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Theorem tgpconncompeqg 21855
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 7705 . . . . 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 3672 . . . . . . . . . 10 {𝑥 ∈ 𝒫 𝑋 ∣ ( 0𝑥 ∧ (𝐽t 𝑥) ∈ Conn)} ⊆ 𝒫 𝑋
5 sspwuni 4584 . . . . . . . . . 10 ({𝑥 ∈ 𝒫 𝑋 ∣ ( 0𝑥 ∧ (𝐽t 𝑥) ∈ Conn)} ⊆ 𝒫 𝑋 {𝑥 ∈ 𝒫 𝑋 ∣ ( 0𝑥 ∧ (𝐽t 𝑥) ∈ Conn)} ⊆ 𝑋)
64, 5mpbi 220 . . . . . . . . 9 {𝑥 ∈ 𝒫 𝑋 ∣ ( 0𝑥 ∧ (𝐽t 𝑥) ∈ Conn)} ⊆ 𝑋
73, 6eqsstri 3620 . . . . . . . 8 𝑆𝑋
87a1i 11 . . . . . . 7 ((𝐺 ∈ TopGrp ∧ 𝐴𝑋) → 𝑆𝑋)
9 tgpconncomp.x . . . . . . . 8 𝑋 = (Base‘𝐺)
10 eqid 2621 . . . . . . . 8 (invg𝐺) = (invg𝐺)
11 eqid 2621 . . . . . . . 8 (+g𝐺) = (+g𝐺)
12 tgpconncompeqg.r . . . . . . . 8 = (𝐺 ~QG 𝑆)
139, 10, 11, 12eqgval 17583 . . . . . . 7 ((𝐺 ∈ TopGrp ∧ 𝑆𝑋) → (𝐴 𝑧 ↔ (𝐴𝑋𝑧𝑋 ∧ (((invg𝐺)‘𝐴)(+g𝐺)𝑧) ∈ 𝑆)))
148, 13syldan 487 . . . . . 6 ((𝐺 ∈ TopGrp ∧ 𝐴𝑋) → (𝐴 𝑧 ↔ (𝐴𝑋𝑧𝑋 ∧ (((invg𝐺)‘𝐴)(+g𝐺)𝑧) ∈ 𝑆)))
15 simp2 1060 . . . . . 6 ((𝐴𝑋𝑧𝑋 ∧ (((invg𝐺)‘𝐴)(+g𝐺)𝑧) ∈ 𝑆) → 𝑧𝑋)
1614, 15syl6bi 243 . . . . 5 ((𝐺 ∈ TopGrp ∧ 𝐴𝑋) → (𝐴 𝑧𝑧𝑋))
1716abssdv 3661 . . . 4 ((𝐺 ∈ TopGrp ∧ 𝐴𝑋) → {𝑧𝐴 𝑧} ⊆ 𝑋)
182, 17eqsstrd 3624 . . 3 ((𝐺 ∈ TopGrp ∧ 𝐴𝑋) → [𝐴] 𝑋)
19 simpr 477 . . . . 5 ((𝐺 ∈ TopGrp ∧ 𝐴𝑋) → 𝐴𝑋)
20 tgpgrp 21822 . . . . . . 7 (𝐺 ∈ TopGrp → 𝐺 ∈ Grp)
21 tgpconncomp.z . . . . . . . 8 0 = (0g𝐺)
229, 11, 21, 10grplinv 17408 . . . . . . 7 ((𝐺 ∈ Grp ∧ 𝐴𝑋) → (((invg𝐺)‘𝐴)(+g𝐺)𝐴) = 0 )
2320, 22sylan 488 . . . . . 6 ((𝐺 ∈ TopGrp ∧ 𝐴𝑋) → (((invg𝐺)‘𝐴)(+g𝐺)𝐴) = 0 )
24 tgpconncomp.j . . . . . . . . 9 𝐽 = (TopOpen‘𝐺)
2524, 9tgptopon 21826 . . . . . . . 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 17390 . . . . . . . 8 (𝐺 ∈ Grp → 0𝑋)
2927, 28syl 17 . . . . . . 7 ((𝐺 ∈ TopGrp ∧ 𝐴𝑋) → 0𝑋)
303conncompid 21174 . . . . . . 7 ((𝐽 ∈ (TopOn‘𝑋) ∧ 0𝑋) → 0𝑆)
3126, 29, 30syl2anc 692 . . . . . 6 ((𝐺 ∈ TopGrp ∧ 𝐴𝑋) → 0𝑆)
3223, 31eqeltrd 2698 . . . . 5 ((𝐺 ∈ TopGrp ∧ 𝐴𝑋) → (((invg𝐺)‘𝐴)(+g𝐺)𝐴) ∈ 𝑆)
339, 10, 11, 12eqgval 17583 . . . . . 6 ((𝐺 ∈ TopGrp ∧ 𝑆𝑋) → (𝐴 𝐴 ↔ (𝐴𝑋𝐴𝑋 ∧ (((invg𝐺)‘𝐴)(+g𝐺)𝐴) ∈ 𝑆)))
348, 33syldan 487 . . . . 5 ((𝐺 ∈ TopGrp ∧ 𝐴𝑋) → (𝐴 𝐴 ↔ (𝐴𝑋𝐴𝑋 ∧ (((invg𝐺)‘𝐴)(+g𝐺)𝐴) ∈ 𝑆)))
3519, 19, 32, 34mpbir3and 1243 . . . 4 ((𝐺 ∈ TopGrp ∧ 𝐴𝑋) → 𝐴 𝐴)
36 elecg 7745 . . . . 5 ((𝐴𝑋𝐴𝑋) → (𝐴 ∈ [𝐴] 𝐴 𝐴))
3719, 19, 36syl2anc 692 . . . 4 ((𝐺 ∈ TopGrp ∧ 𝐴𝑋) → (𝐴 ∈ [𝐴] 𝐴 𝐴))
3835, 37mpbird 247 . . 3 ((𝐺 ∈ TopGrp ∧ 𝐴𝑋) → 𝐴 ∈ [𝐴] )
399, 12, 11eqglact 17585 . . . . . . 7 ((𝐺 ∈ Grp ∧ 𝑆𝑋𝐴𝑋) → [𝐴] = ((𝑧𝑋 ↦ (𝐴(+g𝐺)𝑧)) “ 𝑆))
407, 39mp3an2 1409 . . . . . 6 ((𝐺 ∈ Grp ∧ 𝐴𝑋) → [𝐴] = ((𝑧𝑋 ↦ (𝐴(+g𝐺)𝑧)) “ 𝑆))
4120, 40sylan 488 . . . . 5 ((𝐺 ∈ TopGrp ∧ 𝐴𝑋) → [𝐴] = ((𝑧𝑋 ↦ (𝐴(+g𝐺)𝑧)) “ 𝑆))
4241oveq2d 6631 . . . 4 ((𝐺 ∈ TopGrp ∧ 𝐴𝑋) → (𝐽t [𝐴] ) = (𝐽t ((𝑧𝑋 ↦ (𝐴(+g𝐺)𝑧)) “ 𝑆)))
43 eqid 2621 . . . . 5 𝐽 = 𝐽
44 eqid 2621 . . . . . . 7 (𝑧𝑋 ↦ (𝐴(+g𝐺)𝑧)) = (𝑧𝑋 ↦ (𝐴(+g𝐺)𝑧))
4544, 9, 11, 24tgplacthmeo 21847 . . . . . 6 ((𝐺 ∈ TopGrp ∧ 𝐴𝑋) → (𝑧𝑋 ↦ (𝐴(+g𝐺)𝑧)) ∈ (𝐽Homeo𝐽))
46 hmeocn 21503 . . . . . 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 20659 . . . . . . 7 (𝐽 ∈ (TopOn‘𝑋) → 𝑋 = 𝐽)
4926, 48syl 17 . . . . . 6 ((𝐺 ∈ TopGrp ∧ 𝐴𝑋) → 𝑋 = 𝐽)
507, 49syl5sseq 3638 . . . . 5 ((𝐺 ∈ TopGrp ∧ 𝐴𝑋) → 𝑆 𝐽)
513conncompconn 21175 . . . . . 6 ((𝐽 ∈ (TopOn‘𝑋) ∧ 0𝑋) → (𝐽t 𝑆) ∈ Conn)
5226, 29, 51syl2anc 692 . . . . 5 ((𝐺 ∈ TopGrp ∧ 𝐴𝑋) → (𝐽t 𝑆) ∈ Conn)
5343, 47, 50, 52connima 21168 . . . 4 ((𝐺 ∈ TopGrp ∧ 𝐴𝑋) → (𝐽t ((𝑧𝑋 ↦ (𝐴(+g𝐺)𝑧)) “ 𝑆)) ∈ Conn)
5442, 53eqeltrd 2698 . . 3 ((𝐺 ∈ TopGrp ∧ 𝐴𝑋) → (𝐽t [𝐴] ) ∈ Conn)
55 eqid 2621 . . . 4 {𝑥 ∈ 𝒫 𝑋 ∣ (𝐴𝑥 ∧ (𝐽t 𝑥) ∈ Conn)} = {𝑥 ∈ 𝒫 𝑋 ∣ (𝐴𝑥 ∧ (𝐽t 𝑥) ∈ Conn)}
5655conncompss 21176 . . 3 (([𝐴] 𝑋𝐴 ∈ [𝐴] ∧ (𝐽t [𝐴] ) ∈ Conn) → [𝐴] {𝑥 ∈ 𝒫 𝑋 ∣ (𝐴𝑥 ∧ (𝐽t 𝑥) ∈ Conn)})
5718, 38, 54, 56syl3anc 1323 . 2 ((𝐺 ∈ TopGrp ∧ 𝐴𝑋) → [𝐴] {𝑥 ∈ 𝒫 𝑋 ∣ (𝐴𝑥 ∧ (𝐽t 𝑥) ∈ Conn)})
58 elpwi 4146 . . . . . 6 (𝑦 ∈ 𝒫 𝑋𝑦𝑋)
5944mptpreima 5597 . . . . . . . . . . . 12 ((𝑧𝑋 ↦ (𝐴(+g𝐺)𝑧)) “ 𝑦) = {𝑧𝑋 ∣ (𝐴(+g𝐺)𝑧) ∈ 𝑦}
60 ssrab2 3672 . . . . . . . . . . . 12 {𝑧𝑋 ∣ (𝐴(+g𝐺)𝑧) ∈ 𝑦} ⊆ 𝑋
6159, 60eqsstri 3620 . . . . . . . . . . 11 ((𝑧𝑋 ↦ (𝐴(+g𝐺)𝑧)) “ 𝑦) ⊆ 𝑋
6261a1i 11 . . . . . . . . . 10 (((𝐺 ∈ TopGrp ∧ 𝐴𝑋) ∧ (𝑦𝑋 ∧ (𝐴𝑦 ∧ (𝐽t 𝑦) ∈ Conn))) → ((𝑧𝑋 ↦ (𝐴(+g𝐺)𝑧)) “ 𝑦) ⊆ 𝑋)
6329adantr 481 . . . . . . . . . . 11 (((𝐺 ∈ TopGrp ∧ 𝐴𝑋) ∧ (𝑦𝑋 ∧ (𝐴𝑦 ∧ (𝐽t 𝑦) ∈ Conn))) → 0𝑋)
649, 11, 21grprid 17393 . . . . . . . . . . . . . 14 ((𝐺 ∈ Grp ∧ 𝐴𝑋) → (𝐴(+g𝐺) 0 ) = 𝐴)
6520, 64sylan 488 . . . . . . . . . . . . 13 ((𝐺 ∈ TopGrp ∧ 𝐴𝑋) → (𝐴(+g𝐺) 0 ) = 𝐴)
6665adantr 481 . . . . . . . . . . . 12 (((𝐺 ∈ TopGrp ∧ 𝐴𝑋) ∧ (𝑦𝑋 ∧ (𝐴𝑦 ∧ (𝐽t 𝑦) ∈ Conn))) → (𝐴(+g𝐺) 0 ) = 𝐴)
67 simprrl 803 . . . . . . . . . . . 12 (((𝐺 ∈ TopGrp ∧ 𝐴𝑋) ∧ (𝑦𝑋 ∧ (𝐴𝑦 ∧ (𝐽t 𝑦) ∈ Conn))) → 𝐴𝑦)
6866, 67eqeltrd 2698 . . . . . . . . . . 11 (((𝐺 ∈ TopGrp ∧ 𝐴𝑋) ∧ (𝑦𝑋 ∧ (𝐴𝑦 ∧ (𝐽t 𝑦) ∈ Conn))) → (𝐴(+g𝐺) 0 ) ∈ 𝑦)
69 oveq2 6623 . . . . . . . . . . . . 13 (𝑧 = 0 → (𝐴(+g𝐺)𝑧) = (𝐴(+g𝐺) 0 ))
7069eleq1d 2683 . . . . . . . . . . . 12 (𝑧 = 0 → ((𝐴(+g𝐺)𝑧) ∈ 𝑦 ↔ (𝐴(+g𝐺) 0 ) ∈ 𝑦))
7170, 59elrab2 3353 . . . . . . . . . . 11 ( 0 ∈ ((𝑧𝑋 ↦ (𝐴(+g𝐺)𝑧)) “ 𝑦) ↔ ( 0𝑋 ∧ (𝐴(+g𝐺) 0 ) ∈ 𝑦))
7263, 68, 71sylanbrc 697 . . . . . . . . . 10 (((𝐺 ∈ TopGrp ∧ 𝐴𝑋) ∧ (𝑦𝑋 ∧ (𝐴𝑦 ∧ (𝐽t 𝑦) ∈ Conn))) → 0 ∈ ((𝑧𝑋 ↦ (𝐴(+g𝐺)𝑧)) “ 𝑦))
73 hmeocnvcn 21504 . . . . . . . . . . . . 13 ((𝑧𝑋 ↦ (𝐴(+g𝐺)𝑧)) ∈ (𝐽Homeo𝐽) → (𝑧𝑋 ↦ (𝐴(+g𝐺)𝑧)) ∈ (𝐽 Cn 𝐽))
7445, 73syl 17 . . . . . . . . . . . 12 ((𝐺 ∈ TopGrp ∧ 𝐴𝑋) → (𝑧𝑋 ↦ (𝐴(+g𝐺)𝑧)) ∈ (𝐽 Cn 𝐽))
7574adantr 481 . . . . . . . . . . 11 (((𝐺 ∈ TopGrp ∧ 𝐴𝑋) ∧ (𝑦𝑋 ∧ (𝐴𝑦 ∧ (𝐽t 𝑦) ∈ Conn))) → (𝑧𝑋 ↦ (𝐴(+g𝐺)𝑧)) ∈ (𝐽 Cn 𝐽))
76 simprl 793 . . . . . . . . . . . 12 (((𝐺 ∈ TopGrp ∧ 𝐴𝑋) ∧ (𝑦𝑋 ∧ (𝐴𝑦 ∧ (𝐽t 𝑦) ∈ Conn))) → 𝑦𝑋)
7749adantr 481 . . . . . . . . . . . 12 (((𝐺 ∈ TopGrp ∧ 𝐴𝑋) ∧ (𝑦𝑋 ∧ (𝐴𝑦 ∧ (𝐽t 𝑦) ∈ Conn))) → 𝑋 = 𝐽)
7876, 77sseqtrd 3626 . . . . . . . . . . 11 (((𝐺 ∈ TopGrp ∧ 𝐴𝑋) ∧ (𝑦𝑋 ∧ (𝐴𝑦 ∧ (𝐽t 𝑦) ∈ Conn))) → 𝑦 𝐽)
79 simprrr 804 . . . . . . . . . . 11 (((𝐺 ∈ TopGrp ∧ 𝐴𝑋) ∧ (𝑦𝑋 ∧ (𝐴𝑦 ∧ (𝐽t 𝑦) ∈ Conn))) → (𝐽t 𝑦) ∈ Conn)
8043, 75, 78, 79connima 21168 . . . . . . . . . 10 (((𝐺 ∈ TopGrp ∧ 𝐴𝑋) ∧ (𝑦𝑋 ∧ (𝐴𝑦 ∧ (𝐽t 𝑦) ∈ Conn))) → (𝐽t ((𝑧𝑋 ↦ (𝐴(+g𝐺)𝑧)) “ 𝑦)) ∈ Conn)
813conncompss 21176 . . . . . . . . . 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)
8262, 72, 80, 81syl3anc 1323 . . . . . . . . 9 (((𝐺 ∈ TopGrp ∧ 𝐴𝑋) ∧ (𝑦𝑋 ∧ (𝐴𝑦 ∧ (𝐽t 𝑦) ∈ Conn))) → ((𝑧𝑋 ↦ (𝐴(+g𝐺)𝑧)) “ 𝑦) ⊆ 𝑆)
83 eqid 2621 . . . . . . . . . . . . . . . 16 (𝑔𝑋 ↦ (𝑧𝑋 ↦ (𝑔(+g𝐺)𝑧))) = (𝑔𝑋 ↦ (𝑧𝑋 ↦ (𝑔(+g𝐺)𝑧)))
8483, 9, 11, 10grplactcnv 17458 . . . . . . . . . . . . . . 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))))
8520, 84sylan 488 . . . . . . . . . . . . . 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))))
8685simpld 475 . . . . . . . . . . . . 13 ((𝐺 ∈ TopGrp ∧ 𝐴𝑋) → ((𝑔𝑋 ↦ (𝑧𝑋 ↦ (𝑔(+g𝐺)𝑧)))‘𝐴):𝑋1-1-onto𝑋)
8783, 9grplactfval 17456 . . . . . . . . . . . . . . 15 (𝐴𝑋 → ((𝑔𝑋 ↦ (𝑧𝑋 ↦ (𝑔(+g𝐺)𝑧)))‘𝐴) = (𝑧𝑋 ↦ (𝐴(+g𝐺)𝑧)))
8887adantl 482 . . . . . . . . . . . . . 14 ((𝐺 ∈ TopGrp ∧ 𝐴𝑋) → ((𝑔𝑋 ↦ (𝑧𝑋 ↦ (𝑔(+g𝐺)𝑧)))‘𝐴) = (𝑧𝑋 ↦ (𝐴(+g𝐺)𝑧)))
89 f1oeq1 6094 . . . . . . . . . . . . . 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))
9088, 89syl 17 . . . . . . . . . . . . 13 ((𝐺 ∈ TopGrp ∧ 𝐴𝑋) → (((𝑔𝑋 ↦ (𝑧𝑋 ↦ (𝑔(+g𝐺)𝑧)))‘𝐴):𝑋1-1-onto𝑋 ↔ (𝑧𝑋 ↦ (𝐴(+g𝐺)𝑧)):𝑋1-1-onto𝑋))
9186, 90mpbid 222 . . . . . . . . . . . 12 ((𝐺 ∈ TopGrp ∧ 𝐴𝑋) → (𝑧𝑋 ↦ (𝐴(+g𝐺)𝑧)):𝑋1-1-onto𝑋)
9291adantr 481 . . . . . . . . . . 11 (((𝐺 ∈ TopGrp ∧ 𝐴𝑋) ∧ (𝑦𝑋 ∧ (𝐴𝑦 ∧ (𝐽t 𝑦) ∈ Conn))) → (𝑧𝑋 ↦ (𝐴(+g𝐺)𝑧)):𝑋1-1-onto𝑋)
93 f1ocnv 6116 . . . . . . . . . . 11 ((𝑧𝑋 ↦ (𝐴(+g𝐺)𝑧)):𝑋1-1-onto𝑋(𝑧𝑋 ↦ (𝐴(+g𝐺)𝑧)):𝑋1-1-onto𝑋)
94 f1ofun 6106 . . . . . . . . . . 11 ((𝑧𝑋 ↦ (𝐴(+g𝐺)𝑧)):𝑋1-1-onto𝑋 → Fun (𝑧𝑋 ↦ (𝐴(+g𝐺)𝑧)))
9592, 93, 943syl 18 . . . . . . . . . 10 (((𝐺 ∈ TopGrp ∧ 𝐴𝑋) ∧ (𝑦𝑋 ∧ (𝐴𝑦 ∧ (𝐽t 𝑦) ∈ Conn))) → Fun (𝑧𝑋 ↦ (𝐴(+g𝐺)𝑧)))
96 f1odm 6108 . . . . . . . . . . . 12 ((𝑧𝑋 ↦ (𝐴(+g𝐺)𝑧)):𝑋1-1-onto𝑋 → dom (𝑧𝑋 ↦ (𝐴(+g𝐺)𝑧)) = 𝑋)
9792, 93, 963syl 18 . . . . . . . . . . 11 (((𝐺 ∈ TopGrp ∧ 𝐴𝑋) ∧ (𝑦𝑋 ∧ (𝐴𝑦 ∧ (𝐽t 𝑦) ∈ Conn))) → dom (𝑧𝑋 ↦ (𝐴(+g𝐺)𝑧)) = 𝑋)
9876, 97sseqtr4d 3627 . . . . . . . . . 10 (((𝐺 ∈ TopGrp ∧ 𝐴𝑋) ∧ (𝑦𝑋 ∧ (𝐴𝑦 ∧ (𝐽t 𝑦) ∈ Conn))) → 𝑦 ⊆ dom (𝑧𝑋 ↦ (𝐴(+g𝐺)𝑧)))
99 funimass3 6299 . . . . . . . . . 10 ((Fun (𝑧𝑋 ↦ (𝐴(+g𝐺)𝑧)) ∧ 𝑦 ⊆ dom (𝑧𝑋 ↦ (𝐴(+g𝐺)𝑧))) → (((𝑧𝑋 ↦ (𝐴(+g𝐺)𝑧)) “ 𝑦) ⊆ 𝑆𝑦 ⊆ ((𝑧𝑋 ↦ (𝐴(+g𝐺)𝑧)) “ 𝑆)))
10095, 98, 99syl2anc 692 . . . . . . . . 9 (((𝐺 ∈ TopGrp ∧ 𝐴𝑋) ∧ (𝑦𝑋 ∧ (𝐴𝑦 ∧ (𝐽t 𝑦) ∈ Conn))) → (((𝑧𝑋 ↦ (𝐴(+g𝐺)𝑧)) “ 𝑦) ⊆ 𝑆𝑦 ⊆ ((𝑧𝑋 ↦ (𝐴(+g𝐺)𝑧)) “ 𝑆)))
10182, 100mpbid 222 . . . . . . . 8 (((𝐺 ∈ TopGrp ∧ 𝐴𝑋) ∧ (𝑦𝑋 ∧ (𝐴𝑦 ∧ (𝐽t 𝑦) ∈ Conn))) → 𝑦 ⊆ ((𝑧𝑋 ↦ (𝐴(+g𝐺)𝑧)) “ 𝑆))
10241adantr 481 . . . . . . . . 9 (((𝐺 ∈ TopGrp ∧ 𝐴𝑋) ∧ (𝑦𝑋 ∧ (𝐴𝑦 ∧ (𝐽t 𝑦) ∈ Conn))) → [𝐴] = ((𝑧𝑋 ↦ (𝐴(+g𝐺)𝑧)) “ 𝑆))
103 imacnvcnv 5568 . . . . . . . . 9 ((𝑧𝑋 ↦ (𝐴(+g𝐺)𝑧)) “ 𝑆) = ((𝑧𝑋 ↦ (𝐴(+g𝐺)𝑧)) “ 𝑆)
104102, 103syl6eqr 2673 . . . . . . . 8 (((𝐺 ∈ TopGrp ∧ 𝐴𝑋) ∧ (𝑦𝑋 ∧ (𝐴𝑦 ∧ (𝐽t 𝑦) ∈ Conn))) → [𝐴] = ((𝑧𝑋 ↦ (𝐴(+g𝐺)𝑧)) “ 𝑆))
105101, 104sseqtr4d 3627 . . . . . . 7 (((𝐺 ∈ TopGrp ∧ 𝐴𝑋) ∧ (𝑦𝑋 ∧ (𝐴𝑦 ∧ (𝐽t 𝑦) ∈ Conn))) → 𝑦 ⊆ [𝐴] )
106105expr 642 . . . . . 6 (((𝐺 ∈ TopGrp ∧ 𝐴𝑋) ∧ 𝑦𝑋) → ((𝐴𝑦 ∧ (𝐽t 𝑦) ∈ Conn) → 𝑦 ⊆ [𝐴] ))
10758, 106sylan2 491 . . . . 5 (((𝐺 ∈ TopGrp ∧ 𝐴𝑋) ∧ 𝑦 ∈ 𝒫 𝑋) → ((𝐴𝑦 ∧ (𝐽t 𝑦) ∈ Conn) → 𝑦 ⊆ [𝐴] ))
108107ralrimiva 2962 . . . 4 ((𝐺 ∈ TopGrp ∧ 𝐴𝑋) → ∀𝑦 ∈ 𝒫 𝑋((𝐴𝑦 ∧ (𝐽t 𝑦) ∈ Conn) → 𝑦 ⊆ [𝐴] ))
109 eleq2 2687 . . . . . 6 (𝑥 = 𝑦 → (𝐴𝑥𝐴𝑦))
110 oveq2 6623 . . . . . . 7 (𝑥 = 𝑦 → (𝐽t 𝑥) = (𝐽t 𝑦))
111110eleq1d 2683 . . . . . 6 (𝑥 = 𝑦 → ((𝐽t 𝑥) ∈ Conn ↔ (𝐽t 𝑦) ∈ Conn))
112109, 111anbi12d 746 . . . . 5 (𝑥 = 𝑦 → ((𝐴𝑥 ∧ (𝐽t 𝑥) ∈ Conn) ↔ (𝐴𝑦 ∧ (𝐽t 𝑦) ∈ Conn)))
113112ralrab 3355 . . . 4 (∀𝑦 ∈ {𝑥 ∈ 𝒫 𝑋 ∣ (𝐴𝑥 ∧ (𝐽t 𝑥) ∈ Conn)}𝑦 ⊆ [𝐴] ↔ ∀𝑦 ∈ 𝒫 𝑋((𝐴𝑦 ∧ (𝐽t 𝑦) ∈ Conn) → 𝑦 ⊆ [𝐴] ))
114108, 113sylibr 224 . . 3 ((𝐺 ∈ TopGrp ∧ 𝐴𝑋) → ∀𝑦 ∈ {𝑥 ∈ 𝒫 𝑋 ∣ (𝐴𝑥 ∧ (𝐽t 𝑥) ∈ Conn)}𝑦 ⊆ [𝐴] )
115 unissb 4442 . . 3 ( {𝑥 ∈ 𝒫 𝑋 ∣ (𝐴𝑥 ∧ (𝐽t 𝑥) ∈ Conn)} ⊆ [𝐴] ↔ ∀𝑦 ∈ {𝑥 ∈ 𝒫 𝑋 ∣ (𝐴𝑥 ∧ (𝐽t 𝑥) ∈ Conn)}𝑦 ⊆ [𝐴] )
116114, 115sylibr 224 . 2 ((𝐺 ∈ TopGrp ∧ 𝐴𝑋) → {𝑥 ∈ 𝒫 𝑋 ∣ (𝐴𝑥 ∧ (𝐽t 𝑥) ∈ Conn)} ⊆ [𝐴] )
11757, 116eqssd 3605 1 ((𝐺 ∈ TopGrp ∧ 𝐴𝑋) → [𝐴] = {𝑥 ∈ 𝒫 𝑋 ∣ (𝐴𝑥 ∧ (𝐽t 𝑥) ∈ Conn)})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 384  w3a 1036   = wceq 1480  wcel 1987  {cab 2607  wral 2908  {crab 2912  wss 3560  𝒫 cpw 4136   cuni 4409   class class class wbr 4623  cmpt 4683  ccnv 5083  dom cdm 5084  cima 5087  Fun wfun 5851  1-1-ontowf1o 5856  cfv 5857  (class class class)co 6615  [cec 7700  Basecbs 15800  +gcplusg 15881  t crest 16021  TopOpenctopn 16022  0gc0g 16040  Grpcgrp 17362  invgcminusg 17363   ~QG cqg 17530  TopOnctopon 20655   Cn ccn 20968  Conncconn 21154  Homeochmeo 21496  TopGrpctgp 21815
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4741  ax-sep 4751  ax-nul 4759  ax-pow 4813  ax-pr 4877  ax-un 6914
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2913  df-rex 2914  df-reu 2915  df-rmo 2916  df-rab 2917  df-v 3192  df-sbc 3423  df-csb 3520  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-pss 3576  df-nul 3898  df-if 4065  df-pw 4138  df-sn 4156  df-pr 4158  df-tp 4160  df-op 4162  df-uni 4410  df-int 4448  df-iun 4494  df-br 4624  df-opab 4684  df-mpt 4685  df-tr 4723  df-eprel 4995  df-id 4999  df-po 5005  df-so 5006  df-fr 5043  df-we 5045  df-xp 5090  df-rel 5091  df-cnv 5092  df-co 5093  df-dm 5094  df-rn 5095  df-res 5096  df-ima 5097  df-pred 5649  df-ord 5695  df-on 5696  df-lim 5697  df-suc 5698  df-iota 5820  df-fun 5859  df-fn 5860  df-f 5861  df-f1 5862  df-fo 5863  df-f1o 5864  df-fv 5865  df-riota 6576  df-ov 6618  df-oprab 6619  df-mpt2 6620  df-om 7028  df-1st 7128  df-2nd 7129  df-wrecs 7367  df-recs 7428  df-rdg 7466  df-oadd 7524  df-er 7702  df-ec 7704  df-map 7819  df-en 7916  df-fin 7919  df-fi 8277  df-rest 16023  df-0g 16042  df-topgen 16044  df-plusf 17181  df-mgm 17182  df-sgrp 17224  df-mnd 17235  df-grp 17365  df-minusg 17366  df-eqg 17533  df-top 20639  df-topon 20656  df-topsp 20677  df-bases 20690  df-cld 20763  df-cn 20971  df-cnp 20972  df-conn 21155  df-tx 21305  df-hmeo 21498  df-tmd 21816  df-tgp 21817
This theorem is referenced by:  tgpconncomp  21856
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