Theorem opnsubg 24095

Description: An open subgroup of a topological group is also closed. (Contributed by Mario Carneiro, 17-Sep-2015.)

Hypothesis

Ref	Expression
subgntr.h	⊢ 𝐽 = (TopOpen‘𝐺)

Assertion

Ref	Expression
opnsubg	⊢ ((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑆 ∈ 𝐽) → 𝑆 ∈ (Clsd‘𝐽))

Proof of Theorem opnsubg

Dummy variables 𝑥 𝑢 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqid 2741	. . . . 5 ⊢ (Base‘𝐺) = (Base‘𝐺)
2	1	subgss 19098	. . . 4 ⊢ (𝑆 ∈ (SubGrp‘𝐺) → 𝑆 ⊆ (Base‘𝐺))
3	2	3ad2ant2 1141	. . 3 ⊢ ((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑆 ∈ 𝐽) → 𝑆 ⊆ (Base‘𝐺))
4		subgntr.h	. . . . . 6 ⊢ 𝐽 = (TopOpen‘𝐺)
5	4, 1	tgptopon 24069	. . . . 5 ⊢ (𝐺 ∈ TopGrp → 𝐽 ∈ (TopOn‘(Base‘𝐺)))
6	5	3ad2ant1 1140	. . . 4 ⊢ ((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑆 ∈ 𝐽) → 𝐽 ∈ (TopOn‘(Base‘𝐺)))
7		toponuni 22901	. . . 4 ⊢ (𝐽 ∈ (TopOn‘(Base‘𝐺)) → (Base‘𝐺) = ∪ 𝐽)
8	6, 7	syl 17	. . 3 ⊢ ((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑆 ∈ 𝐽) → (Base‘𝐺) = ∪ 𝐽)
9	3, 8	sseqtrd 3953	. 2 ⊢ ((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑆 ∈ 𝐽) → 𝑆 ⊆ ∪ 𝐽)
10	8	difeq1d 4059	. . 3 ⊢ ((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑆 ∈ 𝐽) → ((Base‘𝐺) ∖ 𝑆) = (∪ 𝐽 ∖ 𝑆))
11		df-ima 5634	. . . . . . . 8 ⊢ ((𝑦 ∈ (Base‘𝐺) ↦ (𝑥(+g‘𝐺)𝑦)) “ 𝑆) = ran ((𝑦 ∈ (Base‘𝐺) ↦ (𝑥(+g‘𝐺)𝑦)) ↾ 𝑆)
12	3	adantr 482	. . . . . . . . . 10 ⊢ (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑆 ∈ 𝐽) ∧ 𝑥 ∈ ((Base‘𝐺) ∖ 𝑆)) → 𝑆 ⊆ (Base‘𝐺))
13	12	resmptd 5999	. . . . . . . . 9 ⊢ (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑆 ∈ 𝐽) ∧ 𝑥 ∈ ((Base‘𝐺) ∖ 𝑆)) → ((𝑦 ∈ (Base‘𝐺) ↦ (𝑥(+g‘𝐺)𝑦)) ↾ 𝑆) = (𝑦 ∈ 𝑆 ↦ (𝑥(+g‘𝐺)𝑦)))
14	13	rneqd 5887	. . . . . . . 8 ⊢ (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑆 ∈ 𝐽) ∧ 𝑥 ∈ ((Base‘𝐺) ∖ 𝑆)) → ran ((𝑦 ∈ (Base‘𝐺) ↦ (𝑥(+g‘𝐺)𝑦)) ↾ 𝑆) = ran (𝑦 ∈ 𝑆 ↦ (𝑥(+g‘𝐺)𝑦)))
15	11, 14	eqtrid 2788	. . . . . . 7 ⊢ (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑆 ∈ 𝐽) ∧ 𝑥 ∈ ((Base‘𝐺) ∖ 𝑆)) → ((𝑦 ∈ (Base‘𝐺) ↦ (𝑥(+g‘𝐺)𝑦)) “ 𝑆) = ran (𝑦 ∈ 𝑆 ↦ (𝑥(+g‘𝐺)𝑦)))
16		simpl1 1199	. . . . . . . . 9 ⊢ (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑆 ∈ 𝐽) ∧ 𝑥 ∈ ((Base‘𝐺) ∖ 𝑆)) → 𝐺 ∈ TopGrp)
17		eldifi 4064	. . . . . . . . . 10 ⊢ (𝑥 ∈ ((Base‘𝐺) ∖ 𝑆) → 𝑥 ∈ (Base‘𝐺))
18	17	adantl 483	. . . . . . . . 9 ⊢ (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑆 ∈ 𝐽) ∧ 𝑥 ∈ ((Base‘𝐺) ∖ 𝑆)) → 𝑥 ∈ (Base‘𝐺))
19		eqid 2741	. . . . . . . . . 10 ⊢ (𝑦 ∈ (Base‘𝐺) ↦ (𝑥(+g‘𝐺)𝑦)) = (𝑦 ∈ (Base‘𝐺) ↦ (𝑥(+g‘𝐺)𝑦))
20		eqid 2741	. . . . . . . . . 10 ⊢ (+g‘𝐺) = (+g‘𝐺)
21	19, 1, 20, 4	tgplacthmeo 24090	. . . . . . . . 9 ⊢ ((𝐺 ∈ TopGrp ∧ 𝑥 ∈ (Base‘𝐺)) → (𝑦 ∈ (Base‘𝐺) ↦ (𝑥(+g‘𝐺)𝑦)) ∈ (𝐽Homeo𝐽))
22	16, 18, 21	syl2anc 591	. . . . . . . 8 ⊢ (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑆 ∈ 𝐽) ∧ 𝑥 ∈ ((Base‘𝐺) ∖ 𝑆)) → (𝑦 ∈ (Base‘𝐺) ↦ (𝑥(+g‘𝐺)𝑦)) ∈ (𝐽Homeo𝐽))
23		simpl3 1201	. . . . . . . 8 ⊢ (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑆 ∈ 𝐽) ∧ 𝑥 ∈ ((Base‘𝐺) ∖ 𝑆)) → 𝑆 ∈ 𝐽)
24		hmeoima 23752	. . . . . . . 8 ⊢ (((𝑦 ∈ (Base‘𝐺) ↦ (𝑥(+g‘𝐺)𝑦)) ∈ (𝐽Homeo𝐽) ∧ 𝑆 ∈ 𝐽) → ((𝑦 ∈ (Base‘𝐺) ↦ (𝑥(+g‘𝐺)𝑦)) “ 𝑆) ∈ 𝐽)
25	22, 23, 24	syl2anc 591	. . . . . . 7 ⊢ (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑆 ∈ 𝐽) ∧ 𝑥 ∈ ((Base‘𝐺) ∖ 𝑆)) → ((𝑦 ∈ (Base‘𝐺) ↦ (𝑥(+g‘𝐺)𝑦)) “ 𝑆) ∈ 𝐽)
26	15, 25	eqeltrrd 2842	. . . . . 6 ⊢ (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑆 ∈ 𝐽) ∧ 𝑥 ∈ ((Base‘𝐺) ∖ 𝑆)) → ran (𝑦 ∈ 𝑆 ↦ (𝑥(+g‘𝐺)𝑦)) ∈ 𝐽)
27		tgpgrp 24065	. . . . . . . . 9 ⊢ (𝐺 ∈ TopGrp → 𝐺 ∈ Grp)
28	16, 27	syl 17	. . . . . . . 8 ⊢ (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑆 ∈ 𝐽) ∧ 𝑥 ∈ ((Base‘𝐺) ∖ 𝑆)) → 𝐺 ∈ Grp)
29		eqid 2741	. . . . . . . . 9 ⊢ (0g‘𝐺) = (0g‘𝐺)
30	1, 20, 29	grprid 18939	. . . . . . . 8 ⊢ ((𝐺 ∈ Grp ∧ 𝑥 ∈ (Base‘𝐺)) → (𝑥(+g‘𝐺)(0g‘𝐺)) = 𝑥)
31	28, 18, 30	syl2anc 591	. . . . . . 7 ⊢ (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑆 ∈ 𝐽) ∧ 𝑥 ∈ ((Base‘𝐺) ∖ 𝑆)) → (𝑥(+g‘𝐺)(0g‘𝐺)) = 𝑥)
32		simpl2 1200	. . . . . . . . 9 ⊢ (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑆 ∈ 𝐽) ∧ 𝑥 ∈ ((Base‘𝐺) ∖ 𝑆)) → 𝑆 ∈ (SubGrp‘𝐺))
33	29	subg0cl 19105	. . . . . . . . 9 ⊢ (𝑆 ∈ (SubGrp‘𝐺) → (0g‘𝐺) ∈ 𝑆)
34	32, 33	syl 17	. . . . . . . 8 ⊢ (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑆 ∈ 𝐽) ∧ 𝑥 ∈ ((Base‘𝐺) ∖ 𝑆)) → (0g‘𝐺) ∈ 𝑆)
35		ovex 7393	. . . . . . . 8 ⊢ (𝑥(+g‘𝐺)(0g‘𝐺)) ∈ V
36		eqid 2741	. . . . . . . . 9 ⊢ (𝑦 ∈ 𝑆 ↦ (𝑥(+g‘𝐺)𝑦)) = (𝑦 ∈ 𝑆 ↦ (𝑥(+g‘𝐺)𝑦))
37		oveq2 7368	. . . . . . . . 9 ⊢ (𝑦 = (0g‘𝐺) → (𝑥(+g‘𝐺)𝑦) = (𝑥(+g‘𝐺)(0g‘𝐺)))
38	36, 37	elrnmpt1s 5908	. . . . . . . 8 ⊢ (((0g‘𝐺) ∈ 𝑆 ∧ (𝑥(+g‘𝐺)(0g‘𝐺)) ∈ V) → (𝑥(+g‘𝐺)(0g‘𝐺)) ∈ ran (𝑦 ∈ 𝑆 ↦ (𝑥(+g‘𝐺)𝑦)))
39	34, 35, 38	sylancl 593	. . . . . . 7 ⊢ (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑆 ∈ 𝐽) ∧ 𝑥 ∈ ((Base‘𝐺) ∖ 𝑆)) → (𝑥(+g‘𝐺)(0g‘𝐺)) ∈ ran (𝑦 ∈ 𝑆 ↦ (𝑥(+g‘𝐺)𝑦)))
40	31, 39	eqeltrrd 2842	. . . . . 6 ⊢ (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑆 ∈ 𝐽) ∧ 𝑥 ∈ ((Base‘𝐺) ∖ 𝑆)) → 𝑥 ∈ ran (𝑦 ∈ 𝑆 ↦ (𝑥(+g‘𝐺)𝑦)))
41	28	adantr 482	. . . . . . . . . 10 ⊢ ((((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑆 ∈ 𝐽) ∧ 𝑥 ∈ ((Base‘𝐺) ∖ 𝑆)) ∧ 𝑦 ∈ 𝑆) → 𝐺 ∈ Grp)
42	18	adantr 482	. . . . . . . . . 10 ⊢ ((((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑆 ∈ 𝐽) ∧ 𝑥 ∈ ((Base‘𝐺) ∖ 𝑆)) ∧ 𝑦 ∈ 𝑆) → 𝑥 ∈ (Base‘𝐺))
43	12	sselda 3917	. . . . . . . . . 10 ⊢ ((((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑆 ∈ 𝐽) ∧ 𝑥 ∈ ((Base‘𝐺) ∖ 𝑆)) ∧ 𝑦 ∈ 𝑆) → 𝑦 ∈ (Base‘𝐺))
44	1, 20	grpcl 18912	. . . . . . . . . 10 ⊢ ((𝐺 ∈ Grp ∧ 𝑥 ∈ (Base‘𝐺) ∧ 𝑦 ∈ (Base‘𝐺)) → (𝑥(+g‘𝐺)𝑦) ∈ (Base‘𝐺))
45	41, 42, 43, 44	syl3anc 1380	. . . . . . . . 9 ⊢ ((((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑆 ∈ 𝐽) ∧ 𝑥 ∈ ((Base‘𝐺) ∖ 𝑆)) ∧ 𝑦 ∈ 𝑆) → (𝑥(+g‘𝐺)𝑦) ∈ (Base‘𝐺))
46		eldifn 4065	. . . . . . . . . . 11 ⊢ (𝑥 ∈ ((Base‘𝐺) ∖ 𝑆) → ¬ 𝑥 ∈ 𝑆)
47	46	ad2antlr 734	. . . . . . . . . 10 ⊢ ((((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑆 ∈ 𝐽) ∧ 𝑥 ∈ ((Base‘𝐺) ∖ 𝑆)) ∧ 𝑦 ∈ 𝑆) → ¬ 𝑥 ∈ 𝑆)
48		eqid 2741	. . . . . . . . . . . . . . 15 ⊢ (-g‘𝐺) = (-g‘𝐺)
49	48	subgsubcl 19108	. . . . . . . . . . . . . 14 ⊢ ((𝑆 ∈ (SubGrp‘𝐺) ∧ (𝑥(+g‘𝐺)𝑦) ∈ 𝑆 ∧ 𝑦 ∈ 𝑆) → ((𝑥(+g‘𝐺)𝑦)(-g‘𝐺)𝑦) ∈ 𝑆)
50	49	3com23 1133	. . . . . . . . . . . . 13 ⊢ ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑦 ∈ 𝑆 ∧ (𝑥(+g‘𝐺)𝑦) ∈ 𝑆) → ((𝑥(+g‘𝐺)𝑦)(-g‘𝐺)𝑦) ∈ 𝑆)
51	50	3expia 1128	. . . . . . . . . . . 12 ⊢ ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑦 ∈ 𝑆) → ((𝑥(+g‘𝐺)𝑦) ∈ 𝑆 → ((𝑥(+g‘𝐺)𝑦)(-g‘𝐺)𝑦) ∈ 𝑆))
52	32, 51	sylan 587	. . . . . . . . . . 11 ⊢ ((((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑆 ∈ 𝐽) ∧ 𝑥 ∈ ((Base‘𝐺) ∖ 𝑆)) ∧ 𝑦 ∈ 𝑆) → ((𝑥(+g‘𝐺)𝑦) ∈ 𝑆 → ((𝑥(+g‘𝐺)𝑦)(-g‘𝐺)𝑦) ∈ 𝑆))
53	1, 20, 48	grppncan 19002	. . . . . . . . . . . . 13 ⊢ ((𝐺 ∈ Grp ∧ 𝑥 ∈ (Base‘𝐺) ∧ 𝑦 ∈ (Base‘𝐺)) → ((𝑥(+g‘𝐺)𝑦)(-g‘𝐺)𝑦) = 𝑥)
54	41, 42, 43, 53	syl3anc 1380	. . . . . . . . . . . 12 ⊢ ((((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑆 ∈ 𝐽) ∧ 𝑥 ∈ ((Base‘𝐺) ∖ 𝑆)) ∧ 𝑦 ∈ 𝑆) → ((𝑥(+g‘𝐺)𝑦)(-g‘𝐺)𝑦) = 𝑥)
55	54	eleq1d 2826	. . . . . . . . . . 11 ⊢ ((((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑆 ∈ 𝐽) ∧ 𝑥 ∈ ((Base‘𝐺) ∖ 𝑆)) ∧ 𝑦 ∈ 𝑆) → (((𝑥(+g‘𝐺)𝑦)(-g‘𝐺)𝑦) ∈ 𝑆 ↔ 𝑥 ∈ 𝑆))
56	52, 55	sylibd 241	. . . . . . . . . 10 ⊢ ((((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑆 ∈ 𝐽) ∧ 𝑥 ∈ ((Base‘𝐺) ∖ 𝑆)) ∧ 𝑦 ∈ 𝑆) → ((𝑥(+g‘𝐺)𝑦) ∈ 𝑆 → 𝑥 ∈ 𝑆))
57	47, 56	mtod 200	. . . . . . . . 9 ⊢ ((((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑆 ∈ 𝐽) ∧ 𝑥 ∈ ((Base‘𝐺) ∖ 𝑆)) ∧ 𝑦 ∈ 𝑆) → ¬ (𝑥(+g‘𝐺)𝑦) ∈ 𝑆)
58	45, 57	eldifd 3896	. . . . . . . 8 ⊢ ((((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑆 ∈ 𝐽) ∧ 𝑥 ∈ ((Base‘𝐺) ∖ 𝑆)) ∧ 𝑦 ∈ 𝑆) → (𝑥(+g‘𝐺)𝑦) ∈ ((Base‘𝐺) ∖ 𝑆))
59	58	fmpttd 7060	. . . . . . 7 ⊢ (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑆 ∈ 𝐽) ∧ 𝑥 ∈ ((Base‘𝐺) ∖ 𝑆)) → (𝑦 ∈ 𝑆 ↦ (𝑥(+g‘𝐺)𝑦)):𝑆⟶((Base‘𝐺) ∖ 𝑆))
60	59	frnd 6667	. . . . . 6 ⊢ (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑆 ∈ 𝐽) ∧ 𝑥 ∈ ((Base‘𝐺) ∖ 𝑆)) → ran (𝑦 ∈ 𝑆 ↦ (𝑥(+g‘𝐺)𝑦)) ⊆ ((Base‘𝐺) ∖ 𝑆))
61		eleq2 2830	. . . . . . . 8 ⊢ (𝑢 = ran (𝑦 ∈ 𝑆 ↦ (𝑥(+g‘𝐺)𝑦)) → (𝑥 ∈ 𝑢 ↔ 𝑥 ∈ ran (𝑦 ∈ 𝑆 ↦ (𝑥(+g‘𝐺)𝑦))))
62		sseq1 3942	. . . . . . . 8 ⊢ (𝑢 = ran (𝑦 ∈ 𝑆 ↦ (𝑥(+g‘𝐺)𝑦)) → (𝑢 ⊆ ((Base‘𝐺) ∖ 𝑆) ↔ ran (𝑦 ∈ 𝑆 ↦ (𝑥(+g‘𝐺)𝑦)) ⊆ ((Base‘𝐺) ∖ 𝑆)))
63	61, 62	anbi12d 639	. . . . . . 7 ⊢ (𝑢 = ran (𝑦 ∈ 𝑆 ↦ (𝑥(+g‘𝐺)𝑦)) → ((𝑥 ∈ 𝑢 ∧ 𝑢 ⊆ ((Base‘𝐺) ∖ 𝑆)) ↔ (𝑥 ∈ ran (𝑦 ∈ 𝑆 ↦ (𝑥(+g‘𝐺)𝑦)) ∧ ran (𝑦 ∈ 𝑆 ↦ (𝑥(+g‘𝐺)𝑦)) ⊆ ((Base‘𝐺) ∖ 𝑆))))
64	63	rspcev 3562	. . . . . 6 ⊢ ((ran (𝑦 ∈ 𝑆 ↦ (𝑥(+g‘𝐺)𝑦)) ∈ 𝐽 ∧ (𝑥 ∈ ran (𝑦 ∈ 𝑆 ↦ (𝑥(+g‘𝐺)𝑦)) ∧ ran (𝑦 ∈ 𝑆 ↦ (𝑥(+g‘𝐺)𝑦)) ⊆ ((Base‘𝐺) ∖ 𝑆))) → ∃𝑢 ∈ 𝐽 (𝑥 ∈ 𝑢 ∧ 𝑢 ⊆ ((Base‘𝐺) ∖ 𝑆)))
65	26, 40, 60, 64	syl12anc 843	. . . . 5 ⊢ (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑆 ∈ 𝐽) ∧ 𝑥 ∈ ((Base‘𝐺) ∖ 𝑆)) → ∃𝑢 ∈ 𝐽 (𝑥 ∈ 𝑢 ∧ 𝑢 ⊆ ((Base‘𝐺) ∖ 𝑆)))
66	65	ralrimiva 3133	. . . 4 ⊢ ((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑆 ∈ 𝐽) → ∀𝑥 ∈ ((Base‘𝐺) ∖ 𝑆)∃𝑢 ∈ 𝐽 (𝑥 ∈ 𝑢 ∧ 𝑢 ⊆ ((Base‘𝐺) ∖ 𝑆)))
67		topontop 22900	. . . . . 6 ⊢ (𝐽 ∈ (TopOn‘(Base‘𝐺)) → 𝐽 ∈ Top)
68	6, 67	syl 17	. . . . 5 ⊢ ((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑆 ∈ 𝐽) → 𝐽 ∈ Top)
69		eltop2 22962	. . . . 5 ⊢ (𝐽 ∈ Top → (((Base‘𝐺) ∖ 𝑆) ∈ 𝐽 ↔ ∀𝑥 ∈ ((Base‘𝐺) ∖ 𝑆)∃𝑢 ∈ 𝐽 (𝑥 ∈ 𝑢 ∧ 𝑢 ⊆ ((Base‘𝐺) ∖ 𝑆))))
70	68, 69	syl 17	. . . 4 ⊢ ((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑆 ∈ 𝐽) → (((Base‘𝐺) ∖ 𝑆) ∈ 𝐽 ↔ ∀𝑥 ∈ ((Base‘𝐺) ∖ 𝑆)∃𝑢 ∈ 𝐽 (𝑥 ∈ 𝑢 ∧ 𝑢 ⊆ ((Base‘𝐺) ∖ 𝑆))))
71	66, 70	mpbird 259	. . 3 ⊢ ((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑆 ∈ 𝐽) → ((Base‘𝐺) ∖ 𝑆) ∈ 𝐽)
72	10, 71	eqeltrrd 2842	. 2 ⊢ ((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑆 ∈ 𝐽) → (∪ 𝐽 ∖ 𝑆) ∈ 𝐽)
73		eqid 2741	. . . 4 ⊢ ∪ 𝐽 = ∪ 𝐽
74	73	iscld 23014	. . 3 ⊢ (𝐽 ∈ Top → (𝑆 ∈ (Clsd‘𝐽) ↔ (𝑆 ⊆ ∪ 𝐽 ∧ (∪ 𝐽 ∖ 𝑆) ∈ 𝐽)))
75	68, 74	syl 17	. 2 ⊢ ((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑆 ∈ 𝐽) → (𝑆 ∈ (Clsd‘𝐽) ↔ (𝑆 ⊆ ∪ 𝐽 ∧ (∪ 𝐽 ∖ 𝑆) ∈ 𝐽)))
76	9, 72, 75	mpbir2and 720	1 ⊢ ((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑆 ∈ 𝐽) → 𝑆 ∈ (Clsd‘𝐽))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 208   ∧ wa 397   ∧ w3a 1093   = wceq 1548   ∈ wcel 2121  ∀wral 3055  ∃wrex 3065  Vcvv 3433   ∖ cdif 3882   ⊆ wss 3885  ∪ cuni 4841   ↦ cmpt 5156  ran crn 5622   ↾ cres 5623   “ cima 5624  ‘cfv 6489  (class class class)co 7360  Basecbs 17174  +_gcplusg 17215  TopOpenctopn 17379  0_gc0g 17397  Grpcgrp 18904  -_gcsg 18906  SubGrpcsubg 19091  Topctop 22880  TopOnctopon 22897  Clsdccld 23003  Homeochmeo 23740  TopGrpctgp 24058

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1975  ax-7 2016  ax-8 2123  ax-9 2131  ax-10 2154  ax-11 2170  ax-12 2191  ax-ext 2713  ax-rep 5202  ax-sep 5221  ax-nul 5231  ax-pow 5297  ax-pr 5365  ax-un 7682  ax-cnex 11089  ax-resscn 11090  ax-1cn 11091  ax-icn 11092  ax-addcl 11093  ax-addrcl 11094  ax-mulcl 11095  ax-mulrcl 11096  ax-mulcom 11097  ax-addass 11098  ax-mulass 11099  ax-distr 11100  ax-i2m1 11101  ax-1ne0 11102  ax-1rid 11103  ax-rnegex 11104  ax-rrecex 11105  ax-cnre 11106  ax-pre-lttri 11107  ax-pre-lttrn 11108  ax-pre-ltadd 11109  ax-pre-mulgt0 11110

This theorem depends on definitions:  df-bi 209  df-an 398  df-or 855  df-3or 1094  df-3an 1095  df-tru 1551  df-fal 1561  df-ex 1788  df-nf 1792  df-sb 2075  df-mo 2545  df-eu 2575  df-clab 2720  df-cleq 2733  df-clel 2816  df-nfc 2890  df-ne 2937  df-nel 3041  df-ral 3056  df-rex 3066  df-rmo 3346  df-reu 3347  df-rab 3394  df-v 3435  df-sbc 3726  df-csb 3834  df-dif 3888  df-un 3890  df-in 3892  df-ss 3902  df-pss 3905  df-nul 4265  df-if 4458  df-pw 4534  df-sn 4559  df-pr 4561  df-op 4565  df-uni 4842  df-iun 4926  df-br 5076  df-opab 5138  df-mpt 5157  df-tr 5183  df-id 5516  df-eprel 5521  df-po 5529  df-so 5530  df-fr 5574  df-we 5576  df-xp 5627  df-rel 5628  df-cnv 5629  df-co 5630  df-dm 5631  df-rn 5632  df-res 5633  df-ima 5634  df-pred 6256  df-ord 6317  df-on 6318  df-lim 6319  df-suc 6320  df-iota 6445  df-fun 6491  df-fn 6492  df-f 6493  df-f1 6494  df-fo 6495  df-f1o 6496  df-fv 6497  df-riota 7317  df-ov 7363  df-oprab 7364  df-mpo 7365  df-om 7811  df-1st 7935  df-2nd 7936  df-frecs 8225  df-wrecs 8256  df-recs 8305  df-rdg 8343  df-er 8637  df-map 8769  df-en 8888  df-dom 8889  df-sdom 8890  df-pnf 11176  df-mnf 11177  df-xr 11178  df-ltxr 11179  df-le 11180  df-sub 11374  df-neg 11375  df-nn 12170  df-2 12239  df-sets 17129  df-slot 17147  df-ndx 17159  df-base 17175  df-ress 17196  df-plusg 17228  df-0g 17399  df-topgen 17401  df-plusf 18602  df-mgm 18603  df-sgrp 18682  df-mnd 18698  df-grp 18907  df-minusg 18908  df-sbg 18909  df-subg 19094  df-top 22881  df-topon 22898  df-topsp 22920  df-bases 22933  df-cld 23006  df-cn 23214  df-cnp 23215  df-tx 23549  df-hmeo 23742  df-tmd 24059  df-tgp 24060

This theorem is referenced by:  cldsubg  24098  tgpconncompss  24101