_1(D,k1_zfmisc_1(u1_struct_0(B)))
=> ( D = k3_vectsp_8(A,B,C)
<=> ? [E] :
( m1_vectsp_4(E,A,B)
& C = E
& D = u1_struct_0(E) ) ) ) ) ) ) ).
fof(d4_vectsp_8,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_group_1(A)
& v7_vectsp_1(A)
& v8_vectsp_1(A)
& v9_vectsp_1(A)
& ~ v10_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)
& v11_vectsp_1(B,A)
& v12_vectsp_1(B,A)
& l4_vectsp_1(B,A) )
=> ! [C] :
( ( v1_funct_1(C)
& v1_funct_2(C,k2_vectsp_8(A,B),k1_zfmisc_1(u1_struct_0(B)))
& m2_relset_1(C,k2_vectsp_8(A,B),k1_zfmisc_1(u1_struct_0(B))) )
=> ( C = k4_vectsp_8(A,B)
<=> ! [D] :
( m2_vectsp_8(D,A,B,k2_vectsp_8(A,B))
=> ! [E] :
( m1_vectsp_4(E,A,B)
=> ( D = E
=> k8_funct_2(k2_vectsp_8(A,B),k1_zfmisc_1(u1_struct_0(B)),C,D) = u1_struct_0(E) ) ) ) ) ) ) ) ).
fof(t1_vectsp_8,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_group_1(A)
& v7_vectsp_1(A)
& v8_vectsp_1(A)
& v9_vectsp_1(A)
& ~ v10_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)
& v11_vectsp_1(B,A)
& v12_vectsp_1(B,A)
& l4_vectsp_1(B,A) )
=> ! [C] :
( ( ~ v1_xboole_0(C)
& m1_subset_1(C,k1_zfmisc_1(k2_vectsp_8(A,B))) )
=> ~ v1_xboole_0(k2_funct_2(k2_vectsp_8(A,B),k1_zfmisc_1(u1_struct_0(B)),k4_vectsp_8(A,B),C)) ) ) ) ).
fof(t2_vectsp_8,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_group_1(A)
& v7_vectsp_1(A)
& v8_vectsp_1(A)
& v9_vectsp_1(A)
& ~ v10_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)
& v11_vectsp_1(B,A)
& v12_vectsp_1(B,A)
& l4_vectsp_1(B,A) )
=> ! [C] :
( ( v11_vectsp_1(C,A)
& m1_vectsp_4(C,A,B) )
=> r2_hidden(k1_rlvect_1(B),k1_funct_1(k4_vectsp_8(A,B),C)) ) ) ) ).
fof(d5_vectsp_8,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_group_1(A)
& v7_vectsp_1(A)
& v8_vectsp_1(A)
& v9_vectsp_1(A)
& ~ v10_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)
& v11_vectsp_1(B,A)
& v12_vectsp_1(B,A)
& l4_vectsp_1(B,A) )
=> ! [C] :
( ( ~ v1_xboole_0(C)
& m1_subset_1(C,k1_zfmisc_1(k2_vectsp_8(A,B))) )
=> ! [D] :
( ( v11_vectsp_1(D,A)
& m1_vectsp_4(D,A,B) )
=> ( D = k5_vectsp_8(A,B,C)
<=> u1_struct_0(D) = k6_setfam_1(u1_struct_0(B),k2_funct_2(k2_vectsp_8(A,B),k1_zfmisc_1(u1_struct_0(B)),k4_vectsp_8(A,B),C)) ) ) ) ) ) ).
fof(t3_vectsp_8,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_group_1(A)
& v7_vectsp_1(A)
& v8_vectsp_1(A)
& v9_vectsp_1(A)
& ~ v10_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)
& v11_vectsp_1(B,A)
& v12_vectsp_1(B,A)
& l4_vectsp_1(B,A) )
=> k2_vectsp_8(A,B) = u1_struct_0(k1_vectsp_8(A,B)) ) ) ).
fof(t4_vectsp_8,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_group_1(A)
& v7_vectsp_1(A)
& v8_vectsp_1(A)
& v9_vectsp_1(A)
& ~ v10_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)
& v11_vectsp_1(B,A)
& v12_vectsp_1(B,A)
& l4_vectsp_1(B,A) )
=> u1_lattices(k1_vectsp_8(A,B)) = k6_vectsp_5(A,B) ) ) ).
fof(t5_vectsp_8,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_group_1(A)
& v7_vectsp_1(A)
& v8_vectsp_1(A)
& v9_vectsp_1(A)
& ~ v10_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)
& v11_vectsp_1(B,A)
& v12_vectsp_1(B,A)
& l4_vectsp_1(B,A) )
=> u2_lattices(k1_vectsp_8(A,B)) = k5_vectsp_5(A,B) ) ) ).
fof(t6_vectsp_8,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_group_1(A)
& v7_vectsp_1(A)
& v8_vectsp_1(A)
& v9_vectsp_1(A)
& ~ v10_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)
& v11_vectsp_1(B,A)
& v12_vectsp_1(B,A)
& l4_vectsp_1(B,A) )
=> ! [C] :
( m1_subset_1(C,u1_struct_0(k1_vectsp_8(A,B)))
=> ! [D] :
( m1_subset_1(D,u1_struct_0(k1_vectsp_8(A,B)))
=> ! [E] :
( ( v11_vectsp_1(E,A)
& m1_vectsp_4(E,A,B) )
=> ! [F] :
( ( v11_vectsp_1(F,A)
& m1_vectsp_4(F,A,B) )
=> ( ( C = E
& D = F )
=> ( r3_lattices(k1_vectsp_8(A,B),C,D)
<=> r1_tarski(u1_struct_0(E),u1_struct_0(F)) ) ) ) ) ) ) ) ) ).
fof(t7_vectsp_8,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_group_1(A)
& v7_vectsp_1(A)
& v8_vectsp_1(A)
& v9_vectsp_1(A)
& ~ v10_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)
& v11_vectsp_1(B,A)
& v12_vectsp_1(B,A)
& l4_vectsp_1(B,A) )
=> ! [C] :
( m1_subset_1(C,u1_struct_0(k1_vectsp_8(A,B)))
=> ! [D] :
( m1_subset_1(D,u1_struct_0(k1_vectsp_8(A,B)))
=> ! [E] :
( m1_vectsp_4(E,A,B)
=> ! [F] :
( m1_vectsp_4(F,A,B)
=> ( ( C = E
& D = F )
=> k3_lattices(k1_vectsp_8(A,B),C,D) = k1_vectsp_5(A,B,E,F) ) ) ) ) ) ) ) ).
fof(t8_vectsp_8,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_group_1(A)
& v7_vectsp_1(A)
& v8_vectsp_1(A)
& v9_vectsp_1(A)
& ~ v10_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)
& v11_vectsp_1(B,A)
& v12_vectsp_1(B,A)
& l4_vectsp_1(B,A) )
=> ! [C] :
( m1_subset_1(C,u1_struct_0(k1_vectsp_8(A,B)))
=> ! [D] :
( m1_subset_1(D,u1_struct_0(k1_vectsp_8(A,B)))
=> ! [E] :
( m1_vectsp_4(E,A,B)
=> ! [F] :
( m1_vectsp_4(F,A,B)
=> ( ( C = E
& D = F )
=> k4_lattices(k1_vectsp_8(A,B),C,D) = k2_vectsp_5(A,B,E,F) ) ) ) ) ) ) ) ).
fof(d6_vectsp_8,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l3_lattices(A) )
=> ( v4_lattice3(A)
<=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
=> ? [C] :
( m1_subset_1(C,u1_struct_0(A))
& r3_lattice3(A,C,B)
& ! [D] :
( m1_subset_1(D,u1_struct_0(A))
=> ( r3_lattice3(A,D,B)
=> r1_lattices(A,D,C) ) ) ) ) ) ) ).
fof(t9_vectsp_8,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_group_1(A)
& v7_vectsp_1(A)
& v8_vectsp_1(A)
& v9_vectsp_1(A)
& ~ v10_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)
& v11_vectsp_1(B,A)
& v12_vectsp_1(B,A)
& l4_vectsp_1(B,A) )
=> v4_lattice3(k1_vectsp_8(A,B)) ) ) ).
fof(t10_vectsp_8,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_group_1(A)
& v7_vectsp_1(A)
& v8_vectsp_1(A)
& v9_vectsp_1(A)
& ~ v10_vectsp_1(A)
& l3_vectsp_1(A) )
=> ! [B,C] :
( ( ~ v3_struct_0(C)
& v3_rlvect_1(C)
& v4_rlvect_1(C)
& v5_rlvect_1(C)
& v6_rlvect_1(C)
& v11_vectsp_1(C,A)
& v12_vectsp_1(C,A)
& l4_vectsp_1(C,A) )
=> ! [D] :
( m1_subset_1(D,k1_zfmisc_1(u1_struct_0(C)))
=> ( ( v1_vectsp_4(D,A,C)
& r1_rlvect_1(k1_vectsp_7(A,C,D),B) )
=> ( v1_xboole_0(D)
| r2_hidden(B,D) ) ) ) ) ) ).
fof(d7_vectsp_8,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_group_1(A)
& v7_vectsp_1(A)
& v8_vectsp_1(A)
& v9_vectsp_1(A)
& ~ v10_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)
& v11_vectsp_1(B,A)
& v12_vectsp_1(B,A)
& l4_vectsp_1(B,A) )
=> ! [C] :
( ( ~ v3_struct_0(C)
& v3_rlvect_1(C)
& v4_rlvect_1(C)
& v5_rlvect_1(C)
& v6_rlvect_1(C)
& v11_vectsp_1(C,A)
& v12_vectsp_1(C,A)
& l4_vectsp_1(C,A) )
=> ! [D] :
( ( v1_funct_1(D)
& v1_funct_2(D,u1_struct_0(B),u1_struct_0(C))
& m2_relset_1(D,u1_struct_0(B),u1_struct_0(C)) )
=> ! [E] :
( ( v1_funct_1(E)
& v1_funct_2(E,u1_struct_0(k1_vectsp_8(A,B)),u1_struct_0(k1_vectsp_8(A,C)))
& m2_relset_1(E,u1_struct_0(k1_vectsp_8(A,B)),u1_struct_0(k1_vectsp_8(A,C))) )
=> ( E = k6_vectsp_8(A,B,C,D)
<=> ! [F] :
( ( v11_vectsp_1(F,A)
& m1_vectsp_4(F,A,B) )
=> ! [G] :
( m1_subset_1(G,k1_zfmisc_1(u1_struct_0(C)))
=> ( G = k2_funct_2(u1_struct_0(B),u1_struct_0(C),D,u1_struct_0(F))
=> k1_funct_1(E,F) = k1_vectsp_7(A,C,G) ) ) ) ) ) ) ) ) ) ).
fof(d8_vectsp_8,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v10_lattices(A)
& l3_lattices(A) )
=> ! [B] :
( ( ~ v3_struct_0(B)
& v10_lattices(B)
& l3_lattices(B) )
=> ! [C] :
( ( v1_funct_1(C)
& v1_funct_2(C,u1_struct_0(A),u1_struct_0(B))
& m2_relset_1(C,u1_struct_0(A),u1_struct_0(B)) )
=> ( m3_vectsp_8(C,A,B)
<=> ! [D] :
( m1_subset_1(D,u1_struct_0(A))
=> ! [E] :
( m1_subset_1(E,u1_struct_0(A))
=> k8_funct_2(u1_struct_0(A),u1_struct_0(B),C,k4_lattices(A,D,E)) = k4_lattices(B,k8_funct_2(u1_struct_0(A),u1_struct_0(B),C,D),k8_funct_2(u1_struct_0(A),u1_struct_0(B),C,E)) ) ) ) ) ) ) ).
fof(d9_vectsp_8,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v10_lattices(A)
& l3_lattices(A) )
=> ! [B] :
( ( ~ v3_struct_0(B)
& v10_lattices(B)
& l3_lattices(B) )
=> ! [C] :
( ( v1_funct_1(C)
& v1_funct_2(C,u1_struct_0(A),u1_struct_0(B))
& m2_relset_1(C,u1_struct_0(A),u1_struct_0(B)) )
=> ( m4_vectsp_8(C,A,B)
<=> ! [D] :
( m1_subset_1(D,u1_struct_0(A))
=> ! [E] :
( m1_subset_1(E,u1_struct_0(A))
=> k8_funct_2(u1_struct_0(A),u1_struct_0(B),C,k3_lattices(A,D,E)) = k3_lattices(B,k8_funct_2(u1_struct_0(A),u1_struct_0(B),C,D),k8_funct_2(u1_struct_0(A),u1_struct_0(B),C,E)) ) ) ) ) ) ) ).
fof(t11_vectsp_8,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v10_lattices(A)
& l3_lattices(A) )
=> ! [B] :
( ( ~ v3_struct_0(B)
& v10_lattices(B)
& l3_lattices(B) )
=> ! [C] :
( ( v1_funct_1(C)
& v1_funct_2(C,u1_struct_0(A),u1_struct_0(B))
& m2_relset_1(C,u1_struct_0(A),u1_struct_0(B)) )
=> ( m1_lattice4(C,A,B)
<=> ( m4_vectsp_8(C,A,B)
& m3_vectsp_8(C,A,B) ) ) ) ) ) ).
fof(t12_vectsp_8,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_group_1(A)
& v7_vectsp_1(A)
& v8_vectsp_1(A)
& v9_vectsp_1(A)
& ~ v10_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)
& v11_vectsp_1(B,A)
& v12_vectsp_1(B,A)
& l4_vectsp_1(B,A) )
=> ! [C] :
( ( ~ v3_struct_0(C)
& v3_rlvect_1(C)
& v4_rlvect_1(C)
& v5_rlvect_1(C)
& v6_rlvect_1(C)
& v11_vectsp_1(C,A)
& v12_vectsp_1(C,A)
& l4_vectsp_1(C,A) )
=> ! [D] :
( ( v1_funct_1(D)
& v1_funct_2(D,u1_struct_0(B),u1_struct_0(C))
& m2_relset_1(D,u1_struct_0(B),u1_struct_0(C)) )
=> ( v1_mod_2(D,A,B,C)
=> m4_vectsp_8(k6_vectsp_8(A,B,C,D),k1_vectsp_8(A,B),k1_vectsp_8(A,C)) ) ) ) ) ) ).
fof(t13_vectsp_8,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_group_1(A)
& v7_vectsp_1(A)
& v8_vectsp_1(A)
& v9_vectsp_1(A)
& ~ v10_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)
& v11_vectsp_1(B,A)
& v12_vectsp_1(B,A)
& l4_vectsp_1(B,A) )
=> ! [C] :
( ( ~ v3_struct_0(C)
& v3_rlvect_1(C)
& v4_rlvect_1(C)
& v5_rlvect_1(C)
& v6_rlvect_1(C)
& v11_vectsp_1(C,A)
& v12_vectsp_1(C,A)
& l4_vectsp_1(C,A) )
=> ! [D] :
( ( v1_funct_1(D)
& v1_funct_2(D,u1_struct_0(B),u1_struct_0(C))
& m2_relset_1(D,u1_struct_0(B),u1_struct_0(C)) )
=> ( ( v2_funct_1(D)
& v1_mod_2(D,A,B,C) )
=> m1_lattice4(k6_vectsp_8(A,B,C,D),k1_vectsp_8(A,B),k1_vectsp_8(A,C)) ) ) ) ) ) ).
fof(t14_vectsp_8,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_group_1(A)
& v7_vectsp_1(A)
& v8_vectsp_1(A)
& v9_vectsp_1(A)
& ~ v10_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)
& v11_vectsp_1(B,A)
& v12_vectsp_1(B,A)
& l4_vectsp_1(B,A) )
=> ! [C] :
( ( ~ v3_struct_0(C)
& v3_rlvect_1(C)
& v4_rlvect_1(C)
& v5_rlvect_1(C)
& v6_rlvect_1(C)
& v11_vectsp_1(C,A)
& v12_vectsp_1(C,A)
& l4_vectsp_1(C,A) )
=> ! [D] :
( ( v1_funct_1(D)
& v1_funct_2(D,u1_struct_0(B),u1_struct_0(C))
& m2_relset_1(D,u1_struct_0(B),u1_struct_0(C)) )
=> ( ( v1_mod_2(D,A,B,C)
& v2_funct_1(D) )
=> v2_funct_1(k6_vectsp_8(A,B,C,D)) ) ) ) ) ) ).
fof(t15_vectsp_8,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_group_1(A)
& v7_vectsp_1(A)
& v8_vectsp_1(A)
& v9_vectsp_1(A)
& ~ v10_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)
& v11_vectsp_1(B,A)
& v12_vectsp_1(B,A)
& l4_vectsp_1(B,A) )
=> k6_vectsp_8(A,B,B,k6_partfun1(u1_struct_0(B))) = k6_partfun1(u1_struct_0(k1_vectsp_8(A,B))) ) ) ).
fof(dt_m1_vectsp_8,axiom,
$true ).
fof(existence_m1_vectsp_8,axiom,
! [A,B] :
( ( ~ 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_group_1(A)
& v7_vectsp_1(A)
& v8_vectsp_1(A)
& v9_vectsp_1(A)
& ~ v10_vectsp_1(A)
& l3_vectsp_1(A)
& ~ v3_struct_0(B)
& v3_rlvect_1(B)
& v4_rlvect_1(B)
& v5_rlvect_1(B)
& v6_rlvect_1(B)
& v11_vectsp_1(B,A)
& v12_vectsp_1(B,A)
& l4_vectsp_1(B,A) )
=> ? [C] : m1_vectsp_8(C,A,B) ) ).
fof(dt_m2_vectsp_8,axiom,
! [A,B,C] :
( ( ~ 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_group_1(A)
& v7_vectsp_1(A)
& v8_vectsp_1(A)
& v9_vectsp_1(A)
& ~ v10_vectsp_1(A)
& l3_vectsp_1(A)
& ~ v3_struct_0(B)
& v3_rlvect_1(B)
& v4_rlvect_1(B)
& v5_rlvect_1(B)
& v6_rlvect_1(B)
& v11_vectsp_1(B,A)
& v12_vectsp_1(B,A)
& l4_vectsp_1(B,A)
& ~ v1_xboole_0(C)
& m1_vectsp_8(C,A,B) )
=> ! [D] :
( m2_vectsp_8(D,A,B,C)
=> m1_vectsp_4(D,A,B) ) ) ).
fof(existence_m2_vectsp_8,axiom,
! [A,B,C] :
( ( ~ 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_group_1(A)
& v7_vectsp_1(A)
& v8_vectsp_1(A)
& v9_vectsp_1(A)
& ~ v10_vectsp_1(A)
& l3_vectsp_1(A)
& ~ v3_struct_0(B)
& v3_rlvect_1(B)
& v4_rlvect_1(B)
& v5_rlvect_1(B)
& v6_rlvect_1(B)
& v11_vectsp_1(B,A)
& v12_vectsp_1(B,A)
& l4_vectsp_1(B,A)
& ~ v1_xboole_0(C)
& m1_vectsp_8(C,A,B) )
=> ? [D] : m2_vectsp_8(D,A,B,C) ) ).
fof(redefinition_m2_vectsp_8,axiom,
! [A,B,C] :
( ( ~ 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_group_1(A)
& v7_vectsp_1(A)
& v8_vectsp_1(A)
& v9_vectsp_1(A)
& ~ v10_vectsp_1(A)
& l3_vectsp_1(A)
& ~ v3_struct_0(B)
& v3_rlvect_1(B)
& v4_rlvect_1(B)
& v5_rlvect_1(B)
& v6_rlvect_1(B)
& v11_vectsp_1(B,A)
& v12_vectsp_1(B,A)
& l4_vectsp_1(B,A)
& ~ v1_xboole_0(C)
& m1_vectsp_8(C,A,B) )
=> ! [D] :
( m2_vectsp_8(D,A,B,C)
<=> m1_subset_1(D,C) ) ) ).
fof(dt_m3_vectsp_8,axiom,
! [A,B] :
( ( ~ v3_struct_0(A)
& v10_lattices(A)
& l3_lattices(A)
& ~ v3_struct_0(B)
& v10_lattices(B)
& l3_lattices(B) )
=> ! [C] :
( m3_vectsp_8(C,A,B)
=> ( v1_funct_1(C)
& v1_funct_2(C,u1_struct_0(A),u1_struct_0(B))
& m2_relset_1(C,u1_struct_0(A),u1_struct_0(B)) ) ) ) ).
fof(existence_m3_vectsp_8,axiom,
! [A,B] :
( ( ~ v3_struct_0(A)
& v10_lattices(A)
& l3_lattices(A)
& ~ v3_struct_0(B)
& v10_lattices(B)
& l3_lattices(B) )
=> ? [C] : m3_vectsp_8(C,A,B) ) ).
fof(dt_m4_vectsp_8,axiom,
! [A,B] :
( ( ~ v3_struct_0(A)
& v10_lattices(A)
& l3_lattices(A)
& ~ v3_struct_0(B)
& v10_lattices(B)
& l3_lattices(B) )
=> ! [C] :
( m4_vectsp_8(C,A,B)
=> ( v1_funct_1(C)
& v1_funct_2(C,u1_struct_0(A),u1_struct_0(B))
& m2_relset_1(C,u1_struct_0(A),u1_struct_0(B)) ) ) ) ).
fof(existence_m4_vectsp_8,axiom,
! [A,B] :
( ( ~ v3_struct_0(A)
& v10_lattices(A)
& l3_lattices(A)
& ~ v3_struct_0(B)
& v10_lattices(B)
& l3_lattices(B) )
=> ? [C] : m4_vectsp_8(C,A,B) ) ).
fof(dt_k1_vectsp_8,axiom,
! [A,B] :
( ( ~ 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_group_1(A)
& v7_vectsp_1(A)
& v8_vectsp_1(A)
& v9_vectsp_1(A)
& ~ v10_vectsp_1(A)
& l3_vectsp_1(A)
& ~ v3_struct_0(B)
& v3_rlvect_1(B)
& v4_rlvect_1(B)
& v5_rlvect_1(B)
& v6_rlvect_1(B)
& v11_vectsp_1(B,A)
& v12_vectsp_1(B,A)
& l4_vectsp_1(B,A) )
=> ( ~ v3_struct_0(k1_vectsp_8(A,B))
& v3_lattices(k1_vectsp_8(A,B))
& v10_lattices(k1_vectsp_8(A,B))
& v15_lattices(k1_vectsp_8(A,B))
& l3_lattices(k1_vectsp_8(A,B)) ) ) ).
fof(dt_k2_vectsp_8,axiom,
! [A,B] :
( ( ~ 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_group_1(A)
& v7_vectsp_1(A)
& v8_vectsp_1(A)
& v9_vectsp_1(A)
& ~ v10_vectsp_1(A)
& l3_vectsp_1(A)
& ~ v3_struct_0(B)
& v3_rlvect_1(B)
& v4_rlvect_1(B)
& v5_rlvect_1(B)
& v6_rlvect_1(B)
& v11_vectsp_1(B,A)
& v12_vectsp_1(B,A)
& l4_vectsp_1(B,A) )
=> ( ~ v1_xboole_0(k2_vectsp_8(A,B))
& m1_vectsp_8(k2_vectsp_8(A,B),A,B) ) ) ).
fof(redefinition_k2_vectsp_8,axiom,
! [A,B] :
( ( ~ 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_group_1(A)
& v7_vectsp_1(A)
& v8_vectsp_1(A)
& v9_vectsp_1(A)
& ~ v10_vectsp_1(A)
& l3_vectsp_1(A)
& ~ v3_struct_0(B)
& v3_rlvect_1(B)
& v4_rlvect_1(B)
& v5_rlvect_1(B)
& v6_rlvect_1(B)
& v11_vectsp_1(B,A)
& v12_vectsp_1(B,A)
& l4_vectsp_1(B,A) )
=> k2_vectsp_8(A,B) = k3_vectsp_5(A,B) ) ).
fof(dt_k3_vectsp_8,axiom,
! [A,B,C] :
( ( ~ 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_group_1(A)
& v7_vectsp_1(A)
& v8_vectsp_1(A)
& v9_vectsp_1(A)
& ~ v10_vectsp_1(A)
& l3_vectsp_1(A)
& ~ v3_struct_0(B)
& v3_rlvect_1(B)
& v4_rlvect_1(B)
& v5_rlvect_1(B)
& v6_rlvect_1(B)
& v11_vectsp_1(B,A)
& v12_vectsp_1(B,A)
& l4_vectsp_1(B,A)
& m1_subset_1(C,k2_vectsp_8(A,B)) )
=> m1_subset_1(k3_vectsp_8(A,B,C),k1_zfmisc_1(u1_struct_0(B))) ) ).
fof(dt_k4_vectsp_8,axiom,
! [A,B] :
( ( ~ 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_group_1(A)
& v7_vectsp_1(A)
& v8_vectsp_1(A)
& v9_vectsp_1(A)
& ~ v10_vectsp_1(A)
& l3_vectsp_1(A)
& ~ v3_struct_0(B)
& v3_rlvect_1(B)
& v4_rlvect_1(B)
& v5_rlvect_1(B)
& v6_rlvect_1(B)
& v11_vectsp_1(B,A)
& v12_vectsp_1(B,A)
& l4_vectsp_1(B,A) )
=> ( v1_funct_1(k4_vectsp_8(A,B))
& v1_funct_2(k4_vectsp_8(A,B),k2_vectsp_8(A,B),k1_zfmisc_1(u1_struct_0(B)))
& m2_relset_1(k4_vectsp_8(A,B),k2_vectsp_8(A,B),k1_zfmisc_1(u1_struct_0(B))) ) ) ).
fof(dt_k5_vectsp_8,axiom,
! [A,B,C] :
( ( ~ 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_group_1(A)
& v7_vectsp_1(A)
& v8_vectsp_1(A)
& v9_vectsp_1(A)
& ~ v10_vectsp_1(A)
& l3_vectsp_1(A)
& ~ v3_struct_0(B)
& v3_rlvect_1(B)
& v4_rlvect_1(B)
& v5_rlvect_1(B)
& v6_rlvect_1(B)
& v11_vectsp_1(B,A)
& v12_vectsp_1(B,A)
& l4_vectsp_1(B,A)
& ~ v1_xboole_0(C)
& m1_subset_1(C,k1_zfmisc_1(k2_vectsp_8(A,B))) )
=> ( v11_vectsp_1(k5_vectsp_8(A,B,C),A)
& m1_vectsp_4(k5_vectsp_8(A,B,C),A,B) ) ) ).
fof(dt_k6_vectsp_8,axiom,
! [A,B,C,D] :
( ( ~ 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_group_1(A)
& v7_vectsp_1(A)
& v8_vectsp_1(A)
& v9_vectsp_1(A)
& ~ v10_vectsp_1(A)
& l3_vectsp_1(A)
& ~ v3_struct_0(B)
& v3_rlvect_1(B)
& v4_rlvect_1(B)
& v5_rlvect_1(B)
& v6_rlvect_1(B)
& v11_vectsp_1(B,A)
& v12_vectsp_1(B,A)
& l4_vectsp_1(B,A)
& ~ v3_struct_0(C)
& v3_rlvect_1(C)
& v4_rlvect_1(C)
& v5_rlvect_1(C)
& v6_rlvect_1(C)
& v11_vectsp_1(C,A)
& v12_vectsp_1(C,A)
& l4_vectsp_1(C,A)
& v1_funct_1(D)
& v1_funct_2(D,u1_struct_0(B),u1_struct_0(C))
& m1_relset_1(D,u1_struct_0(B),u1_struct_0(C)) )
=> ( v1_funct_1(k6_vectsp_8(A,B,C,D))
& v1_funct_2(k6_vectsp_8(A,B,C,D),u1_struct_0(k1_vectsp_8(A,B)),u1_struct_0(k1_vectsp_8(A,C)))
& m2_relset_1(k6_vectsp_8(A,B,C,D),u1_struct_0(k1_vectsp_8(A,B)),u1_struct_0(k1_vectsp_8(A,C))) ) ) ).
%------------------------------------------------------------------------------