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Theorem cldsubg 24094

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 1198	. . . . . . . 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 24065	. . . . . . . 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 22897	. . . . . . 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 1199	. . . . . . . . 9 ⊢ (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ (𝑋 / 𝑅) ∈ Fin) ∧ 𝑆 ∈ (Clsd‘𝐽)) → 𝑆 ∈ (SubGrp‘𝐺))
10		unisng 4856	. . . . . . . . 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 4098	. . . . . . 7 ⊢ (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ (𝑋 / 𝑅) ∈ Fin) ∧ 𝑆 ∈ (Clsd‘𝐽)) → (∪ ((𝑋 / 𝑅) ∖ {𝑆}) ∪ ∪ {𝑆}) = (∪ ((𝑋 / 𝑅) ∖ {𝑆}) ∪ 𝑆))
13		uniun 4861	. . . . . . . 8 ⊢ ∪ (((𝑋 / 𝑅) ∖ {𝑆}) ∪ {𝑆}) = (∪ ((𝑋 / 𝑅) ∖ {𝑆}) ∪ ∪ {𝑆})
14		undif1 4404	. . . . . . . . . . 11 ⊢ (((𝑋 / 𝑅) ∖ {𝑆}) ∪ {𝑆}) = ((𝑋 / 𝑅) ∪ {𝑆})
15		cldsubg.1	. . . . . . . . . . . . . . . 16 ⊢ 𝑅 = (𝐺 ~_QG 𝑆)
16		eqid 2739	. . . . . . . . . . . . . . . 16 ⊢ (0_g‘𝐺) = (0_g‘𝐺)
17	3, 15, 16	eqgid 19146	. . . . . . . . . . . . . . 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 7390	. . . . . . . . . . . . . . 15 ⊢ 𝑅 ∈ V
20		tgpgrp 24061	. . . . . . . . . . . . . . . . 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 18932	. . . . . . . . . . . . . . . 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 8705	. . . . . . . . . . . . . . 15 ⊢ ((𝑅 ∈ V ∧ (0_g‘𝐺) ∈ 𝑋) → [(0_g‘𝐺)]𝑅 ∈ (𝑋 / 𝑅))
25	19, 23, 24	sylancr 593	. . . . . . . . . . . . . 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 4718	. . . . . . . . . . . 12 ⊢ (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ (𝑋 / 𝑅) ∈ Fin) ∧ 𝑆 ∈ (Clsd‘𝐽)) → {𝑆} ⊆ (𝑋 / 𝑅))
28		ssequn2 4118	. . . . . . . . . . . 12 ⊢ ({𝑆} ⊆ (𝑋 / 𝑅) ↔ ((𝑋 / 𝑅) ∪ {𝑆}) = (𝑋 / 𝑅))
29	27, 28	sylib 219	. . . . . . . . . . 11 ⊢ (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ (𝑋 / 𝑅) ∈ Fin) ∧ 𝑆 ∈ (Clsd‘𝐽)) → ((𝑋 / 𝑅) ∪ {𝑆}) = (𝑋 / 𝑅))
30	14, 29	eqtrid 2786	. . . . . . . . . 10 ⊢ (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ (𝑋 / 𝑅) ∈ Fin) ∧ 𝑆 ∈ (Clsd‘𝐽)) → (((𝑋 / 𝑅) ∖ {𝑆}) ∪ {𝑆}) = (𝑋 / 𝑅))
31	30	unieqd 4851	. . . . . . . . 9 ⊢ (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ (𝑋 / 𝑅) ∈ Fin) ∧ 𝑆 ∈ (Clsd‘𝐽)) → ∪ (((𝑋 / 𝑅) ∖ {𝑆}) ∪ {𝑆}) = ∪ (𝑋 / 𝑅))
32	3, 15	eqger 19144	. . . . . . . . . . 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 8713	. . . . . . . . 9 ⊢ (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ (𝑋 / 𝑅) ∈ Fin) ∧ 𝑆 ∈ (Clsd‘𝐽)) → ∪ (𝑋 / 𝑅) = 𝑋)
36	31, 35	eqtrd 2774	. . . . . . . 8 ⊢ (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ (𝑋 / 𝑅) ∈ Fin) ∧ 𝑆 ∈ (Clsd‘𝐽)) → ∪ (((𝑋 / 𝑅) ∖ {𝑆}) ∪ {𝑆}) = 𝑋)
37	13, 36	eqtr3id 2788	. . . . . . 7 ⊢ (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ (𝑋 / 𝑅) ∈ Fin) ∧ 𝑆 ∈ (Clsd‘𝐽)) → (∪ ((𝑋 / 𝑅) ∖ {𝑆}) ∪ ∪ {𝑆}) = 𝑋)
38	12, 37	eqtr3d 2776	. . . . . 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 4847	. . . . . . . 8 ⊢ ∪ ((𝑋 / 𝑅) ∖ {𝑆}) ⊆ ∪ (𝑋 / 𝑅)
41	40, 35	sseqtrid 3957	. . . . . . 7 ⊢ (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ (𝑋 / 𝑅) ∈ Fin) ∧ 𝑆 ∈ (Clsd‘𝐽)) → ∪ ((𝑋 / 𝑅) ∖ {𝑆}) ⊆ 𝑋)
42		df-ne 2935	. . . . . . . . . . . . 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 8731	. . . . . . . . . . . . . . 15 ⊢ ((((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ (𝑋 / 𝑅) ∈ Fin) ∧ 𝑆 ∈ (Clsd‘𝐽)) ∧ 𝑥 ∈ (𝑋 / 𝑅)) → (𝑥 = 𝑆 ∨ (𝑥 ∩ 𝑆) = ∅))
47	46	ord 870	. . . . . . . . . . . . . 14 ⊢ ((((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ (𝑋 / 𝑅) ∈ Fin) ∧ 𝑆 ∈ (Clsd‘𝐽)) ∧ 𝑥 ∈ (𝑋 / 𝑅)) → (¬ 𝑥 = 𝑆 → (𝑥 ∩ 𝑆) = ∅))
48		disj2 4386	. . . . . . . . . . . . . 14 ⊢ ((𝑥 ∩ 𝑆) = ∅ ↔ 𝑥 ⊆ (V ∖ 𝑆))
49	47, 48	imbitrdi 252	. . . . . . . . . . . . 13 ⊢ ((((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ (𝑋 / 𝑅) ∈ Fin) ∧ 𝑆 ∈ (Clsd‘𝐽)) ∧ 𝑥 ∈ (𝑋 / 𝑅)) → (¬ 𝑥 = 𝑆 → 𝑥 ⊆ (V ∖ 𝑆)))
50	42, 49	biimtrid 243	. . . . . . . . . . . 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 4719	. . . . . . . . . . 11 ⊢ (𝑥 ∈ ((𝑋 / 𝑅) ∖ {𝑆}) ↔ (𝑥 ∈ (𝑋 / 𝑅) ∧ 𝑥 ≠ 𝑆))
53		velpw 4534	. . . . . . . . . . 11 ⊢ (𝑥 ∈ 𝒫 (V ∖ 𝑆) ↔ 𝑥 ⊆ (V ∖ 𝑆))
54	51, 52, 53	3imtr4g 297	. . . . . . . . . 10 ⊢ (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ (𝑋 / 𝑅) ∈ Fin) ∧ 𝑆 ∈ (Clsd‘𝐽)) → (𝑥 ∈ ((𝑋 / 𝑅) ∖ {𝑆}) → 𝑥 ∈ 𝒫 (V ∖ 𝑆)))
55	54	ssrdv 3921	. . . . . . . . 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 219	. . . . . . . 8 ⊢ (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ (𝑋 / 𝑅) ∈ Fin) ∧ 𝑆 ∈ (Clsd‘𝐽)) → ∪ ((𝑋 / 𝑅) ∖ {𝑆}) ⊆ (V ∖ 𝑆))
58		disj2 4386	. . . . . . . 8 ⊢ ((∪ ((𝑋 / 𝑅) ∖ {𝑆}) ∩ 𝑆) = ∅ ↔ ∪ ((𝑋 / 𝑅) ∖ {𝑆}) ⊆ (V ∖ 𝑆))
59	57, 58	sylibr 235	. . . . . . 7 ⊢ (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ (𝑋 / 𝑅) ∈ Fin) ∧ 𝑆 ∈ (Clsd‘𝐽)) → (∪ ((𝑋 / 𝑅) ∖ {𝑆}) ∩ 𝑆) = ∅)
60		uneqdifeq 4420	. . . . . . 7 ⊢ ((∪ ((𝑋 / 𝑅) ∖ {𝑆}) ⊆ 𝑋 ∧ (∪ ((𝑋 / 𝑅) ∖ {𝑆}) ∩ 𝑆) = ∅) → ((∪ ((𝑋 / 𝑅) ∖ {𝑆}) ∪ 𝑆) = 𝑋 ↔ (𝑋 ∖ ∪ ((𝑋 / 𝑅) ∖ {𝑆})) = 𝑆))
61	41, 59, 60	syl2anc 590	. . . . . 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 233	. . . . 5 ⊢ (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ (𝑋 / 𝑅) ∈ Fin) ∧ 𝑆 ∈ (Clsd‘𝐽)) → (𝑋 ∖ ∪ ((𝑋 / 𝑅) ∖ {𝑆})) = 𝑆)
63	8, 62	eqtr3d 2776	. . . 4 ⊢ (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ (𝑋 / 𝑅) ∈ Fin) ∧ 𝑆 ∈ (Clsd‘𝐽)) → (∪ 𝐽 ∖ ∪ ((𝑋 / 𝑅) ∖ {𝑆})) = 𝑆)
64		topontop 22896	. . . . . . 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 1200	. . . . . . 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 3435	. . . . . . . . . 10 ⊢ 𝑥 ∈ V
70	69	elqs 8701	. . . . . . . . 9 ⊢ (𝑥 ∈ (𝑋 / 𝑅) ↔ ∃𝑦 ∈ 𝑋 𝑥 = [𝑦]𝑅)
71		simpll2 1220	. . . . . . . . . . . . . 14 ⊢ ((((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ (𝑋 / 𝑅) ∈ Fin) ∧ 𝑆 ∈ (Clsd‘𝐽)) ∧ 𝑦 ∈ 𝑋) → 𝑆 ∈ (SubGrp‘𝐺))
72		subgrcl 19098	. . . . . . . . . . . . . 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 19094	. . . . . . . . . . . . . . 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 2739	. . . . . . . . . . . . . 14 ⊢ (+_g‘𝐺) = (+_g‘𝐺)
79	3, 15, 78	eqglact 19145	. . . . . . . . . . . . 13 ⊢ ((𝐺 ∈ Grp ∧ 𝑆 ⊆ 𝑋 ∧ 𝑦 ∈ 𝑋) → [𝑦]𝑅 = ((𝑧 ∈ 𝑋 ↦ (𝑦(+_g‘𝐺)𝑧)) “ 𝑆))
80	73, 76, 77, 79	syl3anc 1379	. . . . . . . . . . . 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 774	. . . . . . . . . . . . 13 ⊢ ((((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ (𝑋 / 𝑅) ∈ Fin) ∧ 𝑆 ∈ (Clsd‘𝐽)) ∧ 𝑦 ∈ 𝑋) → 𝑆 ∈ (Clsd‘𝐽))
82		eqid 2739	. . . . . . . . . . . . . . . 16 ⊢ (𝑧 ∈ 𝑋 ↦ (𝑦(+_g‘𝐺)𝑧)) = (𝑧 ∈ 𝑋 ↦ (𝑦(+_g‘𝐺)𝑧))
83	82, 3, 78, 2	tgplacthmeo 24086	. . . . . . . . . . . . . . 15 ⊢ ((𝐺 ∈ TopGrp ∧ 𝑦 ∈ 𝑋) → (𝑧 ∈ 𝑋 ↦ (𝑦(+_g‘𝐺)𝑧)) ∈ (𝐽Homeo𝐽))
84	1, 83	sylan 586	. . . . . . . . . . . . . 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 3951	. . . . . . . . . . . . . . 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 2739	. . . . . . . . . . . . . . 15 ⊢ ∪ 𝐽 = ∪ 𝐽
88	87	hmeocld 23750	. . . . . . . . . . . . . 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 590	. . . . . . . . . . . . 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 233	. . . . . . . . . . . 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 2827	. . . . . . . . . . 11 ⊢ (𝑥 = [𝑦]𝑅 → (𝑥 ∈ (Clsd‘𝐽) ↔ [𝑦]𝑅 ∈ (Clsd‘𝐽)))
93	91, 92	syl5ibrcom 248	. . . . . . . . . 10 ⊢ ((((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ (𝑋 / 𝑅) ∈ Fin) ∧ 𝑆 ∈ (Clsd‘𝐽)) ∧ 𝑦 ∈ 𝑋) → (𝑥 = [𝑦]𝑅 → 𝑥 ∈ (Clsd‘𝐽)))
94	93	rexlimdva 3140	. . . . . . . . 9 ⊢ (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ (𝑋 / 𝑅) ∈ Fin) ∧ 𝑆 ∈ (Clsd‘𝐽)) → (∃𝑦 ∈ 𝑋 𝑥 = [𝑦]𝑅 → 𝑥 ∈ (Clsd‘𝐽)))
95	70, 94	biimtrid 243	. . . . . . . 8 ⊢ (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ (𝑋 / 𝑅) ∈ Fin) ∧ 𝑆 ∈ (Clsd‘𝐽)) → (𝑥 ∈ (𝑋 / 𝑅) → 𝑥 ∈ (Clsd‘𝐽)))
96	95	ssrdv 3921	. . . . . . 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 23029	. . . . . 6 ⊢ ((𝐽 ∈ Top ∧ ((𝑋 / 𝑅) ∖ {𝑆}) ∈ Fin ∧ ((𝑋 / 𝑅) ∖ {𝑆}) ⊆ (Clsd‘𝐽)) → ∪ ((𝑋 / 𝑅) ∖ {𝑆}) ∈ (Clsd‘𝐽))
99	65, 68, 97, 98	syl3anc 1379	. . . . 5 ⊢ (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ (𝑋 / 𝑅) ∈ Fin) ∧ 𝑆 ∈ (Clsd‘𝐽)) → ∪ ((𝑋 / 𝑅) ∖ {𝑆}) ∈ (Clsd‘𝐽))
100	87	cldopn 23014	. . . . 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 24091	. . . 4 ⊢ ((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑆 ∈ 𝐽) → 𝑆 ∈ (Clsd‘𝐽))
105	104	3expia 1127	. . 3 ⊢ ((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺)) → (𝑆 ∈ 𝐽 → 𝑆 ∈ (Clsd‘𝐽)))
106	105	3adant3 1138	. 2 ⊢ ((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ (𝑋 / 𝑅) ∈ Fin) → (𝑆 ∈ 𝐽 → 𝑆 ∈ (Clsd‘𝐽)))
107	103, 106	impbid 213	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 207 ∧ wa 396 ∧ w3a 1092 = wceq 1547 ∈ wcel 2119 ≠ wne 2934 ∃wrex 3063 Vcvv 3431 ∖ cdif 3880 ∪ cun 3881 ∩ cin 3882 ⊆ wss 3883 ∅c0 4261 𝒫 cpw 4529 {csn 4555 ∪ cuni 4838 ↦ cmpt 5153 “ cima 5621 ‘cfv 6485 (class class class)co 7356 Er wer 8630 [cec 8631 / cqs 8632 Fincfn 8883 Basecbs 17170 +_gcplusg 17211 TopOpenctopn 17375 0_gc0g 17393 Grpcgrp 18900 SubGrpcsubg 19087 ~_QG cqg 19089 Topctop 22876 TopOnctopon 22893 Clsdccld 22999 Homeochmeo 23736 TopGrpctgp 24054

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1974 ax-7 2015 ax-8 2121 ax-9 2129 ax-10 2152 ax-11 2168 ax-12 2189 ax-ext 2711 ax-rep 5199 ax-sep 5218 ax-nul 5228 ax-pow 5294 ax-pr 5362 ax-un 7678 ax-cnex 11085 ax-resscn 11086 ax-1cn 11087 ax-icn 11088 ax-addcl 11089 ax-addrcl 11090 ax-mulcl 11091 ax-mulrcl 11092 ax-mulcom 11093 ax-addass 11094 ax-mulass 11095 ax-distr 11096 ax-i2m1 11097 ax-1ne0 11098 ax-1rid 11099 ax-rnegex 11100 ax-rrecex 11101 ax-cnre 11102 ax-pre-lttri 11103 ax-pre-lttrn 11104 ax-pre-ltadd 11105 ax-pre-mulgt0 11106

This theorem depends on definitions: df-bi 208 df-an 397 df-or 854 df-3or 1093 df-3an 1094 df-tru 1550 df-fal 1560 df-ex 1787 df-nf 1791 df-sb 2074 df-mo 2543 df-eu 2573 df-clab 2718 df-cleq 2731 df-clel 2814 df-nfc 2888 df-ne 2935 df-nel 3039 df-ral 3054 df-rex 3064 df-rmo 3344 df-reu 3345 df-rab 3392 df-v 3433 df-sbc 3724 df-csb 3832 df-dif 3886 df-un 3888 df-in 3890 df-ss 3900 df-pss 3903 df-nul 4262 df-if 4455 df-pw 4531 df-sn 4556 df-pr 4558 df-op 4562 df-uni 4839 df-int 4878 df-iun 4923 df-iin 4924 df-br 5073 df-opab 5135 df-mpt 5154 df-tr 5180 df-id 5513 df-eprel 5518 df-po 5526 df-so 5527 df-fr 5571 df-we 5573 df-xp 5624 df-rel 5625 df-cnv 5626 df-co 5627 df-dm 5628 df-rn 5629 df-res 5630 df-ima 5631 df-pred 6252 df-ord 6313 df-on 6314 df-lim 6315 df-suc 6316 df-iota 6441 df-fun 6487 df-fn 6488 df-f 6489 df-f1 6490 df-fo 6491 df-f1o 6492 df-fv 6493 df-riota 7313 df-ov 7359 df-oprab 7360 df-mpo 7361 df-om 7807 df-1st 7931 df-2nd 7932 df-frecs 8221 df-wrecs 8252 df-recs 8301 df-rdg 8339 df-1o 8395 df-2o 8396 df-er 8633 df-ec 8635 df-qs 8639 df-map 8765 df-en 8884 df-dom 8885 df-sdom 8886 df-fin 8887 df-pnf 11172 df-mnf 11173 df-xr 11174 df-ltxr 11175 df-le 11176 df-sub 11370 df-neg 11371 df-nn 12166 df-2 12235 df-sets 17125 df-slot 17143 df-ndx 17155 df-base 17171 df-ress 17192 df-plusg 17224 df-0g 17395 df-topgen 17397 df-plusf 18598 df-mgm 18599 df-sgrp 18678 df-mnd 18694 df-grp 18903 df-minusg 18904 df-sbg 18905 df-subg 19090 df-eqg 19092 df-top 22877 df-topon 22894 df-topsp 22916 df-bases 22929 df-cld 23002 df-cn 23210 df-cnp 23211 df-tx 23545 df-hmeo 23738 df-tmd 24055 df-tgp 24056

This theorem is referenced by: (None)

W3C validator