Theorem subgntr 24072

Description: A subgroup of a topological group with nonempty interior is open. Alternatively, dual to clssubg 24074, the interior of a subgroup is either a subgroup, or empty. (Contributed by Mario Carneiro, 19-Sep-2015.)

Hypothesis

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

Assertion

Ref	Expression
subgntr	⊢ ((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴 ∈ ((int‘𝐽)‘𝑆)) → 𝑆 ∈ 𝐽)

Proof of Theorem subgntr

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

Step	Hyp	Ref	Expression
1		df-ima 5644	. . . . . 6 ⊢ ((𝑦 ∈ (Base‘𝐺) ↦ ((𝑥(-g‘𝐺)𝐴)(+g‘𝐺)𝑦)) “ ((int‘𝐽)‘𝑆)) = ran ((𝑦 ∈ (Base‘𝐺) ↦ ((𝑥(-g‘𝐺)𝐴)(+g‘𝐺)𝑦)) ↾ ((int‘𝐽)‘𝑆))
2		subgntr.h	. . . . . . . . . . . 12 ⊢ 𝐽 = (TopOpen‘𝐺)
3		eqid 2736	. . . . . . . . . . . 12 ⊢ (Base‘𝐺) = (Base‘𝐺)
4	2, 3	tgptopon 24047	. . . . . . . . . . 11 ⊢ (𝐺 ∈ TopGrp → 𝐽 ∈ (TopOn‘(Base‘𝐺)))
5	4	3ad2ant1 1134	. . . . . . . . . 10 ⊢ ((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴 ∈ ((int‘𝐽)‘𝑆)) → 𝐽 ∈ (TopOn‘(Base‘𝐺)))
6	5	adantr 480	. . . . . . . . 9 ⊢ (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴 ∈ ((int‘𝐽)‘𝑆)) ∧ 𝑥 ∈ 𝑆) → 𝐽 ∈ (TopOn‘(Base‘𝐺)))
7		topontop 22878	. . . . . . . . . . . 12 ⊢ (𝐽 ∈ (TopOn‘(Base‘𝐺)) → 𝐽 ∈ Top)
8	5, 7	syl 17	. . . . . . . . . . 11 ⊢ ((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴 ∈ ((int‘𝐽)‘𝑆)) → 𝐽 ∈ Top)
9	8	adantr 480	. . . . . . . . . 10 ⊢ (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴 ∈ ((int‘𝐽)‘𝑆)) ∧ 𝑥 ∈ 𝑆) → 𝐽 ∈ Top)
10		simpl2 1194	. . . . . . . . . . . 12 ⊢ (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴 ∈ ((int‘𝐽)‘𝑆)) ∧ 𝑥 ∈ 𝑆) → 𝑆 ∈ (SubGrp‘𝐺))
11	3	subgss 19103	. . . . . . . . . . . 12 ⊢ (𝑆 ∈ (SubGrp‘𝐺) → 𝑆 ⊆ (Base‘𝐺))
12	10, 11	syl 17	. . . . . . . . . . 11 ⊢ (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴 ∈ ((int‘𝐽)‘𝑆)) ∧ 𝑥 ∈ 𝑆) → 𝑆 ⊆ (Base‘𝐺))
13		toponuni 22879	. . . . . . . . . . . 12 ⊢ (𝐽 ∈ (TopOn‘(Base‘𝐺)) → (Base‘𝐺) = ∪ 𝐽)
14	6, 13	syl 17	. . . . . . . . . . 11 ⊢ (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴 ∈ ((int‘𝐽)‘𝑆)) ∧ 𝑥 ∈ 𝑆) → (Base‘𝐺) = ∪ 𝐽)
15	12, 14	sseqtrd 3958	. . . . . . . . . 10 ⊢ (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴 ∈ ((int‘𝐽)‘𝑆)) ∧ 𝑥 ∈ 𝑆) → 𝑆 ⊆ ∪ 𝐽)
16		eqid 2736	. . . . . . . . . . 11 ⊢ ∪ 𝐽 = ∪ 𝐽
17	16	ntropn 23014	. . . . . . . . . 10 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ ∪ 𝐽) → ((int‘𝐽)‘𝑆) ∈ 𝐽)
18	9, 15, 17	syl2anc 585	. . . . . . . . 9 ⊢ (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴 ∈ ((int‘𝐽)‘𝑆)) ∧ 𝑥 ∈ 𝑆) → ((int‘𝐽)‘𝑆) ∈ 𝐽)
19		toponss 22892	. . . . . . . . 9 ⊢ ((𝐽 ∈ (TopOn‘(Base‘𝐺)) ∧ ((int‘𝐽)‘𝑆) ∈ 𝐽) → ((int‘𝐽)‘𝑆) ⊆ (Base‘𝐺))
20	6, 18, 19	syl2anc 585	. . . . . . . 8 ⊢ (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴 ∈ ((int‘𝐽)‘𝑆)) ∧ 𝑥 ∈ 𝑆) → ((int‘𝐽)‘𝑆) ⊆ (Base‘𝐺))
21	20	resmptd 6005	. . . . . . 7 ⊢ (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴 ∈ ((int‘𝐽)‘𝑆)) ∧ 𝑥 ∈ 𝑆) → ((𝑦 ∈ (Base‘𝐺) ↦ ((𝑥(-g‘𝐺)𝐴)(+g‘𝐺)𝑦)) ↾ ((int‘𝐽)‘𝑆)) = (𝑦 ∈ ((int‘𝐽)‘𝑆) ↦ ((𝑥(-g‘𝐺)𝐴)(+g‘𝐺)𝑦)))
22	21	rneqd 5893	. . . . . 6 ⊢ (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴 ∈ ((int‘𝐽)‘𝑆)) ∧ 𝑥 ∈ 𝑆) → ran ((𝑦 ∈ (Base‘𝐺) ↦ ((𝑥(-g‘𝐺)𝐴)(+g‘𝐺)𝑦)) ↾ ((int‘𝐽)‘𝑆)) = ran (𝑦 ∈ ((int‘𝐽)‘𝑆) ↦ ((𝑥(-g‘𝐺)𝐴)(+g‘𝐺)𝑦)))
23	1, 22	eqtrid 2783	. . . . 5 ⊢ (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴 ∈ ((int‘𝐽)‘𝑆)) ∧ 𝑥 ∈ 𝑆) → ((𝑦 ∈ (Base‘𝐺) ↦ ((𝑥(-g‘𝐺)𝐴)(+g‘𝐺)𝑦)) “ ((int‘𝐽)‘𝑆)) = ran (𝑦 ∈ ((int‘𝐽)‘𝑆) ↦ ((𝑥(-g‘𝐺)𝐴)(+g‘𝐺)𝑦)))
24		simpl1 1193	. . . . . . 7 ⊢ (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴 ∈ ((int‘𝐽)‘𝑆)) ∧ 𝑥 ∈ 𝑆) → 𝐺 ∈ TopGrp)
25		simpr 484	. . . . . . . . 9 ⊢ (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴 ∈ ((int‘𝐽)‘𝑆)) ∧ 𝑥 ∈ 𝑆) → 𝑥 ∈ 𝑆)
26	16	ntrss2 23022	. . . . . . . . . . 11 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ ∪ 𝐽) → ((int‘𝐽)‘𝑆) ⊆ 𝑆)
27	9, 15, 26	syl2anc 585	. . . . . . . . . 10 ⊢ (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴 ∈ ((int‘𝐽)‘𝑆)) ∧ 𝑥 ∈ 𝑆) → ((int‘𝐽)‘𝑆) ⊆ 𝑆)
28		simpl3 1195	. . . . . . . . . 10 ⊢ (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴 ∈ ((int‘𝐽)‘𝑆)) ∧ 𝑥 ∈ 𝑆) → 𝐴 ∈ ((int‘𝐽)‘𝑆))
29	27, 28	sseldd 3922	. . . . . . . . 9 ⊢ (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴 ∈ ((int‘𝐽)‘𝑆)) ∧ 𝑥 ∈ 𝑆) → 𝐴 ∈ 𝑆)
30		eqid 2736	. . . . . . . . . 10 ⊢ (-g‘𝐺) = (-g‘𝐺)
31	30	subgsubcl 19113	. . . . . . . . 9 ⊢ ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑥 ∈ 𝑆 ∧ 𝐴 ∈ 𝑆) → (𝑥(-g‘𝐺)𝐴) ∈ 𝑆)
32	10, 25, 29, 31	syl3anc 1374	. . . . . . . 8 ⊢ (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴 ∈ ((int‘𝐽)‘𝑆)) ∧ 𝑥 ∈ 𝑆) → (𝑥(-g‘𝐺)𝐴) ∈ 𝑆)
33	12, 32	sseldd 3922	. . . . . . 7 ⊢ (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴 ∈ ((int‘𝐽)‘𝑆)) ∧ 𝑥 ∈ 𝑆) → (𝑥(-g‘𝐺)𝐴) ∈ (Base‘𝐺))
34		eqid 2736	. . . . . . . 8 ⊢ (𝑦 ∈ (Base‘𝐺) ↦ ((𝑥(-g‘𝐺)𝐴)(+g‘𝐺)𝑦)) = (𝑦 ∈ (Base‘𝐺) ↦ ((𝑥(-g‘𝐺)𝐴)(+g‘𝐺)𝑦))
35		eqid 2736	. . . . . . . 8 ⊢ (+g‘𝐺) = (+g‘𝐺)
36	34, 3, 35, 2	tgplacthmeo 24068	. . . . . . 7 ⊢ ((𝐺 ∈ TopGrp ∧ (𝑥(-g‘𝐺)𝐴) ∈ (Base‘𝐺)) → (𝑦 ∈ (Base‘𝐺) ↦ ((𝑥(-g‘𝐺)𝐴)(+g‘𝐺)𝑦)) ∈ (𝐽Homeo𝐽))
37	24, 33, 36	syl2anc 585	. . . . . 6 ⊢ (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴 ∈ ((int‘𝐽)‘𝑆)) ∧ 𝑥 ∈ 𝑆) → (𝑦 ∈ (Base‘𝐺) ↦ ((𝑥(-g‘𝐺)𝐴)(+g‘𝐺)𝑦)) ∈ (𝐽Homeo𝐽))
38		hmeoima 23730	. . . . . 6 ⊢ (((𝑦 ∈ (Base‘𝐺) ↦ ((𝑥(-g‘𝐺)𝐴)(+g‘𝐺)𝑦)) ∈ (𝐽Homeo𝐽) ∧ ((int‘𝐽)‘𝑆) ∈ 𝐽) → ((𝑦 ∈ (Base‘𝐺) ↦ ((𝑥(-g‘𝐺)𝐴)(+g‘𝐺)𝑦)) “ ((int‘𝐽)‘𝑆)) ∈ 𝐽)
39	37, 18, 38	syl2anc 585	. . . . 5 ⊢ (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴 ∈ ((int‘𝐽)‘𝑆)) ∧ 𝑥 ∈ 𝑆) → ((𝑦 ∈ (Base‘𝐺) ↦ ((𝑥(-g‘𝐺)𝐴)(+g‘𝐺)𝑦)) “ ((int‘𝐽)‘𝑆)) ∈ 𝐽)
40	23, 39	eqeltrrd 2837	. . . 4 ⊢ (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴 ∈ ((int‘𝐽)‘𝑆)) ∧ 𝑥 ∈ 𝑆) → ran (𝑦 ∈ ((int‘𝐽)‘𝑆) ↦ ((𝑥(-g‘𝐺)𝐴)(+g‘𝐺)𝑦)) ∈ 𝐽)
41		tgpgrp 24043	. . . . . . 7 ⊢ (𝐺 ∈ TopGrp → 𝐺 ∈ Grp)
42	24, 41	syl 17	. . . . . 6 ⊢ (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴 ∈ ((int‘𝐽)‘𝑆)) ∧ 𝑥 ∈ 𝑆) → 𝐺 ∈ Grp)
43	11	3ad2ant2 1135	. . . . . . 7 ⊢ ((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴 ∈ ((int‘𝐽)‘𝑆)) → 𝑆 ⊆ (Base‘𝐺))
44	43	sselda 3921	. . . . . 6 ⊢ (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴 ∈ ((int‘𝐽)‘𝑆)) ∧ 𝑥 ∈ 𝑆) → 𝑥 ∈ (Base‘𝐺))
45	20, 28	sseldd 3922	. . . . . 6 ⊢ (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴 ∈ ((int‘𝐽)‘𝑆)) ∧ 𝑥 ∈ 𝑆) → 𝐴 ∈ (Base‘𝐺))
46	3, 35, 30	grpnpcan 19008	. . . . . 6 ⊢ ((𝐺 ∈ Grp ∧ 𝑥 ∈ (Base‘𝐺) ∧ 𝐴 ∈ (Base‘𝐺)) → ((𝑥(-g‘𝐺)𝐴)(+g‘𝐺)𝐴) = 𝑥)
47	42, 44, 45, 46	syl3anc 1374	. . . . 5 ⊢ (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴 ∈ ((int‘𝐽)‘𝑆)) ∧ 𝑥 ∈ 𝑆) → ((𝑥(-g‘𝐺)𝐴)(+g‘𝐺)𝐴) = 𝑥)
48		ovex 7400	. . . . . 6 ⊢ ((𝑥(-g‘𝐺)𝐴)(+g‘𝐺)𝐴) ∈ V
49		eqid 2736	. . . . . . 7 ⊢ (𝑦 ∈ ((int‘𝐽)‘𝑆) ↦ ((𝑥(-g‘𝐺)𝐴)(+g‘𝐺)𝑦)) = (𝑦 ∈ ((int‘𝐽)‘𝑆) ↦ ((𝑥(-g‘𝐺)𝐴)(+g‘𝐺)𝑦))
50		oveq2 7375	. . . . . . 7 ⊢ (𝑦 = 𝐴 → ((𝑥(-g‘𝐺)𝐴)(+g‘𝐺)𝑦) = ((𝑥(-g‘𝐺)𝐴)(+g‘𝐺)𝐴))
51	49, 50	elrnmpt1s 5914	. . . . . 6 ⊢ ((𝐴 ∈ ((int‘𝐽)‘𝑆) ∧ ((𝑥(-g‘𝐺)𝐴)(+g‘𝐺)𝐴) ∈ V) → ((𝑥(-g‘𝐺)𝐴)(+g‘𝐺)𝐴) ∈ ran (𝑦 ∈ ((int‘𝐽)‘𝑆) ↦ ((𝑥(-g‘𝐺)𝐴)(+g‘𝐺)𝑦)))
52	28, 48, 51	sylancl 587	. . . . 5 ⊢ (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴 ∈ ((int‘𝐽)‘𝑆)) ∧ 𝑥 ∈ 𝑆) → ((𝑥(-g‘𝐺)𝐴)(+g‘𝐺)𝐴) ∈ ran (𝑦 ∈ ((int‘𝐽)‘𝑆) ↦ ((𝑥(-g‘𝐺)𝐴)(+g‘𝐺)𝑦)))
53	47, 52	eqeltrrd 2837	. . . 4 ⊢ (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴 ∈ ((int‘𝐽)‘𝑆)) ∧ 𝑥 ∈ 𝑆) → 𝑥 ∈ ran (𝑦 ∈ ((int‘𝐽)‘𝑆) ↦ ((𝑥(-g‘𝐺)𝐴)(+g‘𝐺)𝑦)))
54	10	adantr 480	. . . . . . 7 ⊢ ((((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴 ∈ ((int‘𝐽)‘𝑆)) ∧ 𝑥 ∈ 𝑆) ∧ 𝑦 ∈ ((int‘𝐽)‘𝑆)) → 𝑆 ∈ (SubGrp‘𝐺))
55	32	adantr 480	. . . . . . 7 ⊢ ((((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴 ∈ ((int‘𝐽)‘𝑆)) ∧ 𝑥 ∈ 𝑆) ∧ 𝑦 ∈ ((int‘𝐽)‘𝑆)) → (𝑥(-g‘𝐺)𝐴) ∈ 𝑆)
56	27	sselda 3921	. . . . . . 7 ⊢ ((((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴 ∈ ((int‘𝐽)‘𝑆)) ∧ 𝑥 ∈ 𝑆) ∧ 𝑦 ∈ ((int‘𝐽)‘𝑆)) → 𝑦 ∈ 𝑆)
57	35	subgcl 19112	. . . . . . 7 ⊢ ((𝑆 ∈ (SubGrp‘𝐺) ∧ (𝑥(-g‘𝐺)𝐴) ∈ 𝑆 ∧ 𝑦 ∈ 𝑆) → ((𝑥(-g‘𝐺)𝐴)(+g‘𝐺)𝑦) ∈ 𝑆)
58	54, 55, 56, 57	syl3anc 1374	. . . . . 6 ⊢ ((((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴 ∈ ((int‘𝐽)‘𝑆)) ∧ 𝑥 ∈ 𝑆) ∧ 𝑦 ∈ ((int‘𝐽)‘𝑆)) → ((𝑥(-g‘𝐺)𝐴)(+g‘𝐺)𝑦) ∈ 𝑆)
59	58	fmpttd 7067	. . . . 5 ⊢ (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴 ∈ ((int‘𝐽)‘𝑆)) ∧ 𝑥 ∈ 𝑆) → (𝑦 ∈ ((int‘𝐽)‘𝑆) ↦ ((𝑥(-g‘𝐺)𝐴)(+g‘𝐺)𝑦)):((int‘𝐽)‘𝑆)⟶𝑆)
60	59	frnd 6676	. . . 4 ⊢ (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴 ∈ ((int‘𝐽)‘𝑆)) ∧ 𝑥 ∈ 𝑆) → ran (𝑦 ∈ ((int‘𝐽)‘𝑆) ↦ ((𝑥(-g‘𝐺)𝐴)(+g‘𝐺)𝑦)) ⊆ 𝑆)
61		eleq2 2825	. . . . . 6 ⊢ (𝑢 = ran (𝑦 ∈ ((int‘𝐽)‘𝑆) ↦ ((𝑥(-g‘𝐺)𝐴)(+g‘𝐺)𝑦)) → (𝑥 ∈ 𝑢 ↔ 𝑥 ∈ ran (𝑦 ∈ ((int‘𝐽)‘𝑆) ↦ ((𝑥(-g‘𝐺)𝐴)(+g‘𝐺)𝑦))))
62		sseq1 3947	. . . . . 6 ⊢ (𝑢 = ran (𝑦 ∈ ((int‘𝐽)‘𝑆) ↦ ((𝑥(-g‘𝐺)𝐴)(+g‘𝐺)𝑦)) → (𝑢 ⊆ 𝑆 ↔ ran (𝑦 ∈ ((int‘𝐽)‘𝑆) ↦ ((𝑥(-g‘𝐺)𝐴)(+g‘𝐺)𝑦)) ⊆ 𝑆))
63	61, 62	anbi12d 633	. . . . 5 ⊢ (𝑢 = ran (𝑦 ∈ ((int‘𝐽)‘𝑆) ↦ ((𝑥(-g‘𝐺)𝐴)(+g‘𝐺)𝑦)) → ((𝑥 ∈ 𝑢 ∧ 𝑢 ⊆ 𝑆) ↔ (𝑥 ∈ ran (𝑦 ∈ ((int‘𝐽)‘𝑆) ↦ ((𝑥(-g‘𝐺)𝐴)(+g‘𝐺)𝑦)) ∧ ran (𝑦 ∈ ((int‘𝐽)‘𝑆) ↦ ((𝑥(-g‘𝐺)𝐴)(+g‘𝐺)𝑦)) ⊆ 𝑆)))
64	63	rspcev 3564	. . . 4 ⊢ ((ran (𝑦 ∈ ((int‘𝐽)‘𝑆) ↦ ((𝑥(-g‘𝐺)𝐴)(+g‘𝐺)𝑦)) ∈ 𝐽 ∧ (𝑥 ∈ ran (𝑦 ∈ ((int‘𝐽)‘𝑆) ↦ ((𝑥(-g‘𝐺)𝐴)(+g‘𝐺)𝑦)) ∧ ran (𝑦 ∈ ((int‘𝐽)‘𝑆) ↦ ((𝑥(-g‘𝐺)𝐴)(+g‘𝐺)𝑦)) ⊆ 𝑆)) → ∃𝑢 ∈ 𝐽 (𝑥 ∈ 𝑢 ∧ 𝑢 ⊆ 𝑆))
65	40, 53, 60, 64	syl12anc 837	. . 3 ⊢ (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴 ∈ ((int‘𝐽)‘𝑆)) ∧ 𝑥 ∈ 𝑆) → ∃𝑢 ∈ 𝐽 (𝑥 ∈ 𝑢 ∧ 𝑢 ⊆ 𝑆))
66	65	ralrimiva 3129	. 2 ⊢ ((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴 ∈ ((int‘𝐽)‘𝑆)) → ∀𝑥 ∈ 𝑆 ∃𝑢 ∈ 𝐽 (𝑥 ∈ 𝑢 ∧ 𝑢 ⊆ 𝑆))
67		eltop2 22940	. . 3 ⊢ (𝐽 ∈ Top → (𝑆 ∈ 𝐽 ↔ ∀𝑥 ∈ 𝑆 ∃𝑢 ∈ 𝐽 (𝑥 ∈ 𝑢 ∧ 𝑢 ⊆ 𝑆)))
68	8, 67	syl 17	. 2 ⊢ ((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴 ∈ ((int‘𝐽)‘𝑆)) → (𝑆 ∈ 𝐽 ↔ ∀𝑥 ∈ 𝑆 ∃𝑢 ∈ 𝐽 (𝑥 ∈ 𝑢 ∧ 𝑢 ⊆ 𝑆)))
69	66, 68	mpbird 257	1 ⊢ ((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴 ∈ ((int‘𝐽)‘𝑆)) → 𝑆 ∈ 𝐽)

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1087   = wceq 1542   ∈ wcel 2114  ∀wral 3051  ∃wrex 3061  Vcvv 3429   ⊆ wss 3889  ∪ cuni 4850   ↦ cmpt 5166  ran crn 5632   ↾ cres 5633   “ cima 5634  ‘cfv 6498  (class class class)co 7367  Basecbs 17179  +_gcplusg 17220  TopOpenctopn 17384  Grpcgrp 18909  -_gcsg 18911  SubGrpcsubg 19096  Topctop 22858  TopOnctopon 22875  intcnt 22982  Homeochmeo 23718  TopGrpctgp 24036

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 2708  ax-rep 5212  ax-sep 5231  ax-nul 5241  ax-pow 5307  ax-pr 5375  ax-un 7689  ax-cnex 11094  ax-resscn 11095  ax-1cn 11096  ax-icn 11097  ax-addcl 11098  ax-addrcl 11099  ax-mulcl 11100  ax-mulrcl 11101  ax-mulcom 11102  ax-addass 11103  ax-mulass 11104  ax-distr 11105  ax-i2m1 11106  ax-1ne0 11107  ax-1rid 11108  ax-rnegex 11109  ax-rrecex 11110  ax-cnre 11111  ax-pre-lttri 11112  ax-pre-lttrn 11113  ax-pre-ltadd 11114  ax-pre-mulgt0 11115

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 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-nel 3037  df-ral 3052  df-rex 3062  df-rmo 3342  df-reu 3343  df-rab 3390  df-v 3431  df-sbc 3729  df-csb 3838  df-dif 3892  df-un 3894  df-in 3896  df-ss 3906  df-pss 3909  df-nul 4274  df-if 4467  df-pw 4543  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4851  df-iun 4935  df-br 5086  df-opab 5148  df-mpt 5167  df-tr 5193  df-id 5526  df-eprel 5531  df-po 5539  df-so 5540  df-fr 5584  df-we 5586  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-pred 6265  df-ord 6326  df-on 6327  df-lim 6328  df-suc 6329  df-iota 6454  df-fun 6500  df-fn 6501  df-f 6502  df-f1 6503  df-fo 6504  df-f1o 6505  df-fv 6506  df-riota 7324  df-ov 7370  df-oprab 7371  df-mpo 7372  df-om 7818  df-1st 7942  df-2nd 7943  df-frecs 8231  df-wrecs 8262  df-recs 8311  df-rdg 8349  df-er 8643  df-map 8775  df-en 8894  df-dom 8895  df-sdom 8896  df-pnf 11181  df-mnf 11182  df-xr 11183  df-ltxr 11184  df-le 11185  df-sub 11379  df-neg 11380  df-nn 12175  df-2 12244  df-sets 17134  df-slot 17152  df-ndx 17164  df-base 17180  df-ress 17201  df-plusg 17233  df-0g 17404  df-topgen 17406  df-plusf 18607  df-mgm 18608  df-sgrp 18687  df-mnd 18703  df-grp 18912  df-minusg 18913  df-sbg 18914  df-subg 19099  df-top 22859  df-topon 22876  df-topsp 22898  df-bases 22911  df-ntr 22985  df-cn 23192  df-cnp 23193  df-tx 23527  df-hmeo 23720  df-tmd 24037  df-tgp 24038

This theorem is referenced by: (None)