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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 = (0g𝐺)
snclseqg.r = (𝐺 ~QG 𝑆)
snclseqg.s 𝑆 = ((cls‘𝐽)‘{ 0 })
Assertion
Ref Expression
snclseqg ((𝐺 ∈ TopGrp ∧ 𝐴𝑋) → [𝐴] = ((cls‘𝐽)‘{𝐴}))

Proof of Theorem snclseqg
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 snclseqg.s . . . 4 𝑆 = ((cls‘𝐽)‘{ 0 })
21imaeq2i 6076 . . 3 ((𝑥𝑋 ↦ (𝐴(+g𝐺)𝑥)) “ 𝑆) = ((𝑥𝑋 ↦ (𝐴(+g𝐺)𝑥)) “ ((cls‘𝐽)‘{ 0 }))
3 tgpgrp 24086 . . . . 5 (𝐺 ∈ TopGrp → 𝐺 ∈ Grp)
43adantr 480 . . . 4 ((𝐺 ∈ TopGrp ∧ 𝐴𝑋) → 𝐺 ∈ Grp)
5 snclseqg.j . . . . . . . . . 10 𝐽 = (TopOpen‘𝐺)
6 snclseqg.x . . . . . . . . . 10 𝑋 = (Base‘𝐺)
75, 6tgptopon 24090 . . . . . . . . 9 (𝐺 ∈ TopGrp → 𝐽 ∈ (TopOn‘𝑋))
87adantr 480 . . . . . . . 8 ((𝐺 ∈ TopGrp ∧ 𝐴𝑋) → 𝐽 ∈ (TopOn‘𝑋))
9 topontop 22919 . . . . . . . 8 (𝐽 ∈ (TopOn‘𝑋) → 𝐽 ∈ Top)
108, 9syl 17 . . . . . . 7 ((𝐺 ∈ TopGrp ∧ 𝐴𝑋) → 𝐽 ∈ Top)
11 snclseqg.z . . . . . . . . . . 11 0 = (0g𝐺)
126, 11grpidcl 18983 . . . . . . . . . 10 (𝐺 ∈ Grp → 0𝑋)
134, 12syl 17 . . . . . . . . 9 ((𝐺 ∈ TopGrp ∧ 𝐴𝑋) → 0𝑋)
1413snssd 4809 . . . . . . . 8 ((𝐺 ∈ TopGrp ∧ 𝐴𝑋) → { 0 } ⊆ 𝑋)
15 toponuni 22920 . . . . . . . . 9 (𝐽 ∈ (TopOn‘𝑋) → 𝑋 = 𝐽)
168, 15syl 17 . . . . . . . 8 ((𝐺 ∈ TopGrp ∧ 𝐴𝑋) → 𝑋 = 𝐽)
1714, 16sseqtrd 4020 . . . . . . 7 ((𝐺 ∈ TopGrp ∧ 𝐴𝑋) → { 0 } ⊆ 𝐽)
18 eqid 2737 . . . . . . . 8 𝐽 = 𝐽
1918clsss3 23067 . . . . . . 7 ((𝐽 ∈ Top ∧ { 0 } ⊆ 𝐽) → ((cls‘𝐽)‘{ 0 }) ⊆ 𝐽)
2010, 17, 19syl2anc 584 . . . . . 6 ((𝐺 ∈ TopGrp ∧ 𝐴𝑋) → ((cls‘𝐽)‘{ 0 }) ⊆ 𝐽)
2120, 16sseqtrrd 4021 . . . . 5 ((𝐺 ∈ TopGrp ∧ 𝐴𝑋) → ((cls‘𝐽)‘{ 0 }) ⊆ 𝑋)
221, 21eqsstrid 4022 . . . 4 ((𝐺 ∈ TopGrp ∧ 𝐴𝑋) → 𝑆𝑋)
23 simpr 484 . . . 4 ((𝐺 ∈ TopGrp ∧ 𝐴𝑋) → 𝐴𝑋)
24 snclseqg.r . . . . 5 = (𝐺 ~QG 𝑆)
25 eqid 2737 . . . . 5 (+g𝐺) = (+g𝐺)
266, 24, 25eqglact 19197 . . . 4 ((𝐺 ∈ Grp ∧ 𝑆𝑋𝐴𝑋) → [𝐴] = ((𝑥𝑋 ↦ (𝐴(+g𝐺)𝑥)) “ 𝑆))
274, 22, 23, 26syl3anc 1373 . . 3 ((𝐺 ∈ TopGrp ∧ 𝐴𝑋) → [𝐴] = ((𝑥𝑋 ↦ (𝐴(+g𝐺)𝑥)) “ 𝑆))
28 eqid 2737 . . . . 5 (𝑥𝑋 ↦ (𝐴(+g𝐺)𝑥)) = (𝑥𝑋 ↦ (𝐴(+g𝐺)𝑥))
2928, 6, 25, 5tgplacthmeo 24111 . . . 4 ((𝐺 ∈ TopGrp ∧ 𝐴𝑋) → (𝑥𝑋 ↦ (𝐴(+g𝐺)𝑥)) ∈ (𝐽Homeo𝐽))
3018hmeocls 23776 . . . 4 (((𝑥𝑋 ↦ (𝐴(+g𝐺)𝑥)) ∈ (𝐽Homeo𝐽) ∧ { 0 } ⊆ 𝐽) → ((cls‘𝐽)‘((𝑥𝑋 ↦ (𝐴(+g𝐺)𝑥)) “ { 0 })) = ((𝑥𝑋 ↦ (𝐴(+g𝐺)𝑥)) “ ((cls‘𝐽)‘{ 0 })))
3129, 17, 30syl2anc 584 . . 3 ((𝐺 ∈ TopGrp ∧ 𝐴𝑋) → ((cls‘𝐽)‘((𝑥𝑋 ↦ (𝐴(+g𝐺)𝑥)) “ { 0 })) = ((𝑥𝑋 ↦ (𝐴(+g𝐺)𝑥)) “ ((cls‘𝐽)‘{ 0 })))
322, 27, 313eqtr4a 2803 . 2 ((𝐺 ∈ TopGrp ∧ 𝐴𝑋) → [𝐴] = ((cls‘𝐽)‘((𝑥𝑋 ↦ (𝐴(+g𝐺)𝑥)) “ { 0 })))
33 df-ima 5698 . . . . 5 ((𝑥𝑋 ↦ (𝐴(+g𝐺)𝑥)) “ { 0 }) = ran ((𝑥𝑋 ↦ (𝐴(+g𝐺)𝑥)) ↾ { 0 })
3414resmptd 6058 . . . . . 6 ((𝐺 ∈ TopGrp ∧ 𝐴𝑋) → ((𝑥𝑋 ↦ (𝐴(+g𝐺)𝑥)) ↾ { 0 }) = (𝑥 ∈ { 0 } ↦ (𝐴(+g𝐺)𝑥)))
3534rneqd 5949 . . . . 5 ((𝐺 ∈ TopGrp ∧ 𝐴𝑋) → ran ((𝑥𝑋 ↦ (𝐴(+g𝐺)𝑥)) ↾ { 0 }) = ran (𝑥 ∈ { 0 } ↦ (𝐴(+g𝐺)𝑥)))
3633, 35eqtrid 2789 . . . 4 ((𝐺 ∈ TopGrp ∧ 𝐴𝑋) → ((𝑥𝑋 ↦ (𝐴(+g𝐺)𝑥)) “ { 0 }) = ran (𝑥 ∈ { 0 } ↦ (𝐴(+g𝐺)𝑥)))
3711fvexi 6920 . . . . . . . 8 0 ∈ V
38 oveq2 7439 . . . . . . . . 9 (𝑥 = 0 → (𝐴(+g𝐺)𝑥) = (𝐴(+g𝐺) 0 ))
3938eqeq2d 2748 . . . . . . . 8 (𝑥 = 0 → (𝑦 = (𝐴(+g𝐺)𝑥) ↔ 𝑦 = (𝐴(+g𝐺) 0 )))
4037, 39rexsn 4682 . . . . . . 7 (∃𝑥 ∈ { 0 }𝑦 = (𝐴(+g𝐺)𝑥) ↔ 𝑦 = (𝐴(+g𝐺) 0 ))
416, 25, 11grprid 18986 . . . . . . . . 9 ((𝐺 ∈ Grp ∧ 𝐴𝑋) → (𝐴(+g𝐺) 0 ) = 𝐴)
423, 41sylan 580 . . . . . . . 8 ((𝐺 ∈ TopGrp ∧ 𝐴𝑋) → (𝐴(+g𝐺) 0 ) = 𝐴)
4342eqeq2d 2748 . . . . . . 7 ((𝐺 ∈ TopGrp ∧ 𝐴𝑋) → (𝑦 = (𝐴(+g𝐺) 0 ) ↔ 𝑦 = 𝐴))
4440, 43bitrid 283 . . . . . 6 ((𝐺 ∈ TopGrp ∧ 𝐴𝑋) → (∃𝑥 ∈ { 0 }𝑦 = (𝐴(+g𝐺)𝑥) ↔ 𝑦 = 𝐴))
4544abbidv 2808 . . . . 5 ((𝐺 ∈ TopGrp ∧ 𝐴𝑋) → {𝑦 ∣ ∃𝑥 ∈ { 0 }𝑦 = (𝐴(+g𝐺)𝑥)} = {𝑦𝑦 = 𝐴})
46 eqid 2737 . . . . . 6 (𝑥 ∈ { 0 } ↦ (𝐴(+g𝐺)𝑥)) = (𝑥 ∈ { 0 } ↦ (𝐴(+g𝐺)𝑥))
4746rnmpt 5968 . . . . 5 ran (𝑥 ∈ { 0 } ↦ (𝐴(+g𝐺)𝑥)) = {𝑦 ∣ ∃𝑥 ∈ { 0 }𝑦 = (𝐴(+g𝐺)𝑥)}
48 df-sn 4627 . . . . 5 {𝐴} = {𝑦𝑦 = 𝐴}
4945, 47, 483eqtr4g 2802 . . . 4 ((𝐺 ∈ TopGrp ∧ 𝐴𝑋) → ran (𝑥 ∈ { 0 } ↦ (𝐴(+g𝐺)𝑥)) = {𝐴})
5036, 49eqtrd 2777 . . 3 ((𝐺 ∈ TopGrp ∧ 𝐴𝑋) → ((𝑥𝑋 ↦ (𝐴(+g𝐺)𝑥)) “ { 0 }) = {𝐴})
5150fveq2d 6910 . 2 ((𝐺 ∈ TopGrp ∧ 𝐴𝑋) → ((cls‘𝐽)‘((𝑥𝑋 ↦ (𝐴(+g𝐺)𝑥)) “ { 0 })) = ((cls‘𝐽)‘{𝐴}))
5232, 51eqtrd 2777 1 ((𝐺 ∈ TopGrp ∧ 𝐴𝑋) → [𝐴] = ((cls‘𝐽)‘{𝐴}))
Colors of variables: wff setvar class
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  0gc0g 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
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