Theorem opnsubg 23167

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 2738	. . . . 5 ⊢ (Base‘𝐺) = (Base‘𝐺)
2	1	subgss 18671	. . . 4 ⊢ (𝑆 ∈ (SubGrp‘𝐺) → 𝑆 ⊆ (Base‘𝐺))
3	2	3ad2ant2 1132	. . 3 ⊢ ((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑆 ∈ 𝐽) → 𝑆 ⊆ (Base‘𝐺))
4		subgntr.h	. . . . . 6 ⊢ 𝐽 = (TopOpen‘𝐺)
5	4, 1	tgptopon 23141	. . . . 5 ⊢ (𝐺 ∈ TopGrp → 𝐽 ∈ (TopOn‘(Base‘𝐺)))
6	5	3ad2ant1 1131	. . . 4 ⊢ ((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑆 ∈ 𝐽) → 𝐽 ∈ (TopOn‘(Base‘𝐺)))
7		toponuni 21971	. . . 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 3957	. 2 ⊢ ((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑆 ∈ 𝐽) → 𝑆 ⊆ ∪ 𝐽)
10	8	difeq1d 4052	. . 3 ⊢ ((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑆 ∈ 𝐽) → ((Base‘𝐺) ∖ 𝑆) = (∪ 𝐽 ∖ 𝑆))
11		df-ima 5593	. . . . . . . 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 5937	. . . . . . . . 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 5836	. . . . . . . 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 2790	. . . . . . 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 1189	. . . . . . . . 9 ⊢ (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑆 ∈ 𝐽) ∧ 𝑥 ∈ ((Base‘𝐺) ∖ 𝑆)) → 𝐺 ∈ TopGrp)
17		eldifi 4057	. . . . . . . . . 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 2738	. . . . . . . . . 10 ⊢ (𝑦 ∈ (Base‘𝐺) ↦ (𝑥(+g‘𝐺)𝑦)) = (𝑦 ∈ (Base‘𝐺) ↦ (𝑥(+g‘𝐺)𝑦))
20		eqid 2738	. . . . . . . . . 10 ⊢ (+g‘𝐺) = (+g‘𝐺)
21	19, 1, 20, 4	tgplacthmeo 23162	. . . . . . . . 9 ⊢ ((𝐺 ∈ TopGrp ∧ 𝑥 ∈ (Base‘𝐺)) → (𝑦 ∈ (Base‘𝐺) ↦ (𝑥(+g‘𝐺)𝑦)) ∈ (𝐽Homeo𝐽))
22	16, 18, 21	syl2anc 583	. . . . . . . 8 ⊢ (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑆 ∈ 𝐽) ∧ 𝑥 ∈ ((Base‘𝐺) ∖ 𝑆)) → (𝑦 ∈ (Base‘𝐺) ↦ (𝑥(+g‘𝐺)𝑦)) ∈ (𝐽Homeo𝐽))
23		simpl3 1191	. . . . . . . 8 ⊢ (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑆 ∈ 𝐽) ∧ 𝑥 ∈ ((Base‘𝐺) ∖ 𝑆)) → 𝑆 ∈ 𝐽)
24		hmeoima 22824	. . . . . . . 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 583	. . . . . . 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 2840	. . . . . 6 ⊢ (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑆 ∈ 𝐽) ∧ 𝑥 ∈ ((Base‘𝐺) ∖ 𝑆)) → ran (𝑦 ∈ 𝑆 ↦ (𝑥(+g‘𝐺)𝑦)) ∈ 𝐽)
27		tgpgrp 23137	. . . . . . . . 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 2738	. . . . . . . . 9 ⊢ (0g‘𝐺) = (0g‘𝐺)
30	1, 20, 29	grprid 18525	. . . . . . . 8 ⊢ ((𝐺 ∈ Grp ∧ 𝑥 ∈ (Base‘𝐺)) → (𝑥(+g‘𝐺)(0g‘𝐺)) = 𝑥)
31	28, 18, 30	syl2anc 583	. . . . . . 7 ⊢ (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑆 ∈ 𝐽) ∧ 𝑥 ∈ ((Base‘𝐺) ∖ 𝑆)) → (𝑥(+g‘𝐺)(0g‘𝐺)) = 𝑥)
32		simpl2 1190	. . . . . . . . 9 ⊢ (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑆 ∈ 𝐽) ∧ 𝑥 ∈ ((Base‘𝐺) ∖ 𝑆)) → 𝑆 ∈ (SubGrp‘𝐺))
33	29	subg0cl 18678	. . . . . . . . 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 7288	. . . . . . . 8 ⊢ (𝑥(+g‘𝐺)(0g‘𝐺)) ∈ V
36		eqid 2738	. . . . . . . . 9 ⊢ (𝑦 ∈ 𝑆 ↦ (𝑥(+g‘𝐺)𝑦)) = (𝑦 ∈ 𝑆 ↦ (𝑥(+g‘𝐺)𝑦))
37		oveq2 7263	. . . . . . . . 9 ⊢ (𝑦 = (0g‘𝐺) → (𝑥(+g‘𝐺)𝑦) = (𝑥(+g‘𝐺)(0g‘𝐺)))
38	36, 37	elrnmpt1s 5855	. . . . . . . 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 585	. . . . . . 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 2840	. . . . . 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 3917	. . . . . . . . . 10 ⊢ ((((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑆 ∈ 𝐽) ∧ 𝑥 ∈ ((Base‘𝐺) ∖ 𝑆)) ∧ 𝑦 ∈ 𝑆) → 𝑦 ∈ (Base‘𝐺))
44	1, 20	grpcl 18500	. . . . . . . . . 10 ⊢ ((𝐺 ∈ Grp ∧ 𝑥 ∈ (Base‘𝐺) ∧ 𝑦 ∈ (Base‘𝐺)) → (𝑥(+g‘𝐺)𝑦) ∈ (Base‘𝐺))
45	41, 42, 43, 44	syl3anc 1369	. . . . . . . . 9 ⊢ ((((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑆 ∈ 𝐽) ∧ 𝑥 ∈ ((Base‘𝐺) ∖ 𝑆)) ∧ 𝑦 ∈ 𝑆) → (𝑥(+g‘𝐺)𝑦) ∈ (Base‘𝐺))
46		eldifn 4058	. . . . . . . . . . 11 ⊢ (𝑥 ∈ ((Base‘𝐺) ∖ 𝑆) → ¬ 𝑥 ∈ 𝑆)
47	46	ad2antlr 723	. . . . . . . . . 10 ⊢ ((((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑆 ∈ 𝐽) ∧ 𝑥 ∈ ((Base‘𝐺) ∖ 𝑆)) ∧ 𝑦 ∈ 𝑆) → ¬ 𝑥 ∈ 𝑆)
48		eqid 2738	. . . . . . . . . . . . . . 15 ⊢ (-g‘𝐺) = (-g‘𝐺)
49	48	subgsubcl 18681	. . . . . . . . . . . . . 14 ⊢ ((𝑆 ∈ (SubGrp‘𝐺) ∧ (𝑥(+g‘𝐺)𝑦) ∈ 𝑆 ∧ 𝑦 ∈ 𝑆) → ((𝑥(+g‘𝐺)𝑦)(-g‘𝐺)𝑦) ∈ 𝑆)
50	49	3com23 1124	. . . . . . . . . . . . 13 ⊢ ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑦 ∈ 𝑆 ∧ (𝑥(+g‘𝐺)𝑦) ∈ 𝑆) → ((𝑥(+g‘𝐺)𝑦)(-g‘𝐺)𝑦) ∈ 𝑆)
51	50	3expia 1119	. . . . . . . . . . . 12 ⊢ ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑦 ∈ 𝑆) → ((𝑥(+g‘𝐺)𝑦) ∈ 𝑆 → ((𝑥(+g‘𝐺)𝑦)(-g‘𝐺)𝑦) ∈ 𝑆))
52	32, 51	sylan 579	. . . . . . . . . . 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 18581	. . . . . . . . . . . . 13 ⊢ ((𝐺 ∈ Grp ∧ 𝑥 ∈ (Base‘𝐺) ∧ 𝑦 ∈ (Base‘𝐺)) → ((𝑥(+g‘𝐺)𝑦)(-g‘𝐺)𝑦) = 𝑥)
54	41, 42, 43, 53	syl3anc 1369	. . . . . . . . . . . 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 2823	. . . . . . . . . . 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 238	. . . . . . . . . 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 197	. . . . . . . . 9 ⊢ ((((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑆 ∈ 𝐽) ∧ 𝑥 ∈ ((Base‘𝐺) ∖ 𝑆)) ∧ 𝑦 ∈ 𝑆) → ¬ (𝑥(+g‘𝐺)𝑦) ∈ 𝑆)
58	45, 57	eldifd 3894	. . . . . . . 8 ⊢ ((((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑆 ∈ 𝐽) ∧ 𝑥 ∈ ((Base‘𝐺) ∖ 𝑆)) ∧ 𝑦 ∈ 𝑆) → (𝑥(+g‘𝐺)𝑦) ∈ ((Base‘𝐺) ∖ 𝑆))
59	58	fmpttd 6971	. . . . . . 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 6592	. . . . . 6 ⊢ (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑆 ∈ 𝐽) ∧ 𝑥 ∈ ((Base‘𝐺) ∖ 𝑆)) → ran (𝑦 ∈ 𝑆 ↦ (𝑥(+g‘𝐺)𝑦)) ⊆ ((Base‘𝐺) ∖ 𝑆))
61		eleq2 2827	. . . . . . . 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 630	. . . . . . 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 3552	. . . . . 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 833	. . . . 5 ⊢ (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑆 ∈ 𝐽) ∧ 𝑥 ∈ ((Base‘𝐺) ∖ 𝑆)) → ∃𝑢 ∈ 𝐽 (𝑥 ∈ 𝑢 ∧ 𝑢 ⊆ ((Base‘𝐺) ∖ 𝑆)))
66	65	ralrimiva 3107	. . . 4 ⊢ ((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑆 ∈ 𝐽) → ∀𝑥 ∈ ((Base‘𝐺) ∖ 𝑆)∃𝑢 ∈ 𝐽 (𝑥 ∈ 𝑢 ∧ 𝑢 ⊆ ((Base‘𝐺) ∖ 𝑆)))
67		topontop 21970	. . . . . 6 ⊢ (𝐽 ∈ (TopOn‘(Base‘𝐺)) → 𝐽 ∈ Top)
68	6, 67	syl 17	. . . . 5 ⊢ ((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑆 ∈ 𝐽) → 𝐽 ∈ Top)
69		eltop2 22033	. . . . 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 256	. . 3 ⊢ ((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑆 ∈ 𝐽) → ((Base‘𝐺) ∖ 𝑆) ∈ 𝐽)
72	10, 71	eqeltrrd 2840	. 2 ⊢ ((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑆 ∈ 𝐽) → (∪ 𝐽 ∖ 𝑆) ∈ 𝐽)
73		eqid 2738	. . . 4 ⊢ ∪ 𝐽 = ∪ 𝐽
74	73	iscld 22086	. . 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 709	1 ⊢ ((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑆 ∈ 𝐽) → 𝑆 ∈ (Clsd‘𝐽))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 205   ∧ wa 395   ∧ w3a 1085   = wceq 1539   ∈ wcel 2108  ∀wral 3063  ∃wrex 3064  Vcvv 3422   ∖ cdif 3880   ⊆ wss 3883  ∪ cuni 4836   ↦ cmpt 5153  ran crn 5581   ↾ cres 5582   “ cima 5583  ‘cfv 6418  (class class class)co 7255  Basecbs 16840  +_gcplusg 16888  TopOpenctopn 17049  0_gc0g 17067  Grpcgrp 18492  -_gcsg 18494  SubGrpcsubg 18664  Topctop 21950  TopOnctopon 21967  Clsdccld 22075  Homeochmeo 22812  TopGrpctgp 23130

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709  ax-rep 5205  ax-sep 5218  ax-nul 5225  ax-pow 5283  ax-pr 5347  ax-un 7566  ax-cnex 10858  ax-resscn 10859  ax-1cn 10860  ax-icn 10861  ax-addcl 10862  ax-addrcl 10863  ax-mulcl 10864  ax-mulrcl 10865  ax-mulcom 10866  ax-addass 10867  ax-mulass 10868  ax-distr 10869  ax-i2m1 10870  ax-1ne0 10871  ax-1rid 10872  ax-rnegex 10873  ax-rrecex 10874  ax-cnre 10875  ax-pre-lttri 10876  ax-pre-lttrn 10877  ax-pre-ltadd 10878  ax-pre-mulgt0 10879

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3or 1086  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-nf 1788  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2888  df-ne 2943  df-nel 3049  df-ral 3068  df-rex 3069  df-reu 3070  df-rmo 3071  df-rab 3072  df-v 3424  df-sbc 3712  df-csb 3829  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-pss 3902  df-nul 4254  df-if 4457  df-pw 4532  df-sn 4559  df-pr 4561  df-tp 4563  df-op 4565  df-uni 4837  df-iun 4923  df-br 5071  df-opab 5133  df-mpt 5154  df-tr 5188  df-id 5480  df-eprel 5486  df-po 5494  df-so 5495  df-fr 5535  df-we 5537  df-xp 5586  df-rel 5587  df-cnv 5588  df-co 5589  df-dm 5590  df-rn 5591  df-res 5592  df-ima 5593  df-pred 6191  df-ord 6254  df-on 6255  df-lim 6256  df-suc 6257  df-iota 6376  df-fun 6420  df-fn 6421  df-f 6422  df-f1 6423  df-fo 6424  df-f1o 6425  df-fv 6426  df-riota 7212  df-ov 7258  df-oprab 7259  df-mpo 7260  df-om 7688  df-1st 7804  df-2nd 7805  df-frecs 8068  df-wrecs 8099  df-recs 8173  df-rdg 8212  df-er 8456  df-map 8575  df-en 8692  df-dom 8693  df-sdom 8694  df-pnf 10942  df-mnf 10943  df-xr 10944  df-ltxr 10945  df-le 10946  df-sub 11137  df-neg 11138  df-nn 11904  df-2 11966  df-sets 16793  df-slot 16811  df-ndx 16823  df-base 16841  df-ress 16868  df-plusg 16901  df-0g 17069  df-topgen 17071  df-plusf 18240  df-mgm 18241  df-sgrp 18290  df-mnd 18301  df-grp 18495  df-minusg 18496  df-sbg 18497  df-subg 18667  df-top 21951  df-topon 21968  df-topsp 21990  df-bases 22004  df-cld 22078  df-cn 22286  df-cnp 22287  df-tx 22621  df-hmeo 22814  df-tmd 23131  df-tgp 23132

This theorem is referenced by:  cldsubg  23170  tgpconncompss  23173