Theorem clssubg 24421

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 2761	. . . . . . 7 ⊢ (Base‘𝐺) = (Base‘𝐺)
3	1, 2	tgptopon 24394	. . . . . 6 ⊢ (𝐺 ∈ TopGrp → 𝐽 ∈ (TopOn‘(Base‘𝐺)))
4	3	adantr 486	. . . . 5 ⊢ ((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺)) → 𝐽 ∈ (TopOn‘(Base‘𝐺)))
5		topontop 23224	. . . . 5 ⊢ (𝐽 ∈ (TopOn‘(Base‘𝐺)) → 𝐽 ∈ Top)
6	4, 5	syl 18	. . . 4 ⊢ ((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺)) → 𝐽 ∈ Top)
7	2	subgss 19330	. . . . . 6 ⊢ (𝑆 ∈ (SubGrp‘𝐺) → 𝑆 ⊆ (Base‘𝐺))
8	7	adantl 487	. . . . 5 ⊢ ((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺)) → 𝑆 ⊆ (Base‘𝐺))
9		toponuni 23225	. . . . . 6 ⊢ (𝐽 ∈ (TopOn‘(Base‘𝐺)) → (Base‘𝐺) = ∪ 𝐽)
10	4, 9	syl 18	. . . . 5 ⊢ ((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺)) → (Base‘𝐺) = ∪ 𝐽)
11	8, 10	sseqtrd 3967	. . . 4 ⊢ ((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺)) → 𝑆 ⊆ ∪ 𝐽)
12		eqid 2761	. . . . 5 ⊢ ∪ 𝐽 = ∪ 𝐽
13	12	clsss3 23370	. . . 4 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ ∪ 𝐽) → ((cls‘𝐽)‘𝑆) ⊆ ∪ 𝐽)
14	6, 11, 13	syl2anc 596	. . 3 ⊢ ((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺)) → ((cls‘𝐽)‘𝑆) ⊆ ∪ 𝐽)
15	14, 10	sseqtrrd 3968	. 2 ⊢ ((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺)) → ((cls‘𝐽)‘𝑆) ⊆ (Base‘𝐺))
16	12	sscls 23367	. . . 4 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ ∪ 𝐽) → 𝑆 ⊆ ((cls‘𝐽)‘𝑆))
17	6, 11, 16	syl2anc 596	. . 3 ⊢ ((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺)) → 𝑆 ⊆ ((cls‘𝐽)‘𝑆))
18		eqid 2761	. . . . . 6 ⊢ (0g‘𝐺) = (0g‘𝐺)
19	18	subg0cl 19337	. . . . 5 ⊢ (𝑆 ∈ (SubGrp‘𝐺) → (0g‘𝐺) ∈ 𝑆)
20	19	adantl 487	. . . 4 ⊢ ((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺)) → (0g‘𝐺) ∈ 𝑆)
21	20	ne0d 4288	. . 3 ⊢ ((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺)) → 𝑆 ≠ ∅)
22		ssn0 4355	. . 3 ⊢ ((𝑆 ⊆ ((cls‘𝐽)‘𝑆) ∧ 𝑆 ≠ ∅) → ((cls‘𝐽)‘𝑆) ≠ ∅)
23	17, 21, 22	syl2anc 596	. 2 ⊢ ((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺)) → ((cls‘𝐽)‘𝑆) ≠ ∅)
24		df-ov 7421	. . . 4 ⊢ (𝑥(-g‘𝐺)𝑦) = ((-g‘𝐺)‘⟨𝑥, 𝑦⟩)
25		opelxpi 5688	. . . . . . 7 ⊢ ((𝑥 ∈ ((cls‘𝐽)‘𝑆) ∧ 𝑦 ∈ ((cls‘𝐽)‘𝑆)) → ⟨𝑥, 𝑦⟩ ∈ (((cls‘𝐽)‘𝑆) × ((cls‘𝐽)‘𝑆)))
26		txcls 23916	. . . . . . . . . 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 852	. . . . . . . . 9 ⊢ ((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺)) → ((cls‘(𝐽 ×t 𝐽))‘(𝑆 × 𝑆)) = (((cls‘𝐽)‘𝑆) × ((cls‘𝐽)‘𝑆)))
28		txtopon 23903	. . . . . . . . . . . . 13 ⊢ ((𝐽 ∈ (TopOn‘(Base‘𝐺)) ∧ 𝐽 ∈ (TopOn‘(Base‘𝐺))) → (𝐽 ×t 𝐽) ∈ (TopOn‘((Base‘𝐺) × (Base‘𝐺))))
29	4, 4, 28	syl2anc 596	. . . . . . . . . . . 12 ⊢ ((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺)) → (𝐽 ×t 𝐽) ∈ (TopOn‘((Base‘𝐺) × (Base‘𝐺))))
30		topontop 23224	. . . . . . . . . . . 12 ⊢ ((𝐽 ×t 𝐽) ∈ (TopOn‘((Base‘𝐺) × (Base‘𝐺))) → (𝐽 ×t 𝐽) ∈ Top)
31	29, 30	syl 18	. . . . . . . . . . 11 ⊢ ((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺)) → (𝐽 ×t 𝐽) ∈ Top)
32		cnvimass 6197	. . . . . . . . . . . . 13 ⊢ (◡(-g‘𝐺) “ 𝑆) ⊆ dom (-g‘𝐺)
33		tgpgrp 24390	. . . . . . . . . . . . . . 15 ⊢ (𝐺 ∈ TopGrp → 𝐺 ∈ Grp)
34	33	adantr 486	. . . . . . . . . . . . . 14 ⊢ ((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺)) → 𝐺 ∈ Grp)
35		eqid 2761	. . . . . . . . . . . . . . 15 ⊢ (-g‘𝐺) = (-g‘𝐺)
36	2, 35	grpsubf 19222	. . . . . . . . . . . . . 14 ⊢ (𝐺 ∈ Grp → (-g‘𝐺):((Base‘𝐺) × (Base‘𝐺))⟶(Base‘𝐺))
37	34, 36	syl 18	. . . . . . . . . . . . 13 ⊢ ((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺)) → (-g‘𝐺):((Base‘𝐺) × (Base‘𝐺))⟶(Base‘𝐺))
38	32, 37	fssdm 6727	. . . . . . . . . . . 12 ⊢ ((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺)) → (◡(-g‘𝐺) “ 𝑆) ⊆ ((Base‘𝐺) × (Base‘𝐺)))
39		toponuni 23225	. . . . . . . . . . . . 13 ⊢ ((𝐽 ×t 𝐽) ∈ (TopOn‘((Base‘𝐺) × (Base‘𝐺))) → ((Base‘𝐺) × (Base‘𝐺)) = ∪ (𝐽 ×t 𝐽))
40	29, 39	syl 18	. . . . . . . . . . . 12 ⊢ ((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺)) → ((Base‘𝐺) × (Base‘𝐺)) = ∪ (𝐽 ×t 𝐽))
41	38, 40	sseqtrd 3967	. . . . . . . . . . 11 ⊢ ((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺)) → (◡(-g‘𝐺) “ 𝑆) ⊆ ∪ (𝐽 ×t 𝐽))
42	35	subgsubcl 19341	. . . . . . . . . . . . . . . 16 ⊢ ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆) → (𝑥(-g‘𝐺)𝑦) ∈ 𝑆)
43	42	3expb 1138	. . . . . . . . . . . . . . 15 ⊢ ((𝑆 ∈ (SubGrp‘𝐺) ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥(-g‘𝐺)𝑦) ∈ 𝑆)
44	43	ralrimivva 3206	. . . . . . . . . . . . . 14 ⊢ (𝑆 ∈ (SubGrp‘𝐺) → ∀𝑥 ∈ 𝑆 ∀𝑦 ∈ 𝑆 (𝑥(-g‘𝐺)𝑦) ∈ 𝑆)
45		fveq2 6883	. . . . . . . . . . . . . . . . 17 ⊢ (𝑧 = ⟨𝑥, 𝑦⟩ → ((-g‘𝐺)‘𝑧) = ((-g‘𝐺)‘⟨𝑥, 𝑦⟩))
46	45, 24	eqtr4di 2814	. . . . . . . . . . . . . . . 16 ⊢ (𝑧 = ⟨𝑥, 𝑦⟩ → ((-g‘𝐺)‘𝑧) = (𝑥(-g‘𝐺)𝑦))
47	46	eleq1d 2846	. . . . . . . . . . . . . . 15 ⊢ (𝑧 = ⟨𝑥, 𝑦⟩ → (((-g‘𝐺)‘𝑧) ∈ 𝑆 ↔ (𝑥(-g‘𝐺)𝑦) ∈ 𝑆))
48	47	ralxp 5818	. . . . . . . . . . . . . 14 ⊢ (∀𝑧 ∈ (𝑆 × 𝑆)((-g‘𝐺)‘𝑧) ∈ 𝑆 ↔ ∀𝑥 ∈ 𝑆 ∀𝑦 ∈ 𝑆 (𝑥(-g‘𝐺)𝑦) ∈ 𝑆)
49	44, 48	sylibr 237	. . . . . . . . . . . . 13 ⊢ (𝑆 ∈ (SubGrp‘𝐺) → ∀𝑧 ∈ (𝑆 × 𝑆)((-g‘𝐺)‘𝑧) ∈ 𝑆)
50	49	adantl 487	. . . . . . . . . . . 12 ⊢ ((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺)) → ∀𝑧 ∈ (𝑆 × 𝑆)((-g‘𝐺)‘𝑧) ∈ 𝑆)
51	37	ffund 6712	. . . . . . . . . . . . 13 ⊢ ((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺)) → Fun (-g‘𝐺))
52		xpss12 5666	. . . . . . . . . . . . . . 15 ⊢ ((𝑆 ⊆ (Base‘𝐺) ∧ 𝑆 ⊆ (Base‘𝐺)) → (𝑆 × 𝑆) ⊆ ((Base‘𝐺) × (Base‘𝐺)))
53	8, 8, 52	syl2anc 596	. . . . . . . . . . . . . 14 ⊢ ((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺)) → (𝑆 × 𝑆) ⊆ ((Base‘𝐺) × (Base‘𝐺)))
54	37	fdmd 6718	. . . . . . . . . . . . . 14 ⊢ ((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺)) → dom (-g‘𝐺) = ((Base‘𝐺) × (Base‘𝐺)))
55	53, 54	sseqtrrd 3968	. . . . . . . . . . . . 13 ⊢ ((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺)) → (𝑆 × 𝑆) ⊆ dom (-g‘𝐺))
56		funimass5 7052	. . . . . . . . . . . . 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 596	. . . . . . . . . . . 12 ⊢ ((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺)) → ((𝑆 × 𝑆) ⊆ (◡(-g‘𝐺) “ 𝑆) ↔ ∀𝑧 ∈ (𝑆 × 𝑆)((-g‘𝐺)‘𝑧) ∈ 𝑆))
58	50, 57	mpbird 260	. . . . . . . . . . 11 ⊢ ((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺)) → (𝑆 × 𝑆) ⊆ (◡(-g‘𝐺) “ 𝑆))
59		eqid 2761	. . . . . . . . . . . 12 ⊢ ∪ (𝐽 ×t 𝐽) = ∪ (𝐽 ×t 𝐽)
60	59	clsss 23365	. . . . . . . . . . 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 1398	. . . . . . . . . 10 ⊢ ((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺)) → ((cls‘(𝐽 ×t 𝐽))‘(𝑆 × 𝑆)) ⊆ ((cls‘(𝐽 ×t 𝐽))‘(◡(-g‘𝐺) “ 𝑆)))
62	1, 35	tgpsubcn 24402	. . . . . . . . . . . 12 ⊢ (𝐺 ∈ TopGrp → (-g‘𝐺) ∈ ((𝐽 ×t 𝐽) Cn 𝐽))
63	62	adantr 486	. . . . . . . . . . 11 ⊢ ((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺)) → (-g‘𝐺) ∈ ((𝐽 ×t 𝐽) Cn 𝐽))
64	12	cncls2i 23581	. . . . . . . . . . 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 596	. . . . . . . . . 10 ⊢ ((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺)) → ((cls‘(𝐽 ×t 𝐽))‘(◡(-g‘𝐺) “ 𝑆)) ⊆ (◡(-g‘𝐺) “ ((cls‘𝐽)‘𝑆)))
66	61, 65	sstrd 3941	. . . . . . . . 9 ⊢ ((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺)) → ((cls‘(𝐽 ×t 𝐽))‘(𝑆 × 𝑆)) ⊆ (◡(-g‘𝐺) “ ((cls‘𝐽)‘𝑆)))
67	27, 66	eqsstrrd 3966	. . . . . . . 8 ⊢ ((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺)) → (((cls‘𝐽)‘𝑆) × ((cls‘𝐽)‘𝑆)) ⊆ (◡(-g‘𝐺) “ ((cls‘𝐽)‘𝑆)))
68	67	sselda 3931	. . . . . . 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 605	. . . . . 6 ⊢ (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺)) ∧ (𝑥 ∈ ((cls‘𝐽)‘𝑆) ∧ 𝑦 ∈ ((cls‘𝐽)‘𝑆))) → ⟨𝑥, 𝑦⟩ ∈ (◡(-g‘𝐺) “ ((cls‘𝐽)‘𝑆)))
70	33	ad2antrr 739	. . . . . . 7 ⊢ (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺)) ∧ (𝑥 ∈ ((cls‘𝐽)‘𝑆) ∧ 𝑦 ∈ ((cls‘𝐽)‘𝑆))) → 𝐺 ∈ Grp)
71		ffn 6707	. . . . . . 7 ⊢ ((-g‘𝐺):((Base‘𝐺) × (Base‘𝐺))⟶(Base‘𝐺) → (-g‘𝐺) Fn ((Base‘𝐺) × (Base‘𝐺)))
72		elpreima 7055	. . . . . . 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 20	. . . . . 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 235	. . . . 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 501	. . . 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 2865	. . 3 ⊢ (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺)) ∧ (𝑥 ∈ ((cls‘𝐽)‘𝑆) ∧ 𝑦 ∈ ((cls‘𝐽)‘𝑆))) → (𝑥(-g‘𝐺)𝑦) ∈ ((cls‘𝐽)‘𝑆))
77	76	ralrimivva 3206	. 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 19349	. . 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 18	. 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 1361	1 ⊢ ((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺)) → ((cls‘𝐽)‘𝑆) ∈ (SubGrp‘𝐺))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   ∧ w3a 1103   = wceq 1570   ∈ wcel 2145   ≠ wne 2956  ∀wral 3077   ⊆ wss 3899  ∅c0 4279  ⟨cop 4590  ∪ cuni 4867   × cxp 5649  ^◡ccnv 5650  dom cdm 5651   “ cima 5654  Fun wfun 6531   Fn wfn 6532  ⟶wf 6533  ‘cfv 6537  (class class class)co 7418  Basecbs 17380  TopOpenctopn 17585  0_gc0g 17603  Grpcgrp 19137  -_gcsg 19139  SubGrpcsubg 19323  Topctop 23204  TopOnctopon 23221  clsccl 23329   Cn ccn 23535   ×_t ctx 23872  TopGrpctgp 24383

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 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2733  ax-rep 5232  ax-sep 5249  ax-nul 5260  ax-pow 5327  ax-pr 5391  ax-un 7749  ax-cnex 11249  ax-resscn 11250  ax-1cn 11251  ax-icn 11252  ax-addcl 11253  ax-addrcl 11254  ax-mulcl 11255  ax-mulrcl 11256  ax-mulcom 11257  ax-addass 11258  ax-mulass 11259  ax-distr 11260  ax-i2m1 11261  ax-1ne0 11262  ax-1rid 11263  ax-rnegex 11264  ax-rrecex 11265  ax-cnre 11266  ax-pre-lttri 11267  ax-pre-lttrn 11268  ax-pre-ltadd 11269  ax-pre-mulgt0 11270

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 2100  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-nel 3063  df-ral 3078  df-rex 3088  df-rmo 3366  df-reu 3367  df-rab 3414  df-v 3453  df-sbc 3740  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-pss 3919  df-nul 4280  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-int 4908  df-iun 4953  df-iin 4954  df-br 5104  df-opab 5168  df-mpt 5187  df-tr 5213  df-id 5546  df-eprel 5551  df-po 5559  df-so 5560  df-fr 5604  df-we 5606  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-rn 5662  df-res 5663  df-ima 5664  df-pred 6303  df-ord 6364  df-on 6365  df-lim 6366  df-suc 6367  df-iota 6493  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-riota 7375  df-ov 7421  df-oprab 7422  df-mpo 7423  df-om 7876  df-1st 7999  df-2nd 8000  df-frecs 8292  df-wrecs 8323  df-recs 8372  df-rdg 8411  df-er 8710  df-map 8842  df-en 8967  df-dom 8968  df-sdom 8969  df-pnf 11338  df-mnf 11339  df-xr 11340  df-ltxr 11341  df-le 11342  df-sub 11536  df-neg 11537  df-nn 12329  df-2 12398  df-sets 17335  df-slot 17353  df-ndx 17365  df-base 17381  df-ress 17402  df-plusg 17434  df-0g 17605  df-topgen 17607  df-plusf 18808  df-mgm 18809  df-sgrp 18901  df-mnd 18917  df-grp 19140  df-minusg 19141  df-sbg 19142  df-subg 19326  df-top 23205  df-topon 23222  df-topsp 23244  df-bases 23257  df-cld 23330  df-ntr 23331  df-cls 23332  df-cn 23538  df-tx 23874  df-tmd 24384  df-tgp 24385

This theorem is used by:  clsnsg  24422  tgptsmscls  24462