Theorem opnsubg 22713

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 2798	. . . . 5 ⊢ (Base‘𝐺) = (Base‘𝐺)
2	1	subgss 18272	. . . 4 ⊢ (𝑆 ∈ (SubGrp‘𝐺) → 𝑆 ⊆ (Base‘𝐺))
3	2	3ad2ant2 1131	. . 3 ⊢ ((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑆 ∈ 𝐽) → 𝑆 ⊆ (Base‘𝐺))
4		subgntr.h	. . . . . 6 ⊢ 𝐽 = (TopOpen‘𝐺)
5	4, 1	tgptopon 22687	. . . . 5 ⊢ (𝐺 ∈ TopGrp → 𝐽 ∈ (TopOn‘(Base‘𝐺)))
6	5	3ad2ant1 1130	. . . 4 ⊢ ((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑆 ∈ 𝐽) → 𝐽 ∈ (TopOn‘(Base‘𝐺)))
7		toponuni 21519	. . . 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 3955	. 2 ⊢ ((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑆 ∈ 𝐽) → 𝑆 ⊆ ∪ 𝐽)
10	8	difeq1d 4049	. . 3 ⊢ ((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑆 ∈ 𝐽) → ((Base‘𝐺) ∖ 𝑆) = (∪ 𝐽 ∖ 𝑆))
11		df-ima 5532	. . . . . . . 8 ⊢ ((𝑦 ∈ (Base‘𝐺) ↦ (𝑥(+g‘𝐺)𝑦)) “ 𝑆) = ran ((𝑦 ∈ (Base‘𝐺) ↦ (𝑥(+g‘𝐺)𝑦)) ↾ 𝑆)
12	3	adantr 484	. . . . . . . . . 10 ⊢ (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑆 ∈ 𝐽) ∧ 𝑥 ∈ ((Base‘𝐺) ∖ 𝑆)) → 𝑆 ⊆ (Base‘𝐺))
13	12	resmptd 5875	. . . . . . . . 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 5772	. . . . . . . 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 2845	. . . . . . 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 1188	. . . . . . . . 9 ⊢ (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑆 ∈ 𝐽) ∧ 𝑥 ∈ ((Base‘𝐺) ∖ 𝑆)) → 𝐺 ∈ TopGrp)
17		eldifi 4054	. . . . . . . . . 10 ⊢ (𝑥 ∈ ((Base‘𝐺) ∖ 𝑆) → 𝑥 ∈ (Base‘𝐺))
18	17	adantl 485	. . . . . . . . 9 ⊢ (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑆 ∈ 𝐽) ∧ 𝑥 ∈ ((Base‘𝐺) ∖ 𝑆)) → 𝑥 ∈ (Base‘𝐺))
19		eqid 2798	. . . . . . . . . 10 ⊢ (𝑦 ∈ (Base‘𝐺) ↦ (𝑥(+g‘𝐺)𝑦)) = (𝑦 ∈ (Base‘𝐺) ↦ (𝑥(+g‘𝐺)𝑦))
20		eqid 2798	. . . . . . . . . 10 ⊢ (+g‘𝐺) = (+g‘𝐺)
21	19, 1, 20, 4	tgplacthmeo 22708	. . . . . . . . 9 ⊢ ((𝐺 ∈ TopGrp ∧ 𝑥 ∈ (Base‘𝐺)) → (𝑦 ∈ (Base‘𝐺) ↦ (𝑥(+g‘𝐺)𝑦)) ∈ (𝐽Homeo𝐽))
22	16, 18, 21	syl2anc 587	. . . . . . . 8 ⊢ (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑆 ∈ 𝐽) ∧ 𝑥 ∈ ((Base‘𝐺) ∖ 𝑆)) → (𝑦 ∈ (Base‘𝐺) ↦ (𝑥(+g‘𝐺)𝑦)) ∈ (𝐽Homeo𝐽))
23		simpl3 1190	. . . . . . . 8 ⊢ (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑆 ∈ 𝐽) ∧ 𝑥 ∈ ((Base‘𝐺) ∖ 𝑆)) → 𝑆 ∈ 𝐽)
24		hmeoima 22370	. . . . . . . 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 587	. . . . . . 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 2891	. . . . . 6 ⊢ (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑆 ∈ 𝐽) ∧ 𝑥 ∈ ((Base‘𝐺) ∖ 𝑆)) → ran (𝑦 ∈ 𝑆 ↦ (𝑥(+g‘𝐺)𝑦)) ∈ 𝐽)
27		tgpgrp 22683	. . . . . . . . 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 2798	. . . . . . . . 9 ⊢ (0g‘𝐺) = (0g‘𝐺)
30	1, 20, 29	grprid 18126	. . . . . . . 8 ⊢ ((𝐺 ∈ Grp ∧ 𝑥 ∈ (Base‘𝐺)) → (𝑥(+g‘𝐺)(0g‘𝐺)) = 𝑥)
31	28, 18, 30	syl2anc 587	. . . . . . 7 ⊢ (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑆 ∈ 𝐽) ∧ 𝑥 ∈ ((Base‘𝐺) ∖ 𝑆)) → (𝑥(+g‘𝐺)(0g‘𝐺)) = 𝑥)
32		simpl2 1189	. . . . . . . . 9 ⊢ (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑆 ∈ 𝐽) ∧ 𝑥 ∈ ((Base‘𝐺) ∖ 𝑆)) → 𝑆 ∈ (SubGrp‘𝐺))
33	29	subg0cl 18279	. . . . . . . . 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 7168	. . . . . . . 8 ⊢ (𝑥(+g‘𝐺)(0g‘𝐺)) ∈ V
36		eqid 2798	. . . . . . . . 9 ⊢ (𝑦 ∈ 𝑆 ↦ (𝑥(+g‘𝐺)𝑦)) = (𝑦 ∈ 𝑆 ↦ (𝑥(+g‘𝐺)𝑦))
37		oveq2 7143	. . . . . . . . 9 ⊢ (𝑦 = (0g‘𝐺) → (𝑥(+g‘𝐺)𝑦) = (𝑥(+g‘𝐺)(0g‘𝐺)))
38	36, 37	elrnmpt1s 5793	. . . . . . . 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 589	. . . . . . 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 2891	. . . . . 6 ⊢ (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑆 ∈ 𝐽) ∧ 𝑥 ∈ ((Base‘𝐺) ∖ 𝑆)) → 𝑥 ∈ ran (𝑦 ∈ 𝑆 ↦ (𝑥(+g‘𝐺)𝑦)))
41	28	adantr 484	. . . . . . . . . 10 ⊢ ((((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑆 ∈ 𝐽) ∧ 𝑥 ∈ ((Base‘𝐺) ∖ 𝑆)) ∧ 𝑦 ∈ 𝑆) → 𝐺 ∈ Grp)
42	18	adantr 484	. . . . . . . . . 10 ⊢ ((((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑆 ∈ 𝐽) ∧ 𝑥 ∈ ((Base‘𝐺) ∖ 𝑆)) ∧ 𝑦 ∈ 𝑆) → 𝑥 ∈ (Base‘𝐺))
43	12	sselda 3915	. . . . . . . . . 10 ⊢ ((((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑆 ∈ 𝐽) ∧ 𝑥 ∈ ((Base‘𝐺) ∖ 𝑆)) ∧ 𝑦 ∈ 𝑆) → 𝑦 ∈ (Base‘𝐺))
44	1, 20	grpcl 18103	. . . . . . . . . 10 ⊢ ((𝐺 ∈ Grp ∧ 𝑥 ∈ (Base‘𝐺) ∧ 𝑦 ∈ (Base‘𝐺)) → (𝑥(+g‘𝐺)𝑦) ∈ (Base‘𝐺))
45	41, 42, 43, 44	syl3anc 1368	. . . . . . . . 9 ⊢ ((((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑆 ∈ 𝐽) ∧ 𝑥 ∈ ((Base‘𝐺) ∖ 𝑆)) ∧ 𝑦 ∈ 𝑆) → (𝑥(+g‘𝐺)𝑦) ∈ (Base‘𝐺))
46		eldifn 4055	. . . . . . . . . . 11 ⊢ (𝑥 ∈ ((Base‘𝐺) ∖ 𝑆) → ¬ 𝑥 ∈ 𝑆)
47	46	ad2antlr 726	. . . . . . . . . 10 ⊢ ((((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑆 ∈ 𝐽) ∧ 𝑥 ∈ ((Base‘𝐺) ∖ 𝑆)) ∧ 𝑦 ∈ 𝑆) → ¬ 𝑥 ∈ 𝑆)
48		eqid 2798	. . . . . . . . . . . . . . 15 ⊢ (-g‘𝐺) = (-g‘𝐺)
49	48	subgsubcl 18282	. . . . . . . . . . . . . 14 ⊢ ((𝑆 ∈ (SubGrp‘𝐺) ∧ (𝑥(+g‘𝐺)𝑦) ∈ 𝑆 ∧ 𝑦 ∈ 𝑆) → ((𝑥(+g‘𝐺)𝑦)(-g‘𝐺)𝑦) ∈ 𝑆)
50	49	3com23 1123	. . . . . . . . . . . . 13 ⊢ ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑦 ∈ 𝑆 ∧ (𝑥(+g‘𝐺)𝑦) ∈ 𝑆) → ((𝑥(+g‘𝐺)𝑦)(-g‘𝐺)𝑦) ∈ 𝑆)
51	50	3expia 1118	. . . . . . . . . . . 12 ⊢ ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑦 ∈ 𝑆) → ((𝑥(+g‘𝐺)𝑦) ∈ 𝑆 → ((𝑥(+g‘𝐺)𝑦)(-g‘𝐺)𝑦) ∈ 𝑆))
52	32, 51	sylan 583	. . . . . . . . . . 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 18182	. . . . . . . . . . . . 13 ⊢ ((𝐺 ∈ Grp ∧ 𝑥 ∈ (Base‘𝐺) ∧ 𝑦 ∈ (Base‘𝐺)) → ((𝑥(+g‘𝐺)𝑦)(-g‘𝐺)𝑦) = 𝑥)
54	41, 42, 43, 53	syl3anc 1368	. . . . . . . . . . . 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 2874	. . . . . . . . . . 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 242	. . . . . . . . . 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 201	. . . . . . . . 9 ⊢ ((((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑆 ∈ 𝐽) ∧ 𝑥 ∈ ((Base‘𝐺) ∖ 𝑆)) ∧ 𝑦 ∈ 𝑆) → ¬ (𝑥(+g‘𝐺)𝑦) ∈ 𝑆)
58	45, 57	eldifd 3892	. . . . . . . 8 ⊢ ((((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑆 ∈ 𝐽) ∧ 𝑥 ∈ ((Base‘𝐺) ∖ 𝑆)) ∧ 𝑦 ∈ 𝑆) → (𝑥(+g‘𝐺)𝑦) ∈ ((Base‘𝐺) ∖ 𝑆))
59	58	fmpttd 6856	. . . . . . 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 6494	. . . . . 6 ⊢ (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑆 ∈ 𝐽) ∧ 𝑥 ∈ ((Base‘𝐺) ∖ 𝑆)) → ran (𝑦 ∈ 𝑆 ↦ (𝑥(+g‘𝐺)𝑦)) ⊆ ((Base‘𝐺) ∖ 𝑆))
61		eleq2 2878	. . . . . . . 8 ⊢ (𝑢 = ran (𝑦 ∈ 𝑆 ↦ (𝑥(+g‘𝐺)𝑦)) → (𝑥 ∈ 𝑢 ↔ 𝑥 ∈ ran (𝑦 ∈ 𝑆 ↦ (𝑥(+g‘𝐺)𝑦))))
62		sseq1 3940	. . . . . . . 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 3571	. . . . . 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 835	. . . . 5 ⊢ (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑆 ∈ 𝐽) ∧ 𝑥 ∈ ((Base‘𝐺) ∖ 𝑆)) → ∃𝑢 ∈ 𝐽 (𝑥 ∈ 𝑢 ∧ 𝑢 ⊆ ((Base‘𝐺) ∖ 𝑆)))
66	65	ralrimiva 3149	. . . 4 ⊢ ((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑆 ∈ 𝐽) → ∀𝑥 ∈ ((Base‘𝐺) ∖ 𝑆)∃𝑢 ∈ 𝐽 (𝑥 ∈ 𝑢 ∧ 𝑢 ⊆ ((Base‘𝐺) ∖ 𝑆)))
67		topontop 21518	. . . . . 6 ⊢ (𝐽 ∈ (TopOn‘(Base‘𝐺)) → 𝐽 ∈ Top)
68	6, 67	syl 17	. . . . 5 ⊢ ((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑆 ∈ 𝐽) → 𝐽 ∈ Top)
69		eltop2 21580	. . . . 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 260	. . 3 ⊢ ((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑆 ∈ 𝐽) → ((Base‘𝐺) ∖ 𝑆) ∈ 𝐽)
72	10, 71	eqeltrrd 2891	. 2 ⊢ ((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑆 ∈ 𝐽) → (∪ 𝐽 ∖ 𝑆) ∈ 𝐽)
73		eqid 2798	. . . 4 ⊢ ∪ 𝐽 = ∪ 𝐽
74	73	iscld 21632	. . 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 712	1 ⊢ ((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑆 ∈ 𝐽) → 𝑆 ∈ (Clsd‘𝐽))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 399   ∧ w3a 1084   = wceq 1538   ∈ wcel 2111  ∀wral 3106  ∃wrex 3107  Vcvv 3441   ∖ cdif 3878   ⊆ wss 3881  ∪ cuni 4800   ↦ cmpt 5110  ran crn 5520   ↾ cres 5521   “ cima 5522  ‘cfv 6324  (class class class)co 7135  Basecbs 16475  +_gcplusg 16557  TopOpenctopn 16687  0_gc0g 16705  Grpcgrp 18095  -_gcsg 18097  SubGrpcsubg 18265  Topctop 21498  TopOnctopon 21515  Clsdccld 21621  Homeochmeo 22358  TopGrpctgp 22676

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 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-rep 5154  ax-sep 5167  ax-nul 5174  ax-pow 5231  ax-pr 5295  ax-un 7441  ax-cnex 10582  ax-resscn 10583  ax-1cn 10584  ax-icn 10585  ax-addcl 10586  ax-addrcl 10587  ax-mulcl 10588  ax-mulrcl 10589  ax-mulcom 10590  ax-addass 10591  ax-mulass 10592  ax-distr 10593  ax-i2m1 10594  ax-1ne0 10595  ax-1rid 10596  ax-rnegex 10597  ax-rrecex 10598  ax-cnre 10599  ax-pre-lttri 10600  ax-pre-lttrn 10601  ax-pre-ltadd 10602  ax-pre-mulgt0 10603

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-nel 3092  df-ral 3111  df-rex 3112  df-reu 3113  df-rmo 3114  df-rab 3115  df-v 3443  df-sbc 3721  df-csb 3829  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-pss 3900  df-nul 4244  df-if 4426  df-pw 4499  df-sn 4526  df-pr 4528  df-tp 4530  df-op 4532  df-uni 4801  df-iun 4883  df-br 5031  df-opab 5093  df-mpt 5111  df-tr 5137  df-id 5425  df-eprel 5430  df-po 5438  df-so 5439  df-fr 5478  df-we 5480  df-xp 5525  df-rel 5526  df-cnv 5527  df-co 5528  df-dm 5529  df-rn 5530  df-res 5531  df-ima 5532  df-pred 6116  df-ord 6162  df-on 6163  df-lim 6164  df-suc 6165  df-iota 6283  df-fun 6326  df-fn 6327  df-f 6328  df-f1 6329  df-fo 6330  df-f1o 6331  df-fv 6332  df-riota 7093  df-ov 7138  df-oprab 7139  df-mpo 7140  df-om 7561  df-1st 7671  df-2nd 7672  df-wrecs 7930  df-recs 7991  df-rdg 8029  df-er 8272  df-map 8391  df-en 8493  df-dom 8494  df-sdom 8495  df-pnf 10666  df-mnf 10667  df-xr 10668  df-ltxr 10669  df-le 10670  df-sub 10861  df-neg 10862  df-nn 11626  df-2 11688  df-ndx 16478  df-slot 16479  df-base 16481  df-sets 16482  df-ress 16483  df-plusg 16570  df-0g 16707  df-topgen 16709  df-plusf 17843  df-mgm 17844  df-sgrp 17893  df-mnd 17904  df-grp 18098  df-minusg 18099  df-sbg 18100  df-subg 18268  df-top 21499  df-topon 21516  df-topsp 21538  df-bases 21551  df-cld 21624  df-cn 21832  df-cnp 21833  df-tx 22167  df-hmeo 22360  df-tmd 22677  df-tgp 22678

This theorem is referenced by:  cldsubg  22716  tgpconncompss  22719