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Theorem tgphaus 23014
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 22975 . . . . 5 (𝐺 ∈ TopGrp → 𝐺 ∈ Grp)
2 eqid 2737 . . . . . 6 (Base‘𝐺) = (Base‘𝐺)
3 tgphaus.1 . . . . . 6 0 = (0g𝐺)
42, 3grpidcl 18395 . . . . 5 (𝐺 ∈ Grp → 0 ∈ (Base‘𝐺))
51, 4syl 17 . . . 4 (𝐺 ∈ TopGrp → 0 ∈ (Base‘𝐺))
6 tgphaus.j . . . . . 6 𝐽 = (TopOpen‘𝐺)
76, 2tgptopon 22979 . . . . 5 (𝐺 ∈ TopGrp → 𝐽 ∈ (TopOn‘(Base‘𝐺)))
8 toponuni 21811 . . . . 5 (𝐽 ∈ (TopOn‘(Base‘𝐺)) → (Base‘𝐺) = 𝐽)
97, 8syl 17 . . . 4 (𝐺 ∈ TopGrp → (Base‘𝐺) = 𝐽)
105, 9eleqtrd 2840 . . 3 (𝐺 ∈ TopGrp → 0 𝐽)
11 eqid 2737 . . . . 5 𝐽 = 𝐽
1211sncld 22268 . . . 4 ((𝐽 ∈ Haus ∧ 0 𝐽) → { 0 } ∈ (Clsd‘𝐽))
1312expcom 417 . . 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 22987 . . . . 5 (𝐺 ∈ TopGrp → (-g𝐺) ∈ ((𝐽 ×t 𝐽) Cn 𝐽))
17 cnclima 22165 . . . . . 6 (((-g𝐺) ∈ ((𝐽 ×t 𝐽) Cn 𝐽) ∧ { 0 } ∈ (Clsd‘𝐽)) → ((-g𝐺) “ { 0 }) ∈ (Clsd‘(𝐽 ×t 𝐽)))
1817ex 416 . . . . 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 5949 . . . . . . . . 9 ((-g𝐺) “ { 0 }) ⊆ dom (-g𝐺)
212, 15grpsubf 18442 . . . . . . . . . 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 6565 . . . . . . . 8 (𝐺 ∈ TopGrp → ((-g𝐺) “ { 0 }) ⊆ ((Base‘𝐺) × (Base‘𝐺)))
24 relxp 5569 . . . . . . . 8 Rel ((Base‘𝐺) × (Base‘𝐺))
25 relss 5653 . . . . . . . 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 6053 . . . . . . 7 (Rel ((-g𝐺) “ { 0 }) ↔ ((-g𝐺) “ { 0 }) = {⟨𝑥, 𝑦⟩ ∣ 𝑥((-g𝐺) “ { 0 })𝑦})
2826, 27sylib 221 . . . . . 6 (𝐺 ∈ TopGrp → ((-g𝐺) “ { 0 }) = {⟨𝑥, 𝑦⟩ ∣ 𝑥((-g𝐺) “ { 0 })𝑦})
2922ffnd 6546 . . . . . . . . . . 11 (𝐺 ∈ TopGrp → (-g𝐺) Fn ((Base‘𝐺) × (Base‘𝐺)))
30 elpreima 6878 . . . . . . . . . . 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 5587 . . . . . . . . . . . 12 (⟨𝑥, 𝑦⟩ ∈ ((Base‘𝐺) × (Base‘𝐺)) ↔ (𝑥 ∈ (Base‘𝐺) ∧ 𝑦 ∈ (Base‘𝐺)))
3332anbi1i 627 . . . . . . . . . . 11 ((⟨𝑥, 𝑦⟩ ∈ ((Base‘𝐺) × (Base‘𝐺)) ∧ ((-g𝐺)‘⟨𝑥, 𝑦⟩) ∈ { 0 }) ↔ ((𝑥 ∈ (Base‘𝐺) ∧ 𝑦 ∈ (Base‘𝐺)) ∧ ((-g𝐺)‘⟨𝑥, 𝑦⟩) ∈ { 0 }))
342, 3, 15grpsubeq0 18449 . . . . . . . . . . . . . . 15 ((𝐺 ∈ Grp ∧ 𝑥 ∈ (Base‘𝐺) ∧ 𝑦 ∈ (Base‘𝐺)) → ((𝑥(-g𝐺)𝑦) = 0𝑥 = 𝑦))
35343expb 1122 . . . . . . . . . . . . . 14 ((𝐺 ∈ Grp ∧ (𝑥 ∈ (Base‘𝐺) ∧ 𝑦 ∈ (Base‘𝐺))) → ((𝑥(-g𝐺)𝑦) = 0𝑥 = 𝑦))
361, 35sylan 583 . . . . . . . . . . . . 13 ((𝐺 ∈ TopGrp ∧ (𝑥 ∈ (Base‘𝐺) ∧ 𝑦 ∈ (Base‘𝐺))) → ((𝑥(-g𝐺)𝑦) = 0𝑥 = 𝑦))
37 df-ov 7216 . . . . . . . . . . . . . . 15 (𝑥(-g𝐺)𝑦) = ((-g𝐺)‘⟨𝑥, 𝑦⟩)
3837eleq1i 2828 . . . . . . . . . . . . . 14 ((𝑥(-g𝐺)𝑦) ∈ { 0 } ↔ ((-g𝐺)‘⟨𝑥, 𝑦⟩) ∈ { 0 })
39 ovex 7246 . . . . . . . . . . . . . . 15 (𝑥(-g𝐺)𝑦) ∈ V
4039elsn 4556 . . . . . . . . . . . . . 14 ((𝑥(-g𝐺)𝑦) ∈ { 0 } ↔ (𝑥(-g𝐺)𝑦) = 0 )
4138, 40bitr3i 280 . . . . . . . . . . . . 13 (((-g𝐺)‘⟨𝑥, 𝑦⟩) ∈ { 0 } ↔ (𝑥(-g𝐺)𝑦) = 0 )
42 equcom 2026 . . . . . . . . . . . . 13 (𝑦 = 𝑥𝑥 = 𝑦)
4336, 41, 423bitr4g 317 . . . . . . . . . . . 12 ((𝐺 ∈ TopGrp ∧ (𝑥 ∈ (Base‘𝐺) ∧ 𝑦 ∈ (Base‘𝐺))) → (((-g𝐺)‘⟨𝑥, 𝑦⟩) ∈ { 0 } ↔ 𝑦 = 𝑥))
4443pm5.32da 582 . . . . . . . . . . 11 (𝐺 ∈ TopGrp → (((𝑥 ∈ (Base‘𝐺) ∧ 𝑦 ∈ (Base‘𝐺)) ∧ ((-g𝐺)‘⟨𝑥, 𝑦⟩) ∈ { 0 }) ↔ ((𝑥 ∈ (Base‘𝐺) ∧ 𝑦 ∈ (Base‘𝐺)) ∧ 𝑦 = 𝑥)))
4533, 44syl5bb 286 . . . . . . . . . 10 (𝐺 ∈ TopGrp → ((⟨𝑥, 𝑦⟩ ∈ ((Base‘𝐺) × (Base‘𝐺)) ∧ ((-g𝐺)‘⟨𝑥, 𝑦⟩) ∈ { 0 }) ↔ ((𝑥 ∈ (Base‘𝐺) ∧ 𝑦 ∈ (Base‘𝐺)) ∧ 𝑦 = 𝑥)))
4631, 45bitrd 282 . . . . . . . . 9 (𝐺 ∈ TopGrp → (⟨𝑥, 𝑦⟩ ∈ ((-g𝐺) “ { 0 }) ↔ ((𝑥 ∈ (Base‘𝐺) ∧ 𝑦 ∈ (Base‘𝐺)) ∧ 𝑦 = 𝑥)))
47 df-br 5054 . . . . . . . . 9 (𝑥((-g𝐺) “ { 0 })𝑦 ↔ ⟨𝑥, 𝑦⟩ ∈ ((-g𝐺) “ { 0 }))
48 eleq1w 2820 . . . . . . . . . . . 12 (𝑦 = 𝑥 → (𝑦 ∈ (Base‘𝐺) ↔ 𝑥 ∈ (Base‘𝐺)))
4948biimparc 483 . . . . . . . . . . 11 ((𝑥 ∈ (Base‘𝐺) ∧ 𝑦 = 𝑥) → 𝑦 ∈ (Base‘𝐺))
5049pm4.71i 563 . . . . . . . . . 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 281 . . . . . . . . 9 ((𝑥 ∈ (Base‘𝐺) ∧ 𝑦 = 𝑥) ↔ ((𝑥 ∈ (Base‘𝐺) ∧ 𝑦 ∈ (Base‘𝐺)) ∧ 𝑦 = 𝑥))
5346, 47, 523bitr4g 317 . . . . . . . 8 (𝐺 ∈ TopGrp → (𝑥((-g𝐺) “ { 0 })𝑦 ↔ (𝑥 ∈ (Base‘𝐺) ∧ 𝑦 = 𝑥)))
5453opabbidv 5119 . . . . . . 7 (𝐺 ∈ TopGrp → {⟨𝑥, 𝑦⟩ ∣ 𝑥((-g𝐺) “ { 0 })𝑦} = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ (Base‘𝐺) ∧ 𝑦 = 𝑥)})
55 opabresid 5917 . . . . . . 7 ( I ↾ (Base‘𝐺)) = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ (Base‘𝐺) ∧ 𝑦 = 𝑥)}
5654, 55eqtr4di 2796 . . . . . 6 (𝐺 ∈ TopGrp → {⟨𝑥, 𝑦⟩ ∣ 𝑥((-g𝐺) “ { 0 })𝑦} = ( I ↾ (Base‘𝐺)))
579reseq2d 5851 . . . . . 6 (𝐺 ∈ TopGrp → ( I ↾ (Base‘𝐺)) = ( I ↾ 𝐽))
5828, 56, 573eqtrd 2781 . . . . 5 (𝐺 ∈ TopGrp → ((-g𝐺) “ { 0 }) = ( I ↾ 𝐽))
5958eleq1d 2822 . . . 4 (𝐺 ∈ TopGrp → (((-g𝐺) “ { 0 }) ∈ (Clsd‘(𝐽 ×t 𝐽)) ↔ ( I ↾ 𝐽) ∈ (Clsd‘(𝐽 ×t 𝐽))))
6019, 59sylibd 242 . . 3 (𝐺 ∈ TopGrp → ({ 0 } ∈ (Clsd‘𝐽) → ( I ↾ 𝐽) ∈ (Clsd‘(𝐽 ×t 𝐽))))
61 topontop 21810 . . . . 5 (𝐽 ∈ (TopOn‘(Base‘𝐺)) → 𝐽 ∈ Top)
627, 61syl 17 . . . 4 (𝐺 ∈ TopGrp → 𝐽 ∈ Top)
6311hausdiag 22542 . . . . 5 (𝐽 ∈ Haus ↔ (𝐽 ∈ Top ∧ ( I ↾ 𝐽) ∈ (Clsd‘(𝐽 ×t 𝐽))))
6463baib 539 . . . 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 262 . 2 (𝐺 ∈ TopGrp → ({ 0 } ∈ (Clsd‘𝐽) → 𝐽 ∈ Haus))
6714, 66impbid 215 1 (𝐺 ∈ TopGrp → (𝐽 ∈ Haus ↔ { 0 } ∈ (Clsd‘𝐽)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wa 399   = wceq 1543  wcel 2110  wss 3866  {csn 4541  cop 4547   cuni 4819   class class class wbr 5053  {copab 5115   I cid 5454   × cxp 5549  ccnv 5550  cres 5553  cima 5554  Rel wrel 5556   Fn wfn 6375  wf 6376  cfv 6380  (class class class)co 7213  Basecbs 16760  TopOpenctopn 16926  0gc0g 16944  Grpcgrp 18365  -gcsg 18367  Topctop 21790  TopOnctopon 21807  Clsdccld 21913   Cn ccn 22121  Hauscha 22205   ×t ctx 22457  TopGrpctgp 22968
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2158  ax-12 2175  ax-ext 2708  ax-sep 5192  ax-nul 5199  ax-pow 5258  ax-pr 5322  ax-un 7523
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-nf 1792  df-sb 2071  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2886  df-ne 2941  df-ral 3066  df-rex 3067  df-reu 3068  df-rmo 3069  df-rab 3070  df-v 3410  df-sbc 3695  df-csb 3812  df-dif 3869  df-un 3871  df-in 3873  df-ss 3883  df-nul 4238  df-if 4440  df-pw 4515  df-sn 4542  df-pr 4544  df-op 4548  df-uni 4820  df-iun 4906  df-br 5054  df-opab 5116  df-mpt 5136  df-id 5455  df-xp 5557  df-rel 5558  df-cnv 5559  df-co 5560  df-dm 5561  df-rn 5562  df-res 5563  df-ima 5564  df-iota 6338  df-fun 6382  df-fn 6383  df-f 6384  df-fo 6386  df-fv 6388  df-riota 7170  df-ov 7216  df-oprab 7217  df-mpo 7218  df-1st 7761  df-2nd 7762  df-map 8510  df-0g 16946  df-topgen 16948  df-plusf 18113  df-mgm 18114  df-sgrp 18163  df-mnd 18174  df-grp 18368  df-minusg 18369  df-sbg 18370  df-top 21791  df-topon 21808  df-topsp 21830  df-bases 21843  df-cld 21916  df-cn 22124  df-t1 22211  df-haus 22212  df-tx 22459  df-tmd 22969  df-tgp 22970
This theorem is referenced by:  tgpt1  23015  qustgphaus  23020
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