Theorem subgntr 24456

Description: A subgroup of a topological group with nonempty interior is open. Alternatively, dual to clssubg 24458, 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 5668	. . . . . 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 2763	. . . . . . . . . . . 12 ⊢ (Base‘𝐺) = (Base‘𝐺)
4	2, 3	tgptopon 24431	. . . . . . . . . . 11 ⊢ (𝐺 ∈ TopGrp → 𝐽 ∈ (TopOn‘(Base‘𝐺)))
5	4	3ad2ant1 1151	. . . . . . . . . 10 ⊢ ((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴 ∈ ((int‘𝐽)‘𝑆)) → 𝐽 ∈ (TopOn‘(Base‘𝐺)))
6	5	adantr 486	. . . . . . . . 9 ⊢ (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴 ∈ ((int‘𝐽)‘𝑆)) ∧ 𝑥 ∈ 𝑆) → 𝐽 ∈ (TopOn‘(Base‘𝐺)))
7		topontop 23261	. . . . . . . . . . . 12 ⊢ (𝐽 ∈ (TopOn‘(Base‘𝐺)) → 𝐽 ∈ Top)
8	5, 7	syl 18	. . . . . . . . . . 11 ⊢ ((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴 ∈ ((int‘𝐽)‘𝑆)) → 𝐽 ∈ Top)
9	8	adantr 486	. . . . . . . . . 10 ⊢ (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴 ∈ ((int‘𝐽)‘𝑆)) ∧ 𝑥 ∈ 𝑆) → 𝐽 ∈ Top)
10		simpl2 1211	. . . . . . . . . . . 12 ⊢ (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴 ∈ ((int‘𝐽)‘𝑆)) ∧ 𝑥 ∈ 𝑆) → 𝑆 ∈ (SubGrp‘𝐺))
11	3	subgss 19366	. . . . . . . . . . . 12 ⊢ (𝑆 ∈ (SubGrp‘𝐺) → 𝑆 ⊆ (Base‘𝐺))
12	10, 11	syl 18	. . . . . . . . . . 11 ⊢ (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴 ∈ ((int‘𝐽)‘𝑆)) ∧ 𝑥 ∈ 𝑆) → 𝑆 ⊆ (Base‘𝐺))
13		toponuni 23262	. . . . . . . . . . . 12 ⊢ (𝐽 ∈ (TopOn‘(Base‘𝐺)) → (Base‘𝐺) = ∪ 𝐽)
14	6, 13	syl 18	. . . . . . . . . . 11 ⊢ (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴 ∈ ((int‘𝐽)‘𝑆)) ∧ 𝑥 ∈ 𝑆) → (Base‘𝐺) = ∪ 𝐽)
15	12, 14	sseqtrd 3949	. . . . . . . . . 10 ⊢ (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴 ∈ ((int‘𝐽)‘𝑆)) ∧ 𝑥 ∈ 𝑆) → 𝑆 ⊆ ∪ 𝐽)
16		eqid 2763	. . . . . . . . . . 11 ⊢ ∪ 𝐽 = ∪ 𝐽
17	16	ntropn 23397	. . . . . . . . . 10 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ ∪ 𝐽) → ((int‘𝐽)‘𝑆) ∈ 𝐽)
18	9, 15, 17	syl2anc 596	. . . . . . . . 9 ⊢ (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴 ∈ ((int‘𝐽)‘𝑆)) ∧ 𝑥 ∈ 𝑆) → ((int‘𝐽)‘𝑆) ∈ 𝐽)
19		toponss 23275	. . . . . . . . 9 ⊢ ((𝐽 ∈ (TopOn‘(Base‘𝐺)) ∧ ((int‘𝐽)‘𝑆) ∈ 𝐽) → ((int‘𝐽)‘𝑆) ⊆ (Base‘𝐺))
20	6, 18, 19	syl2anc 596	. . . . . . . 8 ⊢ (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴 ∈ ((int‘𝐽)‘𝑆)) ∧ 𝑥 ∈ 𝑆) → ((int‘𝐽)‘𝑆) ⊆ (Base‘𝐺))
21	20	resmptd 6044	. . . . . . 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 5927	. . . . . 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 2810	. . . . 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 1210	. . . . . . 7 ⊢ (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴 ∈ ((int‘𝐽)‘𝑆)) ∧ 𝑥 ∈ 𝑆) → 𝐺 ∈ TopGrp)
25		simpr 490	. . . . . . . . 9 ⊢ (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴 ∈ ((int‘𝐽)‘𝑆)) ∧ 𝑥 ∈ 𝑆) → 𝑥 ∈ 𝑆)
26	16	ntrss2 23405	. . . . . . . . . . 11 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ ∪ 𝐽) → ((int‘𝐽)‘𝑆) ⊆ 𝑆)
27	9, 15, 26	syl2anc 596	. . . . . . . . . 10 ⊢ (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴 ∈ ((int‘𝐽)‘𝑆)) ∧ 𝑥 ∈ 𝑆) → ((int‘𝐽)‘𝑆) ⊆ 𝑆)
28		simpl3 1212	. . . . . . . . . 10 ⊢ (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴 ∈ ((int‘𝐽)‘𝑆)) ∧ 𝑥 ∈ 𝑆) → 𝐴 ∈ ((int‘𝐽)‘𝑆))
29	27, 28	sseldd 3914	. . . . . . . . 9 ⊢ (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴 ∈ ((int‘𝐽)‘𝑆)) ∧ 𝑥 ∈ 𝑆) → 𝐴 ∈ 𝑆)
30		eqid 2763	. . . . . . . . . 10 ⊢ (-g‘𝐺) = (-g‘𝐺)
31	30	subgsubcl 19377	. . . . . . . . 9 ⊢ ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑥 ∈ 𝑆 ∧ 𝐴 ∈ 𝑆) → (𝑥(-g‘𝐺)𝐴) ∈ 𝑆)
32	10, 25, 29, 31	syl3anc 1399	. . . . . . . 8 ⊢ (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴 ∈ ((int‘𝐽)‘𝑆)) ∧ 𝑥 ∈ 𝑆) → (𝑥(-g‘𝐺)𝐴) ∈ 𝑆)
33	12, 32	sseldd 3914	. . . . . . 7 ⊢ (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴 ∈ ((int‘𝐽)‘𝑆)) ∧ 𝑥 ∈ 𝑆) → (𝑥(-g‘𝐺)𝐴) ∈ (Base‘𝐺))
34		eqid 2763	. . . . . . . 8 ⊢ (𝑦 ∈ (Base‘𝐺) ↦ ((𝑥(-g‘𝐺)𝐴)(+g‘𝐺)𝑦)) = (𝑦 ∈ (Base‘𝐺) ↦ ((𝑥(-g‘𝐺)𝐴)(+g‘𝐺)𝑦))
35		eqid 2763	. . . . . . . 8 ⊢ (+g‘𝐺) = (+g‘𝐺)
36	34, 3, 35, 2	tgplacthmeo 24452	. . . . . . 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 596	. . . . . 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 24114	. . . . . 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 596	. . . . 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 2865	. . . 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 24427	. . . . . . 7 ⊢ (𝐺 ∈ TopGrp → 𝐺 ∈ Grp)
42	24, 41	syl 18	. . . . . 6 ⊢ (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴 ∈ ((int‘𝐽)‘𝑆)) ∧ 𝑥 ∈ 𝑆) → 𝐺 ∈ Grp)
43	11	3ad2ant2 1152	. . . . . . 7 ⊢ ((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴 ∈ ((int‘𝐽)‘𝑆)) → 𝑆 ⊆ (Base‘𝐺))
44	43	sselda 3913	. . . . . 6 ⊢ (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴 ∈ ((int‘𝐽)‘𝑆)) ∧ 𝑥 ∈ 𝑆) → 𝑥 ∈ (Base‘𝐺))
45	20, 28	sseldd 3914	. . . . . 6 ⊢ (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴 ∈ ((int‘𝐽)‘𝑆)) ∧ 𝑥 ∈ 𝑆) → 𝐴 ∈ (Base‘𝐺))
46	3, 35, 30	grpnpcan 19271	. . . . . 6 ⊢ ((𝐺 ∈ Grp ∧ 𝑥 ∈ (Base‘𝐺) ∧ 𝐴 ∈ (Base‘𝐺)) → ((𝑥(-g‘𝐺)𝐴)(+g‘𝐺)𝐴) = 𝑥)
47	42, 44, 45, 46	syl3anc 1399	. . . . 5 ⊢ (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴 ∈ ((int‘𝐽)‘𝑆)) ∧ 𝑥 ∈ 𝑆) → ((𝑥(-g‘𝐺)𝐴)(+g‘𝐺)𝐴) = 𝑥)
48		ovex 7476	. . . . . 6 ⊢ ((𝑥(-g‘𝐺)𝐴)(+g‘𝐺)𝐴) ∈ V
49		eqid 2763	. . . . . . 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 7451	. . . . . . 7 ⊢ (𝑦 = 𝐴 → ((𝑥(-g‘𝐺)𝐴)(+g‘𝐺)𝑦) = ((𝑥(-g‘𝐺)𝐴)(+g‘𝐺)𝐴))
51	49, 50	elrnmpt1s 5948	. . . . . 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 598	. . . . 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 2865	. . . 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 486	. . . . . . 7 ⊢ ((((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴 ∈ ((int‘𝐽)‘𝑆)) ∧ 𝑥 ∈ 𝑆) ∧ 𝑦 ∈ ((int‘𝐽)‘𝑆)) → 𝑆 ∈ (SubGrp‘𝐺))
55	32	adantr 486	. . . . . . 7 ⊢ ((((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴 ∈ ((int‘𝐽)‘𝑆)) ∧ 𝑥 ∈ 𝑆) ∧ 𝑦 ∈ ((int‘𝐽)‘𝑆)) → (𝑥(-g‘𝐺)𝐴) ∈ 𝑆)
56	27	sselda 3913	. . . . . . 7 ⊢ ((((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴 ∈ ((int‘𝐽)‘𝑆)) ∧ 𝑥 ∈ 𝑆) ∧ 𝑦 ∈ ((int‘𝐽)‘𝑆)) → 𝑦 ∈ 𝑆)
57	35	subgcl 19375	. . . . . . 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 1399	. . . . . 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 7139	. . . . 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 6740	. . . 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 2853	. . . . . 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 3938	. . . . . 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 644	. . . . 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 3580	. . . 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 850	. . 3 ⊢ (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴 ∈ ((int‘𝐽)‘𝑆)) ∧ 𝑥 ∈ 𝑆) → ∃𝑢 ∈ 𝐽 (𝑥 ∈ 𝑢 ∧ 𝑢 ⊆ 𝑆))
66	65	ralrimiva 3158	. 2 ⊢ ((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴 ∈ ((int‘𝐽)‘𝑆)) → ∀𝑥 ∈ 𝑆 ∃𝑢 ∈ 𝐽 (𝑥 ∈ 𝑢 ∧ 𝑢 ⊆ 𝑆))
67		eltop2 23323	. . 3 ⊢ (𝐽 ∈ Top → (𝑆 ∈ 𝐽 ↔ ∀𝑥 ∈ 𝑆 ∃𝑢 ∈ 𝐽 (𝑥 ∈ 𝑢 ∧ 𝑢 ⊆ 𝑆)))
68	8, 67	syl 18	. 2 ⊢ ((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴 ∈ ((int‘𝐽)‘𝑆)) → (𝑆 ∈ 𝐽 ↔ ∀𝑥 ∈ 𝑆 ∃𝑢 ∈ 𝐽 (𝑥 ∈ 𝑢 ∧ 𝑢 ⊆ 𝑆)))
69	66, 68	mpbird 260	1 ⊢ ((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴 ∈ ((int‘𝐽)‘𝑆)) → 𝑆 ∈ 𝐽)

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   ∧ w3a 1103   = wceq 1570   ∈ wcel 2147  ∀wral 3080  ∃wrex 3090  Vcvv 3454   ⊆ wss 3900  ∪ cuni 4870   ↦ cmpt 5189  ran crn 5656   ↾ cres 5657   “ cima 5658  ‘cfv 6558  (class class class)co 7443  Basecbs 17416  +_gcplusg 17457  TopOpenctopn 17621  Grpcgrp 19173  -_gcsg 19175  SubGrpcsubg 19359  Topctop 23241  TopOnctopon 23258  intcnt 23365  Homeochmeo 24102  TopGrpctgp 24420

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-rep 5235  ax-sep 5252  ax-nul 5263  ax-pow 5331  ax-pr 5395  ax-un 7775  ax-cnex 11283  ax-resscn 11284  ax-1cn 11285  ax-icn 11286  ax-addcl 11287  ax-addrcl 11288  ax-mulcl 11289  ax-mulrcl 11290  ax-mulcom 11291  ax-addass 11292  ax-mulass 11293  ax-distr 11294  ax-i2m1 11295  ax-1ne0 11296  ax-1rid 11297  ax-rnegex 11298  ax-rrecex 11299  ax-cnre 11300  ax-pre-lttri 11301  ax-pre-lttrn 11302  ax-pre-ltadd 11303  ax-pre-mulgt0 11304

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-nel 3066  df-ral 3081  df-rex 3091  df-rmo 3369  df-reu 3370  df-rab 3417  df-v 3456  df-sbc 3743  df-csb 3852  df-ss 3901  df-pss 4017  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-op 4839  df-uni 4871  df-iun 4956  df-br 5107  df-opab 5171  df-mpt 5190  df-tr 5216  df-id 5550  df-eprel 5555  df-po 5563  df-so 5564  df-fr 5608  df-we 5610  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-rn 5666  df-res 5667  df-ima 5668  df-pred 6322  df-ord 6383  df-on 6384  df-lim 6385  df-suc 6386  df-iota 6512  df-fun 6560  df-fn 6561  df-f 6562  df-f1 6563  df-fo 6564  df-f1o 6565  df-fv 6566  df-riota 7400  df-ov 7446  df-oprab 7447  df-mpo 7448  df-om 7902  df-1st 8025  df-2nd 8026  df-frecs 8327  df-wrecs 8358  df-recs 8407  df-rdg 8446  df-er 8745  df-map 8877  df-en 9002  df-dom 9003  df-sdom 9004  df-pnf 11372  df-mnf 11373  df-xr 11374  df-ltxr 11375  df-le 11376  df-sub 11570  df-neg 11571  df-nn 12363  df-2 12432  df-sets 17371  df-slot 17389  df-ndx 17401  df-base 17417  df-ress 17438  df-plusg 17470  df-0g 17641  df-topgen 17643  df-plusf 18844  df-mgm 18845  df-sgrp 18937  df-mnd 18953  df-grp 19176  df-minusg 19177  df-sbg 19178  df-subg 19362  df-top 23242  df-topon 23259  df-topsp 23281  df-bases 23294  df-ntr 23368  df-cn 23575  df-cnp 23576  df-tx 23911  df-hmeo 24104  df-tmd 24421  df-tgp 24422

This theorem is used by: (None)