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 = (0g‘𝐺)
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.

Step	Hyp	Ref	Expression
1		tgpgrp 24086	. . . . 5 ⊢ (𝐺 ∈ TopGrp → 𝐺 ∈ Grp)
2		eqid 2737	. . . . . 6 ⊢ (Base‘𝐺) = (Base‘𝐺)
3		tgphaus.1	. . . . . 6 ⊢ 0 = (0g‘𝐺)
4	2, 3	grpidcl 18983	. . . . 5 ⊢ (𝐺 ∈ Grp → 0 ∈ (Base‘𝐺))
5	1, 4	syl 17	. . . 4 ⊢ (𝐺 ∈ TopGrp → 0 ∈ (Base‘𝐺))
6		tgphaus.j	. . . . . 6 ⊢ 𝐽 = (TopOpen‘𝐺)
7	6, 2	tgptopon 24090	. . . . 5 ⊢ (𝐺 ∈ TopGrp → 𝐽 ∈ (TopOn‘(Base‘𝐺)))
8		toponuni 22920	. . . . 5 ⊢ (𝐽 ∈ (TopOn‘(Base‘𝐺)) → (Base‘𝐺) = ∪ 𝐽)
9	7, 8	syl 17	. . . 4 ⊢ (𝐺 ∈ TopGrp → (Base‘𝐺) = ∪ 𝐽)
10	5, 9	eleqtrd 2843	. . 3 ⊢ (𝐺 ∈ TopGrp → 0 ∈ ∪ 𝐽)
11		eqid 2737	. . . . 5 ⊢ ∪ 𝐽 = ∪ 𝐽
12	11	sncld 23379	. . . 4 ⊢ ((𝐽 ∈ Haus ∧ 0 ∈ ∪ 𝐽) → { 0 } ∈ (Clsd‘𝐽))
13	12	expcom 413	. . 3 ⊢ ( 0 ∈ ∪ 𝐽 → (𝐽 ∈ Haus → { 0 } ∈ (Clsd‘𝐽)))
14	10, 13	syl 17	. 2 ⊢ (𝐺 ∈ TopGrp → (𝐽 ∈ Haus → { 0 } ∈ (Clsd‘𝐽)))
15		eqid 2737	. . . . . 6 ⊢ (-g‘𝐺) = (-g‘𝐺)
16	6, 15	tgpsubcn 24098	. . . . 5 ⊢ (𝐺 ∈ TopGrp → (-g‘𝐺) ∈ ((𝐽 ×t 𝐽) Cn 𝐽))
17		cnclima 23276	. . . . . 6 ⊢ (((-g‘𝐺) ∈ ((𝐽 ×t 𝐽) Cn 𝐽) ∧ { 0 } ∈ (Clsd‘𝐽)) → (◡(-g‘𝐺) “ { 0 }) ∈ (Clsd‘(𝐽 ×t 𝐽)))
18	17	ex 412	. . . . 5 ⊢ ((-g‘𝐺) ∈ ((𝐽 ×t 𝐽) Cn 𝐽) → ({ 0 } ∈ (Clsd‘𝐽) → (◡(-g‘𝐺) “ { 0 }) ∈ (Clsd‘(𝐽 ×t 𝐽))))
19	16, 18	syl 17	. . . 4 ⊢ (𝐺 ∈ TopGrp → ({ 0 } ∈ (Clsd‘𝐽) → (◡(-g‘𝐺) “ { 0 }) ∈ (Clsd‘(𝐽 ×t 𝐽))))
20		cnvimass 6100	. . . . . . . . 9 ⊢ (◡(-g‘𝐺) “ { 0 }) ⊆ dom (-g‘𝐺)
21	2, 15	grpsubf 19037	. . . . . . . . . 10 ⊢ (𝐺 ∈ Grp → (-g‘𝐺):((Base‘𝐺) × (Base‘𝐺))⟶(Base‘𝐺))
22	1, 21	syl 17	. . . . . . . . 9 ⊢ (𝐺 ∈ TopGrp → (-g‘𝐺):((Base‘𝐺) × (Base‘𝐺))⟶(Base‘𝐺))
23	20, 22	fssdm 6755	. . . . . . . 8 ⊢ (𝐺 ∈ TopGrp → (◡(-g‘𝐺) “ { 0 }) ⊆ ((Base‘𝐺) × (Base‘𝐺)))
24		relxp 5703	. . . . . . . 8 ⊢ Rel ((Base‘𝐺) × (Base‘𝐺))
25		relss 5791	. . . . . . . 8 ⊢ ((◡(-g‘𝐺) “ { 0 }) ⊆ ((Base‘𝐺) × (Base‘𝐺)) → (Rel ((Base‘𝐺) × (Base‘𝐺)) → Rel (◡(-g‘𝐺) “ { 0 })))
26	23, 24, 25	mpisyl 21	. . . . . . 7 ⊢ (𝐺 ∈ TopGrp → Rel (◡(-g‘𝐺) “ { 0 }))
27		dfrel4v 6210	. . . . . . 7 ⊢ (Rel (◡(-g‘𝐺) “ { 0 }) ↔ (◡(-g‘𝐺) “ { 0 }) = {⟨𝑥, 𝑦⟩ ∣ 𝑥(◡(-g‘𝐺) “ { 0 })𝑦})
28	26, 27	sylib 218	. . . . . 6 ⊢ (𝐺 ∈ TopGrp → (◡(-g‘𝐺) “ { 0 }) = {⟨𝑥, 𝑦⟩ ∣ 𝑥(◡(-g‘𝐺) “ { 0 })𝑦})
29	22	ffnd 6737	. . . . . . . . . . 11 ⊢ (𝐺 ∈ TopGrp → (-g‘𝐺) Fn ((Base‘𝐺) × (Base‘𝐺)))
30		elpreima 7078	. . . . . . . . . . 11 ⊢ ((-g‘𝐺) Fn ((Base‘𝐺) × (Base‘𝐺)) → (⟨𝑥, 𝑦⟩ ∈ (◡(-g‘𝐺) “ { 0 }) ↔ (⟨𝑥, 𝑦⟩ ∈ ((Base‘𝐺) × (Base‘𝐺)) ∧ ((-g‘𝐺)‘⟨𝑥, 𝑦⟩) ∈ { 0 })))
31	29, 30	syl 17	. . . . . . . . . 10 ⊢ (𝐺 ∈ TopGrp → (⟨𝑥, 𝑦⟩ ∈ (◡(-g‘𝐺) “ { 0 }) ↔ (⟨𝑥, 𝑦⟩ ∈ ((Base‘𝐺) × (Base‘𝐺)) ∧ ((-g‘𝐺)‘⟨𝑥, 𝑦⟩) ∈ { 0 })))
32		opelxp 5721	. . . . . . . . . . . 12 ⊢ (⟨𝑥, 𝑦⟩ ∈ ((Base‘𝐺) × (Base‘𝐺)) ↔ (𝑥 ∈ (Base‘𝐺) ∧ 𝑦 ∈ (Base‘𝐺)))
33	32	anbi1i 624	. . . . . . . . . . 11 ⊢ ((⟨𝑥, 𝑦⟩ ∈ ((Base‘𝐺) × (Base‘𝐺)) ∧ ((-g‘𝐺)‘⟨𝑥, 𝑦⟩) ∈ { 0 }) ↔ ((𝑥 ∈ (Base‘𝐺) ∧ 𝑦 ∈ (Base‘𝐺)) ∧ ((-g‘𝐺)‘⟨𝑥, 𝑦⟩) ∈ { 0 }))
34	2, 3, 15	grpsubeq0 19044	. . . . . . . . . . . . . . 15 ⊢ ((𝐺 ∈ Grp ∧ 𝑥 ∈ (Base‘𝐺) ∧ 𝑦 ∈ (Base‘𝐺)) → ((𝑥(-g‘𝐺)𝑦) = 0 ↔ 𝑥 = 𝑦))
35	34	3expb 1121	. . . . . . . . . . . . . 14 ⊢ ((𝐺 ∈ Grp ∧ (𝑥 ∈ (Base‘𝐺) ∧ 𝑦 ∈ (Base‘𝐺))) → ((𝑥(-g‘𝐺)𝑦) = 0 ↔ 𝑥 = 𝑦))
36	1, 35	sylan 580	. . . . . . . . . . . . 13 ⊢ ((𝐺 ∈ TopGrp ∧ (𝑥 ∈ (Base‘𝐺) ∧ 𝑦 ∈ (Base‘𝐺))) → ((𝑥(-g‘𝐺)𝑦) = 0 ↔ 𝑥 = 𝑦))
37		df-ov 7434	. . . . . . . . . . . . . . 15 ⊢ (𝑥(-g‘𝐺)𝑦) = ((-g‘𝐺)‘⟨𝑥, 𝑦⟩)
38	37	eleq1i 2832	. . . . . . . . . . . . . 14 ⊢ ((𝑥(-g‘𝐺)𝑦) ∈ { 0 } ↔ ((-g‘𝐺)‘⟨𝑥, 𝑦⟩) ∈ { 0 })
39		ovex 7464	. . . . . . . . . . . . . . 15 ⊢ (𝑥(-g‘𝐺)𝑦) ∈ V
40	39	elsn 4641	. . . . . . . . . . . . . 14 ⊢ ((𝑥(-g‘𝐺)𝑦) ∈ { 0 } ↔ (𝑥(-g‘𝐺)𝑦) = 0 )
41	38, 40	bitr3i 277	. . . . . . . . . . . . 13 ⊢ (((-g‘𝐺)‘⟨𝑥, 𝑦⟩) ∈ { 0 } ↔ (𝑥(-g‘𝐺)𝑦) = 0 )
42		equcom 2017	. . . . . . . . . . . . 13 ⊢ (𝑦 = 𝑥 ↔ 𝑥 = 𝑦)
43	36, 41, 42	3bitr4g 314	. . . . . . . . . . . 12 ⊢ ((𝐺 ∈ TopGrp ∧ (𝑥 ∈ (Base‘𝐺) ∧ 𝑦 ∈ (Base‘𝐺))) → (((-g‘𝐺)‘⟨𝑥, 𝑦⟩) ∈ { 0 } ↔ 𝑦 = 𝑥))
44	43	pm5.32da 579	. . . . . . . . . . 11 ⊢ (𝐺 ∈ TopGrp → (((𝑥 ∈ (Base‘𝐺) ∧ 𝑦 ∈ (Base‘𝐺)) ∧ ((-g‘𝐺)‘⟨𝑥, 𝑦⟩) ∈ { 0 }) ↔ ((𝑥 ∈ (Base‘𝐺) ∧ 𝑦 ∈ (Base‘𝐺)) ∧ 𝑦 = 𝑥)))
45	33, 44	bitrid 283	. . . . . . . . . 10 ⊢ (𝐺 ∈ TopGrp → ((⟨𝑥, 𝑦⟩ ∈ ((Base‘𝐺) × (Base‘𝐺)) ∧ ((-g‘𝐺)‘⟨𝑥, 𝑦⟩) ∈ { 0 }) ↔ ((𝑥 ∈ (Base‘𝐺) ∧ 𝑦 ∈ (Base‘𝐺)) ∧ 𝑦 = 𝑥)))
46	31, 45	bitrd 279	. . . . . . . . 9 ⊢ (𝐺 ∈ TopGrp → (⟨𝑥, 𝑦⟩ ∈ (◡(-g‘𝐺) “ { 0 }) ↔ ((𝑥 ∈ (Base‘𝐺) ∧ 𝑦 ∈ (Base‘𝐺)) ∧ 𝑦 = 𝑥)))
47		df-br 5144	. . . . . . . . 9 ⊢ (𝑥(◡(-g‘𝐺) “ { 0 })𝑦 ↔ ⟨𝑥, 𝑦⟩ ∈ (◡(-g‘𝐺) “ { 0 }))
48		eleq1w 2824	. . . . . . . . . . . 12 ⊢ (𝑦 = 𝑥 → (𝑦 ∈ (Base‘𝐺) ↔ 𝑥 ∈ (Base‘𝐺)))
49	48	biimparc 479	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ (Base‘𝐺) ∧ 𝑦 = 𝑥) → 𝑦 ∈ (Base‘𝐺))
50	49	pm4.71i 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)))
52	50, 51	bitr4i 278	. . . . . . . . 9 ⊢ ((𝑥 ∈ (Base‘𝐺) ∧ 𝑦 = 𝑥) ↔ ((𝑥 ∈ (Base‘𝐺) ∧ 𝑦 ∈ (Base‘𝐺)) ∧ 𝑦 = 𝑥))
53	46, 47, 52	3bitr4g 314	. . . . . . . 8 ⊢ (𝐺 ∈ TopGrp → (𝑥(◡(-g‘𝐺) “ { 0 })𝑦 ↔ (𝑥 ∈ (Base‘𝐺) ∧ 𝑦 = 𝑥)))
54	53	opabbidv 5209	. . . . . . 7 ⊢ (𝐺 ∈ TopGrp → {⟨𝑥, 𝑦⟩ ∣ 𝑥(◡(-g‘𝐺) “ { 0 })𝑦} = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ (Base‘𝐺) ∧ 𝑦 = 𝑥)})
55		opabresid 6068	. . . . . . 7 ⊢ ( I ↾ (Base‘𝐺)) = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ (Base‘𝐺) ∧ 𝑦 = 𝑥)}
56	54, 55	eqtr4di 2795	. . . . . 6 ⊢ (𝐺 ∈ TopGrp → {⟨𝑥, 𝑦⟩ ∣ 𝑥(◡(-g‘𝐺) “ { 0 })𝑦} = ( I ↾ (Base‘𝐺)))
57	9	reseq2d 5997	. . . . . 6 ⊢ (𝐺 ∈ TopGrp → ( I ↾ (Base‘𝐺)) = ( I ↾ ∪ 𝐽))
58	28, 56, 57	3eqtrd 2781	. . . . 5 ⊢ (𝐺 ∈ TopGrp → (◡(-g‘𝐺) “ { 0 }) = ( I ↾ ∪ 𝐽))
59	58	eleq1d 2826	. . . 4 ⊢ (𝐺 ∈ TopGrp → ((◡(-g‘𝐺) “ { 0 }) ∈ (Clsd‘(𝐽 ×t 𝐽)) ↔ ( I ↾ ∪ 𝐽) ∈ (Clsd‘(𝐽 ×t 𝐽))))
60	19, 59	sylibd 239	. . 3 ⊢ (𝐺 ∈ TopGrp → ({ 0 } ∈ (Clsd‘𝐽) → ( I ↾ ∪ 𝐽) ∈ (Clsd‘(𝐽 ×t 𝐽))))
61		topontop 22919	. . . . 5 ⊢ (𝐽 ∈ (TopOn‘(Base‘𝐺)) → 𝐽 ∈ Top)
62	7, 61	syl 17	. . . 4 ⊢ (𝐺 ∈ TopGrp → 𝐽 ∈ Top)
63	11	hausdiag 23653	. . . . 5 ⊢ (𝐽 ∈ Haus ↔ (𝐽 ∈ Top ∧ ( I ↾ ∪ 𝐽) ∈ (Clsd‘(𝐽 ×t 𝐽))))
64	63	baib 535	. . . 4 ⊢ (𝐽 ∈ Top → (𝐽 ∈ Haus ↔ ( I ↾ ∪ 𝐽) ∈ (Clsd‘(𝐽 ×t 𝐽))))
65	62, 64	syl 17	. . 3 ⊢ (𝐺 ∈ TopGrp → (𝐽 ∈ Haus ↔ ( I ↾ ∪ 𝐽) ∈ (Clsd‘(𝐽 ×t 𝐽))))
66	60, 65	sylibrd 259	. 2 ⊢ (𝐺 ∈ TopGrp → ({ 0 } ∈ (Clsd‘𝐽) → 𝐽 ∈ Haus))
67	14, 66	impbid 212	1 ⊢ (𝐺 ∈ TopGrp → (𝐽 ∈ Haus ↔ { 0 } ∈ (Clsd‘𝐽)))

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  0_gc0g 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