Theorem opnsubg 24087

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 2737	. . . . 5 ⊢ (Base‘𝐺) = (Base‘𝐺)
2	1	subgss 19098	. . . 4 ⊢ (𝑆 ∈ (SubGrp‘𝐺) → 𝑆 ⊆ (Base‘𝐺))
3	2	3ad2ant2 1135	. . 3 ⊢ ((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑆 ∈ 𝐽) → 𝑆 ⊆ (Base‘𝐺))
4		subgntr.h	. . . . . 6 ⊢ 𝐽 = (TopOpen‘𝐺)
5	4, 1	tgptopon 24061	. . . . 5 ⊢ (𝐺 ∈ TopGrp → 𝐽 ∈ (TopOn‘(Base‘𝐺)))
6	5	3ad2ant1 1134	. . . 4 ⊢ ((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑆 ∈ 𝐽) → 𝐽 ∈ (TopOn‘(Base‘𝐺)))
7		toponuni 22893	. . . 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 3959	. 2 ⊢ ((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑆 ∈ 𝐽) → 𝑆 ⊆ ∪ 𝐽)
10	8	difeq1d 4066	. . 3 ⊢ ((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑆 ∈ 𝐽) → ((Base‘𝐺) ∖ 𝑆) = (∪ 𝐽 ∖ 𝑆))
11		df-ima 5639	. . . . . . . 8 ⊢ ((𝑦 ∈ (Base‘𝐺) ↦ (𝑥(+g‘𝐺)𝑦)) “ 𝑆) = ran ((𝑦 ∈ (Base‘𝐺) ↦ (𝑥(+g‘𝐺)𝑦)) ↾ 𝑆)
12	3	adantr 480	. . . . . . . . . 10 ⊢ (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑆 ∈ 𝐽) ∧ 𝑥 ∈ ((Base‘𝐺) ∖ 𝑆)) → 𝑆 ⊆ (Base‘𝐺))
13	12	resmptd 6001	. . . . . . . . 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 5889	. . . . . . . 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 2784	. . . . . . 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 1193	. . . . . . . . 9 ⊢ (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑆 ∈ 𝐽) ∧ 𝑥 ∈ ((Base‘𝐺) ∖ 𝑆)) → 𝐺 ∈ TopGrp)
17		eldifi 4072	. . . . . . . . . 10 ⊢ (𝑥 ∈ ((Base‘𝐺) ∖ 𝑆) → 𝑥 ∈ (Base‘𝐺))
18	17	adantl 481	. . . . . . . . 9 ⊢ (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑆 ∈ 𝐽) ∧ 𝑥 ∈ ((Base‘𝐺) ∖ 𝑆)) → 𝑥 ∈ (Base‘𝐺))
19		eqid 2737	. . . . . . . . . 10 ⊢ (𝑦 ∈ (Base‘𝐺) ↦ (𝑥(+g‘𝐺)𝑦)) = (𝑦 ∈ (Base‘𝐺) ↦ (𝑥(+g‘𝐺)𝑦))
20		eqid 2737	. . . . . . . . . 10 ⊢ (+g‘𝐺) = (+g‘𝐺)
21	19, 1, 20, 4	tgplacthmeo 24082	. . . . . . . . 9 ⊢ ((𝐺 ∈ TopGrp ∧ 𝑥 ∈ (Base‘𝐺)) → (𝑦 ∈ (Base‘𝐺) ↦ (𝑥(+g‘𝐺)𝑦)) ∈ (𝐽Homeo𝐽))
22	16, 18, 21	syl2anc 585	. . . . . . . 8 ⊢ (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑆 ∈ 𝐽) ∧ 𝑥 ∈ ((Base‘𝐺) ∖ 𝑆)) → (𝑦 ∈ (Base‘𝐺) ↦ (𝑥(+g‘𝐺)𝑦)) ∈ (𝐽Homeo𝐽))
23		simpl3 1195	. . . . . . . 8 ⊢ (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑆 ∈ 𝐽) ∧ 𝑥 ∈ ((Base‘𝐺) ∖ 𝑆)) → 𝑆 ∈ 𝐽)
24		hmeoima 23744	. . . . . . . 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 585	. . . . . . 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 2838	. . . . . 6 ⊢ (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑆 ∈ 𝐽) ∧ 𝑥 ∈ ((Base‘𝐺) ∖ 𝑆)) → ran (𝑦 ∈ 𝑆 ↦ (𝑥(+g‘𝐺)𝑦)) ∈ 𝐽)
27		tgpgrp 24057	. . . . . . . . 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 2737	. . . . . . . . 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 585	. . . . . . 7 ⊢ (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑆 ∈ 𝐽) ∧ 𝑥 ∈ ((Base‘𝐺) ∖ 𝑆)) → (𝑥(+g‘𝐺)(0g‘𝐺)) = 𝑥)
32		simpl2 1194	. . . . . . . . 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 7395	. . . . . . . 8 ⊢ (𝑥(+g‘𝐺)(0g‘𝐺)) ∈ V
36		eqid 2737	. . . . . . . . 9 ⊢ (𝑦 ∈ 𝑆 ↦ (𝑥(+g‘𝐺)𝑦)) = (𝑦 ∈ 𝑆 ↦ (𝑥(+g‘𝐺)𝑦))
37		oveq2 7370	. . . . . . . . 9 ⊢ (𝑦 = (0g‘𝐺) → (𝑥(+g‘𝐺)𝑦) = (𝑥(+g‘𝐺)(0g‘𝐺)))
38	36, 37	elrnmpt1s 5910	. . . . . . . 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 587	. . . . . . 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 2838	. . . . . 6 ⊢ (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑆 ∈ 𝐽) ∧ 𝑥 ∈ ((Base‘𝐺) ∖ 𝑆)) → 𝑥 ∈ ran (𝑦 ∈ 𝑆 ↦ (𝑥(+g‘𝐺)𝑦)))
41	28	adantr 480	. . . . . . . . . 10 ⊢ ((((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑆 ∈ 𝐽) ∧ 𝑥 ∈ ((Base‘𝐺) ∖ 𝑆)) ∧ 𝑦 ∈ 𝑆) → 𝐺 ∈ Grp)
42	18	adantr 480	. . . . . . . . . 10 ⊢ ((((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑆 ∈ 𝐽) ∧ 𝑥 ∈ ((Base‘𝐺) ∖ 𝑆)) ∧ 𝑦 ∈ 𝑆) → 𝑥 ∈ (Base‘𝐺))
43	12	sselda 3922	. . . . . . . . . 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 1374	. . . . . . . . 9 ⊢ ((((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑆 ∈ 𝐽) ∧ 𝑥 ∈ ((Base‘𝐺) ∖ 𝑆)) ∧ 𝑦 ∈ 𝑆) → (𝑥(+g‘𝐺)𝑦) ∈ (Base‘𝐺))
46		eldifn 4073	. . . . . . . . . . 11 ⊢ (𝑥 ∈ ((Base‘𝐺) ∖ 𝑆) → ¬ 𝑥 ∈ 𝑆)
47	46	ad2antlr 728	. . . . . . . . . 10 ⊢ ((((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑆 ∈ 𝐽) ∧ 𝑥 ∈ ((Base‘𝐺) ∖ 𝑆)) ∧ 𝑦 ∈ 𝑆) → ¬ 𝑥 ∈ 𝑆)
48		eqid 2737	. . . . . . . . . . . . . . 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 1127	. . . . . . . . . . . . 13 ⊢ ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑦 ∈ 𝑆 ∧ (𝑥(+g‘𝐺)𝑦) ∈ 𝑆) → ((𝑥(+g‘𝐺)𝑦)(-g‘𝐺)𝑦) ∈ 𝑆)
51	50	3expia 1122	. . . . . . . . . . . 12 ⊢ ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑦 ∈ 𝑆) → ((𝑥(+g‘𝐺)𝑦) ∈ 𝑆 → ((𝑥(+g‘𝐺)𝑦)(-g‘𝐺)𝑦) ∈ 𝑆))
52	32, 51	sylan 581	. . . . . . . . . . 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 1374	. . . . . . . . . . . 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 2822	. . . . . . . . . . 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 239	. . . . . . . . . 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 198	. . . . . . . . 9 ⊢ ((((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑆 ∈ 𝐽) ∧ 𝑥 ∈ ((Base‘𝐺) ∖ 𝑆)) ∧ 𝑦 ∈ 𝑆) → ¬ (𝑥(+g‘𝐺)𝑦) ∈ 𝑆)
58	45, 57	eldifd 3901	. . . . . . . 8 ⊢ ((((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑆 ∈ 𝐽) ∧ 𝑥 ∈ ((Base‘𝐺) ∖ 𝑆)) ∧ 𝑦 ∈ 𝑆) → (𝑥(+g‘𝐺)𝑦) ∈ ((Base‘𝐺) ∖ 𝑆))
59	58	fmpttd 7063	. . . . . . 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 6672	. . . . . 6 ⊢ (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑆 ∈ 𝐽) ∧ 𝑥 ∈ ((Base‘𝐺) ∖ 𝑆)) → ran (𝑦 ∈ 𝑆 ↦ (𝑥(+g‘𝐺)𝑦)) ⊆ ((Base‘𝐺) ∖ 𝑆))
61		eleq2 2826	. . . . . . . 8 ⊢ (𝑢 = ran (𝑦 ∈ 𝑆 ↦ (𝑥(+g‘𝐺)𝑦)) → (𝑥 ∈ 𝑢 ↔ 𝑥 ∈ ran (𝑦 ∈ 𝑆 ↦ (𝑥(+g‘𝐺)𝑦))))
62		sseq1 3948	. . . . . . . 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 633	. . . . . . 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 3565	. . . . . 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 837	. . . . 5 ⊢ (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑆 ∈ 𝐽) ∧ 𝑥 ∈ ((Base‘𝐺) ∖ 𝑆)) → ∃𝑢 ∈ 𝐽 (𝑥 ∈ 𝑢 ∧ 𝑢 ⊆ ((Base‘𝐺) ∖ 𝑆)))
66	65	ralrimiva 3130	. . . 4 ⊢ ((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑆 ∈ 𝐽) → ∀𝑥 ∈ ((Base‘𝐺) ∖ 𝑆)∃𝑢 ∈ 𝐽 (𝑥 ∈ 𝑢 ∧ 𝑢 ⊆ ((Base‘𝐺) ∖ 𝑆)))
67		topontop 22892	. . . . . 6 ⊢ (𝐽 ∈ (TopOn‘(Base‘𝐺)) → 𝐽 ∈ Top)
68	6, 67	syl 17	. . . . 5 ⊢ ((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑆 ∈ 𝐽) → 𝐽 ∈ Top)
69		eltop2 22954	. . . . 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 257	. . 3 ⊢ ((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑆 ∈ 𝐽) → ((Base‘𝐺) ∖ 𝑆) ∈ 𝐽)
72	10, 71	eqeltrrd 2838	. 2 ⊢ ((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑆 ∈ 𝐽) → (∪ 𝐽 ∖ 𝑆) ∈ 𝐽)
73		eqid 2737	. . . 4 ⊢ ∪ 𝐽 = ∪ 𝐽
74	73	iscld 23006	. . 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 714	1 ⊢ ((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑆 ∈ 𝐽) → 𝑆 ∈ (Clsd‘𝐽))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1087   = wceq 1542   ∈ wcel 2114  ∀wral 3052  ∃wrex 3062  Vcvv 3430   ∖ cdif 3887   ⊆ wss 3890  ∪ cuni 4851   ↦ cmpt 5167  ran crn 5627   ↾ cres 5628   “ cima 5629  ‘cfv 6494  (class class class)co 7362  Basecbs 17174  +_gcplusg 17215  TopOpenctopn 17379  0_gc0g 17397  Grpcgrp 18904  -_gcsg 18906  SubGrpcsubg 19091  Topctop 22872  TopOnctopon 22889  Clsdccld 22995  Homeochmeo 23732  TopGrpctgp 24050

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-rep 5213  ax-sep 5232  ax-nul 5242  ax-pow 5304  ax-pr 5372  ax-un 7684  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 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-nel 3038  df-ral 3053  df-rex 3063  df-rmo 3343  df-reu 3344  df-rab 3391  df-v 3432  df-sbc 3730  df-csb 3839  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-pss 3910  df-nul 4275  df-if 4468  df-pw 4544  df-sn 4569  df-pr 4571  df-op 4575  df-uni 4852  df-iun 4936  df-br 5087  df-opab 5149  df-mpt 5168  df-tr 5194  df-id 5521  df-eprel 5526  df-po 5534  df-so 5535  df-fr 5579  df-we 5581  df-xp 5632  df-rel 5633  df-cnv 5634  df-co 5635  df-dm 5636  df-rn 5637  df-res 5638  df-ima 5639  df-pred 6261  df-ord 6322  df-on 6323  df-lim 6324  df-suc 6325  df-iota 6450  df-fun 6496  df-fn 6497  df-f 6498  df-f1 6499  df-fo 6500  df-f1o 6501  df-fv 6502  df-riota 7319  df-ov 7365  df-oprab 7366  df-mpo 7367  df-om 7813  df-1st 7937  df-2nd 7938  df-frecs 8226  df-wrecs 8257  df-recs 8306  df-rdg 8344  df-er 8638  df-map 8770  df-en 8889  df-dom 8890  df-sdom 8891  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 22873  df-topon 22890  df-topsp 22912  df-bases 22925  df-cld 22998  df-cn 23206  df-cnp 23207  df-tx 23541  df-hmeo 23734  df-tmd 24051  df-tgp 24052

This theorem is referenced by:  cldsubg  24090  tgpconncompss  24093