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Theorem snclseqg 22120
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 5622 . . 3 ((𝑥𝑋 ↦ (𝐴(+g𝐺)𝑥)) “ 𝑆) = ((𝑥𝑋 ↦ (𝐴(+g𝐺)𝑥)) “ ((cls‘𝐽)‘{ 0 }))
3 tgpgrp 22083 . . . . 5 (𝐺 ∈ TopGrp → 𝐺 ∈ Grp)
43adantr 472 . . . 4 ((𝐺 ∈ TopGrp ∧ 𝐴𝑋) → 𝐺 ∈ Grp)
5 snclseqg.j . . . . . . . . . 10 𝐽 = (TopOpen‘𝐺)
6 snclseqg.x . . . . . . . . . 10 𝑋 = (Base‘𝐺)
75, 6tgptopon 22087 . . . . . . . . 9 (𝐺 ∈ TopGrp → 𝐽 ∈ (TopOn‘𝑋))
87adantr 472 . . . . . . . 8 ((𝐺 ∈ TopGrp ∧ 𝐴𝑋) → 𝐽 ∈ (TopOn‘𝑋))
9 topontop 20920 . . . . . . . 8 (𝐽 ∈ (TopOn‘𝑋) → 𝐽 ∈ Top)
108, 9syl 17 . . . . . . 7 ((𝐺 ∈ TopGrp ∧ 𝐴𝑋) → 𝐽 ∈ Top)
11 snclseqg.z . . . . . . . . . . 11 0 = (0g𝐺)
126, 11grpidcl 17651 . . . . . . . . . 10 (𝐺 ∈ Grp → 0𝑋)
134, 12syl 17 . . . . . . . . 9 ((𝐺 ∈ TopGrp ∧ 𝐴𝑋) → 0𝑋)
1413snssd 4485 . . . . . . . 8 ((𝐺 ∈ TopGrp ∧ 𝐴𝑋) → { 0 } ⊆ 𝑋)
15 toponuni 20921 . . . . . . . . 9 (𝐽 ∈ (TopOn‘𝑋) → 𝑋 = 𝐽)
168, 15syl 17 . . . . . . . 8 ((𝐺 ∈ TopGrp ∧ 𝐴𝑋) → 𝑋 = 𝐽)
1714, 16sseqtrd 3782 . . . . . . 7 ((𝐺 ∈ TopGrp ∧ 𝐴𝑋) → { 0 } ⊆ 𝐽)
18 eqid 2760 . . . . . . . 8 𝐽 = 𝐽
1918clsss3 21065 . . . . . . 7 ((𝐽 ∈ Top ∧ { 0 } ⊆ 𝐽) → ((cls‘𝐽)‘{ 0 }) ⊆ 𝐽)
2010, 17, 19syl2anc 696 . . . . . 6 ((𝐺 ∈ TopGrp ∧ 𝐴𝑋) → ((cls‘𝐽)‘{ 0 }) ⊆ 𝐽)
2120, 16sseqtr4d 3783 . . . . 5 ((𝐺 ∈ TopGrp ∧ 𝐴𝑋) → ((cls‘𝐽)‘{ 0 }) ⊆ 𝑋)
221, 21syl5eqss 3790 . . . 4 ((𝐺 ∈ TopGrp ∧ 𝐴𝑋) → 𝑆𝑋)
23 simpr 479 . . . 4 ((𝐺 ∈ TopGrp ∧ 𝐴𝑋) → 𝐴𝑋)
24 snclseqg.r . . . . 5 = (𝐺 ~QG 𝑆)
25 eqid 2760 . . . . 5 (+g𝐺) = (+g𝐺)
266, 24, 25eqglact 17846 . . . 4 ((𝐺 ∈ Grp ∧ 𝑆𝑋𝐴𝑋) → [𝐴] = ((𝑥𝑋 ↦ (𝐴(+g𝐺)𝑥)) “ 𝑆))
274, 22, 23, 26syl3anc 1477 . . 3 ((𝐺 ∈ TopGrp ∧ 𝐴𝑋) → [𝐴] = ((𝑥𝑋 ↦ (𝐴(+g𝐺)𝑥)) “ 𝑆))
28 eqid 2760 . . . . 5 (𝑥𝑋 ↦ (𝐴(+g𝐺)𝑥)) = (𝑥𝑋 ↦ (𝐴(+g𝐺)𝑥))
2928, 6, 25, 5tgplacthmeo 22108 . . . 4 ((𝐺 ∈ TopGrp ∧ 𝐴𝑋) → (𝑥𝑋 ↦ (𝐴(+g𝐺)𝑥)) ∈ (𝐽Homeo𝐽))
3018hmeocls 21773 . . . 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 696 . . 3 ((𝐺 ∈ TopGrp ∧ 𝐴𝑋) → ((cls‘𝐽)‘((𝑥𝑋 ↦ (𝐴(+g𝐺)𝑥)) “ { 0 })) = ((𝑥𝑋 ↦ (𝐴(+g𝐺)𝑥)) “ ((cls‘𝐽)‘{ 0 })))
322, 27, 313eqtr4a 2820 . 2 ((𝐺 ∈ TopGrp ∧ 𝐴𝑋) → [𝐴] = ((cls‘𝐽)‘((𝑥𝑋 ↦ (𝐴(+g𝐺)𝑥)) “ { 0 })))
33 df-ima 5279 . . . . 5 ((𝑥𝑋 ↦ (𝐴(+g𝐺)𝑥)) “ { 0 }) = ran ((𝑥𝑋 ↦ (𝐴(+g𝐺)𝑥)) ↾ { 0 })
3414resmptd 5610 . . . . . 6 ((𝐺 ∈ TopGrp ∧ 𝐴𝑋) → ((𝑥𝑋 ↦ (𝐴(+g𝐺)𝑥)) ↾ { 0 }) = (𝑥 ∈ { 0 } ↦ (𝐴(+g𝐺)𝑥)))
3534rneqd 5508 . . . . 5 ((𝐺 ∈ TopGrp ∧ 𝐴𝑋) → ran ((𝑥𝑋 ↦ (𝐴(+g𝐺)𝑥)) ↾ { 0 }) = ran (𝑥 ∈ { 0 } ↦ (𝐴(+g𝐺)𝑥)))
3633, 35syl5eq 2806 . . . 4 ((𝐺 ∈ TopGrp ∧ 𝐴𝑋) → ((𝑥𝑋 ↦ (𝐴(+g𝐺)𝑥)) “ { 0 }) = ran (𝑥 ∈ { 0 } ↦ (𝐴(+g𝐺)𝑥)))
37 fvex 6362 . . . . . . . . 9 (0g𝐺) ∈ V
3811, 37eqeltri 2835 . . . . . . . 8 0 ∈ V
39 oveq2 6821 . . . . . . . . 9 (𝑥 = 0 → (𝐴(+g𝐺)𝑥) = (𝐴(+g𝐺) 0 ))
4039eqeq2d 2770 . . . . . . . 8 (𝑥 = 0 → (𝑦 = (𝐴(+g𝐺)𝑥) ↔ 𝑦 = (𝐴(+g𝐺) 0 )))
4138, 40rexsn 4367 . . . . . . 7 (∃𝑥 ∈ { 0 }𝑦 = (𝐴(+g𝐺)𝑥) ↔ 𝑦 = (𝐴(+g𝐺) 0 ))
426, 25, 11grprid 17654 . . . . . . . . 9 ((𝐺 ∈ Grp ∧ 𝐴𝑋) → (𝐴(+g𝐺) 0 ) = 𝐴)
433, 42sylan 489 . . . . . . . 8 ((𝐺 ∈ TopGrp ∧ 𝐴𝑋) → (𝐴(+g𝐺) 0 ) = 𝐴)
4443eqeq2d 2770 . . . . . . 7 ((𝐺 ∈ TopGrp ∧ 𝐴𝑋) → (𝑦 = (𝐴(+g𝐺) 0 ) ↔ 𝑦 = 𝐴))
4541, 44syl5bb 272 . . . . . 6 ((𝐺 ∈ TopGrp ∧ 𝐴𝑋) → (∃𝑥 ∈ { 0 }𝑦 = (𝐴(+g𝐺)𝑥) ↔ 𝑦 = 𝐴))
4645abbidv 2879 . . . . 5 ((𝐺 ∈ TopGrp ∧ 𝐴𝑋) → {𝑦 ∣ ∃𝑥 ∈ { 0 }𝑦 = (𝐴(+g𝐺)𝑥)} = {𝑦𝑦 = 𝐴})
47 eqid 2760 . . . . . 6 (𝑥 ∈ { 0 } ↦ (𝐴(+g𝐺)𝑥)) = (𝑥 ∈ { 0 } ↦ (𝐴(+g𝐺)𝑥))
4847rnmpt 5526 . . . . 5 ran (𝑥 ∈ { 0 } ↦ (𝐴(+g𝐺)𝑥)) = {𝑦 ∣ ∃𝑥 ∈ { 0 }𝑦 = (𝐴(+g𝐺)𝑥)}
49 df-sn 4322 . . . . 5 {𝐴} = {𝑦𝑦 = 𝐴}
5046, 48, 493eqtr4g 2819 . . . 4 ((𝐺 ∈ TopGrp ∧ 𝐴𝑋) → ran (𝑥 ∈ { 0 } ↦ (𝐴(+g𝐺)𝑥)) = {𝐴})
5136, 50eqtrd 2794 . . 3 ((𝐺 ∈ TopGrp ∧ 𝐴𝑋) → ((𝑥𝑋 ↦ (𝐴(+g𝐺)𝑥)) “ { 0 }) = {𝐴})
5251fveq2d 6356 . 2 ((𝐺 ∈ TopGrp ∧ 𝐴𝑋) → ((cls‘𝐽)‘((𝑥𝑋 ↦ (𝐴(+g𝐺)𝑥)) “ { 0 })) = ((cls‘𝐽)‘{𝐴}))
5332, 52eqtrd 2794 1 ((𝐺 ∈ TopGrp ∧ 𝐴𝑋) → [𝐴] = ((cls‘𝐽)‘{𝐴}))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383   = wceq 1632  wcel 2139  {cab 2746  wrex 3051  Vcvv 3340  wss 3715  {csn 4321   cuni 4588  cmpt 4881  ran crn 5267  cres 5268  cima 5269  cfv 6049  (class class class)co 6813  [cec 7909  Basecbs 16059  +gcplusg 16143  TopOpenctopn 16284  0gc0g 16302  Grpcgrp 17623   ~QG cqg 17791  Topctop 20900  TopOnctopon 20917  clsccl 21024  Homeochmeo 21758  TopGrpctgp 22076
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-8 2141  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-rep 4923  ax-sep 4933  ax-nul 4941  ax-pow 4992  ax-pr 5055  ax-un 7114
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-ral 3055  df-rex 3056  df-reu 3057  df-rmo 3058  df-rab 3059  df-v 3342  df-sbc 3577  df-csb 3675  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-pw 4304  df-sn 4322  df-pr 4324  df-op 4328  df-uni 4589  df-int 4628  df-iun 4674  df-iin 4675  df-br 4805  df-opab 4865  df-mpt 4882  df-id 5174  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-rn 5277  df-res 5278  df-ima 5279  df-iota 6012  df-fun 6051  df-fn 6052  df-f 6053  df-f1 6054  df-fo 6055  df-f1o 6056  df-fv 6057  df-riota 6774  df-ov 6816  df-oprab 6817  df-mpt2 6818  df-1st 7333  df-2nd 7334  df-ec 7913  df-map 8025  df-0g 16304  df-topgen 16306  df-plusf 17442  df-mgm 17443  df-sgrp 17485  df-mnd 17496  df-grp 17626  df-minusg 17627  df-eqg 17794  df-top 20901  df-topon 20918  df-topsp 20939  df-bases 20952  df-cld 21025  df-cls 21027  df-cn 21233  df-cnp 21234  df-tx 21567  df-hmeo 21760  df-tmd 22077  df-tgp 22078
This theorem is referenced by:  tgptsmscls  22154
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