:: WAYBEL25 semantic presentation
theorem Th1: :: WAYBEL25:1
theorem Th2: :: WAYBEL25:2
theorem Th3: :: WAYBEL25:3
theorem Th4: :: WAYBEL25:4
theorem Th5: :: WAYBEL25:5
theorem Th6: :: WAYBEL25:6
theorem Th7: :: WAYBEL25:7
:: deftheorem Def1 defines is_Retract_of WAYBEL25:def 1 :
theorem Th8: :: WAYBEL25:8
theorem Th9: :: WAYBEL25:9
theorem Th10: :: WAYBEL25:10
theorem Th11: :: WAYBEL25:11
theorem Th12: :: WAYBEL25:12
definition
let T be
TopStruct ;
func Omega c1 -> strict TopRelStr means :
Def2:
:: WAYBEL25:def 2
(
TopStruct(# the
carrier of
it,the
topology of
it #)
= TopStruct(# the
carrier of
T,the
topology of
T #) & ( for
x,
y being
Element of
it holds
(
x <= y iff ex
Y being
Subset of
T st
(
Y = {y} &
x in Cl Y ) ) ) );
existence
ex b1 being strict TopRelStr st
( TopStruct(# the carrier of b1,the topology of b1 #) = TopStruct(# the carrier of T,the topology of T #) & ( for x, y being Element of b1 holds
( x <= y iff ex Y being Subset of T st
( Y = {y} & x in Cl Y ) ) ) )
uniqueness
for b1, b2 being strict TopRelStr st TopStruct(# the carrier of b1,the topology of b1 #) = TopStruct(# the carrier of T,the topology of T #) & ( for x, y being Element of b1 holds
( x <= y iff ex Y being Subset of T st
( Y = {y} & x in Cl Y ) ) ) & TopStruct(# the carrier of b2,the topology of b2 #) = TopStruct(# the carrier of T,the topology of T #) & ( for x, y being Element of b2 holds
( x <= y iff ex Y being Subset of T st
( Y = {y} & x in Cl Y ) ) ) holds
b1 = b2
end;
:: deftheorem Def2 defines Omega WAYBEL25:def 2 :
Lemma142:
for T being TopStruct holds the carrier of T = the carrier of (Omega T)
then Lemma143:
for T being TopStruct
for a being set holds
( a is Subset of T iff a is Subset of (Omega T) )
;
Lemma146:
for T being TopStruct
for x, y being Element of T
for X, Y being Subset of T st X = {x} & Y = {y} holds
( x in Cl Y iff Cl X c= Cl Y )
theorem Th13: :: WAYBEL25:13
theorem Th14: :: WAYBEL25:14
theorem Th15: :: WAYBEL25:15
theorem Th16: :: WAYBEL25:16
theorem Th17: :: WAYBEL25:17
theorem Th18: :: WAYBEL25:18
theorem Th19: :: WAYBEL25:19
Lemma179:
for S, T being non empty RelStr
for f being Function of S,S
for g being Function of T,T st RelStr(# the carrier of S,the InternalRel of S #) = RelStr(# the carrier of T,the InternalRel of T #) & f = g & f is projection holds
g is projection
theorem Th20: :: WAYBEL25:20
theorem Th21: :: WAYBEL25:21
theorem Th22: :: WAYBEL25:22
theorem Th23: :: WAYBEL25:23
E186:
now
let X be non
empty TopSpace,
Y be non
empty TopSpace;
let N be
net of
ContMaps X,
(Omega Y);
E42:
the
mapping of
N in Funcs the
carrier of
N,the
carrier of
(ContMaps X,(Omega Y))
by FUNCT_2:11;
E43:
the
carrier of
Y = the
carrier of
(Omega Y)
by ;
the
carrier of
(ContMaps X,(Omega Y)) c= Funcs the
carrier of
X,the
carrier of
Y
then
Funcs the
carrier of
N,the
carrier of
(ContMaps X,(Omega Y)) c= Funcs the
carrier of
N,
(Funcs the carrier of X,the carrier of Y)
by FUNCT_5:63;
hence
the
mapping of
N in Funcs the
carrier of
N,
(Funcs the carrier of X,the carrier of Y)
by ;
end;
definition
let I be non
empty set ;
let S be non
empty RelStr ,
T be non
empty RelStr ;
let N be
net of
T;
let i be
Element of
I;
assume E42:
the
carrier of
T c= the
carrier of
(S |^ I)
;
func commute c4,
c5,
c2 -> strict net of
a2 means :
Def3:
:: WAYBEL25:def 3
(
RelStr(# the
carrier of
it,the
InternalRel of
it #)
= RelStr(# the
carrier of
N,the
InternalRel of
N #) & the
mapping of
it = (commute the mapping of N) . i );
existence
ex b1 being strict net of S st
( RelStr(# the carrier of b1,the InternalRel of b1 #) = RelStr(# the carrier of N,the InternalRel of N #) & the mapping of b1 = (commute the mapping of N) . i )
uniqueness
for b1, b2 being strict net of S st RelStr(# the carrier of b1,the InternalRel of b1 #) = RelStr(# the carrier of N,the InternalRel of N #) & the mapping of b1 = (commute the mapping of N) . i & RelStr(# the carrier of b2,the InternalRel of b2 #) = RelStr(# the carrier of N,the InternalRel of N #) & the mapping of b2 = (commute the mapping of N) . i holds
b1 = b2
;
end;
:: deftheorem Def3 defines commute WAYBEL25:def 3 :
theorem Th24: :: WAYBEL25:24
theorem Th25: :: WAYBEL25:25
E195:
now
let X be non
empty TopSpace,
Y be non
empty TopSpace;
let N be
net of
ContMaps X,
(Omega Y);
let x be
Point of
X;
ContMaps X,
(Omega Y) is
SubRelStr of
(Omega Y) |^ the
carrier of
X
by WAYBEL24:def 3;
then
the
carrier of
(ContMaps X,(Omega Y)) c= the
carrier of
((Omega Y) |^ the carrier of X)
by YELLOW_0:def 13;
then E42:
RelStr(# the
carrier of
N,the
InternalRel of
N #)
= RelStr(# the
carrier of
(commute N,x,(Omega Y)),the
InternalRel of
(commute N,x,(Omega Y)) #)
by ;
thus dom the
mapping of
N =
the
carrier of
N
by FUNCT_2:def 1
.=
dom the
mapping of
(commute N,x,(Omega Y))
by , FUNCT_2:def 1
;
end;
theorem Th26: :: WAYBEL25:26
:: deftheorem Def4 defines monotone-convergence WAYBEL25:def 4 :
theorem Th27: :: WAYBEL25:27
theorem Th28: :: WAYBEL25:28
theorem Th29: :: WAYBEL25:29
theorem Th30: :: WAYBEL25:30
theorem Th31: :: WAYBEL25:31
theorem Th32: :: WAYBEL25:32
theorem Th33: :: WAYBEL25:33
theorem Th34: :: WAYBEL25:34
theorem Th35: :: WAYBEL25:35
theorem Th36: :: WAYBEL25:36
theorem Th37: :: WAYBEL25:37
theorem Th38: :: WAYBEL25:38
theorem Th39: :: WAYBEL25:39
theorem Th40: :: WAYBEL25:40
theorem Th41: :: WAYBEL25:41
theorem Th42: :: WAYBEL25:42
theorem Th43: :: WAYBEL25:43