Theorem subgntr 23258

Description: A subgroup of a topological group with nonempty interior is open. Alternatively, dual to clssubg 23260, 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 5602	. . . . . 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 2738	. . . . . . . . . . . 12 ⊢ (Base‘𝐺) = (Base‘𝐺)
4	2, 3	tgptopon 23233	. . . . . . . . . . 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 481	. . . . . . . . 9 ⊢ (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴 ∈ ((int‘𝐽)‘𝑆)) ∧ 𝑥 ∈ 𝑆) → 𝐽 ∈ (TopOn‘(Base‘𝐺)))
7		topontop 22062	. . . . . . . . . . . 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 481	. . . . . . . . . 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 18756	. . . . . . . . . . . 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 22063	. . . . . . . . . . . 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 3961	. . . . . . . . . 10 ⊢ (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴 ∈ ((int‘𝐽)‘𝑆)) ∧ 𝑥 ∈ 𝑆) → 𝑆 ⊆ ∪ 𝐽)
16		eqid 2738	. . . . . . . . . . 11 ⊢ ∪ 𝐽 = ∪ 𝐽
17	16	ntropn 22200	. . . . . . . . . 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 22076	. . . . . . . . 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 5948	. . . . . . 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 5847	. . . . . 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 2790	. . . . 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 485	. . . . . . . . 9 ⊢ (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴 ∈ ((int‘𝐽)‘𝑆)) ∧ 𝑥 ∈ 𝑆) → 𝑥 ∈ 𝑆)
26	16	ntrss2 22208	. . . . . . . . . . 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 1192	. . . . . . . . . 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 2738	. . . . . . . . . 10 ⊢ (-g‘𝐺) = (-g‘𝐺)
31	30	subgsubcl 18766	. . . . . . . . 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 3922	. . . . . . 7 ⊢ (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴 ∈ ((int‘𝐽)‘𝑆)) ∧ 𝑥 ∈ 𝑆) → (𝑥(-g‘𝐺)𝐴) ∈ (Base‘𝐺))
34		eqid 2738	. . . . . . . 8 ⊢ (𝑦 ∈ (Base‘𝐺) ↦ ((𝑥(-g‘𝐺)𝐴)(+g‘𝐺)𝑦)) = (𝑦 ∈ (Base‘𝐺) ↦ ((𝑥(-g‘𝐺)𝐴)(+g‘𝐺)𝑦))
35		eqid 2738	. . . . . . . 8 ⊢ (+g‘𝐺) = (+g‘𝐺)
36	34, 3, 35, 2	tgplacthmeo 23254	. . . . . . 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 22916	. . . . . 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 2840	. . . 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 23229	. . . . . . 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 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 18667	. . . . . 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 7308	. . . . . 6 ⊢ ((𝑥(-g‘𝐺)𝐴)(+g‘𝐺)𝐴) ∈ V
49		eqid 2738	. . . . . . 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 7283	. . . . . . 7 ⊢ (𝑦 = 𝐴 → ((𝑥(-g‘𝐺)𝐴)(+g‘𝐺)𝑦) = ((𝑥(-g‘𝐺)𝐴)(+g‘𝐺)𝐴))
51	49, 50	elrnmpt1s 5866	. . . . . 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 2840	. . . 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 481	. . . . . . 7 ⊢ ((((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴 ∈ ((int‘𝐽)‘𝑆)) ∧ 𝑥 ∈ 𝑆) ∧ 𝑦 ∈ ((int‘𝐽)‘𝑆)) → 𝑆 ∈ (SubGrp‘𝐺))
55	32	adantr 481	. . . . . . 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 18765	. . . . . . 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 6989	. . . . 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 6608	. . . 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 2827	. . . . . 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 3946	. . . . . 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 631	. . . . 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 3561	. . . 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 3103	. 2 ⊢ ((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴 ∈ ((int‘𝐽)‘𝑆)) → ∀𝑥 ∈ 𝑆 ∃𝑢 ∈ 𝐽 (𝑥 ∈ 𝑢 ∧ 𝑢 ⊆ 𝑆))
67		eltop2 22125	. . 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 256	1 ⊢ ((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴 ∈ ((int‘𝐽)‘𝑆)) → 𝑆 ∈ 𝐽)

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 396   ∧ w3a 1086   = wceq 1539   ∈ wcel 2106  ∀wral 3064  ∃wrex 3065  Vcvv 3432   ⊆ wss 3887  ∪ cuni 4839   ↦ cmpt 5157  ran crn 5590   ↾ cres 5591   “ cima 5592  ‘cfv 6433  (class class class)co 7275  Basecbs 16912  +_gcplusg 16962  TopOpenctopn 17132  Grpcgrp 18577  -_gcsg 18579  SubGrpcsubg 18749  Topctop 22042  TopOnctopon 22059  intcnt 22168  Homeochmeo 22904  TopGrpctgp 23222

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-rep 5209  ax-sep 5223  ax-nul 5230  ax-pow 5288  ax-pr 5352  ax-un 7588  ax-cnex 10927  ax-resscn 10928  ax-1cn 10929  ax-icn 10930  ax-addcl 10931  ax-addrcl 10932  ax-mulcl 10933  ax-mulrcl 10934  ax-mulcom 10935  ax-addass 10936  ax-mulass 10937  ax-distr 10938  ax-i2m1 10939  ax-1ne0 10940  ax-1rid 10941  ax-rnegex 10942  ax-rrecex 10943  ax-cnre 10944  ax-pre-lttri 10945  ax-pre-lttrn 10946  ax-pre-ltadd 10947  ax-pre-mulgt0 10948

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ne 2944  df-nel 3050  df-ral 3069  df-rex 3070  df-rmo 3071  df-reu 3072  df-rab 3073  df-v 3434  df-sbc 3717  df-csb 3833  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-pss 3906  df-nul 4257  df-if 4460  df-pw 4535  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-iun 4926  df-br 5075  df-opab 5137  df-mpt 5158  df-tr 5192  df-id 5489  df-eprel 5495  df-po 5503  df-so 5504  df-fr 5544  df-we 5546  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-res 5601  df-ima 5602  df-pred 6202  df-ord 6269  df-on 6270  df-lim 6271  df-suc 6272  df-iota 6391  df-fun 6435  df-fn 6436  df-f 6437  df-f1 6438  df-fo 6439  df-f1o 6440  df-fv 6441  df-riota 7232  df-ov 7278  df-oprab 7279  df-mpo 7280  df-om 7713  df-1st 7831  df-2nd 7832  df-frecs 8097  df-wrecs 8128  df-recs 8202  df-rdg 8241  df-er 8498  df-map 8617  df-en 8734  df-dom 8735  df-sdom 8736  df-pnf 11011  df-mnf 11012  df-xr 11013  df-ltxr 11014  df-le 11015  df-sub 11207  df-neg 11208  df-nn 11974  df-2 12036  df-sets 16865  df-slot 16883  df-ndx 16895  df-base 16913  df-ress 16942  df-plusg 16975  df-0g 17152  df-topgen 17154  df-plusf 18325  df-mgm 18326  df-sgrp 18375  df-mnd 18386  df-grp 18580  df-minusg 18581  df-sbg 18582  df-subg 18752  df-top 22043  df-topon 22060  df-topsp 22082  df-bases 22096  df-ntr 22171  df-cn 22378  df-cnp 22379  df-tx 22713  df-hmeo 22906  df-tmd 23223  df-tgp 23224

This theorem is referenced by: (None)