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Theorem tgphaus 21967
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 21929 . . . . 5 (𝐺 ∈ TopGrp → 𝐺 ∈ Grp)
2 eqid 2651 . . . . . 6 (Base‘𝐺) = (Base‘𝐺)
3 tgphaus.1 . . . . . 6 0 = (0g𝐺)
42, 3grpidcl 17497 . . . . 5 (𝐺 ∈ Grp → 0 ∈ (Base‘𝐺))
51, 4syl 17 . . . 4 (𝐺 ∈ TopGrp → 0 ∈ (Base‘𝐺))
6 tgphaus.j . . . . . 6 𝐽 = (TopOpen‘𝐺)
76, 2tgptopon 21933 . . . . 5 (𝐺 ∈ TopGrp → 𝐽 ∈ (TopOn‘(Base‘𝐺)))
8 toponuni 20767 . . . . 5 (𝐽 ∈ (TopOn‘(Base‘𝐺)) → (Base‘𝐺) = 𝐽)
97, 8syl 17 . . . 4 (𝐺 ∈ TopGrp → (Base‘𝐺) = 𝐽)
105, 9eleqtrd 2732 . . 3 (𝐺 ∈ TopGrp → 0 𝐽)
11 eqid 2651 . . . . 5 𝐽 = 𝐽
1211sncld 21223 . . . 4 ((𝐽 ∈ Haus ∧ 0 𝐽) → { 0 } ∈ (Clsd‘𝐽))
1312expcom 450 . . 3 ( 0 𝐽 → (𝐽 ∈ Haus → { 0 } ∈ (Clsd‘𝐽)))
1410, 13syl 17 . 2 (𝐺 ∈ TopGrp → (𝐽 ∈ Haus → { 0 } ∈ (Clsd‘𝐽)))
15 eqid 2651 . . . . . 6 (-g𝐺) = (-g𝐺)
166, 15tgpsubcn 21941 . . . . 5 (𝐺 ∈ TopGrp → (-g𝐺) ∈ ((𝐽 ×t 𝐽) Cn 𝐽))
17 cnclima 21120 . . . . . 6 (((-g𝐺) ∈ ((𝐽 ×t 𝐽) Cn 𝐽) ∧ { 0 } ∈ (Clsd‘𝐽)) → ((-g𝐺) “ { 0 }) ∈ (Clsd‘(𝐽 ×t 𝐽)))
1817ex 449 . . . . 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 5520 . . . . . . . . 9 ((-g𝐺) “ { 0 }) ⊆ dom (-g𝐺)
212, 15grpsubf 17541 . . . . . . . . . . 11 (𝐺 ∈ Grp → (-g𝐺):((Base‘𝐺) × (Base‘𝐺))⟶(Base‘𝐺))
221, 21syl 17 . . . . . . . . . 10 (𝐺 ∈ TopGrp → (-g𝐺):((Base‘𝐺) × (Base‘𝐺))⟶(Base‘𝐺))
23 fdm 6089 . . . . . . . . . 10 ((-g𝐺):((Base‘𝐺) × (Base‘𝐺))⟶(Base‘𝐺) → dom (-g𝐺) = ((Base‘𝐺) × (Base‘𝐺)))
2422, 23syl 17 . . . . . . . . 9 (𝐺 ∈ TopGrp → dom (-g𝐺) = ((Base‘𝐺) × (Base‘𝐺)))
2520, 24syl5sseq 3686 . . . . . . . 8 (𝐺 ∈ TopGrp → ((-g𝐺) “ { 0 }) ⊆ ((Base‘𝐺) × (Base‘𝐺)))
26 relxp 5160 . . . . . . . 8 Rel ((Base‘𝐺) × (Base‘𝐺))
27 relss 5240 . . . . . . . 8 (((-g𝐺) “ { 0 }) ⊆ ((Base‘𝐺) × (Base‘𝐺)) → (Rel ((Base‘𝐺) × (Base‘𝐺)) → Rel ((-g𝐺) “ { 0 })))
2825, 26, 27mpisyl 21 . . . . . . 7 (𝐺 ∈ TopGrp → Rel ((-g𝐺) “ { 0 }))
29 dfrel4v 5619 . . . . . . 7 (Rel ((-g𝐺) “ { 0 }) ↔ ((-g𝐺) “ { 0 }) = {⟨𝑥, 𝑦⟩ ∣ 𝑥((-g𝐺) “ { 0 })𝑦})
3028, 29sylib 208 . . . . . 6 (𝐺 ∈ TopGrp → ((-g𝐺) “ { 0 }) = {⟨𝑥, 𝑦⟩ ∣ 𝑥((-g𝐺) “ { 0 })𝑦})
31 ffn 6083 . . . . . . . . . . . 12 ((-g𝐺):((Base‘𝐺) × (Base‘𝐺))⟶(Base‘𝐺) → (-g𝐺) Fn ((Base‘𝐺) × (Base‘𝐺)))
3222, 31syl 17 . . . . . . . . . . 11 (𝐺 ∈ TopGrp → (-g𝐺) Fn ((Base‘𝐺) × (Base‘𝐺)))
33 elpreima 6377 . . . . . . . . . . 11 ((-g𝐺) Fn ((Base‘𝐺) × (Base‘𝐺)) → (⟨𝑥, 𝑦⟩ ∈ ((-g𝐺) “ { 0 }) ↔ (⟨𝑥, 𝑦⟩ ∈ ((Base‘𝐺) × (Base‘𝐺)) ∧ ((-g𝐺)‘⟨𝑥, 𝑦⟩) ∈ { 0 })))
3432, 33syl 17 . . . . . . . . . 10 (𝐺 ∈ TopGrp → (⟨𝑥, 𝑦⟩ ∈ ((-g𝐺) “ { 0 }) ↔ (⟨𝑥, 𝑦⟩ ∈ ((Base‘𝐺) × (Base‘𝐺)) ∧ ((-g𝐺)‘⟨𝑥, 𝑦⟩) ∈ { 0 })))
35 opelxp 5180 . . . . . . . . . . . 12 (⟨𝑥, 𝑦⟩ ∈ ((Base‘𝐺) × (Base‘𝐺)) ↔ (𝑥 ∈ (Base‘𝐺) ∧ 𝑦 ∈ (Base‘𝐺)))
3635anbi1i 731 . . . . . . . . . . 11 ((⟨𝑥, 𝑦⟩ ∈ ((Base‘𝐺) × (Base‘𝐺)) ∧ ((-g𝐺)‘⟨𝑥, 𝑦⟩) ∈ { 0 }) ↔ ((𝑥 ∈ (Base‘𝐺) ∧ 𝑦 ∈ (Base‘𝐺)) ∧ ((-g𝐺)‘⟨𝑥, 𝑦⟩) ∈ { 0 }))
372, 3, 15grpsubeq0 17548 . . . . . . . . . . . . . . 15 ((𝐺 ∈ Grp ∧ 𝑥 ∈ (Base‘𝐺) ∧ 𝑦 ∈ (Base‘𝐺)) → ((𝑥(-g𝐺)𝑦) = 0𝑥 = 𝑦))
38373expb 1285 . . . . . . . . . . . . . 14 ((𝐺 ∈ Grp ∧ (𝑥 ∈ (Base‘𝐺) ∧ 𝑦 ∈ (Base‘𝐺))) → ((𝑥(-g𝐺)𝑦) = 0𝑥 = 𝑦))
391, 38sylan 487 . . . . . . . . . . . . 13 ((𝐺 ∈ TopGrp ∧ (𝑥 ∈ (Base‘𝐺) ∧ 𝑦 ∈ (Base‘𝐺))) → ((𝑥(-g𝐺)𝑦) = 0𝑥 = 𝑦))
40 df-ov 6693 . . . . . . . . . . . . . . 15 (𝑥(-g𝐺)𝑦) = ((-g𝐺)‘⟨𝑥, 𝑦⟩)
4140eleq1i 2721 . . . . . . . . . . . . . 14 ((𝑥(-g𝐺)𝑦) ∈ { 0 } ↔ ((-g𝐺)‘⟨𝑥, 𝑦⟩) ∈ { 0 })
42 ovex 6718 . . . . . . . . . . . . . . 15 (𝑥(-g𝐺)𝑦) ∈ V
4342elsn 4225 . . . . . . . . . . . . . 14 ((𝑥(-g𝐺)𝑦) ∈ { 0 } ↔ (𝑥(-g𝐺)𝑦) = 0 )
4441, 43bitr3i 266 . . . . . . . . . . . . 13 (((-g𝐺)‘⟨𝑥, 𝑦⟩) ∈ { 0 } ↔ (𝑥(-g𝐺)𝑦) = 0 )
45 equcom 1991 . . . . . . . . . . . . 13 (𝑦 = 𝑥𝑥 = 𝑦)
4639, 44, 453bitr4g 303 . . . . . . . . . . . 12 ((𝐺 ∈ TopGrp ∧ (𝑥 ∈ (Base‘𝐺) ∧ 𝑦 ∈ (Base‘𝐺))) → (((-g𝐺)‘⟨𝑥, 𝑦⟩) ∈ { 0 } ↔ 𝑦 = 𝑥))
4746pm5.32da 674 . . . . . . . . . . 11 (𝐺 ∈ TopGrp → (((𝑥 ∈ (Base‘𝐺) ∧ 𝑦 ∈ (Base‘𝐺)) ∧ ((-g𝐺)‘⟨𝑥, 𝑦⟩) ∈ { 0 }) ↔ ((𝑥 ∈ (Base‘𝐺) ∧ 𝑦 ∈ (Base‘𝐺)) ∧ 𝑦 = 𝑥)))
4836, 47syl5bb 272 . . . . . . . . . 10 (𝐺 ∈ TopGrp → ((⟨𝑥, 𝑦⟩ ∈ ((Base‘𝐺) × (Base‘𝐺)) ∧ ((-g𝐺)‘⟨𝑥, 𝑦⟩) ∈ { 0 }) ↔ ((𝑥 ∈ (Base‘𝐺) ∧ 𝑦 ∈ (Base‘𝐺)) ∧ 𝑦 = 𝑥)))
4934, 48bitrd 268 . . . . . . . . 9 (𝐺 ∈ TopGrp → (⟨𝑥, 𝑦⟩ ∈ ((-g𝐺) “ { 0 }) ↔ ((𝑥 ∈ (Base‘𝐺) ∧ 𝑦 ∈ (Base‘𝐺)) ∧ 𝑦 = 𝑥)))
50 df-br 4686 . . . . . . . . 9 (𝑥((-g𝐺) “ { 0 })𝑦 ↔ ⟨𝑥, 𝑦⟩ ∈ ((-g𝐺) “ { 0 }))
51 eleq1 2718 . . . . . . . . . . . 12 (𝑦 = 𝑥 → (𝑦 ∈ (Base‘𝐺) ↔ 𝑥 ∈ (Base‘𝐺)))
5251biimparc 503 . . . . . . . . . . 11 ((𝑥 ∈ (Base‘𝐺) ∧ 𝑦 = 𝑥) → 𝑦 ∈ (Base‘𝐺))
5352pm4.71i 665 . . . . . . . . . 10 ((𝑥 ∈ (Base‘𝐺) ∧ 𝑦 = 𝑥) ↔ ((𝑥 ∈ (Base‘𝐺) ∧ 𝑦 = 𝑥) ∧ 𝑦 ∈ (Base‘𝐺)))
54 an32 856 . . . . . . . . . 10 (((𝑥 ∈ (Base‘𝐺) ∧ 𝑦 ∈ (Base‘𝐺)) ∧ 𝑦 = 𝑥) ↔ ((𝑥 ∈ (Base‘𝐺) ∧ 𝑦 = 𝑥) ∧ 𝑦 ∈ (Base‘𝐺)))
5553, 54bitr4i 267 . . . . . . . . 9 ((𝑥 ∈ (Base‘𝐺) ∧ 𝑦 = 𝑥) ↔ ((𝑥 ∈ (Base‘𝐺) ∧ 𝑦 ∈ (Base‘𝐺)) ∧ 𝑦 = 𝑥))
5649, 50, 553bitr4g 303 . . . . . . . 8 (𝐺 ∈ TopGrp → (𝑥((-g𝐺) “ { 0 })𝑦 ↔ (𝑥 ∈ (Base‘𝐺) ∧ 𝑦 = 𝑥)))
5756opabbidv 4749 . . . . . . 7 (𝐺 ∈ TopGrp → {⟨𝑥, 𝑦⟩ ∣ 𝑥((-g𝐺) “ { 0 })𝑦} = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ (Base‘𝐺) ∧ 𝑦 = 𝑥)})
58 opabresid 5490 . . . . . . 7 {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ (Base‘𝐺) ∧ 𝑦 = 𝑥)} = ( I ↾ (Base‘𝐺))
5957, 58syl6eq 2701 . . . . . 6 (𝐺 ∈ TopGrp → {⟨𝑥, 𝑦⟩ ∣ 𝑥((-g𝐺) “ { 0 })𝑦} = ( I ↾ (Base‘𝐺)))
609reseq2d 5428 . . . . . 6 (𝐺 ∈ TopGrp → ( I ↾ (Base‘𝐺)) = ( I ↾ 𝐽))
6130, 59, 603eqtrd 2689 . . . . 5 (𝐺 ∈ TopGrp → ((-g𝐺) “ { 0 }) = ( I ↾ 𝐽))
6261eleq1d 2715 . . . 4 (𝐺 ∈ TopGrp → (((-g𝐺) “ { 0 }) ∈ (Clsd‘(𝐽 ×t 𝐽)) ↔ ( I ↾ 𝐽) ∈ (Clsd‘(𝐽 ×t 𝐽))))
6319, 62sylibd 229 . . 3 (𝐺 ∈ TopGrp → ({ 0 } ∈ (Clsd‘𝐽) → ( I ↾ 𝐽) ∈ (Clsd‘(𝐽 ×t 𝐽))))
64 topontop 20766 . . . . 5 (𝐽 ∈ (TopOn‘(Base‘𝐺)) → 𝐽 ∈ Top)
657, 64syl 17 . . . 4 (𝐺 ∈ TopGrp → 𝐽 ∈ Top)
6611hausdiag 21496 . . . . 5 (𝐽 ∈ Haus ↔ (𝐽 ∈ Top ∧ ( I ↾ 𝐽) ∈ (Clsd‘(𝐽 ×t 𝐽))))
6766baib 964 . . . 4 (𝐽 ∈ Top → (𝐽 ∈ Haus ↔ ( I ↾ 𝐽) ∈ (Clsd‘(𝐽 ×t 𝐽))))
6865, 67syl 17 . . 3 (𝐺 ∈ TopGrp → (𝐽 ∈ Haus ↔ ( I ↾ 𝐽) ∈ (Clsd‘(𝐽 ×t 𝐽))))
6963, 68sylibrd 249 . 2 (𝐺 ∈ TopGrp → ({ 0 } ∈ (Clsd‘𝐽) → 𝐽 ∈ Haus))
7014, 69impbid 202 1 (𝐺 ∈ TopGrp → (𝐽 ∈ Haus ↔ { 0 } ∈ (Clsd‘𝐽)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 383   = wceq 1523  wcel 2030  wss 3607  {csn 4210  cop 4216   cuni 4468   class class class wbr 4685  {copab 4745   I cid 5052   × cxp 5141  ccnv 5142  dom cdm 5143  cres 5145  cima 5146  Rel wrel 5148   Fn wfn 5921  wf 5922  cfv 5926  (class class class)co 6690  Basecbs 15904  TopOpenctopn 16129  0gc0g 16147  Grpcgrp 17469  -gcsg 17471  Topctop 20746  TopOnctopon 20763  Clsdccld 20868   Cn ccn 21076  Hauscha 21160   ×t ctx 21411  TopGrpctgp 21922
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-rep 4804  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-reu 2948  df-rmo 2949  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-iun 4554  df-br 4686  df-opab 4746  df-mpt 4763  df-id 5053  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-riota 6651  df-ov 6693  df-oprab 6694  df-mpt2 6695  df-1st 7210  df-2nd 7211  df-map 7901  df-0g 16149  df-topgen 16151  df-plusf 17288  df-mgm 17289  df-sgrp 17331  df-mnd 17342  df-grp 17472  df-minusg 17473  df-sbg 17474  df-top 20747  df-topon 20764  df-topsp 20785  df-bases 20798  df-cld 20871  df-cn 21079  df-t1 21166  df-haus 21167  df-tx 21413  df-tmd 21923  df-tgp 21924
This theorem is referenced by:  tgpt1  21968  qustgphaus  21973
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