:: YELLOW_4 semantic presentation

theorem Th1: :: YELLOW_4:1

for L being RelStr
for X being set
for a being Element of L st a in X & ex_sup_of X,L holds
a <= "\/" X,L

proof end;

theorem Th2: :: YELLOW_4:2

for L being RelStr
for X being set
for a being Element of L st a in X & ex_inf_of X,L holds
"/\" X,L <= a

proof end;

definition

let L be RelStr ;
let A, B be Subset of L;
pred A is_finer_than B means :Def1: :: YELLOW_4:def 1
for a being Element of L st a in A holds
ex b being Element of L st
( b in B & a <= b );
pred B is_coarser_than A means :Def2: :: YELLOW_4:def 2
for b being Element of L st b in B holds
ex a being Element of L st
( a in A & a <= b );

end;

:: deftheorem Def1 defines is_finer_than YELLOW_4:def 1 :

:: deftheorem Def2 defines is_coarser_than YELLOW_4:def 2 :

definition

let L be non empty reflexive RelStr ;
let A, B be Subset of L;
:: original: is_finer_than
redefine pred A is_finer_than B;
reflexivity

proof end;

:: original: is_coarser_than
redefine pred B is_coarser_than A;
reflexivity

proof end;

end;

theorem :: YELLOW_4:3

for L being RelStr
for B being Subset of L holds {} L is_finer_than B

proof end;

theorem :: YELLOW_4:4

for L being transitive RelStr
for A, B, C being Subset of L st A is_finer_than B & B is_finer_than C holds
A is_finer_than C

proof end;

theorem :: YELLOW_4:5

for L being RelStr
for A, B being Subset of L st B is_finer_than A & A is lower holds
B c= A

proof end;

theorem :: YELLOW_4:6

for L being RelStr
for A being Subset of L holds {} L is_coarser_than A

proof end;

theorem :: YELLOW_4:7

for L being transitive RelStr
for A, B, C being Subset of L st C is_coarser_than B & B is_coarser_than A holds
C is_coarser_than A

proof end;

theorem :: YELLOW_4:8

for L being RelStr
for A, B being Subset of L st A is_coarser_than B & B is upper holds
A c= B

proof end;

definition

let L be non empty RelStr ;
let D1, D2 be Subset of L;
func D1 "\/" D2 -> Subset of L equals :: YELLOW_4:def 3
{ (x "\/" y) where x, y is Element of L : ( x in D1 & y in D2 ) } ;
coherence

proof end;

end;

:: deftheorem defines "\/" YELLOW_4:def 3 :

definition

let L be antisymmetric with_suprema RelStr ;
let D1, D2 be Subset of L;
:: original: "\/"
redefine func D1 "\/" D2 -> Subset of L;
commutativity

proof end;

end;

theorem :: YELLOW_4:9

for L being non empty RelStr
for X being Subset of L holds X "\/" ({} L) = {}

proof end;

theorem :: YELLOW_4:10

for L being non empty RelStr
for X, Y being Subset of L
for x, y being Element of L st x in X & y in Y holds
x "\/" y in X "\/" Y ;

theorem :: YELLOW_4:11

for L being antisymmetric with_suprema RelStr
for A being Subset of L
for B being non empty Subset of L holds A is_finer_than A "\/" B

proof end;

theorem :: YELLOW_4:12

for L being antisymmetric with_suprema RelStr
for A, B being Subset of L holds A "\/" B is_coarser_than A

proof end;

theorem :: YELLOW_4:13

for L being reflexive antisymmetric with_suprema RelStr
for A being Subset of L holds A c= A "\/" A

proof end;

theorem :: YELLOW_4:14

for L being transitive antisymmetric with_suprema RelStr
for D1, D2, D3 being Subset of L holds (D1 "\/" D2) "\/" D3 = D1 "\/" (D2 "\/" D3)

proof end;

registration

let L be non empty RelStr ;
let D1, D2 be non empty Subset of L;
cluster D1 "\/" D2 -> non empty ;
coherence

proof end;

end;

registration

let L be transitive antisymmetric with_suprema RelStr ;
let D1, D2 be directed Subset of L;
cluster D1 "\/" D2 -> directed ;
coherence

proof end;

end;

registration

let L be transitive antisymmetric with_suprema RelStr ;
let D1, D2 be filtered Subset of L;
cluster D1 "\/" D2 -> filtered ;
coherence

proof end;

end;

registration

let L be with_suprema Poset;
let D1, D2 be upper Subset of L;
cluster D1 "\/" D2 -> upper ;
coherence

proof end;

end;

theorem Th15: :: YELLOW_4:15

for L being non empty RelStr
for Y being Subset of L
for x being Element of L holds {x} "\/" Y = { (x "\/" y) where y is Element of L : y in Y }

proof end;

theorem :: YELLOW_4:16

for L being non empty RelStr
for A, B, C being Subset of L holds A "\/" (B \/ C) = (A "\/" B) \/ (A "\/" C)

proof end;

theorem :: YELLOW_4:17

for L being reflexive antisymmetric with_suprema RelStr
for A, B, C being Subset of L holds A \/ (B "\/" C) c= (A \/ B) "\/" (A \/ C)

proof end;

theorem :: YELLOW_4:18

for L being antisymmetric with_suprema RelStr
for A being upper Subset of L
for B, C being Subset of L holds (A \/ B) "\/" (A \/ C) c= A \/ (B "\/" C)

proof end;

Lm1: now

let L be non empty RelStr ; :: thesis:
let x, y be Element of L; :: thesis:
thus { (a "\/" b) where a, b is Element of L : ( a in {x} & b in {y} ) } = {(x "\/" y)} :: thesis:

proof

thus { (a "\/" b) where a, b is Element of L : ( a in {x} & b in {y} ) } c= {(x "\/" y)} :: according to XBOOLE_0:def 10 :: thesis:

proof

let q be set ; :: according to TARSKI:def 3 :: thesis:
assume q in { (a "\/" b) where a, b is Element of L : ( a in {x} & b in {y} ) } ; :: thesis:
then consider u, v being Element of L such that
A1: q = u "\/" v and
A2: ( u in {x} & v in {y} ) ;
( u = x & v = y ) by A2, TARSKI:def 1;
hence q in {(x "\/" y)} by A1, TARSKI:def 1; :: thesis:

end;

let q be set ; :: according to TARSKI:def 3 :: thesis:
assume q in {(x "\/" y)} ; :: thesis:
then ( q = x "\/" y & x in {x} & y in {y} ) by TARSKI:def 1;
hence q in { (a "\/" b) where a, b is Element of L : ( a in {x} & b in {y} ) } ; :: thesis:

end;

Lm2: now

let L be non empty RelStr ; :: thesis:
let x, y, z be Element of L; :: thesis:
thus { (a "\/" b) where a, b is Element of L : ( a in {x} & b in {y,z} ) } = {(x "\/" y),(x "\/" z)} :: thesis:

proof

thus { (a "\/" b) where a, b is Element of L : ( a in {x} & b in {y,z} ) } c= {(x "\/" y),(x "\/" z)} :: according to XBOOLE_0:def 10 :: thesis:

proof

let q be set ; :: according to TARSKI:def 3 :: thesis:
assume q in { (a "\/" b) where a, b is Element of L : ( a in {x} & b in {y,z} ) } ; :: thesis:
then consider u, v being Element of L such that
A1: q = u "\/" v and
A2: ( u in {x} & v in {y,z} ) ;
( u = x & ( v = y or v = z ) ) by A2, TARSKI:def 1, TARSKI:def 2;
hence q in {(x "\/" y),(x "\/" z)} by A1, TARSKI:def 2; :: thesis:

end;

let q be set ; :: according to TARSKI:def 3 :: thesis:
assume q in {(x "\/" y),(x "\/" z)} ; :: thesis:
then A3: ( q = x "\/" y or q = x "\/" z ) by TARSKI:def 2;
( x in {x} & y in {y,z} & z in {y,z} ) by TARSKI:def 1, TARSKI:def 2;
hence q in { (a "\/" b) where a, b is Element of L : ( a in {x} & b in {y,z} ) } by A3; :: thesis:

end;

theorem :: YELLOW_4:19

for L being non empty RelStr
for x, y being Element of L holds {x} "\/" {y} = {(x "\/" y)} by Lm1;

theorem :: YELLOW_4:20

for L being non empty RelStr
for x, y, z being Element of L holds {x} "\/" {y,z} = {(x "\/" y),(x "\/" z)} by Lm2;

theorem :: YELLOW_4:21

for L being non empty RelStr
for X1, X2, Y1, Y2 being Subset of L st X1 c= Y1 & X2 c= Y2 holds
X1 "\/" X2 c= Y1 "\/" Y2

proof end;

theorem :: YELLOW_4:22

for L being reflexive antisymmetric with_suprema RelStr
for D being Subset of L
for x being Element of L st x is_<=_than D holds
{x} "\/" D = D

proof end;

theorem :: YELLOW_4:23

for L being antisymmetric with_suprema RelStr
for D being Subset of L
for x being Element of L holds x is_<=_than {x} "\/" D

proof end;

theorem :: YELLOW_4:24

for L being with_suprema Poset
for X being Subset of L
for x being Element of L st ex_inf_of {x} "\/" X,L & ex_inf_of X,L holds
x "\/" (inf X) <= inf ({x} "\/" X)

proof end;

theorem Th25: :: YELLOW_4:25

for L being non empty transitive antisymmetric complete RelStr
for A being Subset of L
for B being non empty Subset of L holds A is_<=_than sup (A "\/" B)

proof end;

theorem :: YELLOW_4:26

for L being transitive antisymmetric with_suprema RelStr
for a, b being Element of L
for A, B being Subset of L st a is_<=_than A & b is_<=_than B holds
a "\/" b is_<=_than A "\/" B

proof end;

theorem Th27: :: YELLOW_4:27

for L being transitive antisymmetric with_suprema RelStr
for a, b being Element of L
for A, B being Subset of L st a is_>=_than A & b is_>=_than B holds
a "\/" b is_>=_than A "\/" B

proof end;

theorem :: YELLOW_4:28

for L being non empty complete Poset
for A, B being non empty Subset of L holds sup (A "\/" B) = (sup A) "\/" (sup B)

proof end;

theorem Th29: :: YELLOW_4:29

for L being antisymmetric with_suprema RelStr
for X being Subset of L
for Y being non empty Subset of L holds X c= downarrow (X "\/" Y)

proof end;

theorem :: YELLOW_4:30

for L being with_suprema Poset
for x, y being Element of (InclPoset (Ids L))
for X, Y being Subset of L st x = X & y = Y holds
x "\/" y = downarrow (X "\/" Y)

proof end;

theorem :: YELLOW_4:31

for L being non empty RelStr
for D being Subset of [:L,L:] holds union { X where X is Subset of L : ex x being Element of L st
( X = {x} "\/" (proj2 D) & x in proj1 D ) } = (proj1 D) "\/" (proj2 D)

proof end;

theorem Th32: :: YELLOW_4:32

for L being transitive antisymmetric with_suprema RelStr
for D1, D2 being Subset of L holds downarrow ((downarrow D1) "\/" (downarrow D2)) c= downarrow (D1 "\/" D2)

proof end;

theorem :: YELLOW_4:33

for L being with_suprema Poset
for D1, D2 being Subset of L holds downarrow ((downarrow D1) "\/" (downarrow D2)) = downarrow (D1 "\/" D2)

proof end;

theorem Th34: :: YELLOW_4:34

for L being transitive antisymmetric with_suprema RelStr
for D1, D2 being Subset of L holds uparrow ((uparrow D1) "\/" (uparrow D2)) c= uparrow (D1 "\/" D2)

proof end;

theorem :: YELLOW_4:35

for L being with_suprema Poset
for D1, D2 being Subset of L holds uparrow ((uparrow D1) "\/" (uparrow D2)) = uparrow (D1 "\/" D2)

proof end;

definition

let L be non empty RelStr ;
let D1, D2 be Subset of L;
func D1 "/\" D2 -> Subset of L equals :: YELLOW_4:def 4
{ (x "/\" y) where x, y is Element of L : ( x in D1 & y in D2 ) } ;
coherence

proof end;

end;

:: deftheorem defines "/\" YELLOW_4:def 4 :

definition

let L be antisymmetric with_infima RelStr ;
let D1, D2 be Subset of L;
:: original: "/\"
redefine func D1 "/\" D2 -> Subset of L;
commutativity

proof end;

end;

theorem :: YELLOW_4:36

for L being non empty RelStr
for X being Subset of L holds X "/\" ({} L) = {}

proof end;

theorem :: YELLOW_4:37

for L being non empty RelStr
for X, Y being Subset of L
for x, y being Element of L st x in X & y in Y holds
x "/\" y in X "/\" Y ;

theorem :: YELLOW_4:38

for L being antisymmetric with_infima RelStr
for A being Subset of L
for B being non empty Subset of L holds A is_coarser_than A "/\" B

proof end;

theorem :: YELLOW_4:39

for L being antisymmetric with_infima RelStr
for A, B being Subset of L holds A "/\" B is_finer_than A

proof end;

theorem :: YELLOW_4:40

for L being reflexive antisymmetric with_infima RelStr
for A being Subset of L holds A c= A "/\" A

proof end;

theorem :: YELLOW_4:41

for L being transitive antisymmetric with_infima RelStr
for D1, D2, D3 being Subset of L holds (D1 "/\" D2) "/\" D3 = D1 "/\" (D2 "/\" D3)

proof end;

registration

let L be non empty RelStr ;
let D1, D2 be non empty Subset of L;
cluster D1 "/\" D2 -> non empty ;
coherence

proof end;

end;

registration

let L be transitive antisymmetric with_infima RelStr ;
let D1, D2 be directed Subset of L;
cluster D1 "/\" D2 -> directed ;
coherence

proof end;

end;

registration

let L be transitive antisymmetric with_infima RelStr ;
let D1, D2 be filtered Subset of L;
cluster D1 "/\" D2 -> filtered ;
coherence

proof end;

end;

registration

let L be Semilattice;
let D1, D2 be lower Subset of L;
cluster D1 "/\" D2 -> lower ;
coherence

proof end;

end;

theorem Th42: :: YELLOW_4:42

for L being non empty RelStr
for Y being Subset of L
for x being Element of L holds {x} "/\" Y = { (x "/\" y) where y is Element of L : y in Y }

proof end;

theorem :: YELLOW_4:43

for L being non empty RelStr
for A, B, C being Subset of L holds A "/\" (B \/ C) = (A "/\" B) \/ (A "/\" C)

proof end;

theorem :: YELLOW_4:44

for L being reflexive antisymmetric with_infima RelStr
for A, B, C being Subset of L holds A \/ (B "/\" C) c= (A \/ B) "/\" (A \/ C)

proof end;

theorem :: YELLOW_4:45

for L being antisymmetric with_infima RelStr
for A being lower Subset of L
for B, C being Subset of L holds (A \/ B) "/\" (A \/ C) c= A \/ (B "/\" C)

proof end;

Lm3: now

let L be non empty RelStr ; :: thesis:
let x, y be Element of L; :: thesis:
thus { (a "/\" b) where a, b is Element of L : ( a in {x} & b in {y} ) } = {(x "/\" y)} :: thesis:

proof

thus { (a "/\" b) where a, b is Element of L : ( a in {x} & b in {y} ) } c= {(x "/\" y)} :: according to XBOOLE_0:def 10 :: thesis:

proof

let q be set ; :: according to TARSKI:def 3 :: thesis:
assume q in { (a "/\" b) where a, b is Element of L : ( a in {x} & b in {y} ) } ; :: thesis:
then consider u, v being Element of L such that
A1: q = u "/\" v and
A2: ( u in {x} & v in {y} ) ;
( u = x & v = y ) by A2, TARSKI:def 1;
hence q in {(x "/\" y)} by A1, TARSKI:def 1; :: thesis:

end;

let q be set ; :: according to TARSKI:def 3 :: thesis:
assume q in {(x "/\" y)} ; :: thesis:
then ( q = x "/\" y & x in {x} & y in {y} ) by TARSKI:def 1;
hence q in { (a "/\" b) where a, b is Element of L : ( a in {x} & b in {y} ) } ; :: thesis:

end;

Lm4: now

let L be non empty RelStr ; :: thesis:
let x, y, z be Element of L; :: thesis:
thus { (a "/\" b) where a, b is Element of L : ( a in {x} & b in {y,z} ) } = {(x "/\" y),(x "/\" z)} :: thesis:

proof

thus { (a "/\" b) where a, b is Element of L : ( a in {x} & b in {y,z} ) } c= {(x "/\" y),(x "/\" z)} :: according to XBOOLE_0:def 10 :: thesis:

proof

let q be set ; :: according to TARSKI:def 3 :: thesis:
assume q in { (a "/\" b) where a, b is Element of L : ( a in {x} & b in {y,z} ) } ; :: thesis:
then consider u, v being Element of L such that
A1: q = u "/\" v and
A2: ( u in {x} & v in {y,z} ) ;
( u = x & ( v = y or v = z ) ) by A2, TARSKI:def 1, TARSKI:def 2;
hence q in {(x "/\" y),(x "/\" z)} by A1, TARSKI:def 2; :: thesis:

end;

let q be set ; :: according to TARSKI:def 3 :: thesis:
assume q in {(x "/\" y),(x "/\" z)} ; :: thesis:
then A3: ( q = x "/\" y or q = x "/\" z ) by TARSKI:def 2;
( x in {x} & y in {y,z} & z in {y,z} ) by TARSKI:def 1, TARSKI:def 2;
hence q in { (a "/\" b) where a, b is Element of L : ( a in {x} & b in {y,z} ) } by A3; :: thesis:

end;

theorem :: YELLOW_4:46

for L being non empty RelStr
for x, y being Element of L holds {x} "/\" {y} = {(x "/\" y)} by Lm3;

theorem :: YELLOW_4:47

for L being non empty RelStr
for x, y, z being Element of L holds {x} "/\" {y,z} = {(x "/\" y),(x "/\" z)} by Lm4;

theorem :: YELLOW_4:48

for L being non empty RelStr
for X1, X2, Y1, Y2 being Subset of L st X1 c= Y1 & X2 c= Y2 holds
X1 "/\" X2 c= Y1 "/\" Y2

proof end;

theorem Th49: :: YELLOW_4:49

for L being reflexive antisymmetric with_infima RelStr
for A, B being Subset of L holds A /\ B c= A "/\" B

proof end;

theorem Th50: :: YELLOW_4:50

for L being reflexive antisymmetric with_infima RelStr
for A, B being lower Subset of L holds A "/\" B = A /\ B

proof end;

theorem :: YELLOW_4:51

for L being reflexive antisymmetric with_infima RelStr
for D being Subset of L
for x being Element of L st x is_>=_than D holds
{x} "/\" D = D

proof end;

theorem :: YELLOW_4:52

for L being antisymmetric with_infima RelStr
for D being Subset of L
for x being Element of L holds {x} "/\" D is_<=_than x

proof end;

theorem :: YELLOW_4:53

for L being Semilattice
for X being Subset of L
for x being Element of L st ex_sup_of {x} "/\" X,L & ex_sup_of X,L holds
sup ({x} "/\" X) <= x "/\" (sup X)

proof end;

theorem Th54: :: YELLOW_4:54

for L being non empty transitive antisymmetric complete RelStr
for A being Subset of L
for B being non empty Subset of L holds A is_>=_than inf (A "/\" B)

proof end;

theorem Th55: :: YELLOW_4:55

for L being transitive antisymmetric with_infima RelStr
for a, b being Element of L
for A, B being Subset of L st a is_<=_than A & b is_<=_than B holds
a "/\" b is_<=_than A "/\" B

proof end;

theorem :: YELLOW_4:56

for L being transitive antisymmetric with_infima RelStr
for a, b being Element of L
for A, B being Subset of L st a is_>=_than A & b is_>=_than B holds
a "/\" b is_>=_than A "/\" B

proof end;

theorem :: YELLOW_4:57

for L being non empty complete Poset
for A, B being non empty Subset of L holds inf (A "/\" B) = (inf A) "/\" (inf B)

proof end;

theorem :: YELLOW_4:58

for L being Semilattice
for x, y being Element of (InclPoset (Ids L))
for x1, y1 being Subset of L st x = x1 & y = y1 holds
x "/\" y = x1 "/\" y1

proof end;

theorem :: YELLOW_4:59

for L being antisymmetric with_infima RelStr
for X being Subset of L
for Y being non empty Subset of L holds X c= uparrow (X "/\" Y)

proof end;

theorem :: YELLOW_4:60

for L being non empty RelStr
for D being Subset of [:L,L:] holds union { X where X is Subset of L : ex x being Element of L st
( X = {x} "/\" (proj2 D) & x in proj1 D ) } = (proj1 D) "/\" (proj2 D)

proof end;

theorem Th61: :: YELLOW_4:61

for L being transitive antisymmetric with_infima RelStr
for D1, D2 being Subset of L holds downarrow ((downarrow D1) "/\" (downarrow D2)) c= downarrow (D1 "/\" D2)

proof end;

theorem :: YELLOW_4:62

for L being Semilattice
for D1, D2 being Subset of L holds downarrow ((downarrow D1) "/\" (downarrow D2)) = downarrow (D1 "/\" D2)

proof end;

theorem Th63: :: YELLOW_4:63

for L being transitive antisymmetric with_infima RelStr
for D1, D2 being Subset of L holds uparrow ((uparrow D1) "/\" (uparrow D2)) c= uparrow (D1 "/\" D2)

proof end;

theorem :: YELLOW_4:64

for L being Semilattice
for D1, D2 being Subset of L holds uparrow ((uparrow D1) "/\" (uparrow D2)) = uparrow (D1 "/\" D2)

proof end;