Theorem opnsubg 21911

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 2622	. . . . 5 ⊢ (Base‘𝐺) = (Base‘𝐺)
2	1	subgss 17595	. . . 4 ⊢ (𝑆 ∈ (SubGrp‘𝐺) → 𝑆 ⊆ (Base‘𝐺))
3	2	3ad2ant2 1083	. . 3 ⊢ ((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑆 ∈ 𝐽) → 𝑆 ⊆ (Base‘𝐺))
4		subgntr.h	. . . . . 6 ⊢ 𝐽 = (TopOpen‘𝐺)
5	4, 1	tgptopon 21886	. . . . 5 ⊢ (𝐺 ∈ TopGrp → 𝐽 ∈ (TopOn‘(Base‘𝐺)))
6	5	3ad2ant1 1082	. . . 4 ⊢ ((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑆 ∈ 𝐽) → 𝐽 ∈ (TopOn‘(Base‘𝐺)))
7		toponuni 20719	. . . 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 3641	. 2 ⊢ ((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑆 ∈ 𝐽) → 𝑆 ⊆ ∪ 𝐽)
10	8	difeq1d 3727	. . 3 ⊢ ((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑆 ∈ 𝐽) → ((Base‘𝐺) ∖ 𝑆) = (∪ 𝐽 ∖ 𝑆))
11		df-ima 5127	. . . . . . . 8 ⊢ ((𝑦 ∈ (Base‘𝐺) ↦ (𝑥(+g‘𝐺)𝑦)) “ 𝑆) = ran ((𝑦 ∈ (Base‘𝐺) ↦ (𝑥(+g‘𝐺)𝑦)) ↾ 𝑆)
12	3	adantr 481	. . . . . . . . . 10 ⊢ (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑆 ∈ 𝐽) ∧ 𝑥 ∈ ((Base‘𝐺) ∖ 𝑆)) → 𝑆 ⊆ (Base‘𝐺))
13	12	resmptd 5452	. . . . . . . . 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 5353	. . . . . . . 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	syl5eq 2668	. . . . . . 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 1064	. . . . . . . . 9 ⊢ (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑆 ∈ 𝐽) ∧ 𝑥 ∈ ((Base‘𝐺) ∖ 𝑆)) → 𝐺 ∈ TopGrp)
17		eldifi 3732	. . . . . . . . . 10 ⊢ (𝑥 ∈ ((Base‘𝐺) ∖ 𝑆) → 𝑥 ∈ (Base‘𝐺))
18	17	adantl 482	. . . . . . . . 9 ⊢ (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑆 ∈ 𝐽) ∧ 𝑥 ∈ ((Base‘𝐺) ∖ 𝑆)) → 𝑥 ∈ (Base‘𝐺))
19		eqid 2622	. . . . . . . . . 10 ⊢ (𝑦 ∈ (Base‘𝐺) ↦ (𝑥(+g‘𝐺)𝑦)) = (𝑦 ∈ (Base‘𝐺) ↦ (𝑥(+g‘𝐺)𝑦))
20		eqid 2622	. . . . . . . . . 10 ⊢ (+g‘𝐺) = (+g‘𝐺)
21	19, 1, 20, 4	tgplacthmeo 21907	. . . . . . . . 9 ⊢ ((𝐺 ∈ TopGrp ∧ 𝑥 ∈ (Base‘𝐺)) → (𝑦 ∈ (Base‘𝐺) ↦ (𝑥(+g‘𝐺)𝑦)) ∈ (𝐽Homeo𝐽))
22	16, 18, 21	syl2anc 693	. . . . . . . 8 ⊢ (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑆 ∈ 𝐽) ∧ 𝑥 ∈ ((Base‘𝐺) ∖ 𝑆)) → (𝑦 ∈ (Base‘𝐺) ↦ (𝑥(+g‘𝐺)𝑦)) ∈ (𝐽Homeo𝐽))
23		simpl3 1066	. . . . . . . 8 ⊢ (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑆 ∈ 𝐽) ∧ 𝑥 ∈ ((Base‘𝐺) ∖ 𝑆)) → 𝑆 ∈ 𝐽)
24		hmeoima 21568	. . . . . . . 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 693	. . . . . . 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 2702	. . . . . 6 ⊢ (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑆 ∈ 𝐽) ∧ 𝑥 ∈ ((Base‘𝐺) ∖ 𝑆)) → ran (𝑦 ∈ 𝑆 ↦ (𝑥(+g‘𝐺)𝑦)) ∈ 𝐽)
27		tgpgrp 21882	. . . . . . . . 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 2622	. . . . . . . . 9 ⊢ (0g‘𝐺) = (0g‘𝐺)
30	1, 20, 29	grprid 17453	. . . . . . . 8 ⊢ ((𝐺 ∈ Grp ∧ 𝑥 ∈ (Base‘𝐺)) → (𝑥(+g‘𝐺)(0g‘𝐺)) = 𝑥)
31	28, 18, 30	syl2anc 693	. . . . . . 7 ⊢ (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑆 ∈ 𝐽) ∧ 𝑥 ∈ ((Base‘𝐺) ∖ 𝑆)) → (𝑥(+g‘𝐺)(0g‘𝐺)) = 𝑥)
32		simpl2 1065	. . . . . . . . 9 ⊢ (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑆 ∈ 𝐽) ∧ 𝑥 ∈ ((Base‘𝐺) ∖ 𝑆)) → 𝑆 ∈ (SubGrp‘𝐺))
33	29	subg0cl 17602	. . . . . . . . 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 6678	. . . . . . . 8 ⊢ (𝑥(+g‘𝐺)(0g‘𝐺)) ∈ V
36		eqid 2622	. . . . . . . . 9 ⊢ (𝑦 ∈ 𝑆 ↦ (𝑥(+g‘𝐺)𝑦)) = (𝑦 ∈ 𝑆 ↦ (𝑥(+g‘𝐺)𝑦))
37		oveq2 6658	. . . . . . . . 9 ⊢ (𝑦 = (0g‘𝐺) → (𝑥(+g‘𝐺)𝑦) = (𝑥(+g‘𝐺)(0g‘𝐺)))
38	36, 37	elrnmpt1s 5373	. . . . . . . 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 694	. . . . . . 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 2702	. . . . . 6 ⊢ (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑆 ∈ 𝐽) ∧ 𝑥 ∈ ((Base‘𝐺) ∖ 𝑆)) → 𝑥 ∈ ran (𝑦 ∈ 𝑆 ↦ (𝑥(+g‘𝐺)𝑦)))
41	28	adantr 481	. . . . . . . . . 10 ⊢ ((((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑆 ∈ 𝐽) ∧ 𝑥 ∈ ((Base‘𝐺) ∖ 𝑆)) ∧ 𝑦 ∈ 𝑆) → 𝐺 ∈ Grp)
42	18	adantr 481	. . . . . . . . . 10 ⊢ ((((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑆 ∈ 𝐽) ∧ 𝑥 ∈ ((Base‘𝐺) ∖ 𝑆)) ∧ 𝑦 ∈ 𝑆) → 𝑥 ∈ (Base‘𝐺))
43	12	sselda 3603	. . . . . . . . . 10 ⊢ ((((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑆 ∈ 𝐽) ∧ 𝑥 ∈ ((Base‘𝐺) ∖ 𝑆)) ∧ 𝑦 ∈ 𝑆) → 𝑦 ∈ (Base‘𝐺))
44	1, 20	grpcl 17430	. . . . . . . . . 10 ⊢ ((𝐺 ∈ Grp ∧ 𝑥 ∈ (Base‘𝐺) ∧ 𝑦 ∈ (Base‘𝐺)) → (𝑥(+g‘𝐺)𝑦) ∈ (Base‘𝐺))
45	41, 42, 43, 44	syl3anc 1326	. . . . . . . . 9 ⊢ ((((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑆 ∈ 𝐽) ∧ 𝑥 ∈ ((Base‘𝐺) ∖ 𝑆)) ∧ 𝑦 ∈ 𝑆) → (𝑥(+g‘𝐺)𝑦) ∈ (Base‘𝐺))
46		eldifn 3733	. . . . . . . . . . 11 ⊢ (𝑥 ∈ ((Base‘𝐺) ∖ 𝑆) → ¬ 𝑥 ∈ 𝑆)
47	46	ad2antlr 763	. . . . . . . . . 10 ⊢ ((((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑆 ∈ 𝐽) ∧ 𝑥 ∈ ((Base‘𝐺) ∖ 𝑆)) ∧ 𝑦 ∈ 𝑆) → ¬ 𝑥 ∈ 𝑆)
48		eqid 2622	. . . . . . . . . . . . . . 15 ⊢ (-g‘𝐺) = (-g‘𝐺)
49	48	subgsubcl 17605	. . . . . . . . . . . . . 14 ⊢ ((𝑆 ∈ (SubGrp‘𝐺) ∧ (𝑥(+g‘𝐺)𝑦) ∈ 𝑆 ∧ 𝑦 ∈ 𝑆) → ((𝑥(+g‘𝐺)𝑦)(-g‘𝐺)𝑦) ∈ 𝑆)
50	49	3com23 1271	. . . . . . . . . . . . 13 ⊢ ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑦 ∈ 𝑆 ∧ (𝑥(+g‘𝐺)𝑦) ∈ 𝑆) → ((𝑥(+g‘𝐺)𝑦)(-g‘𝐺)𝑦) ∈ 𝑆)
51	50	3expia 1267	. . . . . . . . . . . 12 ⊢ ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑦 ∈ 𝑆) → ((𝑥(+g‘𝐺)𝑦) ∈ 𝑆 → ((𝑥(+g‘𝐺)𝑦)(-g‘𝐺)𝑦) ∈ 𝑆))
52	32, 51	sylan 488	. . . . . . . . . . 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 17506	. . . . . . . . . . . . 13 ⊢ ((𝐺 ∈ Grp ∧ 𝑥 ∈ (Base‘𝐺) ∧ 𝑦 ∈ (Base‘𝐺)) → ((𝑥(+g‘𝐺)𝑦)(-g‘𝐺)𝑦) = 𝑥)
54	41, 42, 43, 53	syl3anc 1326	. . . . . . . . . . . 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 2686	. . . . . . . . . . 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 229	. . . . . . . . . 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 189	. . . . . . . . 9 ⊢ ((((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑆 ∈ 𝐽) ∧ 𝑥 ∈ ((Base‘𝐺) ∖ 𝑆)) ∧ 𝑦 ∈ 𝑆) → ¬ (𝑥(+g‘𝐺)𝑦) ∈ 𝑆)
58	45, 57	eldifd 3585	. . . . . . . 8 ⊢ ((((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑆 ∈ 𝐽) ∧ 𝑥 ∈ ((Base‘𝐺) ∖ 𝑆)) ∧ 𝑦 ∈ 𝑆) → (𝑥(+g‘𝐺)𝑦) ∈ ((Base‘𝐺) ∖ 𝑆))
59	58, 36	fmptd 6385	. . . . . . 7 ⊢ (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑆 ∈ 𝐽) ∧ 𝑥 ∈ ((Base‘𝐺) ∖ 𝑆)) → (𝑦 ∈ 𝑆 ↦ (𝑥(+g‘𝐺)𝑦)):𝑆⟶((Base‘𝐺) ∖ 𝑆))
60		frn 6053	. . . . . . 7 ⊢ ((𝑦 ∈ 𝑆 ↦ (𝑥(+g‘𝐺)𝑦)):𝑆⟶((Base‘𝐺) ∖ 𝑆) → ran (𝑦 ∈ 𝑆 ↦ (𝑥(+g‘𝐺)𝑦)) ⊆ ((Base‘𝐺) ∖ 𝑆))
61	59, 60	syl 17	. . . . . 6 ⊢ (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑆 ∈ 𝐽) ∧ 𝑥 ∈ ((Base‘𝐺) ∖ 𝑆)) → ran (𝑦 ∈ 𝑆 ↦ (𝑥(+g‘𝐺)𝑦)) ⊆ ((Base‘𝐺) ∖ 𝑆))
62		eleq2 2690	. . . . . . . 8 ⊢ (𝑢 = ran (𝑦 ∈ 𝑆 ↦ (𝑥(+g‘𝐺)𝑦)) → (𝑥 ∈ 𝑢 ↔ 𝑥 ∈ ran (𝑦 ∈ 𝑆 ↦ (𝑥(+g‘𝐺)𝑦))))
63		sseq1 3626	. . . . . . . 8 ⊢ (𝑢 = ran (𝑦 ∈ 𝑆 ↦ (𝑥(+g‘𝐺)𝑦)) → (𝑢 ⊆ ((Base‘𝐺) ∖ 𝑆) ↔ ran (𝑦 ∈ 𝑆 ↦ (𝑥(+g‘𝐺)𝑦)) ⊆ ((Base‘𝐺) ∖ 𝑆)))
64	62, 63	anbi12d 747	. . . . . . 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))))
65	64	rspcev 3309	. . . . . 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)))
66	26, 40, 61, 65	syl12anc 1324	. . . . 5 ⊢ (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑆 ∈ 𝐽) ∧ 𝑥 ∈ ((Base‘𝐺) ∖ 𝑆)) → ∃𝑢 ∈ 𝐽 (𝑥 ∈ 𝑢 ∧ 𝑢 ⊆ ((Base‘𝐺) ∖ 𝑆)))
67	66	ralrimiva 2966	. . . 4 ⊢ ((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑆 ∈ 𝐽) → ∀𝑥 ∈ ((Base‘𝐺) ∖ 𝑆)∃𝑢 ∈ 𝐽 (𝑥 ∈ 𝑢 ∧ 𝑢 ⊆ ((Base‘𝐺) ∖ 𝑆)))
68		topontop 20718	. . . . . 6 ⊢ (𝐽 ∈ (TopOn‘(Base‘𝐺)) → 𝐽 ∈ Top)
69	6, 68	syl 17	. . . . 5 ⊢ ((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑆 ∈ 𝐽) → 𝐽 ∈ Top)
70		eltop2 20779	. . . . 5 ⊢ (𝐽 ∈ Top → (((Base‘𝐺) ∖ 𝑆) ∈ 𝐽 ↔ ∀𝑥 ∈ ((Base‘𝐺) ∖ 𝑆)∃𝑢 ∈ 𝐽 (𝑥 ∈ 𝑢 ∧ 𝑢 ⊆ ((Base‘𝐺) ∖ 𝑆))))
71	69, 70	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))))
72	67, 71	mpbird 247	. . 3 ⊢ ((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑆 ∈ 𝐽) → ((Base‘𝐺) ∖ 𝑆) ∈ 𝐽)
73	10, 72	eqeltrrd 2702	. 2 ⊢ ((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑆 ∈ 𝐽) → (∪ 𝐽 ∖ 𝑆) ∈ 𝐽)
74		eqid 2622	. . . 4 ⊢ ∪ 𝐽 = ∪ 𝐽
75	74	iscld 20831	. . 3 ⊢ (𝐽 ∈ Top → (𝑆 ∈ (Clsd‘𝐽) ↔ (𝑆 ⊆ ∪ 𝐽 ∧ (∪ 𝐽 ∖ 𝑆) ∈ 𝐽)))
76	69, 75	syl 17	. 2 ⊢ ((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑆 ∈ 𝐽) → (𝑆 ∈ (Clsd‘𝐽) ↔ (𝑆 ⊆ ∪ 𝐽 ∧ (∪ 𝐽 ∖ 𝑆) ∈ 𝐽)))
77	9, 73, 76	mpbir2and 957	1 ⊢ ((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑆 ∈ 𝐽) → 𝑆 ∈ (Clsd‘𝐽))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 196   ∧ wa 384   ∧ w3a 1037   = wceq 1483   ∈ wcel 1990  ∀wral 2912  ∃wrex 2913  Vcvv 3200   ∖ cdif 3571   ⊆ wss 3574  ∪ cuni 4436   ↦ cmpt 4729  ran crn 5115   ↾ cres 5116   “ cima 5117  ⟶wf 5884  ‘cfv 5888  (class class class)co 6650  Basecbs 15857  +_gcplusg 15941  TopOpenctopn 16082  0_gc0g 16100  Grpcgrp 17422  -_gcsg 17424  SubGrpcsubg 17588  Topctop 20698  TopOnctopon 20715  Clsdccld 20820  Homeochmeo 21556  TopGrpctgp 21875

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-er 7742  df-map 7859  df-en 7956  df-dom 7957  df-sdom 7958  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-nn 11021  df-2 11079  df-ndx 15860  df-slot 15861  df-base 15863  df-sets 15864  df-ress 15865  df-plusg 15954  df-0g 16102  df-topgen 16104  df-plusf 17241  df-mgm 17242  df-sgrp 17284  df-mnd 17295  df-grp 17425  df-minusg 17426  df-sbg 17427  df-subg 17591  df-top 20699  df-topon 20716  df-topsp 20737  df-bases 20750  df-cld 20823  df-cn 21031  df-cnp 21032  df-tx 21365  df-hmeo 21558  df-tmd 21876  df-tgp 21877

This theorem is referenced by:  cldsubg  21914  tgpconncompss  21917