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Theorem cldsubg 23996

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 23967	. . . . . . . 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 22799	. . . . . . 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 4076	. . . . 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 4876	. . . . . . . . 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 4119	. . . . . . 7 ⊢ (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ (𝑋 / 𝑅) ∈ Fin) ∧ 𝑆 ∈ (Clsd‘𝐽)) → (∪ ((𝑋 / 𝑅) ∖ {𝑆}) ∪ ∪ {𝑆}) = (∪ ((𝑋 / 𝑅) ∖ {𝑆}) ∪ 𝑆))
13		uniun 4881	. . . . . . . 8 ⊢ ∪ (((𝑋 / 𝑅) ∖ {𝑆}) ∪ {𝑆}) = (∪ ((𝑋 / 𝑅) ∖ {𝑆}) ∪ ∪ {𝑆})
14		undif1 4427	. . . . . . . . . . 11 ⊢ (((𝑋 / 𝑅) ∖ {𝑆}) ∪ {𝑆}) = ((𝑋 / 𝑅) ∪ {𝑆})
15		cldsubg.1	. . . . . . . . . . . . . . . 16 ⊢ 𝑅 = (𝐺 ~_QG 𝑆)
16		eqid 2729	. . . . . . . . . . . . . . . 16 ⊢ (0_g‘𝐺) = (0_g‘𝐺)
17	3, 15, 16	eqgid 19059	. . . . . . . . . . . . . . 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 7383	. . . . . . . . . . . . . . 15 ⊢ 𝑅 ∈ V
20		tgpgrp 23963	. . . . . . . . . . . . . . . . 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 18844	. . . . . . . . . . . . . . . 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 8696	. . . . . . . . . . . . . . 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 2829	. . . . . . . . . . . . 13 ⊢ (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ (𝑋 / 𝑅) ∈ Fin) ∧ 𝑆 ∈ (Clsd‘𝐽)) → 𝑆 ∈ (𝑋 / 𝑅))
27	26	snssd 4760	. . . . . . . . . . . 12 ⊢ (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ (𝑋 / 𝑅) ∈ Fin) ∧ 𝑆 ∈ (Clsd‘𝐽)) → {𝑆} ⊆ (𝑋 / 𝑅))
28		ssequn2 4140	. . . . . . . . . . . 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 2776	. . . . . . . . . 10 ⊢ (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ (𝑋 / 𝑅) ∈ Fin) ∧ 𝑆 ∈ (Clsd‘𝐽)) → (((𝑋 / 𝑅) ∖ {𝑆}) ∪ {𝑆}) = (𝑋 / 𝑅))
31	30	unieqd 4871	. . . . . . . . 9 ⊢ (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ (𝑋 / 𝑅) ∈ Fin) ∧ 𝑆 ∈ (Clsd‘𝐽)) → ∪ (((𝑋 / 𝑅) ∖ {𝑆}) ∪ {𝑆}) = ∪ (𝑋 / 𝑅))
32	3, 15	eqger 19057	. . . . . . . . . . 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 8704	. . . . . . . . 9 ⊢ (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ (𝑋 / 𝑅) ∈ Fin) ∧ 𝑆 ∈ (Clsd‘𝐽)) → ∪ (𝑋 / 𝑅) = 𝑋)
36	31, 35	eqtrd 2764	. . . . . . . 8 ⊢ (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ (𝑋 / 𝑅) ∈ Fin) ∧ 𝑆 ∈ (Clsd‘𝐽)) → ∪ (((𝑋 / 𝑅) ∖ {𝑆}) ∪ {𝑆}) = 𝑋)
37	13, 36	eqtr3id 2778	. . . . . . 7 ⊢ (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ (𝑋 / 𝑅) ∈ Fin) ∧ 𝑆 ∈ (Clsd‘𝐽)) → (∪ ((𝑋 / 𝑅) ∖ {𝑆}) ∪ ∪ {𝑆}) = 𝑋)
38	12, 37	eqtr3d 2766	. . . . . 6 ⊢ (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ (𝑋 / 𝑅) ∈ Fin) ∧ 𝑆 ∈ (Clsd‘𝐽)) → (∪ ((𝑋 / 𝑅) ∖ {𝑆}) ∪ 𝑆) = 𝑋)
39		difss 4087	. . . . . . . . 9 ⊢ ((𝑋 / 𝑅) ∖ {𝑆}) ⊆ (𝑋 / 𝑅)
40	39	unissi 4867	. . . . . . . 8 ⊢ ∪ ((𝑋 / 𝑅) ∖ {𝑆}) ⊆ ∪ (𝑋 / 𝑅)
41	40, 35	sseqtrid 3978	. . . . . . 7 ⊢ (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ (𝑋 / 𝑅) ∈ Fin) ∧ 𝑆 ∈ (Clsd‘𝐽)) → ∪ ((𝑋 / 𝑅) ∖ {𝑆}) ⊆ 𝑋)
42		df-ne 2926	. . . . . . . . . . . . 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 8721	. . . . . . . . . . . . . . 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 4409	. . . . . . . . . . . . . 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 4737	. . . . . . . . . . 11 ⊢ (𝑥 ∈ ((𝑋 / 𝑅) ∖ {𝑆}) ↔ (𝑥 ∈ (𝑋 / 𝑅) ∧ 𝑥 ≠ 𝑆))
53		velpw 4556	. . . . . . . . . . 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 3941	. . . . . . . . 9 ⊢ (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ (𝑋 / 𝑅) ∈ Fin) ∧ 𝑆 ∈ (Clsd‘𝐽)) → ((𝑋 / 𝑅) ∖ {𝑆}) ⊆ 𝒫 (V ∖ 𝑆))
56		sspwuni 5049	. . . . . . . . 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 4409	. . . . . . . 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 4444	. . . . . . 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 2766	. . . 4 ⊢ (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ (𝑋 / 𝑅) ∈ Fin) ∧ 𝑆 ∈ (Clsd‘𝐽)) → (∪ 𝐽 ∖ ∪ ((𝑋 / 𝑅) ∖ {𝑆})) = 𝑆)
64		topontop 22798	. . . . . . 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 9089	. . . . . . 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 3440	. . . . . . . . . 10 ⊢ 𝑥 ∈ V
70	69	elqs 8692	. . . . . . . . 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 19010	. . . . . . . . . . . . . 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 19006	. . . . . . . . . . . . . . 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 2729	. . . . . . . . . . . . . 14 ⊢ (+_g‘𝐺) = (+_g‘𝐺)
79	3, 15, 78	eqglact 19058	. . . . . . . . . . . . 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 2729	. . . . . . . . . . . . . . . 16 ⊢ (𝑧 ∈ 𝑋 ↦ (𝑦(+_g‘𝐺)𝑧)) = (𝑧 ∈ 𝑋 ↦ (𝑦(+_g‘𝐺)𝑧))
83	82, 3, 78, 2	tgplacthmeo 23988	. . . . . . . . . . . . . . 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 3972	. . . . . . . . . . . . . . 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 2729	. . . . . . . . . . . . . . 15 ⊢ ∪ 𝐽 = ∪ 𝐽
88	87	hmeocld 23652	. . . . . . . . . . . . . 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 2828	. . . . . . . . . . 11 ⊢ ((((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ (𝑋 / 𝑅) ∈ Fin) ∧ 𝑆 ∈ (Clsd‘𝐽)) ∧ 𝑦 ∈ 𝑋) → [𝑦]𝑅 ∈ (Clsd‘𝐽))
92		eleq1 2816	. . . . . . . . . . 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 3130	. . . . . . . . 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 3941	. . . . . . 7 ⊢ (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ (𝑋 / 𝑅) ∈ Fin) ∧ 𝑆 ∈ (Clsd‘𝐽)) → (𝑋 / 𝑅) ⊆ (Clsd‘𝐽))
97	96	ssdifssd 4098	. . . . . 6 ⊢ (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ (𝑋 / 𝑅) ∈ Fin) ∧ 𝑆 ∈ (Clsd‘𝐽)) → ((𝑋 / 𝑅) ∖ {𝑆}) ⊆ (Clsd‘𝐽))
98	87	unicld 22931	. . . . . 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 22916	. . . . 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 2829	. . 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 23993	. . . 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 1540 ∈ wcel 2109 ≠ wne 2925 ∃wrex 3053 Vcvv 3436 ∖ cdif 3900 ∪ cun 3901 ∩ cin 3902 ⊆ wss 3903 ∅c0 4284 𝒫 cpw 4551 {csn 4577 ∪ cuni 4858 ↦ cmpt 5173 “ cima 5622 ‘cfv 6482 (class class class)co 7349 Er wer 8622 [cec 8623 / cqs 8624 Fincfn 8872 Basecbs 17120 +_gcplusg 17161 TopOpenctopn 17325 0_gc0g 17343 Grpcgrp 18812 SubGrpcsubg 18999 ~_QG cqg 19001 Topctop 22778 TopOnctopon 22795 Clsdccld 22901 Homeochmeo 23638 TopGrpctgp 23956

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 2008 ax-8 2111 ax-9 2119 ax-10 2142 ax-11 2158 ax-12 2178 ax-ext 2701 ax-rep 5218 ax-sep 5235 ax-nul 5245 ax-pow 5304 ax-pr 5371 ax-un 7671 ax-cnex 11065 ax-resscn 11066 ax-1cn 11067 ax-icn 11068 ax-addcl 11069 ax-addrcl 11070 ax-mulcl 11071 ax-mulrcl 11072 ax-mulcom 11073 ax-addass 11074 ax-mulass 11075 ax-distr 11076 ax-i2m1 11077 ax-1ne0 11078 ax-1rid 11079 ax-rnegex 11080 ax-rrecex 11081 ax-cnre 11082 ax-pre-lttri 11083 ax-pre-lttrn 11084 ax-pre-ltadd 11085 ax-pre-mulgt0 11086

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2533 df-eu 2562 df-clab 2708 df-cleq 2721 df-clel 2803 df-nfc 2878 df-ne 2926 df-nel 3030 df-ral 3045 df-rex 3054 df-rmo 3343 df-reu 3344 df-rab 3395 df-v 3438 df-sbc 3743 df-csb 3852 df-dif 3906 df-un 3908 df-in 3910 df-ss 3920 df-pss 3923 df-nul 4285 df-if 4477 df-pw 4553 df-sn 4578 df-pr 4580 df-op 4584 df-uni 4859 df-int 4897 df-iun 4943 df-iin 4944 df-br 5093 df-opab 5155 df-mpt 5174 df-tr 5200 df-id 5514 df-eprel 5519 df-po 5527 df-so 5528 df-fr 5572 df-we 5574 df-xp 5625 df-rel 5626 df-cnv 5627 df-co 5628 df-dm 5629 df-rn 5630 df-res 5631 df-ima 5632 df-pred 6249 df-ord 6310 df-on 6311 df-lim 6312 df-suc 6313 df-iota 6438 df-fun 6484 df-fn 6485 df-f 6486 df-f1 6487 df-fo 6488 df-f1o 6489 df-fv 6490 df-riota 7306 df-ov 7352 df-oprab 7353 df-mpo 7354 df-om 7800 df-1st 7924 df-2nd 7925 df-frecs 8214 df-wrecs 8245 df-recs 8294 df-rdg 8332 df-1o 8388 df-2o 8389 df-er 8625 df-ec 8627 df-qs 8631 df-map 8755 df-en 8873 df-dom 8874 df-sdom 8875 df-fin 8876 df-pnf 11151 df-mnf 11152 df-xr 11153 df-ltxr 11154 df-le 11155 df-sub 11349 df-neg 11350 df-nn 12129 df-2 12191 df-sets 17075 df-slot 17093 df-ndx 17105 df-base 17121 df-ress 17142 df-plusg 17174 df-0g 17345 df-topgen 17347 df-plusf 18513 df-mgm 18514 df-sgrp 18593 df-mnd 18609 df-grp 18815 df-minusg 18816 df-sbg 18817 df-subg 19002 df-eqg 19004 df-top 22779 df-topon 22796 df-topsp 22818 df-bases 22831 df-cld 22904 df-cn 23112 df-cnp 23113 df-tx 23447 df-hmeo 23640 df-tmd 23957 df-tgp 23958

This theorem is referenced by: (None)

W3C validator