:: TOPGEN_5 semantic presentation
theorem Th1: :: TOPGEN_5:1
theorem Th2: :: TOPGEN_5:2
theorem Th3: :: TOPGEN_5:3
theorem Th4: :: TOPGEN_5:4
theorem Th5: :: TOPGEN_5:5
theorem Th6: :: TOPGEN_5:6
theorem Th7: :: TOPGEN_5:7
theorem Th8: :: TOPGEN_5:8
theorem Th9: :: TOPGEN_5:9
theorem Th10: :: TOPGEN_5:10
theorem Th11: :: TOPGEN_5:11
theorem Th12: :: TOPGEN_5:12
scheme :: TOPGEN_5:sch 1
s1{ P
1[
set ], F
1()
-> non
empty set , F
2()
-> non
empty set , F
3()
-> non
empty set , F
4(
set )
-> set , F
5(
set )
-> set } :
ex b
1 being
Function of F
3(),F
2() st
for b
2 being
Element of F
1() holds
( b
2 in F
3() implies ( ( P
1[b
2] implies b
1 . b
2 = F
4(b
2) ) & ( not P
1[b
2] implies b
1 . b
2 = F
5(b
2) ) ) )
provided
E11:
F
3()
c= F
1()
and
E12:
for b
1 being
Element of F
1() holds
( b
1 in F
3() implies ( ( P
1[b
1] implies F
4(b
1)
in F
2() ) & ( not P
1[b
1] implies F
5(b
1)
in F
2() ) ) )
scheme :: TOPGEN_5:sch 2
s2{ P
1[
set ], P
2[
set ], F
1()
-> non
empty set , F
2()
-> non
empty set , F
3()
-> non
empty set , F
4(
set )
-> set , F
5(
set )
-> set , F
6(
set )
-> set } :
ex b
1 being
Function of F
3(),F
2() st
for b
2 being
Element of F
1() holds
( b
2 in F
3() implies ( ( P
1[b
2] implies b
1 . b
2 = F
4(b
2) ) & ( not P
1[b
2] & P
2[b
2] implies b
1 . b
2 = F
5(b
2) ) & ( not P
1[b
2] & not P
2[b
2] implies b
1 . b
2 = F
6(b
2) ) ) )
provided
E11:
F
3()
c= F
1()
and
E12:
for b
1 being
Element of F
1() holds
( b
1 in F
3() implies ( ( P
1[b
1] implies F
4(b
1)
in F
2() ) & ( not P
1[b
1] & P
2[b
1] implies F
5(b
1)
in F
2() ) & ( not P
1[b
1] & not P
2[b
1] implies F
6(b
1)
in F
2() ) ) )
theorem Th13: :: TOPGEN_5:13
theorem Th14: :: TOPGEN_5:14
theorem Th15: :: TOPGEN_5:15
theorem Th16: :: TOPGEN_5:16
theorem Th17: :: TOPGEN_5:17
:: deftheorem Def1 defines y=0-line TOPGEN_5:def 1 :
:: deftheorem Def2 defines y>=0-plane TOPGEN_5:def 2 :
theorem Th18: :: TOPGEN_5:18
theorem Th19: :: TOPGEN_5:19
theorem Th20: :: TOPGEN_5:20
theorem Th21: :: TOPGEN_5:21
theorem Th22: :: TOPGEN_5:22
theorem Th23: :: TOPGEN_5:23
theorem Th24: :: TOPGEN_5:24
theorem Th25: :: TOPGEN_5:25
theorem Th26: :: TOPGEN_5:26
theorem Th27: :: TOPGEN_5:27
theorem Th28: :: TOPGEN_5:28
definition
func Niemytzki-plane -> non
empty strict TopSpace means :
Def3:
:: TOPGEN_5:def 3
( the
carrier of a
1 = y>=0-plane & ex b
1 being
Neighborhood_System of a
1 st
( ( for b
2 being
Element of
REAL holds b
1 . |[b2,0]| = { ((Ball |[b2,b3]|,b3) \/ {|[b2,0]|}) where B is Element of REAL : b3 > 0 } ) & ( for b
2, b
3 being
Element of
REAL holds
( b
3 > 0 implies b
1 . |[b2,b3]| = { ((Ball |[b2,b3]|,b4) /\ y>=0-plane ) where B is Element of REAL : b4 > 0 } ) ) ) );
existence
ex b1 being non empty strict TopSpace st
( the carrier of b1 = y>=0-plane & ex b2 being Neighborhood_System of b1 st
( ( for b3 being Element of REAL holds b2 . |[b3,0]| = { ((Ball |[b3,b4]|,b4) \/ {|[b3,0]|}) where B is Element of REAL : b4 > 0 } ) & ( for b3, b4 being Element of REAL holds
( b4 > 0 implies b2 . |[b3,b4]| = { ((Ball |[b3,b4]|,b5) /\ y>=0-plane ) where B is Element of REAL : b5 > 0 } ) ) ) )
uniqueness
for b1, b2 being non empty strict TopSpace holds
( the carrier of b1 = y>=0-plane & ex b3 being Neighborhood_System of b1 st
( ( for b4 being Element of REAL holds b3 . |[b4,0]| = { ((Ball |[b4,b5]|,b5) \/ {|[b4,0]|}) where B is Element of REAL : b5 > 0 } ) & ( for b4, b5 being Element of REAL holds
( b5 > 0 implies b3 . |[b4,b5]| = { ((Ball |[b4,b5]|,b6) /\ y>=0-plane ) where B is Element of REAL : b6 > 0 } ) ) ) & the carrier of b2 = y>=0-plane & ex b3 being Neighborhood_System of b2 st
( ( for b4 being Element of REAL holds b3 . |[b4,0]| = { ((Ball |[b4,b5]|,b5) \/ {|[b4,0]|}) where B is Element of REAL : b5 > 0 } ) & ( for b4, b5 being Element of REAL holds
( b5 > 0 implies b3 . |[b4,b5]| = { ((Ball |[b4,b5]|,b6) /\ y>=0-plane ) where B is Element of REAL : b6 > 0 } ) ) ) implies b1 = b2 )
end;
:: deftheorem Def3 defines Niemytzki-plane TOPGEN_5:def 3 :
theorem Th29: :: TOPGEN_5:29
Lemma27:
the carrier of Niemytzki-plane = y>=0-plane
by Def3;
theorem Th30: :: TOPGEN_5:30
theorem Th31: :: TOPGEN_5:31
theorem Th32: :: TOPGEN_5:32
theorem Th33: :: TOPGEN_5:33
theorem Th34: :: TOPGEN_5:34
theorem Th35: :: TOPGEN_5:35
theorem Th36: :: TOPGEN_5:36
theorem Th37: :: TOPGEN_5:37
theorem Th38: :: TOPGEN_5:38
theorem Th39: :: TOPGEN_5:39
theorem Th40: :: TOPGEN_5:40
theorem Th41: :: TOPGEN_5:41
theorem Th42: :: TOPGEN_5:42
theorem Th43: :: TOPGEN_5:43
theorem Th44: :: TOPGEN_5:44
theorem Th45: :: TOPGEN_5:45
theorem Th46: :: TOPGEN_5:46
theorem Th47: :: TOPGEN_5:47
theorem Th48: :: TOPGEN_5:48
theorem Th49: :: TOPGEN_5:49
theorem Th50: :: TOPGEN_5:50
theorem Th51: :: TOPGEN_5:51
:: deftheorem Def4 defines Tychonoff TOPGEN_5:def 4 :
theorem Th52: :: TOPGEN_5:52
theorem Th53: :: TOPGEN_5:53
theorem Th54: :: TOPGEN_5:54
theorem Th55: :: TOPGEN_5:55
theorem Th56: :: TOPGEN_5:56
theorem Th57: :: TOPGEN_5:57
theorem Th58: :: TOPGEN_5:58
theorem Th59: :: TOPGEN_5:59
theorem Th60: :: TOPGEN_5:60
theorem Th61: :: TOPGEN_5:61
theorem Th62: :: TOPGEN_5:62
theorem Th63: :: TOPGEN_5:63
definition
let c
1 be
real number ;
let c
2 be
positive real number ;
func + c
1,c
2 -> Function of
Niemytzki-plane ,
I[01] means :
Def5:
:: TOPGEN_5:def 5
( a
3 . |[a1,0]| = 0 & ( for b
1 being
real number for b
2 being non
negative real number holds
( ( not ( not b
1 <> a
1 & not b
2 <> 0 ) & not
|[b1,b2]| in Ball |[a1,a2]|,a
2 implies a
3 . |[b1,b2]| = 1 ) & (
|[b1,b2]| in Ball |[a1,a2]|,a
2 implies a
3 . |[b1,b2]| = (|.(|[a1,0]| - |[b1,b2]|).| ^2 ) / ((2 * a2) * b2) ) ) ) );
existence
ex b1 being Function of Niemytzki-plane ,I[01] st
( b1 . |[c1,0]| = 0 & ( for b2 being real number
for b3 being non negative real number holds
( ( not ( not b2 <> c1 & not b3 <> 0 ) & not |[b2,b3]| in Ball |[c1,c2]|,c2 implies b1 . |[b2,b3]| = 1 ) & ( |[b2,b3]| in Ball |[c1,c2]|,c2 implies b1 . |[b2,b3]| = (|.(|[c1,0]| - |[b2,b3]|).| ^2 ) / ((2 * c2) * b3) ) ) ) )
uniqueness
for b1, b2 being Function of Niemytzki-plane ,I[01] holds
( b1 . |[c1,0]| = 0 & ( for b3 being real number
for b4 being non negative real number holds
( ( not ( not b3 <> c1 & not b4 <> 0 ) & not |[b3,b4]| in Ball |[c1,c2]|,c2 implies b1 . |[b3,b4]| = 1 ) & ( |[b3,b4]| in Ball |[c1,c2]|,c2 implies b1 . |[b3,b4]| = (|.(|[c1,0]| - |[b3,b4]|).| ^2 ) / ((2 * c2) * b4) ) ) ) & b2 . |[c1,0]| = 0 & ( for b3 being real number
for b4 being non negative real number holds
( ( not ( not b3 <> c1 & not b4 <> 0 ) & not |[b3,b4]| in Ball |[c1,c2]|,c2 implies b2 . |[b3,b4]| = 1 ) & ( |[b3,b4]| in Ball |[c1,c2]|,c2 implies b2 . |[b3,b4]| = (|.(|[c1,0]| - |[b3,b4]|).| ^2 ) / ((2 * c2) * b4) ) ) ) implies b1 = b2 )
end;
:: deftheorem Def5 defines + TOPGEN_5:def 5 :
for b
1 being
real number for b
2 being
positive real number for b
3 being
Function of
Niemytzki-plane ,
I[01] holds
( b
3 = + b
1,b
2 iff ( b
3 . |[b1,0]| = 0 & ( for b
4 being
real number for b
5 being non
negative real number holds
( ( not ( not b
4 <> b
1 & not b
5 <> 0 ) & not
|[b4,b5]| in Ball |[b1,b2]|,b
2 implies b
3 . |[b4,b5]| = 1 ) & (
|[b4,b5]| in Ball |[b1,b2]|,b
2 implies b
3 . |[b4,b5]| = (|.(|[b1,0]| - |[b4,b5]|).| ^2 ) / ((2 * b2) * b5) ) ) ) ) );
theorem Th64: :: TOPGEN_5:64
theorem Th65: :: TOPGEN_5:65
theorem Th66: :: TOPGEN_5:66
theorem Th67: :: TOPGEN_5:67
theorem Th68: :: TOPGEN_5:68
theorem Th69: :: TOPGEN_5:69
theorem Th70: :: TOPGEN_5:70
theorem Th71: :: TOPGEN_5:71
theorem Th72: :: TOPGEN_5:72
theorem Th73: :: TOPGEN_5:73
theorem Th74: :: TOPGEN_5:74
theorem Th75: :: TOPGEN_5:75
theorem Th76: :: TOPGEN_5:76
theorem Th77: :: TOPGEN_5:77
theorem Th78: :: TOPGEN_5:78
theorem Th79: :: TOPGEN_5:79
theorem Th80: :: TOPGEN_5:80
for b
1 being
Subset of
Niemytzki-plane for b
2 being
Element of
REAL for b
3 being
positive real number holds
not ( b
1 = (Ball |[b2,b3]|,b3) \/ {|[b2,0]|} & ( for b
4 being
continuous Function of
Niemytzki-plane ,
I[01] holds
not ( b
4 . |[b2,0]| = 0 & ( for b
5, b
6 being
real number holds
( (
|[b5,b6]| in b
1 ` implies b
4 . |[b5,b6]| = 1 ) & (
|[b5,b6]| in b
1 \ {|[b2,0]|} implies b
4 . |[b5,b6]| = (|.(|[b2,0]| - |[b5,b6]|).| ^2 ) / ((2 * b3) * b6) ) ) ) ) ) )
definition
let c
1, c
2 be
real number ;
let c
3 be
positive real number ;
func + c
1,c
2,c
3 -> Function of
Niemytzki-plane ,
I[01] means :
Def6:
:: TOPGEN_5:def 6
for b
1 being
real number for b
2 being non
negative real number holds
( ( not
|[b1,b2]| in Ball |[a1,a2]|,a
3 implies a
4 . |[b1,b2]| = 1 ) & (
|[b1,b2]| in Ball |[a1,a2]|,a
3 implies a
4 . |[b1,b2]| = |.(|[a1,a2]| - |[b1,b2]|).| / a
3 ) );
existence
ex b1 being Function of Niemytzki-plane ,I[01] st
for b2 being real number
for b3 being non negative real number holds
( ( not |[b2,b3]| in Ball |[c1,c2]|,c3 implies b1 . |[b2,b3]| = 1 ) & ( |[b2,b3]| in Ball |[c1,c2]|,c3 implies b1 . |[b2,b3]| = |.(|[c1,c2]| - |[b2,b3]|).| / c3 ) )
uniqueness
for b1, b2 being Function of Niemytzki-plane ,I[01] holds
( ( for b3 being real number
for b4 being non negative real number holds
( ( not |[b3,b4]| in Ball |[c1,c2]|,c3 implies b1 . |[b3,b4]| = 1 ) & ( |[b3,b4]| in Ball |[c1,c2]|,c3 implies b1 . |[b3,b4]| = |.(|[c1,c2]| - |[b3,b4]|).| / c3 ) ) ) & ( for b3 being real number
for b4 being non negative real number holds
( ( not |[b3,b4]| in Ball |[c1,c2]|,c3 implies b2 . |[b3,b4]| = 1 ) & ( |[b3,b4]| in Ball |[c1,c2]|,c3 implies b2 . |[b3,b4]| = |.(|[c1,c2]| - |[b3,b4]|).| / c3 ) ) ) implies b1 = b2 )
end;
:: deftheorem Def6 defines + TOPGEN_5:def 6 :
for b
1, b
2 being
real number for b
3 being
positive real number for b
4 being
Function of
Niemytzki-plane ,
I[01] holds
( b
4 = + b
1,b
2,b
3 iff for b
5 being
real number for b
6 being non
negative real number holds
( ( not
|[b5,b6]| in Ball |[b1,b2]|,b
3 implies b
4 . |[b5,b6]| = 1 ) & (
|[b5,b6]| in Ball |[b1,b2]|,b
3 implies b
4 . |[b5,b6]| = |.(|[b1,b2]| - |[b5,b6]|).| / b
3 ) ) );
theorem Th81: :: TOPGEN_5:81
theorem Th82: :: TOPGEN_5:82
theorem Th83: :: TOPGEN_5:83
theorem Th84: :: TOPGEN_5:84
for b
1 being
Point of
(TOP-REAL 2) holds
( b
1 `2 = 0 implies for b
2 being
real number for b
3 being non
negative real number for b
4, b
5 being
positive real number holds
(
(+ b2,b4,b5) . b
1 > b
3 implies (
|.(|[b2,b4]| - b1).| > b
5 * b
3 & ex b
6 being
positive real number st
( b
6 = (|.(|[b2,b4]| - b1).| - (b5 * b3)) / 2 &
(Ball |[(b1 `1 ),b6]|,b6) \/ {b1} c= (+ b2,b4,b5) " ].b3,1.] ) ) ) )
theorem Th85: :: TOPGEN_5:85
for b
1 being
Subset of
Niemytzki-plane for b
2, b
3 being
Element of
REAL for b
4 being
positive real number holds
not ( b
3 > 0 & b
1 = (Ball |[b2,b3]|,b4) /\ y>=0-plane & ( for b
5 being
continuous Function of
Niemytzki-plane ,
I[01] holds
not ( b
5 . |[b2,b3]| = 0 & ( for b
6, b
7 being
real number holds
( (
|[b6,b7]| in b
1 ` implies b
5 . |[b6,b7]| = 1 ) & (
|[b6,b7]| in b
1 implies b
5 . |[b6,b7]| = |.(|[b2,b3]| - |[b6,b7]|).| / b
4 ) ) ) ) ) )
theorem Th86: :: TOPGEN_5:86
theorem Th87: :: TOPGEN_5:87