Theorem subgntr 24010

Description: A subgroup of a topological group with nonempty interior is open. Alternatively, dual to clssubg 24012, 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 5636	. . . . . 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 2729	. . . . . . . . . . . 12 ⊢ (Base‘𝐺) = (Base‘𝐺)
4	2, 3	tgptopon 23985	. . . . . . . . . . 11 ⊢ (𝐺 ∈ TopGrp → 𝐽 ∈ (TopOn‘(Base‘𝐺)))
5	4	3ad2ant1 1133	. . . . . . . . . 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 22816	. . . . . . . . . . . 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 1193	. . . . . . . . . . . 12 ⊢ (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴 ∈ ((int‘𝐽)‘𝑆)) ∧ 𝑥 ∈ 𝑆) → 𝑆 ∈ (SubGrp‘𝐺))
11	3	subgss 19024	. . . . . . . . . . . 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 22817	. . . . . . . . . . . 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 3974	. . . . . . . . . 10 ⊢ (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴 ∈ ((int‘𝐽)‘𝑆)) ∧ 𝑥 ∈ 𝑆) → 𝑆 ⊆ ∪ 𝐽)
16		eqid 2729	. . . . . . . . . . 11 ⊢ ∪ 𝐽 = ∪ 𝐽
17	16	ntropn 22952	. . . . . . . . . 10 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ ∪ 𝐽) → ((int‘𝐽)‘𝑆) ∈ 𝐽)
18	9, 15, 17	syl2anc 584	. . . . . . . . 9 ⊢ (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴 ∈ ((int‘𝐽)‘𝑆)) ∧ 𝑥 ∈ 𝑆) → ((int‘𝐽)‘𝑆) ∈ 𝐽)
19		toponss 22830	. . . . . . . . 9 ⊢ ((𝐽 ∈ (TopOn‘(Base‘𝐺)) ∧ ((int‘𝐽)‘𝑆) ∈ 𝐽) → ((int‘𝐽)‘𝑆) ⊆ (Base‘𝐺))
20	6, 18, 19	syl2anc 584	. . . . . . . 8 ⊢ (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴 ∈ ((int‘𝐽)‘𝑆)) ∧ 𝑥 ∈ 𝑆) → ((int‘𝐽)‘𝑆) ⊆ (Base‘𝐺))
21	20	resmptd 5995	. . . . . . 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 5884	. . . . . 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 2776	. . . . 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 1192	. . . . . . 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 22960	. . . . . . . . . . 11 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ ∪ 𝐽) → ((int‘𝐽)‘𝑆) ⊆ 𝑆)
27	9, 15, 26	syl2anc 584	. . . . . . . . . 10 ⊢ (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴 ∈ ((int‘𝐽)‘𝑆)) ∧ 𝑥 ∈ 𝑆) → ((int‘𝐽)‘𝑆) ⊆ 𝑆)
28		simpl3 1194	. . . . . . . . . 10 ⊢ (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴 ∈ ((int‘𝐽)‘𝑆)) ∧ 𝑥 ∈ 𝑆) → 𝐴 ∈ ((int‘𝐽)‘𝑆))
29	27, 28	sseldd 3938	. . . . . . . . 9 ⊢ (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴 ∈ ((int‘𝐽)‘𝑆)) ∧ 𝑥 ∈ 𝑆) → 𝐴 ∈ 𝑆)
30		eqid 2729	. . . . . . . . . 10 ⊢ (-g‘𝐺) = (-g‘𝐺)
31	30	subgsubcl 19034	. . . . . . . . 9 ⊢ ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑥 ∈ 𝑆 ∧ 𝐴 ∈ 𝑆) → (𝑥(-g‘𝐺)𝐴) ∈ 𝑆)
32	10, 25, 29, 31	syl3anc 1373	. . . . . . . 8 ⊢ (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴 ∈ ((int‘𝐽)‘𝑆)) ∧ 𝑥 ∈ 𝑆) → (𝑥(-g‘𝐺)𝐴) ∈ 𝑆)
33	12, 32	sseldd 3938	. . . . . . 7 ⊢ (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴 ∈ ((int‘𝐽)‘𝑆)) ∧ 𝑥 ∈ 𝑆) → (𝑥(-g‘𝐺)𝐴) ∈ (Base‘𝐺))
34		eqid 2729	. . . . . . . 8 ⊢ (𝑦 ∈ (Base‘𝐺) ↦ ((𝑥(-g‘𝐺)𝐴)(+g‘𝐺)𝑦)) = (𝑦 ∈ (Base‘𝐺) ↦ ((𝑥(-g‘𝐺)𝐴)(+g‘𝐺)𝑦))
35		eqid 2729	. . . . . . . 8 ⊢ (+g‘𝐺) = (+g‘𝐺)
36	34, 3, 35, 2	tgplacthmeo 24006	. . . . . . 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 584	. . . . . 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 23668	. . . . . 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 584	. . . . 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 2829	. . . 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 23981	. . . . . . 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 1134	. . . . . . 7 ⊢ ((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴 ∈ ((int‘𝐽)‘𝑆)) → 𝑆 ⊆ (Base‘𝐺))
44	43	sselda 3937	. . . . . 6 ⊢ (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴 ∈ ((int‘𝐽)‘𝑆)) ∧ 𝑥 ∈ 𝑆) → 𝑥 ∈ (Base‘𝐺))
45	20, 28	sseldd 3938	. . . . . 6 ⊢ (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴 ∈ ((int‘𝐽)‘𝑆)) ∧ 𝑥 ∈ 𝑆) → 𝐴 ∈ (Base‘𝐺))
46	3, 35, 30	grpnpcan 18929	. . . . . 6 ⊢ ((𝐺 ∈ Grp ∧ 𝑥 ∈ (Base‘𝐺) ∧ 𝐴 ∈ (Base‘𝐺)) → ((𝑥(-g‘𝐺)𝐴)(+g‘𝐺)𝐴) = 𝑥)
47	42, 44, 45, 46	syl3anc 1373	. . . . 5 ⊢ (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴 ∈ ((int‘𝐽)‘𝑆)) ∧ 𝑥 ∈ 𝑆) → ((𝑥(-g‘𝐺)𝐴)(+g‘𝐺)𝐴) = 𝑥)
48		ovex 7386	. . . . . 6 ⊢ ((𝑥(-g‘𝐺)𝐴)(+g‘𝐺)𝐴) ∈ V
49		eqid 2729	. . . . . . 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 7361	. . . . . . 7 ⊢ (𝑦 = 𝐴 → ((𝑥(-g‘𝐺)𝐴)(+g‘𝐺)𝑦) = ((𝑥(-g‘𝐺)𝐴)(+g‘𝐺)𝐴))
51	49, 50	elrnmpt1s 5905	. . . . . 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 586	. . . . 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 2829	. . . 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 3937	. . . . . . 7 ⊢ ((((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴 ∈ ((int‘𝐽)‘𝑆)) ∧ 𝑥 ∈ 𝑆) ∧ 𝑦 ∈ ((int‘𝐽)‘𝑆)) → 𝑦 ∈ 𝑆)
57	35	subgcl 19033	. . . . . . 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 1373	. . . . . 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 7053	. . . . 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 6664	. . . 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 2817	. . . . . 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 3963	. . . . . 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 632	. . . . 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 3579	. . . 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 836	. . 3 ⊢ (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴 ∈ ((int‘𝐽)‘𝑆)) ∧ 𝑥 ∈ 𝑆) → ∃𝑢 ∈ 𝐽 (𝑥 ∈ 𝑢 ∧ 𝑢 ⊆ 𝑆))
66	65	ralrimiva 3121	. 2 ⊢ ((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴 ∈ ((int‘𝐽)‘𝑆)) → ∀𝑥 ∈ 𝑆 ∃𝑢 ∈ 𝐽 (𝑥 ∈ 𝑢 ∧ 𝑢 ⊆ 𝑆))
67		eltop2 22878	. . 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 1086   = wceq 1540   ∈ wcel 2109  ∀wral 3044  ∃wrex 3053  Vcvv 3438   ⊆ wss 3905  ∪ cuni 4861   ↦ cmpt 5176  ran crn 5624   ↾ cres 5625   “ cima 5626  ‘cfv 6486  (class class class)co 7353  Basecbs 17138  +_gcplusg 17179  TopOpenctopn 17343  Grpcgrp 18830  -_gcsg 18832  SubGrpcsubg 19017  Topctop 22796  TopOnctopon 22813  intcnt 22920  Homeochmeo 23656  TopGrpctgp 23974

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-rep 5221  ax-sep 5238  ax-nul 5248  ax-pow 5307  ax-pr 5374  ax-un 7675  ax-cnex 11084  ax-resscn 11085  ax-1cn 11086  ax-icn 11087  ax-addcl 11088  ax-addrcl 11089  ax-mulcl 11090  ax-mulrcl 11091  ax-mulcom 11092  ax-addass 11093  ax-mulass 11094  ax-distr 11095  ax-i2m1 11096  ax-1ne0 11097  ax-1rid 11098  ax-rnegex 11099  ax-rrecex 11100  ax-cnre 11101  ax-pre-lttri 11102  ax-pre-lttrn 11103  ax-pre-ltadd 11104  ax-pre-mulgt0 11105

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-nel 3030  df-ral 3045  df-rex 3054  df-rmo 3345  df-reu 3346  df-rab 3397  df-v 3440  df-sbc 3745  df-csb 3854  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-pss 3925  df-nul 4287  df-if 4479  df-pw 4555  df-sn 4580  df-pr 4582  df-op 4586  df-uni 4862  df-iun 4946  df-br 5096  df-opab 5158  df-mpt 5177  df-tr 5203  df-id 5518  df-eprel 5523  df-po 5531  df-so 5532  df-fr 5576  df-we 5578  df-xp 5629  df-rel 5630  df-cnv 5631  df-co 5632  df-dm 5633  df-rn 5634  df-res 5635  df-ima 5636  df-pred 6253  df-ord 6314  df-on 6315  df-lim 6316  df-suc 6317  df-iota 6442  df-fun 6488  df-fn 6489  df-f 6490  df-f1 6491  df-fo 6492  df-f1o 6493  df-fv 6494  df-riota 7310  df-ov 7356  df-oprab 7357  df-mpo 7358  df-om 7807  df-1st 7931  df-2nd 7932  df-frecs 8221  df-wrecs 8252  df-recs 8301  df-rdg 8339  df-er 8632  df-map 8762  df-en 8880  df-dom 8881  df-sdom 8882  df-pnf 11170  df-mnf 11171  df-xr 11172  df-ltxr 11173  df-le 11174  df-sub 11367  df-neg 11368  df-nn 12147  df-2 12209  df-sets 17093  df-slot 17111  df-ndx 17123  df-base 17139  df-ress 17160  df-plusg 17192  df-0g 17363  df-topgen 17365  df-plusf 18531  df-mgm 18532  df-sgrp 18611  df-mnd 18627  df-grp 18833  df-minusg 18834  df-sbg 18835  df-subg 19020  df-top 22797  df-topon 22814  df-topsp 22836  df-bases 22849  df-ntr 22923  df-cn 23130  df-cnp 23131  df-tx 23465  df-hmeo 23658  df-tmd 23975  df-tgp 23976

This theorem is referenced by: (None)