Theorem tgpconncomp 22286

Description: The identity component, the connected component containing the identity element, is a closed (conncompcld 21608) 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 3912	. . . . . 6 ⊢ {𝑥 ∈ 𝒫 𝑋 ∣ ( 0 ∈ 𝑥 ∧ (𝐽 ↾t 𝑥) ∈ Conn)} ⊆ 𝒫 𝑋
3		sspwuni 4832	. . . . . 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 222	. . . . 5 ⊢ ∪ {𝑥 ∈ 𝒫 𝑋 ∣ ( 0 ∈ 𝑥 ∧ (𝐽 ↾t 𝑥) ∈ Conn)} ⊆ 𝑋
5	1, 4	eqsstri 3860	. . . 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 22256	. . . . 5 ⊢ (𝐺 ∈ TopGrp → 𝐽 ∈ (TopOn‘𝑋))
10		tgpgrp 22252	. . . . . 6 ⊢ (𝐺 ∈ TopGrp → 𝐺 ∈ Grp)
11		tgpconncomp.z	. . . . . . 7 ⊢ 0 = (0g‘𝐺)
12	8, 11	grpidcl 17804	. . . . . 6 ⊢ (𝐺 ∈ Grp → 0 ∈ 𝑋)
13	10, 12	syl 17	. . . . 5 ⊢ (𝐺 ∈ TopGrp → 0 ∈ 𝑋)
14	1	conncompid 21605	. . . . 5 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 0 ∈ 𝑋) → 0 ∈ 𝑆)
15	9, 13, 14	syl2anc 579	. . . 4 ⊢ (𝐺 ∈ TopGrp → 0 ∈ 𝑆)
16	15	ne0d 4151	. . 3 ⊢ (𝐺 ∈ TopGrp → 𝑆 ≠ ∅)
17		df-ima 5355	. . . . . . . 8 ⊢ ((𝑧 ∈ 𝑋 ↦ (𝑦(-g‘𝐺)𝑧)) “ 𝑆) = ran ((𝑧 ∈ 𝑋 ↦ (𝑦(-g‘𝐺)𝑧)) ↾ 𝑆)
18		resmpt 5686	. . . . . . . . . 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 5584	. . . . . . . 8 ⊢ ran ((𝑧 ∈ 𝑋 ↦ (𝑦(-g‘𝐺)𝑧)) ↾ 𝑆) = ran (𝑧 ∈ 𝑆 ↦ (𝑦(-g‘𝐺)𝑧))
21	17, 20	eqtri 2849	. . . . . . 7 ⊢ ((𝑧 ∈ 𝑋 ↦ (𝑦(-g‘𝐺)𝑧)) “ 𝑆) = ran (𝑧 ∈ 𝑆 ↦ (𝑦(-g‘𝐺)𝑧))
22		imassrn 5718	. . . . . . . . 9 ⊢ ((𝑧 ∈ 𝑋 ↦ (𝑦(-g‘𝐺)𝑧)) “ 𝑆) ⊆ ran (𝑧 ∈ 𝑋 ↦ (𝑦(-g‘𝐺)𝑧))
23	10	adantr 474	. . . . . . . . . . . . 13 ⊢ ((𝐺 ∈ TopGrp ∧ 𝑦 ∈ 𝑆) → 𝐺 ∈ Grp)
24	23	adantr 474	. . . . . . . . . . . 12 ⊢ (((𝐺 ∈ TopGrp ∧ 𝑦 ∈ 𝑆) ∧ 𝑧 ∈ 𝑋) → 𝐺 ∈ Grp)
25	6	sselda 3827	. . . . . . . . . . . . 13 ⊢ ((𝐺 ∈ TopGrp ∧ 𝑦 ∈ 𝑆) → 𝑦 ∈ 𝑋)
26	25	adantr 474	. . . . . . . . . . . 12 ⊢ (((𝐺 ∈ TopGrp ∧ 𝑦 ∈ 𝑆) ∧ 𝑧 ∈ 𝑋) → 𝑦 ∈ 𝑋)
27		simpr 479	. . . . . . . . . . . 12 ⊢ (((𝐺 ∈ TopGrp ∧ 𝑦 ∈ 𝑆) ∧ 𝑧 ∈ 𝑋) → 𝑧 ∈ 𝑋)
28		eqid 2825	. . . . . . . . . . . . 13 ⊢ (-g‘𝐺) = (-g‘𝐺)
29	8, 28	grpsubcl 17849	. . . . . . . . . . . 12 ⊢ ((𝐺 ∈ Grp ∧ 𝑦 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋) → (𝑦(-g‘𝐺)𝑧) ∈ 𝑋)
30	24, 26, 27, 29	syl3anc 1494	. . . . . . . . . . 11 ⊢ (((𝐺 ∈ TopGrp ∧ 𝑦 ∈ 𝑆) ∧ 𝑧 ∈ 𝑋) → (𝑦(-g‘𝐺)𝑧) ∈ 𝑋)
31	30	fmpttd 6634	. . . . . . . . . 10 ⊢ ((𝐺 ∈ TopGrp ∧ 𝑦 ∈ 𝑆) → (𝑧 ∈ 𝑋 ↦ (𝑦(-g‘𝐺)𝑧)):𝑋⟶𝑋)
32	31	frnd 6285	. . . . . . . . 9 ⊢ ((𝐺 ∈ TopGrp ∧ 𝑦 ∈ 𝑆) → ran (𝑧 ∈ 𝑋 ↦ (𝑦(-g‘𝐺)𝑧)) ⊆ 𝑋)
33	22, 32	syl5ss 3838	. . . . . . . 8 ⊢ ((𝐺 ∈ TopGrp ∧ 𝑦 ∈ 𝑆) → ((𝑧 ∈ 𝑋 ↦ (𝑦(-g‘𝐺)𝑧)) “ 𝑆) ⊆ 𝑋)
34	8, 11, 28	grpsubid 17853	. . . . . . . . . . 11 ⊢ ((𝐺 ∈ Grp ∧ 𝑦 ∈ 𝑋) → (𝑦(-g‘𝐺)𝑦) = 0 )
35	23, 25, 34	syl2anc 579	. . . . . . . . . 10 ⊢ ((𝐺 ∈ TopGrp ∧ 𝑦 ∈ 𝑆) → (𝑦(-g‘𝐺)𝑦) = 0 )
36		simpr 479	. . . . . . . . . . 11 ⊢ ((𝐺 ∈ TopGrp ∧ 𝑦 ∈ 𝑆) → 𝑦 ∈ 𝑆)
37		ovex 6937	. . . . . . . . . . 11 ⊢ (𝑦(-g‘𝐺)𝑦) ∈ V
38		eqid 2825	. . . . . . . . . . . 12 ⊢ (𝑧 ∈ 𝑆 ↦ (𝑦(-g‘𝐺)𝑧)) = (𝑧 ∈ 𝑆 ↦ (𝑦(-g‘𝐺)𝑧))
39		oveq2 6913	. . . . . . . . . . . 12 ⊢ (𝑧 = 𝑦 → (𝑦(-g‘𝐺)𝑧) = (𝑦(-g‘𝐺)𝑦))
40	38, 39	elrnmpt1s 5606	. . . . . . . . . . 11 ⊢ ((𝑦 ∈ 𝑆 ∧ (𝑦(-g‘𝐺)𝑦) ∈ V) → (𝑦(-g‘𝐺)𝑦) ∈ ran (𝑧 ∈ 𝑆 ↦ (𝑦(-g‘𝐺)𝑧)))
41	36, 37, 40	sylancl 580	. . . . . . . . . 10 ⊢ ((𝐺 ∈ TopGrp ∧ 𝑦 ∈ 𝑆) → (𝑦(-g‘𝐺)𝑦) ∈ ran (𝑧 ∈ 𝑆 ↦ (𝑦(-g‘𝐺)𝑧)))
42	35, 41	eqeltrrd 2907	. . . . . . . . 9 ⊢ ((𝐺 ∈ TopGrp ∧ 𝑦 ∈ 𝑆) → 0 ∈ ran (𝑧 ∈ 𝑆 ↦ (𝑦(-g‘𝐺)𝑧)))
43	42, 21	syl6eleqr 2917	. . . . . . . 8 ⊢ ((𝐺 ∈ TopGrp ∧ 𝑦 ∈ 𝑆) → 0 ∈ ((𝑧 ∈ 𝑋 ↦ (𝑦(-g‘𝐺)𝑧)) “ 𝑆))
44		eqid 2825	. . . . . . . . 9 ⊢ ∪ 𝐽 = ∪ 𝐽
45		eqid 2825	. . . . . . . . . . . . . . 15 ⊢ (+g‘𝐺) = (+g‘𝐺)
46		eqid 2825	. . . . . . . . . . . . . . 15 ⊢ (invg‘𝐺) = (invg‘𝐺)
47	8, 45, 46, 28	grpsubval 17819	. . . . . . . . . . . . . 14 ⊢ ((𝑦 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋) → (𝑦(-g‘𝐺)𝑧) = (𝑦(+g‘𝐺)((invg‘𝐺)‘𝑧)))
48	25, 47	sylan 575	. . . . . . . . . . . . 13 ⊢ (((𝐺 ∈ TopGrp ∧ 𝑦 ∈ 𝑆) ∧ 𝑧 ∈ 𝑋) → (𝑦(-g‘𝐺)𝑧) = (𝑦(+g‘𝐺)((invg‘𝐺)‘𝑧)))
49	48	mpteq2dva 4967	. . . . . . . . . . . 12 ⊢ ((𝐺 ∈ TopGrp ∧ 𝑦 ∈ 𝑆) → (𝑧 ∈ 𝑋 ↦ (𝑦(-g‘𝐺)𝑧)) = (𝑧 ∈ 𝑋 ↦ (𝑦(+g‘𝐺)((invg‘𝐺)‘𝑧))))
50	8, 46	grpinvcl 17821	. . . . . . . . . . . . . 14 ⊢ ((𝐺 ∈ Grp ∧ 𝑧 ∈ 𝑋) → ((invg‘𝐺)‘𝑧) ∈ 𝑋)
51	23, 50	sylan 575	. . . . . . . . . . . . 13 ⊢ (((𝐺 ∈ TopGrp ∧ 𝑦 ∈ 𝑆) ∧ 𝑧 ∈ 𝑋) → ((invg‘𝐺)‘𝑧) ∈ 𝑋)
52	8, 46	grpinvf 17820	. . . . . . . . . . . . . . . 16 ⊢ (𝐺 ∈ Grp → (invg‘𝐺):𝑋⟶𝑋)
53	10, 52	syl 17	. . . . . . . . . . . . . . 15 ⊢ (𝐺 ∈ TopGrp → (invg‘𝐺):𝑋⟶𝑋)
54	53	adantr 474	. . . . . . . . . . . . . 14 ⊢ ((𝐺 ∈ TopGrp ∧ 𝑦 ∈ 𝑆) → (invg‘𝐺):𝑋⟶𝑋)
55	54	feqmptd 6496	. . . . . . . . . . . . 13 ⊢ ((𝐺 ∈ TopGrp ∧ 𝑦 ∈ 𝑆) → (invg‘𝐺) = (𝑧 ∈ 𝑋 ↦ ((invg‘𝐺)‘𝑧)))
56		eqidd 2826	. . . . . . . . . . . . 13 ⊢ ((𝐺 ∈ TopGrp ∧ 𝑦 ∈ 𝑆) → (𝑤 ∈ 𝑋 ↦ (𝑦(+g‘𝐺)𝑤)) = (𝑤 ∈ 𝑋 ↦ (𝑦(+g‘𝐺)𝑤)))
57		oveq2 6913	. . . . . . . . . . . . 13 ⊢ (𝑤 = ((invg‘𝐺)‘𝑧) → (𝑦(+g‘𝐺)𝑤) = (𝑦(+g‘𝐺)((invg‘𝐺)‘𝑧)))
58	51, 55, 56, 57	fmptco 6646	. . . . . . . . . . . 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 2864	. . . . . . . . . . 11 ⊢ ((𝐺 ∈ TopGrp ∧ 𝑦 ∈ 𝑆) → (𝑧 ∈ 𝑋 ↦ (𝑦(-g‘𝐺)𝑧)) = ((𝑤 ∈ 𝑋 ↦ (𝑦(+g‘𝐺)𝑤)) ∘ (invg‘𝐺)))
60	7, 46	grpinvhmeo 22260	. . . . . . . . . . . . 13 ⊢ (𝐺 ∈ TopGrp → (invg‘𝐺) ∈ (𝐽Homeo𝐽))
61	60	adantr 474	. . . . . . . . . . . 12 ⊢ ((𝐺 ∈ TopGrp ∧ 𝑦 ∈ 𝑆) → (invg‘𝐺) ∈ (𝐽Homeo𝐽))
62		eqid 2825	. . . . . . . . . . . . . 14 ⊢ (𝑤 ∈ 𝑋 ↦ (𝑦(+g‘𝐺)𝑤)) = (𝑤 ∈ 𝑋 ↦ (𝑦(+g‘𝐺)𝑤))
63	62, 8, 45, 7	tgplacthmeo 22277	. . . . . . . . . . . . 13 ⊢ ((𝐺 ∈ TopGrp ∧ 𝑦 ∈ 𝑋) → (𝑤 ∈ 𝑋 ↦ (𝑦(+g‘𝐺)𝑤)) ∈ (𝐽Homeo𝐽))
64	25, 63	syldan 585	. . . . . . . . . . . 12 ⊢ ((𝐺 ∈ TopGrp ∧ 𝑦 ∈ 𝑆) → (𝑤 ∈ 𝑋 ↦ (𝑦(+g‘𝐺)𝑤)) ∈ (𝐽Homeo𝐽))
65		hmeoco 21946	. . . . . . . . . . . 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 579	. . . . . . . . . . 11 ⊢ ((𝐺 ∈ TopGrp ∧ 𝑦 ∈ 𝑆) → ((𝑤 ∈ 𝑋 ↦ (𝑦(+g‘𝐺)𝑤)) ∘ (invg‘𝐺)) ∈ (𝐽Homeo𝐽))
67	59, 66	eqeltrd 2906	. . . . . . . . . 10 ⊢ ((𝐺 ∈ TopGrp ∧ 𝑦 ∈ 𝑆) → (𝑧 ∈ 𝑋 ↦ (𝑦(-g‘𝐺)𝑧)) ∈ (𝐽Homeo𝐽))
68		hmeocn 21934	. . . . . . . . . 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 21089	. . . . . . . . . . . 12 ⊢ (𝐽 ∈ (TopOn‘𝑋) → 𝑋 = ∪ 𝐽)
71	9, 70	syl 17	. . . . . . . . . . 11 ⊢ (𝐺 ∈ TopGrp → 𝑋 = ∪ 𝐽)
72	71	adantr 474	. . . . . . . . . 10 ⊢ ((𝐺 ∈ TopGrp ∧ 𝑦 ∈ 𝑆) → 𝑋 = ∪ 𝐽)
73	5, 72	syl5sseq 3878	. . . . . . . . 9 ⊢ ((𝐺 ∈ TopGrp ∧ 𝑦 ∈ 𝑆) → 𝑆 ⊆ ∪ 𝐽)
74	1	conncompconn 21606	. . . . . . . . . . 11 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 0 ∈ 𝑋) → (𝐽 ↾t 𝑆) ∈ Conn)
75	9, 13, 74	syl2anc 579	. . . . . . . . . 10 ⊢ (𝐺 ∈ TopGrp → (𝐽 ↾t 𝑆) ∈ Conn)
76	75	adantr 474	. . . . . . . . 9 ⊢ ((𝐺 ∈ TopGrp ∧ 𝑦 ∈ 𝑆) → (𝐽 ↾t 𝑆) ∈ Conn)
77	44, 69, 73, 76	connima 21599	. . . . . . . 8 ⊢ ((𝐺 ∈ TopGrp ∧ 𝑦 ∈ 𝑆) → (𝐽 ↾t ((𝑧 ∈ 𝑋 ↦ (𝑦(-g‘𝐺)𝑧)) “ 𝑆)) ∈ Conn)
78	1	conncompss 21607	. . . . . . . 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 1494	. . . . . . 7 ⊢ ((𝐺 ∈ TopGrp ∧ 𝑦 ∈ 𝑆) → ((𝑧 ∈ 𝑋 ↦ (𝑦(-g‘𝐺)𝑧)) “ 𝑆) ⊆ 𝑆)
80	21, 79	syl5eqssr 3875	. . . . . 6 ⊢ ((𝐺 ∈ TopGrp ∧ 𝑦 ∈ 𝑆) → ran (𝑧 ∈ 𝑆 ↦ (𝑦(-g‘𝐺)𝑧)) ⊆ 𝑆)
81		ovex 6937	. . . . . . . 8 ⊢ (𝑦(-g‘𝐺)𝑧) ∈ V
82	81, 38	fnmpti 6255	. . . . . . 7 ⊢ (𝑧 ∈ 𝑆 ↦ (𝑦(-g‘𝐺)𝑧)) Fn 𝑆
83		df-f 6127	. . . . . . 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 700	. . . . . 6 ⊢ ((𝑧 ∈ 𝑆 ↦ (𝑦(-g‘𝐺)𝑧)):𝑆⟶𝑆 ↔ ran (𝑧 ∈ 𝑆 ↦ (𝑦(-g‘𝐺)𝑧)) ⊆ 𝑆)
85	80, 84	sylibr 226	. . . . 5 ⊢ ((𝐺 ∈ TopGrp ∧ 𝑦 ∈ 𝑆) → (𝑧 ∈ 𝑆 ↦ (𝑦(-g‘𝐺)𝑧)):𝑆⟶𝑆)
86	38	fmpt 6629	. . . . 5 ⊢ (∀𝑧 ∈ 𝑆 (𝑦(-g‘𝐺)𝑧) ∈ 𝑆 ↔ (𝑧 ∈ 𝑆 ↦ (𝑦(-g‘𝐺)𝑧)):𝑆⟶𝑆)
87	85, 86	sylibr 226	. . . 4 ⊢ ((𝐺 ∈ TopGrp ∧ 𝑦 ∈ 𝑆) → ∀𝑧 ∈ 𝑆 (𝑦(-g‘𝐺)𝑧) ∈ 𝑆)
88	87	ralrimiva 3175	. . 3 ⊢ (𝐺 ∈ TopGrp → ∀𝑦 ∈ 𝑆 ∀𝑧 ∈ 𝑆 (𝑦(-g‘𝐺)𝑧) ∈ 𝑆)
89	8, 28	issubg4 17964	. . . 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 1446	. 2 ⊢ (𝐺 ∈ TopGrp → 𝑆 ∈ (SubGrp‘𝐺))
92	10	adantr 474	. . . . . . . . . 10 ⊢ ((𝐺 ∈ TopGrp ∧ ((𝑦 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋) ∧ (𝑦(+g‘𝐺)𝑧) ∈ 𝑆)) → 𝐺 ∈ Grp)
93		eqid 2825	. . . . . . . . . . 11 ⊢ (oppg‘𝐺) = (oppg‘𝐺)
94	93, 46	oppginv 18139	. . . . . . . . . 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 6435	. . . . . . . 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 797	. . . . . . . . 9 ⊢ ((𝐺 ∈ TopGrp ∧ ((𝑦 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋) ∧ (𝑦(+g‘𝐺)𝑧) ∈ 𝑆)) → 𝑦 ∈ 𝑋)
98	8, 46	grpinvinv 17836	. . . . . . . . 9 ⊢ ((𝐺 ∈ Grp ∧ 𝑦 ∈ 𝑋) → ((invg‘𝐺)‘((invg‘𝐺)‘𝑦)) = 𝑦)
99	92, 97, 98	syl2anc 579	. . . . . . . 8 ⊢ ((𝐺 ∈ TopGrp ∧ ((𝑦 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋) ∧ (𝑦(+g‘𝐺)𝑧) ∈ 𝑆)) → ((invg‘𝐺)‘((invg‘𝐺)‘𝑦)) = 𝑦)
100	96, 99	eqtr3d 2863	. . . . . . 7 ⊢ ((𝐺 ∈ TopGrp ∧ ((𝑦 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋) ∧ (𝑦(+g‘𝐺)𝑧) ∈ 𝑆)) → ((invg‘(oppg‘𝐺))‘((invg‘𝐺)‘𝑦)) = 𝑦)
101	100	oveq1d 6920	. . . . . 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 2825	. . . . . . 7 ⊢ (+g‘(oppg‘𝐺)) = (+g‘(oppg‘𝐺))
103	45, 93, 102	oppgplus 18129	. . . . . 6 ⊢ (𝑦(+g‘(oppg‘𝐺))𝑧) = (𝑧(+g‘𝐺)𝑦)
104	101, 103	syl6eq 2877	. . . . 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 17821	. . . . . . . . . 10 ⊢ ((𝐺 ∈ Grp ∧ 𝑦 ∈ 𝑋) → ((invg‘𝐺)‘𝑦) ∈ 𝑋)
106	92, 97, 105	syl2anc 579	. . . . . . . . 9 ⊢ ((𝐺 ∈ TopGrp ∧ ((𝑦 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋) ∧ (𝑦(+g‘𝐺)𝑧) ∈ 𝑆)) → ((invg‘𝐺)‘𝑦) ∈ 𝑋)
107		simprlr 798	. . . . . . . . 9 ⊢ ((𝐺 ∈ TopGrp ∧ ((𝑦 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋) ∧ (𝑦(+g‘𝐺)𝑧) ∈ 𝑆)) → 𝑧 ∈ 𝑋)
108	99	oveq1d 6920	. . . . . . . . . 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 789	. . . . . . . . . 10 ⊢ ((𝐺 ∈ TopGrp ∧ ((𝑦 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋) ∧ (𝑦(+g‘𝐺)𝑧) ∈ 𝑆)) → (𝑦(+g‘𝐺)𝑧) ∈ 𝑆)
110	108, 109	eqeltrd 2906	. . . . . . . . 9 ⊢ ((𝐺 ∈ TopGrp ∧ ((𝑦 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋) ∧ (𝑦(+g‘𝐺)𝑧) ∈ 𝑆)) → (((invg‘𝐺)‘((invg‘𝐺)‘𝑦))(+g‘𝐺)𝑧) ∈ 𝑆)
111		eqid 2825	. . . . . . . . . . 11 ⊢ (𝐺 ~QG 𝑆) = (𝐺 ~QG 𝑆)
112	8, 46, 45, 111	eqgval 17994	. . . . . . . . . 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 580	. . . . . . . . 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 1446	. . . . . . . 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 22285	. . . . . . . . . . . 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 585	. . . . . . . . . . 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 22272	. . . . . . . . . . . . 13 ⊢ (𝐺 ∈ TopGrp → (oppg‘𝐺) ∈ TopGrp)
118	117	adantr 474	. . . . . . . . . . . 12 ⊢ ((𝐺 ∈ TopGrp ∧ ((𝑦 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋) ∧ (𝑦(+g‘𝐺)𝑧) ∈ 𝑆)) → (oppg‘𝐺) ∈ TopGrp)
119	93, 8	oppgbas 18131	. . . . . . . . . . . . 13 ⊢ 𝑋 = (Base‘(oppg‘𝐺))
120	93, 11	oppgid 18136	. . . . . . . . . . . . 13 ⊢ 0 = (0g‘(oppg‘𝐺))
121	93, 7	oppgtopn 18133	. . . . . . . . . . . . 13 ⊢ 𝐽 = (TopOpen‘(oppg‘𝐺))
122		eqid 2825	. . . . . . . . . . . . 13 ⊢ ((oppg‘𝐺) ~QG 𝑆) = ((oppg‘𝐺) ~QG 𝑆)
123	119, 120, 121, 1, 122	tgpconncompeqg 22285	. . . . . . . . . . . 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 579	. . . . . . . . . . 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 2864	. . . . . . . . . 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 2892	. . . . . . . . 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 3417	. . . . . . . . . 10 ⊢ 𝑧 ∈ V
128		fvex 6446	. . . . . . . . . 10 ⊢ ((invg‘𝐺)‘𝑦) ∈ V
129	127, 128	elec 8051	. . . . . . . . 9 ⊢ (𝑧 ∈ [((invg‘𝐺)‘𝑦)](𝐺 ~QG 𝑆) ↔ ((invg‘𝐺)‘𝑦)(𝐺 ~QG 𝑆)𝑧)
130	127, 128	elec 8051	. . . . . . . . 9 ⊢ (𝑧 ∈ [((invg‘𝐺)‘𝑦)]((oppg‘𝐺) ~QG 𝑆) ↔ ((invg‘𝐺)‘𝑦)((oppg‘𝐺) ~QG 𝑆)𝑧)
131	126, 129, 130	3bitr3g 305	. . . . . . . 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 224	. . . . . . 7 ⊢ ((𝐺 ∈ TopGrp ∧ ((𝑦 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋) ∧ (𝑦(+g‘𝐺)𝑧) ∈ 𝑆)) → ((invg‘𝐺)‘𝑦)((oppg‘𝐺) ~QG 𝑆)𝑧)
133		eqid 2825	. . . . . . . . 9 ⊢ (invg‘(oppg‘𝐺)) = (invg‘(oppg‘𝐺))
134	119, 133, 102, 122	eqgval 17994	. . . . . . . 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 580	. . . . . . 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 224	. . . . . 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 1178	. . . . 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 2907	. . . 4 ⊢ ((𝐺 ∈ TopGrp ∧ ((𝑦 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋) ∧ (𝑦(+g‘𝐺)𝑧) ∈ 𝑆)) → (𝑧(+g‘𝐺)𝑦) ∈ 𝑆)
139	138	expr 450	. . 3 ⊢ ((𝐺 ∈ TopGrp ∧ (𝑦 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋)) → ((𝑦(+g‘𝐺)𝑧) ∈ 𝑆 → (𝑧(+g‘𝐺)𝑦) ∈ 𝑆))
140	139	ralrimivva 3180	. 2 ⊢ (𝐺 ∈ TopGrp → ∀𝑦 ∈ 𝑋 ∀𝑧 ∈ 𝑋 ((𝑦(+g‘𝐺)𝑧) ∈ 𝑆 → (𝑧(+g‘𝐺)𝑦) ∈ 𝑆))
141	8, 45	isnsg2 17975	. 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 578	1 ⊢ (𝐺 ∈ TopGrp → 𝑆 ∈ (NrmSGrp‘𝐺))

Syntax hints:   → wi 4   ↔ wb 198   ∧ wa 386   ∧ w3a 1111   = wceq 1656   ∈ wcel 2164   ≠ wne 2999  ∀wral 3117  {crab 3121  Vcvv 3414   ⊆ wss 3798  ∅c0 4144  𝒫 cpw 4378  ∪ cuni 4658   class class class wbr 4873   ↦ cmpt 4952  ran crn 5343   ↾ cres 5344   “ cima 5345   ∘ ccom 5346   Fn wfn 6118  ⟶wf 6119  ‘cfv 6123  (class class class)co 6905  [cec 8007  Basecbs 16222  +_gcplusg 16305   ↾_t crest 16434  TopOpenctopn 16435  0_gc0g 16453  Grpcgrp 17776  inv_gcminusg 17777  -_gcsg 17778  SubGrpcsubg 17939  NrmSGrpcnsg 17940   ~_QG cqg 17941  opp_gcoppg 18125  TopOnctopon 21085   Cn ccn 21399  Conncconn 21585  Homeochmeo 21927  TopGrpctgp 22245

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1894  ax-4 1908  ax-5 2009  ax-6 2075  ax-7 2112  ax-8 2166  ax-9 2173  ax-10 2192  ax-11 2207  ax-12 2220  ax-13 2389  ax-ext 2803  ax-rep 4994  ax-sep 5005  ax-nul 5013  ax-pow 5065  ax-pr 5127  ax-un 7209  ax-cnex 10308  ax-resscn 10309  ax-1cn 10310  ax-icn 10311  ax-addcl 10312  ax-addrcl 10313  ax-mulcl 10314  ax-mulrcl 10315  ax-mulcom 10316  ax-addass 10317  ax-mulass 10318  ax-distr 10319  ax-i2m1 10320  ax-1ne0 10321  ax-1rid 10322  ax-rnegex 10323  ax-rrecex 10324  ax-cnre 10325  ax-pre-lttri 10326  ax-pre-lttrn 10327  ax-pre-ltadd 10328  ax-pre-mulgt0 10329

This theorem depends on definitions:  df-bi 199  df-an 387  df-or 879  df-3or 1112  df-3an 1113  df-tru 1660  df-ex 1879  df-nf 1883  df-sb 2068  df-mo 2605  df-eu 2640  df-clab 2812  df-cleq 2818  df-clel 2821  df-nfc 2958  df-ne 3000  df-nel 3103  df-ral 3122  df-rex 3123  df-reu 3124  df-rmo 3125  df-rab 3126  df-v 3416  df-sbc 3663  df-csb 3758  df-dif 3801  df-un 3803  df-in 3805  df-ss 3812  df-pss 3814  df-nul 4145  df-if 4307  df-pw 4380  df-sn 4398  df-pr 4400  df-tp 4402  df-op 4404  df-uni 4659  df-int 4698  df-iun 4742  df-br 4874  df-opab 4936  df-mpt 4953  df-tr 4976  df-id 5250  df-eprel 5255  df-po 5263  df-so 5264  df-fr 5301  df-we 5303  df-xp 5348  df-rel 5349  df-cnv 5350  df-co 5351  df-dm 5352  df-rn 5353  df-res 5354  df-ima 5355  df-pred 5920  df-ord 5966  df-on 5967  df-lim 5968  df-suc 5969  df-iota 6086  df-fun 6125  df-fn 6126  df-f 6127  df-f1 6128  df-fo 6129  df-f1o 6130  df-fv 6131  df-riota 6866  df-ov 6908  df-oprab 6909  df-mpt2 6910  df-om 7327  df-1st 7428  df-2nd 7429  df-tpos 7617  df-wrecs 7672  df-recs 7734  df-rdg 7772  df-oadd 7830  df-er 8009  df-ec 8011  df-map 8124  df-en 8223  df-dom 8224  df-sdom 8225  df-fin 8226  df-fi 8586  df-pnf 10393  df-mnf 10394  df-xr 10395  df-ltxr 10396  df-le 10397  df-sub 10587  df-neg 10588  df-nn 11351  df-2 11414  df-3 11415  df-4 11416  df-5 11417  df-6 11418  df-7 11419  df-8 11420  df-9 11421  df-ndx 16225  df-slot 16226  df-base 16228  df-sets 16229  df-ress 16230  df-plusg 16318  df-tset 16324  df-rest 16436  df-topn 16437  df-0g 16455  df-topgen 16457  df-plusf 17594  df-mgm 17595  df-sgrp 17637  df-mnd 17648  df-grp 17779  df-minusg 17780  df-sbg 17781  df-subg 17942  df-nsg 17943  df-eqg 17944  df-oppg 18126  df-top 21069  df-topon 21086  df-topsp 21108  df-bases 21121  df-cld 21194  df-cn 21402  df-cnp 21403  df-conn 21586  df-tx 21736  df-hmeo 21929  df-tmd 22246  df-tgp 22247

This theorem is referenced by: (None)