Theorem subgntr 23931

Description: A subgroup of a topological group with nonempty interior is open. Alternatively, dual to clssubg 23933, 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 5689	. . . . . 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 2731	. . . . . . . . . . . 12 ⊢ (Base‘𝐺) = (Base‘𝐺)
4	2, 3	tgptopon 23906	. . . . . . . . . . 11 ⊢ (𝐺 ∈ TopGrp → 𝐽 ∈ (TopOn‘(Base‘𝐺)))
5	4	3ad2ant1 1132	. . . . . . . . . 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 22735	. . . . . . . . . . . 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 1191	. . . . . . . . . . . 12 ⊢ (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴 ∈ ((int‘𝐽)‘𝑆)) ∧ 𝑥 ∈ 𝑆) → 𝑆 ∈ (SubGrp‘𝐺))
11	3	subgss 19050	. . . . . . . . . . . 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 22736	. . . . . . . . . . . 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 4022	. . . . . . . . . 10 ⊢ (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴 ∈ ((int‘𝐽)‘𝑆)) ∧ 𝑥 ∈ 𝑆) → 𝑆 ⊆ ∪ 𝐽)
16		eqid 2731	. . . . . . . . . . 11 ⊢ ∪ 𝐽 = ∪ 𝐽
17	16	ntropn 22873	. . . . . . . . . 10 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ ∪ 𝐽) → ((int‘𝐽)‘𝑆) ∈ 𝐽)
18	9, 15, 17	syl2anc 583	. . . . . . . . 9 ⊢ (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴 ∈ ((int‘𝐽)‘𝑆)) ∧ 𝑥 ∈ 𝑆) → ((int‘𝐽)‘𝑆) ∈ 𝐽)
19		toponss 22749	. . . . . . . . 9 ⊢ ((𝐽 ∈ (TopOn‘(Base‘𝐺)) ∧ ((int‘𝐽)‘𝑆) ∈ 𝐽) → ((int‘𝐽)‘𝑆) ⊆ (Base‘𝐺))
20	6, 18, 19	syl2anc 583	. . . . . . . 8 ⊢ (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴 ∈ ((int‘𝐽)‘𝑆)) ∧ 𝑥 ∈ 𝑆) → ((int‘𝐽)‘𝑆) ⊆ (Base‘𝐺))
21	20	resmptd 6040	. . . . . . 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 5937	. . . . . 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 1190	. . . . . . 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 22881	. . . . . . . . . . 11 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ ∪ 𝐽) → ((int‘𝐽)‘𝑆) ⊆ 𝑆)
27	9, 15, 26	syl2anc 583	. . . . . . . . . 10 ⊢ (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴 ∈ ((int‘𝐽)‘𝑆)) ∧ 𝑥 ∈ 𝑆) → ((int‘𝐽)‘𝑆) ⊆ 𝑆)
28		simpl3 1192	. . . . . . . . . 10 ⊢ (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴 ∈ ((int‘𝐽)‘𝑆)) ∧ 𝑥 ∈ 𝑆) → 𝐴 ∈ ((int‘𝐽)‘𝑆))
29	27, 28	sseldd 3983	. . . . . . . . 9 ⊢ (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴 ∈ ((int‘𝐽)‘𝑆)) ∧ 𝑥 ∈ 𝑆) → 𝐴 ∈ 𝑆)
30		eqid 2731	. . . . . . . . . 10 ⊢ (-g‘𝐺) = (-g‘𝐺)
31	30	subgsubcl 19060	. . . . . . . . 9 ⊢ ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑥 ∈ 𝑆 ∧ 𝐴 ∈ 𝑆) → (𝑥(-g‘𝐺)𝐴) ∈ 𝑆)
32	10, 25, 29, 31	syl3anc 1370	. . . . . . . 8 ⊢ (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴 ∈ ((int‘𝐽)‘𝑆)) ∧ 𝑥 ∈ 𝑆) → (𝑥(-g‘𝐺)𝐴) ∈ 𝑆)
33	12, 32	sseldd 3983	. . . . . . 7 ⊢ (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴 ∈ ((int‘𝐽)‘𝑆)) ∧ 𝑥 ∈ 𝑆) → (𝑥(-g‘𝐺)𝐴) ∈ (Base‘𝐺))
34		eqid 2731	. . . . . . . 8 ⊢ (𝑦 ∈ (Base‘𝐺) ↦ ((𝑥(-g‘𝐺)𝐴)(+g‘𝐺)𝑦)) = (𝑦 ∈ (Base‘𝐺) ↦ ((𝑥(-g‘𝐺)𝐴)(+g‘𝐺)𝑦))
35		eqid 2731	. . . . . . . 8 ⊢ (+g‘𝐺) = (+g‘𝐺)
36	34, 3, 35, 2	tgplacthmeo 23927	. . . . . . 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 583	. . . . . 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 23589	. . . . . 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 583	. . . . 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 2833	. . . 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 23902	. . . . . . 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 1133	. . . . . . 7 ⊢ ((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴 ∈ ((int‘𝐽)‘𝑆)) → 𝑆 ⊆ (Base‘𝐺))
44	43	sselda 3982	. . . . . 6 ⊢ (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴 ∈ ((int‘𝐽)‘𝑆)) ∧ 𝑥 ∈ 𝑆) → 𝑥 ∈ (Base‘𝐺))
45	20, 28	sseldd 3983	. . . . . 6 ⊢ (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴 ∈ ((int‘𝐽)‘𝑆)) ∧ 𝑥 ∈ 𝑆) → 𝐴 ∈ (Base‘𝐺))
46	3, 35, 30	grpnpcan 18958	. . . . . 6 ⊢ ((𝐺 ∈ Grp ∧ 𝑥 ∈ (Base‘𝐺) ∧ 𝐴 ∈ (Base‘𝐺)) → ((𝑥(-g‘𝐺)𝐴)(+g‘𝐺)𝐴) = 𝑥)
47	42, 44, 45, 46	syl3anc 1370	. . . . 5 ⊢ (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴 ∈ ((int‘𝐽)‘𝑆)) ∧ 𝑥 ∈ 𝑆) → ((𝑥(-g‘𝐺)𝐴)(+g‘𝐺)𝐴) = 𝑥)
48		ovex 7445	. . . . . 6 ⊢ ((𝑥(-g‘𝐺)𝐴)(+g‘𝐺)𝐴) ∈ V
49		eqid 2731	. . . . . . 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 7420	. . . . . . 7 ⊢ (𝑦 = 𝐴 → ((𝑥(-g‘𝐺)𝐴)(+g‘𝐺)𝑦) = ((𝑥(-g‘𝐺)𝐴)(+g‘𝐺)𝐴))
51	49, 50	elrnmpt1s 5956	. . . . . 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 585	. . . . 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 2833	. . . 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 3982	. . . . . . 7 ⊢ ((((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴 ∈ ((int‘𝐽)‘𝑆)) ∧ 𝑥 ∈ 𝑆) ∧ 𝑦 ∈ ((int‘𝐽)‘𝑆)) → 𝑦 ∈ 𝑆)
57	35	subgcl 19059	. . . . . . 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 1370	. . . . . 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 7116	. . . . 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 6725	. . . 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 2821	. . . . . 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 4007	. . . . . 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 630	. . . . 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 3612	. . . 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 834	. . 3 ⊢ (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴 ∈ ((int‘𝐽)‘𝑆)) ∧ 𝑥 ∈ 𝑆) → ∃𝑢 ∈ 𝐽 (𝑥 ∈ 𝑢 ∧ 𝑢 ⊆ 𝑆))
66	65	ralrimiva 3145	. 2 ⊢ ((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴 ∈ ((int‘𝐽)‘𝑆)) → ∀𝑥 ∈ 𝑆 ∃𝑢 ∈ 𝐽 (𝑥 ∈ 𝑢 ∧ 𝑢 ⊆ 𝑆))
67		eltop2 22798	. . 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 205   ∧ wa 395   ∧ w3a 1086   = wceq 1540   ∈ wcel 2105  ∀wral 3060  ∃wrex 3069  Vcvv 3473   ⊆ wss 3948  ∪ cuni 4908   ↦ cmpt 5231  ran crn 5677   ↾ cres 5678   “ cima 5679  ‘cfv 6543  (class class class)co 7412  Basecbs 17151  +_gcplusg 17204  TopOpenctopn 17374  Grpcgrp 18861  -_gcsg 18863  SubGrpcsubg 19043  Topctop 22715  TopOnctopon 22732  intcnt 22841  Homeochmeo 23577  TopGrpctgp 23895

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2702  ax-rep 5285  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7729  ax-cnex 11172  ax-resscn 11173  ax-1cn 11174  ax-icn 11175  ax-addcl 11176  ax-addrcl 11177  ax-mulcl 11178  ax-mulrcl 11179  ax-mulcom 11180  ax-addass 11181  ax-mulass 11182  ax-distr 11183  ax-i2m1 11184  ax-1ne0 11185  ax-1rid 11186  ax-rnegex 11187  ax-rrecex 11188  ax-cnre 11189  ax-pre-lttri 11190  ax-pre-lttrn 11191  ax-pre-ltadd 11192  ax-pre-mulgt0 11193

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ne 2940  df-nel 3046  df-ral 3061  df-rex 3070  df-rmo 3375  df-reu 3376  df-rab 3432  df-v 3475  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-pss 3967  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-iun 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5574  df-eprel 5580  df-po 5588  df-so 5589  df-fr 5631  df-we 5633  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-pred 6300  df-ord 6367  df-on 6368  df-lim 6369  df-suc 6370  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-riota 7368  df-ov 7415  df-oprab 7416  df-mpo 7417  df-om 7860  df-1st 7979  df-2nd 7980  df-frecs 8272  df-wrecs 8303  df-recs 8377  df-rdg 8416  df-er 8709  df-map 8828  df-en 8946  df-dom 8947  df-sdom 8948  df-pnf 11257  df-mnf 11258  df-xr 11259  df-ltxr 11260  df-le 11261  df-sub 11453  df-neg 11454  df-nn 12220  df-2 12282  df-sets 17104  df-slot 17122  df-ndx 17134  df-base 17152  df-ress 17181  df-plusg 17217  df-0g 17394  df-topgen 17396  df-plusf 18570  df-mgm 18571  df-sgrp 18650  df-mnd 18666  df-grp 18864  df-minusg 18865  df-sbg 18866  df-subg 19046  df-top 22716  df-topon 22733  df-topsp 22755  df-bases 22769  df-ntr 22844  df-cn 23051  df-cnp 23052  df-tx 23386  df-hmeo 23579  df-tmd 23896  df-tgp 23897

This theorem is referenced by: (None)