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Theorem snclseqg 23840
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 = (Baseβ€˜πΊ)
snclseqg.j 𝐽 = (TopOpenβ€˜πΊ)
snclseqg.z 0 = (0gβ€˜πΊ)
snclseqg.r ∼ = (𝐺 ~QG 𝑆)
snclseqg.s 𝑆 = ((clsβ€˜π½)β€˜{ 0 })
Assertion
Ref Expression
snclseqg ((𝐺 ∈ TopGrp ∧ 𝐴 ∈ 𝑋) β†’ [𝐴] ∼ = ((clsβ€˜π½)β€˜{𝐴}))

Proof of Theorem snclseqg
Dummy variables π‘₯ 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 snclseqg.s . . . 4 𝑆 = ((clsβ€˜π½)β€˜{ 0 })
21imaeq2i 6056 . . 3 ((π‘₯ ∈ 𝑋 ↦ (𝐴(+gβ€˜πΊ)π‘₯)) β€œ 𝑆) = ((π‘₯ ∈ 𝑋 ↦ (𝐴(+gβ€˜πΊ)π‘₯)) β€œ ((clsβ€˜π½)β€˜{ 0 }))
3 tgpgrp 23802 . . . . 5 (𝐺 ∈ TopGrp β†’ 𝐺 ∈ Grp)
43adantr 479 . . . 4 ((𝐺 ∈ TopGrp ∧ 𝐴 ∈ 𝑋) β†’ 𝐺 ∈ Grp)
5 snclseqg.j . . . . . . . . . 10 𝐽 = (TopOpenβ€˜πΊ)
6 snclseqg.x . . . . . . . . . 10 𝑋 = (Baseβ€˜πΊ)
75, 6tgptopon 23806 . . . . . . . . 9 (𝐺 ∈ TopGrp β†’ 𝐽 ∈ (TopOnβ€˜π‘‹))
87adantr 479 . . . . . . . 8 ((𝐺 ∈ TopGrp ∧ 𝐴 ∈ 𝑋) β†’ 𝐽 ∈ (TopOnβ€˜π‘‹))
9 topontop 22635 . . . . . . . 8 (𝐽 ∈ (TopOnβ€˜π‘‹) β†’ 𝐽 ∈ Top)
108, 9syl 17 . . . . . . 7 ((𝐺 ∈ TopGrp ∧ 𝐴 ∈ 𝑋) β†’ 𝐽 ∈ Top)
11 snclseqg.z . . . . . . . . . . 11 0 = (0gβ€˜πΊ)
126, 11grpidcl 18886 . . . . . . . . . 10 (𝐺 ∈ Grp β†’ 0 ∈ 𝑋)
134, 12syl 17 . . . . . . . . 9 ((𝐺 ∈ TopGrp ∧ 𝐴 ∈ 𝑋) β†’ 0 ∈ 𝑋)
1413snssd 4811 . . . . . . . 8 ((𝐺 ∈ TopGrp ∧ 𝐴 ∈ 𝑋) β†’ { 0 } βŠ† 𝑋)
15 toponuni 22636 . . . . . . . . 9 (𝐽 ∈ (TopOnβ€˜π‘‹) β†’ 𝑋 = βˆͺ 𝐽)
168, 15syl 17 . . . . . . . 8 ((𝐺 ∈ TopGrp ∧ 𝐴 ∈ 𝑋) β†’ 𝑋 = βˆͺ 𝐽)
1714, 16sseqtrd 4021 . . . . . . 7 ((𝐺 ∈ TopGrp ∧ 𝐴 ∈ 𝑋) β†’ { 0 } βŠ† βˆͺ 𝐽)
18 eqid 2730 . . . . . . . 8 βˆͺ 𝐽 = βˆͺ 𝐽
1918clsss3 22783 . . . . . . 7 ((𝐽 ∈ Top ∧ { 0 } βŠ† βˆͺ 𝐽) β†’ ((clsβ€˜π½)β€˜{ 0 }) βŠ† βˆͺ 𝐽)
2010, 17, 19syl2anc 582 . . . . . 6 ((𝐺 ∈ TopGrp ∧ 𝐴 ∈ 𝑋) β†’ ((clsβ€˜π½)β€˜{ 0 }) βŠ† βˆͺ 𝐽)
2120, 16sseqtrrd 4022 . . . . 5 ((𝐺 ∈ TopGrp ∧ 𝐴 ∈ 𝑋) β†’ ((clsβ€˜π½)β€˜{ 0 }) βŠ† 𝑋)
221, 21eqsstrid 4029 . . . 4 ((𝐺 ∈ TopGrp ∧ 𝐴 ∈ 𝑋) β†’ 𝑆 βŠ† 𝑋)
23 simpr 483 . . . 4 ((𝐺 ∈ TopGrp ∧ 𝐴 ∈ 𝑋) β†’ 𝐴 ∈ 𝑋)
24 snclseqg.r . . . . 5 ∼ = (𝐺 ~QG 𝑆)
25 eqid 2730 . . . . 5 (+gβ€˜πΊ) = (+gβ€˜πΊ)
266, 24, 25eqglact 19095 . . . 4 ((𝐺 ∈ Grp ∧ 𝑆 βŠ† 𝑋 ∧ 𝐴 ∈ 𝑋) β†’ [𝐴] ∼ = ((π‘₯ ∈ 𝑋 ↦ (𝐴(+gβ€˜πΊ)π‘₯)) β€œ 𝑆))
274, 22, 23, 26syl3anc 1369 . . 3 ((𝐺 ∈ TopGrp ∧ 𝐴 ∈ 𝑋) β†’ [𝐴] ∼ = ((π‘₯ ∈ 𝑋 ↦ (𝐴(+gβ€˜πΊ)π‘₯)) β€œ 𝑆))
28 eqid 2730 . . . . 5 (π‘₯ ∈ 𝑋 ↦ (𝐴(+gβ€˜πΊ)π‘₯)) = (π‘₯ ∈ 𝑋 ↦ (𝐴(+gβ€˜πΊ)π‘₯))
2928, 6, 25, 5tgplacthmeo 23827 . . . 4 ((𝐺 ∈ TopGrp ∧ 𝐴 ∈ 𝑋) β†’ (π‘₯ ∈ 𝑋 ↦ (𝐴(+gβ€˜πΊ)π‘₯)) ∈ (𝐽Homeo𝐽))
3018hmeocls 23492 . . . 4 (((π‘₯ ∈ 𝑋 ↦ (𝐴(+gβ€˜πΊ)π‘₯)) ∈ (𝐽Homeo𝐽) ∧ { 0 } βŠ† βˆͺ 𝐽) β†’ ((clsβ€˜π½)β€˜((π‘₯ ∈ 𝑋 ↦ (𝐴(+gβ€˜πΊ)π‘₯)) β€œ { 0 })) = ((π‘₯ ∈ 𝑋 ↦ (𝐴(+gβ€˜πΊ)π‘₯)) β€œ ((clsβ€˜π½)β€˜{ 0 })))
3129, 17, 30syl2anc 582 . . 3 ((𝐺 ∈ TopGrp ∧ 𝐴 ∈ 𝑋) β†’ ((clsβ€˜π½)β€˜((π‘₯ ∈ 𝑋 ↦ (𝐴(+gβ€˜πΊ)π‘₯)) β€œ { 0 })) = ((π‘₯ ∈ 𝑋 ↦ (𝐴(+gβ€˜πΊ)π‘₯)) β€œ ((clsβ€˜π½)β€˜{ 0 })))
322, 27, 313eqtr4a 2796 . 2 ((𝐺 ∈ TopGrp ∧ 𝐴 ∈ 𝑋) β†’ [𝐴] ∼ = ((clsβ€˜π½)β€˜((π‘₯ ∈ 𝑋 ↦ (𝐴(+gβ€˜πΊ)π‘₯)) β€œ { 0 })))
33 df-ima 5688 . . . . 5 ((π‘₯ ∈ 𝑋 ↦ (𝐴(+gβ€˜πΊ)π‘₯)) β€œ { 0 }) = ran ((π‘₯ ∈ 𝑋 ↦ (𝐴(+gβ€˜πΊ)π‘₯)) β†Ύ { 0 })
3414resmptd 6039 . . . . . 6 ((𝐺 ∈ TopGrp ∧ 𝐴 ∈ 𝑋) β†’ ((π‘₯ ∈ 𝑋 ↦ (𝐴(+gβ€˜πΊ)π‘₯)) β†Ύ { 0 }) = (π‘₯ ∈ { 0 } ↦ (𝐴(+gβ€˜πΊ)π‘₯)))
3534rneqd 5936 . . . . 5 ((𝐺 ∈ TopGrp ∧ 𝐴 ∈ 𝑋) β†’ ran ((π‘₯ ∈ 𝑋 ↦ (𝐴(+gβ€˜πΊ)π‘₯)) β†Ύ { 0 }) = ran (π‘₯ ∈ { 0 } ↦ (𝐴(+gβ€˜πΊ)π‘₯)))
3633, 35eqtrid 2782 . . . 4 ((𝐺 ∈ TopGrp ∧ 𝐴 ∈ 𝑋) β†’ ((π‘₯ ∈ 𝑋 ↦ (𝐴(+gβ€˜πΊ)π‘₯)) β€œ { 0 }) = ran (π‘₯ ∈ { 0 } ↦ (𝐴(+gβ€˜πΊ)π‘₯)))
3711fvexi 6904 . . . . . . . 8 0 ∈ V
38 oveq2 7419 . . . . . . . . 9 (π‘₯ = 0 β†’ (𝐴(+gβ€˜πΊ)π‘₯) = (𝐴(+gβ€˜πΊ) 0 ))
3938eqeq2d 2741 . . . . . . . 8 (π‘₯ = 0 β†’ (𝑦 = (𝐴(+gβ€˜πΊ)π‘₯) ↔ 𝑦 = (𝐴(+gβ€˜πΊ) 0 )))
4037, 39rexsn 4685 . . . . . . 7 (βˆƒπ‘₯ ∈ { 0 }𝑦 = (𝐴(+gβ€˜πΊ)π‘₯) ↔ 𝑦 = (𝐴(+gβ€˜πΊ) 0 ))
416, 25, 11grprid 18889 . . . . . . . . 9 ((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋) β†’ (𝐴(+gβ€˜πΊ) 0 ) = 𝐴)
423, 41sylan 578 . . . . . . . 8 ((𝐺 ∈ TopGrp ∧ 𝐴 ∈ 𝑋) β†’ (𝐴(+gβ€˜πΊ) 0 ) = 𝐴)
4342eqeq2d 2741 . . . . . . 7 ((𝐺 ∈ TopGrp ∧ 𝐴 ∈ 𝑋) β†’ (𝑦 = (𝐴(+gβ€˜πΊ) 0 ) ↔ 𝑦 = 𝐴))
4440, 43bitrid 282 . . . . . 6 ((𝐺 ∈ TopGrp ∧ 𝐴 ∈ 𝑋) β†’ (βˆƒπ‘₯ ∈ { 0 }𝑦 = (𝐴(+gβ€˜πΊ)π‘₯) ↔ 𝑦 = 𝐴))
4544abbidv 2799 . . . . 5 ((𝐺 ∈ TopGrp ∧ 𝐴 ∈ 𝑋) β†’ {𝑦 ∣ βˆƒπ‘₯ ∈ { 0 }𝑦 = (𝐴(+gβ€˜πΊ)π‘₯)} = {𝑦 ∣ 𝑦 = 𝐴})
46 eqid 2730 . . . . . 6 (π‘₯ ∈ { 0 } ↦ (𝐴(+gβ€˜πΊ)π‘₯)) = (π‘₯ ∈ { 0 } ↦ (𝐴(+gβ€˜πΊ)π‘₯))
4746rnmpt 5953 . . . . 5 ran (π‘₯ ∈ { 0 } ↦ (𝐴(+gβ€˜πΊ)π‘₯)) = {𝑦 ∣ βˆƒπ‘₯ ∈ { 0 }𝑦 = (𝐴(+gβ€˜πΊ)π‘₯)}
48 df-sn 4628 . . . . 5 {𝐴} = {𝑦 ∣ 𝑦 = 𝐴}
4945, 47, 483eqtr4g 2795 . . . 4 ((𝐺 ∈ TopGrp ∧ 𝐴 ∈ 𝑋) β†’ ran (π‘₯ ∈ { 0 } ↦ (𝐴(+gβ€˜πΊ)π‘₯)) = {𝐴})
5036, 49eqtrd 2770 . . 3 ((𝐺 ∈ TopGrp ∧ 𝐴 ∈ 𝑋) β†’ ((π‘₯ ∈ 𝑋 ↦ (𝐴(+gβ€˜πΊ)π‘₯)) β€œ { 0 }) = {𝐴})
5150fveq2d 6894 . 2 ((𝐺 ∈ TopGrp ∧ 𝐴 ∈ 𝑋) β†’ ((clsβ€˜π½)β€˜((π‘₯ ∈ 𝑋 ↦ (𝐴(+gβ€˜πΊ)π‘₯)) β€œ { 0 })) = ((clsβ€˜π½)β€˜{𝐴}))
5232, 51eqtrd 2770 1 ((𝐺 ∈ TopGrp ∧ 𝐴 ∈ 𝑋) β†’ [𝐴] ∼ = ((clsβ€˜π½)β€˜{𝐴}))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ wa 394   = wceq 1539   ∈ wcel 2104  {cab 2707  βˆƒwrex 3068   βŠ† wss 3947  {csn 4627  βˆͺ cuni 4907   ↦ cmpt 5230  ran crn 5676   β†Ύ cres 5677   β€œ cima 5678  β€˜cfv 6542  (class class class)co 7411  [cec 8703  Basecbs 17148  +gcplusg 17201  TopOpenctopn 17371  0gc0g 17389  Grpcgrp 18855   ~QG cqg 19038  Topctop 22615  TopOnctopon 22632  clsccl 22742  Homeochmeo 23477  TopGrpctgp 23795
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 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-10 2135  ax-11 2152  ax-12 2169  ax-ext 2701  ax-rep 5284  ax-sep 5298  ax-nul 5305  ax-pow 5362  ax-pr 5426  ax-un 7727
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2532  df-eu 2561  df-clab 2708  df-cleq 2722  df-clel 2808  df-nfc 2883  df-ne 2939  df-ral 3060  df-rex 3069  df-rmo 3374  df-reu 3375  df-rab 3431  df-v 3474  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4322  df-if 4528  df-pw 4603  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-int 4950  df-iun 4998  df-iin 4999  df-br 5148  df-opab 5210  df-mpt 5231  df-id 5573  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-res 5687  df-ima 5688  df-iota 6494  df-fun 6544  df-fn 6545  df-f 6546  df-f1 6547  df-fo 6548  df-f1o 6549  df-fv 6550  df-riota 7367  df-ov 7414  df-oprab 7415  df-mpo 7416  df-1st 7977  df-2nd 7978  df-ec 8707  df-map 8824  df-0g 17391  df-topgen 17393  df-plusf 18564  df-mgm 18565  df-sgrp 18644  df-mnd 18660  df-grp 18858  df-minusg 18859  df-eqg 19041  df-top 22616  df-topon 22633  df-topsp 22655  df-bases 22669  df-cld 22743  df-cls 22745  df-cn 22951  df-cnp 22952  df-tx 23286  df-hmeo 23479  df-tmd 23796  df-tgp 23797
This theorem is referenced by:  tgptsmscls  23874
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