Theorem clssubg 22194

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 2765	. . . . . . 7 ⊢ (Base‘𝐺) = (Base‘𝐺)
3	1, 2	tgptopon 22168	. . . . . 6 ⊢ (𝐺 ∈ TopGrp → 𝐽 ∈ (TopOn‘(Base‘𝐺)))
4	3	adantr 472	. . . . 5 ⊢ ((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺)) → 𝐽 ∈ (TopOn‘(Base‘𝐺)))
5		topontop 21000	. . . . 5 ⊢ (𝐽 ∈ (TopOn‘(Base‘𝐺)) → 𝐽 ∈ Top)
6	4, 5	syl 17	. . . 4 ⊢ ((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺)) → 𝐽 ∈ Top)
7	2	subgss 17862	. . . . . 6 ⊢ (𝑆 ∈ (SubGrp‘𝐺) → 𝑆 ⊆ (Base‘𝐺))
8	7	adantl 473	. . . . 5 ⊢ ((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺)) → 𝑆 ⊆ (Base‘𝐺))
9		toponuni 21001	. . . . . 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 3803	. . . 4 ⊢ ((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺)) → 𝑆 ⊆ ∪ 𝐽)
12		eqid 2765	. . . . 5 ⊢ ∪ 𝐽 = ∪ 𝐽
13	12	clsss3 21146	. . . 4 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ ∪ 𝐽) → ((cls‘𝐽)‘𝑆) ⊆ ∪ 𝐽)
14	6, 11, 13	syl2anc 579	. . 3 ⊢ ((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺)) → ((cls‘𝐽)‘𝑆) ⊆ ∪ 𝐽)
15	14, 10	sseqtr4d 3804	. 2 ⊢ ((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺)) → ((cls‘𝐽)‘𝑆) ⊆ (Base‘𝐺))
16	12	sscls 21143	. . . 4 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ ∪ 𝐽) → 𝑆 ⊆ ((cls‘𝐽)‘𝑆))
17	6, 11, 16	syl2anc 579	. . 3 ⊢ ((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺)) → 𝑆 ⊆ ((cls‘𝐽)‘𝑆))
18		eqid 2765	. . . . . 6 ⊢ (0g‘𝐺) = (0g‘𝐺)
19	18	subg0cl 17869	. . . . 5 ⊢ (𝑆 ∈ (SubGrp‘𝐺) → (0g‘𝐺) ∈ 𝑆)
20	19	adantl 473	. . . 4 ⊢ ((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺)) → (0g‘𝐺) ∈ 𝑆)
21	20	ne0d 4088	. . 3 ⊢ ((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺)) → 𝑆 ≠ ∅)
22		ssn0 4140	. . 3 ⊢ ((𝑆 ⊆ ((cls‘𝐽)‘𝑆) ∧ 𝑆 ≠ ∅) → ((cls‘𝐽)‘𝑆) ≠ ∅)
23	17, 21, 22	syl2anc 579	. 2 ⊢ ((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺)) → ((cls‘𝐽)‘𝑆) ≠ ∅)
24		df-ov 6847	. . . 4 ⊢ (𝑥(-g‘𝐺)𝑦) = ((-g‘𝐺)‘⟨𝑥, 𝑦⟩)
25		opelxpi 5316	. . . . . . 7 ⊢ ((𝑥 ∈ ((cls‘𝐽)‘𝑆) ∧ 𝑦 ∈ ((cls‘𝐽)‘𝑆)) → ⟨𝑥, 𝑦⟩ ∈ (((cls‘𝐽)‘𝑆) × ((cls‘𝐽)‘𝑆)))
26		txcls 21690	. . . . . . . . . 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 867	. . . . . . . . 9 ⊢ ((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺)) → ((cls‘(𝐽 ×t 𝐽))‘(𝑆 × 𝑆)) = (((cls‘𝐽)‘𝑆) × ((cls‘𝐽)‘𝑆)))
28		txtopon 21677	. . . . . . . . . . . . 13 ⊢ ((𝐽 ∈ (TopOn‘(Base‘𝐺)) ∧ 𝐽 ∈ (TopOn‘(Base‘𝐺))) → (𝐽 ×t 𝐽) ∈ (TopOn‘((Base‘𝐺) × (Base‘𝐺))))
29	4, 4, 28	syl2anc 579	. . . . . . . . . . . 12 ⊢ ((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺)) → (𝐽 ×t 𝐽) ∈ (TopOn‘((Base‘𝐺) × (Base‘𝐺))))
30		topontop 21000	. . . . . . . . . . . 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 5669	. . . . . . . . . . . . 13 ⊢ (◡(-g‘𝐺) “ 𝑆) ⊆ dom (-g‘𝐺)
33		tgpgrp 22164	. . . . . . . . . . . . . . 15 ⊢ (𝐺 ∈ TopGrp → 𝐺 ∈ Grp)
34	33	adantr 472	. . . . . . . . . . . . . 14 ⊢ ((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺)) → 𝐺 ∈ Grp)
35		eqid 2765	. . . . . . . . . . . . . . 15 ⊢ (-g‘𝐺) = (-g‘𝐺)
36	2, 35	grpsubf 17764	. . . . . . . . . . . . . 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 6241	. . . . . . . . . . . 12 ⊢ ((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺)) → (◡(-g‘𝐺) “ 𝑆) ⊆ ((Base‘𝐺) × (Base‘𝐺)))
39		toponuni 21001	. . . . . . . . . . . . 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 3803	. . . . . . . . . . 11 ⊢ ((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺)) → (◡(-g‘𝐺) “ 𝑆) ⊆ ∪ (𝐽 ×t 𝐽))
42	35	subgsubcl 17872	. . . . . . . . . . . . . . . 16 ⊢ ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆) → (𝑥(-g‘𝐺)𝑦) ∈ 𝑆)
43	42	3expb 1149	. . . . . . . . . . . . . . 15 ⊢ ((𝑆 ∈ (SubGrp‘𝐺) ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥(-g‘𝐺)𝑦) ∈ 𝑆)
44	43	ralrimivva 3118	. . . . . . . . . . . . . 14 ⊢ (𝑆 ∈ (SubGrp‘𝐺) → ∀𝑥 ∈ 𝑆 ∀𝑦 ∈ 𝑆 (𝑥(-g‘𝐺)𝑦) ∈ 𝑆)
45		fveq2 6377	. . . . . . . . . . . . . . . . 17 ⊢ (𝑧 = ⟨𝑥, 𝑦⟩ → ((-g‘𝐺)‘𝑧) = ((-g‘𝐺)‘⟨𝑥, 𝑦⟩))
46	45, 24	syl6eqr 2817	. . . . . . . . . . . . . . . 16 ⊢ (𝑧 = ⟨𝑥, 𝑦⟩ → ((-g‘𝐺)‘𝑧) = (𝑥(-g‘𝐺)𝑦))
47	46	eleq1d 2829	. . . . . . . . . . . . . . 15 ⊢ (𝑧 = ⟨𝑥, 𝑦⟩ → (((-g‘𝐺)‘𝑧) ∈ 𝑆 ↔ (𝑥(-g‘𝐺)𝑦) ∈ 𝑆))
48	47	ralxp 5434	. . . . . . . . . . . . . 14 ⊢ (∀𝑧 ∈ (𝑆 × 𝑆)((-g‘𝐺)‘𝑧) ∈ 𝑆 ↔ ∀𝑥 ∈ 𝑆 ∀𝑦 ∈ 𝑆 (𝑥(-g‘𝐺)𝑦) ∈ 𝑆)
49	44, 48	sylibr 225	. . . . . . . . . . . . 13 ⊢ (𝑆 ∈ (SubGrp‘𝐺) → ∀𝑧 ∈ (𝑆 × 𝑆)((-g‘𝐺)‘𝑧) ∈ 𝑆)
50	49	adantl 473	. . . . . . . . . . . 12 ⊢ ((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺)) → ∀𝑧 ∈ (𝑆 × 𝑆)((-g‘𝐺)‘𝑧) ∈ 𝑆)
51	37	ffund 6229	. . . . . . . . . . . . 13 ⊢ ((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺)) → Fun (-g‘𝐺))
52		xpss12 5294	. . . . . . . . . . . . . . 15 ⊢ ((𝑆 ⊆ (Base‘𝐺) ∧ 𝑆 ⊆ (Base‘𝐺)) → (𝑆 × 𝑆) ⊆ ((Base‘𝐺) × (Base‘𝐺)))
53	8, 8, 52	syl2anc 579	. . . . . . . . . . . . . 14 ⊢ ((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺)) → (𝑆 × 𝑆) ⊆ ((Base‘𝐺) × (Base‘𝐺)))
54	37	fdmd 6234	. . . . . . . . . . . . . 14 ⊢ ((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺)) → dom (-g‘𝐺) = ((Base‘𝐺) × (Base‘𝐺)))
55	53, 54	sseqtr4d 3804	. . . . . . . . . . . . 13 ⊢ ((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺)) → (𝑆 × 𝑆) ⊆ dom (-g‘𝐺))
56		funimass5 6526	. . . . . . . . . . . . 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 579	. . . . . . . . . . . 12 ⊢ ((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺)) → ((𝑆 × 𝑆) ⊆ (◡(-g‘𝐺) “ 𝑆) ↔ ∀𝑧 ∈ (𝑆 × 𝑆)((-g‘𝐺)‘𝑧) ∈ 𝑆))
58	50, 57	mpbird 248	. . . . . . . . . . 11 ⊢ ((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺)) → (𝑆 × 𝑆) ⊆ (◡(-g‘𝐺) “ 𝑆))
59		eqid 2765	. . . . . . . . . . . 12 ⊢ ∪ (𝐽 ×t 𝐽) = ∪ (𝐽 ×t 𝐽)
60	59	clsss 21141	. . . . . . . . . . 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 1490	. . . . . . . . . 10 ⊢ ((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺)) → ((cls‘(𝐽 ×t 𝐽))‘(𝑆 × 𝑆)) ⊆ ((cls‘(𝐽 ×t 𝐽))‘(◡(-g‘𝐺) “ 𝑆)))
62	1, 35	tgpsubcn 22176	. . . . . . . . . . . 12 ⊢ (𝐺 ∈ TopGrp → (-g‘𝐺) ∈ ((𝐽 ×t 𝐽) Cn 𝐽))
63	62	adantr 472	. . . . . . . . . . 11 ⊢ ((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺)) → (-g‘𝐺) ∈ ((𝐽 ×t 𝐽) Cn 𝐽))
64	12	cncls2i 21357	. . . . . . . . . . 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 579	. . . . . . . . . 10 ⊢ ((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺)) → ((cls‘(𝐽 ×t 𝐽))‘(◡(-g‘𝐺) “ 𝑆)) ⊆ (◡(-g‘𝐺) “ ((cls‘𝐽)‘𝑆)))
66	61, 65	sstrd 3773	. . . . . . . . 9 ⊢ ((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺)) → ((cls‘(𝐽 ×t 𝐽))‘(𝑆 × 𝑆)) ⊆ (◡(-g‘𝐺) “ ((cls‘𝐽)‘𝑆)))
67	27, 66	eqsstr3d 3802	. . . . . . . 8 ⊢ ((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺)) → (((cls‘𝐽)‘𝑆) × ((cls‘𝐽)‘𝑆)) ⊆ (◡(-g‘𝐺) “ ((cls‘𝐽)‘𝑆)))
68	67	sselda 3763	. . . . . . 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 586	. . . . . 6 ⊢ (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺)) ∧ (𝑥 ∈ ((cls‘𝐽)‘𝑆) ∧ 𝑦 ∈ ((cls‘𝐽)‘𝑆))) → ⟨𝑥, 𝑦⟩ ∈ (◡(-g‘𝐺) “ ((cls‘𝐽)‘𝑆)))
70	33	ad2antrr 717	. . . . . . 7 ⊢ (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺)) ∧ (𝑥 ∈ ((cls‘𝐽)‘𝑆) ∧ 𝑦 ∈ ((cls‘𝐽)‘𝑆))) → 𝐺 ∈ Grp)
71		ffn 6225	. . . . . . 7 ⊢ ((-g‘𝐺):((Base‘𝐺) × (Base‘𝐺))⟶(Base‘𝐺) → (-g‘𝐺) Fn ((Base‘𝐺) × (Base‘𝐺)))
72		elpreima 6529	. . . . . . 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 223	. . . . 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 489	. . . 4 ⊢ (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺)) ∧ (𝑥 ∈ ((cls‘𝐽)‘𝑆) ∧ 𝑦 ∈ ((cls‘𝐽)‘𝑆))) → ((-g‘𝐺)‘⟨𝑥, 𝑦⟩) ∈ ((cls‘𝐽)‘𝑆))
76	24, 75	syl5eqel 2848	. . 3 ⊢ (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺)) ∧ (𝑥 ∈ ((cls‘𝐽)‘𝑆) ∧ 𝑦 ∈ ((cls‘𝐽)‘𝑆))) → (𝑥(-g‘𝐺)𝑦) ∈ ((cls‘𝐽)‘𝑆))
77	76	ralrimivva 3118	. 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 17880	. . 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 1442	1 ⊢ ((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺)) → ((cls‘𝐽)‘𝑆) ∈ (SubGrp‘𝐺))

Syntax hints:   → wi 4   ↔ wb 197   ∧ wa 384   ∧ w3a 1107   = wceq 1652   ∈ wcel 2155   ≠ wne 2937  ∀wral 3055   ⊆ wss 3734  ∅c0 4081  ⟨cop 4342  ∪ cuni 4596   × cxp 5277  ^◡ccnv 5278  dom cdm 5279   “ cima 5282  Fun wfun 6064   Fn wfn 6065  ⟶wf 6066  ‘cfv 6070  (class class class)co 6844  Basecbs 16133  TopOpenctopn 16351  0_gc0g 16369  Grpcgrp 17692  -_gcsg 17694  SubGrpcsubg 17855  Topctop 20980  TopOnctopon 20997  clsccl 21105   Cn ccn 21311   ×_t ctx 21646  TopGrpctgp 22157

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2070  ax-7 2105  ax-8 2157  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-13 2352  ax-ext 2743  ax-rep 4932  ax-sep 4943  ax-nul 4951  ax-pow 5003  ax-pr 5064  ax-un 7149  ax-cnex 10247  ax-resscn 10248  ax-1cn 10249  ax-icn 10250  ax-addcl 10251  ax-addrcl 10252  ax-mulcl 10253  ax-mulrcl 10254  ax-mulcom 10255  ax-addass 10256  ax-mulass 10257  ax-distr 10258  ax-i2m1 10259  ax-1ne0 10260  ax-1rid 10261  ax-rnegex 10262  ax-rrecex 10263  ax-cnre 10264  ax-pre-lttri 10265  ax-pre-lttrn 10266  ax-pre-ltadd 10267  ax-pre-mulgt0 10268

This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-3or 1108  df-3an 1109  df-tru 1656  df-ex 1875  df-nf 1879  df-sb 2063  df-mo 2565  df-eu 2582  df-clab 2752  df-cleq 2758  df-clel 2761  df-nfc 2896  df-ne 2938  df-nel 3041  df-ral 3060  df-rex 3061  df-reu 3062  df-rmo 3063  df-rab 3064  df-v 3352  df-sbc 3599  df-csb 3694  df-dif 3737  df-un 3739  df-in 3741  df-ss 3748  df-pss 3750  df-nul 4082  df-if 4246  df-pw 4319  df-sn 4337  df-pr 4339  df-tp 4341  df-op 4343  df-uni 4597  df-int 4636  df-iun 4680  df-iin 4681  df-br 4812  df-opab 4874  df-mpt 4891  df-tr 4914  df-id 5187  df-eprel 5192  df-po 5200  df-so 5201  df-fr 5238  df-we 5240  df-xp 5285  df-rel 5286  df-cnv 5287  df-co 5288  df-dm 5289  df-rn 5290  df-res 5291  df-ima 5292  df-pred 5867  df-ord 5913  df-on 5914  df-lim 5915  df-suc 5916  df-iota 6033  df-fun 6072  df-fn 6073  df-f 6074  df-f1 6075  df-fo 6076  df-f1o 6077  df-fv 6078  df-riota 6805  df-ov 6847  df-oprab 6848  df-mpt2 6849  df-om 7266  df-1st 7368  df-2nd 7369  df-wrecs 7612  df-recs 7674  df-rdg 7712  df-er 7949  df-map 8064  df-en 8163  df-dom 8164  df-sdom 8165  df-pnf 10332  df-mnf 10333  df-xr 10334  df-ltxr 10335  df-le 10336  df-sub 10524  df-neg 10525  df-nn 11277  df-2 11337  df-ndx 16136  df-slot 16137  df-base 16139  df-sets 16140  df-ress 16141  df-plusg 16230  df-0g 16371  df-topgen 16373  df-plusf 17510  df-mgm 17511  df-sgrp 17553  df-mnd 17564  df-grp 17695  df-minusg 17696  df-sbg 17697  df-subg 17858  df-top 20981  df-topon 20998  df-topsp 21020  df-bases 21033  df-cld 21106  df-ntr 21107  df-cls 21108  df-cn 21314  df-tx 21648  df-tmd 22158  df-tgp 22159

This theorem is referenced by:  clsnsg  22195  tgptsmscls  22235