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Theorem cldsubg 24055

Description: A subgroup of finite index is closed iff it is open. (Contributed by Mario Carneiro, 20-Sep-2015.)

**Hypotheses**
Ref	Expression
subgntr.h	⊢ 𝐽 = (TopOpen‘𝐺)
cldsubg.1	⊢ 𝑅 = (𝐺 ~_QG 𝑆)
cldsubg.2	⊢ 𝑋 = (Base‘𝐺)

**Assertion**
Ref	Expression
cldsubg	⊢ ((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ (𝑋 / 𝑅) ∈ Fin) → (𝑆 ∈ (Clsd‘𝐽) ↔ 𝑆 ∈ 𝐽))

Proof of Theorem cldsubg Dummy variables 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		simpl1 1192	. . . . . . . 8 ⊢ (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ (𝑋 / 𝑅) ∈ Fin) ∧ 𝑆 ∈ (Clsd‘𝐽)) → 𝐺 ∈ TopGrp)
2		subgntr.h	. . . . . . . . 9 ⊢ 𝐽 = (TopOpen‘𝐺)
3		cldsubg.2	. . . . . . . . 9 ⊢ 𝑋 = (Base‘𝐺)
4	2, 3	tgptopon 24026	. . . . . . . 8 ⊢ (𝐺 ∈ TopGrp → 𝐽 ∈ (TopOn‘𝑋))
5	1, 4	syl 17	. . . . . . 7 ⊢ (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ (𝑋 / 𝑅) ∈ Fin) ∧ 𝑆 ∈ (Clsd‘𝐽)) → 𝐽 ∈ (TopOn‘𝑋))
6		toponuni 22858	. . . . . . 7 ⊢ (𝐽 ∈ (TopOn‘𝑋) → 𝑋 = ∪ 𝐽)
7	5, 6	syl 17	. . . . . 6 ⊢ (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ (𝑋 / 𝑅) ∈ Fin) ∧ 𝑆 ∈ (Clsd‘𝐽)) → 𝑋 = ∪ 𝐽)
8	7	difeq1d 4077	. . . . 5 ⊢ (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ (𝑋 / 𝑅) ∈ Fin) ∧ 𝑆 ∈ (Clsd‘𝐽)) → (𝑋 ∖ ∪ ((𝑋 / 𝑅) ∖ {𝑆})) = (∪ 𝐽 ∖ ∪ ((𝑋 / 𝑅) ∖ {𝑆})))
9		simpl2 1193	. . . . . . . . 9 ⊢ (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ (𝑋 / 𝑅) ∈ Fin) ∧ 𝑆 ∈ (Clsd‘𝐽)) → 𝑆 ∈ (SubGrp‘𝐺))
10		unisng 4881	. . . . . . . . 9 ⊢ (𝑆 ∈ (SubGrp‘𝐺) → ∪ {𝑆} = 𝑆)
11	9, 10	syl 17	. . . . . . . 8 ⊢ (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ (𝑋 / 𝑅) ∈ Fin) ∧ 𝑆 ∈ (Clsd‘𝐽)) → ∪ {𝑆} = 𝑆)
12	11	uneq2d 4120	. . . . . . 7 ⊢ (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ (𝑋 / 𝑅) ∈ Fin) ∧ 𝑆 ∈ (Clsd‘𝐽)) → (∪ ((𝑋 / 𝑅) ∖ {𝑆}) ∪ ∪ {𝑆}) = (∪ ((𝑋 / 𝑅) ∖ {𝑆}) ∪ 𝑆))
13		uniun 4886	. . . . . . . 8 ⊢ ∪ (((𝑋 / 𝑅) ∖ {𝑆}) ∪ {𝑆}) = (∪ ((𝑋 / 𝑅) ∖ {𝑆}) ∪ ∪ {𝑆})
14		undif1 4428	. . . . . . . . . . 11 ⊢ (((𝑋 / 𝑅) ∖ {𝑆}) ∪ {𝑆}) = ((𝑋 / 𝑅) ∪ {𝑆})
15		cldsubg.1	. . . . . . . . . . . . . . . 16 ⊢ 𝑅 = (𝐺 ~_QG 𝑆)
16		eqid 2736	. . . . . . . . . . . . . . . 16 ⊢ (0_g‘𝐺) = (0_g‘𝐺)
17	3, 15, 16	eqgid 19109	. . . . . . . . . . . . . . 15 ⊢ (𝑆 ∈ (SubGrp‘𝐺) → [(0_g‘𝐺)]𝑅 = 𝑆)
18	9, 17	syl 17	. . . . . . . . . . . . . 14 ⊢ (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ (𝑋 / 𝑅) ∈ Fin) ∧ 𝑆 ∈ (Clsd‘𝐽)) → [(0_g‘𝐺)]𝑅 = 𝑆)
19	15	ovexi 7392	. . . . . . . . . . . . . . 15 ⊢ 𝑅 ∈ V
20		tgpgrp 24022	. . . . . . . . . . . . . . . . 17 ⊢ (𝐺 ∈ TopGrp → 𝐺 ∈ Grp)
21	1, 20	syl 17	. . . . . . . . . . . . . . . 16 ⊢ (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ (𝑋 / 𝑅) ∈ Fin) ∧ 𝑆 ∈ (Clsd‘𝐽)) → 𝐺 ∈ Grp)
22	3, 16	grpidcl 18895	. . . . . . . . . . . . . . . 16 ⊢ (𝐺 ∈ Grp → (0_g‘𝐺) ∈ 𝑋)
23	21, 22	syl 17	. . . . . . . . . . . . . . 15 ⊢ (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ (𝑋 / 𝑅) ∈ Fin) ∧ 𝑆 ∈ (Clsd‘𝐽)) → (0_g‘𝐺) ∈ 𝑋)
24		ecelqsw 8706	. . . . . . . . . . . . . . 15 ⊢ ((𝑅 ∈ V ∧ (0_g‘𝐺) ∈ 𝑋) → [(0_g‘𝐺)]𝑅 ∈ (𝑋 / 𝑅))
25	19, 23, 24	sylancr 587	. . . . . . . . . . . . . 14 ⊢ (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ (𝑋 / 𝑅) ∈ Fin) ∧ 𝑆 ∈ (Clsd‘𝐽)) → [(0_g‘𝐺)]𝑅 ∈ (𝑋 / 𝑅))
26	18, 25	eqeltrrd 2837	. . . . . . . . . . . . 13 ⊢ (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ (𝑋 / 𝑅) ∈ Fin) ∧ 𝑆 ∈ (Clsd‘𝐽)) → 𝑆 ∈ (𝑋 / 𝑅))
27	26	snssd 4765	. . . . . . . . . . . 12 ⊢ (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ (𝑋 / 𝑅) ∈ Fin) ∧ 𝑆 ∈ (Clsd‘𝐽)) → {𝑆} ⊆ (𝑋 / 𝑅))
28		ssequn2 4141	. . . . . . . . . . . 12 ⊢ ({𝑆} ⊆ (𝑋 / 𝑅) ↔ ((𝑋 / 𝑅) ∪ {𝑆}) = (𝑋 / 𝑅))
29	27, 28	sylib 218	. . . . . . . . . . 11 ⊢ (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ (𝑋 / 𝑅) ∈ Fin) ∧ 𝑆 ∈ (Clsd‘𝐽)) → ((𝑋 / 𝑅) ∪ {𝑆}) = (𝑋 / 𝑅))
30	14, 29	eqtrid 2783	. . . . . . . . . 10 ⊢ (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ (𝑋 / 𝑅) ∈ Fin) ∧ 𝑆 ∈ (Clsd‘𝐽)) → (((𝑋 / 𝑅) ∖ {𝑆}) ∪ {𝑆}) = (𝑋 / 𝑅))
31	30	unieqd 4876	. . . . . . . . 9 ⊢ (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ (𝑋 / 𝑅) ∈ Fin) ∧ 𝑆 ∈ (Clsd‘𝐽)) → ∪ (((𝑋 / 𝑅) ∖ {𝑆}) ∪ {𝑆}) = ∪ (𝑋 / 𝑅))
32	3, 15	eqger 19107	. . . . . . . . . . 11 ⊢ (𝑆 ∈ (SubGrp‘𝐺) → 𝑅 Er 𝑋)
33	9, 32	syl 17	. . . . . . . . . 10 ⊢ (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ (𝑋 / 𝑅) ∈ Fin) ∧ 𝑆 ∈ (Clsd‘𝐽)) → 𝑅 Er 𝑋)
34	19	a1i 11	. . . . . . . . . 10 ⊢ (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ (𝑋 / 𝑅) ∈ Fin) ∧ 𝑆 ∈ (Clsd‘𝐽)) → 𝑅 ∈ V)
35	33, 34	uniqs2 8714	. . . . . . . . 9 ⊢ (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ (𝑋 / 𝑅) ∈ Fin) ∧ 𝑆 ∈ (Clsd‘𝐽)) → ∪ (𝑋 / 𝑅) = 𝑋)
36	31, 35	eqtrd 2771	. . . . . . . 8 ⊢ (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ (𝑋 / 𝑅) ∈ Fin) ∧ 𝑆 ∈ (Clsd‘𝐽)) → ∪ (((𝑋 / 𝑅) ∖ {𝑆}) ∪ {𝑆}) = 𝑋)
37	13, 36	eqtr3id 2785	. . . . . . 7 ⊢ (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ (𝑋 / 𝑅) ∈ Fin) ∧ 𝑆 ∈ (Clsd‘𝐽)) → (∪ ((𝑋 / 𝑅) ∖ {𝑆}) ∪ ∪ {𝑆}) = 𝑋)
38	12, 37	eqtr3d 2773	. . . . . 6 ⊢ (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ (𝑋 / 𝑅) ∈ Fin) ∧ 𝑆 ∈ (Clsd‘𝐽)) → (∪ ((𝑋 / 𝑅) ∖ {𝑆}) ∪ 𝑆) = 𝑋)
39		difss 4088	. . . . . . . . 9 ⊢ ((𝑋 / 𝑅) ∖ {𝑆}) ⊆ (𝑋 / 𝑅)
40	39	unissi 4872	. . . . . . . 8 ⊢ ∪ ((𝑋 / 𝑅) ∖ {𝑆}) ⊆ ∪ (𝑋 / 𝑅)
41	40, 35	sseqtrid 3976	. . . . . . 7 ⊢ (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ (𝑋 / 𝑅) ∈ Fin) ∧ 𝑆 ∈ (Clsd‘𝐽)) → ∪ ((𝑋 / 𝑅) ∖ {𝑆}) ⊆ 𝑋)
42		df-ne 2933	. . . . . . . . . . . . 13 ⊢ (𝑥 ≠ 𝑆 ↔ ¬ 𝑥 = 𝑆)
43	33	adantr 480	. . . . . . . . . . . . . . . 16 ⊢ ((((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ (𝑋 / 𝑅) ∈ Fin) ∧ 𝑆 ∈ (Clsd‘𝐽)) ∧ 𝑥 ∈ (𝑋 / 𝑅)) → 𝑅 Er 𝑋)
44		simpr 484	. . . . . . . . . . . . . . . 16 ⊢ ((((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ (𝑋 / 𝑅) ∈ Fin) ∧ 𝑆 ∈ (Clsd‘𝐽)) ∧ 𝑥 ∈ (𝑋 / 𝑅)) → 𝑥 ∈ (𝑋 / 𝑅))
45	26	adantr 480	. . . . . . . . . . . . . . . 16 ⊢ ((((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ (𝑋 / 𝑅) ∈ Fin) ∧ 𝑆 ∈ (Clsd‘𝐽)) ∧ 𝑥 ∈ (𝑋 / 𝑅)) → 𝑆 ∈ (𝑋 / 𝑅))
46	43, 44, 45	qsdisj 8731	. . . . . . . . . . . . . . 15 ⊢ ((((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ (𝑋 / 𝑅) ∈ Fin) ∧ 𝑆 ∈ (Clsd‘𝐽)) ∧ 𝑥 ∈ (𝑋 / 𝑅)) → (𝑥 = 𝑆 ∨ (𝑥 ∩ 𝑆) = ∅))
47	46	ord 864	. . . . . . . . . . . . . 14 ⊢ ((((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ (𝑋 / 𝑅) ∈ Fin) ∧ 𝑆 ∈ (Clsd‘𝐽)) ∧ 𝑥 ∈ (𝑋 / 𝑅)) → (¬ 𝑥 = 𝑆 → (𝑥 ∩ 𝑆) = ∅))
48		disj2 4410	. . . . . . . . . . . . . 14 ⊢ ((𝑥 ∩ 𝑆) = ∅ ↔ 𝑥 ⊆ (V ∖ 𝑆))
49	47, 48	imbitrdi 251	. . . . . . . . . . . . 13 ⊢ ((((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ (𝑋 / 𝑅) ∈ Fin) ∧ 𝑆 ∈ (Clsd‘𝐽)) ∧ 𝑥 ∈ (𝑋 / 𝑅)) → (¬ 𝑥 = 𝑆 → 𝑥 ⊆ (V ∖ 𝑆)))
50	42, 49	biimtrid 242	. . . . . . . . . . . 12 ⊢ ((((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ (𝑋 / 𝑅) ∈ Fin) ∧ 𝑆 ∈ (Clsd‘𝐽)) ∧ 𝑥 ∈ (𝑋 / 𝑅)) → (𝑥 ≠ 𝑆 → 𝑥 ⊆ (V ∖ 𝑆)))
51	50	expimpd 453	. . . . . . . . . . 11 ⊢ (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ (𝑋 / 𝑅) ∈ Fin) ∧ 𝑆 ∈ (Clsd‘𝐽)) → ((𝑥 ∈ (𝑋 / 𝑅) ∧ 𝑥 ≠ 𝑆) → 𝑥 ⊆ (V ∖ 𝑆)))
52		eldifsn 4742	. . . . . . . . . . 11 ⊢ (𝑥 ∈ ((𝑋 / 𝑅) ∖ {𝑆}) ↔ (𝑥 ∈ (𝑋 / 𝑅) ∧ 𝑥 ≠ 𝑆))
53		velpw 4559	. . . . . . . . . . 11 ⊢ (𝑥 ∈ 𝒫 (V ∖ 𝑆) ↔ 𝑥 ⊆ (V ∖ 𝑆))
54	51, 52, 53	3imtr4g 296	. . . . . . . . . 10 ⊢ (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ (𝑋 / 𝑅) ∈ Fin) ∧ 𝑆 ∈ (Clsd‘𝐽)) → (𝑥 ∈ ((𝑋 / 𝑅) ∖ {𝑆}) → 𝑥 ∈ 𝒫 (V ∖ 𝑆)))
55	54	ssrdv 3939	. . . . . . . . 9 ⊢ (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ (𝑋 / 𝑅) ∈ Fin) ∧ 𝑆 ∈ (Clsd‘𝐽)) → ((𝑋 / 𝑅) ∖ {𝑆}) ⊆ 𝒫 (V ∖ 𝑆))
56		sspwuni 5055	. . . . . . . . 9 ⊢ (((𝑋 / 𝑅) ∖ {𝑆}) ⊆ 𝒫 (V ∖ 𝑆) ↔ ∪ ((𝑋 / 𝑅) ∖ {𝑆}) ⊆ (V ∖ 𝑆))
57	55, 56	sylib 218	. . . . . . . 8 ⊢ (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ (𝑋 / 𝑅) ∈ Fin) ∧ 𝑆 ∈ (Clsd‘𝐽)) → ∪ ((𝑋 / 𝑅) ∖ {𝑆}) ⊆ (V ∖ 𝑆))
58		disj2 4410	. . . . . . . 8 ⊢ ((∪ ((𝑋 / 𝑅) ∖ {𝑆}) ∩ 𝑆) = ∅ ↔ ∪ ((𝑋 / 𝑅) ∖ {𝑆}) ⊆ (V ∖ 𝑆))
59	57, 58	sylibr 234	. . . . . . 7 ⊢ (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ (𝑋 / 𝑅) ∈ Fin) ∧ 𝑆 ∈ (Clsd‘𝐽)) → (∪ ((𝑋 / 𝑅) ∖ {𝑆}) ∩ 𝑆) = ∅)
60		uneqdifeq 4445	. . . . . . 7 ⊢ ((∪ ((𝑋 / 𝑅) ∖ {𝑆}) ⊆ 𝑋 ∧ (∪ ((𝑋 / 𝑅) ∖ {𝑆}) ∩ 𝑆) = ∅) → ((∪ ((𝑋 / 𝑅) ∖ {𝑆}) ∪ 𝑆) = 𝑋 ↔ (𝑋 ∖ ∪ ((𝑋 / 𝑅) ∖ {𝑆})) = 𝑆))
61	41, 59, 60	syl2anc 584	. . . . . 6 ⊢ (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ (𝑋 / 𝑅) ∈ Fin) ∧ 𝑆 ∈ (Clsd‘𝐽)) → ((∪ ((𝑋 / 𝑅) ∖ {𝑆}) ∪ 𝑆) = 𝑋 ↔ (𝑋 ∖ ∪ ((𝑋 / 𝑅) ∖ {𝑆})) = 𝑆))
62	38, 61	mpbid 232	. . . . 5 ⊢ (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ (𝑋 / 𝑅) ∈ Fin) ∧ 𝑆 ∈ (Clsd‘𝐽)) → (𝑋 ∖ ∪ ((𝑋 / 𝑅) ∖ {𝑆})) = 𝑆)
63	8, 62	eqtr3d 2773	. . . 4 ⊢ (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ (𝑋 / 𝑅) ∈ Fin) ∧ 𝑆 ∈ (Clsd‘𝐽)) → (∪ 𝐽 ∖ ∪ ((𝑋 / 𝑅) ∖ {𝑆})) = 𝑆)
64		topontop 22857	. . . . . . 7 ⊢ (𝐽 ∈ (TopOn‘𝑋) → 𝐽 ∈ Top)
65	5, 64	syl 17	. . . . . 6 ⊢ (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ (𝑋 / 𝑅) ∈ Fin) ∧ 𝑆 ∈ (Clsd‘𝐽)) → 𝐽 ∈ Top)
66		simpl3 1194	. . . . . . 7 ⊢ (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ (𝑋 / 𝑅) ∈ Fin) ∧ 𝑆 ∈ (Clsd‘𝐽)) → (𝑋 / 𝑅) ∈ Fin)
67		diffi 9099	. . . . . . 7 ⊢ ((𝑋 / 𝑅) ∈ Fin → ((𝑋 / 𝑅) ∖ {𝑆}) ∈ Fin)
68	66, 67	syl 17	. . . . . 6 ⊢ (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ (𝑋 / 𝑅) ∈ Fin) ∧ 𝑆 ∈ (Clsd‘𝐽)) → ((𝑋 / 𝑅) ∖ {𝑆}) ∈ Fin)
69		vex 3444	. . . . . . . . . 10 ⊢ 𝑥 ∈ V
70	69	elqs 8702	. . . . . . . . 9 ⊢ (𝑥 ∈ (𝑋 / 𝑅) ↔ ∃𝑦 ∈ 𝑋 𝑥 = [𝑦]𝑅)
71		simpll2 1214	. . . . . . . . . . . . . 14 ⊢ ((((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ (𝑋 / 𝑅) ∈ Fin) ∧ 𝑆 ∈ (Clsd‘𝐽)) ∧ 𝑦 ∈ 𝑋) → 𝑆 ∈ (SubGrp‘𝐺))
72		subgrcl 19061	. . . . . . . . . . . . . 14 ⊢ (𝑆 ∈ (SubGrp‘𝐺) → 𝐺 ∈ Grp)
73	71, 72	syl 17	. . . . . . . . . . . . 13 ⊢ ((((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ (𝑋 / 𝑅) ∈ Fin) ∧ 𝑆 ∈ (Clsd‘𝐽)) ∧ 𝑦 ∈ 𝑋) → 𝐺 ∈ Grp)
74	3	subgss 19057	. . . . . . . . . . . . . . 15 ⊢ (𝑆 ∈ (SubGrp‘𝐺) → 𝑆 ⊆ 𝑋)
75	9, 74	syl 17	. . . . . . . . . . . . . 14 ⊢ (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ (𝑋 / 𝑅) ∈ Fin) ∧ 𝑆 ∈ (Clsd‘𝐽)) → 𝑆 ⊆ 𝑋)
76	75	adantr 480	. . . . . . . . . . . . 13 ⊢ ((((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ (𝑋 / 𝑅) ∈ Fin) ∧ 𝑆 ∈ (Clsd‘𝐽)) ∧ 𝑦 ∈ 𝑋) → 𝑆 ⊆ 𝑋)
77		simpr 484	. . . . . . . . . . . . 13 ⊢ ((((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ (𝑋 / 𝑅) ∈ Fin) ∧ 𝑆 ∈ (Clsd‘𝐽)) ∧ 𝑦 ∈ 𝑋) → 𝑦 ∈ 𝑋)
78		eqid 2736	. . . . . . . . . . . . . 14 ⊢ (+_g‘𝐺) = (+_g‘𝐺)
79	3, 15, 78	eqglact 19108	. . . . . . . . . . . . 13 ⊢ ((𝐺 ∈ Grp ∧ 𝑆 ⊆ 𝑋 ∧ 𝑦 ∈ 𝑋) → [𝑦]𝑅 = ((𝑧 ∈ 𝑋 ↦ (𝑦(+_g‘𝐺)𝑧)) “ 𝑆))
80	73, 76, 77, 79	syl3anc 1373	. . . . . . . . . . . 12 ⊢ ((((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ (𝑋 / 𝑅) ∈ Fin) ∧ 𝑆 ∈ (Clsd‘𝐽)) ∧ 𝑦 ∈ 𝑋) → [𝑦]𝑅 = ((𝑧 ∈ 𝑋 ↦ (𝑦(+_g‘𝐺)𝑧)) “ 𝑆))
81		simplr 768	. . . . . . . . . . . . 13 ⊢ ((((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ (𝑋 / 𝑅) ∈ Fin) ∧ 𝑆 ∈ (Clsd‘𝐽)) ∧ 𝑦 ∈ 𝑋) → 𝑆 ∈ (Clsd‘𝐽))
82		eqid 2736	. . . . . . . . . . . . . . . 16 ⊢ (𝑧 ∈ 𝑋 ↦ (𝑦(+_g‘𝐺)𝑧)) = (𝑧 ∈ 𝑋 ↦ (𝑦(+_g‘𝐺)𝑧))
83	82, 3, 78, 2	tgplacthmeo 24047	. . . . . . . . . . . . . . 15 ⊢ ((𝐺 ∈ TopGrp ∧ 𝑦 ∈ 𝑋) → (𝑧 ∈ 𝑋 ↦ (𝑦(+_g‘𝐺)𝑧)) ∈ (𝐽Homeo𝐽))
84	1, 83	sylan 580	. . . . . . . . . . . . . 14 ⊢ ((((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ (𝑋 / 𝑅) ∈ Fin) ∧ 𝑆 ∈ (Clsd‘𝐽)) ∧ 𝑦 ∈ 𝑋) → (𝑧 ∈ 𝑋 ↦ (𝑦(+_g‘𝐺)𝑧)) ∈ (𝐽Homeo𝐽))
85	75, 7	sseqtrd 3970	. . . . . . . . . . . . . . 15 ⊢ (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ (𝑋 / 𝑅) ∈ Fin) ∧ 𝑆 ∈ (Clsd‘𝐽)) → 𝑆 ⊆ ∪ 𝐽)
86	85	adantr 480	. . . . . . . . . . . . . 14 ⊢ ((((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ (𝑋 / 𝑅) ∈ Fin) ∧ 𝑆 ∈ (Clsd‘𝐽)) ∧ 𝑦 ∈ 𝑋) → 𝑆 ⊆ ∪ 𝐽)
87		eqid 2736	. . . . . . . . . . . . . . 15 ⊢ ∪ 𝐽 = ∪ 𝐽
88	87	hmeocld 23711	. . . . . . . . . . . . . 14 ⊢ (((𝑧 ∈ 𝑋 ↦ (𝑦(+_g‘𝐺)𝑧)) ∈ (𝐽Homeo𝐽) ∧ 𝑆 ⊆ ∪ 𝐽) → (𝑆 ∈ (Clsd‘𝐽) ↔ ((𝑧 ∈ 𝑋 ↦ (𝑦(+_g‘𝐺)𝑧)) “ 𝑆) ∈ (Clsd‘𝐽)))
89	84, 86, 88	syl2anc 584	. . . . . . . . . . . . 13 ⊢ ((((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ (𝑋 / 𝑅) ∈ Fin) ∧ 𝑆 ∈ (Clsd‘𝐽)) ∧ 𝑦 ∈ 𝑋) → (𝑆 ∈ (Clsd‘𝐽) ↔ ((𝑧 ∈ 𝑋 ↦ (𝑦(+_g‘𝐺)𝑧)) “ 𝑆) ∈ (Clsd‘𝐽)))
90	81, 89	mpbid 232	. . . . . . . . . . . 12 ⊢ ((((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ (𝑋 / 𝑅) ∈ Fin) ∧ 𝑆 ∈ (Clsd‘𝐽)) ∧ 𝑦 ∈ 𝑋) → ((𝑧 ∈ 𝑋 ↦ (𝑦(+_g‘𝐺)𝑧)) “ 𝑆) ∈ (Clsd‘𝐽))
91	80, 90	eqeltrd 2836	. . . . . . . . . . 11 ⊢ ((((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ (𝑋 / 𝑅) ∈ Fin) ∧ 𝑆 ∈ (Clsd‘𝐽)) ∧ 𝑦 ∈ 𝑋) → [𝑦]𝑅 ∈ (Clsd‘𝐽))
92		eleq1 2824	. . . . . . . . . . 11 ⊢ (𝑥 = [𝑦]𝑅 → (𝑥 ∈ (Clsd‘𝐽) ↔ [𝑦]𝑅 ∈ (Clsd‘𝐽)))
93	91, 92	syl5ibrcom 247	. . . . . . . . . 10 ⊢ ((((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ (𝑋 / 𝑅) ∈ Fin) ∧ 𝑆 ∈ (Clsd‘𝐽)) ∧ 𝑦 ∈ 𝑋) → (𝑥 = [𝑦]𝑅 → 𝑥 ∈ (Clsd‘𝐽)))
94	93	rexlimdva 3137	. . . . . . . . 9 ⊢ (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ (𝑋 / 𝑅) ∈ Fin) ∧ 𝑆 ∈ (Clsd‘𝐽)) → (∃𝑦 ∈ 𝑋 𝑥 = [𝑦]𝑅 → 𝑥 ∈ (Clsd‘𝐽)))
95	70, 94	biimtrid 242	. . . . . . . 8 ⊢ (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ (𝑋 / 𝑅) ∈ Fin) ∧ 𝑆 ∈ (Clsd‘𝐽)) → (𝑥 ∈ (𝑋 / 𝑅) → 𝑥 ∈ (Clsd‘𝐽)))
96	95	ssrdv 3939	. . . . . . 7 ⊢ (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ (𝑋 / 𝑅) ∈ Fin) ∧ 𝑆 ∈ (Clsd‘𝐽)) → (𝑋 / 𝑅) ⊆ (Clsd‘𝐽))
97	96	ssdifssd 4099	. . . . . 6 ⊢ (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ (𝑋 / 𝑅) ∈ Fin) ∧ 𝑆 ∈ (Clsd‘𝐽)) → ((𝑋 / 𝑅) ∖ {𝑆}) ⊆ (Clsd‘𝐽))
98	87	unicld 22990	. . . . . 6 ⊢ ((𝐽 ∈ Top ∧ ((𝑋 / 𝑅) ∖ {𝑆}) ∈ Fin ∧ ((𝑋 / 𝑅) ∖ {𝑆}) ⊆ (Clsd‘𝐽)) → ∪ ((𝑋 / 𝑅) ∖ {𝑆}) ∈ (Clsd‘𝐽))
99	65, 68, 97, 98	syl3anc 1373	. . . . 5 ⊢ (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ (𝑋 / 𝑅) ∈ Fin) ∧ 𝑆 ∈ (Clsd‘𝐽)) → ∪ ((𝑋 / 𝑅) ∖ {𝑆}) ∈ (Clsd‘𝐽))
100	87	cldopn 22975	. . . . 5 ⊢ (∪ ((𝑋 / 𝑅) ∖ {𝑆}) ∈ (Clsd‘𝐽) → (∪ 𝐽 ∖ ∪ ((𝑋 / 𝑅) ∖ {𝑆})) ∈ 𝐽)
101	99, 100	syl 17	. . . 4 ⊢ (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ (𝑋 / 𝑅) ∈ Fin) ∧ 𝑆 ∈ (Clsd‘𝐽)) → (∪ 𝐽 ∖ ∪ ((𝑋 / 𝑅) ∖ {𝑆})) ∈ 𝐽)
102	63, 101	eqeltrrd 2837	. . 3 ⊢ (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ (𝑋 / 𝑅) ∈ Fin) ∧ 𝑆 ∈ (Clsd‘𝐽)) → 𝑆 ∈ 𝐽)
103	102	ex 412	. 2 ⊢ ((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ (𝑋 / 𝑅) ∈ Fin) → (𝑆 ∈ (Clsd‘𝐽) → 𝑆 ∈ 𝐽))
104	2	opnsubg 24052	. . . 4 ⊢ ((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑆 ∈ 𝐽) → 𝑆 ∈ (Clsd‘𝐽))
105	104	3expia 1121	. . 3 ⊢ ((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺)) → (𝑆 ∈ 𝐽 → 𝑆 ∈ (Clsd‘𝐽)))
106	105	3adant3 1132	. 2 ⊢ ((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ (𝑋 / 𝑅) ∈ Fin) → (𝑆 ∈ 𝐽 → 𝑆 ∈ (Clsd‘𝐽)))
107	103, 106	impbid 212	1 ⊢ ((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ (𝑋 / 𝑅) ∈ Fin) → (𝑆 ∈ (Clsd‘𝐽) ↔ 𝑆 ∈ 𝐽))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 206 ∧ wa 395 ∧ w3a 1086 = wceq 1541 ∈ wcel 2113 ≠ wne 2932 ∃wrex 3060 Vcvv 3440 ∖ cdif 3898 ∪ cun 3899 ∩ cin 3900 ⊆ wss 3901 ∅c0 4285 𝒫 cpw 4554 {csn 4580 ∪ cuni 4863 ↦ cmpt 5179 “ cima 5627 ‘cfv 6492 (class class class)co 7358 Er wer 8632 [cec 8633 / cqs 8634 Fincfn 8883 Basecbs 17136 +_gcplusg 17177 TopOpenctopn 17341 0_gc0g 17359 Grpcgrp 18863 SubGrpcsubg 19050 ~_QG cqg 19052 Topctop 22837 TopOnctopon 22854 Clsdccld 22960 Homeochmeo 23697 TopGrpctgp 24015

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1968 ax-7 2009 ax-8 2115 ax-9 2123 ax-10 2146 ax-11 2162 ax-12 2184 ax-ext 2708 ax-rep 5224 ax-sep 5241 ax-nul 5251 ax-pow 5310 ax-pr 5377 ax-un 7680 ax-cnex 11082 ax-resscn 11083 ax-1cn 11084 ax-icn 11085 ax-addcl 11086 ax-addrcl 11087 ax-mulcl 11088 ax-mulrcl 11089 ax-mulcom 11090 ax-addass 11091 ax-mulass 11092 ax-distr 11093 ax-i2m1 11094 ax-1ne0 11095 ax-1rid 11096 ax-rnegex 11097 ax-rrecex 11098 ax-cnre 11099 ax-pre-lttri 11100 ax-pre-lttrn 11101 ax-pre-ltadd 11102 ax-pre-mulgt0 11103

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1544 df-fal 1554 df-ex 1781 df-nf 1785 df-sb 2068 df-mo 2539 df-eu 2569 df-clab 2715 df-cleq 2728 df-clel 2811 df-nfc 2885 df-ne 2933 df-nel 3037 df-ral 3052 df-rex 3061 df-rmo 3350 df-reu 3351 df-rab 3400 df-v 3442 df-sbc 3741 df-csb 3850 df-dif 3904 df-un 3906 df-in 3908 df-ss 3918 df-pss 3921 df-nul 4286 df-if 4480 df-pw 4556 df-sn 4581 df-pr 4583 df-op 4587 df-uni 4864 df-int 4903 df-iun 4948 df-iin 4949 df-br 5099 df-opab 5161 df-mpt 5180 df-tr 5206 df-id 5519 df-eprel 5524 df-po 5532 df-so 5533 df-fr 5577 df-we 5579 df-xp 5630 df-rel 5631 df-cnv 5632 df-co 5633 df-dm 5634 df-rn 5635 df-res 5636 df-ima 5637 df-pred 6259 df-ord 6320 df-on 6321 df-lim 6322 df-suc 6323 df-iota 6448 df-fun 6494 df-fn 6495 df-f 6496 df-f1 6497 df-fo 6498 df-f1o 6499 df-fv 6500 df-riota 7315 df-ov 7361 df-oprab 7362 df-mpo 7363 df-om 7809 df-1st 7933 df-2nd 7934 df-frecs 8223 df-wrecs 8254 df-recs 8303 df-rdg 8341 df-1o 8397 df-2o 8398 df-er 8635 df-ec 8637 df-qs 8641 df-map 8765 df-en 8884 df-dom 8885 df-sdom 8886 df-fin 8887 df-pnf 11168 df-mnf 11169 df-xr 11170 df-ltxr 11171 df-le 11172 df-sub 11366 df-neg 11367 df-nn 12146 df-2 12208 df-sets 17091 df-slot 17109 df-ndx 17121 df-base 17137 df-ress 17158 df-plusg 17190 df-0g 17361 df-topgen 17363 df-plusf 18564 df-mgm 18565 df-sgrp 18644 df-mnd 18660 df-grp 18866 df-minusg 18867 df-sbg 18868 df-subg 19053 df-eqg 19055 df-top 22838 df-topon 22855 df-topsp 22877 df-bases 22890 df-cld 22963 df-cn 23171 df-cnp 23172 df-tx 23506 df-hmeo 23699 df-tmd 24016 df-tgp 24017

This theorem is referenced by: (None)

W3C validator