Theorem tgpconncomp 24010

Description: The identity component, the connected component containing the identity element, is a closed (conncompcld 23331) 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 4073	. . . . . 6 ⊢ {𝑥 ∈ 𝒫 𝑋 ∣ ( 0 ∈ 𝑥 ∧ (𝐽 ↾t 𝑥) ∈ Conn)} ⊆ 𝒫 𝑋
3		sspwuni 5097	. . . . . 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 229	. . . . 5 ⊢ ∪ {𝑥 ∈ 𝒫 𝑋 ∣ ( 0 ∈ 𝑥 ∧ (𝐽 ↾t 𝑥) ∈ Conn)} ⊆ 𝑋
5	1, 4	eqsstri 4012	. . . 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 23979	. . . . 5 ⊢ (𝐺 ∈ TopGrp → 𝐽 ∈ (TopOn‘𝑋))
10		tgpgrp 23975	. . . . . 6 ⊢ (𝐺 ∈ TopGrp → 𝐺 ∈ Grp)
11		tgpconncomp.z	. . . . . . 7 ⊢ 0 = (0g‘𝐺)
12	8, 11	grpidcl 18915	. . . . . 6 ⊢ (𝐺 ∈ Grp → 0 ∈ 𝑋)
13	10, 12	syl 17	. . . . 5 ⊢ (𝐺 ∈ TopGrp → 0 ∈ 𝑋)
14	1	conncompid 23328	. . . . 5 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 0 ∈ 𝑋) → 0 ∈ 𝑆)
15	9, 13, 14	syl2anc 583	. . . 4 ⊢ (𝐺 ∈ TopGrp → 0 ∈ 𝑆)
16	15	ne0d 4331	. . 3 ⊢ (𝐺 ∈ TopGrp → 𝑆 ≠ ∅)
17		df-ima 5685	. . . . . . . 8 ⊢ ((𝑧 ∈ 𝑋 ↦ (𝑦(-g‘𝐺)𝑧)) “ 𝑆) = ran ((𝑧 ∈ 𝑋 ↦ (𝑦(-g‘𝐺)𝑧)) ↾ 𝑆)
18		resmpt 6035	. . . . . . . . . 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 5933	. . . . . . . 8 ⊢ ran ((𝑧 ∈ 𝑋 ↦ (𝑦(-g‘𝐺)𝑧)) ↾ 𝑆) = ran (𝑧 ∈ 𝑆 ↦ (𝑦(-g‘𝐺)𝑧))
21	17, 20	eqtri 2755	. . . . . . 7 ⊢ ((𝑧 ∈ 𝑋 ↦ (𝑦(-g‘𝐺)𝑧)) “ 𝑆) = ran (𝑧 ∈ 𝑆 ↦ (𝑦(-g‘𝐺)𝑧))
22		imassrn 6068	. . . . . . . . 9 ⊢ ((𝑧 ∈ 𝑋 ↦ (𝑦(-g‘𝐺)𝑧)) “ 𝑆) ⊆ ran (𝑧 ∈ 𝑋 ↦ (𝑦(-g‘𝐺)𝑧))
23	10	adantr 480	. . . . . . . . . . . . 13 ⊢ ((𝐺 ∈ TopGrp ∧ 𝑦 ∈ 𝑆) → 𝐺 ∈ Grp)
24	23	adantr 480	. . . . . . . . . . . 12 ⊢ (((𝐺 ∈ TopGrp ∧ 𝑦 ∈ 𝑆) ∧ 𝑧 ∈ 𝑋) → 𝐺 ∈ Grp)
25	6	sselda 3978	. . . . . . . . . . . . 13 ⊢ ((𝐺 ∈ TopGrp ∧ 𝑦 ∈ 𝑆) → 𝑦 ∈ 𝑋)
26	25	adantr 480	. . . . . . . . . . . 12 ⊢ (((𝐺 ∈ TopGrp ∧ 𝑦 ∈ 𝑆) ∧ 𝑧 ∈ 𝑋) → 𝑦 ∈ 𝑋)
27		simpr 484	. . . . . . . . . . . 12 ⊢ (((𝐺 ∈ TopGrp ∧ 𝑦 ∈ 𝑆) ∧ 𝑧 ∈ 𝑋) → 𝑧 ∈ 𝑋)
28		eqid 2727	. . . . . . . . . . . . 13 ⊢ (-g‘𝐺) = (-g‘𝐺)
29	8, 28	grpsubcl 18969	. . . . . . . . . . . 12 ⊢ ((𝐺 ∈ Grp ∧ 𝑦 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋) → (𝑦(-g‘𝐺)𝑧) ∈ 𝑋)
30	24, 26, 27, 29	syl3anc 1369	. . . . . . . . . . 11 ⊢ (((𝐺 ∈ TopGrp ∧ 𝑦 ∈ 𝑆) ∧ 𝑧 ∈ 𝑋) → (𝑦(-g‘𝐺)𝑧) ∈ 𝑋)
31	30	fmpttd 7119	. . . . . . . . . 10 ⊢ ((𝐺 ∈ TopGrp ∧ 𝑦 ∈ 𝑆) → (𝑧 ∈ 𝑋 ↦ (𝑦(-g‘𝐺)𝑧)):𝑋⟶𝑋)
32	31	frnd 6724	. . . . . . . . 9 ⊢ ((𝐺 ∈ TopGrp ∧ 𝑦 ∈ 𝑆) → ran (𝑧 ∈ 𝑋 ↦ (𝑦(-g‘𝐺)𝑧)) ⊆ 𝑋)
33	22, 32	sstrid 3989	. . . . . . . 8 ⊢ ((𝐺 ∈ TopGrp ∧ 𝑦 ∈ 𝑆) → ((𝑧 ∈ 𝑋 ↦ (𝑦(-g‘𝐺)𝑧)) “ 𝑆) ⊆ 𝑋)
34	8, 11, 28	grpsubid 18973	. . . . . . . . . . 11 ⊢ ((𝐺 ∈ Grp ∧ 𝑦 ∈ 𝑋) → (𝑦(-g‘𝐺)𝑦) = 0 )
35	23, 25, 34	syl2anc 583	. . . . . . . . . 10 ⊢ ((𝐺 ∈ TopGrp ∧ 𝑦 ∈ 𝑆) → (𝑦(-g‘𝐺)𝑦) = 0 )
36		simpr 484	. . . . . . . . . . 11 ⊢ ((𝐺 ∈ TopGrp ∧ 𝑦 ∈ 𝑆) → 𝑦 ∈ 𝑆)
37		ovex 7447	. . . . . . . . . . 11 ⊢ (𝑦(-g‘𝐺)𝑦) ∈ V
38		eqid 2727	. . . . . . . . . . . 12 ⊢ (𝑧 ∈ 𝑆 ↦ (𝑦(-g‘𝐺)𝑧)) = (𝑧 ∈ 𝑆 ↦ (𝑦(-g‘𝐺)𝑧))
39		oveq2 7422	. . . . . . . . . . . 12 ⊢ (𝑧 = 𝑦 → (𝑦(-g‘𝐺)𝑧) = (𝑦(-g‘𝐺)𝑦))
40	38, 39	elrnmpt1s 5953	. . . . . . . . . . 11 ⊢ ((𝑦 ∈ 𝑆 ∧ (𝑦(-g‘𝐺)𝑦) ∈ V) → (𝑦(-g‘𝐺)𝑦) ∈ ran (𝑧 ∈ 𝑆 ↦ (𝑦(-g‘𝐺)𝑧)))
41	36, 37, 40	sylancl 585	. . . . . . . . . 10 ⊢ ((𝐺 ∈ TopGrp ∧ 𝑦 ∈ 𝑆) → (𝑦(-g‘𝐺)𝑦) ∈ ran (𝑧 ∈ 𝑆 ↦ (𝑦(-g‘𝐺)𝑧)))
42	35, 41	eqeltrrd 2829	. . . . . . . . 9 ⊢ ((𝐺 ∈ TopGrp ∧ 𝑦 ∈ 𝑆) → 0 ∈ ran (𝑧 ∈ 𝑆 ↦ (𝑦(-g‘𝐺)𝑧)))
43	42, 21	eleqtrrdi 2839	. . . . . . . 8 ⊢ ((𝐺 ∈ TopGrp ∧ 𝑦 ∈ 𝑆) → 0 ∈ ((𝑧 ∈ 𝑋 ↦ (𝑦(-g‘𝐺)𝑧)) “ 𝑆))
44		eqid 2727	. . . . . . . . 9 ⊢ ∪ 𝐽 = ∪ 𝐽
45		eqid 2727	. . . . . . . . . . . . . . 15 ⊢ (+g‘𝐺) = (+g‘𝐺)
46		eqid 2727	. . . . . . . . . . . . . . 15 ⊢ (invg‘𝐺) = (invg‘𝐺)
47	8, 45, 46, 28	grpsubval 18935	. . . . . . . . . . . . . 14 ⊢ ((𝑦 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋) → (𝑦(-g‘𝐺)𝑧) = (𝑦(+g‘𝐺)((invg‘𝐺)‘𝑧)))
48	25, 47	sylan 579	. . . . . . . . . . . . 13 ⊢ (((𝐺 ∈ TopGrp ∧ 𝑦 ∈ 𝑆) ∧ 𝑧 ∈ 𝑋) → (𝑦(-g‘𝐺)𝑧) = (𝑦(+g‘𝐺)((invg‘𝐺)‘𝑧)))
49	48	mpteq2dva 5242	. . . . . . . . . . . 12 ⊢ ((𝐺 ∈ TopGrp ∧ 𝑦 ∈ 𝑆) → (𝑧 ∈ 𝑋 ↦ (𝑦(-g‘𝐺)𝑧)) = (𝑧 ∈ 𝑋 ↦ (𝑦(+g‘𝐺)((invg‘𝐺)‘𝑧))))
50	8, 46	grpinvcl 18937	. . . . . . . . . . . . . 14 ⊢ ((𝐺 ∈ Grp ∧ 𝑧 ∈ 𝑋) → ((invg‘𝐺)‘𝑧) ∈ 𝑋)
51	23, 50	sylan 579	. . . . . . . . . . . . 13 ⊢ (((𝐺 ∈ TopGrp ∧ 𝑦 ∈ 𝑆) ∧ 𝑧 ∈ 𝑋) → ((invg‘𝐺)‘𝑧) ∈ 𝑋)
52	8, 46	grpinvf 18936	. . . . . . . . . . . . . . . 16 ⊢ (𝐺 ∈ Grp → (invg‘𝐺):𝑋⟶𝑋)
53	10, 52	syl 17	. . . . . . . . . . . . . . 15 ⊢ (𝐺 ∈ TopGrp → (invg‘𝐺):𝑋⟶𝑋)
54	53	adantr 480	. . . . . . . . . . . . . 14 ⊢ ((𝐺 ∈ TopGrp ∧ 𝑦 ∈ 𝑆) → (invg‘𝐺):𝑋⟶𝑋)
55	54	feqmptd 6961	. . . . . . . . . . . . 13 ⊢ ((𝐺 ∈ TopGrp ∧ 𝑦 ∈ 𝑆) → (invg‘𝐺) = (𝑧 ∈ 𝑋 ↦ ((invg‘𝐺)‘𝑧)))
56		eqidd 2728	. . . . . . . . . . . . 13 ⊢ ((𝐺 ∈ TopGrp ∧ 𝑦 ∈ 𝑆) → (𝑤 ∈ 𝑋 ↦ (𝑦(+g‘𝐺)𝑤)) = (𝑤 ∈ 𝑋 ↦ (𝑦(+g‘𝐺)𝑤)))
57		oveq2 7422	. . . . . . . . . . . . 13 ⊢ (𝑤 = ((invg‘𝐺)‘𝑧) → (𝑦(+g‘𝐺)𝑤) = (𝑦(+g‘𝐺)((invg‘𝐺)‘𝑧)))
58	51, 55, 56, 57	fmptco 7132	. . . . . . . . . . . 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 2770	. . . . . . . . . . 11 ⊢ ((𝐺 ∈ TopGrp ∧ 𝑦 ∈ 𝑆) → (𝑧 ∈ 𝑋 ↦ (𝑦(-g‘𝐺)𝑧)) = ((𝑤 ∈ 𝑋 ↦ (𝑦(+g‘𝐺)𝑤)) ∘ (invg‘𝐺)))
60	7, 46	grpinvhmeo 23983	. . . . . . . . . . . . 13 ⊢ (𝐺 ∈ TopGrp → (invg‘𝐺) ∈ (𝐽Homeo𝐽))
61	60	adantr 480	. . . . . . . . . . . 12 ⊢ ((𝐺 ∈ TopGrp ∧ 𝑦 ∈ 𝑆) → (invg‘𝐺) ∈ (𝐽Homeo𝐽))
62		eqid 2727	. . . . . . . . . . . . . 14 ⊢ (𝑤 ∈ 𝑋 ↦ (𝑦(+g‘𝐺)𝑤)) = (𝑤 ∈ 𝑋 ↦ (𝑦(+g‘𝐺)𝑤))
63	62, 8, 45, 7	tgplacthmeo 24000	. . . . . . . . . . . . 13 ⊢ ((𝐺 ∈ TopGrp ∧ 𝑦 ∈ 𝑋) → (𝑤 ∈ 𝑋 ↦ (𝑦(+g‘𝐺)𝑤)) ∈ (𝐽Homeo𝐽))
64	25, 63	syldan 590	. . . . . . . . . . . 12 ⊢ ((𝐺 ∈ TopGrp ∧ 𝑦 ∈ 𝑆) → (𝑤 ∈ 𝑋 ↦ (𝑦(+g‘𝐺)𝑤)) ∈ (𝐽Homeo𝐽))
65		hmeoco 23669	. . . . . . . . . . . 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 583	. . . . . . . . . . 11 ⊢ ((𝐺 ∈ TopGrp ∧ 𝑦 ∈ 𝑆) → ((𝑤 ∈ 𝑋 ↦ (𝑦(+g‘𝐺)𝑤)) ∘ (invg‘𝐺)) ∈ (𝐽Homeo𝐽))
67	59, 66	eqeltrd 2828	. . . . . . . . . 10 ⊢ ((𝐺 ∈ TopGrp ∧ 𝑦 ∈ 𝑆) → (𝑧 ∈ 𝑋 ↦ (𝑦(-g‘𝐺)𝑧)) ∈ (𝐽Homeo𝐽))
68		hmeocn 23657	. . . . . . . . . 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 22809	. . . . . . . . . . . 12 ⊢ (𝐽 ∈ (TopOn‘𝑋) → 𝑋 = ∪ 𝐽)
71	9, 70	syl 17	. . . . . . . . . . 11 ⊢ (𝐺 ∈ TopGrp → 𝑋 = ∪ 𝐽)
72	71	adantr 480	. . . . . . . . . 10 ⊢ ((𝐺 ∈ TopGrp ∧ 𝑦 ∈ 𝑆) → 𝑋 = ∪ 𝐽)
73	5, 72	sseqtrid 4030	. . . . . . . . 9 ⊢ ((𝐺 ∈ TopGrp ∧ 𝑦 ∈ 𝑆) → 𝑆 ⊆ ∪ 𝐽)
74	1	conncompconn 23329	. . . . . . . . . . 11 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 0 ∈ 𝑋) → (𝐽 ↾t 𝑆) ∈ Conn)
75	9, 13, 74	syl2anc 583	. . . . . . . . . 10 ⊢ (𝐺 ∈ TopGrp → (𝐽 ↾t 𝑆) ∈ Conn)
76	75	adantr 480	. . . . . . . . 9 ⊢ ((𝐺 ∈ TopGrp ∧ 𝑦 ∈ 𝑆) → (𝐽 ↾t 𝑆) ∈ Conn)
77	44, 69, 73, 76	connima 23322	. . . . . . . 8 ⊢ ((𝐺 ∈ TopGrp ∧ 𝑦 ∈ 𝑆) → (𝐽 ↾t ((𝑧 ∈ 𝑋 ↦ (𝑦(-g‘𝐺)𝑧)) “ 𝑆)) ∈ Conn)
78	1	conncompss 23330	. . . . . . . 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 1369	. . . . . . 7 ⊢ ((𝐺 ∈ TopGrp ∧ 𝑦 ∈ 𝑆) → ((𝑧 ∈ 𝑋 ↦ (𝑦(-g‘𝐺)𝑧)) “ 𝑆) ⊆ 𝑆)
80	21, 79	eqsstrrid 4027	. . . . . 6 ⊢ ((𝐺 ∈ TopGrp ∧ 𝑦 ∈ 𝑆) → ran (𝑧 ∈ 𝑆 ↦ (𝑦(-g‘𝐺)𝑧)) ⊆ 𝑆)
81		ovex 7447	. . . . . . . 8 ⊢ (𝑦(-g‘𝐺)𝑧) ∈ V
82	81, 38	fnmpti 6692	. . . . . . 7 ⊢ (𝑧 ∈ 𝑆 ↦ (𝑦(-g‘𝐺)𝑧)) Fn 𝑆
83		df-f 6546	. . . . . . 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 708	. . . . . 6 ⊢ ((𝑧 ∈ 𝑆 ↦ (𝑦(-g‘𝐺)𝑧)):𝑆⟶𝑆 ↔ ran (𝑧 ∈ 𝑆 ↦ (𝑦(-g‘𝐺)𝑧)) ⊆ 𝑆)
85	80, 84	sylibr 233	. . . . 5 ⊢ ((𝐺 ∈ TopGrp ∧ 𝑦 ∈ 𝑆) → (𝑧 ∈ 𝑆 ↦ (𝑦(-g‘𝐺)𝑧)):𝑆⟶𝑆)
86	38	fmpt 7114	. . . . 5 ⊢ (∀𝑧 ∈ 𝑆 (𝑦(-g‘𝐺)𝑧) ∈ 𝑆 ↔ (𝑧 ∈ 𝑆 ↦ (𝑦(-g‘𝐺)𝑧)):𝑆⟶𝑆)
87	85, 86	sylibr 233	. . . 4 ⊢ ((𝐺 ∈ TopGrp ∧ 𝑦 ∈ 𝑆) → ∀𝑧 ∈ 𝑆 (𝑦(-g‘𝐺)𝑧) ∈ 𝑆)
88	87	ralrimiva 3141	. . 3 ⊢ (𝐺 ∈ TopGrp → ∀𝑦 ∈ 𝑆 ∀𝑧 ∈ 𝑆 (𝑦(-g‘𝐺)𝑧) ∈ 𝑆)
89	8, 28	issubg4 19093	. . . 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 1340	. 2 ⊢ (𝐺 ∈ TopGrp → 𝑆 ∈ (SubGrp‘𝐺))
92	10	adantr 480	. . . . . . . . . 10 ⊢ ((𝐺 ∈ TopGrp ∧ ((𝑦 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋) ∧ (𝑦(+g‘𝐺)𝑧) ∈ 𝑆)) → 𝐺 ∈ Grp)
93		eqid 2727	. . . . . . . . . . 11 ⊢ (oppg‘𝐺) = (oppg‘𝐺)
94	93, 46	oppginv 19306	. . . . . . . . . 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 6893	. . . . . . . 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 778	. . . . . . . . 9 ⊢ ((𝐺 ∈ TopGrp ∧ ((𝑦 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋) ∧ (𝑦(+g‘𝐺)𝑧) ∈ 𝑆)) → 𝑦 ∈ 𝑋)
98	8, 46	grpinvinv 18955	. . . . . . . . 9 ⊢ ((𝐺 ∈ Grp ∧ 𝑦 ∈ 𝑋) → ((invg‘𝐺)‘((invg‘𝐺)‘𝑦)) = 𝑦)
99	92, 97, 98	syl2anc 583	. . . . . . . 8 ⊢ ((𝐺 ∈ TopGrp ∧ ((𝑦 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋) ∧ (𝑦(+g‘𝐺)𝑧) ∈ 𝑆)) → ((invg‘𝐺)‘((invg‘𝐺)‘𝑦)) = 𝑦)
100	96, 99	eqtr3d 2769	. . . . . . 7 ⊢ ((𝐺 ∈ TopGrp ∧ ((𝑦 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋) ∧ (𝑦(+g‘𝐺)𝑧) ∈ 𝑆)) → ((invg‘(oppg‘𝐺))‘((invg‘𝐺)‘𝑦)) = 𝑦)
101	100	oveq1d 7429	. . . . . 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 2727	. . . . . . 7 ⊢ (+g‘(oppg‘𝐺)) = (+g‘(oppg‘𝐺))
103	45, 93, 102	oppgplus 19293	. . . . . 6 ⊢ (𝑦(+g‘(oppg‘𝐺))𝑧) = (𝑧(+g‘𝐺)𝑦)
104	101, 103	eqtrdi 2783	. . . . 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 18937	. . . . . . . . . 10 ⊢ ((𝐺 ∈ Grp ∧ 𝑦 ∈ 𝑋) → ((invg‘𝐺)‘𝑦) ∈ 𝑋)
106	92, 97, 105	syl2anc 583	. . . . . . . . 9 ⊢ ((𝐺 ∈ TopGrp ∧ ((𝑦 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋) ∧ (𝑦(+g‘𝐺)𝑧) ∈ 𝑆)) → ((invg‘𝐺)‘𝑦) ∈ 𝑋)
107		simprlr 779	. . . . . . . . 9 ⊢ ((𝐺 ∈ TopGrp ∧ ((𝑦 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋) ∧ (𝑦(+g‘𝐺)𝑧) ∈ 𝑆)) → 𝑧 ∈ 𝑋)
108	99	oveq1d 7429	. . . . . . . . . 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 772	. . . . . . . . . 10 ⊢ ((𝐺 ∈ TopGrp ∧ ((𝑦 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋) ∧ (𝑦(+g‘𝐺)𝑧) ∈ 𝑆)) → (𝑦(+g‘𝐺)𝑧) ∈ 𝑆)
110	108, 109	eqeltrd 2828	. . . . . . . . 9 ⊢ ((𝐺 ∈ TopGrp ∧ ((𝑦 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋) ∧ (𝑦(+g‘𝐺)𝑧) ∈ 𝑆)) → (((invg‘𝐺)‘((invg‘𝐺)‘𝑦))(+g‘𝐺)𝑧) ∈ 𝑆)
111		eqid 2727	. . . . . . . . . . 11 ⊢ (𝐺 ~QG 𝑆) = (𝐺 ~QG 𝑆)
112	8, 46, 45, 111	eqgval 19125	. . . . . . . . . 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 585	. . . . . . . . 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 1340	. . . . . . . 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 24009	. . . . . . . . . . . 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 590	. . . . . . . . . . 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 23995	. . . . . . . . . . . . 13 ⊢ (𝐺 ∈ TopGrp → (oppg‘𝐺) ∈ TopGrp)
118	117	adantr 480	. . . . . . . . . . . 12 ⊢ ((𝐺 ∈ TopGrp ∧ ((𝑦 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋) ∧ (𝑦(+g‘𝐺)𝑧) ∈ 𝑆)) → (oppg‘𝐺) ∈ TopGrp)
119	93, 8	oppgbas 19296	. . . . . . . . . . . . 13 ⊢ 𝑋 = (Base‘(oppg‘𝐺))
120	93, 11	oppgid 19303	. . . . . . . . . . . . 13 ⊢ 0 = (0g‘(oppg‘𝐺))
121	93, 7	oppgtopn 19300	. . . . . . . . . . . . 13 ⊢ 𝐽 = (TopOpen‘(oppg‘𝐺))
122		eqid 2727	. . . . . . . . . . . . 13 ⊢ ((oppg‘𝐺) ~QG 𝑆) = ((oppg‘𝐺) ~QG 𝑆)
123	119, 120, 121, 1, 122	tgpconncompeqg 24009	. . . . . . . . . . . 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 583	. . . . . . . . . . 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 2770	. . . . . . . . . 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 2814	. . . . . . . . 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 3473	. . . . . . . . . 10 ⊢ 𝑧 ∈ V
128		fvex 6904	. . . . . . . . . 10 ⊢ ((invg‘𝐺)‘𝑦) ∈ V
129	127, 128	elec 8763	. . . . . . . . 9 ⊢ (𝑧 ∈ [((invg‘𝐺)‘𝑦)](𝐺 ~QG 𝑆) ↔ ((invg‘𝐺)‘𝑦)(𝐺 ~QG 𝑆)𝑧)
130	127, 128	elec 8763	. . . . . . . . 9 ⊢ (𝑧 ∈ [((invg‘𝐺)‘𝑦)]((oppg‘𝐺) ~QG 𝑆) ↔ ((invg‘𝐺)‘𝑦)((oppg‘𝐺) ~QG 𝑆)𝑧)
131	126, 129, 130	3bitr3g 313	. . . . . . . 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 231	. . . . . . 7 ⊢ ((𝐺 ∈ TopGrp ∧ ((𝑦 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋) ∧ (𝑦(+g‘𝐺)𝑧) ∈ 𝑆)) → ((invg‘𝐺)‘𝑦)((oppg‘𝐺) ~QG 𝑆)𝑧)
133		eqid 2727	. . . . . . . . 9 ⊢ (invg‘(oppg‘𝐺)) = (invg‘(oppg‘𝐺))
134	119, 133, 102, 122	eqgval 19125	. . . . . . . 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 585	. . . . . . 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 231	. . . . . 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 1142	. . . . 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 2829	. . . 4 ⊢ ((𝐺 ∈ TopGrp ∧ ((𝑦 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋) ∧ (𝑦(+g‘𝐺)𝑧) ∈ 𝑆)) → (𝑧(+g‘𝐺)𝑦) ∈ 𝑆)
139	138	expr 456	. . 3 ⊢ ((𝐺 ∈ TopGrp ∧ (𝑦 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋)) → ((𝑦(+g‘𝐺)𝑧) ∈ 𝑆 → (𝑧(+g‘𝐺)𝑦) ∈ 𝑆))
140	139	ralrimivva 3195	. 2 ⊢ (𝐺 ∈ TopGrp → ∀𝑦 ∈ 𝑋 ∀𝑧 ∈ 𝑋 ((𝑦(+g‘𝐺)𝑧) ∈ 𝑆 → (𝑧(+g‘𝐺)𝑦) ∈ 𝑆))
141	8, 45	isnsg2 19104	. 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 582	1 ⊢ (𝐺 ∈ TopGrp → 𝑆 ∈ (NrmSGrp‘𝐺))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 395   ∧ w3a 1085   = wceq 1534   ∈ wcel 2099   ≠ wne 2935  ∀wral 3056  {crab 3427  Vcvv 3469   ⊆ wss 3944  ∅c0 4318  𝒫 cpw 4598  ∪ cuni 4903   class class class wbr 5142   ↦ cmpt 5225  ran crn 5673   ↾ cres 5674   “ cima 5675   ∘ ccom 5676   Fn wfn 6537  ⟶wf 6538  ‘cfv 6542  (class class class)co 7414  [cec 8716  Basecbs 17173  +_gcplusg 17226   ↾_t crest 17395  TopOpenctopn 17396  0_gc0g 17414  Grpcgrp 18883  inv_gcminusg 18884  -_gcsg 18885  SubGrpcsubg 19068  NrmSGrpcnsg 19069   ~_QG cqg 19070  opp_gcoppg 19289  TopOnctopon 22805   Cn ccn 23121  Conncconn 23308  Homeochmeo 23650  TopGrpctgp 23968

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2164  ax-ext 2698  ax-rep 5279  ax-sep 5293  ax-nul 5300  ax-pow 5359  ax-pr 5423  ax-un 7734  ax-cnex 11188  ax-resscn 11189  ax-1cn 11190  ax-icn 11191  ax-addcl 11192  ax-addrcl 11193  ax-mulcl 11194  ax-mulrcl 11195  ax-mulcom 11196  ax-addass 11197  ax-mulass 11198  ax-distr 11199  ax-i2m1 11200  ax-1ne0 11201  ax-1rid 11202  ax-rnegex 11203  ax-rrecex 11204  ax-cnre 11205  ax-pre-lttri 11206  ax-pre-lttrn 11207  ax-pre-ltadd 11208  ax-pre-mulgt0 11209

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3or 1086  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2705  df-cleq 2719  df-clel 2805  df-nfc 2880  df-ne 2936  df-nel 3042  df-ral 3057  df-rex 3066  df-rmo 3371  df-reu 3372  df-rab 3428  df-v 3471  df-sbc 3775  df-csb 3890  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-pss 3963  df-nul 4319  df-if 4525  df-pw 4600  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4904  df-int 4945  df-iun 4993  df-br 5143  df-opab 5205  df-mpt 5226  df-tr 5260  df-id 5570  df-eprel 5576  df-po 5584  df-so 5585  df-fr 5627  df-we 5629  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-rn 5683  df-res 5684  df-ima 5685  df-pred 6299  df-ord 6366  df-on 6367  df-lim 6368  df-suc 6369  df-iota 6494  df-fun 6544  df-fn 6545  df-f 6546  df-f1 6547  df-fo 6548  df-f1o 6549  df-fv 6550  df-riota 7370  df-ov 7417  df-oprab 7418  df-mpo 7419  df-om 7865  df-1st 7987  df-2nd 7988  df-tpos 8225  df-frecs 8280  df-wrecs 8311  df-recs 8385  df-rdg 8424  df-er 8718  df-ec 8720  df-map 8840  df-en 8958  df-dom 8959  df-sdom 8960  df-fin 8961  df-fi 9428  df-pnf 11274  df-mnf 11275  df-xr 11276  df-ltxr 11277  df-le 11278  df-sub 11470  df-neg 11471  df-nn 12237  df-2 12299  df-3 12300  df-4 12301  df-5 12302  df-6 12303  df-7 12304  df-8 12305  df-9 12306  df-sets 17126  df-slot 17144  df-ndx 17156  df-base 17174  df-ress 17203  df-plusg 17239  df-tset 17245  df-rest 17397  df-topn 17398  df-0g 17416  df-topgen 17418  df-plusf 18592  df-mgm 18593  df-sgrp 18672  df-mnd 18688  df-grp 18886  df-minusg 18887  df-sbg 18888  df-subg 19071  df-nsg 19072  df-eqg 19073  df-oppg 19290  df-top 22789  df-topon 22806  df-topsp 22828  df-bases 22842  df-cld 22916  df-cn 23124  df-cnp 23125  df-conn 23309  df-tx 23459  df-hmeo 23652  df-tmd 23969  df-tgp 23970

This theorem is referenced by: (None)