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Theorem cldsubg 23262

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 1190	. . . . . . . 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 23233	. . . . . . . 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 22063	. . . . . . 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 4056	. . . . 5 ⊢ (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ (𝑋 / 𝑅) ∈ Fin) ∧ 𝑆 ∈ (Clsd‘𝐽)) → (𝑋 ∖ ∪ ((𝑋 / 𝑅) ∖ {𝑆})) = (∪ 𝐽 ∖ ∪ ((𝑋 / 𝑅) ∖ {𝑆})))
9		simpl2 1191	. . . . . . . . 9 ⊢ (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ (𝑋 / 𝑅) ∈ Fin) ∧ 𝑆 ∈ (Clsd‘𝐽)) → 𝑆 ∈ (SubGrp‘𝐺))
10		unisng 4860	. . . . . . . . 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 4097	. . . . . . 7 ⊢ (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ (𝑋 / 𝑅) ∈ Fin) ∧ 𝑆 ∈ (Clsd‘𝐽)) → (∪ ((𝑋 / 𝑅) ∖ {𝑆}) ∪ ∪ {𝑆}) = (∪ ((𝑋 / 𝑅) ∖ {𝑆}) ∪ 𝑆))
13		uniun 4864	. . . . . . . 8 ⊢ ∪ (((𝑋 / 𝑅) ∖ {𝑆}) ∪ {𝑆}) = (∪ ((𝑋 / 𝑅) ∖ {𝑆}) ∪ ∪ {𝑆})
14		undif1 4409	. . . . . . . . . . 11 ⊢ (((𝑋 / 𝑅) ∖ {𝑆}) ∪ {𝑆}) = ((𝑋 / 𝑅) ∪ {𝑆})
15		cldsubg.1	. . . . . . . . . . . . . . . 16 ⊢ 𝑅 = (𝐺 ~_QG 𝑆)
16		eqid 2738	. . . . . . . . . . . . . . . 16 ⊢ (0_g‘𝐺) = (0_g‘𝐺)
17	3, 15, 16	eqgid 18808	. . . . . . . . . . . . . . 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 7309	. . . . . . . . . . . . . . 15 ⊢ 𝑅 ∈ V
20		tgpgrp 23229	. . . . . . . . . . . . . . . . 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 18607	. . . . . . . . . . . . . . . 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		ecelqsg 8561	. . . . . . . . . . . . . . 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 2840	. . . . . . . . . . . . 13 ⊢ (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ (𝑋 / 𝑅) ∈ Fin) ∧ 𝑆 ∈ (Clsd‘𝐽)) → 𝑆 ∈ (𝑋 / 𝑅))
27	26	snssd 4742	. . . . . . . . . . . 12 ⊢ (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ (𝑋 / 𝑅) ∈ Fin) ∧ 𝑆 ∈ (Clsd‘𝐽)) → {𝑆} ⊆ (𝑋 / 𝑅))
28		ssequn2 4117	. . . . . . . . . . . 12 ⊢ ({𝑆} ⊆ (𝑋 / 𝑅) ↔ ((𝑋 / 𝑅) ∪ {𝑆}) = (𝑋 / 𝑅))
29	27, 28	sylib 217	. . . . . . . . . . 11 ⊢ (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ (𝑋 / 𝑅) ∈ Fin) ∧ 𝑆 ∈ (Clsd‘𝐽)) → ((𝑋 / 𝑅) ∪ {𝑆}) = (𝑋 / 𝑅))
30	14, 29	eqtrid 2790	. . . . . . . . . 10 ⊢ (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ (𝑋 / 𝑅) ∈ Fin) ∧ 𝑆 ∈ (Clsd‘𝐽)) → (((𝑋 / 𝑅) ∖ {𝑆}) ∪ {𝑆}) = (𝑋 / 𝑅))
31	30	unieqd 4853	. . . . . . . . 9 ⊢ (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ (𝑋 / 𝑅) ∈ Fin) ∧ 𝑆 ∈ (Clsd‘𝐽)) → ∪ (((𝑋 / 𝑅) ∖ {𝑆}) ∪ {𝑆}) = ∪ (𝑋 / 𝑅))
32	3, 15	eqger 18806	. . . . . . . . . . 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 8568	. . . . . . . . 9 ⊢ (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ (𝑋 / 𝑅) ∈ Fin) ∧ 𝑆 ∈ (Clsd‘𝐽)) → ∪ (𝑋 / 𝑅) = 𝑋)
36	31, 35	eqtrd 2778	. . . . . . . 8 ⊢ (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ (𝑋 / 𝑅) ∈ Fin) ∧ 𝑆 ∈ (Clsd‘𝐽)) → ∪ (((𝑋 / 𝑅) ∖ {𝑆}) ∪ {𝑆}) = 𝑋)
37	13, 36	eqtr3id 2792	. . . . . . 7 ⊢ (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ (𝑋 / 𝑅) ∈ Fin) ∧ 𝑆 ∈ (Clsd‘𝐽)) → (∪ ((𝑋 / 𝑅) ∖ {𝑆}) ∪ ∪ {𝑆}) = 𝑋)
38	12, 37	eqtr3d 2780	. . . . . 6 ⊢ (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ (𝑋 / 𝑅) ∈ Fin) ∧ 𝑆 ∈ (Clsd‘𝐽)) → (∪ ((𝑋 / 𝑅) ∖ {𝑆}) ∪ 𝑆) = 𝑋)
39		difss 4066	. . . . . . . . 9 ⊢ ((𝑋 / 𝑅) ∖ {𝑆}) ⊆ (𝑋 / 𝑅)
40	39	unissi 4848	. . . . . . . 8 ⊢ ∪ ((𝑋 / 𝑅) ∖ {𝑆}) ⊆ ∪ (𝑋 / 𝑅)
41	40, 35	sseqtrid 3973	. . . . . . 7 ⊢ (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ (𝑋 / 𝑅) ∈ Fin) ∧ 𝑆 ∈ (Clsd‘𝐽)) → ∪ ((𝑋 / 𝑅) ∖ {𝑆}) ⊆ 𝑋)
42		df-ne 2944	. . . . . . . . . . . . 13 ⊢ (𝑥 ≠ 𝑆 ↔ ¬ 𝑥 = 𝑆)
43	33	adantr 481	. . . . . . . . . . . . . . . 16 ⊢ ((((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ (𝑋 / 𝑅) ∈ Fin) ∧ 𝑆 ∈ (Clsd‘𝐽)) ∧ 𝑥 ∈ (𝑋 / 𝑅)) → 𝑅 Er 𝑋)
44		simpr 485	. . . . . . . . . . . . . . . 16 ⊢ ((((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ (𝑋 / 𝑅) ∈ Fin) ∧ 𝑆 ∈ (Clsd‘𝐽)) ∧ 𝑥 ∈ (𝑋 / 𝑅)) → 𝑥 ∈ (𝑋 / 𝑅))
45	26	adantr 481	. . . . . . . . . . . . . . . 16 ⊢ ((((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ (𝑋 / 𝑅) ∈ Fin) ∧ 𝑆 ∈ (Clsd‘𝐽)) ∧ 𝑥 ∈ (𝑋 / 𝑅)) → 𝑆 ∈ (𝑋 / 𝑅))
46	43, 44, 45	qsdisj 8583	. . . . . . . . . . . . . . 15 ⊢ ((((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ (𝑋 / 𝑅) ∈ Fin) ∧ 𝑆 ∈ (Clsd‘𝐽)) ∧ 𝑥 ∈ (𝑋 / 𝑅)) → (𝑥 = 𝑆 ∨ (𝑥 ∩ 𝑆) = ∅))
47	46	ord 861	. . . . . . . . . . . . . 14 ⊢ ((((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ (𝑋 / 𝑅) ∈ Fin) ∧ 𝑆 ∈ (Clsd‘𝐽)) ∧ 𝑥 ∈ (𝑋 / 𝑅)) → (¬ 𝑥 = 𝑆 → (𝑥 ∩ 𝑆) = ∅))
48		disj2 4391	. . . . . . . . . . . . . 14 ⊢ ((𝑥 ∩ 𝑆) = ∅ ↔ 𝑥 ⊆ (V ∖ 𝑆))
49	47, 48	syl6ib 250	. . . . . . . . . . . . 13 ⊢ ((((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ (𝑋 / 𝑅) ∈ Fin) ∧ 𝑆 ∈ (Clsd‘𝐽)) ∧ 𝑥 ∈ (𝑋 / 𝑅)) → (¬ 𝑥 = 𝑆 → 𝑥 ⊆ (V ∖ 𝑆)))
50	42, 49	syl5bi 241	. . . . . . . . . . . 12 ⊢ ((((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ (𝑋 / 𝑅) ∈ Fin) ∧ 𝑆 ∈ (Clsd‘𝐽)) ∧ 𝑥 ∈ (𝑋 / 𝑅)) → (𝑥 ≠ 𝑆 → 𝑥 ⊆ (V ∖ 𝑆)))
51	50	expimpd 454	. . . . . . . . . . 11 ⊢ (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ (𝑋 / 𝑅) ∈ Fin) ∧ 𝑆 ∈ (Clsd‘𝐽)) → ((𝑥 ∈ (𝑋 / 𝑅) ∧ 𝑥 ≠ 𝑆) → 𝑥 ⊆ (V ∖ 𝑆)))
52		eldifsn 4720	. . . . . . . . . . 11 ⊢ (𝑥 ∈ ((𝑋 / 𝑅) ∖ {𝑆}) ↔ (𝑥 ∈ (𝑋 / 𝑅) ∧ 𝑥 ≠ 𝑆))
53		velpw 4538	. . . . . . . . . . 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 3927	. . . . . . . . 9 ⊢ (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ (𝑋 / 𝑅) ∈ Fin) ∧ 𝑆 ∈ (Clsd‘𝐽)) → ((𝑋 / 𝑅) ∖ {𝑆}) ⊆ 𝒫 (V ∖ 𝑆))
56		sspwuni 5029	. . . . . . . . 9 ⊢ (((𝑋 / 𝑅) ∖ {𝑆}) ⊆ 𝒫 (V ∖ 𝑆) ↔ ∪ ((𝑋 / 𝑅) ∖ {𝑆}) ⊆ (V ∖ 𝑆))
57	55, 56	sylib 217	. . . . . . . 8 ⊢ (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ (𝑋 / 𝑅) ∈ Fin) ∧ 𝑆 ∈ (Clsd‘𝐽)) → ∪ ((𝑋 / 𝑅) ∖ {𝑆}) ⊆ (V ∖ 𝑆))
58		disj2 4391	. . . . . . . 8 ⊢ ((∪ ((𝑋 / 𝑅) ∖ {𝑆}) ∩ 𝑆) = ∅ ↔ ∪ ((𝑋 / 𝑅) ∖ {𝑆}) ⊆ (V ∖ 𝑆))
59	57, 58	sylibr 233	. . . . . . 7 ⊢ (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ (𝑋 / 𝑅) ∈ Fin) ∧ 𝑆 ∈ (Clsd‘𝐽)) → (∪ ((𝑋 / 𝑅) ∖ {𝑆}) ∩ 𝑆) = ∅)
60		uneqdifeq 4423	. . . . . . 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 231	. . . . 5 ⊢ (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ (𝑋 / 𝑅) ∈ Fin) ∧ 𝑆 ∈ (Clsd‘𝐽)) → (𝑋 ∖ ∪ ((𝑋 / 𝑅) ∖ {𝑆})) = 𝑆)
63	8, 62	eqtr3d 2780	. . . 4 ⊢ (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ (𝑋 / 𝑅) ∈ Fin) ∧ 𝑆 ∈ (Clsd‘𝐽)) → (∪ 𝐽 ∖ ∪ ((𝑋 / 𝑅) ∖ {𝑆})) = 𝑆)
64		topontop 22062	. . . . . . 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 1192	. . . . . . 7 ⊢ (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ (𝑋 / 𝑅) ∈ Fin) ∧ 𝑆 ∈ (Clsd‘𝐽)) → (𝑋 / 𝑅) ∈ Fin)
67		diffi 8962	. . . . . . 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 3436	. . . . . . . . . 10 ⊢ 𝑥 ∈ V
70	69	elqs 8558	. . . . . . . . 9 ⊢ (𝑥 ∈ (𝑋 / 𝑅) ↔ ∃𝑦 ∈ 𝑋 𝑥 = [𝑦]𝑅)
71		simpll2 1212	. . . . . . . . . . . . . 14 ⊢ ((((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ (𝑋 / 𝑅) ∈ Fin) ∧ 𝑆 ∈ (Clsd‘𝐽)) ∧ 𝑦 ∈ 𝑋) → 𝑆 ∈ (SubGrp‘𝐺))
72		subgrcl 18760	. . . . . . . . . . . . . 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 18756	. . . . . . . . . . . . . . 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 481	. . . . . . . . . . . . 13 ⊢ ((((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ (𝑋 / 𝑅) ∈ Fin) ∧ 𝑆 ∈ (Clsd‘𝐽)) ∧ 𝑦 ∈ 𝑋) → 𝑆 ⊆ 𝑋)
77		simpr 485	. . . . . . . . . . . . 13 ⊢ ((((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ (𝑋 / 𝑅) ∈ Fin) ∧ 𝑆 ∈ (Clsd‘𝐽)) ∧ 𝑦 ∈ 𝑋) → 𝑦 ∈ 𝑋)
78		eqid 2738	. . . . . . . . . . . . . 14 ⊢ (+_g‘𝐺) = (+_g‘𝐺)
79	3, 15, 78	eqglact 18807	. . . . . . . . . . . . 13 ⊢ ((𝐺 ∈ Grp ∧ 𝑆 ⊆ 𝑋 ∧ 𝑦 ∈ 𝑋) → [𝑦]𝑅 = ((𝑧 ∈ 𝑋 ↦ (𝑦(+_g‘𝐺)𝑧)) “ 𝑆))
80	73, 76, 77, 79	syl3anc 1370	. . . . . . . . . . . 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 766	. . . . . . . . . . . . 13 ⊢ ((((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ (𝑋 / 𝑅) ∈ Fin) ∧ 𝑆 ∈ (Clsd‘𝐽)) ∧ 𝑦 ∈ 𝑋) → 𝑆 ∈ (Clsd‘𝐽))
82		eqid 2738	. . . . . . . . . . . . . . . 16 ⊢ (𝑧 ∈ 𝑋 ↦ (𝑦(+_g‘𝐺)𝑧)) = (𝑧 ∈ 𝑋 ↦ (𝑦(+_g‘𝐺)𝑧))
83	82, 3, 78, 2	tgplacthmeo 23254	. . . . . . . . . . . . . . 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 3961	. . . . . . . . . . . . . . 15 ⊢ (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ (𝑋 / 𝑅) ∈ Fin) ∧ 𝑆 ∈ (Clsd‘𝐽)) → 𝑆 ⊆ ∪ 𝐽)
86	85	adantr 481	. . . . . . . . . . . . . 14 ⊢ ((((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ (𝑋 / 𝑅) ∈ Fin) ∧ 𝑆 ∈ (Clsd‘𝐽)) ∧ 𝑦 ∈ 𝑋) → 𝑆 ⊆ ∪ 𝐽)
87		eqid 2738	. . . . . . . . . . . . . . 15 ⊢ ∪ 𝐽 = ∪ 𝐽
88	87	hmeocld 22918	. . . . . . . . . . . . . 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 231	. . . . . . . . . . . 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 2839	. . . . . . . . . . 11 ⊢ ((((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ (𝑋 / 𝑅) ∈ Fin) ∧ 𝑆 ∈ (Clsd‘𝐽)) ∧ 𝑦 ∈ 𝑋) → [𝑦]𝑅 ∈ (Clsd‘𝐽))
92		eleq1 2826	. . . . . . . . . . 11 ⊢ (𝑥 = [𝑦]𝑅 → (𝑥 ∈ (Clsd‘𝐽) ↔ [𝑦]𝑅 ∈ (Clsd‘𝐽)))
93	91, 92	syl5ibrcom 246	. . . . . . . . . 10 ⊢ ((((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ (𝑋 / 𝑅) ∈ Fin) ∧ 𝑆 ∈ (Clsd‘𝐽)) ∧ 𝑦 ∈ 𝑋) → (𝑥 = [𝑦]𝑅 → 𝑥 ∈ (Clsd‘𝐽)))
94	93	rexlimdva 3213	. . . . . . . . 9 ⊢ (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ (𝑋 / 𝑅) ∈ Fin) ∧ 𝑆 ∈ (Clsd‘𝐽)) → (∃𝑦 ∈ 𝑋 𝑥 = [𝑦]𝑅 → 𝑥 ∈ (Clsd‘𝐽)))
95	70, 94	syl5bi 241	. . . . . . . 8 ⊢ (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ (𝑋 / 𝑅) ∈ Fin) ∧ 𝑆 ∈ (Clsd‘𝐽)) → (𝑥 ∈ (𝑋 / 𝑅) → 𝑥 ∈ (Clsd‘𝐽)))
96	95	ssrdv 3927	. . . . . . 7 ⊢ (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ (𝑋 / 𝑅) ∈ Fin) ∧ 𝑆 ∈ (Clsd‘𝐽)) → (𝑋 / 𝑅) ⊆ (Clsd‘𝐽))
97	96	ssdifssd 4077	. . . . . 6 ⊢ (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ (𝑋 / 𝑅) ∈ Fin) ∧ 𝑆 ∈ (Clsd‘𝐽)) → ((𝑋 / 𝑅) ∖ {𝑆}) ⊆ (Clsd‘𝐽))
98	87	unicld 22197	. . . . . 6 ⊢ ((𝐽 ∈ Top ∧ ((𝑋 / 𝑅) ∖ {𝑆}) ∈ Fin ∧ ((𝑋 / 𝑅) ∖ {𝑆}) ⊆ (Clsd‘𝐽)) → ∪ ((𝑋 / 𝑅) ∖ {𝑆}) ∈ (Clsd‘𝐽))
99	65, 68, 97, 98	syl3anc 1370	. . . . 5 ⊢ (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ (𝑋 / 𝑅) ∈ Fin) ∧ 𝑆 ∈ (Clsd‘𝐽)) → ∪ ((𝑋 / 𝑅) ∖ {𝑆}) ∈ (Clsd‘𝐽))
100	87	cldopn 22182	. . . . 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 2840	. . 3 ⊢ (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ (𝑋 / 𝑅) ∈ Fin) ∧ 𝑆 ∈ (Clsd‘𝐽)) → 𝑆 ∈ 𝐽)
103	102	ex 413	. 2 ⊢ ((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ (𝑋 / 𝑅) ∈ Fin) → (𝑆 ∈ (Clsd‘𝐽) → 𝑆 ∈ 𝐽))
104	2	opnsubg 23259	. . . 4 ⊢ ((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑆 ∈ 𝐽) → 𝑆 ∈ (Clsd‘𝐽))
105	104	3expia 1120	. . 3 ⊢ ((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺)) → (𝑆 ∈ 𝐽 → 𝑆 ∈ (Clsd‘𝐽)))
106	105	3adant3 1131	. 2 ⊢ ((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ (𝑋 / 𝑅) ∈ Fin) → (𝑆 ∈ 𝐽 → 𝑆 ∈ (Clsd‘𝐽)))
107	103, 106	impbid 211	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 205 ∧ wa 396 ∧ w3a 1086 = wceq 1539 ∈ wcel 2106 ≠ wne 2943 ∃wrex 3065 Vcvv 3432 ∖ cdif 3884 ∪ cun 3885 ∩ cin 3886 ⊆ wss 3887 ∅c0 4256 𝒫 cpw 4533 {csn 4561 ∪ cuni 4839 ↦ cmpt 5157 “ cima 5592 ‘cfv 6433 (class class class)co 7275 Er wer 8495 [cec 8496 / cqs 8497 Fincfn 8733 Basecbs 16912 +_gcplusg 16962 TopOpenctopn 17132 0_gc0g 17150 Grpcgrp 18577 SubGrpcsubg 18749 ~_QG cqg 18751 Topctop 22042 TopOnctopon 22059 Clsdccld 22167 Homeochmeo 22904 TopGrpctgp 23222

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2709 ax-rep 5209 ax-sep 5223 ax-nul 5230 ax-pow 5288 ax-pr 5352 ax-un 7588 ax-cnex 10927 ax-resscn 10928 ax-1cn 10929 ax-icn 10930 ax-addcl 10931 ax-addrcl 10932 ax-mulcl 10933 ax-mulrcl 10934 ax-mulcom 10935 ax-addass 10936 ax-mulass 10937 ax-distr 10938 ax-i2m1 10939 ax-1ne0 10940 ax-1rid 10941 ax-rnegex 10942 ax-rrecex 10943 ax-cnre 10944 ax-pre-lttri 10945 ax-pre-lttrn 10946 ax-pre-ltadd 10947 ax-pre-mulgt0 10948

This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1783 df-nf 1787 df-sb 2068 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2889 df-ne 2944 df-nel 3050 df-ral 3069 df-rex 3070 df-rmo 3071 df-reu 3072 df-rab 3073 df-v 3434 df-sbc 3717 df-csb 3833 df-dif 3890 df-un 3892 df-in 3894 df-ss 3904 df-pss 3906 df-nul 4257 df-if 4460 df-pw 4535 df-sn 4562 df-pr 4564 df-op 4568 df-uni 4840 df-int 4880 df-iun 4926 df-iin 4927 df-br 5075 df-opab 5137 df-mpt 5158 df-tr 5192 df-id 5489 df-eprel 5495 df-po 5503 df-so 5504 df-fr 5544 df-we 5546 df-xp 5595 df-rel 5596 df-cnv 5597 df-co 5598 df-dm 5599 df-rn 5600 df-res 5601 df-ima 5602 df-pred 6202 df-ord 6269 df-on 6270 df-lim 6271 df-suc 6272 df-iota 6391 df-fun 6435 df-fn 6436 df-f 6437 df-f1 6438 df-fo 6439 df-f1o 6440 df-fv 6441 df-riota 7232 df-ov 7278 df-oprab 7279 df-mpo 7280 df-om 7713 df-1st 7831 df-2nd 7832 df-frecs 8097 df-wrecs 8128 df-recs 8202 df-rdg 8241 df-1o 8297 df-er 8498 df-ec 8500 df-qs 8504 df-map 8617 df-en 8734 df-dom 8735 df-sdom 8736 df-fin 8737 df-pnf 11011 df-mnf 11012 df-xr 11013 df-ltxr 11014 df-le 11015 df-sub 11207 df-neg 11208 df-nn 11974 df-2 12036 df-sets 16865 df-slot 16883 df-ndx 16895 df-base 16913 df-ress 16942 df-plusg 16975 df-0g 17152 df-topgen 17154 df-plusf 18325 df-mgm 18326 df-sgrp 18375 df-mnd 18386 df-grp 18580 df-minusg 18581 df-sbg 18582 df-subg 18752 df-eqg 18754 df-top 22043 df-topon 22060 df-topsp 22082 df-bases 22096 df-cld 22170 df-cn 22378 df-cnp 22379 df-tx 22713 df-hmeo 22906 df-tmd 23223 df-tgp 23224

This theorem is referenced by: (None)

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