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Theorem tgphaus 24125
Description: A topological group is Hausdorff iff the identity subgroup is closed. (Contributed by Mario Carneiro, 18-Sep-2015.)
Hypotheses
Ref Expression
tgphaus.1 0 = (0g𝐺)
tgphaus.j 𝐽 = (TopOpen‘𝐺)
Assertion
Ref Expression
tgphaus (𝐺 ∈ TopGrp → (𝐽 ∈ Haus ↔ { 0 } ∈ (Clsd‘𝐽)))

Proof of Theorem tgphaus
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 tgpgrp 24086 . . . . 5 (𝐺 ∈ TopGrp → 𝐺 ∈ Grp)
2 eqid 2737 . . . . . 6 (Base‘𝐺) = (Base‘𝐺)
3 tgphaus.1 . . . . . 6 0 = (0g𝐺)
42, 3grpidcl 18983 . . . . 5 (𝐺 ∈ Grp → 0 ∈ (Base‘𝐺))
51, 4syl 17 . . . 4 (𝐺 ∈ TopGrp → 0 ∈ (Base‘𝐺))
6 tgphaus.j . . . . . 6 𝐽 = (TopOpen‘𝐺)
76, 2tgptopon 24090 . . . . 5 (𝐺 ∈ TopGrp → 𝐽 ∈ (TopOn‘(Base‘𝐺)))
8 toponuni 22920 . . . . 5 (𝐽 ∈ (TopOn‘(Base‘𝐺)) → (Base‘𝐺) = 𝐽)
97, 8syl 17 . . . 4 (𝐺 ∈ TopGrp → (Base‘𝐺) = 𝐽)
105, 9eleqtrd 2843 . . 3 (𝐺 ∈ TopGrp → 0 𝐽)
11 eqid 2737 . . . . 5 𝐽 = 𝐽
1211sncld 23379 . . . 4 ((𝐽 ∈ Haus ∧ 0 𝐽) → { 0 } ∈ (Clsd‘𝐽))
1312expcom 413 . . 3 ( 0 𝐽 → (𝐽 ∈ Haus → { 0 } ∈ (Clsd‘𝐽)))
1410, 13syl 17 . 2 (𝐺 ∈ TopGrp → (𝐽 ∈ Haus → { 0 } ∈ (Clsd‘𝐽)))
15 eqid 2737 . . . . . 6 (-g𝐺) = (-g𝐺)
166, 15tgpsubcn 24098 . . . . 5 (𝐺 ∈ TopGrp → (-g𝐺) ∈ ((𝐽 ×t 𝐽) Cn 𝐽))
17 cnclima 23276 . . . . . 6 (((-g𝐺) ∈ ((𝐽 ×t 𝐽) Cn 𝐽) ∧ { 0 } ∈ (Clsd‘𝐽)) → ((-g𝐺) “ { 0 }) ∈ (Clsd‘(𝐽 ×t 𝐽)))
1817ex 412 . . . . 5 ((-g𝐺) ∈ ((𝐽 ×t 𝐽) Cn 𝐽) → ({ 0 } ∈ (Clsd‘𝐽) → ((-g𝐺) “ { 0 }) ∈ (Clsd‘(𝐽 ×t 𝐽))))
1916, 18syl 17 . . . 4 (𝐺 ∈ TopGrp → ({ 0 } ∈ (Clsd‘𝐽) → ((-g𝐺) “ { 0 }) ∈ (Clsd‘(𝐽 ×t 𝐽))))
20 cnvimass 6100 . . . . . . . . 9 ((-g𝐺) “ { 0 }) ⊆ dom (-g𝐺)
212, 15grpsubf 19037 . . . . . . . . . 10 (𝐺 ∈ Grp → (-g𝐺):((Base‘𝐺) × (Base‘𝐺))⟶(Base‘𝐺))
221, 21syl 17 . . . . . . . . 9 (𝐺 ∈ TopGrp → (-g𝐺):((Base‘𝐺) × (Base‘𝐺))⟶(Base‘𝐺))
2320, 22fssdm 6755 . . . . . . . 8 (𝐺 ∈ TopGrp → ((-g𝐺) “ { 0 }) ⊆ ((Base‘𝐺) × (Base‘𝐺)))
24 relxp 5703 . . . . . . . 8 Rel ((Base‘𝐺) × (Base‘𝐺))
25 relss 5791 . . . . . . . 8 (((-g𝐺) “ { 0 }) ⊆ ((Base‘𝐺) × (Base‘𝐺)) → (Rel ((Base‘𝐺) × (Base‘𝐺)) → Rel ((-g𝐺) “ { 0 })))
2623, 24, 25mpisyl 21 . . . . . . 7 (𝐺 ∈ TopGrp → Rel ((-g𝐺) “ { 0 }))
27 dfrel4v 6210 . . . . . . 7 (Rel ((-g𝐺) “ { 0 }) ↔ ((-g𝐺) “ { 0 }) = {⟨𝑥, 𝑦⟩ ∣ 𝑥((-g𝐺) “ { 0 })𝑦})
2826, 27sylib 218 . . . . . 6 (𝐺 ∈ TopGrp → ((-g𝐺) “ { 0 }) = {⟨𝑥, 𝑦⟩ ∣ 𝑥((-g𝐺) “ { 0 })𝑦})
2922ffnd 6737 . . . . . . . . . . 11 (𝐺 ∈ TopGrp → (-g𝐺) Fn ((Base‘𝐺) × (Base‘𝐺)))
30 elpreima 7078 . . . . . . . . . . 11 ((-g𝐺) Fn ((Base‘𝐺) × (Base‘𝐺)) → (⟨𝑥, 𝑦⟩ ∈ ((-g𝐺) “ { 0 }) ↔ (⟨𝑥, 𝑦⟩ ∈ ((Base‘𝐺) × (Base‘𝐺)) ∧ ((-g𝐺)‘⟨𝑥, 𝑦⟩) ∈ { 0 })))
3129, 30syl 17 . . . . . . . . . 10 (𝐺 ∈ TopGrp → (⟨𝑥, 𝑦⟩ ∈ ((-g𝐺) “ { 0 }) ↔ (⟨𝑥, 𝑦⟩ ∈ ((Base‘𝐺) × (Base‘𝐺)) ∧ ((-g𝐺)‘⟨𝑥, 𝑦⟩) ∈ { 0 })))
32 opelxp 5721 . . . . . . . . . . . 12 (⟨𝑥, 𝑦⟩ ∈ ((Base‘𝐺) × (Base‘𝐺)) ↔ (𝑥 ∈ (Base‘𝐺) ∧ 𝑦 ∈ (Base‘𝐺)))
3332anbi1i 624 . . . . . . . . . . 11 ((⟨𝑥, 𝑦⟩ ∈ ((Base‘𝐺) × (Base‘𝐺)) ∧ ((-g𝐺)‘⟨𝑥, 𝑦⟩) ∈ { 0 }) ↔ ((𝑥 ∈ (Base‘𝐺) ∧ 𝑦 ∈ (Base‘𝐺)) ∧ ((-g𝐺)‘⟨𝑥, 𝑦⟩) ∈ { 0 }))
342, 3, 15grpsubeq0 19044 . . . . . . . . . . . . . . 15 ((𝐺 ∈ Grp ∧ 𝑥 ∈ (Base‘𝐺) ∧ 𝑦 ∈ (Base‘𝐺)) → ((𝑥(-g𝐺)𝑦) = 0𝑥 = 𝑦))
35343expb 1121 . . . . . . . . . . . . . 14 ((𝐺 ∈ Grp ∧ (𝑥 ∈ (Base‘𝐺) ∧ 𝑦 ∈ (Base‘𝐺))) → ((𝑥(-g𝐺)𝑦) = 0𝑥 = 𝑦))
361, 35sylan 580 . . . . . . . . . . . . 13 ((𝐺 ∈ TopGrp ∧ (𝑥 ∈ (Base‘𝐺) ∧ 𝑦 ∈ (Base‘𝐺))) → ((𝑥(-g𝐺)𝑦) = 0𝑥 = 𝑦))
37 df-ov 7434 . . . . . . . . . . . . . . 15 (𝑥(-g𝐺)𝑦) = ((-g𝐺)‘⟨𝑥, 𝑦⟩)
3837eleq1i 2832 . . . . . . . . . . . . . 14 ((𝑥(-g𝐺)𝑦) ∈ { 0 } ↔ ((-g𝐺)‘⟨𝑥, 𝑦⟩) ∈ { 0 })
39 ovex 7464 . . . . . . . . . . . . . . 15 (𝑥(-g𝐺)𝑦) ∈ V
4039elsn 4641 . . . . . . . . . . . . . 14 ((𝑥(-g𝐺)𝑦) ∈ { 0 } ↔ (𝑥(-g𝐺)𝑦) = 0 )
4138, 40bitr3i 277 . . . . . . . . . . . . 13 (((-g𝐺)‘⟨𝑥, 𝑦⟩) ∈ { 0 } ↔ (𝑥(-g𝐺)𝑦) = 0 )
42 equcom 2017 . . . . . . . . . . . . 13 (𝑦 = 𝑥𝑥 = 𝑦)
4336, 41, 423bitr4g 314 . . . . . . . . . . . 12 ((𝐺 ∈ TopGrp ∧ (𝑥 ∈ (Base‘𝐺) ∧ 𝑦 ∈ (Base‘𝐺))) → (((-g𝐺)‘⟨𝑥, 𝑦⟩) ∈ { 0 } ↔ 𝑦 = 𝑥))
4443pm5.32da 579 . . . . . . . . . . 11 (𝐺 ∈ TopGrp → (((𝑥 ∈ (Base‘𝐺) ∧ 𝑦 ∈ (Base‘𝐺)) ∧ ((-g𝐺)‘⟨𝑥, 𝑦⟩) ∈ { 0 }) ↔ ((𝑥 ∈ (Base‘𝐺) ∧ 𝑦 ∈ (Base‘𝐺)) ∧ 𝑦 = 𝑥)))
4533, 44bitrid 283 . . . . . . . . . 10 (𝐺 ∈ TopGrp → ((⟨𝑥, 𝑦⟩ ∈ ((Base‘𝐺) × (Base‘𝐺)) ∧ ((-g𝐺)‘⟨𝑥, 𝑦⟩) ∈ { 0 }) ↔ ((𝑥 ∈ (Base‘𝐺) ∧ 𝑦 ∈ (Base‘𝐺)) ∧ 𝑦 = 𝑥)))
4631, 45bitrd 279 . . . . . . . . 9 (𝐺 ∈ TopGrp → (⟨𝑥, 𝑦⟩ ∈ ((-g𝐺) “ { 0 }) ↔ ((𝑥 ∈ (Base‘𝐺) ∧ 𝑦 ∈ (Base‘𝐺)) ∧ 𝑦 = 𝑥)))
47 df-br 5144 . . . . . . . . 9 (𝑥((-g𝐺) “ { 0 })𝑦 ↔ ⟨𝑥, 𝑦⟩ ∈ ((-g𝐺) “ { 0 }))
48 eleq1w 2824 . . . . . . . . . . . 12 (𝑦 = 𝑥 → (𝑦 ∈ (Base‘𝐺) ↔ 𝑥 ∈ (Base‘𝐺)))
4948biimparc 479 . . . . . . . . . . 11 ((𝑥 ∈ (Base‘𝐺) ∧ 𝑦 = 𝑥) → 𝑦 ∈ (Base‘𝐺))
5049pm4.71i 559 . . . . . . . . . 10 ((𝑥 ∈ (Base‘𝐺) ∧ 𝑦 = 𝑥) ↔ ((𝑥 ∈ (Base‘𝐺) ∧ 𝑦 = 𝑥) ∧ 𝑦 ∈ (Base‘𝐺)))
51 an32 646 . . . . . . . . . 10 (((𝑥 ∈ (Base‘𝐺) ∧ 𝑦 ∈ (Base‘𝐺)) ∧ 𝑦 = 𝑥) ↔ ((𝑥 ∈ (Base‘𝐺) ∧ 𝑦 = 𝑥) ∧ 𝑦 ∈ (Base‘𝐺)))
5250, 51bitr4i 278 . . . . . . . . 9 ((𝑥 ∈ (Base‘𝐺) ∧ 𝑦 = 𝑥) ↔ ((𝑥 ∈ (Base‘𝐺) ∧ 𝑦 ∈ (Base‘𝐺)) ∧ 𝑦 = 𝑥))
5346, 47, 523bitr4g 314 . . . . . . . 8 (𝐺 ∈ TopGrp → (𝑥((-g𝐺) “ { 0 })𝑦 ↔ (𝑥 ∈ (Base‘𝐺) ∧ 𝑦 = 𝑥)))
5453opabbidv 5209 . . . . . . 7 (𝐺 ∈ TopGrp → {⟨𝑥, 𝑦⟩ ∣ 𝑥((-g𝐺) “ { 0 })𝑦} = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ (Base‘𝐺) ∧ 𝑦 = 𝑥)})
55 opabresid 6068 . . . . . . 7 ( I ↾ (Base‘𝐺)) = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ (Base‘𝐺) ∧ 𝑦 = 𝑥)}
5654, 55eqtr4di 2795 . . . . . 6 (𝐺 ∈ TopGrp → {⟨𝑥, 𝑦⟩ ∣ 𝑥((-g𝐺) “ { 0 })𝑦} = ( I ↾ (Base‘𝐺)))
579reseq2d 5997 . . . . . 6 (𝐺 ∈ TopGrp → ( I ↾ (Base‘𝐺)) = ( I ↾ 𝐽))
5828, 56, 573eqtrd 2781 . . . . 5 (𝐺 ∈ TopGrp → ((-g𝐺) “ { 0 }) = ( I ↾ 𝐽))
5958eleq1d 2826 . . . 4 (𝐺 ∈ TopGrp → (((-g𝐺) “ { 0 }) ∈ (Clsd‘(𝐽 ×t 𝐽)) ↔ ( I ↾ 𝐽) ∈ (Clsd‘(𝐽 ×t 𝐽))))
6019, 59sylibd 239 . . 3 (𝐺 ∈ TopGrp → ({ 0 } ∈ (Clsd‘𝐽) → ( I ↾ 𝐽) ∈ (Clsd‘(𝐽 ×t 𝐽))))
61 topontop 22919 . . . . 5 (𝐽 ∈ (TopOn‘(Base‘𝐺)) → 𝐽 ∈ Top)
627, 61syl 17 . . . 4 (𝐺 ∈ TopGrp → 𝐽 ∈ Top)
6311hausdiag 23653 . . . . 5 (𝐽 ∈ Haus ↔ (𝐽 ∈ Top ∧ ( I ↾ 𝐽) ∈ (Clsd‘(𝐽 ×t 𝐽))))
6463baib 535 . . . 4 (𝐽 ∈ Top → (𝐽 ∈ Haus ↔ ( I ↾ 𝐽) ∈ (Clsd‘(𝐽 ×t 𝐽))))
6562, 64syl 17 . . 3 (𝐺 ∈ TopGrp → (𝐽 ∈ Haus ↔ ( I ↾ 𝐽) ∈ (Clsd‘(𝐽 ×t 𝐽))))
6660, 65sylibrd 259 . 2 (𝐺 ∈ TopGrp → ({ 0 } ∈ (Clsd‘𝐽) → 𝐽 ∈ Haus))
6714, 66impbid 212 1 (𝐺 ∈ TopGrp → (𝐽 ∈ Haus ↔ { 0 } ∈ (Clsd‘𝐽)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395   = wceq 1540  wcel 2108  wss 3951  {csn 4626  cop 4632   cuni 4907   class class class wbr 5143  {copab 5205   I cid 5577   × cxp 5683  ccnv 5684  cres 5687  cima 5688  Rel wrel 5690   Fn wfn 6556  wf 6557  cfv 6561  (class class class)co 7431  Basecbs 17247  TopOpenctopn 17466  0gc0g 17484  Grpcgrp 18951  -gcsg 18953  Topctop 22899  TopOnctopon 22916  Clsdccld 23024   Cn ccn 23232  Hauscha 23316   ×t ctx 23568  TopGrpctgp 24079
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 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-rmo 3380  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-fo 6567  df-fv 6569  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-1st 8014  df-2nd 8015  df-map 8868  df-0g 17486  df-topgen 17488  df-plusf 18652  df-mgm 18653  df-sgrp 18732  df-mnd 18748  df-grp 18954  df-minusg 18955  df-sbg 18956  df-top 22900  df-topon 22917  df-topsp 22939  df-bases 22953  df-cld 23027  df-cn 23235  df-t1 23322  df-haus 23323  df-tx 23570  df-tmd 24080  df-tgp 24081
This theorem is referenced by:  tgpt1  24126  qustgphaus  24131
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