:: RLTOPSP1 semantic presentation
theorem Th1: :: RLTOPSP1:1
theorem Th2: :: RLTOPSP1:2
theorem Th3: :: RLTOPSP1:3
Lemma44:
for X being non empty LoopStr
for M being Subset of X
for x, y being Point of X st y in M holds
x + y in x + M
Lemma45:
for X being non empty LoopStr
for M, N being Subset of X holds { (x + N) where x is Point of X : x in M } is Subset-Family of X
theorem Th4: :: RLTOPSP1:4
theorem Th5: :: RLTOPSP1:5
theorem Th6: :: RLTOPSP1:6
theorem Th7: :: RLTOPSP1:7
theorem Th8: :: RLTOPSP1:8
theorem Th9: :: RLTOPSP1:9
theorem Th10: :: RLTOPSP1:10
Lemma73:
for X being non empty LoopStr
for M, N, V being Subset of X st M c= N holds
V + M c= V + N
theorem Th11: :: RLTOPSP1:11
theorem Th12: :: RLTOPSP1:12
theorem Th13: :: RLTOPSP1:13
theorem Th14: :: RLTOPSP1:14
theorem Th15: :: RLTOPSP1:15
theorem Th16: :: RLTOPSP1:16
:: deftheorem Def1 defines convex RLTOPSP1:def 1 :
Lemma87:
for X being RealLinearSpace holds conv ({} X) = {}
theorem Th17: :: RLTOPSP1:17
canceled;
theorem Th18: :: RLTOPSP1:18
theorem Th19: :: RLTOPSP1:19
theorem Th20: :: RLTOPSP1:20
theorem Th21: :: RLTOPSP1:21
theorem Th22: :: RLTOPSP1:22
:: deftheorem Def2 defines LSeg RLTOPSP1:def 2 :
theorem Th23: :: RLTOPSP1:23
:: deftheorem Def3 defines convex-membered RLTOPSP1:def 3 :
theorem Th24: :: RLTOPSP1:24
:: deftheorem Def4 defines - RLTOPSP1:def 4 :
theorem Th25: :: RLTOPSP1:25
:: deftheorem Def5 defines symmetric RLTOPSP1:def 5 :
theorem Th26: :: RLTOPSP1:26
:: deftheorem Def6 defines circled RLTOPSP1:def 6 :
Lemma100:
for X being non empty RLSStruct holds {} X is circled
theorem Th27: :: RLTOPSP1:27
theorem Th28: :: RLTOPSP1:28
Lemma103:
for X being RealLinearSpace
for A, B being circled Subset of X holds A + B is circled
theorem Th29: :: RLTOPSP1:29
Lemma106:
for X being RealLinearSpace
for A being circled Subset of X holds A is symmetric
Lemma107:
for X being RealLinearSpace
for M being circled Subset of X holds conv M is circled
:: deftheorem Def7 defines circled-membered RLTOPSP1:def 7 :
theorem Th30: :: RLTOPSP1:30
theorem Th31: :: RLTOPSP1:31
registration
let X be non
empty set ;
let O be
Element of
X;
let F be
BinOp of
X;
let G be
Function of
[:REAL ,X:],
X;
let T be
Subset-Family of
X;
cluster RLTopStruct(#
a1,
a2,
a3,
a4,
a5 #)
-> non
empty ;
coherence
not RLTopStruct(# X,O,F,G,T #) is empty
by STRUCT_0:def 1;
end;
:: deftheorem Def8 defines add-continuous RLTOPSP1:def 8 :
:: deftheorem Def9 defines Mult-continuous RLTOPSP1:def 9 :
theorem Th32: :: RLTOPSP1:32
theorem Th33: :: RLTOPSP1:33
:: deftheorem Def10 defines transl RLTOPSP1:def 10 :
theorem Th34: :: RLTOPSP1:34
theorem Th35: :: RLTOPSP1:35
Lemma128:
for X being LinearTopSpace
for a being Point of X holds transl a,X is one-to-one
theorem Th36: :: RLTOPSP1:36
Lemma132:
for X being LinearTopSpace
for a being Point of X holds transl a,X is continuous
Lemma135:
for X being LinearTopSpace
for E being Subset of X
for x being Point of X st E is open holds
x + E is open
Lemma137:
for X being LinearTopSpace
for E being open Subset of X
for K being Subset of X holds K + E is open
Lemma139:
for X being LinearTopSpace
for D being closed Subset of X
for x being Point of X holds x + D is closed
theorem Th37: :: RLTOPSP1:37
theorem Th38: :: RLTOPSP1:38
theorem Th39: :: RLTOPSP1:39
theorem Th40: :: RLTOPSP1:40
theorem Th41: :: RLTOPSP1:41
:: deftheorem Def11 defines locally-convex RLTOPSP1:def 11 :
:: deftheorem Def12 defines bounded RLTOPSP1:def 12 :
theorem Th42: :: RLTOPSP1:42
theorem Th43: :: RLTOPSP1:43
theorem Th44: :: RLTOPSP1:44
theorem Th45: :: RLTOPSP1:45
theorem Th46: :: RLTOPSP1:46
:: deftheorem Def13 defines mlt RLTOPSP1:def 13 :
theorem Th47: :: RLTOPSP1:47
theorem Th48: :: RLTOPSP1:48
Lemma152:
for X being LinearTopSpace
for r being non zero Real holds mlt r,X is one-to-one
theorem Th49: :: RLTOPSP1:49
Lemma154:
for X being LinearTopSpace
for r being non zero Real holds mlt r,X is continuous
theorem Th50: :: RLTOPSP1:50
theorem Th51: :: RLTOPSP1:51
theorem Th52: :: RLTOPSP1:52
theorem Th53: :: RLTOPSP1:53
theorem Th54: :: RLTOPSP1:54
theorem Th55: :: RLTOPSP1:55
theorem Th56: :: RLTOPSP1:56
Lemma162:
for X being LinearTopSpace
for V being bounded Subset of X
for r being Real holds r * V is bounded
theorem Th57: :: RLTOPSP1:57
theorem Th58: :: RLTOPSP1:58
theorem Th59: :: RLTOPSP1:59
theorem Th60: :: RLTOPSP1:60
theorem Th61: :: RLTOPSP1:61
theorem Th62: :: RLTOPSP1:62
theorem Th63: :: RLTOPSP1:63
Lemma308:
for X being LinearTopSpace
for C being convex Subset of X holds Cl C is convex
Lemma309:
for X being LinearTopSpace
for C being convex Subset of X holds Int C is convex
Lemma310:
for X being LinearTopSpace
for B being circled Subset of X holds Cl B is circled
theorem Th64: :: RLTOPSP1:64
Lemma311:
for X being LinearTopSpace
for E being bounded Subset of X holds Cl E is bounded
Lemma312:
for X being LinearTopSpace
for U, V being a_neighborhood of 0. X
for F being Subset-Family of X
for r being positive Real st ( for s being Real st abs s < r holds
s * V c= U ) & F = { (a * V) where a is Real : abs a < r } holds
( union F is a_neighborhood of 0. X & union F is circled & union F c= U )
theorem Th65: :: RLTOPSP1:65
theorem Th66: :: RLTOPSP1:66
theorem Th67: :: RLTOPSP1:67
theorem Th68: :: RLTOPSP1:68