Theorem snclseqg 24230

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 6050	. . 3 ⊢ ((𝑥 ∈ 𝑋 ↦ (𝐴(+g‘𝐺)𝑥)) “ 𝑆) = ((𝑥 ∈ 𝑋 ↦ (𝐴(+g‘𝐺)𝑥)) “ ((cls‘𝐽)‘{ 0 }))
3		tgpgrp 24192	. . . . 5 ⊢ (𝐺 ∈ TopGrp → 𝐺 ∈ Grp)
4	3	adantr 485	. . . 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 24196	. . . . . . . . 9 ⊢ (𝐺 ∈ TopGrp → 𝐽 ∈ (TopOn‘𝑋))
8	7	adantr 485	. . . . . . . 8 ⊢ ((𝐺 ∈ TopGrp ∧ 𝐴 ∈ 𝑋) → 𝐽 ∈ (TopOn‘𝑋))
9		topontop 23027	. . . . . . . 8 ⊢ (𝐽 ∈ (TopOn‘𝑋) → 𝐽 ∈ Top)
10	8, 9	syl 18	. . . . . . 7 ⊢ ((𝐺 ∈ TopGrp ∧ 𝐴 ∈ 𝑋) → 𝐽 ∈ Top)
11		snclseqg.z	. . . . . . . . . . 11 ⊢ 0 = (0g‘𝐺)
12	6, 11	grpidcl 19020	. . . . . . . . . 10 ⊢ (𝐺 ∈ Grp → 0 ∈ 𝑋)
13	4, 12	syl 18	. . . . . . . . 9 ⊢ ((𝐺 ∈ TopGrp ∧ 𝐴 ∈ 𝑋) → 0 ∈ 𝑋)
14	13	snssd 4748	. . . . . . . 8 ⊢ ((𝐺 ∈ TopGrp ∧ 𝐴 ∈ 𝑋) → { 0 } ⊆ 𝑋)
15		toponuni 23028	. . . . . . . . 9 ⊢ (𝐽 ∈ (TopOn‘𝑋) → 𝑋 = ∪ 𝐽)
16	8, 15	syl 18	. . . . . . . 8 ⊢ ((𝐺 ∈ TopGrp ∧ 𝐴 ∈ 𝑋) → 𝑋 = ∪ 𝐽)
17	14, 16	sseqtrd 3975	. . . . . . 7 ⊢ ((𝐺 ∈ TopGrp ∧ 𝐴 ∈ 𝑋) → { 0 } ⊆ ∪ 𝐽)
18		eqid 2765	. . . . . . . 8 ⊢ ∪ 𝐽 = ∪ 𝐽
19	18	clsss3 23173	. . . . . . 7 ⊢ ((𝐽 ∈ Top ∧ { 0 } ⊆ ∪ 𝐽) → ((cls‘𝐽)‘{ 0 }) ⊆ ∪ 𝐽)
20	10, 17, 19	syl2anc 595	. . . . . 6 ⊢ ((𝐺 ∈ TopGrp ∧ 𝐴 ∈ 𝑋) → ((cls‘𝐽)‘{ 0 }) ⊆ ∪ 𝐽)
21	20, 16	sseqtrrd 3976	. . . . 5 ⊢ ((𝐺 ∈ TopGrp ∧ 𝐴 ∈ 𝑋) → ((cls‘𝐽)‘{ 0 }) ⊆ 𝑋)
22	1, 21	eqsstrid 3977	. . . 4 ⊢ ((𝐺 ∈ TopGrp ∧ 𝐴 ∈ 𝑋) → 𝑆 ⊆ 𝑋)
23		simpr 489	. . . 4 ⊢ ((𝐺 ∈ TopGrp ∧ 𝐴 ∈ 𝑋) → 𝐴 ∈ 𝑋)
24		snclseqg.r	. . . . 5 ⊢ ∼ = (𝐺 ~QG 𝑆)
25		eqid 2765	. . . . 5 ⊢ (+g‘𝐺) = (+g‘𝐺)
26	6, 24, 25	eqglact 19235	. . . 4 ⊢ ((𝐺 ∈ Grp ∧ 𝑆 ⊆ 𝑋 ∧ 𝐴 ∈ 𝑋) → [𝐴] ∼ = ((𝑥 ∈ 𝑋 ↦ (𝐴(+g‘𝐺)𝑥)) “ 𝑆))
27	4, 22, 23, 26	syl3anc 1394	. . 3 ⊢ ((𝐺 ∈ TopGrp ∧ 𝐴 ∈ 𝑋) → [𝐴] ∼ = ((𝑥 ∈ 𝑋 ↦ (𝐴(+g‘𝐺)𝑥)) “ 𝑆))
28		eqid 2765	. . . . 5 ⊢ (𝑥 ∈ 𝑋 ↦ (𝐴(+g‘𝐺)𝑥)) = (𝑥 ∈ 𝑋 ↦ (𝐴(+g‘𝐺)𝑥))
29	28, 6, 25, 5	tgplacthmeo 24217	. . . 4 ⊢ ((𝐺 ∈ TopGrp ∧ 𝐴 ∈ 𝑋) → (𝑥 ∈ 𝑋 ↦ (𝐴(+g‘𝐺)𝑥)) ∈ (𝐽Homeo𝐽))
30	18	hmeocls 23882	. . . 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 595	. . 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 2826	. 2 ⊢ ((𝐺 ∈ TopGrp ∧ 𝐴 ∈ 𝑋) → [𝐴] ∼ = ((cls‘𝐽)‘((𝑥 ∈ 𝑋 ↦ (𝐴(+g‘𝐺)𝑥)) “ { 0 })))
33		df-ima 5664	. . . . 5 ⊢ ((𝑥 ∈ 𝑋 ↦ (𝐴(+g‘𝐺)𝑥)) “ { 0 }) = ran ((𝑥 ∈ 𝑋 ↦ (𝐴(+g‘𝐺)𝑥)) ↾ { 0 })
34	14	resmptd 6032	. . . . . 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 2812	. . . 4 ⊢ ((𝐺 ∈ TopGrp ∧ 𝐴 ∈ 𝑋) → ((𝑥 ∈ 𝑋 ↦ (𝐴(+g‘𝐺)𝑥)) “ { 0 }) = ran (𝑥 ∈ { 0 } ↦ (𝐴(+g‘𝐺)𝑥)))
37	11	fvexi 6885	. . . . . . . 8 ⊢ 0 ∈ V
38		oveq2 7408	. . . . . . . . 9 ⊢ (𝑥 = 0 → (𝐴(+g‘𝐺)𝑥) = (𝐴(+g‘𝐺) 0 ))
39	38	eqeq2d 2776	. . . . . . . 8 ⊢ (𝑥 = 0 → (𝑦 = (𝐴(+g‘𝐺)𝑥) ↔ 𝑦 = (𝐴(+g‘𝐺) 0 )))
40	37, 39	rexsn 4644	. . . . . . 7 ⊢ (∃𝑥 ∈ { 0 }𝑦 = (𝐴(+g‘𝐺)𝑥) ↔ 𝑦 = (𝐴(+g‘𝐺) 0 ))
41	6, 25, 11	grprid 19023	. . . . . . . . 9 ⊢ ((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋) → (𝐴(+g‘𝐺) 0 ) = 𝐴)
42	3, 41	sylan 591	. . . . . . . 8 ⊢ ((𝐺 ∈ TopGrp ∧ 𝐴 ∈ 𝑋) → (𝐴(+g‘𝐺) 0 ) = 𝐴)
43	42	eqeq2d 2776	. . . . . . 7 ⊢ ((𝐺 ∈ TopGrp ∧ 𝐴 ∈ 𝑋) → (𝑦 = (𝐴(+g‘𝐺) 0 ) ↔ 𝑦 = 𝐴))
44	40, 43	bitrid 286	. . . . . 6 ⊢ ((𝐺 ∈ TopGrp ∧ 𝐴 ∈ 𝑋) → (∃𝑥 ∈ { 0 }𝑦 = (𝐴(+g‘𝐺)𝑥) ↔ 𝑦 = 𝐴))
45	44	abbidv 2831	. . . . 5 ⊢ ((𝐺 ∈ TopGrp ∧ 𝐴 ∈ 𝑋) → {𝑦 ∣ ∃𝑥 ∈ { 0 }𝑦 = (𝐴(+g‘𝐺)𝑥)} = {𝑦 ∣ 𝑦 = 𝐴})
46		eqid 2765	. . . . . 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 4586	. . . . 5 ⊢ {𝐴} = {𝑦 ∣ 𝑦 = 𝐴}
49	45, 47, 48	3eqtr4g 2825	. . . 4 ⊢ ((𝐺 ∈ TopGrp ∧ 𝐴 ∈ 𝑋) → ran (𝑥 ∈ { 0 } ↦ (𝐴(+g‘𝐺)𝑥)) = {𝐴})
50	36, 49	eqtrd 2800	. . 3 ⊢ ((𝐺 ∈ TopGrp ∧ 𝐴 ∈ 𝑋) → ((𝑥 ∈ 𝑋 ↦ (𝐴(+g‘𝐺)𝑥)) “ { 0 }) = {𝐴})
51	50	fveq2d 6875	. 2 ⊢ ((𝐺 ∈ TopGrp ∧ 𝐴 ∈ 𝑋) → ((cls‘𝐽)‘((𝑥 ∈ 𝑋 ↦ (𝐴(+g‘𝐺)𝑥)) “ { 0 })) = ((cls‘𝐽)‘{𝐴}))
52	32, 51	eqtrd 2800	1 ⊢ ((𝐺 ∈ TopGrp ∧ 𝐴 ∈ 𝑋) → [𝐴] ∼ = ((cls‘𝐽)‘{𝐴}))

Syntax hints:   → wi 4   ∧ wa 400   = wceq 1563   ∈ wcel 2145  {cab 2743  ∃wrex 3089   ⊆ wss 3907  {csn 4585  ∪ cuni 4867   ↦ cmpt 5185  ran crn 5652   ↾ cres 5653   “ cima 5654  ‘cfv 6525  (class class class)co 7400  [cec 8680  Basecbs 17257  +_gcplusg 17298  TopOpenctopn 17462  0_gc0g 17480  Grpcgrp 18988   ~_QG cqg 19176  Topctop 23007  TopOnctopon 23024  clsccl 23132  Homeochmeo 23867  TopGrpctgp 24185

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1818  ax-4 1832  ax-5 1933  ax-6 1990  ax-7 2031  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2215  ax-ext 2737  ax-rep 5231  ax-sep 5250  ax-nul 5260  ax-pow 5326  ax-pr 5394  ax-un 7722

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1566  df-fal 1576  df-ex 1803  df-nf 1807  df-sb 2094  df-mo 2569  df-eu 2599  df-clab 2744  df-cleq 2757  df-clel 2840  df-nfc 2914  df-ne 2961  df-ral 3080  df-rex 3090  df-rmo 3370  df-reu 3371  df-rab 3418  df-v 3459  df-sbc 3748  df-csb 3856  df-dif 3910  df-un 3912  df-in 3914  df-ss 3924  df-nul 4289  df-if 4484  df-pw 4560  df-sn 4586  df-pr 4588  df-op 4592  df-uni 4868  df-int 4908  df-iun 4953  df-iin 4954  df-br 5105  df-opab 5167  df-mpt 5186  df-id 5546  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-rn 5662  df-res 5663  df-ima 5664  df-iota 6481  df-fun 6527  df-fn 6528  df-f 6529  df-f1 6530  df-fo 6531  df-f1o 6532  df-fv 6533  df-riota 7357  df-ov 7403  df-oprab 7404  df-mpo 7405  df-1st 7974  df-2nd 7975  df-ec 8684  df-map 8814  df-0g 17482  df-topgen 17484  df-plusf 18685  df-mgm 18686  df-sgrp 18765  df-mnd 18781  df-grp 18991  df-minusg 18992  df-eqg 19179  df-top 23008  df-topon 23025  df-topsp 23047  df-bases 23060  df-cld 23133  df-cls 23135  df-cn 23341  df-cnp 23342  df-tx 23676  df-hmeo 23869  df-tmd 24186  df-tgp 24187

This theorem is referenced by:  tgptsmscls  24264