Theorem snclseqg 24124

Description: The coset of the closure of the identity is the closure of a point. (Contributed by Mario Carneiro, 22-Sep-2015.)

Hypotheses

Ref	Expression
snclseqg.x	⊢ 𝑋 = (Base‘𝐺)
snclseqg.j	⊢ 𝐽 = (TopOpen‘𝐺)
snclseqg.z	⊢ 0 = (0g‘𝐺)
snclseqg.r	⊢ ∼ = (𝐺 ~QG 𝑆)
snclseqg.s	⊢ 𝑆 = ((cls‘𝐽)‘{ 0 })

Assertion

Ref	Expression
snclseqg	⊢ ((𝐺 ∈ TopGrp ∧ 𝐴 ∈ 𝑋) → [𝐴] ∼ = ((cls‘𝐽)‘{𝐴}))

Proof of Theorem snclseqg

Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		snclseqg.s	. . . 4 ⊢ 𝑆 = ((cls‘𝐽)‘{ 0 })
2	1	imaeq2i 6076	. . 3 ⊢ ((𝑥 ∈ 𝑋 ↦ (𝐴(+g‘𝐺)𝑥)) “ 𝑆) = ((𝑥 ∈ 𝑋 ↦ (𝐴(+g‘𝐺)𝑥)) “ ((cls‘𝐽)‘{ 0 }))
3		tgpgrp 24086	. . . . 5 ⊢ (𝐺 ∈ TopGrp → 𝐺 ∈ Grp)
4	3	adantr 480	. . . 4 ⊢ ((𝐺 ∈ TopGrp ∧ 𝐴 ∈ 𝑋) → 𝐺 ∈ Grp)
5		snclseqg.j	. . . . . . . . . 10 ⊢ 𝐽 = (TopOpen‘𝐺)
6		snclseqg.x	. . . . . . . . . 10 ⊢ 𝑋 = (Base‘𝐺)
7	5, 6	tgptopon 24090	. . . . . . . . 9 ⊢ (𝐺 ∈ TopGrp → 𝐽 ∈ (TopOn‘𝑋))
8	7	adantr 480	. . . . . . . 8 ⊢ ((𝐺 ∈ TopGrp ∧ 𝐴 ∈ 𝑋) → 𝐽 ∈ (TopOn‘𝑋))
9		topontop 22919	. . . . . . . 8 ⊢ (𝐽 ∈ (TopOn‘𝑋) → 𝐽 ∈ Top)
10	8, 9	syl 17	. . . . . . 7 ⊢ ((𝐺 ∈ TopGrp ∧ 𝐴 ∈ 𝑋) → 𝐽 ∈ Top)
11		snclseqg.z	. . . . . . . . . . 11 ⊢ 0 = (0g‘𝐺)
12	6, 11	grpidcl 18983	. . . . . . . . . 10 ⊢ (𝐺 ∈ Grp → 0 ∈ 𝑋)
13	4, 12	syl 17	. . . . . . . . 9 ⊢ ((𝐺 ∈ TopGrp ∧ 𝐴 ∈ 𝑋) → 0 ∈ 𝑋)
14	13	snssd 4809	. . . . . . . 8 ⊢ ((𝐺 ∈ TopGrp ∧ 𝐴 ∈ 𝑋) → { 0 } ⊆ 𝑋)
15		toponuni 22920	. . . . . . . . 9 ⊢ (𝐽 ∈ (TopOn‘𝑋) → 𝑋 = ∪ 𝐽)
16	8, 15	syl 17	. . . . . . . 8 ⊢ ((𝐺 ∈ TopGrp ∧ 𝐴 ∈ 𝑋) → 𝑋 = ∪ 𝐽)
17	14, 16	sseqtrd 4020	. . . . . . 7 ⊢ ((𝐺 ∈ TopGrp ∧ 𝐴 ∈ 𝑋) → { 0 } ⊆ ∪ 𝐽)
18		eqid 2737	. . . . . . . 8 ⊢ ∪ 𝐽 = ∪ 𝐽
19	18	clsss3 23067	. . . . . . 7 ⊢ ((𝐽 ∈ Top ∧ { 0 } ⊆ ∪ 𝐽) → ((cls‘𝐽)‘{ 0 }) ⊆ ∪ 𝐽)
20	10, 17, 19	syl2anc 584	. . . . . 6 ⊢ ((𝐺 ∈ TopGrp ∧ 𝐴 ∈ 𝑋) → ((cls‘𝐽)‘{ 0 }) ⊆ ∪ 𝐽)
21	20, 16	sseqtrrd 4021	. . . . 5 ⊢ ((𝐺 ∈ TopGrp ∧ 𝐴 ∈ 𝑋) → ((cls‘𝐽)‘{ 0 }) ⊆ 𝑋)
22	1, 21	eqsstrid 4022	. . . 4 ⊢ ((𝐺 ∈ TopGrp ∧ 𝐴 ∈ 𝑋) → 𝑆 ⊆ 𝑋)
23		simpr 484	. . . 4 ⊢ ((𝐺 ∈ TopGrp ∧ 𝐴 ∈ 𝑋) → 𝐴 ∈ 𝑋)
24		snclseqg.r	. . . . 5 ⊢ ∼ = (𝐺 ~QG 𝑆)
25		eqid 2737	. . . . 5 ⊢ (+g‘𝐺) = (+g‘𝐺)
26	6, 24, 25	eqglact 19197	. . . 4 ⊢ ((𝐺 ∈ Grp ∧ 𝑆 ⊆ 𝑋 ∧ 𝐴 ∈ 𝑋) → [𝐴] ∼ = ((𝑥 ∈ 𝑋 ↦ (𝐴(+g‘𝐺)𝑥)) “ 𝑆))
27	4, 22, 23, 26	syl3anc 1373	. . 3 ⊢ ((𝐺 ∈ TopGrp ∧ 𝐴 ∈ 𝑋) → [𝐴] ∼ = ((𝑥 ∈ 𝑋 ↦ (𝐴(+g‘𝐺)𝑥)) “ 𝑆))
28		eqid 2737	. . . . 5 ⊢ (𝑥 ∈ 𝑋 ↦ (𝐴(+g‘𝐺)𝑥)) = (𝑥 ∈ 𝑋 ↦ (𝐴(+g‘𝐺)𝑥))
29	28, 6, 25, 5	tgplacthmeo 24111	. . . 4 ⊢ ((𝐺 ∈ TopGrp ∧ 𝐴 ∈ 𝑋) → (𝑥 ∈ 𝑋 ↦ (𝐴(+g‘𝐺)𝑥)) ∈ (𝐽Homeo𝐽))
30	18	hmeocls 23776	. . . 4 ⊢ (((𝑥 ∈ 𝑋 ↦ (𝐴(+g‘𝐺)𝑥)) ∈ (𝐽Homeo𝐽) ∧ { 0 } ⊆ ∪ 𝐽) → ((cls‘𝐽)‘((𝑥 ∈ 𝑋 ↦ (𝐴(+g‘𝐺)𝑥)) “ { 0 })) = ((𝑥 ∈ 𝑋 ↦ (𝐴(+g‘𝐺)𝑥)) “ ((cls‘𝐽)‘{ 0 })))
31	29, 17, 30	syl2anc 584	. . 3 ⊢ ((𝐺 ∈ TopGrp ∧ 𝐴 ∈ 𝑋) → ((cls‘𝐽)‘((𝑥 ∈ 𝑋 ↦ (𝐴(+g‘𝐺)𝑥)) “ { 0 })) = ((𝑥 ∈ 𝑋 ↦ (𝐴(+g‘𝐺)𝑥)) “ ((cls‘𝐽)‘{ 0 })))
32	2, 27, 31	3eqtr4a 2803	. 2 ⊢ ((𝐺 ∈ TopGrp ∧ 𝐴 ∈ 𝑋) → [𝐴] ∼ = ((cls‘𝐽)‘((𝑥 ∈ 𝑋 ↦ (𝐴(+g‘𝐺)𝑥)) “ { 0 })))
33		df-ima 5698	. . . . 5 ⊢ ((𝑥 ∈ 𝑋 ↦ (𝐴(+g‘𝐺)𝑥)) “ { 0 }) = ran ((𝑥 ∈ 𝑋 ↦ (𝐴(+g‘𝐺)𝑥)) ↾ { 0 })
34	14	resmptd 6058	. . . . . 6 ⊢ ((𝐺 ∈ TopGrp ∧ 𝐴 ∈ 𝑋) → ((𝑥 ∈ 𝑋 ↦ (𝐴(+g‘𝐺)𝑥)) ↾ { 0 }) = (𝑥 ∈ { 0 } ↦ (𝐴(+g‘𝐺)𝑥)))
35	34	rneqd 5949	. . . . 5 ⊢ ((𝐺 ∈ TopGrp ∧ 𝐴 ∈ 𝑋) → ran ((𝑥 ∈ 𝑋 ↦ (𝐴(+g‘𝐺)𝑥)) ↾ { 0 }) = ran (𝑥 ∈ { 0 } ↦ (𝐴(+g‘𝐺)𝑥)))
36	33, 35	eqtrid 2789	. . . 4 ⊢ ((𝐺 ∈ TopGrp ∧ 𝐴 ∈ 𝑋) → ((𝑥 ∈ 𝑋 ↦ (𝐴(+g‘𝐺)𝑥)) “ { 0 }) = ran (𝑥 ∈ { 0 } ↦ (𝐴(+g‘𝐺)𝑥)))
37	11	fvexi 6920	. . . . . . . 8 ⊢ 0 ∈ V
38		oveq2 7439	. . . . . . . . 9 ⊢ (𝑥 = 0 → (𝐴(+g‘𝐺)𝑥) = (𝐴(+g‘𝐺) 0 ))
39	38	eqeq2d 2748	. . . . . . . 8 ⊢ (𝑥 = 0 → (𝑦 = (𝐴(+g‘𝐺)𝑥) ↔ 𝑦 = (𝐴(+g‘𝐺) 0 )))
40	37, 39	rexsn 4682	. . . . . . 7 ⊢ (∃𝑥 ∈ { 0 }𝑦 = (𝐴(+g‘𝐺)𝑥) ↔ 𝑦 = (𝐴(+g‘𝐺) 0 ))
41	6, 25, 11	grprid 18986	. . . . . . . . 9 ⊢ ((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋) → (𝐴(+g‘𝐺) 0 ) = 𝐴)
42	3, 41	sylan 580	. . . . . . . 8 ⊢ ((𝐺 ∈ TopGrp ∧ 𝐴 ∈ 𝑋) → (𝐴(+g‘𝐺) 0 ) = 𝐴)
43	42	eqeq2d 2748	. . . . . . 7 ⊢ ((𝐺 ∈ TopGrp ∧ 𝐴 ∈ 𝑋) → (𝑦 = (𝐴(+g‘𝐺) 0 ) ↔ 𝑦 = 𝐴))
44	40, 43	bitrid 283	. . . . . 6 ⊢ ((𝐺 ∈ TopGrp ∧ 𝐴 ∈ 𝑋) → (∃𝑥 ∈ { 0 }𝑦 = (𝐴(+g‘𝐺)𝑥) ↔ 𝑦 = 𝐴))
45	44	abbidv 2808	. . . . 5 ⊢ ((𝐺 ∈ TopGrp ∧ 𝐴 ∈ 𝑋) → {𝑦 ∣ ∃𝑥 ∈ { 0 }𝑦 = (𝐴(+g‘𝐺)𝑥)} = {𝑦 ∣ 𝑦 = 𝐴})
46		eqid 2737	. . . . . 6 ⊢ (𝑥 ∈ { 0 } ↦ (𝐴(+g‘𝐺)𝑥)) = (𝑥 ∈ { 0 } ↦ (𝐴(+g‘𝐺)𝑥))
47	46	rnmpt 5968	. . . . 5 ⊢ ran (𝑥 ∈ { 0 } ↦ (𝐴(+g‘𝐺)𝑥)) = {𝑦 ∣ ∃𝑥 ∈ { 0 }𝑦 = (𝐴(+g‘𝐺)𝑥)}
48		df-sn 4627	. . . . 5 ⊢ {𝐴} = {𝑦 ∣ 𝑦 = 𝐴}
49	45, 47, 48	3eqtr4g 2802	. . . 4 ⊢ ((𝐺 ∈ TopGrp ∧ 𝐴 ∈ 𝑋) → ran (𝑥 ∈ { 0 } ↦ (𝐴(+g‘𝐺)𝑥)) = {𝐴})
50	36, 49	eqtrd 2777	. . 3 ⊢ ((𝐺 ∈ TopGrp ∧ 𝐴 ∈ 𝑋) → ((𝑥 ∈ 𝑋 ↦ (𝐴(+g‘𝐺)𝑥)) “ { 0 }) = {𝐴})
51	50	fveq2d 6910	. 2 ⊢ ((𝐺 ∈ TopGrp ∧ 𝐴 ∈ 𝑋) → ((cls‘𝐽)‘((𝑥 ∈ 𝑋 ↦ (𝐴(+g‘𝐺)𝑥)) “ { 0 })) = ((cls‘𝐽)‘{𝐴}))
52	32, 51	eqtrd 2777	1 ⊢ ((𝐺 ∈ TopGrp ∧ 𝐴 ∈ 𝑋) → [𝐴] ∼ = ((cls‘𝐽)‘{𝐴}))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1540   ∈ wcel 2108  {cab 2714  ∃wrex 3070   ⊆ wss 3951  {csn 4626  ∪ cuni 4907   ↦ cmpt 5225  ran crn 5686   ↾ cres 5687   “ cima 5688  ‘cfv 6561  (class class class)co 7431  [cec 8743  Basecbs 17247  +_gcplusg 17297  TopOpenctopn 17466  0_gc0g 17484  Grpcgrp 18951   ~_QG cqg 19140  Topctop 22899  TopOnctopon 22916  clsccl 23026  Homeochmeo 23761  TopGrpctgp 24079

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-rmo 3380  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-int 4947  df-iun 4993  df-iin 4994  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-1st 8014  df-2nd 8015  df-ec 8747  df-map 8868  df-0g 17486  df-topgen 17488  df-plusf 18652  df-mgm 18653  df-sgrp 18732  df-mnd 18748  df-grp 18954  df-minusg 18955  df-eqg 19143  df-top 22900  df-topon 22917  df-topsp 22939  df-bases 22953  df-cld 23027  df-cls 23029  df-cn 23235  df-cnp 23236  df-tx 23570  df-hmeo 23763  df-tmd 24080  df-tgp 24081

This theorem is referenced by:  tgptsmscls  24158