Theorem snclseqg 24089

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 6015	. . 3 ⊢ ((𝑥 ∈ 𝑋 ↦ (𝐴(+g‘𝐺)𝑥)) “ 𝑆) = ((𝑥 ∈ 𝑋 ↦ (𝐴(+g‘𝐺)𝑥)) “ ((cls‘𝐽)‘{ 0 }))
3		tgpgrp 24051	. . . . 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 24055	. . . . . . . . 9 ⊢ (𝐺 ∈ TopGrp → 𝐽 ∈ (TopOn‘𝑋))
8	7	adantr 480	. . . . . . . 8 ⊢ ((𝐺 ∈ TopGrp ∧ 𝐴 ∈ 𝑋) → 𝐽 ∈ (TopOn‘𝑋))
9		topontop 22886	. . . . . . . 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 18930	. . . . . . . . . 10 ⊢ (𝐺 ∈ Grp → 0 ∈ 𝑋)
13	4, 12	syl 17	. . . . . . . . 9 ⊢ ((𝐺 ∈ TopGrp ∧ 𝐴 ∈ 𝑋) → 0 ∈ 𝑋)
14	13	snssd 4753	. . . . . . . 8 ⊢ ((𝐺 ∈ TopGrp ∧ 𝐴 ∈ 𝑋) → { 0 } ⊆ 𝑋)
15		toponuni 22887	. . . . . . . . 9 ⊢ (𝐽 ∈ (TopOn‘𝑋) → 𝑋 = ∪ 𝐽)
16	8, 15	syl 17	. . . . . . . 8 ⊢ ((𝐺 ∈ TopGrp ∧ 𝐴 ∈ 𝑋) → 𝑋 = ∪ 𝐽)
17	14, 16	sseqtrd 3959	. . . . . . 7 ⊢ ((𝐺 ∈ TopGrp ∧ 𝐴 ∈ 𝑋) → { 0 } ⊆ ∪ 𝐽)
18		eqid 2737	. . . . . . . 8 ⊢ ∪ 𝐽 = ∪ 𝐽
19	18	clsss3 23032	. . . . . . 7 ⊢ ((𝐽 ∈ Top ∧ { 0 } ⊆ ∪ 𝐽) → ((cls‘𝐽)‘{ 0 }) ⊆ ∪ 𝐽)
20	10, 17, 19	syl2anc 585	. . . . . 6 ⊢ ((𝐺 ∈ TopGrp ∧ 𝐴 ∈ 𝑋) → ((cls‘𝐽)‘{ 0 }) ⊆ ∪ 𝐽)
21	20, 16	sseqtrrd 3960	. . . . 5 ⊢ ((𝐺 ∈ TopGrp ∧ 𝐴 ∈ 𝑋) → ((cls‘𝐽)‘{ 0 }) ⊆ 𝑋)
22	1, 21	eqsstrid 3961	. . . 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 19143	. . . 4 ⊢ ((𝐺 ∈ Grp ∧ 𝑆 ⊆ 𝑋 ∧ 𝐴 ∈ 𝑋) → [𝐴] ∼ = ((𝑥 ∈ 𝑋 ↦ (𝐴(+g‘𝐺)𝑥)) “ 𝑆))
27	4, 22, 23, 26	syl3anc 1374	. . 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 24076	. . . 4 ⊢ ((𝐺 ∈ TopGrp ∧ 𝐴 ∈ 𝑋) → (𝑥 ∈ 𝑋 ↦ (𝐴(+g‘𝐺)𝑥)) ∈ (𝐽Homeo𝐽))
30	18	hmeocls 23741	. . . 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 585	. . 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 2798	. 2 ⊢ ((𝐺 ∈ TopGrp ∧ 𝐴 ∈ 𝑋) → [𝐴] ∼ = ((cls‘𝐽)‘((𝑥 ∈ 𝑋 ↦ (𝐴(+g‘𝐺)𝑥)) “ { 0 })))
33		df-ima 5635	. . . . 5 ⊢ ((𝑥 ∈ 𝑋 ↦ (𝐴(+g‘𝐺)𝑥)) “ { 0 }) = ran ((𝑥 ∈ 𝑋 ↦ (𝐴(+g‘𝐺)𝑥)) ↾ { 0 })
34	14	resmptd 5997	. . . . . 6 ⊢ ((𝐺 ∈ TopGrp ∧ 𝐴 ∈ 𝑋) → ((𝑥 ∈ 𝑋 ↦ (𝐴(+g‘𝐺)𝑥)) ↾ { 0 }) = (𝑥 ∈ { 0 } ↦ (𝐴(+g‘𝐺)𝑥)))
35	34	rneqd 5885	. . . . 5 ⊢ ((𝐺 ∈ TopGrp ∧ 𝐴 ∈ 𝑋) → ran ((𝑥 ∈ 𝑋 ↦ (𝐴(+g‘𝐺)𝑥)) ↾ { 0 }) = ran (𝑥 ∈ { 0 } ↦ (𝐴(+g‘𝐺)𝑥)))
36	33, 35	eqtrid 2784	. . . 4 ⊢ ((𝐺 ∈ TopGrp ∧ 𝐴 ∈ 𝑋) → ((𝑥 ∈ 𝑋 ↦ (𝐴(+g‘𝐺)𝑥)) “ { 0 }) = ran (𝑥 ∈ { 0 } ↦ (𝐴(+g‘𝐺)𝑥)))
37	11	fvexi 6846	. . . . . . . 8 ⊢ 0 ∈ V
38		oveq2 7366	. . . . . . . . 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 4627	. . . . . . 7 ⊢ (∃𝑥 ∈ { 0 }𝑦 = (𝐴(+g‘𝐺)𝑥) ↔ 𝑦 = (𝐴(+g‘𝐺) 0 ))
41	6, 25, 11	grprid 18933	. . . . . . . . 9 ⊢ ((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋) → (𝐴(+g‘𝐺) 0 ) = 𝐴)
42	3, 41	sylan 581	. . . . . . . 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 2803	. . . . 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 5904	. . . . 5 ⊢ ran (𝑥 ∈ { 0 } ↦ (𝐴(+g‘𝐺)𝑥)) = {𝑦 ∣ ∃𝑥 ∈ { 0 }𝑦 = (𝐴(+g‘𝐺)𝑥)}
48		df-sn 4569	. . . . 5 ⊢ {𝐴} = {𝑦 ∣ 𝑦 = 𝐴}
49	45, 47, 48	3eqtr4g 2797	. . . 4 ⊢ ((𝐺 ∈ TopGrp ∧ 𝐴 ∈ 𝑋) → ran (𝑥 ∈ { 0 } ↦ (𝐴(+g‘𝐺)𝑥)) = {𝐴})
50	36, 49	eqtrd 2772	. . 3 ⊢ ((𝐺 ∈ TopGrp ∧ 𝐴 ∈ 𝑋) → ((𝑥 ∈ 𝑋 ↦ (𝐴(+g‘𝐺)𝑥)) “ { 0 }) = {𝐴})
51	50	fveq2d 6836	. 2 ⊢ ((𝐺 ∈ TopGrp ∧ 𝐴 ∈ 𝑋) → ((cls‘𝐽)‘((𝑥 ∈ 𝑋 ↦ (𝐴(+g‘𝐺)𝑥)) “ { 0 })) = ((cls‘𝐽)‘{𝐴}))
52	32, 51	eqtrd 2772	1 ⊢ ((𝐺 ∈ TopGrp ∧ 𝐴 ∈ 𝑋) → [𝐴] ∼ = ((cls‘𝐽)‘{𝐴}))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1542   ∈ wcel 2114  {cab 2715  ∃wrex 3062   ⊆ wss 3890  {csn 4568  ∪ cuni 4851   ↦ cmpt 5167  ran crn 5623   ↾ cres 5624   “ cima 5625  ‘cfv 6490  (class class class)co 7358  [cec 8632  Basecbs 17168  +_gcplusg 17209  TopOpenctopn 17373  0_gc0g 17391  Grpcgrp 18898   ~_QG cqg 19087  Topctop 22866  TopOnctopon 22883  clsccl 22991  Homeochmeo 23726  TopGrpctgp 24044

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-rep 5212  ax-sep 5231  ax-nul 5241  ax-pow 5300  ax-pr 5368  ax-un 7680

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3063  df-rmo 3343  df-reu 3344  df-rab 3391  df-v 3432  df-sbc 3730  df-csb 3839  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-nul 4275  df-if 4468  df-pw 4544  df-sn 4569  df-pr 4571  df-op 4575  df-uni 4852  df-int 4891  df-iun 4936  df-iin 4937  df-br 5087  df-opab 5149  df-mpt 5168  df-id 5517  df-xp 5628  df-rel 5629  df-cnv 5630  df-co 5631  df-dm 5632  df-rn 5633  df-res 5634  df-ima 5635  df-iota 6446  df-fun 6492  df-fn 6493  df-f 6494  df-f1 6495  df-fo 6496  df-f1o 6497  df-fv 6498  df-riota 7315  df-ov 7361  df-oprab 7362  df-mpo 7363  df-1st 7933  df-2nd 7934  df-ec 8636  df-map 8766  df-0g 17393  df-topgen 17395  df-plusf 18596  df-mgm 18597  df-sgrp 18676  df-mnd 18692  df-grp 18901  df-minusg 18902  df-eqg 19090  df-top 22867  df-topon 22884  df-topsp 22906  df-bases 22919  df-cld 22992  df-cls 22994  df-cn 23200  df-cnp 23201  df-tx 23535  df-hmeo 23728  df-tmd 24045  df-tgp 24046

This theorem is referenced by:  tgptsmscls  24123