Theorem snclseqg 24019

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 6013	. . 3 ⊢ ((𝑥 ∈ 𝑋 ↦ (𝐴(+g‘𝐺)𝑥)) “ 𝑆) = ((𝑥 ∈ 𝑋 ↦ (𝐴(+g‘𝐺)𝑥)) “ ((cls‘𝐽)‘{ 0 }))
3		tgpgrp 23981	. . . . 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 23985	. . . . . . . . 9 ⊢ (𝐺 ∈ TopGrp → 𝐽 ∈ (TopOn‘𝑋))
8	7	adantr 480	. . . . . . . 8 ⊢ ((𝐺 ∈ TopGrp ∧ 𝐴 ∈ 𝑋) → 𝐽 ∈ (TopOn‘𝑋))
9		topontop 22816	. . . . . . . 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 18862	. . . . . . . . . 10 ⊢ (𝐺 ∈ Grp → 0 ∈ 𝑋)
13	4, 12	syl 17	. . . . . . . . 9 ⊢ ((𝐺 ∈ TopGrp ∧ 𝐴 ∈ 𝑋) → 0 ∈ 𝑋)
14	13	snssd 4763	. . . . . . . 8 ⊢ ((𝐺 ∈ TopGrp ∧ 𝐴 ∈ 𝑋) → { 0 } ⊆ 𝑋)
15		toponuni 22817	. . . . . . . . 9 ⊢ (𝐽 ∈ (TopOn‘𝑋) → 𝑋 = ∪ 𝐽)
16	8, 15	syl 17	. . . . . . . 8 ⊢ ((𝐺 ∈ TopGrp ∧ 𝐴 ∈ 𝑋) → 𝑋 = ∪ 𝐽)
17	14, 16	sseqtrd 3974	. . . . . . 7 ⊢ ((𝐺 ∈ TopGrp ∧ 𝐴 ∈ 𝑋) → { 0 } ⊆ ∪ 𝐽)
18		eqid 2729	. . . . . . . 8 ⊢ ∪ 𝐽 = ∪ 𝐽
19	18	clsss3 22962	. . . . . . 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 3975	. . . . 5 ⊢ ((𝐺 ∈ TopGrp ∧ 𝐴 ∈ 𝑋) → ((cls‘𝐽)‘{ 0 }) ⊆ 𝑋)
22	1, 21	eqsstrid 3976	. . . 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 2729	. . . . 5 ⊢ (+g‘𝐺) = (+g‘𝐺)
26	6, 24, 25	eqglact 19076	. . . 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 2729	. . . . 5 ⊢ (𝑥 ∈ 𝑋 ↦ (𝐴(+g‘𝐺)𝑥)) = (𝑥 ∈ 𝑋 ↦ (𝐴(+g‘𝐺)𝑥))
29	28, 6, 25, 5	tgplacthmeo 24006	. . . 4 ⊢ ((𝐺 ∈ TopGrp ∧ 𝐴 ∈ 𝑋) → (𝑥 ∈ 𝑋 ↦ (𝐴(+g‘𝐺)𝑥)) ∈ (𝐽Homeo𝐽))
30	18	hmeocls 23671	. . . 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 2790	. 2 ⊢ ((𝐺 ∈ TopGrp ∧ 𝐴 ∈ 𝑋) → [𝐴] ∼ = ((cls‘𝐽)‘((𝑥 ∈ 𝑋 ↦ (𝐴(+g‘𝐺)𝑥)) “ { 0 })))
33		df-ima 5636	. . . . 5 ⊢ ((𝑥 ∈ 𝑋 ↦ (𝐴(+g‘𝐺)𝑥)) “ { 0 }) = ran ((𝑥 ∈ 𝑋 ↦ (𝐴(+g‘𝐺)𝑥)) ↾ { 0 })
34	14	resmptd 5995	. . . . . 6 ⊢ ((𝐺 ∈ TopGrp ∧ 𝐴 ∈ 𝑋) → ((𝑥 ∈ 𝑋 ↦ (𝐴(+g‘𝐺)𝑥)) ↾ { 0 }) = (𝑥 ∈ { 0 } ↦ (𝐴(+g‘𝐺)𝑥)))
35	34	rneqd 5884	. . . . 5 ⊢ ((𝐺 ∈ TopGrp ∧ 𝐴 ∈ 𝑋) → ran ((𝑥 ∈ 𝑋 ↦ (𝐴(+g‘𝐺)𝑥)) ↾ { 0 }) = ran (𝑥 ∈ { 0 } ↦ (𝐴(+g‘𝐺)𝑥)))
36	33, 35	eqtrid 2776	. . . 4 ⊢ ((𝐺 ∈ TopGrp ∧ 𝐴 ∈ 𝑋) → ((𝑥 ∈ 𝑋 ↦ (𝐴(+g‘𝐺)𝑥)) “ { 0 }) = ran (𝑥 ∈ { 0 } ↦ (𝐴(+g‘𝐺)𝑥)))
37	11	fvexi 6840	. . . . . . . 8 ⊢ 0 ∈ V
38		oveq2 7361	. . . . . . . . 9 ⊢ (𝑥 = 0 → (𝐴(+g‘𝐺)𝑥) = (𝐴(+g‘𝐺) 0 ))
39	38	eqeq2d 2740	. . . . . . . 8 ⊢ (𝑥 = 0 → (𝑦 = (𝐴(+g‘𝐺)𝑥) ↔ 𝑦 = (𝐴(+g‘𝐺) 0 )))
40	37, 39	rexsn 4636	. . . . . . 7 ⊢ (∃𝑥 ∈ { 0 }𝑦 = (𝐴(+g‘𝐺)𝑥) ↔ 𝑦 = (𝐴(+g‘𝐺) 0 ))
41	6, 25, 11	grprid 18865	. . . . . . . . 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 2740	. . . . . . 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 2795	. . . . 5 ⊢ ((𝐺 ∈ TopGrp ∧ 𝐴 ∈ 𝑋) → {𝑦 ∣ ∃𝑥 ∈ { 0 }𝑦 = (𝐴(+g‘𝐺)𝑥)} = {𝑦 ∣ 𝑦 = 𝐴})
46		eqid 2729	. . . . . 6 ⊢ (𝑥 ∈ { 0 } ↦ (𝐴(+g‘𝐺)𝑥)) = (𝑥 ∈ { 0 } ↦ (𝐴(+g‘𝐺)𝑥))
47	46	rnmpt 5903	. . . . 5 ⊢ ran (𝑥 ∈ { 0 } ↦ (𝐴(+g‘𝐺)𝑥)) = {𝑦 ∣ ∃𝑥 ∈ { 0 }𝑦 = (𝐴(+g‘𝐺)𝑥)}
48		df-sn 4580	. . . . 5 ⊢ {𝐴} = {𝑦 ∣ 𝑦 = 𝐴}
49	45, 47, 48	3eqtr4g 2789	. . . 4 ⊢ ((𝐺 ∈ TopGrp ∧ 𝐴 ∈ 𝑋) → ran (𝑥 ∈ { 0 } ↦ (𝐴(+g‘𝐺)𝑥)) = {𝐴})
50	36, 49	eqtrd 2764	. . 3 ⊢ ((𝐺 ∈ TopGrp ∧ 𝐴 ∈ 𝑋) → ((𝑥 ∈ 𝑋 ↦ (𝐴(+g‘𝐺)𝑥)) “ { 0 }) = {𝐴})
51	50	fveq2d 6830	. 2 ⊢ ((𝐺 ∈ TopGrp ∧ 𝐴 ∈ 𝑋) → ((cls‘𝐽)‘((𝑥 ∈ 𝑋 ↦ (𝐴(+g‘𝐺)𝑥)) “ { 0 })) = ((cls‘𝐽)‘{𝐴}))
52	32, 51	eqtrd 2764	1 ⊢ ((𝐺 ∈ TopGrp ∧ 𝐴 ∈ 𝑋) → [𝐴] ∼ = ((cls‘𝐽)‘{𝐴}))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1540   ∈ wcel 2109  {cab 2707  ∃wrex 3053   ⊆ wss 3905  {csn 4579  ∪ cuni 4861   ↦ cmpt 5176  ran crn 5624   ↾ cres 5625   “ cima 5626  ‘cfv 6486  (class class class)co 7353  [cec 8630  Basecbs 17138  +_gcplusg 17179  TopOpenctopn 17343  0_gc0g 17361  Grpcgrp 18830   ~_QG cqg 19019  Topctop 22796  TopOnctopon 22813  clsccl 22921  Homeochmeo 23656  TopGrpctgp 23974

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 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-rep 5221  ax-sep 5238  ax-nul 5248  ax-pow 5307  ax-pr 5374  ax-un 7675

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 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-rmo 3345  df-reu 3346  df-rab 3397  df-v 3440  df-sbc 3745  df-csb 3854  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-nul 4287  df-if 4479  df-pw 4555  df-sn 4580  df-pr 4582  df-op 4586  df-uni 4862  df-int 4900  df-iun 4946  df-iin 4947  df-br 5096  df-opab 5158  df-mpt 5177  df-id 5518  df-xp 5629  df-rel 5630  df-cnv 5631  df-co 5632  df-dm 5633  df-rn 5634  df-res 5635  df-ima 5636  df-iota 6442  df-fun 6488  df-fn 6489  df-f 6490  df-f1 6491  df-fo 6492  df-f1o 6493  df-fv 6494  df-riota 7310  df-ov 7356  df-oprab 7357  df-mpo 7358  df-1st 7931  df-2nd 7932  df-ec 8634  df-map 8762  df-0g 17363  df-topgen 17365  df-plusf 18531  df-mgm 18532  df-sgrp 18611  df-mnd 18627  df-grp 18833  df-minusg 18834  df-eqg 19022  df-top 22797  df-topon 22814  df-topsp 22836  df-bases 22849  df-cld 22922  df-cls 22924  df-cn 23130  df-cnp 23131  df-tx 23465  df-hmeo 23658  df-tmd 23975  df-tgp 23976

This theorem is referenced by:  tgptsmscls  24053