:: WAYBEL_5 semantic presentation
Lemma1:
for b1 being continuous Semilattice
for b2 being Element of b1 holds
( waybelow b2 is Ideal of b1 & b2 <= sup (waybelow b2) & ( for b3 being Ideal of b1 holds
( b2 <= sup b3 implies waybelow b2 c= b3 ) ) )
Lemma2:
for b1 being up-complete Semilattice holds
( ( for b2 being Element of b1 holds
( waybelow b2 is Ideal of b1 & b2 <= sup (waybelow b2) & ( for b3 being Ideal of b1 holds
( b2 <= sup b3 implies waybelow b2 c= b3 ) ) ) ) implies b1 is continuous )
theorem Th1: :: WAYBEL_5:1
Lemma3:
for b1 being up-complete Semilattice holds
( b1 is continuous implies for b2 being Element of b1 holds
ex b3 being Ideal of b1 st
( b2 <= sup b3 & ( for b4 being Ideal of b1 holds
( b2 <= sup b4 implies b3 c= b4 ) ) ) )
Lemma4:
for b1 being up-complete Semilattice holds
( ( for b2 being Element of b1 holds
ex b3 being Ideal of b1 st
( b2 <= sup b3 & ( for b4 being Ideal of b1 holds
( b2 <= sup b4 implies b3 c= b4 ) ) ) ) implies b1 is continuous )
theorem Th2: :: WAYBEL_5:2
theorem Th3: :: WAYBEL_5:3
theorem Th4: :: WAYBEL_5:4
theorem Th5: :: WAYBEL_5:5
theorem Th6: :: WAYBEL_5:6
theorem Th7: :: WAYBEL_5:7
theorem Th8: :: WAYBEL_5:8
theorem Th9: :: WAYBEL_5:9
theorem Th10: :: WAYBEL_5:10
Lemma10:
for b1, b2 being set
for b3 being ManySortedSet of b1
for b4 being DoubleIndexedSet of b3,b2
for b5 being Function holds
( b5 in dom (Frege b4) implies for b6 being set holds
( b6 in b1 implies ( ((Frege b4) . b5) . b6 = (b4 . b6) . (b5 . b6) & (b4 . b6) . (b5 . b6) in rng ((Frege b4) . b5) ) ) )
Lemma11:
for b1 being set
for b2 being ManySortedSet of b1
for b3 being non empty set
for b4 being DoubleIndexedSet of b2,b3
for b5 being Function holds
( b5 in dom (Frege b4) implies for b6 being set holds
( b6 in b1 implies b5 . b6 in b2 . b6 ) )
definition
let c
1 be non
empty RelStr ;
let c
2 be
Function-yielding Function;
func \// c
2,c
1 -> Function of
dom a
2,the
carrier of a
1 means :
Def1:
:: WAYBEL_5:def 1
for b
1 being
set holds
( b
1 in dom a
2 implies a
3 . b
1 = \\/ (a2 . b1),a
1 );
existence
ex b1 being Function of dom c2,the carrier of c1 st
for b2 being set holds
( b2 in dom c2 implies b1 . b2 = \\/ (c2 . b2),c1 )
uniqueness
for b1, b2 being Function of dom c2,the carrier of c1 holds
( ( for b3 being set holds
( b3 in dom c2 implies b1 . b3 = \\/ (c2 . b3),c1 ) ) & ( for b3 being set holds
( b3 in dom c2 implies b2 . b3 = \\/ (c2 . b3),c1 ) ) implies b1 = b2 )
func /\\ c
2,c
1 -> Function of
dom a
2,the
carrier of a
1 means :
Def2:
:: WAYBEL_5:def 2
for b
1 being
set holds
( b
1 in dom a
2 implies a
3 . b
1 = //\ (a2 . b1),a
1 );
existence
ex b1 being Function of dom c2,the carrier of c1 st
for b2 being set holds
( b2 in dom c2 implies b1 . b2 = //\ (c2 . b2),c1 )
uniqueness
for b1, b2 being Function of dom c2,the carrier of c1 holds
( ( for b3 being set holds
( b3 in dom c2 implies b1 . b3 = //\ (c2 . b3),c1 ) ) & ( for b3 being set holds
( b3 in dom c2 implies b2 . b3 = //\ (c2 . b3),c1 ) ) implies b1 = b2 )
end;
:: deftheorem Def1 defines \// WAYBEL_5:def 1 :
:: deftheorem Def2 defines /\\ WAYBEL_5:def 2 :
theorem Th11: :: WAYBEL_5:11
theorem Th12: :: WAYBEL_5:12
theorem Th13: :: WAYBEL_5:13
theorem Th14: :: WAYBEL_5:14
Lemma18:
for b1 being complete LATTICE
for b2 being non empty set
for b3 being V7 ManySortedSet of b2
for b4 being DoubleIndexedSet of b3,b1
for b5 being set holds
( b5 is Element of product (doms b4) iff b5 in dom (Frege b4) )
theorem Th15: :: WAYBEL_5:15
theorem Th16: :: WAYBEL_5:16
theorem Th17: :: WAYBEL_5:17
Lemma22:
for b1 being complete LATTICE holds
( b1 is continuous implies for b2 being non empty set
for b3 being V7 ManySortedSet of b2
for b4 being DoubleIndexedSet of b3,b1 holds
( ( for b5 being Element of b2 holds rng (b4 . b5) is directed ) implies Inf = Sup ) )
theorem Th18: :: WAYBEL_5:18
Lemma24:
for b1 being complete LATTICE holds
( ( for b2 being non empty set
for b3 being V7 ManySortedSet of b2
for b4 being DoubleIndexedSet of b3,b1 holds
( ( for b5 being Element of b2 holds rng (b4 . b5) is directed ) implies Inf = Sup ) ) implies b1 is continuous )
theorem Th19: :: WAYBEL_5:19
theorem Th20: :: WAYBEL_5:20
Lemma26:
for b1 being complete LATTICE holds
( ( for b2, b3 being non empty set
for b4 being Function of [:b2,b3:],the carrier of b1 holds
( ( for b5 being Element of b2 holds rng ((curry b4) . b5) is directed ) implies Inf = Sup ) ) implies b1 is continuous )
theorem Th21: :: WAYBEL_5:21
Lemma27:
for b1, b2 being non empty set
for b3 being Function holds
( b3 in Funcs b1,(Fin b2) implies for b4 being Element of b1 holds
b3 . b4 is finite Subset of b2 )
Lemma28:
for b1 being complete LATTICE
for b2, b3, b4 being non empty set
for b5 being Element of b2
for b6 being Function of [:b2,b3:],b4
for b7 being V7 ManySortedSet of b2 holds
( b7 in Funcs b2,(Fin b3) implies for b8 being DoubleIndexedSet of b7,b1 holds
( ( for b9 being Element of b2
for b10 being set holds
( b10 in b7 . b9 implies (b8 . b9) . b10 = b6 . [b9,b10] ) ) implies rng (b8 . b5) is finite Subset of (rng ((curry b6) . b5)) ) )
theorem Th22: :: WAYBEL_5:22
Lemma30:
for b1 being complete LATTICE holds
( b1 is continuous implies for b2, b3 being non empty set
for b4 being Function of [:b2,b3:],the carrier of b1
for b5 being Subset of b1 holds
( b5 = { b6 where B is Element of b1 : ex b1 being V7 ManySortedSet of b2 st
( b7 in Funcs b2,(Fin b3) & ex b2 being DoubleIndexedSet of b7,b1 st
( ( for b3 being Element of b2
for b4 being set holds
( b10 in b7 . b9 implies (b8 . b9) . b10 = b4 . [b9,b10] ) ) & b6 = Inf ) ) } implies Inf = sup b5 ) )
Lemma31:
for b1 being complete LATTICE holds
( ( for b2, b3 being non empty set
for b4 being Function of [:b2,b3:],the carrier of b1
for b5 being Subset of b1 holds
( b5 = { b6 where B is Element of b1 : ex b1 being V7 ManySortedSet of b2 st
( b7 in Funcs b2,(Fin b3) & ex b2 being DoubleIndexedSet of b7,b1 st
( ( for b3 being Element of b2
for b4 being set holds
( b10 in b7 . b9 implies (b8 . b9) . b10 = b4 . [b9,b10] ) ) & b6 = Inf ) ) } implies Inf = sup b5 ) ) implies b1 is continuous )
theorem Th23: :: WAYBEL_5:23
:: deftheorem Def3 defines completely-distributive WAYBEL_5:def 3 :
theorem Th24: :: WAYBEL_5:24
theorem Th25: :: WAYBEL_5:25
Lemma35:
for b1 being completely-distributive LATTICE
for b2 being non empty Subset of b1
for b3 being Element of b1 holds b3 "/\" (sup b2) = "\/" { (b3 "/\" b4) where B is Element of b1 : b4 in b2 } ,b1
theorem Th26: :: WAYBEL_5:26
theorem Th27: :: WAYBEL_5:27
theorem Th28: :: WAYBEL_5:28
theorem Th29: :: WAYBEL_5:29
theorem Th30: :: WAYBEL_5:30
theorem Th31: :: WAYBEL_5:31
theorem Th32: :: WAYBEL_5:32
theorem Th33: :: WAYBEL_5:33