Theorem clssubg 23972

Description: The closure of a subgroup in a topological group is a subgroup. (Contributed by Mario Carneiro, 17-Sep-2015.)

Hypothesis

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

Assertion

Ref	Expression
clssubg	⊢ ((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺)) → ((cls‘𝐽)‘𝑆) ∈ (SubGrp‘𝐺))

Proof of Theorem clssubg

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

Step	Hyp	Ref	Expression
1		subgntr.h	. . . . . . 7 ⊢ 𝐽 = (TopOpen‘𝐺)
2		eqid 2729	. . . . . . 7 ⊢ (Base‘𝐺) = (Base‘𝐺)
3	1, 2	tgptopon 23945	. . . . . 6 ⊢ (𝐺 ∈ TopGrp → 𝐽 ∈ (TopOn‘(Base‘𝐺)))
4	3	adantr 480	. . . . 5 ⊢ ((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺)) → 𝐽 ∈ (TopOn‘(Base‘𝐺)))
5		topontop 22776	. . . . 5 ⊢ (𝐽 ∈ (TopOn‘(Base‘𝐺)) → 𝐽 ∈ Top)
6	4, 5	syl 17	. . . 4 ⊢ ((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺)) → 𝐽 ∈ Top)
7	2	subgss 19035	. . . . . 6 ⊢ (𝑆 ∈ (SubGrp‘𝐺) → 𝑆 ⊆ (Base‘𝐺))
8	7	adantl 481	. . . . 5 ⊢ ((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺)) → 𝑆 ⊆ (Base‘𝐺))
9		toponuni 22777	. . . . . 6 ⊢ (𝐽 ∈ (TopOn‘(Base‘𝐺)) → (Base‘𝐺) = ∪ 𝐽)
10	4, 9	syl 17	. . . . 5 ⊢ ((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺)) → (Base‘𝐺) = ∪ 𝐽)
11	8, 10	sseqtrd 3980	. . . 4 ⊢ ((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺)) → 𝑆 ⊆ ∪ 𝐽)
12		eqid 2729	. . . . 5 ⊢ ∪ 𝐽 = ∪ 𝐽
13	12	clsss3 22922	. . . 4 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ ∪ 𝐽) → ((cls‘𝐽)‘𝑆) ⊆ ∪ 𝐽)
14	6, 11, 13	syl2anc 584	. . 3 ⊢ ((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺)) → ((cls‘𝐽)‘𝑆) ⊆ ∪ 𝐽)
15	14, 10	sseqtrrd 3981	. 2 ⊢ ((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺)) → ((cls‘𝐽)‘𝑆) ⊆ (Base‘𝐺))
16	12	sscls 22919	. . . 4 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ ∪ 𝐽) → 𝑆 ⊆ ((cls‘𝐽)‘𝑆))
17	6, 11, 16	syl2anc 584	. . 3 ⊢ ((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺)) → 𝑆 ⊆ ((cls‘𝐽)‘𝑆))
18		eqid 2729	. . . . . 6 ⊢ (0g‘𝐺) = (0g‘𝐺)
19	18	subg0cl 19042	. . . . 5 ⊢ (𝑆 ∈ (SubGrp‘𝐺) → (0g‘𝐺) ∈ 𝑆)
20	19	adantl 481	. . . 4 ⊢ ((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺)) → (0g‘𝐺) ∈ 𝑆)
21	20	ne0d 4301	. . 3 ⊢ ((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺)) → 𝑆 ≠ ∅)
22		ssn0 4363	. . 3 ⊢ ((𝑆 ⊆ ((cls‘𝐽)‘𝑆) ∧ 𝑆 ≠ ∅) → ((cls‘𝐽)‘𝑆) ≠ ∅)
23	17, 21, 22	syl2anc 584	. 2 ⊢ ((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺)) → ((cls‘𝐽)‘𝑆) ≠ ∅)
24		df-ov 7372	. . . 4 ⊢ (𝑥(-g‘𝐺)𝑦) = ((-g‘𝐺)‘⟨𝑥, 𝑦⟩)
25		opelxpi 5668	. . . . . . 7 ⊢ ((𝑥 ∈ ((cls‘𝐽)‘𝑆) ∧ 𝑦 ∈ ((cls‘𝐽)‘𝑆)) → ⟨𝑥, 𝑦⟩ ∈ (((cls‘𝐽)‘𝑆) × ((cls‘𝐽)‘𝑆)))
26		txcls 23467	. . . . . . . . . 10 ⊢ (((𝐽 ∈ (TopOn‘(Base‘𝐺)) ∧ 𝐽 ∈ (TopOn‘(Base‘𝐺))) ∧ (𝑆 ⊆ (Base‘𝐺) ∧ 𝑆 ⊆ (Base‘𝐺))) → ((cls‘(𝐽 ×t 𝐽))‘(𝑆 × 𝑆)) = (((cls‘𝐽)‘𝑆) × ((cls‘𝐽)‘𝑆)))
27	4, 4, 8, 8, 26	syl22anc 838	. . . . . . . . 9 ⊢ ((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺)) → ((cls‘(𝐽 ×t 𝐽))‘(𝑆 × 𝑆)) = (((cls‘𝐽)‘𝑆) × ((cls‘𝐽)‘𝑆)))
28		txtopon 23454	. . . . . . . . . . . . 13 ⊢ ((𝐽 ∈ (TopOn‘(Base‘𝐺)) ∧ 𝐽 ∈ (TopOn‘(Base‘𝐺))) → (𝐽 ×t 𝐽) ∈ (TopOn‘((Base‘𝐺) × (Base‘𝐺))))
29	4, 4, 28	syl2anc 584	. . . . . . . . . . . 12 ⊢ ((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺)) → (𝐽 ×t 𝐽) ∈ (TopOn‘((Base‘𝐺) × (Base‘𝐺))))
30		topontop 22776	. . . . . . . . . . . 12 ⊢ ((𝐽 ×t 𝐽) ∈ (TopOn‘((Base‘𝐺) × (Base‘𝐺))) → (𝐽 ×t 𝐽) ∈ Top)
31	29, 30	syl 17	. . . . . . . . . . 11 ⊢ ((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺)) → (𝐽 ×t 𝐽) ∈ Top)
32		cnvimass 6042	. . . . . . . . . . . . 13 ⊢ (◡(-g‘𝐺) “ 𝑆) ⊆ dom (-g‘𝐺)
33		tgpgrp 23941	. . . . . . . . . . . . . . 15 ⊢ (𝐺 ∈ TopGrp → 𝐺 ∈ Grp)
34	33	adantr 480	. . . . . . . . . . . . . 14 ⊢ ((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺)) → 𝐺 ∈ Grp)
35		eqid 2729	. . . . . . . . . . . . . . 15 ⊢ (-g‘𝐺) = (-g‘𝐺)
36	2, 35	grpsubf 18927	. . . . . . . . . . . . . 14 ⊢ (𝐺 ∈ Grp → (-g‘𝐺):((Base‘𝐺) × (Base‘𝐺))⟶(Base‘𝐺))
37	34, 36	syl 17	. . . . . . . . . . . . 13 ⊢ ((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺)) → (-g‘𝐺):((Base‘𝐺) × (Base‘𝐺))⟶(Base‘𝐺))
38	32, 37	fssdm 6689	. . . . . . . . . . . 12 ⊢ ((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺)) → (◡(-g‘𝐺) “ 𝑆) ⊆ ((Base‘𝐺) × (Base‘𝐺)))
39		toponuni 22777	. . . . . . . . . . . . 13 ⊢ ((𝐽 ×t 𝐽) ∈ (TopOn‘((Base‘𝐺) × (Base‘𝐺))) → ((Base‘𝐺) × (Base‘𝐺)) = ∪ (𝐽 ×t 𝐽))
40	29, 39	syl 17	. . . . . . . . . . . 12 ⊢ ((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺)) → ((Base‘𝐺) × (Base‘𝐺)) = ∪ (𝐽 ×t 𝐽))
41	38, 40	sseqtrd 3980	. . . . . . . . . . 11 ⊢ ((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺)) → (◡(-g‘𝐺) “ 𝑆) ⊆ ∪ (𝐽 ×t 𝐽))
42	35	subgsubcl 19045	. . . . . . . . . . . . . . . 16 ⊢ ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆) → (𝑥(-g‘𝐺)𝑦) ∈ 𝑆)
43	42	3expb 1120	. . . . . . . . . . . . . . 15 ⊢ ((𝑆 ∈ (SubGrp‘𝐺) ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥(-g‘𝐺)𝑦) ∈ 𝑆)
44	43	ralrimivva 3178	. . . . . . . . . . . . . 14 ⊢ (𝑆 ∈ (SubGrp‘𝐺) → ∀𝑥 ∈ 𝑆 ∀𝑦 ∈ 𝑆 (𝑥(-g‘𝐺)𝑦) ∈ 𝑆)
45		fveq2 6840	. . . . . . . . . . . . . . . . 17 ⊢ (𝑧 = ⟨𝑥, 𝑦⟩ → ((-g‘𝐺)‘𝑧) = ((-g‘𝐺)‘⟨𝑥, 𝑦⟩))
46	45, 24	eqtr4di 2782	. . . . . . . . . . . . . . . 16 ⊢ (𝑧 = ⟨𝑥, 𝑦⟩ → ((-g‘𝐺)‘𝑧) = (𝑥(-g‘𝐺)𝑦))
47	46	eleq1d 2813	. . . . . . . . . . . . . . 15 ⊢ (𝑧 = ⟨𝑥, 𝑦⟩ → (((-g‘𝐺)‘𝑧) ∈ 𝑆 ↔ (𝑥(-g‘𝐺)𝑦) ∈ 𝑆))
48	47	ralxp 5795	. . . . . . . . . . . . . 14 ⊢ (∀𝑧 ∈ (𝑆 × 𝑆)((-g‘𝐺)‘𝑧) ∈ 𝑆 ↔ ∀𝑥 ∈ 𝑆 ∀𝑦 ∈ 𝑆 (𝑥(-g‘𝐺)𝑦) ∈ 𝑆)
49	44, 48	sylibr 234	. . . . . . . . . . . . 13 ⊢ (𝑆 ∈ (SubGrp‘𝐺) → ∀𝑧 ∈ (𝑆 × 𝑆)((-g‘𝐺)‘𝑧) ∈ 𝑆)
50	49	adantl 481	. . . . . . . . . . . 12 ⊢ ((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺)) → ∀𝑧 ∈ (𝑆 × 𝑆)((-g‘𝐺)‘𝑧) ∈ 𝑆)
51	37	ffund 6674	. . . . . . . . . . . . 13 ⊢ ((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺)) → Fun (-g‘𝐺))
52		xpss12 5646	. . . . . . . . . . . . . . 15 ⊢ ((𝑆 ⊆ (Base‘𝐺) ∧ 𝑆 ⊆ (Base‘𝐺)) → (𝑆 × 𝑆) ⊆ ((Base‘𝐺) × (Base‘𝐺)))
53	8, 8, 52	syl2anc 584	. . . . . . . . . . . . . 14 ⊢ ((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺)) → (𝑆 × 𝑆) ⊆ ((Base‘𝐺) × (Base‘𝐺)))
54	37	fdmd 6680	. . . . . . . . . . . . . 14 ⊢ ((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺)) → dom (-g‘𝐺) = ((Base‘𝐺) × (Base‘𝐺)))
55	53, 54	sseqtrrd 3981	. . . . . . . . . . . . 13 ⊢ ((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺)) → (𝑆 × 𝑆) ⊆ dom (-g‘𝐺))
56		funimass5 7009	. . . . . . . . . . . . 13 ⊢ ((Fun (-g‘𝐺) ∧ (𝑆 × 𝑆) ⊆ dom (-g‘𝐺)) → ((𝑆 × 𝑆) ⊆ (◡(-g‘𝐺) “ 𝑆) ↔ ∀𝑧 ∈ (𝑆 × 𝑆)((-g‘𝐺)‘𝑧) ∈ 𝑆))
57	51, 55, 56	syl2anc 584	. . . . . . . . . . . 12 ⊢ ((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺)) → ((𝑆 × 𝑆) ⊆ (◡(-g‘𝐺) “ 𝑆) ↔ ∀𝑧 ∈ (𝑆 × 𝑆)((-g‘𝐺)‘𝑧) ∈ 𝑆))
58	50, 57	mpbird 257	. . . . . . . . . . 11 ⊢ ((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺)) → (𝑆 × 𝑆) ⊆ (◡(-g‘𝐺) “ 𝑆))
59		eqid 2729	. . . . . . . . . . . 12 ⊢ ∪ (𝐽 ×t 𝐽) = ∪ (𝐽 ×t 𝐽)
60	59	clsss 22917	. . . . . . . . . . 11 ⊢ (((𝐽 ×t 𝐽) ∈ Top ∧ (◡(-g‘𝐺) “ 𝑆) ⊆ ∪ (𝐽 ×t 𝐽) ∧ (𝑆 × 𝑆) ⊆ (◡(-g‘𝐺) “ 𝑆)) → ((cls‘(𝐽 ×t 𝐽))‘(𝑆 × 𝑆)) ⊆ ((cls‘(𝐽 ×t 𝐽))‘(◡(-g‘𝐺) “ 𝑆)))
61	31, 41, 58, 60	syl3anc 1373	. . . . . . . . . 10 ⊢ ((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺)) → ((cls‘(𝐽 ×t 𝐽))‘(𝑆 × 𝑆)) ⊆ ((cls‘(𝐽 ×t 𝐽))‘(◡(-g‘𝐺) “ 𝑆)))
62	1, 35	tgpsubcn 23953	. . . . . . . . . . . 12 ⊢ (𝐺 ∈ TopGrp → (-g‘𝐺) ∈ ((𝐽 ×t 𝐽) Cn 𝐽))
63	62	adantr 480	. . . . . . . . . . 11 ⊢ ((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺)) → (-g‘𝐺) ∈ ((𝐽 ×t 𝐽) Cn 𝐽))
64	12	cncls2i 23133	. . . . . . . . . . 11 ⊢ (((-g‘𝐺) ∈ ((𝐽 ×t 𝐽) Cn 𝐽) ∧ 𝑆 ⊆ ∪ 𝐽) → ((cls‘(𝐽 ×t 𝐽))‘(◡(-g‘𝐺) “ 𝑆)) ⊆ (◡(-g‘𝐺) “ ((cls‘𝐽)‘𝑆)))
65	63, 11, 64	syl2anc 584	. . . . . . . . . 10 ⊢ ((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺)) → ((cls‘(𝐽 ×t 𝐽))‘(◡(-g‘𝐺) “ 𝑆)) ⊆ (◡(-g‘𝐺) “ ((cls‘𝐽)‘𝑆)))
66	61, 65	sstrd 3954	. . . . . . . . 9 ⊢ ((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺)) → ((cls‘(𝐽 ×t 𝐽))‘(𝑆 × 𝑆)) ⊆ (◡(-g‘𝐺) “ ((cls‘𝐽)‘𝑆)))
67	27, 66	eqsstrrd 3979	. . . . . . . 8 ⊢ ((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺)) → (((cls‘𝐽)‘𝑆) × ((cls‘𝐽)‘𝑆)) ⊆ (◡(-g‘𝐺) “ ((cls‘𝐽)‘𝑆)))
68	67	sselda 3943	. . . . . . 7 ⊢ (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺)) ∧ ⟨𝑥, 𝑦⟩ ∈ (((cls‘𝐽)‘𝑆) × ((cls‘𝐽)‘𝑆))) → ⟨𝑥, 𝑦⟩ ∈ (◡(-g‘𝐺) “ ((cls‘𝐽)‘𝑆)))
69	25, 68	sylan2 593	. . . . . 6 ⊢ (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺)) ∧ (𝑥 ∈ ((cls‘𝐽)‘𝑆) ∧ 𝑦 ∈ ((cls‘𝐽)‘𝑆))) → ⟨𝑥, 𝑦⟩ ∈ (◡(-g‘𝐺) “ ((cls‘𝐽)‘𝑆)))
70	33	ad2antrr 726	. . . . . . 7 ⊢ (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺)) ∧ (𝑥 ∈ ((cls‘𝐽)‘𝑆) ∧ 𝑦 ∈ ((cls‘𝐽)‘𝑆))) → 𝐺 ∈ Grp)
71		ffn 6670	. . . . . . 7 ⊢ ((-g‘𝐺):((Base‘𝐺) × (Base‘𝐺))⟶(Base‘𝐺) → (-g‘𝐺) Fn ((Base‘𝐺) × (Base‘𝐺)))
72		elpreima 7012	. . . . . . 7 ⊢ ((-g‘𝐺) Fn ((Base‘𝐺) × (Base‘𝐺)) → (⟨𝑥, 𝑦⟩ ∈ (◡(-g‘𝐺) “ ((cls‘𝐽)‘𝑆)) ↔ (⟨𝑥, 𝑦⟩ ∈ ((Base‘𝐺) × (Base‘𝐺)) ∧ ((-g‘𝐺)‘⟨𝑥, 𝑦⟩) ∈ ((cls‘𝐽)‘𝑆))))
73	70, 36, 71, 72	4syl 19	. . . . . 6 ⊢ (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺)) ∧ (𝑥 ∈ ((cls‘𝐽)‘𝑆) ∧ 𝑦 ∈ ((cls‘𝐽)‘𝑆))) → (⟨𝑥, 𝑦⟩ ∈ (◡(-g‘𝐺) “ ((cls‘𝐽)‘𝑆)) ↔ (⟨𝑥, 𝑦⟩ ∈ ((Base‘𝐺) × (Base‘𝐺)) ∧ ((-g‘𝐺)‘⟨𝑥, 𝑦⟩) ∈ ((cls‘𝐽)‘𝑆))))
74	69, 73	mpbid 232	. . . . 5 ⊢ (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺)) ∧ (𝑥 ∈ ((cls‘𝐽)‘𝑆) ∧ 𝑦 ∈ ((cls‘𝐽)‘𝑆))) → (⟨𝑥, 𝑦⟩ ∈ ((Base‘𝐺) × (Base‘𝐺)) ∧ ((-g‘𝐺)‘⟨𝑥, 𝑦⟩) ∈ ((cls‘𝐽)‘𝑆)))
75	74	simprd 495	. . . 4 ⊢ (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺)) ∧ (𝑥 ∈ ((cls‘𝐽)‘𝑆) ∧ 𝑦 ∈ ((cls‘𝐽)‘𝑆))) → ((-g‘𝐺)‘⟨𝑥, 𝑦⟩) ∈ ((cls‘𝐽)‘𝑆))
76	24, 75	eqeltrid 2832	. . 3 ⊢ (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺)) ∧ (𝑥 ∈ ((cls‘𝐽)‘𝑆) ∧ 𝑦 ∈ ((cls‘𝐽)‘𝑆))) → (𝑥(-g‘𝐺)𝑦) ∈ ((cls‘𝐽)‘𝑆))
77	76	ralrimivva 3178	. 2 ⊢ ((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺)) → ∀𝑥 ∈ ((cls‘𝐽)‘𝑆)∀𝑦 ∈ ((cls‘𝐽)‘𝑆)(𝑥(-g‘𝐺)𝑦) ∈ ((cls‘𝐽)‘𝑆))
78	2, 35	issubg4 19053	. . 3 ⊢ (𝐺 ∈ Grp → (((cls‘𝐽)‘𝑆) ∈ (SubGrp‘𝐺) ↔ (((cls‘𝐽)‘𝑆) ⊆ (Base‘𝐺) ∧ ((cls‘𝐽)‘𝑆) ≠ ∅ ∧ ∀𝑥 ∈ ((cls‘𝐽)‘𝑆)∀𝑦 ∈ ((cls‘𝐽)‘𝑆)(𝑥(-g‘𝐺)𝑦) ∈ ((cls‘𝐽)‘𝑆))))
79	34, 78	syl 17	. 2 ⊢ ((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺)) → (((cls‘𝐽)‘𝑆) ∈ (SubGrp‘𝐺) ↔ (((cls‘𝐽)‘𝑆) ⊆ (Base‘𝐺) ∧ ((cls‘𝐽)‘𝑆) ≠ ∅ ∧ ∀𝑥 ∈ ((cls‘𝐽)‘𝑆)∀𝑦 ∈ ((cls‘𝐽)‘𝑆)(𝑥(-g‘𝐺)𝑦) ∈ ((cls‘𝐽)‘𝑆))))
80	15, 23, 77, 79	mpbir3and 1343	1 ⊢ ((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺)) → ((cls‘𝐽)‘𝑆) ∈ (SubGrp‘𝐺))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1086   = wceq 1540   ∈ wcel 2109   ≠ wne 2925  ∀wral 3044   ⊆ wss 3911  ∅c0 4292  ⟨cop 4591  ∪ cuni 4867   × cxp 5629  ^◡ccnv 5630  dom cdm 5631   “ cima 5634  Fun wfun 6493   Fn wfn 6494  ⟶wf 6495  ‘cfv 6499  (class class class)co 7369  Basecbs 17155  TopOpenctopn 17360  0_gc0g 17378  Grpcgrp 18841  -_gcsg 18843  SubGrpcsubg 19028  Topctop 22756  TopOnctopon 22773  clsccl 22881   Cn ccn 23087   ×_t ctx 23423  TopGrpctgp 23934

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 5229  ax-sep 5246  ax-nul 5256  ax-pow 5315  ax-pr 5382  ax-un 7691  ax-cnex 11100  ax-resscn 11101  ax-1cn 11102  ax-icn 11103  ax-addcl 11104  ax-addrcl 11105  ax-mulcl 11106  ax-mulrcl 11107  ax-mulcom 11108  ax-addass 11109  ax-mulass 11110  ax-distr 11111  ax-i2m1 11112  ax-1ne0 11113  ax-1rid 11114  ax-rnegex 11115  ax-rrecex 11116  ax-cnre 11117  ax-pre-lttri 11118  ax-pre-lttrn 11119  ax-pre-ltadd 11120  ax-pre-mulgt0 11121

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 3351  df-reu 3352  df-rab 3403  df-v 3446  df-sbc 3751  df-csb 3860  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-pss 3931  df-nul 4293  df-if 4485  df-pw 4561  df-sn 4586  df-pr 4588  df-op 4592  df-uni 4868  df-int 4907  df-iun 4953  df-iin 4954  df-br 5103  df-opab 5165  df-mpt 5184  df-tr 5210  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 6262  df-ord 6323  df-on 6324  df-lim 6325  df-suc 6326  df-iota 6452  df-fun 6501  df-fn 6502  df-f 6503  df-f1 6504  df-fo 6505  df-f1o 6506  df-fv 6507  df-riota 7326  df-ov 7372  df-oprab 7373  df-mpo 7374  df-om 7823  df-1st 7947  df-2nd 7948  df-frecs 8237  df-wrecs 8268  df-recs 8317  df-rdg 8355  df-er 8648  df-map 8778  df-en 8896  df-dom 8897  df-sdom 8898  df-pnf 11186  df-mnf 11187  df-xr 11188  df-ltxr 11189  df-le 11190  df-sub 11383  df-neg 11384  df-nn 12163  df-2 12225  df-sets 17110  df-slot 17128  df-ndx 17140  df-base 17156  df-ress 17177  df-plusg 17209  df-0g 17380  df-topgen 17382  df-plusf 18542  df-mgm 18543  df-sgrp 18622  df-mnd 18638  df-grp 18844  df-minusg 18845  df-sbg 18846  df-subg 19031  df-top 22757  df-topon 22774  df-topsp 22796  df-bases 22809  df-cld 22882  df-ntr 22883  df-cls 22884  df-cn 23090  df-tx 23425  df-tmd 23935  df-tgp 23936

This theorem is referenced by:  clsnsg  23973  tgptsmscls  24013