:: The "Way-Below" Relation
:: by Grzegorz Bancerek
::
:: Copyright (c) 1996-2019 Association of Mizar Users

definition
let L be non empty reflexive RelStr ;
let x, y be Element of L;
pred x is_way_below y means :: WAYBEL_3:def 1
for D being non empty directed Subset of L st y <= sup D holds
ex d being Element of L st
( d in D & x <= d );
end;

:: deftheorem defines is_way_below WAYBEL_3:def 1 :
for L being non empty reflexive RelStr
for x, y being Element of L holds
( x is_way_below y iff for D being non empty directed Subset of L st y <= sup D holds
ex d being Element of L st
( d in D & x <= d ) );

notation
let L be non empty reflexive RelStr ;
let x, y be Element of L;
synonym x << y for x is_way_below y;
synonym y >> x for x is_way_below y;
end;

definition
let L be non empty reflexive RelStr ;
let x be Element of L;
attr x is compact means :: WAYBEL_3:def 2
x is_way_below x;
end;

:: deftheorem defines compact WAYBEL_3:def 2 :
for L being non empty reflexive RelStr
for x being Element of L holds
( x is compact iff x is_way_below x );

notation
let L be non empty reflexive RelStr ;
let x be Element of L;
synonym isolated_from_below x for compact ;
end;

theorem Th1: :: WAYBEL_3:1
for L being non empty reflexive antisymmetric RelStr
for x, y being Element of L st x << y holds
x <= y
proof end;

theorem Th2: :: WAYBEL_3:2
for L being non empty reflexive transitive RelStr
for u, x, y, z being Element of L st u <= x & x << y & y <= z holds
u << z
proof end;

theorem Th3: :: WAYBEL_3:3
for L being non empty Poset st ( L is with_suprema or L is /\-complete ) holds
for x, y, z being Element of L st x << z & y << z holds
( ex_sup_of {x,y},L & x "\/" y << z )
proof end;

theorem Th4: :: WAYBEL_3:4
for L being non empty reflexive antisymmetric lower-bounded RelStr
for x being Element of L holds Bottom L << x
proof end;

theorem :: WAYBEL_3:5
for L being non empty Poset
for x, y, z being Element of L st x << y & y << z holds
x << z
proof end;

theorem :: WAYBEL_3:6
for L being non empty reflexive antisymmetric RelStr
for x, y being Element of L st x << y & x >> y holds
x = y
proof end;

definition
let L be non empty reflexive RelStr ;
let x be Element of L;
A1: { y where y is Element of L : y << x } c= the carrier of L
proof end;
func waybelow x -> Subset of L equals :: WAYBEL_3:def 3
{ y where y is Element of L : y << x } ;
correctness
coherence
{ y where y is Element of L : y << x } is Subset of L
;
by A1;
A2: { y where y is Element of L : y >> x } c= the carrier of L
proof end;
func wayabove x -> Subset of L equals :: WAYBEL_3:def 4
{ y where y is Element of L : y >> x } ;
correctness
coherence
{ y where y is Element of L : y >> x } is Subset of L
;
by A2;
end;

:: deftheorem defines waybelow WAYBEL_3:def 3 :
for L being non empty reflexive RelStr
for x being Element of L holds waybelow x = { y where y is Element of L : y << x } ;

:: deftheorem defines wayabove WAYBEL_3:def 4 :
for L being non empty reflexive RelStr
for x being Element of L holds wayabove x = { y where y is Element of L : y >> x } ;

theorem Th7: :: WAYBEL_3:7
for L being non empty reflexive RelStr
for x, y being Element of L holds
( x in waybelow y iff x << y )
proof end;

theorem Th8: :: WAYBEL_3:8
for L being non empty reflexive RelStr
for x, y being Element of L holds
( x in wayabove y iff x >> y )
proof end;

theorem Th9: :: WAYBEL_3:9
for L being non empty reflexive antisymmetric RelStr
for x being Element of L holds x is_>=_than waybelow x by ;

theorem :: WAYBEL_3:10
for L being non empty reflexive antisymmetric RelStr
for x being Element of L holds x is_<=_than wayabove x by ;

theorem Th11: :: WAYBEL_3:11
for L being non empty reflexive antisymmetric RelStr
for x being Element of L holds
( waybelow x c= downarrow x & wayabove x c= uparrow x )
proof end;

theorem Th12: :: WAYBEL_3:12
for L being non empty reflexive transitive RelStr
for x, y being Element of L st x <= y holds
( waybelow x c= waybelow y & wayabove y c= wayabove x )
proof end;

registration
let L be non empty reflexive antisymmetric lower-bounded RelStr ;
let x be Element of L;
cluster waybelow x -> non empty ;
coherence
not waybelow x is empty
proof end;
end;

registration
let L be non empty reflexive transitive RelStr ;
let x be Element of L;
coherence
proof end;
coherence
proof end;
end;

registration
let L be sup-Semilattice;
let x be Element of L;
coherence
proof end;
end;

registration
let L be non empty /\-complete Poset;
let x be Element of L;
coherence
proof end;
end;

:: EXAMPLES, 1.3, p. 39
registration
let L be non empty connected RelStr ;
cluster -> directed filtered for Element of K10( the carrier of L);
coherence
for b1 being Subset of L holds
( b1 is directed & b1 is filtered )
proof end;
end;

registration
coherence
for b1 being Chain st b1 is up-complete & b1 is lower-bounded holds
b1 is complete
proof end;
end;

registration
existence
ex b1 being Chain st b1 is complete
proof end;
end;

theorem Th13: :: WAYBEL_3:13
for L being up-complete Chain
for x, y being Element of L st x < y holds
x << y
proof end;

theorem :: WAYBEL_3:14
for L being non empty reflexive antisymmetric RelStr
for x, y being Element of L st not x is compact & x << y holds
x < y by Th1;

theorem :: WAYBEL_3:15
for L being non empty reflexive antisymmetric lower-bounded RelStr holds Bottom L is compact by Th4;

theorem Th16: :: WAYBEL_3:16
for L being non empty up-complete Poset
for D being non empty finite directed Subset of L holds sup D in D
proof end;

theorem :: WAYBEL_3:17
for L being non empty up-complete Poset st L is finite holds
for x being Element of L holds x is isolated_from_below
proof end;

theorem Th18: :: WAYBEL_3:18
for L being complete LATTICE
for x, y being Element of L st x << y holds
for X being Subset of L st y <= sup X holds
ex A being finite Subset of L st
( A c= X & x <= sup A )
proof end;

theorem :: WAYBEL_3:19
for L being complete LATTICE
for x, y being Element of L st ( for X being Subset of L st y <= sup X holds
ex A being finite Subset of L st
( A c= X & x <= sup A ) ) holds
x << y
proof end;

theorem :: WAYBEL_3:20
for L being non empty reflexive transitive RelStr
for x, y being Element of L st x << y holds
for I being Ideal of L st y <= sup I holds
x in I
proof end;

theorem Th21: :: WAYBEL_3:21
for L being non empty up-complete Poset
for x, y being Element of L st ( for I being Ideal of L st y <= sup I holds
x in I ) holds
x << y
proof end;

theorem :: WAYBEL_3:22
for L being lower-bounded LATTICE st L is meet-continuous holds
for x, y being Element of L holds
( x << y iff for I being Ideal of L st y = sup I holds
x in I )
proof end;

theorem :: WAYBEL_3:23
for L being complete LATTICE holds
( ( for x being Element of L holds x is compact ) iff for X being non empty Subset of L ex x being Element of L st
( x in X & ( for y being Element of L st y in X holds
not x < y ) ) )
proof end;

definition
let L be non empty reflexive RelStr ;
attr L is satisfying_axiom_of_approximation means :Def5: :: WAYBEL_3:def 5
for x being Element of L holds x = sup ();
end;

:: deftheorem Def5 defines satisfying_axiom_of_approximation WAYBEL_3:def 5 :
for L being non empty reflexive RelStr holds
( L is satisfying_axiom_of_approximation iff for x being Element of L holds x = sup () );

registration
coherence
for b1 being 1 -element reflexive RelStr holds b1 is satisfying_axiom_of_approximation
by ZFMISC_1:def 10;
end;

definition
let L be non empty reflexive RelStr ;
attr L is continuous means :Def6: :: WAYBEL_3:def 6
( ( for x being Element of L holds
( not waybelow x is empty & waybelow x is directed ) ) & L is up-complete & L is satisfying_axiom_of_approximation );
end;

:: deftheorem Def6 defines continuous WAYBEL_3:def 6 :
for L being non empty reflexive RelStr holds
( L is continuous iff ( ( for x being Element of L holds
( not waybelow x is empty & waybelow x is directed ) ) & L is up-complete & L is satisfying_axiom_of_approximation ) );

registration
coherence
for b1 being non empty reflexive RelStr st b1 is continuous holds
( b1 is up-complete & b1 is satisfying_axiom_of_approximation )
;
coherence
for b1 being lower-bounded sup-Semilattice st b1 is up-complete & b1 is satisfying_axiom_of_approximation holds
b1 is continuous
;
end;

registration
existence
ex b1 being LATTICE st
( b1 is continuous & b1 is complete & b1 is strict )
proof end;
end;

registration
let L be non empty reflexive continuous RelStr ;
let x be Element of L;
cluster waybelow x -> non empty directed ;
coherence
( not waybelow x is empty & waybelow x is directed )
by Def6;
end;

theorem :: WAYBEL_3:24
for L being up-complete Semilattice st ( for x being Element of L holds
( not waybelow x is empty & waybelow x is directed ) ) holds
( L is satisfying_axiom_of_approximation iff for x, y being Element of L st not x <= y holds
ex u being Element of L st
( u << x & not u <= y ) )
proof end;

theorem :: WAYBEL_3:25
for L being continuous LATTICE
for x, y being Element of L holds
( x <= y iff waybelow x c= waybelow y )
proof end;

registration
coherence
for b1 being Chain st b1 is complete holds
b1 is satisfying_axiom_of_approximation
proof end;
end;

theorem :: WAYBEL_3:26
for L being complete LATTICE st ( for x being Element of L holds x is compact ) holds
L is satisfying_axiom_of_approximation
proof end;

definition
let f be Relation;
attr f is non-Empty means :Def7: :: WAYBEL_3:def 7
for S being 1-sorted st S in rng f holds
not S is empty ;
attr f is reflexive-yielding means :Def8: :: WAYBEL_3:def 8
for S being RelStr st S in rng f holds
S is reflexive ;
end;

:: deftheorem Def7 defines non-Empty WAYBEL_3:def 7 :
for f being Relation holds
( f is non-Empty iff for S being 1-sorted st S in rng f holds
not S is empty );

:: deftheorem Def8 defines reflexive-yielding WAYBEL_3:def 8 :
for f being Relation holds
( f is reflexive-yielding iff for S being RelStr st S in rng f holds
S is reflexive );

registration
let I be set ;
existence
ex b1 being ManySortedSet of I st
( b1 is RelStr-yielding & b1 is non-Empty & b1 is reflexive-yielding )
proof end;
end;

registration
let I be set ;
let J be RelStr-yielding non-Empty ManySortedSet of I;
cluster product J -> non empty ;
coherence
not product J is empty
proof end;
end;

definition
let I be non empty set ;
let J be RelStr-yielding ManySortedSet of I;
let i be Element of I;
:: original: .
redefine func J . i -> RelStr ;
coherence
J . i is RelStr
proof end;
end;

registration
let I be non empty set ;
let J be RelStr-yielding non-Empty ManySortedSet of I;
let i be Element of I;
cluster J . i -> non empty for RelStr ;
coherence
for b1 being RelStr st b1 = J . i holds
not b1 is empty
proof end;
end;

registration
let I be set ;
let J be RelStr-yielding non-Empty ManySortedSet of I;
coherence
proof end;
end;

definition
let I be non empty set ;
let J be RelStr-yielding non-Empty ManySortedSet of I;
let x be Element of ();
let i be Element of I;
:: original: .
redefine func x . i -> Element of (J . i);
coherence
x . i is Element of (J . i)
proof end;
end;

definition
let I be non empty set ;
let J be RelStr-yielding non-Empty ManySortedSet of I;
let i be Element of I;
let X be Subset of ();
:: original: pi
redefine func pi (X,i) -> Subset of (J . i);
coherence
pi (X,i) is Subset of (J . i)
proof end;
end;

theorem Th27: :: WAYBEL_3:27
for I being non empty set
for J being RelStr-yielding non-Empty ManySortedSet of I
for x being Function holds
( x is Element of () iff ( dom x = I & ( for i being Element of I holds x . i is Element of (J . i) ) ) )
proof end;

theorem Th28: :: WAYBEL_3:28
for I being non empty set
for J being RelStr-yielding non-Empty ManySortedSet of I
for x, y being Element of () holds
( x <= y iff for i being Element of I holds x . i <= y . i )
proof end;

registration
let I be non empty set ;
let J be RelStr-yielding reflexive-yielding ManySortedSet of I;
let i be Element of I;
cluster J . i -> reflexive for RelStr ;
coherence
for b1 being RelStr st b1 = J . i holds
b1 is reflexive
proof end;
end;

registration
let I be non empty set ;
let J be RelStr-yielding non-Empty reflexive-yielding ManySortedSet of I;
coherence
proof end;
end;

theorem Th29: :: WAYBEL_3:29
for I being non empty set
for J being RelStr-yielding non-Empty ManySortedSet of I st ( for i being Element of I holds J . i is transitive ) holds
product J is transitive
proof end;

theorem Th30: :: WAYBEL_3:30
for I being non empty set
for J being RelStr-yielding non-Empty ManySortedSet of I st ( for i being Element of I holds J . i is antisymmetric ) holds
product J is antisymmetric
proof end;

theorem Th31: :: WAYBEL_3:31
for I being non empty set
for J being RelStr-yielding non-Empty reflexive-yielding ManySortedSet of I st ( for i being Element of I holds J . i is complete LATTICE ) holds
product J is complete LATTICE
proof end;

theorem Th32: :: WAYBEL_3:32
for I being non empty set
for J being RelStr-yielding non-Empty reflexive-yielding ManySortedSet of I st ( for i being Element of I holds J . i is complete LATTICE ) holds
for X being Subset of ()
for i being Element of I holds (sup X) . i = sup (pi (X,i))
proof end;

theorem :: WAYBEL_3:33
for I being non empty set
for J being RelStr-yielding non-Empty reflexive-yielding ManySortedSet of I st ( for i being Element of I holds J . i is complete LATTICE ) holds
for x, y being Element of () holds
( x << y iff ( ( for i being Element of I holds x . i << y . i ) & ex K being finite Subset of I st
for i being Element of I st not i in K holds
x . i = Bottom (J . i) ) )
proof end;

theorem Th34: :: WAYBEL_3:34
for T being non empty TopSpace
for x, y being Element of (InclPoset the topology of T) st x is_way_below y holds
for F being Subset-Family of T st F is open & y c= union F holds
ex G being finite Subset of F st x c= union G
proof end;

theorem Th35: :: WAYBEL_3:35
for T being non empty TopSpace
for x, y being Element of (InclPoset the topology of T) st ( for F being Subset-Family of T st F is open & y c= union F holds
ex G being finite Subset of F st x c= union G ) holds
x is_way_below y
proof end;

theorem Th36: :: WAYBEL_3:36
for T being non empty TopSpace
for x being Element of (InclPoset the topology of T)
for X being Subset of T st x = X holds
( x is compact iff X is compact )
proof end;

theorem :: WAYBEL_3:37
for T being non empty TopSpace
for x being Element of (InclPoset the topology of T) st x = the carrier of T holds
( x is compact iff T is compact )
proof end;

definition
let T be non empty TopSpace;
attr T is locally-compact means :: WAYBEL_3:def 9
for x being Point of T
for X being Subset of T st x in X & X is open holds
ex Y being Subset of T st
( x in Int Y & Y c= X & Y is compact );
end;

:: deftheorem defines locally-compact WAYBEL_3:def 9 :
for T being non empty TopSpace holds
( T is locally-compact iff for x being Point of T
for X being Subset of T st x in X & X is open holds
ex Y being Subset of T st
( x in Int Y & Y c= X & Y is compact ) );

registration
coherence
for b1 being non empty TopSpace st b1 is compact & b1 is T_2 holds
( b1 is regular & b1 is normal & b1 is locally-compact )
proof end;
end;

registration
existence
ex b1 being non empty TopSpace st
( b1 is compact & b1 is T_2 )
proof end;
end;

theorem Th38: :: WAYBEL_3:38
for T being non empty TopSpace
for x, y being Element of (InclPoset the topology of T) st ex Z being Subset of T st
( x c= Z & Z c= y & Z is compact ) holds
x << y
proof end;

theorem Th39: :: WAYBEL_3:39
for T being non empty TopSpace st T is locally-compact holds
for x, y being Element of (InclPoset the topology of T) st x << y holds
ex Z being Subset of T st
( x c= Z & Z c= y & Z is compact )
proof end;

theorem :: WAYBEL_3:40
for T being non empty TopSpace st T is locally-compact & T is T_2 holds
for x, y being Element of (InclPoset the topology of T) st x << y holds
ex Z being Subset of T st
( Z = x & Cl Z c= y & Cl Z is compact )
proof end;

theorem :: WAYBEL_3:41
for X being non empty TopSpace st X is regular & InclPoset the topology of X is continuous holds
X is locally-compact
proof end;

theorem :: WAYBEL_3:42
for T being non empty TopSpace st T is locally-compact holds
InclPoset the topology of T is continuous
proof end;