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Theorem snclseqg 23620
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 6058 . . 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 23582 . . . . 5 (𝐺 ∈ TopGrp β†’ 𝐺 ∈ Grp)
43adantr 482 . . . 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 23586 . . . . . . . . 9 (𝐺 ∈ TopGrp β†’ 𝐽 ∈ (TopOnβ€˜π‘‹))
87adantr 482 . . . . . . . 8 ((𝐺 ∈ TopGrp ∧ 𝐴 ∈ 𝑋) β†’ 𝐽 ∈ (TopOnβ€˜π‘‹))
9 topontop 22415 . . . . . . . 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 18850 . . . . . . . . . 10 (𝐺 ∈ Grp β†’ 0 ∈ 𝑋)
134, 12syl 17 . . . . . . . . 9 ((𝐺 ∈ TopGrp ∧ 𝐴 ∈ 𝑋) β†’ 0 ∈ 𝑋)
1413snssd 4813 . . . . . . . 8 ((𝐺 ∈ TopGrp ∧ 𝐴 ∈ 𝑋) β†’ { 0 } βŠ† 𝑋)
15 toponuni 22416 . . . . . . . . 9 (𝐽 ∈ (TopOnβ€˜π‘‹) β†’ 𝑋 = βˆͺ 𝐽)
168, 15syl 17 . . . . . . . 8 ((𝐺 ∈ TopGrp ∧ 𝐴 ∈ 𝑋) β†’ 𝑋 = βˆͺ 𝐽)
1714, 16sseqtrd 4023 . . . . . . 7 ((𝐺 ∈ TopGrp ∧ 𝐴 ∈ 𝑋) β†’ { 0 } βŠ† βˆͺ 𝐽)
18 eqid 2733 . . . . . . . 8 βˆͺ 𝐽 = βˆͺ 𝐽
1918clsss3 22563 . . . . . . 7 ((𝐽 ∈ Top ∧ { 0 } βŠ† βˆͺ 𝐽) β†’ ((clsβ€˜π½)β€˜{ 0 }) βŠ† βˆͺ 𝐽)
2010, 17, 19syl2anc 585 . . . . . 6 ((𝐺 ∈ TopGrp ∧ 𝐴 ∈ 𝑋) β†’ ((clsβ€˜π½)β€˜{ 0 }) βŠ† βˆͺ 𝐽)
2120, 16sseqtrrd 4024 . . . . 5 ((𝐺 ∈ TopGrp ∧ 𝐴 ∈ 𝑋) β†’ ((clsβ€˜π½)β€˜{ 0 }) βŠ† 𝑋)
221, 21eqsstrid 4031 . . . 4 ((𝐺 ∈ TopGrp ∧ 𝐴 ∈ 𝑋) β†’ 𝑆 βŠ† 𝑋)
23 simpr 486 . . . 4 ((𝐺 ∈ TopGrp ∧ 𝐴 ∈ 𝑋) β†’ 𝐴 ∈ 𝑋)
24 snclseqg.r . . . . 5 ∼ = (𝐺 ~QG 𝑆)
25 eqid 2733 . . . . 5 (+gβ€˜πΊ) = (+gβ€˜πΊ)
266, 24, 25eqglact 19059 . . . 4 ((𝐺 ∈ Grp ∧ 𝑆 βŠ† 𝑋 ∧ 𝐴 ∈ 𝑋) β†’ [𝐴] ∼ = ((π‘₯ ∈ 𝑋 ↦ (𝐴(+gβ€˜πΊ)π‘₯)) β€œ 𝑆))
274, 22, 23, 26syl3anc 1372 . . 3 ((𝐺 ∈ TopGrp ∧ 𝐴 ∈ 𝑋) β†’ [𝐴] ∼ = ((π‘₯ ∈ 𝑋 ↦ (𝐴(+gβ€˜πΊ)π‘₯)) β€œ 𝑆))
28 eqid 2733 . . . . 5 (π‘₯ ∈ 𝑋 ↦ (𝐴(+gβ€˜πΊ)π‘₯)) = (π‘₯ ∈ 𝑋 ↦ (𝐴(+gβ€˜πΊ)π‘₯))
2928, 6, 25, 5tgplacthmeo 23607 . . . 4 ((𝐺 ∈ TopGrp ∧ 𝐴 ∈ 𝑋) β†’ (π‘₯ ∈ 𝑋 ↦ (𝐴(+gβ€˜πΊ)π‘₯)) ∈ (𝐽Homeo𝐽))
3018hmeocls 23272 . . . 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 585 . . 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 2799 . 2 ((𝐺 ∈ TopGrp ∧ 𝐴 ∈ 𝑋) β†’ [𝐴] ∼ = ((clsβ€˜π½)β€˜((π‘₯ ∈ 𝑋 ↦ (𝐴(+gβ€˜πΊ)π‘₯)) β€œ { 0 })))
33 df-ima 5690 . . . . 5 ((π‘₯ ∈ 𝑋 ↦ (𝐴(+gβ€˜πΊ)π‘₯)) β€œ { 0 }) = ran ((π‘₯ ∈ 𝑋 ↦ (𝐴(+gβ€˜πΊ)π‘₯)) β†Ύ { 0 })
3414resmptd 6041 . . . . . 6 ((𝐺 ∈ TopGrp ∧ 𝐴 ∈ 𝑋) β†’ ((π‘₯ ∈ 𝑋 ↦ (𝐴(+gβ€˜πΊ)π‘₯)) β†Ύ { 0 }) = (π‘₯ ∈ { 0 } ↦ (𝐴(+gβ€˜πΊ)π‘₯)))
3534rneqd 5938 . . . . 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 2785 . . . 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 6906 . . . . . . . 8 0 ∈ V
38 oveq2 7417 . . . . . . . . 9 (π‘₯ = 0 β†’ (𝐴(+gβ€˜πΊ)π‘₯) = (𝐴(+gβ€˜πΊ) 0 ))
3938eqeq2d 2744 . . . . . . . 8 (π‘₯ = 0 β†’ (𝑦 = (𝐴(+gβ€˜πΊ)π‘₯) ↔ 𝑦 = (𝐴(+gβ€˜πΊ) 0 )))
4037, 39rexsn 4687 . . . . . . 7 (βˆƒπ‘₯ ∈ { 0 }𝑦 = (𝐴(+gβ€˜πΊ)π‘₯) ↔ 𝑦 = (𝐴(+gβ€˜πΊ) 0 ))
416, 25, 11grprid 18853 . . . . . . . . 9 ((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋) β†’ (𝐴(+gβ€˜πΊ) 0 ) = 𝐴)
423, 41sylan 581 . . . . . . . 8 ((𝐺 ∈ TopGrp ∧ 𝐴 ∈ 𝑋) β†’ (𝐴(+gβ€˜πΊ) 0 ) = 𝐴)
4342eqeq2d 2744 . . . . . . 7 ((𝐺 ∈ TopGrp ∧ 𝐴 ∈ 𝑋) β†’ (𝑦 = (𝐴(+gβ€˜πΊ) 0 ) ↔ 𝑦 = 𝐴))
4440, 43bitrid 283 . . . . . 6 ((𝐺 ∈ TopGrp ∧ 𝐴 ∈ 𝑋) β†’ (βˆƒπ‘₯ ∈ { 0 }𝑦 = (𝐴(+gβ€˜πΊ)π‘₯) ↔ 𝑦 = 𝐴))
4544abbidv 2802 . . . . 5 ((𝐺 ∈ TopGrp ∧ 𝐴 ∈ 𝑋) β†’ {𝑦 ∣ βˆƒπ‘₯ ∈ { 0 }𝑦 = (𝐴(+gβ€˜πΊ)π‘₯)} = {𝑦 ∣ 𝑦 = 𝐴})
46 eqid 2733 . . . . . 6 (π‘₯ ∈ { 0 } ↦ (𝐴(+gβ€˜πΊ)π‘₯)) = (π‘₯ ∈ { 0 } ↦ (𝐴(+gβ€˜πΊ)π‘₯))
4746rnmpt 5955 . . . . 5 ran (π‘₯ ∈ { 0 } ↦ (𝐴(+gβ€˜πΊ)π‘₯)) = {𝑦 ∣ βˆƒπ‘₯ ∈ { 0 }𝑦 = (𝐴(+gβ€˜πΊ)π‘₯)}
48 df-sn 4630 . . . . 5 {𝐴} = {𝑦 ∣ 𝑦 = 𝐴}
4945, 47, 483eqtr4g 2798 . . . 4 ((𝐺 ∈ TopGrp ∧ 𝐴 ∈ 𝑋) β†’ ran (π‘₯ ∈ { 0 } ↦ (𝐴(+gβ€˜πΊ)π‘₯)) = {𝐴})
5036, 49eqtrd 2773 . . 3 ((𝐺 ∈ TopGrp ∧ 𝐴 ∈ 𝑋) β†’ ((π‘₯ ∈ 𝑋 ↦ (𝐴(+gβ€˜πΊ)π‘₯)) β€œ { 0 }) = {𝐴})
5150fveq2d 6896 . 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 2773 1 ((𝐺 ∈ TopGrp ∧ 𝐴 ∈ 𝑋) β†’ [𝐴] ∼ = ((clsβ€˜π½)β€˜{𝐴}))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ wa 397   = wceq 1542   ∈ wcel 2107  {cab 2710  βˆƒwrex 3071   βŠ† wss 3949  {csn 4629  βˆͺ cuni 4909   ↦ cmpt 5232  ran crn 5678   β†Ύ cres 5679   β€œ cima 5680  β€˜cfv 6544  (class class class)co 7409  [cec 8701  Basecbs 17144  +gcplusg 17197  TopOpenctopn 17367  0gc0g 17385  Grpcgrp 18819   ~QG cqg 19002  Topctop 22395  TopOnctopon 22412  clsccl 22522  Homeochmeo 23257  TopGrpctgp 23575
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-rep 5286  ax-sep 5300  ax-nul 5307  ax-pow 5364  ax-pr 5428  ax-un 7725
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-ral 3063  df-rex 3072  df-rmo 3377  df-reu 3378  df-rab 3434  df-v 3477  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-int 4952  df-iun 5000  df-iin 5001  df-br 5150  df-opab 5212  df-mpt 5233  df-id 5575  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-iota 6496  df-fun 6546  df-fn 6547  df-f 6548  df-f1 6549  df-fo 6550  df-f1o 6551  df-fv 6552  df-riota 7365  df-ov 7412  df-oprab 7413  df-mpo 7414  df-1st 7975  df-2nd 7976  df-ec 8705  df-map 8822  df-0g 17387  df-topgen 17389  df-plusf 18560  df-mgm 18561  df-sgrp 18610  df-mnd 18626  df-grp 18822  df-minusg 18823  df-eqg 19005  df-top 22396  df-topon 22413  df-topsp 22435  df-bases 22449  df-cld 22523  df-cls 22525  df-cn 22731  df-cnp 22732  df-tx 23066  df-hmeo 23259  df-tmd 23576  df-tgp 23577
This theorem is referenced by:  tgptsmscls  23654
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