MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  snclseqg Structured version   Visualization version   GIF version

Theorem snclseqg 22139
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 5605 . . 3 ((𝑥𝑋 ↦ (𝐴(+g𝐺)𝑥)) “ 𝑆) = ((𝑥𝑋 ↦ (𝐴(+g𝐺)𝑥)) “ ((cls‘𝐽)‘{ 0 }))
3 tgpgrp 22102 . . . . 5 (𝐺 ∈ TopGrp → 𝐺 ∈ Grp)
43adantr 466 . . . 4 ((𝐺 ∈ TopGrp ∧ 𝐴𝑋) → 𝐺 ∈ Grp)
5 snclseqg.j . . . . . . . . . 10 𝐽 = (TopOpen‘𝐺)
6 snclseqg.x . . . . . . . . . 10 𝑋 = (Base‘𝐺)
75, 6tgptopon 22106 . . . . . . . . 9 (𝐺 ∈ TopGrp → 𝐽 ∈ (TopOn‘𝑋))
87adantr 466 . . . . . . . 8 ((𝐺 ∈ TopGrp ∧ 𝐴𝑋) → 𝐽 ∈ (TopOn‘𝑋))
9 topontop 20938 . . . . . . . 8 (𝐽 ∈ (TopOn‘𝑋) → 𝐽 ∈ Top)
108, 9syl 17 . . . . . . 7 ((𝐺 ∈ TopGrp ∧ 𝐴𝑋) → 𝐽 ∈ Top)
11 snclseqg.z . . . . . . . . . . 11 0 = (0g𝐺)
126, 11grpidcl 17658 . . . . . . . . . 10 (𝐺 ∈ Grp → 0𝑋)
134, 12syl 17 . . . . . . . . 9 ((𝐺 ∈ TopGrp ∧ 𝐴𝑋) → 0𝑋)
1413snssd 4475 . . . . . . . 8 ((𝐺 ∈ TopGrp ∧ 𝐴𝑋) → { 0 } ⊆ 𝑋)
15 toponuni 20939 . . . . . . . . 9 (𝐽 ∈ (TopOn‘𝑋) → 𝑋 = 𝐽)
168, 15syl 17 . . . . . . . 8 ((𝐺 ∈ TopGrp ∧ 𝐴𝑋) → 𝑋 = 𝐽)
1714, 16sseqtrd 3790 . . . . . . 7 ((𝐺 ∈ TopGrp ∧ 𝐴𝑋) → { 0 } ⊆ 𝐽)
18 eqid 2771 . . . . . . . 8 𝐽 = 𝐽
1918clsss3 21084 . . . . . . 7 ((𝐽 ∈ Top ∧ { 0 } ⊆ 𝐽) → ((cls‘𝐽)‘{ 0 }) ⊆ 𝐽)
2010, 17, 19syl2anc 573 . . . . . 6 ((𝐺 ∈ TopGrp ∧ 𝐴𝑋) → ((cls‘𝐽)‘{ 0 }) ⊆ 𝐽)
2120, 16sseqtr4d 3791 . . . . 5 ((𝐺 ∈ TopGrp ∧ 𝐴𝑋) → ((cls‘𝐽)‘{ 0 }) ⊆ 𝑋)
221, 21syl5eqss 3798 . . . 4 ((𝐺 ∈ TopGrp ∧ 𝐴𝑋) → 𝑆𝑋)
23 simpr 471 . . . 4 ((𝐺 ∈ TopGrp ∧ 𝐴𝑋) → 𝐴𝑋)
24 snclseqg.r . . . . 5 = (𝐺 ~QG 𝑆)
25 eqid 2771 . . . . 5 (+g𝐺) = (+g𝐺)
266, 24, 25eqglact 17853 . . . 4 ((𝐺 ∈ Grp ∧ 𝑆𝑋𝐴𝑋) → [𝐴] = ((𝑥𝑋 ↦ (𝐴(+g𝐺)𝑥)) “ 𝑆))
274, 22, 23, 26syl3anc 1476 . . 3 ((𝐺 ∈ TopGrp ∧ 𝐴𝑋) → [𝐴] = ((𝑥𝑋 ↦ (𝐴(+g𝐺)𝑥)) “ 𝑆))
28 eqid 2771 . . . . 5 (𝑥𝑋 ↦ (𝐴(+g𝐺)𝑥)) = (𝑥𝑋 ↦ (𝐴(+g𝐺)𝑥))
2928, 6, 25, 5tgplacthmeo 22127 . . . 4 ((𝐺 ∈ TopGrp ∧ 𝐴𝑋) → (𝑥𝑋 ↦ (𝐴(+g𝐺)𝑥)) ∈ (𝐽Homeo𝐽))
3018hmeocls 21792 . . . 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 573 . . 3 ((𝐺 ∈ TopGrp ∧ 𝐴𝑋) → ((cls‘𝐽)‘((𝑥𝑋 ↦ (𝐴(+g𝐺)𝑥)) “ { 0 })) = ((𝑥𝑋 ↦ (𝐴(+g𝐺)𝑥)) “ ((cls‘𝐽)‘{ 0 })))
322, 27, 313eqtr4a 2831 . 2 ((𝐺 ∈ TopGrp ∧ 𝐴𝑋) → [𝐴] = ((cls‘𝐽)‘((𝑥𝑋 ↦ (𝐴(+g𝐺)𝑥)) “ { 0 })))
33 df-ima 5262 . . . . 5 ((𝑥𝑋 ↦ (𝐴(+g𝐺)𝑥)) “ { 0 }) = ran ((𝑥𝑋 ↦ (𝐴(+g𝐺)𝑥)) ↾ { 0 })
3414resmptd 5593 . . . . . 6 ((𝐺 ∈ TopGrp ∧ 𝐴𝑋) → ((𝑥𝑋 ↦ (𝐴(+g𝐺)𝑥)) ↾ { 0 }) = (𝑥 ∈ { 0 } ↦ (𝐴(+g𝐺)𝑥)))
3534rneqd 5491 . . . . 5 ((𝐺 ∈ TopGrp ∧ 𝐴𝑋) → ran ((𝑥𝑋 ↦ (𝐴(+g𝐺)𝑥)) ↾ { 0 }) = ran (𝑥 ∈ { 0 } ↦ (𝐴(+g𝐺)𝑥)))
3633, 35syl5eq 2817 . . . 4 ((𝐺 ∈ TopGrp ∧ 𝐴𝑋) → ((𝑥𝑋 ↦ (𝐴(+g𝐺)𝑥)) “ { 0 }) = ran (𝑥 ∈ { 0 } ↦ (𝐴(+g𝐺)𝑥)))
37 fvex 6342 . . . . . . . . 9 (0g𝐺) ∈ V
3811, 37eqeltri 2846 . . . . . . . 8 0 ∈ V
39 oveq2 6801 . . . . . . . . 9 (𝑥 = 0 → (𝐴(+g𝐺)𝑥) = (𝐴(+g𝐺) 0 ))
4039eqeq2d 2781 . . . . . . . 8 (𝑥 = 0 → (𝑦 = (𝐴(+g𝐺)𝑥) ↔ 𝑦 = (𝐴(+g𝐺) 0 )))
4138, 40rexsn 4361 . . . . . . 7 (∃𝑥 ∈ { 0 }𝑦 = (𝐴(+g𝐺)𝑥) ↔ 𝑦 = (𝐴(+g𝐺) 0 ))
426, 25, 11grprid 17661 . . . . . . . . 9 ((𝐺 ∈ Grp ∧ 𝐴𝑋) → (𝐴(+g𝐺) 0 ) = 𝐴)
433, 42sylan 569 . . . . . . . 8 ((𝐺 ∈ TopGrp ∧ 𝐴𝑋) → (𝐴(+g𝐺) 0 ) = 𝐴)
4443eqeq2d 2781 . . . . . . 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 2890 . . . . 5 ((𝐺 ∈ TopGrp ∧ 𝐴𝑋) → {𝑦 ∣ ∃𝑥 ∈ { 0 }𝑦 = (𝐴(+g𝐺)𝑥)} = {𝑦𝑦 = 𝐴})
47 eqid 2771 . . . . . 6 (𝑥 ∈ { 0 } ↦ (𝐴(+g𝐺)𝑥)) = (𝑥 ∈ { 0 } ↦ (𝐴(+g𝐺)𝑥))
4847rnmpt 5509 . . . . 5 ran (𝑥 ∈ { 0 } ↦ (𝐴(+g𝐺)𝑥)) = {𝑦 ∣ ∃𝑥 ∈ { 0 }𝑦 = (𝐴(+g𝐺)𝑥)}
49 df-sn 4317 . . . . 5 {𝐴} = {𝑦𝑦 = 𝐴}
5046, 48, 493eqtr4g 2830 . . . 4 ((𝐺 ∈ TopGrp ∧ 𝐴𝑋) → ran (𝑥 ∈ { 0 } ↦ (𝐴(+g𝐺)𝑥)) = {𝐴})
5136, 50eqtrd 2805 . . 3 ((𝐺 ∈ TopGrp ∧ 𝐴𝑋) → ((𝑥𝑋 ↦ (𝐴(+g𝐺)𝑥)) “ { 0 }) = {𝐴})
5251fveq2d 6336 . 2 ((𝐺 ∈ TopGrp ∧ 𝐴𝑋) → ((cls‘𝐽)‘((𝑥𝑋 ↦ (𝐴(+g𝐺)𝑥)) “ { 0 })) = ((cls‘𝐽)‘{𝐴}))
5332, 52eqtrd 2805 1 ((𝐺 ∈ TopGrp ∧ 𝐴𝑋) → [𝐴] = ((cls‘𝐽)‘{𝐴}))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 382   = wceq 1631  wcel 2145  {cab 2757  wrex 3062  Vcvv 3351  wss 3723  {csn 4316   cuni 4574  cmpt 4863  ran crn 5250  cres 5251  cima 5252  cfv 6031  (class class class)co 6793  [cec 7894  Basecbs 16064  +gcplusg 16149  TopOpenctopn 16290  0gc0g 16308  Grpcgrp 17630   ~QG cqg 17798  Topctop 20918  TopOnctopon 20935  clsccl 21043  Homeochmeo 21777  TopGrpctgp 22095
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-rep 4904  ax-sep 4915  ax-nul 4923  ax-pow 4974  ax-pr 5034  ax-un 7096
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-ral 3066  df-rex 3067  df-reu 3068  df-rmo 3069  df-rab 3070  df-v 3353  df-sbc 3588  df-csb 3683  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4226  df-pw 4299  df-sn 4317  df-pr 4319  df-op 4323  df-uni 4575  df-int 4612  df-iun 4656  df-iin 4657  df-br 4787  df-opab 4847  df-mpt 4864  df-id 5157  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-res 5261  df-ima 5262  df-iota 5994  df-fun 6033  df-fn 6034  df-f 6035  df-f1 6036  df-fo 6037  df-f1o 6038  df-fv 6039  df-riota 6754  df-ov 6796  df-oprab 6797  df-mpt2 6798  df-1st 7315  df-2nd 7316  df-ec 7898  df-map 8011  df-0g 16310  df-topgen 16312  df-plusf 17449  df-mgm 17450  df-sgrp 17492  df-mnd 17503  df-grp 17633  df-minusg 17634  df-eqg 17801  df-top 20919  df-topon 20936  df-topsp 20958  df-bases 20971  df-cld 21044  df-cls 21046  df-cn 21252  df-cnp 21253  df-tx 21586  df-hmeo 21779  df-tmd 22096  df-tgp 22097
This theorem is referenced by:  tgptsmscls  22173
  Copyright terms: Public domain W3C validator