Theorem snclseqg 24054

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 6045	. . 3 ⊢ ((𝑥 ∈ 𝑋 ↦ (𝐴(+g‘𝐺)𝑥)) “ 𝑆) = ((𝑥 ∈ 𝑋 ↦ (𝐴(+g‘𝐺)𝑥)) “ ((cls‘𝐽)‘{ 0 }))
3		tgpgrp 24016	. . . . 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 24020	. . . . . . . . 9 ⊢ (𝐺 ∈ TopGrp → 𝐽 ∈ (TopOn‘𝑋))
8	7	adantr 480	. . . . . . . 8 ⊢ ((𝐺 ∈ TopGrp ∧ 𝐴 ∈ 𝑋) → 𝐽 ∈ (TopOn‘𝑋))
9		topontop 22851	. . . . . . . 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 18948	. . . . . . . . . 10 ⊢ (𝐺 ∈ Grp → 0 ∈ 𝑋)
13	4, 12	syl 17	. . . . . . . . 9 ⊢ ((𝐺 ∈ TopGrp ∧ 𝐴 ∈ 𝑋) → 0 ∈ 𝑋)
14	13	snssd 4785	. . . . . . . 8 ⊢ ((𝐺 ∈ TopGrp ∧ 𝐴 ∈ 𝑋) → { 0 } ⊆ 𝑋)
15		toponuni 22852	. . . . . . . . 9 ⊢ (𝐽 ∈ (TopOn‘𝑋) → 𝑋 = ∪ 𝐽)
16	8, 15	syl 17	. . . . . . . 8 ⊢ ((𝐺 ∈ TopGrp ∧ 𝐴 ∈ 𝑋) → 𝑋 = ∪ 𝐽)
17	14, 16	sseqtrd 3995	. . . . . . 7 ⊢ ((𝐺 ∈ TopGrp ∧ 𝐴 ∈ 𝑋) → { 0 } ⊆ ∪ 𝐽)
18		eqid 2735	. . . . . . . 8 ⊢ ∪ 𝐽 = ∪ 𝐽
19	18	clsss3 22997	. . . . . . 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 3996	. . . . 5 ⊢ ((𝐺 ∈ TopGrp ∧ 𝐴 ∈ 𝑋) → ((cls‘𝐽)‘{ 0 }) ⊆ 𝑋)
22	1, 21	eqsstrid 3997	. . . 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 2735	. . . . 5 ⊢ (+g‘𝐺) = (+g‘𝐺)
26	6, 24, 25	eqglact 19162	. . . 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 2735	. . . . 5 ⊢ (𝑥 ∈ 𝑋 ↦ (𝐴(+g‘𝐺)𝑥)) = (𝑥 ∈ 𝑋 ↦ (𝐴(+g‘𝐺)𝑥))
29	28, 6, 25, 5	tgplacthmeo 24041	. . . 4 ⊢ ((𝐺 ∈ TopGrp ∧ 𝐴 ∈ 𝑋) → (𝑥 ∈ 𝑋 ↦ (𝐴(+g‘𝐺)𝑥)) ∈ (𝐽Homeo𝐽))
30	18	hmeocls 23706	. . . 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 2796	. 2 ⊢ ((𝐺 ∈ TopGrp ∧ 𝐴 ∈ 𝑋) → [𝐴] ∼ = ((cls‘𝐽)‘((𝑥 ∈ 𝑋 ↦ (𝐴(+g‘𝐺)𝑥)) “ { 0 })))
33		df-ima 5667	. . . . 5 ⊢ ((𝑥 ∈ 𝑋 ↦ (𝐴(+g‘𝐺)𝑥)) “ { 0 }) = ran ((𝑥 ∈ 𝑋 ↦ (𝐴(+g‘𝐺)𝑥)) ↾ { 0 })
34	14	resmptd 6027	. . . . . 6 ⊢ ((𝐺 ∈ TopGrp ∧ 𝐴 ∈ 𝑋) → ((𝑥 ∈ 𝑋 ↦ (𝐴(+g‘𝐺)𝑥)) ↾ { 0 }) = (𝑥 ∈ { 0 } ↦ (𝐴(+g‘𝐺)𝑥)))
35	34	rneqd 5918	. . . . 5 ⊢ ((𝐺 ∈ TopGrp ∧ 𝐴 ∈ 𝑋) → ran ((𝑥 ∈ 𝑋 ↦ (𝐴(+g‘𝐺)𝑥)) ↾ { 0 }) = ran (𝑥 ∈ { 0 } ↦ (𝐴(+g‘𝐺)𝑥)))
36	33, 35	eqtrid 2782	. . . 4 ⊢ ((𝐺 ∈ TopGrp ∧ 𝐴 ∈ 𝑋) → ((𝑥 ∈ 𝑋 ↦ (𝐴(+g‘𝐺)𝑥)) “ { 0 }) = ran (𝑥 ∈ { 0 } ↦ (𝐴(+g‘𝐺)𝑥)))
37	11	fvexi 6890	. . . . . . . 8 ⊢ 0 ∈ V
38		oveq2 7413	. . . . . . . . 9 ⊢ (𝑥 = 0 → (𝐴(+g‘𝐺)𝑥) = (𝐴(+g‘𝐺) 0 ))
39	38	eqeq2d 2746	. . . . . . . 8 ⊢ (𝑥 = 0 → (𝑦 = (𝐴(+g‘𝐺)𝑥) ↔ 𝑦 = (𝐴(+g‘𝐺) 0 )))
40	37, 39	rexsn 4658	. . . . . . 7 ⊢ (∃𝑥 ∈ { 0 }𝑦 = (𝐴(+g‘𝐺)𝑥) ↔ 𝑦 = (𝐴(+g‘𝐺) 0 ))
41	6, 25, 11	grprid 18951	. . . . . . . . 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 2746	. . . . . . 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 2801	. . . . 5 ⊢ ((𝐺 ∈ TopGrp ∧ 𝐴 ∈ 𝑋) → {𝑦 ∣ ∃𝑥 ∈ { 0 }𝑦 = (𝐴(+g‘𝐺)𝑥)} = {𝑦 ∣ 𝑦 = 𝐴})
46		eqid 2735	. . . . . 6 ⊢ (𝑥 ∈ { 0 } ↦ (𝐴(+g‘𝐺)𝑥)) = (𝑥 ∈ { 0 } ↦ (𝐴(+g‘𝐺)𝑥))
47	46	rnmpt 5937	. . . . 5 ⊢ ran (𝑥 ∈ { 0 } ↦ (𝐴(+g‘𝐺)𝑥)) = {𝑦 ∣ ∃𝑥 ∈ { 0 }𝑦 = (𝐴(+g‘𝐺)𝑥)}
48		df-sn 4602	. . . . 5 ⊢ {𝐴} = {𝑦 ∣ 𝑦 = 𝐴}
49	45, 47, 48	3eqtr4g 2795	. . . 4 ⊢ ((𝐺 ∈ TopGrp ∧ 𝐴 ∈ 𝑋) → ran (𝑥 ∈ { 0 } ↦ (𝐴(+g‘𝐺)𝑥)) = {𝐴})
50	36, 49	eqtrd 2770	. . 3 ⊢ ((𝐺 ∈ TopGrp ∧ 𝐴 ∈ 𝑋) → ((𝑥 ∈ 𝑋 ↦ (𝐴(+g‘𝐺)𝑥)) “ { 0 }) = {𝐴})
51	50	fveq2d 6880	. 2 ⊢ ((𝐺 ∈ TopGrp ∧ 𝐴 ∈ 𝑋) → ((cls‘𝐽)‘((𝑥 ∈ 𝑋 ↦ (𝐴(+g‘𝐺)𝑥)) “ { 0 })) = ((cls‘𝐽)‘{𝐴}))
52	32, 51	eqtrd 2770	1 ⊢ ((𝐺 ∈ TopGrp ∧ 𝐴 ∈ 𝑋) → [𝐴] ∼ = ((cls‘𝐽)‘{𝐴}))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1540   ∈ wcel 2108  {cab 2713  ∃wrex 3060   ⊆ wss 3926  {csn 4601  ∪ cuni 4883   ↦ cmpt 5201  ran crn 5655   ↾ cres 5656   “ cima 5657  ‘cfv 6531  (class class class)co 7405  [cec 8717  Basecbs 17228  +_gcplusg 17271  TopOpenctopn 17435  0_gc0g 17453  Grpcgrp 18916   ~_QG cqg 19105  Topctop 22831  TopOnctopon 22848  clsccl 22956  Homeochmeo 23691  TopGrpctgp 24009

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 2707  ax-rep 5249  ax-sep 5266  ax-nul 5276  ax-pow 5335  ax-pr 5402  ax-un 7729

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2727  df-clel 2809  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3061  df-rmo 3359  df-reu 3360  df-rab 3416  df-v 3461  df-sbc 3766  df-csb 3875  df-dif 3929  df-un 3931  df-in 3933  df-ss 3943  df-nul 4309  df-if 4501  df-pw 4577  df-sn 4602  df-pr 4604  df-op 4608  df-uni 4884  df-int 4923  df-iun 4969  df-iin 4970  df-br 5120  df-opab 5182  df-mpt 5202  df-id 5548  df-xp 5660  df-rel 5661  df-cnv 5662  df-co 5663  df-dm 5664  df-rn 5665  df-res 5666  df-ima 5667  df-iota 6484  df-fun 6533  df-fn 6534  df-f 6535  df-f1 6536  df-fo 6537  df-f1o 6538  df-fv 6539  df-riota 7362  df-ov 7408  df-oprab 7409  df-mpo 7410  df-1st 7988  df-2nd 7989  df-ec 8721  df-map 8842  df-0g 17455  df-topgen 17457  df-plusf 18617  df-mgm 18618  df-sgrp 18697  df-mnd 18713  df-grp 18919  df-minusg 18920  df-eqg 19108  df-top 22832  df-topon 22849  df-topsp 22871  df-bases 22884  df-cld 22957  df-cls 22959  df-cn 23165  df-cnp 23166  df-tx 23500  df-hmeo 23693  df-tmd 24010  df-tgp 24011

This theorem is referenced by:  tgptsmscls  24088