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Theorem tgpconncompeqg 22647
Description: The connected component containing 𝐴 is the left coset of the identity component containing 𝐴. (Contributed by Mario Carneiro, 17-Sep-2015.)
Hypotheses
Ref Expression
tgpconncomp.x 𝑋 = (Base‘𝐺)
tgpconncomp.z 0 = (0g𝐺)
tgpconncomp.j 𝐽 = (TopOpen‘𝐺)
tgpconncomp.s 𝑆 = {𝑥 ∈ 𝒫 𝑋 ∣ ( 0𝑥 ∧ (𝐽t 𝑥) ∈ Conn)}
tgpconncompeqg.r = (𝐺 ~QG 𝑆)
Assertion
Ref Expression
tgpconncompeqg ((𝐺 ∈ TopGrp ∧ 𝐴𝑋) → [𝐴] = {𝑥 ∈ 𝒫 𝑋 ∣ (𝐴𝑥 ∧ (𝐽t 𝑥) ∈ Conn)})
Distinct variable groups:   𝑥, 0   𝑥,𝐴   𝑥,𝐽   𝑥,𝐺   𝑥,𝑋
Allowed substitution hints:   (𝑥)   𝑆(𝑥)

Proof of Theorem tgpconncompeqg
Dummy variables 𝑦 𝑧 𝑔 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 dfec2 8281 . . . . 5 (𝐴𝑋 → [𝐴] = {𝑧𝐴 𝑧})
21adantl 482 . . . 4 ((𝐺 ∈ TopGrp ∧ 𝐴𝑋) → [𝐴] = {𝑧𝐴 𝑧})
3 tgpconncomp.s . . . . . . . . 9 𝑆 = {𝑥 ∈ 𝒫 𝑋 ∣ ( 0𝑥 ∧ (𝐽t 𝑥) ∈ Conn)}
4 ssrab2 4053 . . . . . . . . . 10 {𝑥 ∈ 𝒫 𝑋 ∣ ( 0𝑥 ∧ (𝐽t 𝑥) ∈ Conn)} ⊆ 𝒫 𝑋
5 sspwuni 5013 . . . . . . . . . 10 ({𝑥 ∈ 𝒫 𝑋 ∣ ( 0𝑥 ∧ (𝐽t 𝑥) ∈ Conn)} ⊆ 𝒫 𝑋 {𝑥 ∈ 𝒫 𝑋 ∣ ( 0𝑥 ∧ (𝐽t 𝑥) ∈ Conn)} ⊆ 𝑋)
64, 5mpbi 231 . . . . . . . . 9 {𝑥 ∈ 𝒫 𝑋 ∣ ( 0𝑥 ∧ (𝐽t 𝑥) ∈ Conn)} ⊆ 𝑋
73, 6eqsstri 3998 . . . . . . . 8 𝑆𝑋
87a1i 11 . . . . . . 7 ((𝐺 ∈ TopGrp ∧ 𝐴𝑋) → 𝑆𝑋)
9 tgpconncomp.x . . . . . . . 8 𝑋 = (Base‘𝐺)
10 eqid 2818 . . . . . . . 8 (invg𝐺) = (invg𝐺)
11 eqid 2818 . . . . . . . 8 (+g𝐺) = (+g𝐺)
12 tgpconncompeqg.r . . . . . . . 8 = (𝐺 ~QG 𝑆)
139, 10, 11, 12eqgval 18267 . . . . . . 7 ((𝐺 ∈ TopGrp ∧ 𝑆𝑋) → (𝐴 𝑧 ↔ (𝐴𝑋𝑧𝑋 ∧ (((invg𝐺)‘𝐴)(+g𝐺)𝑧) ∈ 𝑆)))
148, 13syldan 591 . . . . . 6 ((𝐺 ∈ TopGrp ∧ 𝐴𝑋) → (𝐴 𝑧 ↔ (𝐴𝑋𝑧𝑋 ∧ (((invg𝐺)‘𝐴)(+g𝐺)𝑧) ∈ 𝑆)))
15 simp2 1129 . . . . . 6 ((𝐴𝑋𝑧𝑋 ∧ (((invg𝐺)‘𝐴)(+g𝐺)𝑧) ∈ 𝑆) → 𝑧𝑋)
1614, 15syl6bi 254 . . . . 5 ((𝐺 ∈ TopGrp ∧ 𝐴𝑋) → (𝐴 𝑧𝑧𝑋))
1716abssdv 4042 . . . 4 ((𝐺 ∈ TopGrp ∧ 𝐴𝑋) → {𝑧𝐴 𝑧} ⊆ 𝑋)
182, 17eqsstrd 4002 . . 3 ((𝐺 ∈ TopGrp ∧ 𝐴𝑋) → [𝐴] 𝑋)
19 simpr 485 . . . . 5 ((𝐺 ∈ TopGrp ∧ 𝐴𝑋) → 𝐴𝑋)
20 tgpgrp 22614 . . . . . . 7 (𝐺 ∈ TopGrp → 𝐺 ∈ Grp)
21 tgpconncomp.z . . . . . . . 8 0 = (0g𝐺)
229, 11, 21, 10grplinv 18090 . . . . . . 7 ((𝐺 ∈ Grp ∧ 𝐴𝑋) → (((invg𝐺)‘𝐴)(+g𝐺)𝐴) = 0 )
2320, 22sylan 580 . . . . . 6 ((𝐺 ∈ TopGrp ∧ 𝐴𝑋) → (((invg𝐺)‘𝐴)(+g𝐺)𝐴) = 0 )
24 tgpconncomp.j . . . . . . . . 9 𝐽 = (TopOpen‘𝐺)
2524, 9tgptopon 22618 . . . . . . . 8 (𝐺 ∈ TopGrp → 𝐽 ∈ (TopOn‘𝑋))
2625adantr 481 . . . . . . 7 ((𝐺 ∈ TopGrp ∧ 𝐴𝑋) → 𝐽 ∈ (TopOn‘𝑋))
2720adantr 481 . . . . . . . 8 ((𝐺 ∈ TopGrp ∧ 𝐴𝑋) → 𝐺 ∈ Grp)
289, 21grpidcl 18069 . . . . . . . 8 (𝐺 ∈ Grp → 0𝑋)
2927, 28syl 17 . . . . . . 7 ((𝐺 ∈ TopGrp ∧ 𝐴𝑋) → 0𝑋)
303conncompid 21967 . . . . . . 7 ((𝐽 ∈ (TopOn‘𝑋) ∧ 0𝑋) → 0𝑆)
3126, 29, 30syl2anc 584 . . . . . 6 ((𝐺 ∈ TopGrp ∧ 𝐴𝑋) → 0𝑆)
3223, 31eqeltrd 2910 . . . . 5 ((𝐺 ∈ TopGrp ∧ 𝐴𝑋) → (((invg𝐺)‘𝐴)(+g𝐺)𝐴) ∈ 𝑆)
339, 10, 11, 12eqgval 18267 . . . . . 6 ((𝐺 ∈ TopGrp ∧ 𝑆𝑋) → (𝐴 𝐴 ↔ (𝐴𝑋𝐴𝑋 ∧ (((invg𝐺)‘𝐴)(+g𝐺)𝐴) ∈ 𝑆)))
348, 33syldan 591 . . . . 5 ((𝐺 ∈ TopGrp ∧ 𝐴𝑋) → (𝐴 𝐴 ↔ (𝐴𝑋𝐴𝑋 ∧ (((invg𝐺)‘𝐴)(+g𝐺)𝐴) ∈ 𝑆)))
3519, 19, 32, 34mpbir3and 1334 . . . 4 ((𝐺 ∈ TopGrp ∧ 𝐴𝑋) → 𝐴 𝐴)
36 elecg 8321 . . . . 5 ((𝐴𝑋𝐴𝑋) → (𝐴 ∈ [𝐴] 𝐴 𝐴))
3719, 19, 36syl2anc 584 . . . 4 ((𝐺 ∈ TopGrp ∧ 𝐴𝑋) → (𝐴 ∈ [𝐴] 𝐴 𝐴))
3835, 37mpbird 258 . . 3 ((𝐺 ∈ TopGrp ∧ 𝐴𝑋) → 𝐴 ∈ [𝐴] )
399, 12, 11eqglact 18269 . . . . . . 7 ((𝐺 ∈ Grp ∧ 𝑆𝑋𝐴𝑋) → [𝐴] = ((𝑧𝑋 ↦ (𝐴(+g𝐺)𝑧)) “ 𝑆))
407, 39mp3an2 1440 . . . . . 6 ((𝐺 ∈ Grp ∧ 𝐴𝑋) → [𝐴] = ((𝑧𝑋 ↦ (𝐴(+g𝐺)𝑧)) “ 𝑆))
4120, 40sylan 580 . . . . 5 ((𝐺 ∈ TopGrp ∧ 𝐴𝑋) → [𝐴] = ((𝑧𝑋 ↦ (𝐴(+g𝐺)𝑧)) “ 𝑆))
4241oveq2d 7161 . . . 4 ((𝐺 ∈ TopGrp ∧ 𝐴𝑋) → (𝐽t [𝐴] ) = (𝐽t ((𝑧𝑋 ↦ (𝐴(+g𝐺)𝑧)) “ 𝑆)))
43 eqid 2818 . . . . 5 𝐽 = 𝐽
44 eqid 2818 . . . . . . 7 (𝑧𝑋 ↦ (𝐴(+g𝐺)𝑧)) = (𝑧𝑋 ↦ (𝐴(+g𝐺)𝑧))
4544, 9, 11, 24tgplacthmeo 22639 . . . . . 6 ((𝐺 ∈ TopGrp ∧ 𝐴𝑋) → (𝑧𝑋 ↦ (𝐴(+g𝐺)𝑧)) ∈ (𝐽Homeo𝐽))
46 hmeocn 22296 . . . . . 6 ((𝑧𝑋 ↦ (𝐴(+g𝐺)𝑧)) ∈ (𝐽Homeo𝐽) → (𝑧𝑋 ↦ (𝐴(+g𝐺)𝑧)) ∈ (𝐽 Cn 𝐽))
4745, 46syl 17 . . . . 5 ((𝐺 ∈ TopGrp ∧ 𝐴𝑋) → (𝑧𝑋 ↦ (𝐴(+g𝐺)𝑧)) ∈ (𝐽 Cn 𝐽))
48 toponuni 21450 . . . . . . 7 (𝐽 ∈ (TopOn‘𝑋) → 𝑋 = 𝐽)
4926, 48syl 17 . . . . . 6 ((𝐺 ∈ TopGrp ∧ 𝐴𝑋) → 𝑋 = 𝐽)
507, 49sseqtrid 4016 . . . . 5 ((𝐺 ∈ TopGrp ∧ 𝐴𝑋) → 𝑆 𝐽)
513conncompconn 21968 . . . . . 6 ((𝐽 ∈ (TopOn‘𝑋) ∧ 0𝑋) → (𝐽t 𝑆) ∈ Conn)
5226, 29, 51syl2anc 584 . . . . 5 ((𝐺 ∈ TopGrp ∧ 𝐴𝑋) → (𝐽t 𝑆) ∈ Conn)
5343, 47, 50, 52connima 21961 . . . 4 ((𝐺 ∈ TopGrp ∧ 𝐴𝑋) → (𝐽t ((𝑧𝑋 ↦ (𝐴(+g𝐺)𝑧)) “ 𝑆)) ∈ Conn)
5442, 53eqeltrd 2910 . . 3 ((𝐺 ∈ TopGrp ∧ 𝐴𝑋) → (𝐽t [𝐴] ) ∈ Conn)
55 eqid 2818 . . . 4 {𝑥 ∈ 𝒫 𝑋 ∣ (𝐴𝑥 ∧ (𝐽t 𝑥) ∈ Conn)} = {𝑥 ∈ 𝒫 𝑋 ∣ (𝐴𝑥 ∧ (𝐽t 𝑥) ∈ Conn)}
5655conncompss 21969 . . 3 (([𝐴] 𝑋𝐴 ∈ [𝐴] ∧ (𝐽t [𝐴] ) ∈ Conn) → [𝐴] {𝑥 ∈ 𝒫 𝑋 ∣ (𝐴𝑥 ∧ (𝐽t 𝑥) ∈ Conn)})
5718, 38, 54, 56syl3anc 1363 . 2 ((𝐺 ∈ TopGrp ∧ 𝐴𝑋) → [𝐴] {𝑥 ∈ 𝒫 𝑋 ∣ (𝐴𝑥 ∧ (𝐽t 𝑥) ∈ Conn)})
58 elpwi 4547 . . . . . 6 (𝑦 ∈ 𝒫 𝑋𝑦𝑋)
5944mptpreima 6085 . . . . . . . . . . 11 ((𝑧𝑋 ↦ (𝐴(+g𝐺)𝑧)) “ 𝑦) = {𝑧𝑋 ∣ (𝐴(+g𝐺)𝑧) ∈ 𝑦}
6059ssrab3 4054 . . . . . . . . . 10 ((𝑧𝑋 ↦ (𝐴(+g𝐺)𝑧)) “ 𝑦) ⊆ 𝑋
6129adantr 481 . . . . . . . . . . 11 (((𝐺 ∈ TopGrp ∧ 𝐴𝑋) ∧ (𝑦𝑋 ∧ (𝐴𝑦 ∧ (𝐽t 𝑦) ∈ Conn))) → 0𝑋)
629, 11, 21grprid 18072 . . . . . . . . . . . . . 14 ((𝐺 ∈ Grp ∧ 𝐴𝑋) → (𝐴(+g𝐺) 0 ) = 𝐴)
6320, 62sylan 580 . . . . . . . . . . . . 13 ((𝐺 ∈ TopGrp ∧ 𝐴𝑋) → (𝐴(+g𝐺) 0 ) = 𝐴)
6463adantr 481 . . . . . . . . . . . 12 (((𝐺 ∈ TopGrp ∧ 𝐴𝑋) ∧ (𝑦𝑋 ∧ (𝐴𝑦 ∧ (𝐽t 𝑦) ∈ Conn))) → (𝐴(+g𝐺) 0 ) = 𝐴)
65 simprrl 777 . . . . . . . . . . . 12 (((𝐺 ∈ TopGrp ∧ 𝐴𝑋) ∧ (𝑦𝑋 ∧ (𝐴𝑦 ∧ (𝐽t 𝑦) ∈ Conn))) → 𝐴𝑦)
6664, 65eqeltrd 2910 . . . . . . . . . . 11 (((𝐺 ∈ TopGrp ∧ 𝐴𝑋) ∧ (𝑦𝑋 ∧ (𝐴𝑦 ∧ (𝐽t 𝑦) ∈ Conn))) → (𝐴(+g𝐺) 0 ) ∈ 𝑦)
67 oveq2 7153 . . . . . . . . . . . . 13 (𝑧 = 0 → (𝐴(+g𝐺)𝑧) = (𝐴(+g𝐺) 0 ))
6867eleq1d 2894 . . . . . . . . . . . 12 (𝑧 = 0 → ((𝐴(+g𝐺)𝑧) ∈ 𝑦 ↔ (𝐴(+g𝐺) 0 ) ∈ 𝑦))
6968, 59elrab2 3680 . . . . . . . . . . 11 ( 0 ∈ ((𝑧𝑋 ↦ (𝐴(+g𝐺)𝑧)) “ 𝑦) ↔ ( 0𝑋 ∧ (𝐴(+g𝐺) 0 ) ∈ 𝑦))
7061, 66, 69sylanbrc 583 . . . . . . . . . 10 (((𝐺 ∈ TopGrp ∧ 𝐴𝑋) ∧ (𝑦𝑋 ∧ (𝐴𝑦 ∧ (𝐽t 𝑦) ∈ Conn))) → 0 ∈ ((𝑧𝑋 ↦ (𝐴(+g𝐺)𝑧)) “ 𝑦))
71 hmeocnvcn 22297 . . . . . . . . . . . . 13 ((𝑧𝑋 ↦ (𝐴(+g𝐺)𝑧)) ∈ (𝐽Homeo𝐽) → (𝑧𝑋 ↦ (𝐴(+g𝐺)𝑧)) ∈ (𝐽 Cn 𝐽))
7245, 71syl 17 . . . . . . . . . . . 12 ((𝐺 ∈ TopGrp ∧ 𝐴𝑋) → (𝑧𝑋 ↦ (𝐴(+g𝐺)𝑧)) ∈ (𝐽 Cn 𝐽))
7372adantr 481 . . . . . . . . . . 11 (((𝐺 ∈ TopGrp ∧ 𝐴𝑋) ∧ (𝑦𝑋 ∧ (𝐴𝑦 ∧ (𝐽t 𝑦) ∈ Conn))) → (𝑧𝑋 ↦ (𝐴(+g𝐺)𝑧)) ∈ (𝐽 Cn 𝐽))
74 simprl 767 . . . . . . . . . . . 12 (((𝐺 ∈ TopGrp ∧ 𝐴𝑋) ∧ (𝑦𝑋 ∧ (𝐴𝑦 ∧ (𝐽t 𝑦) ∈ Conn))) → 𝑦𝑋)
7549adantr 481 . . . . . . . . . . . 12 (((𝐺 ∈ TopGrp ∧ 𝐴𝑋) ∧ (𝑦𝑋 ∧ (𝐴𝑦 ∧ (𝐽t 𝑦) ∈ Conn))) → 𝑋 = 𝐽)
7674, 75sseqtrd 4004 . . . . . . . . . . 11 (((𝐺 ∈ TopGrp ∧ 𝐴𝑋) ∧ (𝑦𝑋 ∧ (𝐴𝑦 ∧ (𝐽t 𝑦) ∈ Conn))) → 𝑦 𝐽)
77 simprrr 778 . . . . . . . . . . 11 (((𝐺 ∈ TopGrp ∧ 𝐴𝑋) ∧ (𝑦𝑋 ∧ (𝐴𝑦 ∧ (𝐽t 𝑦) ∈ Conn))) → (𝐽t 𝑦) ∈ Conn)
7843, 73, 76, 77connima 21961 . . . . . . . . . 10 (((𝐺 ∈ TopGrp ∧ 𝐴𝑋) ∧ (𝑦𝑋 ∧ (𝐴𝑦 ∧ (𝐽t 𝑦) ∈ Conn))) → (𝐽t ((𝑧𝑋 ↦ (𝐴(+g𝐺)𝑧)) “ 𝑦)) ∈ Conn)
793conncompss 21969 . . . . . . . . . 10 ((((𝑧𝑋 ↦ (𝐴(+g𝐺)𝑧)) “ 𝑦) ⊆ 𝑋0 ∈ ((𝑧𝑋 ↦ (𝐴(+g𝐺)𝑧)) “ 𝑦) ∧ (𝐽t ((𝑧𝑋 ↦ (𝐴(+g𝐺)𝑧)) “ 𝑦)) ∈ Conn) → ((𝑧𝑋 ↦ (𝐴(+g𝐺)𝑧)) “ 𝑦) ⊆ 𝑆)
8060, 70, 78, 79mp3an2i 1457 . . . . . . . . 9 (((𝐺 ∈ TopGrp ∧ 𝐴𝑋) ∧ (𝑦𝑋 ∧ (𝐴𝑦 ∧ (𝐽t 𝑦) ∈ Conn))) → ((𝑧𝑋 ↦ (𝐴(+g𝐺)𝑧)) “ 𝑦) ⊆ 𝑆)
81 eqid 2818 . . . . . . . . . . . . . . . 16 (𝑔𝑋 ↦ (𝑧𝑋 ↦ (𝑔(+g𝐺)𝑧))) = (𝑔𝑋 ↦ (𝑧𝑋 ↦ (𝑔(+g𝐺)𝑧)))
8281, 9, 11, 10grplactcnv 18140 . . . . . . . . . . . . . . 15 ((𝐺 ∈ Grp ∧ 𝐴𝑋) → (((𝑔𝑋 ↦ (𝑧𝑋 ↦ (𝑔(+g𝐺)𝑧)))‘𝐴):𝑋1-1-onto𝑋((𝑔𝑋 ↦ (𝑧𝑋 ↦ (𝑔(+g𝐺)𝑧)))‘𝐴) = ((𝑔𝑋 ↦ (𝑧𝑋 ↦ (𝑔(+g𝐺)𝑧)))‘((invg𝐺)‘𝐴))))
8320, 82sylan 580 . . . . . . . . . . . . . 14 ((𝐺 ∈ TopGrp ∧ 𝐴𝑋) → (((𝑔𝑋 ↦ (𝑧𝑋 ↦ (𝑔(+g𝐺)𝑧)))‘𝐴):𝑋1-1-onto𝑋((𝑔𝑋 ↦ (𝑧𝑋 ↦ (𝑔(+g𝐺)𝑧)))‘𝐴) = ((𝑔𝑋 ↦ (𝑧𝑋 ↦ (𝑔(+g𝐺)𝑧)))‘((invg𝐺)‘𝐴))))
8483simpld 495 . . . . . . . . . . . . 13 ((𝐺 ∈ TopGrp ∧ 𝐴𝑋) → ((𝑔𝑋 ↦ (𝑧𝑋 ↦ (𝑔(+g𝐺)𝑧)))‘𝐴):𝑋1-1-onto𝑋)
8581, 9grplactfval 18138 . . . . . . . . . . . . . . 15 (𝐴𝑋 → ((𝑔𝑋 ↦ (𝑧𝑋 ↦ (𝑔(+g𝐺)𝑧)))‘𝐴) = (𝑧𝑋 ↦ (𝐴(+g𝐺)𝑧)))
8685adantl 482 . . . . . . . . . . . . . 14 ((𝐺 ∈ TopGrp ∧ 𝐴𝑋) → ((𝑔𝑋 ↦ (𝑧𝑋 ↦ (𝑔(+g𝐺)𝑧)))‘𝐴) = (𝑧𝑋 ↦ (𝐴(+g𝐺)𝑧)))
87 f1oeq1 6597 . . . . . . . . . . . . . 14 (((𝑔𝑋 ↦ (𝑧𝑋 ↦ (𝑔(+g𝐺)𝑧)))‘𝐴) = (𝑧𝑋 ↦ (𝐴(+g𝐺)𝑧)) → (((𝑔𝑋 ↦ (𝑧𝑋 ↦ (𝑔(+g𝐺)𝑧)))‘𝐴):𝑋1-1-onto𝑋 ↔ (𝑧𝑋 ↦ (𝐴(+g𝐺)𝑧)):𝑋1-1-onto𝑋))
8886, 87syl 17 . . . . . . . . . . . . 13 ((𝐺 ∈ TopGrp ∧ 𝐴𝑋) → (((𝑔𝑋 ↦ (𝑧𝑋 ↦ (𝑔(+g𝐺)𝑧)))‘𝐴):𝑋1-1-onto𝑋 ↔ (𝑧𝑋 ↦ (𝐴(+g𝐺)𝑧)):𝑋1-1-onto𝑋))
8984, 88mpbid 233 . . . . . . . . . . . 12 ((𝐺 ∈ TopGrp ∧ 𝐴𝑋) → (𝑧𝑋 ↦ (𝐴(+g𝐺)𝑧)):𝑋1-1-onto𝑋)
9089adantr 481 . . . . . . . . . . 11 (((𝐺 ∈ TopGrp ∧ 𝐴𝑋) ∧ (𝑦𝑋 ∧ (𝐴𝑦 ∧ (𝐽t 𝑦) ∈ Conn))) → (𝑧𝑋 ↦ (𝐴(+g𝐺)𝑧)):𝑋1-1-onto𝑋)
91 f1ocnv 6620 . . . . . . . . . . 11 ((𝑧𝑋 ↦ (𝐴(+g𝐺)𝑧)):𝑋1-1-onto𝑋(𝑧𝑋 ↦ (𝐴(+g𝐺)𝑧)):𝑋1-1-onto𝑋)
92 f1ofun 6610 . . . . . . . . . . 11 ((𝑧𝑋 ↦ (𝐴(+g𝐺)𝑧)):𝑋1-1-onto𝑋 → Fun (𝑧𝑋 ↦ (𝐴(+g𝐺)𝑧)))
9390, 91, 923syl 18 . . . . . . . . . 10 (((𝐺 ∈ TopGrp ∧ 𝐴𝑋) ∧ (𝑦𝑋 ∧ (𝐴𝑦 ∧ (𝐽t 𝑦) ∈ Conn))) → Fun (𝑧𝑋 ↦ (𝐴(+g𝐺)𝑧)))
94 f1odm 6612 . . . . . . . . . . . 12 ((𝑧𝑋 ↦ (𝐴(+g𝐺)𝑧)):𝑋1-1-onto𝑋 → dom (𝑧𝑋 ↦ (𝐴(+g𝐺)𝑧)) = 𝑋)
9590, 91, 943syl 18 . . . . . . . . . . 11 (((𝐺 ∈ TopGrp ∧ 𝐴𝑋) ∧ (𝑦𝑋 ∧ (𝐴𝑦 ∧ (𝐽t 𝑦) ∈ Conn))) → dom (𝑧𝑋 ↦ (𝐴(+g𝐺)𝑧)) = 𝑋)
9674, 95sseqtrrd 4005 . . . . . . . . . 10 (((𝐺 ∈ TopGrp ∧ 𝐴𝑋) ∧ (𝑦𝑋 ∧ (𝐴𝑦 ∧ (𝐽t 𝑦) ∈ Conn))) → 𝑦 ⊆ dom (𝑧𝑋 ↦ (𝐴(+g𝐺)𝑧)))
97 funimass3 6816 . . . . . . . . . 10 ((Fun (𝑧𝑋 ↦ (𝐴(+g𝐺)𝑧)) ∧ 𝑦 ⊆ dom (𝑧𝑋 ↦ (𝐴(+g𝐺)𝑧))) → (((𝑧𝑋 ↦ (𝐴(+g𝐺)𝑧)) “ 𝑦) ⊆ 𝑆𝑦 ⊆ ((𝑧𝑋 ↦ (𝐴(+g𝐺)𝑧)) “ 𝑆)))
9893, 96, 97syl2anc 584 . . . . . . . . 9 (((𝐺 ∈ TopGrp ∧ 𝐴𝑋) ∧ (𝑦𝑋 ∧ (𝐴𝑦 ∧ (𝐽t 𝑦) ∈ Conn))) → (((𝑧𝑋 ↦ (𝐴(+g𝐺)𝑧)) “ 𝑦) ⊆ 𝑆𝑦 ⊆ ((𝑧𝑋 ↦ (𝐴(+g𝐺)𝑧)) “ 𝑆)))
9980, 98mpbid 233 . . . . . . . 8 (((𝐺 ∈ TopGrp ∧ 𝐴𝑋) ∧ (𝑦𝑋 ∧ (𝐴𝑦 ∧ (𝐽t 𝑦) ∈ Conn))) → 𝑦 ⊆ ((𝑧𝑋 ↦ (𝐴(+g𝐺)𝑧)) “ 𝑆))
10041adantr 481 . . . . . . . . 9 (((𝐺 ∈ TopGrp ∧ 𝐴𝑋) ∧ (𝑦𝑋 ∧ (𝐴𝑦 ∧ (𝐽t 𝑦) ∈ Conn))) → [𝐴] = ((𝑧𝑋 ↦ (𝐴(+g𝐺)𝑧)) “ 𝑆))
101 imacnvcnv 6056 . . . . . . . . 9 ((𝑧𝑋 ↦ (𝐴(+g𝐺)𝑧)) “ 𝑆) = ((𝑧𝑋 ↦ (𝐴(+g𝐺)𝑧)) “ 𝑆)
102100, 101syl6eqr 2871 . . . . . . . 8 (((𝐺 ∈ TopGrp ∧ 𝐴𝑋) ∧ (𝑦𝑋 ∧ (𝐴𝑦 ∧ (𝐽t 𝑦) ∈ Conn))) → [𝐴] = ((𝑧𝑋 ↦ (𝐴(+g𝐺)𝑧)) “ 𝑆))
10399, 102sseqtrrd 4005 . . . . . . 7 (((𝐺 ∈ TopGrp ∧ 𝐴𝑋) ∧ (𝑦𝑋 ∧ (𝐴𝑦 ∧ (𝐽t 𝑦) ∈ Conn))) → 𝑦 ⊆ [𝐴] )
104103expr 457 . . . . . 6 (((𝐺 ∈ TopGrp ∧ 𝐴𝑋) ∧ 𝑦𝑋) → ((𝐴𝑦 ∧ (𝐽t 𝑦) ∈ Conn) → 𝑦 ⊆ [𝐴] ))
10558, 104sylan2 592 . . . . 5 (((𝐺 ∈ TopGrp ∧ 𝐴𝑋) ∧ 𝑦 ∈ 𝒫 𝑋) → ((𝐴𝑦 ∧ (𝐽t 𝑦) ∈ Conn) → 𝑦 ⊆ [𝐴] ))
106105ralrimiva 3179 . . . 4 ((𝐺 ∈ TopGrp ∧ 𝐴𝑋) → ∀𝑦 ∈ 𝒫 𝑋((𝐴𝑦 ∧ (𝐽t 𝑦) ∈ Conn) → 𝑦 ⊆ [𝐴] ))
107 eleq2w 2893 . . . . . 6 (𝑥 = 𝑦 → (𝐴𝑥𝐴𝑦))
108 oveq2 7153 . . . . . . 7 (𝑥 = 𝑦 → (𝐽t 𝑥) = (𝐽t 𝑦))
109108eleq1d 2894 . . . . . 6 (𝑥 = 𝑦 → ((𝐽t 𝑥) ∈ Conn ↔ (𝐽t 𝑦) ∈ Conn))
110107, 109anbi12d 630 . . . . 5 (𝑥 = 𝑦 → ((𝐴𝑥 ∧ (𝐽t 𝑥) ∈ Conn) ↔ (𝐴𝑦 ∧ (𝐽t 𝑦) ∈ Conn)))
111110ralrab 3682 . . . 4 (∀𝑦 ∈ {𝑥 ∈ 𝒫 𝑋 ∣ (𝐴𝑥 ∧ (𝐽t 𝑥) ∈ Conn)}𝑦 ⊆ [𝐴] ↔ ∀𝑦 ∈ 𝒫 𝑋((𝐴𝑦 ∧ (𝐽t 𝑦) ∈ Conn) → 𝑦 ⊆ [𝐴] ))
112106, 111sylibr 235 . . 3 ((𝐺 ∈ TopGrp ∧ 𝐴𝑋) → ∀𝑦 ∈ {𝑥 ∈ 𝒫 𝑋 ∣ (𝐴𝑥 ∧ (𝐽t 𝑥) ∈ Conn)}𝑦 ⊆ [𝐴] )
113 unissb 4861 . . 3 ( {𝑥 ∈ 𝒫 𝑋 ∣ (𝐴𝑥 ∧ (𝐽t 𝑥) ∈ Conn)} ⊆ [𝐴] ↔ ∀𝑦 ∈ {𝑥 ∈ 𝒫 𝑋 ∣ (𝐴𝑥 ∧ (𝐽t 𝑥) ∈ Conn)}𝑦 ⊆ [𝐴] )
114112, 113sylibr 235 . 2 ((𝐺 ∈ TopGrp ∧ 𝐴𝑋) → {𝑥 ∈ 𝒫 𝑋 ∣ (𝐴𝑥 ∧ (𝐽t 𝑥) ∈ Conn)} ⊆ [𝐴] )
11557, 114eqssd 3981 1 ((𝐺 ∈ TopGrp ∧ 𝐴𝑋) → [𝐴] = {𝑥 ∈ 𝒫 𝑋 ∣ (𝐴𝑥 ∧ (𝐽t 𝑥) ∈ Conn)})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 207  wa 396  w3a 1079   = wceq 1528  wcel 2105  {cab 2796  wral 3135  {crab 3139  wss 3933  𝒫 cpw 4535   cuni 4830   class class class wbr 5057  cmpt 5137  ccnv 5547  dom cdm 5548  cima 5551  Fun wfun 6342  1-1-ontowf1o 6347  cfv 6348  (class class class)co 7145  [cec 8276  Basecbs 16471  +gcplusg 16553  t crest 16682  TopOpenctopn 16683  0gc0g 16701  Grpcgrp 18041  invgcminusg 18042   ~QG cqg 18213  TopOnctopon 21446   Cn ccn 21760  Conncconn 21947  Homeochmeo 22289  TopGrpctgp 22607
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-rep 5181  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7450
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3or 1080  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ne 3014  df-ral 3140  df-rex 3141  df-reu 3142  df-rmo 3143  df-rab 3144  df-v 3494  df-sbc 3770  df-csb 3881  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-pss 3951  df-nul 4289  df-if 4464  df-pw 4537  df-sn 4558  df-pr 4560  df-tp 4562  df-op 4564  df-uni 4831  df-int 4868  df-iun 4912  df-br 5058  df-opab 5120  df-mpt 5138  df-tr 5164  df-id 5453  df-eprel 5458  df-po 5467  df-so 5468  df-fr 5507  df-we 5509  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-pred 6141  df-ord 6187  df-on 6188  df-lim 6189  df-suc 6190  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-f1 6353  df-fo 6354  df-f1o 6355  df-fv 6356  df-riota 7103  df-ov 7148  df-oprab 7149  df-mpo 7150  df-om 7570  df-1st 7678  df-2nd 7679  df-wrecs 7936  df-recs 7997  df-rdg 8035  df-oadd 8095  df-er 8278  df-ec 8280  df-map 8397  df-en 8498  df-fin 8501  df-fi 8863  df-rest 16684  df-0g 16703  df-topgen 16705  df-plusf 17839  df-mgm 17840  df-sgrp 17889  df-mnd 17900  df-grp 18044  df-minusg 18045  df-eqg 18216  df-top 21430  df-topon 21447  df-topsp 21469  df-bases 21482  df-cld 21555  df-cn 21763  df-cnp 21764  df-conn 21948  df-tx 22098  df-hmeo 22291  df-tmd 22608  df-tgp 22609
This theorem is referenced by:  tgpconncomp  22648
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