SET007 Axioms: SET007+340.ax
%------------------------------------------------------------------------------
% File : SET007+340 : TPTP v8.2.0. Released v3.4.0.
% Domain : Set Theory
% Axioms : The Lattice of Domains of a Topological 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 : tdlat_1 [Urb08]
% Status : Satisfiable
% Syntax : Number of formulae : 74 ( 0 unt; 0 def)
% Number of atoms : 354 ( 41 equ)
% Maximal formula atoms : 9 ( 4 avg)
% Number of connectives : 289 ( 9 ~; 2 |; 121 &)
% ( 11 <=>; 146 =>; 0 <=; 0 <~>)
% Maximal formula depth : 12 ( 6 avg)
% Maximal term depth : 8 ( 2 avg)
% Number of predicates : 24 ( 23 usr; 0 prp; 1-3 aty)
% Number of functors : 29 ( 29 usr; 0 con; 1-6 aty)
% Number of variables : 142 ( 139 !; 3 ?)
% 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_tdlat_1,axiom,
! [A] :
( ( v2_pre_topc(A)
& l1_pre_topc(A) )
=> ~ v1_xboole_0(k1_tdlat_1(A)) ) ).
fof(fc2_tdlat_1,axiom,
! [A] :
( ( v2_pre_topc(A)
& l1_pre_topc(A) )
=> ~ v1_xboole_0(k5_tdlat_1(A)) ) ).
fof(fc3_tdlat_1,axiom,
! [A] :
( ( v2_pre_topc(A)
& l1_pre_topc(A) )
=> ~ v1_xboole_0(k9_tdlat_1(A)) ) ).
fof(t1_tdlat_1,axiom,
! [A] :
( ( v2_pre_topc(A)
& l1_pre_topc(A) )
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A)))
=> ( k4_subset_1(u1_struct_0(A),B,C) = k2_pre_topc(A)
<=> r1_tarski(k3_subset_1(u1_struct_0(A),B),C) ) ) ) ) ).
fof(t2_tdlat_1,axiom,
! [A] :
( ( v2_pre_topc(A)
& l1_pre_topc(A) )
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A)))
=> ( r1_xboole_0(B,C)
<=> r1_tarski(C,k3_subset_1(u1_struct_0(A),B)) ) ) ) ) ).
fof(t3_tdlat_1,axiom,
! [A] :
( ( v2_pre_topc(A)
& l1_pre_topc(A) )
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
=> r1_tarski(k6_pre_topc(A,k1_tops_1(A,k6_pre_topc(A,B))),k6_pre_topc(A,B)) ) ) ).
fof(t4_tdlat_1,axiom,
! [A] :
( ( v2_pre_topc(A)
& l1_pre_topc(A) )
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
=> r1_tarski(k1_tops_1(A,B),k1_tops_1(A,k6_pre_topc(A,k1_tops_1(A,B)))) ) ) ).
fof(t5_tdlat_1,axiom,
! [A] :
( ( v2_pre_topc(A)
& l1_pre_topc(A) )
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
=> k1_tops_1(A,k6_pre_topc(A,B)) = k1_tops_1(A,k6_pre_topc(A,k1_tops_1(A,k6_pre_topc(A,B)))) ) ) ).
fof(t6_tdlat_1,axiom,
! [A] :
( ( v2_pre_topc(A)
& l1_pre_topc(A) )
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A)))
=> ( ( v4_pre_topc(B,A)
| v4_pre_topc(C,A) )
=> k4_subset_1(u1_struct_0(A),k6_pre_topc(A,k1_tops_1(A,B)),k6_pre_topc(A,k1_tops_1(A,C))) = k6_pre_topc(A,k1_tops_1(A,k4_subset_1(u1_struct_0(A),B,C))) ) ) ) ) ).
fof(t7_tdlat_1,axiom,
! [A] :
( ( v2_pre_topc(A)
& l1_pre_topc(A) )
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A)))
=> ( ( v3_pre_topc(B,A)
| v3_pre_topc(C,A) )
=> k5_subset_1(u1_struct_0(A),k1_tops_1(A,k6_pre_topc(A,B)),k1_tops_1(A,k6_pre_topc(A,C))) = k1_tops_1(A,k6_pre_topc(A,k5_subset_1(u1_struct_0(A),B,C))) ) ) ) ) ).
fof(t8_tdlat_1,axiom,
! [A] :
( ( v2_pre_topc(A)
& l1_pre_topc(A) )
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
=> k1_tops_1(A,k5_subset_1(u1_struct_0(A),B,k6_pre_topc(A,k3_subset_1(u1_struct_0(A),B)))) = k1_pre_topc(A) ) ) ).
fof(t9_tdlat_1,axiom,
! [A] :
( ( v2_pre_topc(A)
& l1_pre_topc(A) )
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
=> k6_pre_topc(A,k4_subset_1(u1_struct_0(A),B,k1_tops_1(A,k3_subset_1(u1_struct_0(A),B)))) = k2_pre_topc(A) ) ) ).
fof(t10_tdlat_1,axiom,
! [A] :
( ( v2_pre_topc(A)
& l1_pre_topc(A) )
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A)))
=> k4_subset_1(u1_struct_0(A),k1_tops_1(A,k6_pre_topc(A,k4_subset_1(u1_struct_0(A),B,k4_subset_1(u1_struct_0(A),k1_tops_1(A,k6_pre_topc(A,C)),C)))),k4_subset_1(u1_struct_0(A),B,k4_subset_1(u1_struct_0(A),k1_tops_1(A,k6_pre_topc(A,C)),C))) = k4_subset_1(u1_struct_0(A),k1_tops_1(A,k6_pre_topc(A,k4_subset_1(u1_struct_0(A),B,C))),k4_subset_1(u1_struct_0(A),B,C)) ) ) ) ).
fof(t11_tdlat_1,axiom,
! [A] :
( ( v2_pre_topc(A)
& l1_pre_topc(A) )
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A)))
=> k4_subset_1(u1_struct_0(A),k1_tops_1(A,k6_pre_topc(A,k4_subset_1(u1_struct_0(A),k4_subset_1(u1_struct_0(A),k1_tops_1(A,k6_pre_topc(A,B)),B),C))),k4_subset_1(u1_struct_0(A),k4_subset_1(u1_struct_0(A),k1_tops_1(A,k6_pre_topc(A,B)),B),C)) = k4_subset_1(u1_struct_0(A),k1_tops_1(A,k6_pre_topc(A,k4_subset_1(u1_struct_0(A),B,C))),k4_subset_1(u1_struct_0(A),B,C)) ) ) ) ).
fof(t12_tdlat_1,axiom,
! [A] :
( ( v2_pre_topc(A)
& l1_pre_topc(A) )
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A)))
=> k5_subset_1(u1_struct_0(A),k6_pre_topc(A,k1_tops_1(A,k5_subset_1(u1_struct_0(A),B,k5_subset_1(u1_struct_0(A),k6_pre_topc(A,k1_tops_1(A,C)),C)))),k5_subset_1(u1_struct_0(A),B,k5_subset_1(u1_struct_0(A),k6_pre_topc(A,k1_tops_1(A,C)),C))) = k5_subset_1(u1_struct_0(A),k6_pre_topc(A,k1_tops_1(A,k5_subset_1(u1_struct_0(A),B,C))),k5_subset_1(u1_struct_0(A),B,C)) ) ) ) ).
fof(t13_tdlat_1,axiom,
! [A] :
( ( v2_pre_topc(A)
& l1_pre_topc(A) )
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A)))
=> k5_subset_1(u1_struct_0(A),k6_pre_topc(A,k1_tops_1(A,k5_subset_1(u1_struct_0(A),k5_subset_1(u1_struct_0(A),k6_pre_topc(A,k1_tops_1(A,B)),B),C))),k5_subset_1(u1_struct_0(A),k5_subset_1(u1_struct_0(A),k6_pre_topc(A,k1_tops_1(A,B)),B),C)) = k5_subset_1(u1_struct_0(A),k6_pre_topc(A,k1_tops_1(A,k5_subset_1(u1_struct_0(A),B,C))),k5_subset_1(u1_struct_0(A),B,C)) ) ) ) ).
fof(t14_tdlat_1,axiom,
! [A] :
( ( v2_pre_topc(A)
& l1_pre_topc(A) )
=> v4_tops_1(k1_pre_topc(A),A) ) ).
fof(t15_tdlat_1,axiom,
! [A] :
( ( v2_pre_topc(A)
& l1_pre_topc(A) )
=> v4_tops_1(k2_pre_topc(A),A) ) ).
fof(t16_tdlat_1,axiom,
! [A] :
( ( v2_pre_topc(A)
& l1_pre_topc(A) )
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
=> ( v4_tops_1(B,A)
=> v4_tops_1(k3_subset_1(u1_struct_0(A),B),A) ) ) ) ).
fof(t17_tdlat_1,axiom,
! [A] :
( ( v2_pre_topc(A)
& l1_pre_topc(A) )
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A)))
=> ( ( v4_tops_1(B,A)
& v4_tops_1(C,A) )
=> ( v4_tops_1(k4_subset_1(u1_struct_0(A),k1_tops_1(A,k6_pre_topc(A,k4_subset_1(u1_struct_0(A),B,C))),k4_subset_1(u1_struct_0(A),B,C)),A)
& v4_tops_1(k5_subset_1(u1_struct_0(A),k6_pre_topc(A,k1_tops_1(A,k5_subset_1(u1_struct_0(A),B,C))),k5_subset_1(u1_struct_0(A),B,C)),A) ) ) ) ) ) ).
fof(t18_tdlat_1,axiom,
! [A] :
( ( v2_pre_topc(A)
& l1_pre_topc(A) )
=> v5_tops_1(k1_pre_topc(A),A) ) ).
fof(t19_tdlat_1,axiom,
! [A] :
( ( v2_pre_topc(A)
& l1_pre_topc(A) )
=> v5_tops_1(k2_pre_topc(A),A) ) ).
fof(t20_tdlat_1,axiom,
! [A] :
( ( v2_pre_topc(A)
& l1_pre_topc(A) )
=> v6_tops_1(k1_pre_topc(A),A) ) ).
fof(t21_tdlat_1,axiom,
! [A] :
( ( v2_pre_topc(A)
& l1_pre_topc(A) )
=> v6_tops_1(k2_pre_topc(A),A) ) ).
fof(t22_tdlat_1,axiom,
! [A] :
( ( v2_pre_topc(A)
& l1_pre_topc(A) )
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
=> v5_tops_1(k6_pre_topc(A,k1_tops_1(A,B)),A) ) ) ).
fof(t23_tdlat_1,axiom,
! [A] :
( ( v2_pre_topc(A)
& l1_pre_topc(A) )
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
=> v6_tops_1(k1_tops_1(A,k6_pre_topc(A,B)),A) ) ) ).
fof(t24_tdlat_1,axiom,
! [A] :
( ( v2_pre_topc(A)
& l1_pre_topc(A) )
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
=> ( v4_tops_1(B,A)
=> v5_tops_1(k6_pre_topc(A,B),A) ) ) ) ).
fof(t25_tdlat_1,axiom,
! [A] :
( ( v2_pre_topc(A)
& l1_pre_topc(A) )
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
=> ( v4_tops_1(B,A)
=> v6_tops_1(k1_tops_1(A,B),A) ) ) ) ).
fof(t26_tdlat_1,axiom,
! [A] :
( ( v2_pre_topc(A)
& l1_pre_topc(A) )
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
=> ( v4_tops_1(B,A)
=> v5_tops_1(k6_pre_topc(A,k3_subset_1(u1_struct_0(A),B)),A) ) ) ) ).
fof(t27_tdlat_1,axiom,
! [A] :
( ( v2_pre_topc(A)
& l1_pre_topc(A) )
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
=> ( v4_tops_1(B,A)
=> v6_tops_1(k1_tops_1(A,k3_subset_1(u1_struct_0(A),B)),A) ) ) ) ).
fof(t28_tdlat_1,axiom,
! [A] :
( ( v2_pre_topc(A)
& l1_pre_topc(A) )
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A)))
=> ! [D] :
( m1_subset_1(D,k1_zfmisc_1(u1_struct_0(A)))
=> ( ( v5_tops_1(B,A)
& v5_tops_1(C,A)
& v5_tops_1(D,A) )
=> k6_pre_topc(A,k1_tops_1(A,k5_subset_1(u1_struct_0(A),B,k6_pre_topc(A,k1_tops_1(A,k5_subset_1(u1_struct_0(A),C,D)))))) = k6_pre_topc(A,k1_tops_1(A,k5_subset_1(u1_struct_0(A),k6_pre_topc(A,k1_tops_1(A,k5_subset_1(u1_struct_0(A),B,C))),D))) ) ) ) ) ) ).
fof(t29_tdlat_1,axiom,
! [A] :
( ( v2_pre_topc(A)
& l1_pre_topc(A) )
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A)))
=> ! [D] :
( m1_subset_1(D,k1_zfmisc_1(u1_struct_0(A)))
=> ( ( v6_tops_1(B,A)
& v6_tops_1(C,A)
& v6_tops_1(D,A) )
=> k1_tops_1(A,k6_pre_topc(A,k4_subset_1(u1_struct_0(A),B,k1_tops_1(A,k6_pre_topc(A,k4_subset_1(u1_struct_0(A),C,D)))))) = k1_tops_1(A,k6_pre_topc(A,k4_subset_1(u1_struct_0(A),k1_tops_1(A,k6_pre_topc(A,k4_subset_1(u1_struct_0(A),B,C))),D))) ) ) ) ) ) ).
fof(d2_tdlat_1,axiom,
! [A] :
( ( v2_pre_topc(A)
& l1_pre_topc(A) )
=> ! [B] :
( ( v1_funct_1(B)
& v1_funct_2(B,k2_zfmisc_1(k1_tdlat_1(A),k1_tdlat_1(A)),k1_tdlat_1(A))
& m2_relset_1(B,k2_zfmisc_1(k1_tdlat_1(A),k1_tdlat_1(A)),k1_tdlat_1(A)) )
=> ( B = k2_tdlat_1(A)
<=> ! [C] :
( m2_subset_1(C,k1_zfmisc_1(u1_struct_0(A)),k1_tdlat_1(A))
=> ! [D] :
( m2_subset_1(D,k1_zfmisc_1(u1_struct_0(A)),k1_tdlat_1(A))
=> k2_binop_1(k1_tdlat_1(A),k1_tdlat_1(A),k1_tdlat_1(A),B,C,D) = k4_subset_1(u1_struct_0(A),k1_tops_1(A,k6_pre_topc(A,k4_subset_1(u1_struct_0(A),C,D))),k4_subset_1(u1_struct_0(A),C,D)) ) ) ) ) ) ).
fof(d3_tdlat_1,axiom,
! [A] :
( ( v2_pre_topc(A)
& l1_pre_topc(A) )
=> ! [B] :
( ( v1_funct_1(B)
& v1_funct_2(B,k2_zfmisc_1(k1_tdlat_1(A),k1_tdlat_1(A)),k1_tdlat_1(A))
& m2_relset_1(B,k2_zfmisc_1(k1_tdlat_1(A),k1_tdlat_1(A)),k1_tdlat_1(A)) )
=> ( B = k3_tdlat_1(A)
<=> ! [C] :
( m2_subset_1(C,k1_zfmisc_1(u1_struct_0(A)),k1_tdlat_1(A))
=> ! [D] :
( m2_subset_1(D,k1_zfmisc_1(u1_struct_0(A)),k1_tdlat_1(A))
=> k2_binop_1(k1_tdlat_1(A),k1_tdlat_1(A),k1_tdlat_1(A),B,C,D) = k5_subset_1(u1_struct_0(A),k6_pre_topc(A,k1_tops_1(A,k5_subset_1(u1_struct_0(A),C,D))),k5_subset_1(u1_struct_0(A),C,D)) ) ) ) ) ) ).
fof(t30_tdlat_1,axiom,
! [A] :
( ( v2_pre_topc(A)
& l1_pre_topc(A) )
=> ( ~ v3_struct_0(g3_lattices(k1_tdlat_1(A),k2_tdlat_1(A),k3_tdlat_1(A)))
& v10_lattices(g3_lattices(k1_tdlat_1(A),k2_tdlat_1(A),k3_tdlat_1(A)))
& v15_lattices(g3_lattices(k1_tdlat_1(A),k2_tdlat_1(A),k3_tdlat_1(A)))
& v16_lattices(g3_lattices(k1_tdlat_1(A),k2_tdlat_1(A),k3_tdlat_1(A)))
& l3_lattices(g3_lattices(k1_tdlat_1(A),k2_tdlat_1(A),k3_tdlat_1(A))) ) ) ).
fof(d4_tdlat_1,axiom,
! [A] :
( ( v2_pre_topc(A)
& l1_pre_topc(A) )
=> k4_tdlat_1(A) = g3_lattices(k1_tdlat_1(A),k2_tdlat_1(A),k3_tdlat_1(A)) ) ).
fof(t31_tdlat_1,axiom,
! [A] :
( ( v2_pre_topc(A)
& l1_pre_topc(A) )
=> r1_tarski(k5_tdlat_1(A),k1_tdlat_1(A)) ) ).
fof(d6_tdlat_1,axiom,
! [A] :
( ( v2_pre_topc(A)
& l1_pre_topc(A) )
=> ! [B] :
( ( v1_funct_1(B)
& v1_funct_2(B,k2_zfmisc_1(k5_tdlat_1(A),k5_tdlat_1(A)),k5_tdlat_1(A))
& m2_relset_1(B,k2_zfmisc_1(k5_tdlat_1(A),k5_tdlat_1(A)),k5_tdlat_1(A)) )
=> ( B = k6_tdlat_1(A)
<=> ! [C] :
( m2_subset_1(C,k1_zfmisc_1(u1_struct_0(A)),k5_tdlat_1(A))
=> ! [D] :
( m2_subset_1(D,k1_zfmisc_1(u1_struct_0(A)),k5_tdlat_1(A))
=> k2_binop_1(k5_tdlat_1(A),k5_tdlat_1(A),k5_tdlat_1(A),B,C,D) = k4_subset_1(u1_struct_0(A),C,D) ) ) ) ) ) ).
fof(t32_tdlat_1,axiom,
! [A] :
( ( v2_pre_topc(A)
& l1_pre_topc(A) )
=> ! [B] :
( m2_subset_1(B,k1_zfmisc_1(u1_struct_0(A)),k5_tdlat_1(A))
=> ! [C] :
( m2_subset_1(C,k1_zfmisc_1(u1_struct_0(A)),k5_tdlat_1(A))
=> k2_binop_1(k5_tdlat_1(A),k5_tdlat_1(A),k5_tdlat_1(A),k6_tdlat_1(A),B,C) = k1_binop_1(k2_tdlat_1(A),B,C) ) ) ) ).
fof(d7_tdlat_1,axiom,
! [A] :
( ( v2_pre_topc(A)
& l1_pre_topc(A) )
=> ! [B] :
( ( v1_funct_1(B)
& v1_funct_2(B,k2_zfmisc_1(k5_tdlat_1(A),k5_tdlat_1(A)),k5_tdlat_1(A))
& m2_relset_1(B,k2_zfmisc_1(k5_tdlat_1(A),k5_tdlat_1(A)),k5_tdlat_1(A)) )
=> ( B = k7_tdlat_1(A)
<=> ! [C] :
( m2_subset_1(C,k1_zfmisc_1(u1_struct_0(A)),k5_tdlat_1(A))
=> ! [D] :
( m2_subset_1(D,k1_zfmisc_1(u1_struct_0(A)),k5_tdlat_1(A))
=> k2_binop_1(k5_tdlat_1(A),k5_tdlat_1(A),k5_tdlat_1(A),B,C,D) = k6_pre_topc(A,k1_tops_1(A,k5_subset_1(u1_struct_0(A),C,D))) ) ) ) ) ) ).
fof(t33_tdlat_1,axiom,
! [A] :
( ( v2_pre_topc(A)
& l1_pre_topc(A) )
=> ! [B] :
( m2_subset_1(B,k1_zfmisc_1(u1_struct_0(A)),k5_tdlat_1(A))
=> ! [C] :
( m2_subset_1(C,k1_zfmisc_1(u1_struct_0(A)),k5_tdlat_1(A))
=> k2_binop_1(k5_tdlat_1(A),k5_tdlat_1(A),k5_tdlat_1(A),k7_tdlat_1(A),B,C) = k1_binop_1(k3_tdlat_1(A),B,C) ) ) ) ).
fof(t34_tdlat_1,axiom,
! [A] :
( ( v2_pre_topc(A)
& l1_pre_topc(A) )
=> ( ~ v3_struct_0(g3_lattices(k5_tdlat_1(A),k6_tdlat_1(A),k7_tdlat_1(A)))
& v10_lattices(g3_lattices(k5_tdlat_1(A),k6_tdlat_1(A),k7_tdlat_1(A)))
& v17_lattices(g3_lattices(k5_tdlat_1(A),k6_tdlat_1(A),k7_tdlat_1(A)))
& l3_lattices(g3_lattices(k5_tdlat_1(A),k6_tdlat_1(A),k7_tdlat_1(A))) ) ) ).
fof(d8_tdlat_1,axiom,
! [A] :
( ( v2_pre_topc(A)
& l1_pre_topc(A) )
=> k8_tdlat_1(A) = g3_lattices(k5_tdlat_1(A),k6_tdlat_1(A),k7_tdlat_1(A)) ) ).
fof(t35_tdlat_1,axiom,
! [A] :
( ( v2_pre_topc(A)
& l1_pre_topc(A) )
=> r1_tarski(k9_tdlat_1(A),k1_tdlat_1(A)) ) ).
fof(d10_tdlat_1,axiom,
! [A] :
( ( v2_pre_topc(A)
& l1_pre_topc(A) )
=> ! [B] :
( ( v1_funct_1(B)
& v1_funct_2(B,k2_zfmisc_1(k9_tdlat_1(A),k9_tdlat_1(A)),k9_tdlat_1(A))
& m2_relset_1(B,k2_zfmisc_1(k9_tdlat_1(A),k9_tdlat_1(A)),k9_tdlat_1(A)) )
=> ( B = k10_tdlat_1(A)
<=> ! [C] :
( m2_subset_1(C,k1_zfmisc_1(u1_struct_0(A)),k9_tdlat_1(A))
=> ! [D] :
( m2_subset_1(D,k1_zfmisc_1(u1_struct_0(A)),k9_tdlat_1(A))
=> k2_binop_1(k9_tdlat_1(A),k9_tdlat_1(A),k9_tdlat_1(A),B,C,D) = k1_tops_1(A,k6_pre_topc(A,k4_subset_1(u1_struct_0(A),C,D))) ) ) ) ) ) ).
fof(t36_tdlat_1,axiom,
! [A] :
( ( v2_pre_topc(A)
& l1_pre_topc(A) )
=> ! [B] :
( m2_subset_1(B,k1_zfmisc_1(u1_struct_0(A)),k9_tdlat_1(A))
=> ! [C] :
( m2_subset_1(C,k1_zfmisc_1(u1_struct_0(A)),k9_tdlat_1(A))
=> k2_binop_1(k9_tdlat_1(A),k9_tdlat_1(A),k9_tdlat_1(A),k10_tdlat_1(A),B,C) = k1_binop_1(k2_tdlat_1(A),B,C) ) ) ) ).
fof(d11_tdlat_1,axiom,
! [A] :
( ( v2_pre_topc(A)
& l1_pre_topc(A) )
=> ! [B] :
( ( v1_funct_1(B)
& v1_funct_2(B,k2_zfmisc_1(k9_tdlat_1(A),k9_tdlat_1(A)),k9_tdlat_1(A))
& m2_relset_1(B,k2_zfmisc_1(k9_tdlat_1(A),k9_tdlat_1(A)),k9_tdlat_1(A)) )
=> ( B = k11_tdlat_1(A)
<=> ! [C] :
( m2_subset_1(C,k1_zfmisc_1(u1_struct_0(A)),k9_tdlat_1(A))
=> ! [D] :
( m2_subset_1(D,k1_zfmisc_1(u1_struct_0(A)),k9_tdlat_1(A))
=> k2_binop_1(k9_tdlat_1(A),k9_tdlat_1(A),k9_tdlat_1(A),B,C,D) = k5_subset_1(u1_struct_0(A),C,D) ) ) ) ) ) ).
fof(t37_tdlat_1,axiom,
! [A] :
( ( v2_pre_topc(A)
& l1_pre_topc(A) )
=> ! [B] :
( m2_subset_1(B,k1_zfmisc_1(u1_struct_0(A)),k9_tdlat_1(A))
=> ! [C] :
( m2_subset_1(C,k1_zfmisc_1(u1_struct_0(A)),k9_tdlat_1(A))
=> k2_binop_1(k9_tdlat_1(A),k9_tdlat_1(A),k9_tdlat_1(A),k11_tdlat_1(A),B,C) = k1_binop_1(k3_tdlat_1(A),B,C) ) ) ) ).
fof(t38_tdlat_1,axiom,
! [A] :
( ( v2_pre_topc(A)
& l1_pre_topc(A) )
=> ( ~ v3_struct_0(g3_lattices(k9_tdlat_1(A),k10_tdlat_1(A),k11_tdlat_1(A)))
& v10_lattices(g3_lattices(k9_tdlat_1(A),k10_tdlat_1(A),k11_tdlat_1(A)))
& v17_lattices(g3_lattices(k9_tdlat_1(A),k10_tdlat_1(A),k11_tdlat_1(A)))
& l3_lattices(g3_lattices(k9_tdlat_1(A),k10_tdlat_1(A),k11_tdlat_1(A))) ) ) ).
fof(d12_tdlat_1,axiom,
! [A] :
( ( v2_pre_topc(A)
& l1_pre_topc(A) )
=> k12_tdlat_1(A) = g3_lattices(k9_tdlat_1(A),k10_tdlat_1(A),k11_tdlat_1(A)) ) ).
fof(t39_tdlat_1,axiom,
! [A] :
( ( v2_pre_topc(A)
& l1_pre_topc(A) )
=> k6_tdlat_1(A) = k1_realset1(k2_tdlat_1(A),k5_tdlat_1(A)) ) ).
fof(t40_tdlat_1,axiom,
! [A] :
( ( v2_pre_topc(A)
& l1_pre_topc(A) )
=> k7_tdlat_1(A) = k1_realset1(k3_tdlat_1(A),k5_tdlat_1(A)) ) ).
fof(t41_tdlat_1,axiom,
! [A] :
( ( v2_pre_topc(A)
& l1_pre_topc(A) )
=> m2_nat_lat(k8_tdlat_1(A),k4_tdlat_1(A)) ) ).
fof(t42_tdlat_1,axiom,
! [A] :
( ( v2_pre_topc(A)
& l1_pre_topc(A) )
=> k10_tdlat_1(A) = k1_realset1(k2_tdlat_1(A),k9_tdlat_1(A)) ) ).
fof(t43_tdlat_1,axiom,
! [A] :
( ( v2_pre_topc(A)
& l1_pre_topc(A) )
=> k11_tdlat_1(A) = k1_realset1(k3_tdlat_1(A),k9_tdlat_1(A)) ) ).
fof(t44_tdlat_1,axiom,
! [A] :
( ( v2_pre_topc(A)
& l1_pre_topc(A) )
=> m2_nat_lat(k12_tdlat_1(A),k4_tdlat_1(A)) ) ).
fof(dt_k1_tdlat_1,axiom,
! [A] :
( l1_pre_topc(A)
=> m1_subset_1(k1_tdlat_1(A),k1_zfmisc_1(k1_zfmisc_1(u1_struct_0(A)))) ) ).
fof(dt_k2_tdlat_1,axiom,
! [A] :
( ( v2_pre_topc(A)
& l1_pre_topc(A) )
=> ( v1_funct_1(k2_tdlat_1(A))
& v1_funct_2(k2_tdlat_1(A),k2_zfmisc_1(k1_tdlat_1(A),k1_tdlat_1(A)),k1_tdlat_1(A))
& m2_relset_1(k2_tdlat_1(A),k2_zfmisc_1(k1_tdlat_1(A),k1_tdlat_1(A)),k1_tdlat_1(A)) ) ) ).
fof(dt_k3_tdlat_1,axiom,
! [A] :
( ( v2_pre_topc(A)
& l1_pre_topc(A) )
=> ( v1_funct_1(k3_tdlat_1(A))
& v1_funct_2(k3_tdlat_1(A),k2_zfmisc_1(k1_tdlat_1(A),k1_tdlat_1(A)),k1_tdlat_1(A))
& m2_relset_1(k3_tdlat_1(A),k2_zfmisc_1(k1_tdlat_1(A),k1_tdlat_1(A)),k1_tdlat_1(A)) ) ) ).
fof(dt_k4_tdlat_1,axiom,
! [A] :
( ( v2_pre_topc(A)
& l1_pre_topc(A) )
=> ( ~ v3_struct_0(k4_tdlat_1(A))
& v10_lattices(k4_tdlat_1(A))
& v15_lattices(k4_tdlat_1(A))
& v16_lattices(k4_tdlat_1(A))
& l3_lattices(k4_tdlat_1(A)) ) ) ).
fof(dt_k5_tdlat_1,axiom,
! [A] :
( l1_pre_topc(A)
=> m1_subset_1(k5_tdlat_1(A),k1_zfmisc_1(k1_zfmisc_1(u1_struct_0(A)))) ) ).
fof(dt_k6_tdlat_1,axiom,
! [A] :
( ( v2_pre_topc(A)
& l1_pre_topc(A) )
=> ( v1_funct_1(k6_tdlat_1(A))
& v1_funct_2(k6_tdlat_1(A),k2_zfmisc_1(k5_tdlat_1(A),k5_tdlat_1(A)),k5_tdlat_1(A))
& m2_relset_1(k6_tdlat_1(A),k2_zfmisc_1(k5_tdlat_1(A),k5_tdlat_1(A)),k5_tdlat_1(A)) ) ) ).
fof(dt_k7_tdlat_1,axiom,
! [A] :
( ( v2_pre_topc(A)
& l1_pre_topc(A) )
=> ( v1_funct_1(k7_tdlat_1(A))
& v1_funct_2(k7_tdlat_1(A),k2_zfmisc_1(k5_tdlat_1(A),k5_tdlat_1(A)),k5_tdlat_1(A))
& m2_relset_1(k7_tdlat_1(A),k2_zfmisc_1(k5_tdlat_1(A),k5_tdlat_1(A)),k5_tdlat_1(A)) ) ) ).
fof(dt_k8_tdlat_1,axiom,
! [A] :
( ( v2_pre_topc(A)
& l1_pre_topc(A) )
=> ( ~ v3_struct_0(k8_tdlat_1(A))
& v10_lattices(k8_tdlat_1(A))
& v17_lattices(k8_tdlat_1(A))
& l3_lattices(k8_tdlat_1(A)) ) ) ).
fof(dt_k9_tdlat_1,axiom,
! [A] :
( l1_pre_topc(A)
=> m1_subset_1(k9_tdlat_1(A),k1_zfmisc_1(k1_zfmisc_1(u1_struct_0(A)))) ) ).
fof(dt_k10_tdlat_1,axiom,
! [A] :
( ( v2_pre_topc(A)
& l1_pre_topc(A) )
=> ( v1_funct_1(k10_tdlat_1(A))
& v1_funct_2(k10_tdlat_1(A),k2_zfmisc_1(k9_tdlat_1(A),k9_tdlat_1(A)),k9_tdlat_1(A))
& m2_relset_1(k10_tdlat_1(A),k2_zfmisc_1(k9_tdlat_1(A),k9_tdlat_1(A)),k9_tdlat_1(A)) ) ) ).
fof(dt_k11_tdlat_1,axiom,
! [A] :
( ( v2_pre_topc(A)
& l1_pre_topc(A) )
=> ( v1_funct_1(k11_tdlat_1(A))
& v1_funct_2(k11_tdlat_1(A),k2_zfmisc_1(k9_tdlat_1(A),k9_tdlat_1(A)),k9_tdlat_1(A))
& m2_relset_1(k11_tdlat_1(A),k2_zfmisc_1(k9_tdlat_1(A),k9_tdlat_1(A)),k9_tdlat_1(A)) ) ) ).
fof(dt_k12_tdlat_1,axiom,
! [A] :
( ( v2_pre_topc(A)
& l1_pre_topc(A) )
=> ( ~ v3_struct_0(k12_tdlat_1(A))
& v10_lattices(k12_tdlat_1(A))
& v17_lattices(k12_tdlat_1(A))
& l3_lattices(k12_tdlat_1(A)) ) ) ).
fof(d1_tdlat_1,axiom,
! [A] :
( l1_pre_topc(A)
=> k1_tdlat_1(A) = a_1_0_tdlat_1(A) ) ).
fof(d5_tdlat_1,axiom,
! [A] :
( l1_pre_topc(A)
=> k5_tdlat_1(A) = a_1_1_tdlat_1(A) ) ).
fof(d9_tdlat_1,axiom,
! [A] :
( l1_pre_topc(A)
=> k9_tdlat_1(A) = a_1_2_tdlat_1(A) ) ).
fof(fraenkel_a_1_0_tdlat_1,axiom,
! [A,B] :
( l1_pre_topc(B)
=> ( r2_hidden(A,a_1_0_tdlat_1(B))
<=> ? [C] :
( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(B)))
& A = C
& v4_tops_1(C,B) ) ) ) ).
fof(fraenkel_a_1_1_tdlat_1,axiom,
! [A,B] :
( l1_pre_topc(B)
=> ( r2_hidden(A,a_1_1_tdlat_1(B))
<=> ? [C] :
( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(B)))
& A = C
& v5_tops_1(C,B) ) ) ) ).
fof(fraenkel_a_1_2_tdlat_1,axiom,
! [A,B] :
( l1_pre_topc(B)
=> ( r2_hidden(A,a_1_2_tdlat_1(B))
<=> ? [C] :
( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(B)))
& A = C
& v6_tops_1(C,B) ) ) ) ).
%------------------------------------------------------------------------------