Theorem tgpconncomp 24153

Description: The identity component, the connected component containing the identity element, is a closed (conncompcld 23474) normal subgroup. (Contributed by Mario Carneiro, 17-Sep-2015.)

Hypotheses

Ref	Expression
tgpconncomp.x	⊢ 𝑋 = (Base‘𝐺)
tgpconncomp.z	⊢ 0 = (0g‘𝐺)
tgpconncomp.j	⊢ 𝐽 = (TopOpen‘𝐺)
tgpconncomp.s	⊢ 𝑆 = ∪ {𝑥 ∈ 𝒫 𝑋 ∣ ( 0 ∈ 𝑥 ∧ (𝐽 ↾t 𝑥) ∈ Conn)}

Assertion

Ref	Expression
tgpconncomp	⊢ (𝐺 ∈ TopGrp → 𝑆 ∈ (NrmSGrp‘𝐺))

Distinct variable groups:   𝑥, 0   𝑥,𝐽   𝑥,𝐺   𝑥,𝑋

Allowed substitution hint:   𝑆(𝑥)

Proof of Theorem tgpconncomp

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

Step	Hyp	Ref	Expression
1		tgpconncomp.s	. . . . 5 ⊢ 𝑆 = ∪ {𝑥 ∈ 𝒫 𝑋 ∣ ( 0 ∈ 𝑥 ∧ (𝐽 ↾t 𝑥) ∈ Conn)}
2		ssrab2 4033	. . . . . 6 ⊢ {𝑥 ∈ 𝒫 𝑋 ∣ ( 0 ∈ 𝑥 ∧ (𝐽 ↾t 𝑥) ∈ Conn)} ⊆ 𝒫 𝑋
3		sspwuni 5056	. . . . . 6 ⊢ ({𝑥 ∈ 𝒫 𝑋 ∣ ( 0 ∈ 𝑥 ∧ (𝐽 ↾t 𝑥) ∈ Conn)} ⊆ 𝒫 𝑋 ↔ ∪ {𝑥 ∈ 𝒫 𝑋 ∣ ( 0 ∈ 𝑥 ∧ (𝐽 ↾t 𝑥) ∈ Conn)} ⊆ 𝑋)
4	2, 3	mpbi 232	. . . . 5 ⊢ ∪ {𝑥 ∈ 𝒫 𝑋 ∣ ( 0 ∈ 𝑥 ∧ (𝐽 ↾t 𝑥) ∈ Conn)} ⊆ 𝑋
5	1, 4	eqsstri 3982	. . . 4 ⊢ 𝑆 ⊆ 𝑋
6	5	a1i 11	. . 3 ⊢ (𝐺 ∈ TopGrp → 𝑆 ⊆ 𝑋)
7		tgpconncomp.j	. . . . . 6 ⊢ 𝐽 = (TopOpen‘𝐺)
8		tgpconncomp.x	. . . . . 6 ⊢ 𝑋 = (Base‘𝐺)
9	7, 8	tgptopon 24122	. . . . 5 ⊢ (𝐺 ∈ TopGrp → 𝐽 ∈ (TopOn‘𝑋))
10		tgpgrp 24118	. . . . . 6 ⊢ (𝐺 ∈ TopGrp → 𝐺 ∈ Grp)
11		tgpconncomp.z	. . . . . . 7 ⊢ 0 = (0g‘𝐺)
12	8, 11	grpidcl 18990	. . . . . 6 ⊢ (𝐺 ∈ Grp → 0 ∈ 𝑋)
13	10, 12	syl 17	. . . . 5 ⊢ (𝐺 ∈ TopGrp → 0 ∈ 𝑋)
14	1	conncompid 23471	. . . . 5 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 0 ∈ 𝑋) → 0 ∈ 𝑆)
15	9, 13, 14	syl2anc 593	. . . 4 ⊢ (𝐺 ∈ TopGrp → 0 ∈ 𝑆)
16	15	ne0d 4294	. . 3 ⊢ (𝐺 ∈ TopGrp → 𝑆 ≠ ∅)
17		df-ima 5658	. . . . . . . 8 ⊢ ((𝑧 ∈ 𝑋 ↦ (𝑦(-g‘𝐺)𝑧)) “ 𝑆) = ran ((𝑧 ∈ 𝑋 ↦ (𝑦(-g‘𝐺)𝑧)) ↾ 𝑆)
18		resmpt 6023	. . . . . . . . . 10 ⊢ (𝑆 ⊆ 𝑋 → ((𝑧 ∈ 𝑋 ↦ (𝑦(-g‘𝐺)𝑧)) ↾ 𝑆) = (𝑧 ∈ 𝑆 ↦ (𝑦(-g‘𝐺)𝑧)))
19	5, 18	ax-mp 5	. . . . . . . . 9 ⊢ ((𝑧 ∈ 𝑋 ↦ (𝑦(-g‘𝐺)𝑧)) ↾ 𝑆) = (𝑧 ∈ 𝑆 ↦ (𝑦(-g‘𝐺)𝑧))
20	19	rneqi 5911	. . . . . . . 8 ⊢ ran ((𝑧 ∈ 𝑋 ↦ (𝑦(-g‘𝐺)𝑧)) ↾ 𝑆) = ran (𝑧 ∈ 𝑆 ↦ (𝑦(-g‘𝐺)𝑧))
21	17, 20	eqtri 2784	. . . . . . 7 ⊢ ((𝑧 ∈ 𝑋 ↦ (𝑦(-g‘𝐺)𝑧)) “ 𝑆) = ran (𝑧 ∈ 𝑆 ↦ (𝑦(-g‘𝐺)𝑧))
22		imassrn 6057	. . . . . . . . 9 ⊢ ((𝑧 ∈ 𝑋 ↦ (𝑦(-g‘𝐺)𝑧)) “ 𝑆) ⊆ ran (𝑧 ∈ 𝑋 ↦ (𝑦(-g‘𝐺)𝑧))
23	10	adantr 484	. . . . . . . . . . . . 13 ⊢ ((𝐺 ∈ TopGrp ∧ 𝑦 ∈ 𝑆) → 𝐺 ∈ Grp)
24	23	adantr 484	. . . . . . . . . . . 12 ⊢ (((𝐺 ∈ TopGrp ∧ 𝑦 ∈ 𝑆) ∧ 𝑧 ∈ 𝑋) → 𝐺 ∈ Grp)
25	6	sselda 3936	. . . . . . . . . . . . 13 ⊢ ((𝐺 ∈ TopGrp ∧ 𝑦 ∈ 𝑆) → 𝑦 ∈ 𝑋)
26	25	adantr 484	. . . . . . . . . . . 12 ⊢ (((𝐺 ∈ TopGrp ∧ 𝑦 ∈ 𝑆) ∧ 𝑧 ∈ 𝑋) → 𝑦 ∈ 𝑋)
27		simpr 488	. . . . . . . . . . . 12 ⊢ (((𝐺 ∈ TopGrp ∧ 𝑦 ∈ 𝑆) ∧ 𝑧 ∈ 𝑋) → 𝑧 ∈ 𝑋)
28		eqid 2761	. . . . . . . . . . . . 13 ⊢ (-g‘𝐺) = (-g‘𝐺)
29	8, 28	grpsubcl 19045	. . . . . . . . . . . 12 ⊢ ((𝐺 ∈ Grp ∧ 𝑦 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋) → (𝑦(-g‘𝐺)𝑧) ∈ 𝑋)
30	24, 26, 27, 29	syl3anc 1389	. . . . . . . . . . 11 ⊢ (((𝐺 ∈ TopGrp ∧ 𝑦 ∈ 𝑆) ∧ 𝑧 ∈ 𝑋) → (𝑦(-g‘𝐺)𝑧) ∈ 𝑋)
31	30	fmpttd 7092	. . . . . . . . . 10 ⊢ ((𝐺 ∈ TopGrp ∧ 𝑦 ∈ 𝑆) → (𝑧 ∈ 𝑋 ↦ (𝑦(-g‘𝐺)𝑧)):𝑋⟶𝑋)
32	31	frnd 6696	. . . . . . . . 9 ⊢ ((𝐺 ∈ TopGrp ∧ 𝑦 ∈ 𝑆) → ran (𝑧 ∈ 𝑋 ↦ (𝑦(-g‘𝐺)𝑧)) ⊆ 𝑋)
33	22, 32	sstrid 3947	. . . . . . . 8 ⊢ ((𝐺 ∈ TopGrp ∧ 𝑦 ∈ 𝑆) → ((𝑧 ∈ 𝑋 ↦ (𝑦(-g‘𝐺)𝑧)) “ 𝑆) ⊆ 𝑋)
34	8, 11, 28	grpsubid 19049	. . . . . . . . . . 11 ⊢ ((𝐺 ∈ Grp ∧ 𝑦 ∈ 𝑋) → (𝑦(-g‘𝐺)𝑦) = 0 )
35	23, 25, 34	syl2anc 593	. . . . . . . . . 10 ⊢ ((𝐺 ∈ TopGrp ∧ 𝑦 ∈ 𝑆) → (𝑦(-g‘𝐺)𝑦) = 0 )
36		simpr 488	. . . . . . . . . . 11 ⊢ ((𝐺 ∈ TopGrp ∧ 𝑦 ∈ 𝑆) → 𝑦 ∈ 𝑆)
37		ovex 7425	. . . . . . . . . . 11 ⊢ (𝑦(-g‘𝐺)𝑦) ∈ V
38		eqid 2761	. . . . . . . . . . . 12 ⊢ (𝑧 ∈ 𝑆 ↦ (𝑦(-g‘𝐺)𝑧)) = (𝑧 ∈ 𝑆 ↦ (𝑦(-g‘𝐺)𝑧))
39		oveq2 7400	. . . . . . . . . . . 12 ⊢ (𝑧 = 𝑦 → (𝑦(-g‘𝐺)𝑧) = (𝑦(-g‘𝐺)𝑦))
40	38, 39	elrnmpt1s 5933	. . . . . . . . . . 11 ⊢ ((𝑦 ∈ 𝑆 ∧ (𝑦(-g‘𝐺)𝑦) ∈ V) → (𝑦(-g‘𝐺)𝑦) ∈ ran (𝑧 ∈ 𝑆 ↦ (𝑦(-g‘𝐺)𝑧)))
41	36, 37, 40	sylancl 595	. . . . . . . . . 10 ⊢ ((𝐺 ∈ TopGrp ∧ 𝑦 ∈ 𝑆) → (𝑦(-g‘𝐺)𝑦) ∈ ran (𝑧 ∈ 𝑆 ↦ (𝑦(-g‘𝐺)𝑧)))
42	35, 41	eqeltrrd 2862	. . . . . . . . 9 ⊢ ((𝐺 ∈ TopGrp ∧ 𝑦 ∈ 𝑆) → 0 ∈ ran (𝑧 ∈ 𝑆 ↦ (𝑦(-g‘𝐺)𝑧)))
43	42, 21	eleqtrrdi 2872	. . . . . . . 8 ⊢ ((𝐺 ∈ TopGrp ∧ 𝑦 ∈ 𝑆) → 0 ∈ ((𝑧 ∈ 𝑋 ↦ (𝑦(-g‘𝐺)𝑧)) “ 𝑆))
44		eqid 2761	. . . . . . . . 9 ⊢ ∪ 𝐽 = ∪ 𝐽
45		eqid 2761	. . . . . . . . . . . . . . 15 ⊢ (+g‘𝐺) = (+g‘𝐺)
46		eqid 2761	. . . . . . . . . . . . . . 15 ⊢ (invg‘𝐺) = (invg‘𝐺)
47	8, 45, 46, 28	grpsubval 19010	. . . . . . . . . . . . . 14 ⊢ ((𝑦 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋) → (𝑦(-g‘𝐺)𝑧) = (𝑦(+g‘𝐺)((invg‘𝐺)‘𝑧)))
48	25, 47	sylan 589	. . . . . . . . . . . . 13 ⊢ (((𝐺 ∈ TopGrp ∧ 𝑦 ∈ 𝑆) ∧ 𝑧 ∈ 𝑋) → (𝑦(-g‘𝐺)𝑧) = (𝑦(+g‘𝐺)((invg‘𝐺)‘𝑧)))
49	48	mpteq2dva 5192	. . . . . . . . . . . 12 ⊢ ((𝐺 ∈ TopGrp ∧ 𝑦 ∈ 𝑆) → (𝑧 ∈ 𝑋 ↦ (𝑦(-g‘𝐺)𝑧)) = (𝑧 ∈ 𝑋 ↦ (𝑦(+g‘𝐺)((invg‘𝐺)‘𝑧))))
50	8, 46	grpinvcl 19012	. . . . . . . . . . . . . 14 ⊢ ((𝐺 ∈ Grp ∧ 𝑧 ∈ 𝑋) → ((invg‘𝐺)‘𝑧) ∈ 𝑋)
51	23, 50	sylan 589	. . . . . . . . . . . . 13 ⊢ (((𝐺 ∈ TopGrp ∧ 𝑦 ∈ 𝑆) ∧ 𝑧 ∈ 𝑋) → ((invg‘𝐺)‘𝑧) ∈ 𝑋)
52	8, 46	grpinvf 19011	. . . . . . . . . . . . . . . 16 ⊢ (𝐺 ∈ Grp → (invg‘𝐺):𝑋⟶𝑋)
53	10, 52	syl 17	. . . . . . . . . . . . . . 15 ⊢ (𝐺 ∈ TopGrp → (invg‘𝐺):𝑋⟶𝑋)
54	53	adantr 484	. . . . . . . . . . . . . 14 ⊢ ((𝐺 ∈ TopGrp ∧ 𝑦 ∈ 𝑆) → (invg‘𝐺):𝑋⟶𝑋)
55	54	feqmptd 6931	. . . . . . . . . . . . 13 ⊢ ((𝐺 ∈ TopGrp ∧ 𝑦 ∈ 𝑆) → (invg‘𝐺) = (𝑧 ∈ 𝑋 ↦ ((invg‘𝐺)‘𝑧)))
56		eqidd 2762	. . . . . . . . . . . . 13 ⊢ ((𝐺 ∈ TopGrp ∧ 𝑦 ∈ 𝑆) → (𝑤 ∈ 𝑋 ↦ (𝑦(+g‘𝐺)𝑤)) = (𝑤 ∈ 𝑋 ↦ (𝑦(+g‘𝐺)𝑤)))
57		oveq2 7400	. . . . . . . . . . . . 13 ⊢ (𝑤 = ((invg‘𝐺)‘𝑧) → (𝑦(+g‘𝐺)𝑤) = (𝑦(+g‘𝐺)((invg‘𝐺)‘𝑧)))
58	51, 55, 56, 57	fmptco 7107	. . . . . . . . . . . 12 ⊢ ((𝐺 ∈ TopGrp ∧ 𝑦 ∈ 𝑆) → ((𝑤 ∈ 𝑋 ↦ (𝑦(+g‘𝐺)𝑤)) ∘ (invg‘𝐺)) = (𝑧 ∈ 𝑋 ↦ (𝑦(+g‘𝐺)((invg‘𝐺)‘𝑧))))
59	49, 58	eqtr4d 2799	. . . . . . . . . . 11 ⊢ ((𝐺 ∈ TopGrp ∧ 𝑦 ∈ 𝑆) → (𝑧 ∈ 𝑋 ↦ (𝑦(-g‘𝐺)𝑧)) = ((𝑤 ∈ 𝑋 ↦ (𝑦(+g‘𝐺)𝑤)) ∘ (invg‘𝐺)))
60	7, 46	grpinvhmeo 24126	. . . . . . . . . . . . 13 ⊢ (𝐺 ∈ TopGrp → (invg‘𝐺) ∈ (𝐽Homeo𝐽))
61	60	adantr 484	. . . . . . . . . . . 12 ⊢ ((𝐺 ∈ TopGrp ∧ 𝑦 ∈ 𝑆) → (invg‘𝐺) ∈ (𝐽Homeo𝐽))
62		eqid 2761	. . . . . . . . . . . . . 14 ⊢ (𝑤 ∈ 𝑋 ↦ (𝑦(+g‘𝐺)𝑤)) = (𝑤 ∈ 𝑋 ↦ (𝑦(+g‘𝐺)𝑤))
63	62, 8, 45, 7	tgplacthmeo 24143	. . . . . . . . . . . . 13 ⊢ ((𝐺 ∈ TopGrp ∧ 𝑦 ∈ 𝑋) → (𝑤 ∈ 𝑋 ↦ (𝑦(+g‘𝐺)𝑤)) ∈ (𝐽Homeo𝐽))
64	25, 63	syldan 600	. . . . . . . . . . . 12 ⊢ ((𝐺 ∈ TopGrp ∧ 𝑦 ∈ 𝑆) → (𝑤 ∈ 𝑋 ↦ (𝑦(+g‘𝐺)𝑤)) ∈ (𝐽Homeo𝐽))
65		hmeoco 23812	. . . . . . . . . . . 12 ⊢ (((invg‘𝐺) ∈ (𝐽Homeo𝐽) ∧ (𝑤 ∈ 𝑋 ↦ (𝑦(+g‘𝐺)𝑤)) ∈ (𝐽Homeo𝐽)) → ((𝑤 ∈ 𝑋 ↦ (𝑦(+g‘𝐺)𝑤)) ∘ (invg‘𝐺)) ∈ (𝐽Homeo𝐽))
66	61, 64, 65	syl2anc 593	. . . . . . . . . . 11 ⊢ ((𝐺 ∈ TopGrp ∧ 𝑦 ∈ 𝑆) → ((𝑤 ∈ 𝑋 ↦ (𝑦(+g‘𝐺)𝑤)) ∘ (invg‘𝐺)) ∈ (𝐽Homeo𝐽))
67	59, 66	eqeltrd 2861	. . . . . . . . . 10 ⊢ ((𝐺 ∈ TopGrp ∧ 𝑦 ∈ 𝑆) → (𝑧 ∈ 𝑋 ↦ (𝑦(-g‘𝐺)𝑧)) ∈ (𝐽Homeo𝐽))
68		hmeocn 23800	. . . . . . . . . 10 ⊢ ((𝑧 ∈ 𝑋 ↦ (𝑦(-g‘𝐺)𝑧)) ∈ (𝐽Homeo𝐽) → (𝑧 ∈ 𝑋 ↦ (𝑦(-g‘𝐺)𝑧)) ∈ (𝐽 Cn 𝐽))
69	67, 68	syl 17	. . . . . . . . 9 ⊢ ((𝐺 ∈ TopGrp ∧ 𝑦 ∈ 𝑆) → (𝑧 ∈ 𝑋 ↦ (𝑦(-g‘𝐺)𝑧)) ∈ (𝐽 Cn 𝐽))
70		toponuni 22954	. . . . . . . . . . . 12 ⊢ (𝐽 ∈ (TopOn‘𝑋) → 𝑋 = ∪ 𝐽)
71	9, 70	syl 17	. . . . . . . . . . 11 ⊢ (𝐺 ∈ TopGrp → 𝑋 = ∪ 𝐽)
72	71	adantr 484	. . . . . . . . . 10 ⊢ ((𝐺 ∈ TopGrp ∧ 𝑦 ∈ 𝑆) → 𝑋 = ∪ 𝐽)
73	5, 72	sseqtrid 3978	. . . . . . . . 9 ⊢ ((𝐺 ∈ TopGrp ∧ 𝑦 ∈ 𝑆) → 𝑆 ⊆ ∪ 𝐽)
74	1	conncompconn 23472	. . . . . . . . . . 11 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 0 ∈ 𝑋) → (𝐽 ↾t 𝑆) ∈ Conn)
75	9, 13, 74	syl2anc 593	. . . . . . . . . 10 ⊢ (𝐺 ∈ TopGrp → (𝐽 ↾t 𝑆) ∈ Conn)
76	75	adantr 484	. . . . . . . . 9 ⊢ ((𝐺 ∈ TopGrp ∧ 𝑦 ∈ 𝑆) → (𝐽 ↾t 𝑆) ∈ Conn)
77	44, 69, 73, 76	connima 23465	. . . . . . . 8 ⊢ ((𝐺 ∈ TopGrp ∧ 𝑦 ∈ 𝑆) → (𝐽 ↾t ((𝑧 ∈ 𝑋 ↦ (𝑦(-g‘𝐺)𝑧)) “ 𝑆)) ∈ Conn)
78	1	conncompss 23473	. . . . . . . 8 ⊢ ((((𝑧 ∈ 𝑋 ↦ (𝑦(-g‘𝐺)𝑧)) “ 𝑆) ⊆ 𝑋 ∧ 0 ∈ ((𝑧 ∈ 𝑋 ↦ (𝑦(-g‘𝐺)𝑧)) “ 𝑆) ∧ (𝐽 ↾t ((𝑧 ∈ 𝑋 ↦ (𝑦(-g‘𝐺)𝑧)) “ 𝑆)) ∈ Conn) → ((𝑧 ∈ 𝑋 ↦ (𝑦(-g‘𝐺)𝑧)) “ 𝑆) ⊆ 𝑆)
79	33, 43, 77, 78	syl3anc 1389	. . . . . . 7 ⊢ ((𝐺 ∈ TopGrp ∧ 𝑦 ∈ 𝑆) → ((𝑧 ∈ 𝑋 ↦ (𝑦(-g‘𝐺)𝑧)) “ 𝑆) ⊆ 𝑆)
80	21, 79	eqsstrrid 3975	. . . . . 6 ⊢ ((𝐺 ∈ TopGrp ∧ 𝑦 ∈ 𝑆) → ran (𝑧 ∈ 𝑆 ↦ (𝑦(-g‘𝐺)𝑧)) ⊆ 𝑆)
81		ovex 7425	. . . . . . . 8 ⊢ (𝑦(-g‘𝐺)𝑧) ∈ V
82	81, 38	fnmpti 6660	. . . . . . 7 ⊢ (𝑧 ∈ 𝑆 ↦ (𝑦(-g‘𝐺)𝑧)) Fn 𝑆
83		df-f 6521	. . . . . . 7 ⊢ ((𝑧 ∈ 𝑆 ↦ (𝑦(-g‘𝐺)𝑧)):𝑆⟶𝑆 ↔ ((𝑧 ∈ 𝑆 ↦ (𝑦(-g‘𝐺)𝑧)) Fn 𝑆 ∧ ran (𝑧 ∈ 𝑆 ↦ (𝑦(-g‘𝐺)𝑧)) ⊆ 𝑆))
84	82, 83	mpbiran 719	. . . . . 6 ⊢ ((𝑧 ∈ 𝑆 ↦ (𝑦(-g‘𝐺)𝑧)):𝑆⟶𝑆 ↔ ran (𝑧 ∈ 𝑆 ↦ (𝑦(-g‘𝐺)𝑧)) ⊆ 𝑆)
85	80, 84	sylibr 236	. . . . 5 ⊢ ((𝐺 ∈ TopGrp ∧ 𝑦 ∈ 𝑆) → (𝑧 ∈ 𝑆 ↦ (𝑦(-g‘𝐺)𝑧)):𝑆⟶𝑆)
86	38	fmpt 7087	. . . . 5 ⊢ (∀𝑧 ∈ 𝑆 (𝑦(-g‘𝐺)𝑧) ∈ 𝑆 ↔ (𝑧 ∈ 𝑆 ↦ (𝑦(-g‘𝐺)𝑧)):𝑆⟶𝑆)
87	85, 86	sylibr 236	. . . 4 ⊢ ((𝐺 ∈ TopGrp ∧ 𝑦 ∈ 𝑆) → ∀𝑧 ∈ 𝑆 (𝑦(-g‘𝐺)𝑧) ∈ 𝑆)
88	87	ralrimiva 3153	. . 3 ⊢ (𝐺 ∈ TopGrp → ∀𝑦 ∈ 𝑆 ∀𝑧 ∈ 𝑆 (𝑦(-g‘𝐺)𝑧) ∈ 𝑆)
89	8, 28	issubg4 19170	. . . 4 ⊢ (𝐺 ∈ Grp → (𝑆 ∈ (SubGrp‘𝐺) ↔ (𝑆 ⊆ 𝑋 ∧ 𝑆 ≠ ∅ ∧ ∀𝑦 ∈ 𝑆 ∀𝑧 ∈ 𝑆 (𝑦(-g‘𝐺)𝑧) ∈ 𝑆)))
90	10, 89	syl 17	. . 3 ⊢ (𝐺 ∈ TopGrp → (𝑆 ∈ (SubGrp‘𝐺) ↔ (𝑆 ⊆ 𝑋 ∧ 𝑆 ≠ ∅ ∧ ∀𝑦 ∈ 𝑆 ∀𝑧 ∈ 𝑆 (𝑦(-g‘𝐺)𝑧) ∈ 𝑆)))
91	6, 16, 88, 90	mpbir3and 1355	. 2 ⊢ (𝐺 ∈ TopGrp → 𝑆 ∈ (SubGrp‘𝐺))
92	10	adantr 484	. . . . . . . . . 10 ⊢ ((𝐺 ∈ TopGrp ∧ ((𝑦 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋) ∧ (𝑦(+g‘𝐺)𝑧) ∈ 𝑆)) → 𝐺 ∈ Grp)
93		eqid 2761	. . . . . . . . . . 11 ⊢ (oppg‘𝐺) = (oppg‘𝐺)
94	93, 46	oppginv 19382	. . . . . . . . . 10 ⊢ (𝐺 ∈ Grp → (invg‘𝐺) = (invg‘(oppg‘𝐺)))
95	92, 94	syl 17	. . . . . . . . 9 ⊢ ((𝐺 ∈ TopGrp ∧ ((𝑦 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋) ∧ (𝑦(+g‘𝐺)𝑧) ∈ 𝑆)) → (invg‘𝐺) = (invg‘(oppg‘𝐺)))
96	95	fveq1d 6865	. . . . . . . 8 ⊢ ((𝐺 ∈ TopGrp ∧ ((𝑦 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋) ∧ (𝑦(+g‘𝐺)𝑧) ∈ 𝑆)) → ((invg‘𝐺)‘((invg‘𝐺)‘𝑦)) = ((invg‘(oppg‘𝐺))‘((invg‘𝐺)‘𝑦)))
97		simprll 788	. . . . . . . . 9 ⊢ ((𝐺 ∈ TopGrp ∧ ((𝑦 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋) ∧ (𝑦(+g‘𝐺)𝑧) ∈ 𝑆)) → 𝑦 ∈ 𝑋)
98	8, 46	grpinvinv 19030	. . . . . . . . 9 ⊢ ((𝐺 ∈ Grp ∧ 𝑦 ∈ 𝑋) → ((invg‘𝐺)‘((invg‘𝐺)‘𝑦)) = 𝑦)
99	92, 97, 98	syl2anc 593	. . . . . . . 8 ⊢ ((𝐺 ∈ TopGrp ∧ ((𝑦 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋) ∧ (𝑦(+g‘𝐺)𝑧) ∈ 𝑆)) → ((invg‘𝐺)‘((invg‘𝐺)‘𝑦)) = 𝑦)
100	96, 99	eqtr3d 2798	. . . . . . 7 ⊢ ((𝐺 ∈ TopGrp ∧ ((𝑦 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋) ∧ (𝑦(+g‘𝐺)𝑧) ∈ 𝑆)) → ((invg‘(oppg‘𝐺))‘((invg‘𝐺)‘𝑦)) = 𝑦)
101	100	oveq1d 7407	. . . . . 6 ⊢ ((𝐺 ∈ TopGrp ∧ ((𝑦 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋) ∧ (𝑦(+g‘𝐺)𝑧) ∈ 𝑆)) → (((invg‘(oppg‘𝐺))‘((invg‘𝐺)‘𝑦))(+g‘(oppg‘𝐺))𝑧) = (𝑦(+g‘(oppg‘𝐺))𝑧))
102		eqid 2761	. . . . . . 7 ⊢ (+g‘(oppg‘𝐺)) = (+g‘(oppg‘𝐺))
103	45, 93, 102	oppgplus 19372	. . . . . 6 ⊢ (𝑦(+g‘(oppg‘𝐺))𝑧) = (𝑧(+g‘𝐺)𝑦)
104	101, 103	eqtrdi 2812	. . . . 5 ⊢ ((𝐺 ∈ TopGrp ∧ ((𝑦 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋) ∧ (𝑦(+g‘𝐺)𝑧) ∈ 𝑆)) → (((invg‘(oppg‘𝐺))‘((invg‘𝐺)‘𝑦))(+g‘(oppg‘𝐺))𝑧) = (𝑧(+g‘𝐺)𝑦))
105	8, 46	grpinvcl 19012	. . . . . . . . . 10 ⊢ ((𝐺 ∈ Grp ∧ 𝑦 ∈ 𝑋) → ((invg‘𝐺)‘𝑦) ∈ 𝑋)
106	92, 97, 105	syl2anc 593	. . . . . . . . 9 ⊢ ((𝐺 ∈ TopGrp ∧ ((𝑦 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋) ∧ (𝑦(+g‘𝐺)𝑧) ∈ 𝑆)) → ((invg‘𝐺)‘𝑦) ∈ 𝑋)
107		simprlr 789	. . . . . . . . 9 ⊢ ((𝐺 ∈ TopGrp ∧ ((𝑦 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋) ∧ (𝑦(+g‘𝐺)𝑧) ∈ 𝑆)) → 𝑧 ∈ 𝑋)
108	99	oveq1d 7407	. . . . . . . . . 10 ⊢ ((𝐺 ∈ TopGrp ∧ ((𝑦 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋) ∧ (𝑦(+g‘𝐺)𝑧) ∈ 𝑆)) → (((invg‘𝐺)‘((invg‘𝐺)‘𝑦))(+g‘𝐺)𝑧) = (𝑦(+g‘𝐺)𝑧))
109		simprr 782	. . . . . . . . . 10 ⊢ ((𝐺 ∈ TopGrp ∧ ((𝑦 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋) ∧ (𝑦(+g‘𝐺)𝑧) ∈ 𝑆)) → (𝑦(+g‘𝐺)𝑧) ∈ 𝑆)
110	108, 109	eqeltrd 2861	. . . . . . . . 9 ⊢ ((𝐺 ∈ TopGrp ∧ ((𝑦 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋) ∧ (𝑦(+g‘𝐺)𝑧) ∈ 𝑆)) → (((invg‘𝐺)‘((invg‘𝐺)‘𝑦))(+g‘𝐺)𝑧) ∈ 𝑆)
111		eqid 2761	. . . . . . . . . . 11 ⊢ (𝐺 ~QG 𝑆) = (𝐺 ~QG 𝑆)
112	8, 46, 45, 111	eqgval 19201	. . . . . . . . . 10 ⊢ ((𝐺 ∈ Grp ∧ 𝑆 ⊆ 𝑋) → (((invg‘𝐺)‘𝑦)(𝐺 ~QG 𝑆)𝑧 ↔ (((invg‘𝐺)‘𝑦) ∈ 𝑋 ∧ 𝑧 ∈ 𝑋 ∧ (((invg‘𝐺)‘((invg‘𝐺)‘𝑦))(+g‘𝐺)𝑧) ∈ 𝑆)))
113	92, 5, 112	sylancl 595	. . . . . . . . 9 ⊢ ((𝐺 ∈ TopGrp ∧ ((𝑦 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋) ∧ (𝑦(+g‘𝐺)𝑧) ∈ 𝑆)) → (((invg‘𝐺)‘𝑦)(𝐺 ~QG 𝑆)𝑧 ↔ (((invg‘𝐺)‘𝑦) ∈ 𝑋 ∧ 𝑧 ∈ 𝑋 ∧ (((invg‘𝐺)‘((invg‘𝐺)‘𝑦))(+g‘𝐺)𝑧) ∈ 𝑆)))
114	106, 107, 110, 113	mpbir3and 1355	. . . . . . . 8 ⊢ ((𝐺 ∈ TopGrp ∧ ((𝑦 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋) ∧ (𝑦(+g‘𝐺)𝑧) ∈ 𝑆)) → ((invg‘𝐺)‘𝑦)(𝐺 ~QG 𝑆)𝑧)
115	8, 11, 7, 1, 111	tgpconncompeqg 24152	. . . . . . . . . . . 12 ⊢ ((𝐺 ∈ TopGrp ∧ ((invg‘𝐺)‘𝑦) ∈ 𝑋) → [((invg‘𝐺)‘𝑦)](𝐺 ~QG 𝑆) = ∪ {𝑥 ∈ 𝒫 𝑋 ∣ (((invg‘𝐺)‘𝑦) ∈ 𝑥 ∧ (𝐽 ↾t 𝑥) ∈ Conn)})
116	106, 115	syldan 600	. . . . . . . . . . 11 ⊢ ((𝐺 ∈ TopGrp ∧ ((𝑦 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋) ∧ (𝑦(+g‘𝐺)𝑧) ∈ 𝑆)) → [((invg‘𝐺)‘𝑦)](𝐺 ~QG 𝑆) = ∪ {𝑥 ∈ 𝒫 𝑋 ∣ (((invg‘𝐺)‘𝑦) ∈ 𝑥 ∧ (𝐽 ↾t 𝑥) ∈ Conn)})
117	93	oppgtgp 24138	. . . . . . . . . . . . 13 ⊢ (𝐺 ∈ TopGrp → (oppg‘𝐺) ∈ TopGrp)
118	117	adantr 484	. . . . . . . . . . . 12 ⊢ ((𝐺 ∈ TopGrp ∧ ((𝑦 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋) ∧ (𝑦(+g‘𝐺)𝑧) ∈ 𝑆)) → (oppg‘𝐺) ∈ TopGrp)
119	93, 8	oppgbas 19374	. . . . . . . . . . . . 13 ⊢ 𝑋 = (Base‘(oppg‘𝐺))
120	93, 11	oppgid 19379	. . . . . . . . . . . . 13 ⊢ 0 = (0g‘(oppg‘𝐺))
121	93, 7	oppgtopn 19376	. . . . . . . . . . . . 13 ⊢ 𝐽 = (TopOpen‘(oppg‘𝐺))
122		eqid 2761	. . . . . . . . . . . . 13 ⊢ ((oppg‘𝐺) ~QG 𝑆) = ((oppg‘𝐺) ~QG 𝑆)
123	119, 120, 121, 1, 122	tgpconncompeqg 24152	. . . . . . . . . . . 12 ⊢ (((oppg‘𝐺) ∈ TopGrp ∧ ((invg‘𝐺)‘𝑦) ∈ 𝑋) → [((invg‘𝐺)‘𝑦)]((oppg‘𝐺) ~QG 𝑆) = ∪ {𝑥 ∈ 𝒫 𝑋 ∣ (((invg‘𝐺)‘𝑦) ∈ 𝑥 ∧ (𝐽 ↾t 𝑥) ∈ Conn)})
124	118, 106, 123	syl2anc 593	. . . . . . . . . . 11 ⊢ ((𝐺 ∈ TopGrp ∧ ((𝑦 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋) ∧ (𝑦(+g‘𝐺)𝑧) ∈ 𝑆)) → [((invg‘𝐺)‘𝑦)]((oppg‘𝐺) ~QG 𝑆) = ∪ {𝑥 ∈ 𝒫 𝑋 ∣ (((invg‘𝐺)‘𝑦) ∈ 𝑥 ∧ (𝐽 ↾t 𝑥) ∈ Conn)})
125	116, 124	eqtr4d 2799	. . . . . . . . . 10 ⊢ ((𝐺 ∈ TopGrp ∧ ((𝑦 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋) ∧ (𝑦(+g‘𝐺)𝑧) ∈ 𝑆)) → [((invg‘𝐺)‘𝑦)](𝐺 ~QG 𝑆) = [((invg‘𝐺)‘𝑦)]((oppg‘𝐺) ~QG 𝑆))
126	125	eleq2d 2847	. . . . . . . . 9 ⊢ ((𝐺 ∈ TopGrp ∧ ((𝑦 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋) ∧ (𝑦(+g‘𝐺)𝑧) ∈ 𝑆)) → (𝑧 ∈ [((invg‘𝐺)‘𝑦)](𝐺 ~QG 𝑆) ↔ 𝑧 ∈ [((invg‘𝐺)‘𝑦)]((oppg‘𝐺) ~QG 𝑆)))
127		vex 3457	. . . . . . . . . 10 ⊢ 𝑧 ∈ V
128		fvex 6876	. . . . . . . . . 10 ⊢ ((invg‘𝐺)‘𝑦) ∈ V
129	127, 128	elec 8720	. . . . . . . . 9 ⊢ (𝑧 ∈ [((invg‘𝐺)‘𝑦)](𝐺 ~QG 𝑆) ↔ ((invg‘𝐺)‘𝑦)(𝐺 ~QG 𝑆)𝑧)
130	127, 128	elec 8720	. . . . . . . . 9 ⊢ (𝑧 ∈ [((invg‘𝐺)‘𝑦)]((oppg‘𝐺) ~QG 𝑆) ↔ ((invg‘𝐺)‘𝑦)((oppg‘𝐺) ~QG 𝑆)𝑧)
131	126, 129, 130	3bitr3g 315	. . . . . . . 8 ⊢ ((𝐺 ∈ TopGrp ∧ ((𝑦 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋) ∧ (𝑦(+g‘𝐺)𝑧) ∈ 𝑆)) → (((invg‘𝐺)‘𝑦)(𝐺 ~QG 𝑆)𝑧 ↔ ((invg‘𝐺)‘𝑦)((oppg‘𝐺) ~QG 𝑆)𝑧))
132	114, 131	mpbid 234	. . . . . . 7 ⊢ ((𝐺 ∈ TopGrp ∧ ((𝑦 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋) ∧ (𝑦(+g‘𝐺)𝑧) ∈ 𝑆)) → ((invg‘𝐺)‘𝑦)((oppg‘𝐺) ~QG 𝑆)𝑧)
133		eqid 2761	. . . . . . . . 9 ⊢ (invg‘(oppg‘𝐺)) = (invg‘(oppg‘𝐺))
134	119, 133, 102, 122	eqgval 19201	. . . . . . . 8 ⊢ (((oppg‘𝐺) ∈ TopGrp ∧ 𝑆 ⊆ 𝑋) → (((invg‘𝐺)‘𝑦)((oppg‘𝐺) ~QG 𝑆)𝑧 ↔ (((invg‘𝐺)‘𝑦) ∈ 𝑋 ∧ 𝑧 ∈ 𝑋 ∧ (((invg‘(oppg‘𝐺))‘((invg‘𝐺)‘𝑦))(+g‘(oppg‘𝐺))𝑧) ∈ 𝑆)))
135	118, 5, 134	sylancl 595	. . . . . . 7 ⊢ ((𝐺 ∈ TopGrp ∧ ((𝑦 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋) ∧ (𝑦(+g‘𝐺)𝑧) ∈ 𝑆)) → (((invg‘𝐺)‘𝑦)((oppg‘𝐺) ~QG 𝑆)𝑧 ↔ (((invg‘𝐺)‘𝑦) ∈ 𝑋 ∧ 𝑧 ∈ 𝑋 ∧ (((invg‘(oppg‘𝐺))‘((invg‘𝐺)‘𝑦))(+g‘(oppg‘𝐺))𝑧) ∈ 𝑆)))
136	132, 135	mpbid 234	. . . . . 6 ⊢ ((𝐺 ∈ TopGrp ∧ ((𝑦 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋) ∧ (𝑦(+g‘𝐺)𝑧) ∈ 𝑆)) → (((invg‘𝐺)‘𝑦) ∈ 𝑋 ∧ 𝑧 ∈ 𝑋 ∧ (((invg‘(oppg‘𝐺))‘((invg‘𝐺)‘𝑦))(+g‘(oppg‘𝐺))𝑧) ∈ 𝑆))
137	136	simp3d 1156	. . . . 5 ⊢ ((𝐺 ∈ TopGrp ∧ ((𝑦 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋) ∧ (𝑦(+g‘𝐺)𝑧) ∈ 𝑆)) → (((invg‘(oppg‘𝐺))‘((invg‘𝐺)‘𝑦))(+g‘(oppg‘𝐺))𝑧) ∈ 𝑆)
138	104, 137	eqeltrrd 2862	. . . 4 ⊢ ((𝐺 ∈ TopGrp ∧ ((𝑦 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋) ∧ (𝑦(+g‘𝐺)𝑧) ∈ 𝑆)) → (𝑧(+g‘𝐺)𝑦) ∈ 𝑆)
139	138	expr 460	. . 3 ⊢ ((𝐺 ∈ TopGrp ∧ (𝑦 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋)) → ((𝑦(+g‘𝐺)𝑧) ∈ 𝑆 → (𝑧(+g‘𝐺)𝑦) ∈ 𝑆))
140	139	ralrimivva 3204	. 2 ⊢ (𝐺 ∈ TopGrp → ∀𝑦 ∈ 𝑋 ∀𝑧 ∈ 𝑋 ((𝑦(+g‘𝐺)𝑧) ∈ 𝑆 → (𝑧(+g‘𝐺)𝑦) ∈ 𝑆))
141	8, 45	isnsg2 19180	. 2 ⊢ (𝑆 ∈ (NrmSGrp‘𝐺) ↔ (𝑆 ∈ (SubGrp‘𝐺) ∧ ∀𝑦 ∈ 𝑋 ∀𝑧 ∈ 𝑋 ((𝑦(+g‘𝐺)𝑧) ∈ 𝑆 → (𝑧(+g‘𝐺)𝑦) ∈ 𝑆)))
142	91, 140, 141	sylanbrc 592	1 ⊢ (𝐺 ∈ TopGrp → 𝑆 ∈ (NrmSGrp‘𝐺))

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 399   ∧ w3a 1097   = wceq 1559   ∈ wcel 2141   ≠ wne 2956  ∀wral 3075  {crab 3413  Vcvv 3453   ⊆ wss 3904  ∅c0 4285  𝒫 cpw 4554  ∪ cuni 4864   class class class wbr 5099   ↦ cmpt 5180  ran crn 5646   ↾ cres 5647   “ cima 5648   ∘ ccom 5649   Fn wfn 6512  ⟶wf 6513  ‘cfv 6517  (class class class)co 7392  [cec 8671  Basecbs 17228  +_gcplusg 17269   ↾_t crest 17432  TopOpenctopn 17433  0_gc0g 17451  Grpcgrp 18958  inv_gcminusg 18959  -_gcsg 18960  SubGrpcsubg 19145  NrmSGrpcnsg 19146   ~_QG cqg 19147  opp_gcoppg 19368  TopOnctopon 22950   Cn ccn 23264  Conncconn 23451  Homeochmeo 23793  TopGrpctgp 24111

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1814  ax-4 1828  ax-5 1929  ax-6 1986  ax-7 2027  ax-8 2143  ax-9 2151  ax-10 2174  ax-11 2190  ax-12 2211  ax-ext 2733  ax-rep 5226  ax-sep 5245  ax-nul 5255  ax-pow 5321  ax-pr 5389  ax-un 7714  ax-cnex 11126  ax-resscn 11127  ax-1cn 11128  ax-icn 11129  ax-addcl 11130  ax-addrcl 11131  ax-mulcl 11132  ax-mulrcl 11133  ax-mulcom 11134  ax-addass 11135  ax-mulass 11136  ax-distr 11137  ax-i2m1 11138  ax-1ne0 11139  ax-1rid 11140  ax-rnegex 11141  ax-rrecex 11142  ax-cnre 11143  ax-pre-lttri 11144  ax-pre-lttrn 11145  ax-pre-ltadd 11146  ax-pre-mulgt0 11147

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3or 1098  df-3an 1099  df-tru 1562  df-fal 1572  df-ex 1799  df-nf 1803  df-sb 2090  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-nel 3061  df-ral 3076  df-rex 3086  df-rmo 3366  df-reu 3367  df-rab 3414  df-v 3455  df-sbc 3745  df-csb 3853  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-pss 3924  df-nul 4286  df-if 4480  df-pw 4556  df-sn 4582  df-pr 4584  df-op 4588  df-uni 4865  df-int 4905  df-iun 4950  df-br 5100  df-opab 5162  df-mpt 5181  df-tr 5207  df-id 5540  df-eprel 5545  df-po 5553  df-so 5554  df-fr 5598  df-we 5600  df-xp 5651  df-rel 5652  df-cnv 5653  df-co 5654  df-dm 5655  df-rn 5656  df-res 5657  df-ima 5658  df-pred 6284  df-ord 6345  df-on 6346  df-lim 6347  df-suc 6348  df-iota 6473  df-fun 6519  df-fn 6520  df-f 6521  df-f1 6522  df-fo 6523  df-f1o 6524  df-fv 6525  df-riota 7349  df-ov 7395  df-oprab 7396  df-mpo 7397  df-om 7843  df-1st 7966  df-2nd 7967  df-tpos 8201  df-frecs 8257  df-wrecs 8288  df-recs 8337  df-rdg 8376  df-er 8673  df-ec 8675  df-map 8805  df-en 8924  df-dom 8925  df-sdom 8926  df-fin 8927  df-fi 9354  df-pnf 11215  df-mnf 11216  df-xr 11217  df-ltxr 11218  df-le 11219  df-sub 11413  df-neg 11414  df-nn 12208  df-2 12277  df-3 12278  df-4 12279  df-5 12280  df-6 12281  df-7 12282  df-8 12283  df-9 12284  df-sets 17183  df-slot 17201  df-ndx 17213  df-base 17229  df-ress 17250  df-plusg 17282  df-tset 17288  df-rest 17434  df-topn 17435  df-0g 17453  df-topgen 17455  df-plusf 18656  df-mgm 18657  df-sgrp 18736  df-mnd 18752  df-grp 18961  df-minusg 18962  df-sbg 18963  df-subg 19148  df-nsg 19149  df-eqg 19150  df-oppg 19369  df-top 22934  df-topon 22951  df-topsp 22973  df-bases 22986  df-cld 23059  df-cn 23267  df-cnp 23268  df-conn 23452  df-tx 23602  df-hmeo 23795  df-tmd 24112  df-tgp 24113

This theorem is referenced by: (None)