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Theorem snclseqg 24140
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 6078 . . 3 ((𝑥𝑋 ↦ (𝐴(+g𝐺)𝑥)) “ 𝑆) = ((𝑥𝑋 ↦ (𝐴(+g𝐺)𝑥)) “ ((cls‘𝐽)‘{ 0 }))
3 tgpgrp 24102 . . . . 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 24106 . . . . . . . . 9 (𝐺 ∈ TopGrp → 𝐽 ∈ (TopOn‘𝑋))
87adantr 480 . . . . . . . 8 ((𝐺 ∈ TopGrp ∧ 𝐴𝑋) → 𝐽 ∈ (TopOn‘𝑋))
9 topontop 22935 . . . . . . . 8 (𝐽 ∈ (TopOn‘𝑋) → 𝐽 ∈ Top)
108, 9syl 17 . . . . . . 7 ((𝐺 ∈ TopGrp ∧ 𝐴𝑋) → 𝐽 ∈ Top)
11 snclseqg.z . . . . . . . . . . 11 0 = (0g𝐺)
126, 11grpidcl 18996 . . . . . . . . . 10 (𝐺 ∈ Grp → 0𝑋)
134, 12syl 17 . . . . . . . . 9 ((𝐺 ∈ TopGrp ∧ 𝐴𝑋) → 0𝑋)
1413snssd 4814 . . . . . . . 8 ((𝐺 ∈ TopGrp ∧ 𝐴𝑋) → { 0 } ⊆ 𝑋)
15 toponuni 22936 . . . . . . . . 9 (𝐽 ∈ (TopOn‘𝑋) → 𝑋 = 𝐽)
168, 15syl 17 . . . . . . . 8 ((𝐺 ∈ TopGrp ∧ 𝐴𝑋) → 𝑋 = 𝐽)
1714, 16sseqtrd 4036 . . . . . . 7 ((𝐺 ∈ TopGrp ∧ 𝐴𝑋) → { 0 } ⊆ 𝐽)
18 eqid 2735 . . . . . . . 8 𝐽 = 𝐽
1918clsss3 23083 . . . . . . 7 ((𝐽 ∈ Top ∧ { 0 } ⊆ 𝐽) → ((cls‘𝐽)‘{ 0 }) ⊆ 𝐽)
2010, 17, 19syl2anc 584 . . . . . 6 ((𝐺 ∈ TopGrp ∧ 𝐴𝑋) → ((cls‘𝐽)‘{ 0 }) ⊆ 𝐽)
2120, 16sseqtrrd 4037 . . . . 5 ((𝐺 ∈ TopGrp ∧ 𝐴𝑋) → ((cls‘𝐽)‘{ 0 }) ⊆ 𝑋)
221, 21eqsstrid 4044 . . . 4 ((𝐺 ∈ TopGrp ∧ 𝐴𝑋) → 𝑆𝑋)
23 simpr 484 . . . 4 ((𝐺 ∈ TopGrp ∧ 𝐴𝑋) → 𝐴𝑋)
24 snclseqg.r . . . . 5 = (𝐺 ~QG 𝑆)
25 eqid 2735 . . . . 5 (+g𝐺) = (+g𝐺)
266, 24, 25eqglact 19210 . . . 4 ((𝐺 ∈ Grp ∧ 𝑆𝑋𝐴𝑋) → [𝐴] = ((𝑥𝑋 ↦ (𝐴(+g𝐺)𝑥)) “ 𝑆))
274, 22, 23, 26syl3anc 1370 . . 3 ((𝐺 ∈ TopGrp ∧ 𝐴𝑋) → [𝐴] = ((𝑥𝑋 ↦ (𝐴(+g𝐺)𝑥)) “ 𝑆))
28 eqid 2735 . . . . 5 (𝑥𝑋 ↦ (𝐴(+g𝐺)𝑥)) = (𝑥𝑋 ↦ (𝐴(+g𝐺)𝑥))
2928, 6, 25, 5tgplacthmeo 24127 . . . 4 ((𝐺 ∈ TopGrp ∧ 𝐴𝑋) → (𝑥𝑋 ↦ (𝐴(+g𝐺)𝑥)) ∈ (𝐽Homeo𝐽))
3018hmeocls 23792 . . . 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 2801 . 2 ((𝐺 ∈ TopGrp ∧ 𝐴𝑋) → [𝐴] = ((cls‘𝐽)‘((𝑥𝑋 ↦ (𝐴(+g𝐺)𝑥)) “ { 0 })))
33 df-ima 5702 . . . . 5 ((𝑥𝑋 ↦ (𝐴(+g𝐺)𝑥)) “ { 0 }) = ran ((𝑥𝑋 ↦ (𝐴(+g𝐺)𝑥)) ↾ { 0 })
3414resmptd 6060 . . . . . 6 ((𝐺 ∈ TopGrp ∧ 𝐴𝑋) → ((𝑥𝑋 ↦ (𝐴(+g𝐺)𝑥)) ↾ { 0 }) = (𝑥 ∈ { 0 } ↦ (𝐴(+g𝐺)𝑥)))
3534rneqd 5952 . . . . 5 ((𝐺 ∈ TopGrp ∧ 𝐴𝑋) → ran ((𝑥𝑋 ↦ (𝐴(+g𝐺)𝑥)) ↾ { 0 }) = ran (𝑥 ∈ { 0 } ↦ (𝐴(+g𝐺)𝑥)))
3633, 35eqtrid 2787 . . . 4 ((𝐺 ∈ TopGrp ∧ 𝐴𝑋) → ((𝑥𝑋 ↦ (𝐴(+g𝐺)𝑥)) “ { 0 }) = ran (𝑥 ∈ { 0 } ↦ (𝐴(+g𝐺)𝑥)))
3711fvexi 6921 . . . . . . . 8 0 ∈ V
38 oveq2 7439 . . . . . . . . 9 (𝑥 = 0 → (𝐴(+g𝐺)𝑥) = (𝐴(+g𝐺) 0 ))
3938eqeq2d 2746 . . . . . . . 8 (𝑥 = 0 → (𝑦 = (𝐴(+g𝐺)𝑥) ↔ 𝑦 = (𝐴(+g𝐺) 0 )))
4037, 39rexsn 4687 . . . . . . 7 (∃𝑥 ∈ { 0 }𝑦 = (𝐴(+g𝐺)𝑥) ↔ 𝑦 = (𝐴(+g𝐺) 0 ))
416, 25, 11grprid 18999 . . . . . . . . 9 ((𝐺 ∈ Grp ∧ 𝐴𝑋) → (𝐴(+g𝐺) 0 ) = 𝐴)
423, 41sylan 580 . . . . . . . 8 ((𝐺 ∈ TopGrp ∧ 𝐴𝑋) → (𝐴(+g𝐺) 0 ) = 𝐴)
4342eqeq2d 2746 . . . . . . 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 2806 . . . . 5 ((𝐺 ∈ TopGrp ∧ 𝐴𝑋) → {𝑦 ∣ ∃𝑥 ∈ { 0 }𝑦 = (𝐴(+g𝐺)𝑥)} = {𝑦𝑦 = 𝐴})
46 eqid 2735 . . . . . 6 (𝑥 ∈ { 0 } ↦ (𝐴(+g𝐺)𝑥)) = (𝑥 ∈ { 0 } ↦ (𝐴(+g𝐺)𝑥))
4746rnmpt 5971 . . . . 5 ran (𝑥 ∈ { 0 } ↦ (𝐴(+g𝐺)𝑥)) = {𝑦 ∣ ∃𝑥 ∈ { 0 }𝑦 = (𝐴(+g𝐺)𝑥)}
48 df-sn 4632 . . . . 5 {𝐴} = {𝑦𝑦 = 𝐴}
4945, 47, 483eqtr4g 2800 . . . 4 ((𝐺 ∈ TopGrp ∧ 𝐴𝑋) → ran (𝑥 ∈ { 0 } ↦ (𝐴(+g𝐺)𝑥)) = {𝐴})
5036, 49eqtrd 2775 . . 3 ((𝐺 ∈ TopGrp ∧ 𝐴𝑋) → ((𝑥𝑋 ↦ (𝐴(+g𝐺)𝑥)) “ { 0 }) = {𝐴})
5150fveq2d 6911 . 2 ((𝐺 ∈ TopGrp ∧ 𝐴𝑋) → ((cls‘𝐽)‘((𝑥𝑋 ↦ (𝐴(+g𝐺)𝑥)) “ { 0 })) = ((cls‘𝐽)‘{𝐴}))
5232, 51eqtrd 2775 1 ((𝐺 ∈ TopGrp ∧ 𝐴𝑋) → [𝐴] = ((cls‘𝐽)‘{𝐴}))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1537  wcel 2106  {cab 2712  wrex 3068  wss 3963  {csn 4631   cuni 4912  cmpt 5231  ran crn 5690  cres 5691  cima 5692  cfv 6563  (class class class)co 7431  [cec 8742  Basecbs 17245  +gcplusg 17298  TopOpenctopn 17468  0gc0g 17486  Grpcgrp 18964   ~QG cqg 19153  Topctop 22915  TopOnctopon 22932  clsccl 23042  Homeochmeo 23777  TopGrpctgp 24095
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rmo 3378  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-int 4952  df-iun 4998  df-iin 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-1st 8013  df-2nd 8014  df-ec 8746  df-map 8867  df-0g 17488  df-topgen 17490  df-plusf 18665  df-mgm 18666  df-sgrp 18745  df-mnd 18761  df-grp 18967  df-minusg 18968  df-eqg 19156  df-top 22916  df-topon 22933  df-topsp 22955  df-bases 22969  df-cld 23043  df-cls 23045  df-cn 23251  df-cnp 23252  df-tx 23586  df-hmeo 23779  df-tmd 24096  df-tgp 24097
This theorem is referenced by:  tgptsmscls  24174
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