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Theorem tgphaus 21833
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 21795 . . . . 5 (𝐺 ∈ TopGrp → 𝐺 ∈ Grp)
2 eqid 2621 . . . . . 6 (Base‘𝐺) = (Base‘𝐺)
3 tgphaus.1 . . . . . 6 0 = (0g𝐺)
42, 3grpidcl 17374 . . . . 5 (𝐺 ∈ Grp → 0 ∈ (Base‘𝐺))
51, 4syl 17 . . . 4 (𝐺 ∈ TopGrp → 0 ∈ (Base‘𝐺))
6 tgphaus.j . . . . . 6 𝐽 = (TopOpen‘𝐺)
76, 2tgptopon 21799 . . . . 5 (𝐺 ∈ TopGrp → 𝐽 ∈ (TopOn‘(Base‘𝐺)))
8 toponuni 20641 . . . . 5 (𝐽 ∈ (TopOn‘(Base‘𝐺)) → (Base‘𝐺) = 𝐽)
97, 8syl 17 . . . 4 (𝐺 ∈ TopGrp → (Base‘𝐺) = 𝐽)
105, 9eleqtrd 2700 . . 3 (𝐺 ∈ TopGrp → 0 𝐽)
11 eqid 2621 . . . . 5 𝐽 = 𝐽
1211sncld 21088 . . . 4 ((𝐽 ∈ Haus ∧ 0 𝐽) → { 0 } ∈ (Clsd‘𝐽))
1312expcom 451 . . 3 ( 0 𝐽 → (𝐽 ∈ Haus → { 0 } ∈ (Clsd‘𝐽)))
1410, 13syl 17 . 2 (𝐺 ∈ TopGrp → (𝐽 ∈ Haus → { 0 } ∈ (Clsd‘𝐽)))
15 eqid 2621 . . . . . 6 (-g𝐺) = (-g𝐺)
166, 15tgpsubcn 21807 . . . . 5 (𝐺 ∈ TopGrp → (-g𝐺) ∈ ((𝐽 ×t 𝐽) Cn 𝐽))
17 cnclima 20985 . . . . . 6 (((-g𝐺) ∈ ((𝐽 ×t 𝐽) Cn 𝐽) ∧ { 0 } ∈ (Clsd‘𝐽)) → ((-g𝐺) “ { 0 }) ∈ (Clsd‘(𝐽 ×t 𝐽)))
1817ex 450 . . . . 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 5446 . . . . . . . . 9 ((-g𝐺) “ { 0 }) ⊆ dom (-g𝐺)
212, 15grpsubf 17418 . . . . . . . . . . 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 6010 . . . . . . . . . 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 3634 . . . . . . . 8 (𝐺 ∈ TopGrp → ((-g𝐺) “ { 0 }) ⊆ ((Base‘𝐺) × (Base‘𝐺)))
26 relxp 5190 . . . . . . . 8 Rel ((Base‘𝐺) × (Base‘𝐺))
27 relss 5169 . . . . . . . 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 5545 . . . . . . 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 6004 . . . . . . . . . . . 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 6295 . . . . . . . . . . 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 5108 . . . . . . . . . . . 12 (⟨𝑥, 𝑦⟩ ∈ ((Base‘𝐺) × (Base‘𝐺)) ↔ (𝑥 ∈ (Base‘𝐺) ∧ 𝑦 ∈ (Base‘𝐺)))
3635anbi1i 730 . . . . . . . . . . 11 ((⟨𝑥, 𝑦⟩ ∈ ((Base‘𝐺) × (Base‘𝐺)) ∧ ((-g𝐺)‘⟨𝑥, 𝑦⟩) ∈ { 0 }) ↔ ((𝑥 ∈ (Base‘𝐺) ∧ 𝑦 ∈ (Base‘𝐺)) ∧ ((-g𝐺)‘⟨𝑥, 𝑦⟩) ∈ { 0 }))
372, 3, 15grpsubeq0 17425 . . . . . . . . . . . . . . 15 ((𝐺 ∈ Grp ∧ 𝑥 ∈ (Base‘𝐺) ∧ 𝑦 ∈ (Base‘𝐺)) → ((𝑥(-g𝐺)𝑦) = 0𝑥 = 𝑦))
38373expb 1263 . . . . . . . . . . . . . 14 ((𝐺 ∈ Grp ∧ (𝑥 ∈ (Base‘𝐺) ∧ 𝑦 ∈ (Base‘𝐺))) → ((𝑥(-g𝐺)𝑦) = 0𝑥 = 𝑦))
391, 38sylan 488 . . . . . . . . . . . . 13 ((𝐺 ∈ TopGrp ∧ (𝑥 ∈ (Base‘𝐺) ∧ 𝑦 ∈ (Base‘𝐺))) → ((𝑥(-g𝐺)𝑦) = 0𝑥 = 𝑦))
40 df-ov 6610 . . . . . . . . . . . . . . 15 (𝑥(-g𝐺)𝑦) = ((-g𝐺)‘⟨𝑥, 𝑦⟩)
4140eleq1i 2689 . . . . . . . . . . . . . 14 ((𝑥(-g𝐺)𝑦) ∈ { 0 } ↔ ((-g𝐺)‘⟨𝑥, 𝑦⟩) ∈ { 0 })
42 ovex 6635 . . . . . . . . . . . . . . 15 (𝑥(-g𝐺)𝑦) ∈ V
4342elsn 4165 . . . . . . . . . . . . . 14 ((𝑥(-g𝐺)𝑦) ∈ { 0 } ↔ (𝑥(-g𝐺)𝑦) = 0 )
4441, 43bitr3i 266 . . . . . . . . . . . . 13 (((-g𝐺)‘⟨𝑥, 𝑦⟩) ∈ { 0 } ↔ (𝑥(-g𝐺)𝑦) = 0 )
45 equcom 1942 . . . . . . . . . . . . 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 672 . . . . . . . . . . 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 4616 . . . . . . . . 9 (𝑥((-g𝐺) “ { 0 })𝑦 ↔ ⟨𝑥, 𝑦⟩ ∈ ((-g𝐺) “ { 0 }))
51 eleq1 2686 . . . . . . . . . . . 12 (𝑦 = 𝑥 → (𝑦 ∈ (Base‘𝐺) ↔ 𝑥 ∈ (Base‘𝐺)))
5251biimparc 504 . . . . . . . . . . 11 ((𝑥 ∈ (Base‘𝐺) ∧ 𝑦 = 𝑥) → 𝑦 ∈ (Base‘𝐺))
5352pm4.71i 663 . . . . . . . . . 10 ((𝑥 ∈ (Base‘𝐺) ∧ 𝑦 = 𝑥) ↔ ((𝑥 ∈ (Base‘𝐺) ∧ 𝑦 = 𝑥) ∧ 𝑦 ∈ (Base‘𝐺)))
54 an32 838 . . . . . . . . . 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 4680 . . . . . . 7 (𝐺 ∈ TopGrp → {⟨𝑥, 𝑦⟩ ∣ 𝑥((-g𝐺) “ { 0 })𝑦} = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ (Base‘𝐺) ∧ 𝑦 = 𝑥)})
58 opabresid 5416 . . . . . . 7 {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ (Base‘𝐺) ∧ 𝑦 = 𝑥)} = ( I ↾ (Base‘𝐺))
5957, 58syl6eq 2671 . . . . . 6 (𝐺 ∈ TopGrp → {⟨𝑥, 𝑦⟩ ∣ 𝑥((-g𝐺) “ { 0 })𝑦} = ( I ↾ (Base‘𝐺)))
609reseq2d 5358 . . . . . 6 (𝐺 ∈ TopGrp → ( I ↾ (Base‘𝐺)) = ( I ↾ 𝐽))
6130, 59, 603eqtrd 2659 . . . . 5 (𝐺 ∈ TopGrp → ((-g𝐺) “ { 0 }) = ( I ↾ 𝐽))
6261eleq1d 2683 . . . 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 20640 . . . . 5 (𝐽 ∈ (TopOn‘(Base‘𝐺)) → 𝐽 ∈ Top)
657, 64syl 17 . . . 4 (𝐺 ∈ TopGrp → 𝐽 ∈ Top)
6611hausdiag 21361 . . . . 5 (𝐽 ∈ Haus ↔ (𝐽 ∈ Top ∧ ( I ↾ 𝐽) ∈ (Clsd‘(𝐽 ×t 𝐽))))
6766baib 943 . . . 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 384   = wceq 1480  wcel 1987  wss 3556  {csn 4150  cop 4156   cuni 4404   class class class wbr 4615  {copab 4674   I cid 4986   × cxp 5074  ccnv 5075  dom cdm 5076  cres 5078  cima 5079  Rel wrel 5081   Fn wfn 5844  wf 5845  cfv 5849  (class class class)co 6607  Basecbs 15784  TopOpenctopn 16006  0gc0g 16024  Grpcgrp 17346  -gcsg 17348  Topctop 20620  TopOnctopon 20637  Clsdccld 20733   Cn ccn 20941  Hauscha 21025   ×t ctx 21276  TopGrpctgp 21788
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4733  ax-sep 4743  ax-nul 4751  ax-pow 4805  ax-pr 4869  ax-un 6905
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-reu 2914  df-rmo 2915  df-rab 2916  df-v 3188  df-sbc 3419  df-csb 3516  df-dif 3559  df-un 3561  df-in 3563  df-ss 3570  df-nul 3894  df-if 4061  df-pw 4134  df-sn 4151  df-pr 4153  df-op 4157  df-uni 4405  df-iun 4489  df-br 4616  df-opab 4676  df-mpt 4677  df-id 4991  df-xp 5082  df-rel 5083  df-cnv 5084  df-co 5085  df-dm 5086  df-rn 5087  df-res 5088  df-ima 5089  df-iota 5812  df-fun 5851  df-fn 5852  df-f 5853  df-f1 5854  df-fo 5855  df-f1o 5856  df-fv 5857  df-riota 6568  df-ov 6610  df-oprab 6611  df-mpt2 6612  df-1st 7116  df-2nd 7117  df-map 7807  df-0g 16026  df-topgen 16028  df-plusf 17165  df-mgm 17166  df-sgrp 17208  df-mnd 17219  df-grp 17349  df-minusg 17350  df-sbg 17351  df-top 20621  df-topon 20638  df-topsp 20651  df-bases 20664  df-cld 20736  df-cn 20944  df-t1 21031  df-haus 21032  df-tx 21278  df-tmd 21789  df-tgp 21790
This theorem is referenced by:  tgpt1  21834  qustgphaus  21839
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