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Theorem qustgpopn 23624

Description: A quotient map in a topological group is an open map. (Contributed by Mario Carneiro, 18-Sep-2015.)

**Hypotheses**
Ref	Expression
qustgp.h	⊢ 𝐻 = (𝐺 /_s (𝐺 ~_QG 𝑌))
qustgpopn.x	⊢ 𝑋 = (Base‘𝐺)
qustgpopn.j	⊢ 𝐽 = (TopOpen‘𝐺)
qustgpopn.k	⊢ 𝐾 = (TopOpen‘𝐻)
qustgpopn.f	⊢ 𝐹 = (𝑥 ∈ 𝑋 ↦ [𝑥](𝐺 ~_QG 𝑌))

**Assertion**
Ref	Expression
qustgpopn	⊢ ((𝐺 ∈ TopGrp ∧ 𝑌 ∈ (NrmSGrp‘𝐺) ∧ 𝑆 ∈ 𝐽) → (𝐹 “ 𝑆) ∈ 𝐾)

Distinct variable groups: 𝑥,𝐺 𝑥,𝐽 𝑥,𝑆 𝑥,𝑋 𝑥,𝐻 𝑥,𝐾 𝑥,𝑌 Allowed substitution hint: 𝐹(𝑥)

Proof of Theorem qustgpopn Dummy variables 𝑎 𝑢 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		imassrn 6071	. . . 4 ⊢ (𝐹 “ 𝑆) ⊆ ran 𝐹
2		qustgp.h	. . . . . . 7 ⊢ 𝐻 = (𝐺 /_s (𝐺 ~_QG 𝑌))
3	2	a1i 11	. . . . . 6 ⊢ ((𝐺 ∈ TopGrp ∧ 𝑌 ∈ (NrmSGrp‘𝐺) ∧ 𝑆 ∈ 𝐽) → 𝐻 = (𝐺 /_s (𝐺 ~_QG 𝑌)))
4		qustgpopn.x	. . . . . . 7 ⊢ 𝑋 = (Base‘𝐺)
5	4	a1i 11	. . . . . 6 ⊢ ((𝐺 ∈ TopGrp ∧ 𝑌 ∈ (NrmSGrp‘𝐺) ∧ 𝑆 ∈ 𝐽) → 𝑋 = (Base‘𝐺))
6		qustgpopn.f	. . . . . 6 ⊢ 𝐹 = (𝑥 ∈ 𝑋 ↦ [𝑥](𝐺 ~_QG 𝑌))
7		ovex 7442	. . . . . . 7 ⊢ (𝐺 ~_QG 𝑌) ∈ V
8	7	a1i 11	. . . . . 6 ⊢ ((𝐺 ∈ TopGrp ∧ 𝑌 ∈ (NrmSGrp‘𝐺) ∧ 𝑆 ∈ 𝐽) → (𝐺 ~_QG 𝑌) ∈ V)
9		simp1 1137	. . . . . 6 ⊢ ((𝐺 ∈ TopGrp ∧ 𝑌 ∈ (NrmSGrp‘𝐺) ∧ 𝑆 ∈ 𝐽) → 𝐺 ∈ TopGrp)
10	3, 5, 6, 8, 9	quslem 17489	. . . . 5 ⊢ ((𝐺 ∈ TopGrp ∧ 𝑌 ∈ (NrmSGrp‘𝐺) ∧ 𝑆 ∈ 𝐽) → 𝐹:𝑋–onto→(𝑋 / (𝐺 ~_QG 𝑌)))
11		forn 6809	. . . . 5 ⊢ (𝐹:𝑋–onto→(𝑋 / (𝐺 ~_QG 𝑌)) → ran 𝐹 = (𝑋 / (𝐺 ~_QG 𝑌)))
12	10, 11	syl 17	. . . 4 ⊢ ((𝐺 ∈ TopGrp ∧ 𝑌 ∈ (NrmSGrp‘𝐺) ∧ 𝑆 ∈ 𝐽) → ran 𝐹 = (𝑋 / (𝐺 ~_QG 𝑌)))
13	1, 12	sseqtrid 4035	. . 3 ⊢ ((𝐺 ∈ TopGrp ∧ 𝑌 ∈ (NrmSGrp‘𝐺) ∧ 𝑆 ∈ 𝐽) → (𝐹 “ 𝑆) ⊆ (𝑋 / (𝐺 ~_QG 𝑌)))
14		eceq1 8741	. . . . . . . . . 10 ⊢ (𝑥 = 𝑦 → [𝑥](𝐺 ~_QG 𝑌) = [𝑦](𝐺 ~_QG 𝑌))
15	14	cbvmptv 5262	. . . . . . . . 9 ⊢ (𝑥 ∈ 𝑋 ↦ [𝑥](𝐺 ~_QG 𝑌)) = (𝑦 ∈ 𝑋 ↦ [𝑦](𝐺 ~_QG 𝑌))
16	6, 15	eqtri 2761	. . . . . . . 8 ⊢ 𝐹 = (𝑦 ∈ 𝑋 ↦ [𝑦](𝐺 ~_QG 𝑌))
17	16	mptpreima 6238	. . . . . . 7 ⊢ (^◡𝐹 “ (𝐹 “ 𝑆)) = {𝑦 ∈ 𝑋 ∣ [𝑦](𝐺 ~_QG 𝑌) ∈ (𝐹 “ 𝑆)}
18	17	reqabi 3455	. . . . . 6 ⊢ (𝑦 ∈ (^◡𝐹 “ (𝐹 “ 𝑆)) ↔ (𝑦 ∈ 𝑋 ∧ [𝑦](𝐺 ~_QG 𝑌) ∈ (𝐹 “ 𝑆)))
19	6	funmpt2 6588	. . . . . . . . 9 ⊢ Fun 𝐹
20		fvelima 6958	. . . . . . . . 9 ⊢ ((Fun 𝐹 ∧ [𝑦](𝐺 ~_QG 𝑌) ∈ (𝐹 “ 𝑆)) → ∃𝑧 ∈ 𝑆 (𝐹‘𝑧) = [𝑦](𝐺 ~_QG 𝑌))
21	19, 20	mpan 689	. . . . . . . 8 ⊢ ([𝑦](𝐺 ~_QG 𝑌) ∈ (𝐹 “ 𝑆) → ∃𝑧 ∈ 𝑆 (𝐹‘𝑧) = [𝑦](𝐺 ~_QG 𝑌))
22		qustgpopn.j	. . . . . . . . . . . . . . . . . . 19 ⊢ 𝐽 = (TopOpen‘𝐺)
23	22, 4	tgptopon 23586	. . . . . . . . . . . . . . . . . 18 ⊢ (𝐺 ∈ TopGrp → 𝐽 ∈ (TopOn‘𝑋))
24	9, 23	syl 17	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐺 ∈ TopGrp ∧ 𝑌 ∈ (NrmSGrp‘𝐺) ∧ 𝑆 ∈ 𝐽) → 𝐽 ∈ (TopOn‘𝑋))
25		simp3 1139	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐺 ∈ TopGrp ∧ 𝑌 ∈ (NrmSGrp‘𝐺) ∧ 𝑆 ∈ 𝐽) → 𝑆 ∈ 𝐽)
26		toponss 22429	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ 𝐽) → 𝑆 ⊆ 𝑋)
27	24, 25, 26	syl2anc 585	. . . . . . . . . . . . . . . 16 ⊢ ((𝐺 ∈ TopGrp ∧ 𝑌 ∈ (NrmSGrp‘𝐺) ∧ 𝑆 ∈ 𝐽) → 𝑆 ⊆ 𝑋)
28	27	adantr 482	. . . . . . . . . . . . . . 15 ⊢ (((𝐺 ∈ TopGrp ∧ 𝑌 ∈ (NrmSGrp‘𝐺) ∧ 𝑆 ∈ 𝐽) ∧ 𝑦 ∈ 𝑋) → 𝑆 ⊆ 𝑋)
29	28	sselda 3983	. . . . . . . . . . . . . 14 ⊢ ((((𝐺 ∈ TopGrp ∧ 𝑌 ∈ (NrmSGrp‘𝐺) ∧ 𝑆 ∈ 𝐽) ∧ 𝑦 ∈ 𝑋) ∧ 𝑧 ∈ 𝑆) → 𝑧 ∈ 𝑋)
30		eceq1 8741	. . . . . . . . . . . . . . 15 ⊢ (𝑥 = 𝑧 → [𝑥](𝐺 ~_QG 𝑌) = [𝑧](𝐺 ~_QG 𝑌))
31		ecexg 8707	. . . . . . . . . . . . . . . 16 ⊢ ((𝐺 ~_QG 𝑌) ∈ V → [𝑧](𝐺 ~_QG 𝑌) ∈ V)
32	7, 31	ax-mp 5	. . . . . . . . . . . . . . 15 ⊢ [𝑧](𝐺 ~_QG 𝑌) ∈ V
33	30, 6, 32	fvmpt 6999	. . . . . . . . . . . . . 14 ⊢ (𝑧 ∈ 𝑋 → (𝐹‘𝑧) = [𝑧](𝐺 ~_QG 𝑌))
34	29, 33	syl 17	. . . . . . . . . . . . 13 ⊢ ((((𝐺 ∈ TopGrp ∧ 𝑌 ∈ (NrmSGrp‘𝐺) ∧ 𝑆 ∈ 𝐽) ∧ 𝑦 ∈ 𝑋) ∧ 𝑧 ∈ 𝑆) → (𝐹‘𝑧) = [𝑧](𝐺 ~_QG 𝑌))
35	34	eqeq1d 2735	. . . . . . . . . . . 12 ⊢ ((((𝐺 ∈ TopGrp ∧ 𝑌 ∈ (NrmSGrp‘𝐺) ∧ 𝑆 ∈ 𝐽) ∧ 𝑦 ∈ 𝑋) ∧ 𝑧 ∈ 𝑆) → ((𝐹‘𝑧) = [𝑦](𝐺 ~_QG 𝑌) ↔ [𝑧](𝐺 ~_QG 𝑌) = [𝑦](𝐺 ~_QG 𝑌)))
36		eqcom 2740	. . . . . . . . . . . 12 ⊢ ([𝑧](𝐺 ~_QG 𝑌) = [𝑦](𝐺 ~_QG 𝑌) ↔ [𝑦](𝐺 ~_QG 𝑌) = [𝑧](𝐺 ~_QG 𝑌))
37	35, 36	bitrdi 287	. . . . . . . . . . 11 ⊢ ((((𝐺 ∈ TopGrp ∧ 𝑌 ∈ (NrmSGrp‘𝐺) ∧ 𝑆 ∈ 𝐽) ∧ 𝑦 ∈ 𝑋) ∧ 𝑧 ∈ 𝑆) → ((𝐹‘𝑧) = [𝑦](𝐺 ~_QG 𝑌) ↔ [𝑦](𝐺 ~_QG 𝑌) = [𝑧](𝐺 ~_QG 𝑌)))
38		nsgsubg 19038	. . . . . . . . . . . . . . 15 ⊢ (𝑌 ∈ (NrmSGrp‘𝐺) → 𝑌 ∈ (SubGrp‘𝐺))
39	38	3ad2ant2 1135	. . . . . . . . . . . . . 14 ⊢ ((𝐺 ∈ TopGrp ∧ 𝑌 ∈ (NrmSGrp‘𝐺) ∧ 𝑆 ∈ 𝐽) → 𝑌 ∈ (SubGrp‘𝐺))
40	39	ad2antrr 725	. . . . . . . . . . . . 13 ⊢ ((((𝐺 ∈ TopGrp ∧ 𝑌 ∈ (NrmSGrp‘𝐺) ∧ 𝑆 ∈ 𝐽) ∧ 𝑦 ∈ 𝑋) ∧ 𝑧 ∈ 𝑆) → 𝑌 ∈ (SubGrp‘𝐺))
41		eqid 2733	. . . . . . . . . . . . . 14 ⊢ (𝐺 ~_QG 𝑌) = (𝐺 ~_QG 𝑌)
42	4, 41	eqger 19058	. . . . . . . . . . . . 13 ⊢ (𝑌 ∈ (SubGrp‘𝐺) → (𝐺 ~_QG 𝑌) Er 𝑋)
43	40, 42	syl 17	. . . . . . . . . . . 12 ⊢ ((((𝐺 ∈ TopGrp ∧ 𝑌 ∈ (NrmSGrp‘𝐺) ∧ 𝑆 ∈ 𝐽) ∧ 𝑦 ∈ 𝑋) ∧ 𝑧 ∈ 𝑆) → (𝐺 ~_QG 𝑌) Er 𝑋)
44		simplr 768	. . . . . . . . . . . 12 ⊢ ((((𝐺 ∈ TopGrp ∧ 𝑌 ∈ (NrmSGrp‘𝐺) ∧ 𝑆 ∈ 𝐽) ∧ 𝑦 ∈ 𝑋) ∧ 𝑧 ∈ 𝑆) → 𝑦 ∈ 𝑋)
45	43, 44	erth 8752	. . . . . . . . . . 11 ⊢ ((((𝐺 ∈ TopGrp ∧ 𝑌 ∈ (NrmSGrp‘𝐺) ∧ 𝑆 ∈ 𝐽) ∧ 𝑦 ∈ 𝑋) ∧ 𝑧 ∈ 𝑆) → (𝑦(𝐺 ~_QG 𝑌)𝑧 ↔ [𝑦](𝐺 ~_QG 𝑌) = [𝑧](𝐺 ~_QG 𝑌)))
46	9	ad2antrr 725	. . . . . . . . . . . 12 ⊢ ((((𝐺 ∈ TopGrp ∧ 𝑌 ∈ (NrmSGrp‘𝐺) ∧ 𝑆 ∈ 𝐽) ∧ 𝑦 ∈ 𝑋) ∧ 𝑧 ∈ 𝑆) → 𝐺 ∈ TopGrp)
47	4	subgss 19007	. . . . . . . . . . . . 13 ⊢ (𝑌 ∈ (SubGrp‘𝐺) → 𝑌 ⊆ 𝑋)
48	40, 47	syl 17	. . . . . . . . . . . 12 ⊢ ((((𝐺 ∈ TopGrp ∧ 𝑌 ∈ (NrmSGrp‘𝐺) ∧ 𝑆 ∈ 𝐽) ∧ 𝑦 ∈ 𝑋) ∧ 𝑧 ∈ 𝑆) → 𝑌 ⊆ 𝑋)
49		eqid 2733	. . . . . . . . . . . . 13 ⊢ (inv_g‘𝐺) = (inv_g‘𝐺)
50		eqid 2733	. . . . . . . . . . . . 13 ⊢ (+_g‘𝐺) = (+_g‘𝐺)
51	4, 49, 50, 41	eqgval 19057	. . . . . . . . . . . 12 ⊢ ((𝐺 ∈ TopGrp ∧ 𝑌 ⊆ 𝑋) → (𝑦(𝐺 ~_QG 𝑌)𝑧 ↔ (𝑦 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋 ∧ (((inv_g‘𝐺)‘𝑦)(+_g‘𝐺)𝑧) ∈ 𝑌)))
52	46, 48, 51	syl2anc 585	. . . . . . . . . . 11 ⊢ ((((𝐺 ∈ TopGrp ∧ 𝑌 ∈ (NrmSGrp‘𝐺) ∧ 𝑆 ∈ 𝐽) ∧ 𝑦 ∈ 𝑋) ∧ 𝑧 ∈ 𝑆) → (𝑦(𝐺 ~_QG 𝑌)𝑧 ↔ (𝑦 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋 ∧ (((inv_g‘𝐺)‘𝑦)(+_g‘𝐺)𝑧) ∈ 𝑌)))
53	37, 45, 52	3bitr2d 307	. . . . . . . . . 10 ⊢ ((((𝐺 ∈ TopGrp ∧ 𝑌 ∈ (NrmSGrp‘𝐺) ∧ 𝑆 ∈ 𝐽) ∧ 𝑦 ∈ 𝑋) ∧ 𝑧 ∈ 𝑆) → ((𝐹‘𝑧) = [𝑦](𝐺 ~_QG 𝑌) ↔ (𝑦 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋 ∧ (((inv_g‘𝐺)‘𝑦)(+_g‘𝐺)𝑧) ∈ 𝑌)))
54		eqid 2733	. . . . . . . . . . . . . . . . . 18 ⊢ (opp_g‘𝐺) = (opp_g‘𝐺)
55		eqid 2733	. . . . . . . . . . . . . . . . . 18 ⊢ (+_g‘(opp_g‘𝐺)) = (+_g‘(opp_g‘𝐺))
56	50, 54, 55	oppgplus 19213	. . . . . . . . . . . . . . . . 17 ⊢ ((((inv_g‘𝐺)‘𝑦)(+_g‘𝐺)𝑧)(+_g‘(opp_g‘𝐺))𝑎) = (𝑎(+_g‘𝐺)(((inv_g‘𝐺)‘𝑦)(+_g‘𝐺)𝑧))
57	56	mpteq2i 5254	. . . . . . . . . . . . . . . 16 ⊢ (𝑎 ∈ 𝑋 ↦ ((((inv_g‘𝐺)‘𝑦)(+_g‘𝐺)𝑧)(+_g‘(opp_g‘𝐺))𝑎)) = (𝑎 ∈ 𝑋 ↦ (𝑎(+_g‘𝐺)(((inv_g‘𝐺)‘𝑦)(+_g‘𝐺)𝑧)))
58	46	adantr 482	. . . . . . . . . . . . . . . . . 18 ⊢ (((((𝐺 ∈ TopGrp ∧ 𝑌 ∈ (NrmSGrp‘𝐺) ∧ 𝑆 ∈ 𝐽) ∧ 𝑦 ∈ 𝑋) ∧ 𝑧 ∈ 𝑆) ∧ (((inv_g‘𝐺)‘𝑦)(+_g‘𝐺)𝑧) ∈ 𝑌) → 𝐺 ∈ TopGrp)
59	54	oppgtgp 23602	. . . . . . . . . . . . . . . . . 18 ⊢ (𝐺 ∈ TopGrp → (opp_g‘𝐺) ∈ TopGrp)
60	58, 59	syl 17	. . . . . . . . . . . . . . . . 17 ⊢ (((((𝐺 ∈ TopGrp ∧ 𝑌 ∈ (NrmSGrp‘𝐺) ∧ 𝑆 ∈ 𝐽) ∧ 𝑦 ∈ 𝑋) ∧ 𝑧 ∈ 𝑆) ∧ (((inv_g‘𝐺)‘𝑦)(+_g‘𝐺)𝑧) ∈ 𝑌) → (opp_g‘𝐺) ∈ TopGrp)
61	48	sselda 3983	. . . . . . . . . . . . . . . . 17 ⊢ (((((𝐺 ∈ TopGrp ∧ 𝑌 ∈ (NrmSGrp‘𝐺) ∧ 𝑆 ∈ 𝐽) ∧ 𝑦 ∈ 𝑋) ∧ 𝑧 ∈ 𝑆) ∧ (((inv_g‘𝐺)‘𝑦)(+_g‘𝐺)𝑧) ∈ 𝑌) → (((inv_g‘𝐺)‘𝑦)(+_g‘𝐺)𝑧) ∈ 𝑋)
62		eqid 2733	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑎 ∈ 𝑋 ↦ ((((inv_g‘𝐺)‘𝑦)(+_g‘𝐺)𝑧)(+_g‘(opp_g‘𝐺))𝑎)) = (𝑎 ∈ 𝑋 ↦ ((((inv_g‘𝐺)‘𝑦)(+_g‘𝐺)𝑧)(+_g‘(opp_g‘𝐺))𝑎))
63	54, 4	oppgbas 19216	. . . . . . . . . . . . . . . . . 18 ⊢ 𝑋 = (Base‘(opp_g‘𝐺))
64	54, 22	oppgtopn 19220	. . . . . . . . . . . . . . . . . 18 ⊢ 𝐽 = (TopOpen‘(opp_g‘𝐺))
65	62, 63, 55, 64	tgplacthmeo 23607	. . . . . . . . . . . . . . . . 17 ⊢ (((opp_g‘𝐺) ∈ TopGrp ∧ (((inv_g‘𝐺)‘𝑦)(+_g‘𝐺)𝑧) ∈ 𝑋) → (𝑎 ∈ 𝑋 ↦ ((((inv_g‘𝐺)‘𝑦)(+_g‘𝐺)𝑧)(+_g‘(opp_g‘𝐺))𝑎)) ∈ (𝐽Homeo𝐽))
66	60, 61, 65	syl2anc 585	. . . . . . . . . . . . . . . 16 ⊢ (((((𝐺 ∈ TopGrp ∧ 𝑌 ∈ (NrmSGrp‘𝐺) ∧ 𝑆 ∈ 𝐽) ∧ 𝑦 ∈ 𝑋) ∧ 𝑧 ∈ 𝑆) ∧ (((inv_g‘𝐺)‘𝑦)(+_g‘𝐺)𝑧) ∈ 𝑌) → (𝑎 ∈ 𝑋 ↦ ((((inv_g‘𝐺)‘𝑦)(+_g‘𝐺)𝑧)(+_g‘(opp_g‘𝐺))𝑎)) ∈ (𝐽Homeo𝐽))
67	57, 66	eqeltrrid 2839	. . . . . . . . . . . . . . 15 ⊢ (((((𝐺 ∈ TopGrp ∧ 𝑌 ∈ (NrmSGrp‘𝐺) ∧ 𝑆 ∈ 𝐽) ∧ 𝑦 ∈ 𝑋) ∧ 𝑧 ∈ 𝑆) ∧ (((inv_g‘𝐺)‘𝑦)(+_g‘𝐺)𝑧) ∈ 𝑌) → (𝑎 ∈ 𝑋 ↦ (𝑎(+_g‘𝐺)(((inv_g‘𝐺)‘𝑦)(+_g‘𝐺)𝑧))) ∈ (𝐽Homeo𝐽))
68		hmeocn 23264	. . . . . . . . . . . . . . 15 ⊢ ((𝑎 ∈ 𝑋 ↦ (𝑎(+_g‘𝐺)(((inv_g‘𝐺)‘𝑦)(+_g‘𝐺)𝑧))) ∈ (𝐽Homeo𝐽) → (𝑎 ∈ 𝑋 ↦ (𝑎(+_g‘𝐺)(((inv_g‘𝐺)‘𝑦)(+_g‘𝐺)𝑧))) ∈ (𝐽 Cn 𝐽))
69	67, 68	syl 17	. . . . . . . . . . . . . 14 ⊢ (((((𝐺 ∈ TopGrp ∧ 𝑌 ∈ (NrmSGrp‘𝐺) ∧ 𝑆 ∈ 𝐽) ∧ 𝑦 ∈ 𝑋) ∧ 𝑧 ∈ 𝑆) ∧ (((inv_g‘𝐺)‘𝑦)(+_g‘𝐺)𝑧) ∈ 𝑌) → (𝑎 ∈ 𝑋 ↦ (𝑎(+_g‘𝐺)(((inv_g‘𝐺)‘𝑦)(+_g‘𝐺)𝑧))) ∈ (𝐽 Cn 𝐽))
70	25	ad3antrrr 729	. . . . . . . . . . . . . 14 ⊢ (((((𝐺 ∈ TopGrp ∧ 𝑌 ∈ (NrmSGrp‘𝐺) ∧ 𝑆 ∈ 𝐽) ∧ 𝑦 ∈ 𝑋) ∧ 𝑧 ∈ 𝑆) ∧ (((inv_g‘𝐺)‘𝑦)(+_g‘𝐺)𝑧) ∈ 𝑌) → 𝑆 ∈ 𝐽)
71		cnima 22769	. . . . . . . . . . . . . 14 ⊢ (((𝑎 ∈ 𝑋 ↦ (𝑎(+_g‘𝐺)(((inv_g‘𝐺)‘𝑦)(+_g‘𝐺)𝑧))) ∈ (𝐽 Cn 𝐽) ∧ 𝑆 ∈ 𝐽) → (^◡(𝑎 ∈ 𝑋 ↦ (𝑎(+_g‘𝐺)(((inv_g‘𝐺)‘𝑦)(+_g‘𝐺)𝑧))) “ 𝑆) ∈ 𝐽)
72	69, 70, 71	syl2anc 585	. . . . . . . . . . . . 13 ⊢ (((((𝐺 ∈ TopGrp ∧ 𝑌 ∈ (NrmSGrp‘𝐺) ∧ 𝑆 ∈ 𝐽) ∧ 𝑦 ∈ 𝑋) ∧ 𝑧 ∈ 𝑆) ∧ (((inv_g‘𝐺)‘𝑦)(+_g‘𝐺)𝑧) ∈ 𝑌) → (^◡(𝑎 ∈ 𝑋 ↦ (𝑎(+_g‘𝐺)(((inv_g‘𝐺)‘𝑦)(+_g‘𝐺)𝑧))) “ 𝑆) ∈ 𝐽)
73	44	adantr 482	. . . . . . . . . . . . . 14 ⊢ (((((𝐺 ∈ TopGrp ∧ 𝑌 ∈ (NrmSGrp‘𝐺) ∧ 𝑆 ∈ 𝐽) ∧ 𝑦 ∈ 𝑋) ∧ 𝑧 ∈ 𝑆) ∧ (((inv_g‘𝐺)‘𝑦)(+_g‘𝐺)𝑧) ∈ 𝑌) → 𝑦 ∈ 𝑋)
74		tgpgrp 23582	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝐺 ∈ TopGrp → 𝐺 ∈ Grp)
75	58, 74	syl 17	. . . . . . . . . . . . . . . . . 18 ⊢ (((((𝐺 ∈ TopGrp ∧ 𝑌 ∈ (NrmSGrp‘𝐺) ∧ 𝑆 ∈ 𝐽) ∧ 𝑦 ∈ 𝑋) ∧ 𝑧 ∈ 𝑆) ∧ (((inv_g‘𝐺)‘𝑦)(+_g‘𝐺)𝑧) ∈ 𝑌) → 𝐺 ∈ Grp)
76		eqid 2733	. . . . . . . . . . . . . . . . . . 19 ⊢ (0_g‘𝐺) = (0_g‘𝐺)
77	4, 50, 76, 49	grprinv 18875	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐺 ∈ Grp ∧ 𝑦 ∈ 𝑋) → (𝑦(+_g‘𝐺)((inv_g‘𝐺)‘𝑦)) = (0_g‘𝐺))
78	75, 73, 77	syl2anc 585	. . . . . . . . . . . . . . . . 17 ⊢ (((((𝐺 ∈ TopGrp ∧ 𝑌 ∈ (NrmSGrp‘𝐺) ∧ 𝑆 ∈ 𝐽) ∧ 𝑦 ∈ 𝑋) ∧ 𝑧 ∈ 𝑆) ∧ (((inv_g‘𝐺)‘𝑦)(+_g‘𝐺)𝑧) ∈ 𝑌) → (𝑦(+_g‘𝐺)((inv_g‘𝐺)‘𝑦)) = (0_g‘𝐺))
79	78	oveq1d 7424	. . . . . . . . . . . . . . . 16 ⊢ (((((𝐺 ∈ TopGrp ∧ 𝑌 ∈ (NrmSGrp‘𝐺) ∧ 𝑆 ∈ 𝐽) ∧ 𝑦 ∈ 𝑋) ∧ 𝑧 ∈ 𝑆) ∧ (((inv_g‘𝐺)‘𝑦)(+_g‘𝐺)𝑧) ∈ 𝑌) → ((𝑦(+_g‘𝐺)((inv_g‘𝐺)‘𝑦))(+_g‘𝐺)𝑧) = ((0_g‘𝐺)(+_g‘𝐺)𝑧))
80	4, 49	grpinvcl 18872	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐺 ∈ Grp ∧ 𝑦 ∈ 𝑋) → ((inv_g‘𝐺)‘𝑦) ∈ 𝑋)
81	75, 73, 80	syl2anc 585	. . . . . . . . . . . . . . . . 17 ⊢ (((((𝐺 ∈ TopGrp ∧ 𝑌 ∈ (NrmSGrp‘𝐺) ∧ 𝑆 ∈ 𝐽) ∧ 𝑦 ∈ 𝑋) ∧ 𝑧 ∈ 𝑆) ∧ (((inv_g‘𝐺)‘𝑦)(+_g‘𝐺)𝑧) ∈ 𝑌) → ((inv_g‘𝐺)‘𝑦) ∈ 𝑋)
82	29	adantr 482	. . . . . . . . . . . . . . . . 17 ⊢ (((((𝐺 ∈ TopGrp ∧ 𝑌 ∈ (NrmSGrp‘𝐺) ∧ 𝑆 ∈ 𝐽) ∧ 𝑦 ∈ 𝑋) ∧ 𝑧 ∈ 𝑆) ∧ (((inv_g‘𝐺)‘𝑦)(+_g‘𝐺)𝑧) ∈ 𝑌) → 𝑧 ∈ 𝑋)
83	4, 50	grpass 18828	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐺 ∈ Grp ∧ (𝑦 ∈ 𝑋 ∧ ((inv_g‘𝐺)‘𝑦) ∈ 𝑋 ∧ 𝑧 ∈ 𝑋)) → ((𝑦(+_g‘𝐺)((inv_g‘𝐺)‘𝑦))(+_g‘𝐺)𝑧) = (𝑦(+_g‘𝐺)(((inv_g‘𝐺)‘𝑦)(+_g‘𝐺)𝑧)))
84	75, 73, 81, 82, 83	syl13anc 1373	. . . . . . . . . . . . . . . 16 ⊢ (((((𝐺 ∈ TopGrp ∧ 𝑌 ∈ (NrmSGrp‘𝐺) ∧ 𝑆 ∈ 𝐽) ∧ 𝑦 ∈ 𝑋) ∧ 𝑧 ∈ 𝑆) ∧ (((inv_g‘𝐺)‘𝑦)(+_g‘𝐺)𝑧) ∈ 𝑌) → ((𝑦(+_g‘𝐺)((inv_g‘𝐺)‘𝑦))(+_g‘𝐺)𝑧) = (𝑦(+_g‘𝐺)(((inv_g‘𝐺)‘𝑦)(+_g‘𝐺)𝑧)))
85	4, 50, 76	grplid 18852	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐺 ∈ Grp ∧ 𝑧 ∈ 𝑋) → ((0_g‘𝐺)(+_g‘𝐺)𝑧) = 𝑧)
86	75, 82, 85	syl2anc 585	. . . . . . . . . . . . . . . 16 ⊢ (((((𝐺 ∈ TopGrp ∧ 𝑌 ∈ (NrmSGrp‘𝐺) ∧ 𝑆 ∈ 𝐽) ∧ 𝑦 ∈ 𝑋) ∧ 𝑧 ∈ 𝑆) ∧ (((inv_g‘𝐺)‘𝑦)(+_g‘𝐺)𝑧) ∈ 𝑌) → ((0_g‘𝐺)(+_g‘𝐺)𝑧) = 𝑧)
87	79, 84, 86	3eqtr3d 2781	. . . . . . . . . . . . . . 15 ⊢ (((((𝐺 ∈ TopGrp ∧ 𝑌 ∈ (NrmSGrp‘𝐺) ∧ 𝑆 ∈ 𝐽) ∧ 𝑦 ∈ 𝑋) ∧ 𝑧 ∈ 𝑆) ∧ (((inv_g‘𝐺)‘𝑦)(+_g‘𝐺)𝑧) ∈ 𝑌) → (𝑦(+_g‘𝐺)(((inv_g‘𝐺)‘𝑦)(+_g‘𝐺)𝑧)) = 𝑧)
88		simplr 768	. . . . . . . . . . . . . . 15 ⊢ (((((𝐺 ∈ TopGrp ∧ 𝑌 ∈ (NrmSGrp‘𝐺) ∧ 𝑆 ∈ 𝐽) ∧ 𝑦 ∈ 𝑋) ∧ 𝑧 ∈ 𝑆) ∧ (((inv_g‘𝐺)‘𝑦)(+_g‘𝐺)𝑧) ∈ 𝑌) → 𝑧 ∈ 𝑆)
89	87, 88	eqeltrd 2834	. . . . . . . . . . . . . 14 ⊢ (((((𝐺 ∈ TopGrp ∧ 𝑌 ∈ (NrmSGrp‘𝐺) ∧ 𝑆 ∈ 𝐽) ∧ 𝑦 ∈ 𝑋) ∧ 𝑧 ∈ 𝑆) ∧ (((inv_g‘𝐺)‘𝑦)(+_g‘𝐺)𝑧) ∈ 𝑌) → (𝑦(+_g‘𝐺)(((inv_g‘𝐺)‘𝑦)(+_g‘𝐺)𝑧)) ∈ 𝑆)
90		oveq1 7416	. . . . . . . . . . . . . . . 16 ⊢ (𝑎 = 𝑦 → (𝑎(+_g‘𝐺)(((inv_g‘𝐺)‘𝑦)(+_g‘𝐺)𝑧)) = (𝑦(+_g‘𝐺)(((inv_g‘𝐺)‘𝑦)(+_g‘𝐺)𝑧)))
91	90	eleq1d 2819	. . . . . . . . . . . . . . 15 ⊢ (𝑎 = 𝑦 → ((𝑎(+_g‘𝐺)(((inv_g‘𝐺)‘𝑦)(+_g‘𝐺)𝑧)) ∈ 𝑆 ↔ (𝑦(+_g‘𝐺)(((inv_g‘𝐺)‘𝑦)(+_g‘𝐺)𝑧)) ∈ 𝑆))
92		eqid 2733	. . . . . . . . . . . . . . . 16 ⊢ (𝑎 ∈ 𝑋 ↦ (𝑎(+_g‘𝐺)(((inv_g‘𝐺)‘𝑦)(+_g‘𝐺)𝑧))) = (𝑎 ∈ 𝑋 ↦ (𝑎(+_g‘𝐺)(((inv_g‘𝐺)‘𝑦)(+_g‘𝐺)𝑧)))
93	92	mptpreima 6238	. . . . . . . . . . . . . . 15 ⊢ (^◡(𝑎 ∈ 𝑋 ↦ (𝑎(+_g‘𝐺)(((inv_g‘𝐺)‘𝑦)(+_g‘𝐺)𝑧))) “ 𝑆) = {𝑎 ∈ 𝑋 ∣ (𝑎(+_g‘𝐺)(((inv_g‘𝐺)‘𝑦)(+_g‘𝐺)𝑧)) ∈ 𝑆}
94	91, 93	elrab2 3687	. . . . . . . . . . . . . 14 ⊢ (𝑦 ∈ (^◡(𝑎 ∈ 𝑋 ↦ (𝑎(+_g‘𝐺)(((inv_g‘𝐺)‘𝑦)(+_g‘𝐺)𝑧))) “ 𝑆) ↔ (𝑦 ∈ 𝑋 ∧ (𝑦(+_g‘𝐺)(((inv_g‘𝐺)‘𝑦)(+_g‘𝐺)𝑧)) ∈ 𝑆))
95	73, 89, 94	sylanbrc 584	. . . . . . . . . . . . 13 ⊢ (((((𝐺 ∈ TopGrp ∧ 𝑌 ∈ (NrmSGrp‘𝐺) ∧ 𝑆 ∈ 𝐽) ∧ 𝑦 ∈ 𝑋) ∧ 𝑧 ∈ 𝑆) ∧ (((inv_g‘𝐺)‘𝑦)(+_g‘𝐺)𝑧) ∈ 𝑌) → 𝑦 ∈ (^◡(𝑎 ∈ 𝑋 ↦ (𝑎(+_g‘𝐺)(((inv_g‘𝐺)‘𝑦)(+_g‘𝐺)𝑧))) “ 𝑆))
96		ecexg 8707	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝐺 ~_QG 𝑌) ∈ V → [𝑥](𝐺 ~_QG 𝑌) ∈ V)
97	7, 96	ax-mp 5	. . . . . . . . . . . . . . . . . 18 ⊢ [𝑥](𝐺 ~_QG 𝑌) ∈ V
98	97, 6	fnmpti 6694	. . . . . . . . . . . . . . . . 17 ⊢ 𝐹 Fn 𝑋
99	28	ad3antrrr 729	. . . . . . . . . . . . . . . . 17 ⊢ ((((((𝐺 ∈ TopGrp ∧ 𝑌 ∈ (NrmSGrp‘𝐺) ∧ 𝑆 ∈ 𝐽) ∧ 𝑦 ∈ 𝑋) ∧ 𝑧 ∈ 𝑆) ∧ (((inv_g‘𝐺)‘𝑦)(+_g‘𝐺)𝑧) ∈ 𝑌) ∧ 𝑎 ∈ 𝑋) → 𝑆 ⊆ 𝑋)
100		fnfvima 7235	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐹 Fn 𝑋 ∧ 𝑆 ⊆ 𝑋 ∧ (𝑎(+_g‘𝐺)(((inv_g‘𝐺)‘𝑦)(+_g‘𝐺)𝑧)) ∈ 𝑆) → (𝐹‘(𝑎(+_g‘𝐺)(((inv_g‘𝐺)‘𝑦)(+_g‘𝐺)𝑧))) ∈ (𝐹 “ 𝑆))
101	100	3expia 1122	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐹 Fn 𝑋 ∧ 𝑆 ⊆ 𝑋) → ((𝑎(+_g‘𝐺)(((inv_g‘𝐺)‘𝑦)(+_g‘𝐺)𝑧)) ∈ 𝑆 → (𝐹‘(𝑎(+_g‘𝐺)(((inv_g‘𝐺)‘𝑦)(+_g‘𝐺)𝑧))) ∈ (𝐹 “ 𝑆)))
102	98, 99, 101	sylancr 588	. . . . . . . . . . . . . . . 16 ⊢ ((((((𝐺 ∈ TopGrp ∧ 𝑌 ∈ (NrmSGrp‘𝐺) ∧ 𝑆 ∈ 𝐽) ∧ 𝑦 ∈ 𝑋) ∧ 𝑧 ∈ 𝑆) ∧ (((inv_g‘𝐺)‘𝑦)(+_g‘𝐺)𝑧) ∈ 𝑌) ∧ 𝑎 ∈ 𝑋) → ((𝑎(+_g‘𝐺)(((inv_g‘𝐺)‘𝑦)(+_g‘𝐺)𝑧)) ∈ 𝑆 → (𝐹‘(𝑎(+_g‘𝐺)(((inv_g‘𝐺)‘𝑦)(+_g‘𝐺)𝑧))) ∈ (𝐹 “ 𝑆)))
103	75	adantr 482	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((((𝐺 ∈ TopGrp ∧ 𝑌 ∈ (NrmSGrp‘𝐺) ∧ 𝑆 ∈ 𝐽) ∧ 𝑦 ∈ 𝑋) ∧ 𝑧 ∈ 𝑆) ∧ (((inv_g‘𝐺)‘𝑦)(+_g‘𝐺)𝑧) ∈ 𝑌) ∧ 𝑎 ∈ 𝑋) → 𝐺 ∈ Grp)
104		simpr 486	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((((𝐺 ∈ TopGrp ∧ 𝑌 ∈ (NrmSGrp‘𝐺) ∧ 𝑆 ∈ 𝐽) ∧ 𝑦 ∈ 𝑋) ∧ 𝑧 ∈ 𝑆) ∧ (((inv_g‘𝐺)‘𝑦)(+_g‘𝐺)𝑧) ∈ 𝑌) ∧ 𝑎 ∈ 𝑋) → 𝑎 ∈ 𝑋)
105	61	adantr 482	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((((𝐺 ∈ TopGrp ∧ 𝑌 ∈ (NrmSGrp‘𝐺) ∧ 𝑆 ∈ 𝐽) ∧ 𝑦 ∈ 𝑋) ∧ 𝑧 ∈ 𝑆) ∧ (((inv_g‘𝐺)‘𝑦)(+_g‘𝐺)𝑧) ∈ 𝑌) ∧ 𝑎 ∈ 𝑋) → (((inv_g‘𝐺)‘𝑦)(+_g‘𝐺)𝑧) ∈ 𝑋)
106	4, 50	grpcl 18827	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝐺 ∈ Grp ∧ 𝑎 ∈ 𝑋 ∧ (((inv_g‘𝐺)‘𝑦)(+_g‘𝐺)𝑧) ∈ 𝑋) → (𝑎(+_g‘𝐺)(((inv_g‘𝐺)‘𝑦)(+_g‘𝐺)𝑧)) ∈ 𝑋)
107	103, 104, 105, 106	syl3anc 1372	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((((𝐺 ∈ TopGrp ∧ 𝑌 ∈ (NrmSGrp‘𝐺) ∧ 𝑆 ∈ 𝐽) ∧ 𝑦 ∈ 𝑋) ∧ 𝑧 ∈ 𝑆) ∧ (((inv_g‘𝐺)‘𝑦)(+_g‘𝐺)𝑧) ∈ 𝑌) ∧ 𝑎 ∈ 𝑋) → (𝑎(+_g‘𝐺)(((inv_g‘𝐺)‘𝑦)(+_g‘𝐺)𝑧)) ∈ 𝑋)
108		eceq1 8741	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑥 = (𝑎(+_g‘𝐺)(((inv_g‘𝐺)‘𝑦)(+_g‘𝐺)𝑧)) → [𝑥](𝐺 ~_QG 𝑌) = [(𝑎(+_g‘𝐺)(((inv_g‘𝐺)‘𝑦)(+_g‘𝐺)𝑧))](𝐺 ~_QG 𝑌))
109	108, 6, 97	fvmpt3i 7004	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑎(+_g‘𝐺)(((inv_g‘𝐺)‘𝑦)(+_g‘𝐺)𝑧)) ∈ 𝑋 → (𝐹‘(𝑎(+_g‘𝐺)(((inv_g‘𝐺)‘𝑦)(+_g‘𝐺)𝑧))) = [(𝑎(+_g‘𝐺)(((inv_g‘𝐺)‘𝑦)(+_g‘𝐺)𝑧))](𝐺 ~_QG 𝑌))
110	107, 109	syl 17	. . . . . . . . . . . . . . . . . 18 ⊢ ((((((𝐺 ∈ TopGrp ∧ 𝑌 ∈ (NrmSGrp‘𝐺) ∧ 𝑆 ∈ 𝐽) ∧ 𝑦 ∈ 𝑋) ∧ 𝑧 ∈ 𝑆) ∧ (((inv_g‘𝐺)‘𝑦)(+_g‘𝐺)𝑧) ∈ 𝑌) ∧ 𝑎 ∈ 𝑋) → (𝐹‘(𝑎(+_g‘𝐺)(((inv_g‘𝐺)‘𝑦)(+_g‘𝐺)𝑧))) = [(𝑎(+_g‘𝐺)(((inv_g‘𝐺)‘𝑦)(+_g‘𝐺)𝑧))](𝐺 ~_QG 𝑌))
111	43	ad2antrr 725	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((((𝐺 ∈ TopGrp ∧ 𝑌 ∈ (NrmSGrp‘𝐺) ∧ 𝑆 ∈ 𝐽) ∧ 𝑦 ∈ 𝑋) ∧ 𝑧 ∈ 𝑆) ∧ (((inv_g‘𝐺)‘𝑦)(+_g‘𝐺)𝑧) ∈ 𝑌) ∧ 𝑎 ∈ 𝑋) → (𝐺 ~_QG 𝑌) Er 𝑋)
112	4, 50, 76, 49	grplinv 18874	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝐺 ∈ Grp ∧ 𝑎 ∈ 𝑋) → (((inv_g‘𝐺)‘𝑎)(+_g‘𝐺)𝑎) = (0_g‘𝐺))
113	103, 104, 112	syl2anc 585	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((((((𝐺 ∈ TopGrp ∧ 𝑌 ∈ (NrmSGrp‘𝐺) ∧ 𝑆 ∈ 𝐽) ∧ 𝑦 ∈ 𝑋) ∧ 𝑧 ∈ 𝑆) ∧ (((inv_g‘𝐺)‘𝑦)(+_g‘𝐺)𝑧) ∈ 𝑌) ∧ 𝑎 ∈ 𝑋) → (((inv_g‘𝐺)‘𝑎)(+_g‘𝐺)𝑎) = (0_g‘𝐺))
114	113	oveq1d 7424	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((((((𝐺 ∈ TopGrp ∧ 𝑌 ∈ (NrmSGrp‘𝐺) ∧ 𝑆 ∈ 𝐽) ∧ 𝑦 ∈ 𝑋) ∧ 𝑧 ∈ 𝑆) ∧ (((inv_g‘𝐺)‘𝑦)(+_g‘𝐺)𝑧) ∈ 𝑌) ∧ 𝑎 ∈ 𝑋) → ((((inv_g‘𝐺)‘𝑎)(+_g‘𝐺)𝑎)(+_g‘𝐺)(((inv_g‘𝐺)‘𝑦)(+_g‘𝐺)𝑧)) = ((0_g‘𝐺)(+_g‘𝐺)(((inv_g‘𝐺)‘𝑦)(+_g‘𝐺)𝑧)))
115	4, 49	grpinvcl 18872	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝐺 ∈ Grp ∧ 𝑎 ∈ 𝑋) → ((inv_g‘𝐺)‘𝑎) ∈ 𝑋)
116	103, 104, 115	syl2anc 585	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((((((𝐺 ∈ TopGrp ∧ 𝑌 ∈ (NrmSGrp‘𝐺) ∧ 𝑆 ∈ 𝐽) ∧ 𝑦 ∈ 𝑋) ∧ 𝑧 ∈ 𝑆) ∧ (((inv_g‘𝐺)‘𝑦)(+_g‘𝐺)𝑧) ∈ 𝑌) ∧ 𝑎 ∈ 𝑋) → ((inv_g‘𝐺)‘𝑎) ∈ 𝑋)
117	4, 50	grpass 18828	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝐺 ∈ Grp ∧ (((inv_g‘𝐺)‘𝑎) ∈ 𝑋 ∧ 𝑎 ∈ 𝑋 ∧ (((inv_g‘𝐺)‘𝑦)(+_g‘𝐺)𝑧) ∈ 𝑋)) → ((((inv_g‘𝐺)‘𝑎)(+_g‘𝐺)𝑎)(+_g‘𝐺)(((inv_g‘𝐺)‘𝑦)(+_g‘𝐺)𝑧)) = (((inv_g‘𝐺)‘𝑎)(+_g‘𝐺)(𝑎(+_g‘𝐺)(((inv_g‘𝐺)‘𝑦)(+_g‘𝐺)𝑧))))
118	103, 116, 104, 105, 117	syl13anc 1373	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((((((𝐺 ∈ TopGrp ∧ 𝑌 ∈ (NrmSGrp‘𝐺) ∧ 𝑆 ∈ 𝐽) ∧ 𝑦 ∈ 𝑋) ∧ 𝑧 ∈ 𝑆) ∧ (((inv_g‘𝐺)‘𝑦)(+_g‘𝐺)𝑧) ∈ 𝑌) ∧ 𝑎 ∈ 𝑋) → ((((inv_g‘𝐺)‘𝑎)(+_g‘𝐺)𝑎)(+_g‘𝐺)(((inv_g‘𝐺)‘𝑦)(+_g‘𝐺)𝑧)) = (((inv_g‘𝐺)‘𝑎)(+_g‘𝐺)(𝑎(+_g‘𝐺)(((inv_g‘𝐺)‘𝑦)(+_g‘𝐺)𝑧))))
119	4, 50, 76	grplid 18852	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝐺 ∈ Grp ∧ (((inv_g‘𝐺)‘𝑦)(+_g‘𝐺)𝑧) ∈ 𝑋) → ((0_g‘𝐺)(+_g‘𝐺)(((inv_g‘𝐺)‘𝑦)(+_g‘𝐺)𝑧)) = (((inv_g‘𝐺)‘𝑦)(+_g‘𝐺)𝑧))
120	103, 105, 119	syl2anc 585	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((((((𝐺 ∈ TopGrp ∧ 𝑌 ∈ (NrmSGrp‘𝐺) ∧ 𝑆 ∈ 𝐽) ∧ 𝑦 ∈ 𝑋) ∧ 𝑧 ∈ 𝑆) ∧ (((inv_g‘𝐺)‘𝑦)(+_g‘𝐺)𝑧) ∈ 𝑌) ∧ 𝑎 ∈ 𝑋) → ((0_g‘𝐺)(+_g‘𝐺)(((inv_g‘𝐺)‘𝑦)(+_g‘𝐺)𝑧)) = (((inv_g‘𝐺)‘𝑦)(+_g‘𝐺)𝑧))
121	114, 118, 120	3eqtr3d 2781	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((((((𝐺 ∈ TopGrp ∧ 𝑌 ∈ (NrmSGrp‘𝐺) ∧ 𝑆 ∈ 𝐽) ∧ 𝑦 ∈ 𝑋) ∧ 𝑧 ∈ 𝑆) ∧ (((inv_g‘𝐺)‘𝑦)(+_g‘𝐺)𝑧) ∈ 𝑌) ∧ 𝑎 ∈ 𝑋) → (((inv_g‘𝐺)‘𝑎)(+_g‘𝐺)(𝑎(+_g‘𝐺)(((inv_g‘𝐺)‘𝑦)(+_g‘𝐺)𝑧))) = (((inv_g‘𝐺)‘𝑦)(+_g‘𝐺)𝑧))
122		simplr 768	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((((((𝐺 ∈ TopGrp ∧ 𝑌 ∈ (NrmSGrp‘𝐺) ∧ 𝑆 ∈ 𝐽) ∧ 𝑦 ∈ 𝑋) ∧ 𝑧 ∈ 𝑆) ∧ (((inv_g‘𝐺)‘𝑦)(+_g‘𝐺)𝑧) ∈ 𝑌) ∧ 𝑎 ∈ 𝑋) → (((inv_g‘𝐺)‘𝑦)(+_g‘𝐺)𝑧) ∈ 𝑌)
123	121, 122	eqeltrd 2834	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((((𝐺 ∈ TopGrp ∧ 𝑌 ∈ (NrmSGrp‘𝐺) ∧ 𝑆 ∈ 𝐽) ∧ 𝑦 ∈ 𝑋) ∧ 𝑧 ∈ 𝑆) ∧ (((inv_g‘𝐺)‘𝑦)(+_g‘𝐺)𝑧) ∈ 𝑌) ∧ 𝑎 ∈ 𝑋) → (((inv_g‘𝐺)‘𝑎)(+_g‘𝐺)(𝑎(+_g‘𝐺)(((inv_g‘𝐺)‘𝑦)(+_g‘𝐺)𝑧))) ∈ 𝑌)
124	48	ad2antrr 725	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((((((𝐺 ∈ TopGrp ∧ 𝑌 ∈ (NrmSGrp‘𝐺) ∧ 𝑆 ∈ 𝐽) ∧ 𝑦 ∈ 𝑋) ∧ 𝑧 ∈ 𝑆) ∧ (((inv_g‘𝐺)‘𝑦)(+_g‘𝐺)𝑧) ∈ 𝑌) ∧ 𝑎 ∈ 𝑋) → 𝑌 ⊆ 𝑋)
125	4, 49, 50, 41	eqgval 19057	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝐺 ∈ Grp ∧ 𝑌 ⊆ 𝑋) → (𝑎(𝐺 ~_QG 𝑌)(𝑎(+_g‘𝐺)(((inv_g‘𝐺)‘𝑦)(+_g‘𝐺)𝑧)) ↔ (𝑎 ∈ 𝑋 ∧ (𝑎(+_g‘𝐺)(((inv_g‘𝐺)‘𝑦)(+_g‘𝐺)𝑧)) ∈ 𝑋 ∧ (((inv_g‘𝐺)‘𝑎)(+_g‘𝐺)(𝑎(+_g‘𝐺)(((inv_g‘𝐺)‘𝑦)(+_g‘𝐺)𝑧))) ∈ 𝑌)))
126	103, 124, 125	syl2anc 585	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((((𝐺 ∈ TopGrp ∧ 𝑌 ∈ (NrmSGrp‘𝐺) ∧ 𝑆 ∈ 𝐽) ∧ 𝑦 ∈ 𝑋) ∧ 𝑧 ∈ 𝑆) ∧ (((inv_g‘𝐺)‘𝑦)(+_g‘𝐺)𝑧) ∈ 𝑌) ∧ 𝑎 ∈ 𝑋) → (𝑎(𝐺 ~_QG 𝑌)(𝑎(+_g‘𝐺)(((inv_g‘𝐺)‘𝑦)(+_g‘𝐺)𝑧)) ↔ (𝑎 ∈ 𝑋 ∧ (𝑎(+_g‘𝐺)(((inv_g‘𝐺)‘𝑦)(+_g‘𝐺)𝑧)) ∈ 𝑋 ∧ (((inv_g‘𝐺)‘𝑎)(+_g‘𝐺)(𝑎(+_g‘𝐺)(((inv_g‘𝐺)‘𝑦)(+_g‘𝐺)𝑧))) ∈ 𝑌)))
127	104, 107, 123, 126	mpbir3and 1343	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((((𝐺 ∈ TopGrp ∧ 𝑌 ∈ (NrmSGrp‘𝐺) ∧ 𝑆 ∈ 𝐽) ∧ 𝑦 ∈ 𝑋) ∧ 𝑧 ∈ 𝑆) ∧ (((inv_g‘𝐺)‘𝑦)(+_g‘𝐺)𝑧) ∈ 𝑌) ∧ 𝑎 ∈ 𝑋) → 𝑎(𝐺 ~_QG 𝑌)(𝑎(+_g‘𝐺)(((inv_g‘𝐺)‘𝑦)(+_g‘𝐺)𝑧)))
128	111, 127	erthi 8754	. . . . . . . . . . . . . . . . . 18 ⊢ ((((((𝐺 ∈ TopGrp ∧ 𝑌 ∈ (NrmSGrp‘𝐺) ∧ 𝑆 ∈ 𝐽) ∧ 𝑦 ∈ 𝑋) ∧ 𝑧 ∈ 𝑆) ∧ (((inv_g‘𝐺)‘𝑦)(+_g‘𝐺)𝑧) ∈ 𝑌) ∧ 𝑎 ∈ 𝑋) → [𝑎](𝐺 ~_QG 𝑌) = [(𝑎(+_g‘𝐺)(((inv_g‘𝐺)‘𝑦)(+_g‘𝐺)𝑧))](𝐺 ~_QG 𝑌))
129	110, 128	eqtr4d 2776	. . . . . . . . . . . . . . . . 17 ⊢ ((((((𝐺 ∈ TopGrp ∧ 𝑌 ∈ (NrmSGrp‘𝐺) ∧ 𝑆 ∈ 𝐽) ∧ 𝑦 ∈ 𝑋) ∧ 𝑧 ∈ 𝑆) ∧ (((inv_g‘𝐺)‘𝑦)(+_g‘𝐺)𝑧) ∈ 𝑌) ∧ 𝑎 ∈ 𝑋) → (𝐹‘(𝑎(+_g‘𝐺)(((inv_g‘𝐺)‘𝑦)(+_g‘𝐺)𝑧))) = [𝑎](𝐺 ~_QG 𝑌))
130	129	eleq1d 2819	. . . . . . . . . . . . . . . 16 ⊢ ((((((𝐺 ∈ TopGrp ∧ 𝑌 ∈ (NrmSGrp‘𝐺) ∧ 𝑆 ∈ 𝐽) ∧ 𝑦 ∈ 𝑋) ∧ 𝑧 ∈ 𝑆) ∧ (((inv_g‘𝐺)‘𝑦)(+_g‘𝐺)𝑧) ∈ 𝑌) ∧ 𝑎 ∈ 𝑋) → ((𝐹‘(𝑎(+_g‘𝐺)(((inv_g‘𝐺)‘𝑦)(+_g‘𝐺)𝑧))) ∈ (𝐹 “ 𝑆) ↔ [𝑎](𝐺 ~_QG 𝑌) ∈ (𝐹 “ 𝑆)))
131	102, 130	sylibd 238	. . . . . . . . . . . . . . 15 ⊢ ((((((𝐺 ∈ TopGrp ∧ 𝑌 ∈ (NrmSGrp‘𝐺) ∧ 𝑆 ∈ 𝐽) ∧ 𝑦 ∈ 𝑋) ∧ 𝑧 ∈ 𝑆) ∧ (((inv_g‘𝐺)‘𝑦)(+_g‘𝐺)𝑧) ∈ 𝑌) ∧ 𝑎 ∈ 𝑋) → ((𝑎(+_g‘𝐺)(((inv_g‘𝐺)‘𝑦)(+_g‘𝐺)𝑧)) ∈ 𝑆 → [𝑎](𝐺 ~_QG 𝑌) ∈ (𝐹 “ 𝑆)))
132	131	ss2rabdv 4074	. . . . . . . . . . . . . 14 ⊢ (((((𝐺 ∈ TopGrp ∧ 𝑌 ∈ (NrmSGrp‘𝐺) ∧ 𝑆 ∈ 𝐽) ∧ 𝑦 ∈ 𝑋) ∧ 𝑧 ∈ 𝑆) ∧ (((inv_g‘𝐺)‘𝑦)(+_g‘𝐺)𝑧) ∈ 𝑌) → {𝑎 ∈ 𝑋 ∣ (𝑎(+_g‘𝐺)(((inv_g‘𝐺)‘𝑦)(+_g‘𝐺)𝑧)) ∈ 𝑆} ⊆ {𝑎 ∈ 𝑋 ∣ [𝑎](𝐺 ~_QG 𝑌) ∈ (𝐹 “ 𝑆)})
133		eceq1 8741	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 = 𝑎 → [𝑥](𝐺 ~_QG 𝑌) = [𝑎](𝐺 ~_QG 𝑌))
134	133	cbvmptv 5262	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 ∈ 𝑋 ↦ [𝑥](𝐺 ~_QG 𝑌)) = (𝑎 ∈ 𝑋 ↦ [𝑎](𝐺 ~_QG 𝑌))
135	6, 134	eqtri 2761	. . . . . . . . . . . . . . 15 ⊢ 𝐹 = (𝑎 ∈ 𝑋 ↦ [𝑎](𝐺 ~_QG 𝑌))
136	135	mptpreima 6238	. . . . . . . . . . . . . 14 ⊢ (^◡𝐹 “ (𝐹 “ 𝑆)) = {𝑎 ∈ 𝑋 ∣ [𝑎](𝐺 ~_QG 𝑌) ∈ (𝐹 “ 𝑆)}
137	132, 93, 136	3sstr4g 4028	. . . . . . . . . . . . 13 ⊢ (((((𝐺 ∈ TopGrp ∧ 𝑌 ∈ (NrmSGrp‘𝐺) ∧ 𝑆 ∈ 𝐽) ∧ 𝑦 ∈ 𝑋) ∧ 𝑧 ∈ 𝑆) ∧ (((inv_g‘𝐺)‘𝑦)(+_g‘𝐺)𝑧) ∈ 𝑌) → (^◡(𝑎 ∈ 𝑋 ↦ (𝑎(+_g‘𝐺)(((inv_g‘𝐺)‘𝑦)(+_g‘𝐺)𝑧))) “ 𝑆) ⊆ (^◡𝐹 “ (𝐹 “ 𝑆)))
138		eleq2 2823	. . . . . . . . . . . . . . 15 ⊢ (𝑢 = (^◡(𝑎 ∈ 𝑋 ↦ (𝑎(+_g‘𝐺)(((inv_g‘𝐺)‘𝑦)(+_g‘𝐺)𝑧))) “ 𝑆) → (𝑦 ∈ 𝑢 ↔ 𝑦 ∈ (^◡(𝑎 ∈ 𝑋 ↦ (𝑎(+_g‘𝐺)(((inv_g‘𝐺)‘𝑦)(+_g‘𝐺)𝑧))) “ 𝑆)))
139		sseq1 4008	. . . . . . . . . . . . . . 15 ⊢ (𝑢 = (^◡(𝑎 ∈ 𝑋 ↦ (𝑎(+_g‘𝐺)(((inv_g‘𝐺)‘𝑦)(+_g‘𝐺)𝑧))) “ 𝑆) → (𝑢 ⊆ (^◡𝐹 “ (𝐹 “ 𝑆)) ↔ (^◡(𝑎 ∈ 𝑋 ↦ (𝑎(+_g‘𝐺)(((inv_g‘𝐺)‘𝑦)(+_g‘𝐺)𝑧))) “ 𝑆) ⊆ (^◡𝐹 “ (𝐹 “ 𝑆))))
140	138, 139	anbi12d 632	. . . . . . . . . . . . . 14 ⊢ (𝑢 = (^◡(𝑎 ∈ 𝑋 ↦ (𝑎(+_g‘𝐺)(((inv_g‘𝐺)‘𝑦)(+_g‘𝐺)𝑧))) “ 𝑆) → ((𝑦 ∈ 𝑢 ∧ 𝑢 ⊆ (^◡𝐹 “ (𝐹 “ 𝑆))) ↔ (𝑦 ∈ (^◡(𝑎 ∈ 𝑋 ↦ (𝑎(+_g‘𝐺)(((inv_g‘𝐺)‘𝑦)(+_g‘𝐺)𝑧))) “ 𝑆) ∧ (^◡(𝑎 ∈ 𝑋 ↦ (𝑎(+_g‘𝐺)(((inv_g‘𝐺)‘𝑦)(+_g‘𝐺)𝑧))) “ 𝑆) ⊆ (^◡𝐹 “ (𝐹 “ 𝑆)))))
141	140	rspcev 3613	. . . . . . . . . . . . 13 ⊢ (((^◡(𝑎 ∈ 𝑋 ↦ (𝑎(+_g‘𝐺)(((inv_g‘𝐺)‘𝑦)(+_g‘𝐺)𝑧))) “ 𝑆) ∈ 𝐽 ∧ (𝑦 ∈ (^◡(𝑎 ∈ 𝑋 ↦ (𝑎(+_g‘𝐺)(((inv_g‘𝐺)‘𝑦)(+_g‘𝐺)𝑧))) “ 𝑆) ∧ (^◡(𝑎 ∈ 𝑋 ↦ (𝑎(+_g‘𝐺)(((inv_g‘𝐺)‘𝑦)(+_g‘𝐺)𝑧))) “ 𝑆) ⊆ (^◡𝐹 “ (𝐹 “ 𝑆)))) → ∃𝑢 ∈ 𝐽 (𝑦 ∈ 𝑢 ∧ 𝑢 ⊆ (^◡𝐹 “ (𝐹 “ 𝑆))))
142	72, 95, 137, 141	syl12anc 836	. . . . . . . . . . . 12 ⊢ (((((𝐺 ∈ TopGrp ∧ 𝑌 ∈ (NrmSGrp‘𝐺) ∧ 𝑆 ∈ 𝐽) ∧ 𝑦 ∈ 𝑋) ∧ 𝑧 ∈ 𝑆) ∧ (((inv_g‘𝐺)‘𝑦)(+_g‘𝐺)𝑧) ∈ 𝑌) → ∃𝑢 ∈ 𝐽 (𝑦 ∈ 𝑢 ∧ 𝑢 ⊆ (^◡𝐹 “ (𝐹 “ 𝑆))))
143	142	3ad2antr3 1191	. . . . . . . . . . 11 ⊢ (((((𝐺 ∈ TopGrp ∧ 𝑌 ∈ (NrmSGrp‘𝐺) ∧ 𝑆 ∈ 𝐽) ∧ 𝑦 ∈ 𝑋) ∧ 𝑧 ∈ 𝑆) ∧ (𝑦 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋 ∧ (((inv_g‘𝐺)‘𝑦)(+_g‘𝐺)𝑧) ∈ 𝑌)) → ∃𝑢 ∈ 𝐽 (𝑦 ∈ 𝑢 ∧ 𝑢 ⊆ (^◡𝐹 “ (𝐹 “ 𝑆))))
144	143	ex 414	. . . . . . . . . 10 ⊢ ((((𝐺 ∈ TopGrp ∧ 𝑌 ∈ (NrmSGrp‘𝐺) ∧ 𝑆 ∈ 𝐽) ∧ 𝑦 ∈ 𝑋) ∧ 𝑧 ∈ 𝑆) → ((𝑦 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋 ∧ (((inv_g‘𝐺)‘𝑦)(+_g‘𝐺)𝑧) ∈ 𝑌) → ∃𝑢 ∈ 𝐽 (𝑦 ∈ 𝑢 ∧ 𝑢 ⊆ (^◡𝐹 “ (𝐹 “ 𝑆)))))
145	53, 144	sylbid 239	. . . . . . . . 9 ⊢ ((((𝐺 ∈ TopGrp ∧ 𝑌 ∈ (NrmSGrp‘𝐺) ∧ 𝑆 ∈ 𝐽) ∧ 𝑦 ∈ 𝑋) ∧ 𝑧 ∈ 𝑆) → ((𝐹‘𝑧) = [𝑦](𝐺 ~_QG 𝑌) → ∃𝑢 ∈ 𝐽 (𝑦 ∈ 𝑢 ∧ 𝑢 ⊆ (^◡𝐹 “ (𝐹 “ 𝑆)))))
146	145	rexlimdva 3156	. . . . . . . 8 ⊢ (((𝐺 ∈ TopGrp ∧ 𝑌 ∈ (NrmSGrp‘𝐺) ∧ 𝑆 ∈ 𝐽) ∧ 𝑦 ∈ 𝑋) → (∃𝑧 ∈ 𝑆 (𝐹‘𝑧) = [𝑦](𝐺 ~_QG 𝑌) → ∃𝑢 ∈ 𝐽 (𝑦 ∈ 𝑢 ∧ 𝑢 ⊆ (^◡𝐹 “ (𝐹 “ 𝑆)))))
147	21, 146	syl5 34	. . . . . . 7 ⊢ (((𝐺 ∈ TopGrp ∧ 𝑌 ∈ (NrmSGrp‘𝐺) ∧ 𝑆 ∈ 𝐽) ∧ 𝑦 ∈ 𝑋) → ([𝑦](𝐺 ~_QG 𝑌) ∈ (𝐹 “ 𝑆) → ∃𝑢 ∈ 𝐽 (𝑦 ∈ 𝑢 ∧ 𝑢 ⊆ (^◡𝐹 “ (𝐹 “ 𝑆)))))
148	147	expimpd 455	. . . . . 6 ⊢ ((𝐺 ∈ TopGrp ∧ 𝑌 ∈ (NrmSGrp‘𝐺) ∧ 𝑆 ∈ 𝐽) → ((𝑦 ∈ 𝑋 ∧ [𝑦](𝐺 ~_QG 𝑌) ∈ (𝐹 “ 𝑆)) → ∃𝑢 ∈ 𝐽 (𝑦 ∈ 𝑢 ∧ 𝑢 ⊆ (^◡𝐹 “ (𝐹 “ 𝑆)))))
149	18, 148	biimtrid 241	. . . . 5 ⊢ ((𝐺 ∈ TopGrp ∧ 𝑌 ∈ (NrmSGrp‘𝐺) ∧ 𝑆 ∈ 𝐽) → (𝑦 ∈ (^◡𝐹 “ (𝐹 “ 𝑆)) → ∃𝑢 ∈ 𝐽 (𝑦 ∈ 𝑢 ∧ 𝑢 ⊆ (^◡𝐹 “ (𝐹 “ 𝑆)))))
150	149	ralrimiv 3146	. . . 4 ⊢ ((𝐺 ∈ TopGrp ∧ 𝑌 ∈ (NrmSGrp‘𝐺) ∧ 𝑆 ∈ 𝐽) → ∀𝑦 ∈ (^◡𝐹 “ (𝐹 “ 𝑆))∃𝑢 ∈ 𝐽 (𝑦 ∈ 𝑢 ∧ 𝑢 ⊆ (^◡𝐹 “ (𝐹 “ 𝑆))))
151		topontop 22415	. . . . 5 ⊢ (𝐽 ∈ (TopOn‘𝑋) → 𝐽 ∈ Top)
152		eltop2 22478	. . . . 5 ⊢ (𝐽 ∈ Top → ((^◡𝐹 “ (𝐹 “ 𝑆)) ∈ 𝐽 ↔ ∀𝑦 ∈ (^◡𝐹 “ (𝐹 “ 𝑆))∃𝑢 ∈ 𝐽 (𝑦 ∈ 𝑢 ∧ 𝑢 ⊆ (^◡𝐹 “ (𝐹 “ 𝑆)))))
153	24, 151, 152	3syl 18	. . . 4 ⊢ ((𝐺 ∈ TopGrp ∧ 𝑌 ∈ (NrmSGrp‘𝐺) ∧ 𝑆 ∈ 𝐽) → ((^◡𝐹 “ (𝐹 “ 𝑆)) ∈ 𝐽 ↔ ∀𝑦 ∈ (^◡𝐹 “ (𝐹 “ 𝑆))∃𝑢 ∈ 𝐽 (𝑦 ∈ 𝑢 ∧ 𝑢 ⊆ (^◡𝐹 “ (𝐹 “ 𝑆)))))
154	150, 153	mpbird 257	. . 3 ⊢ ((𝐺 ∈ TopGrp ∧ 𝑌 ∈ (NrmSGrp‘𝐺) ∧ 𝑆 ∈ 𝐽) → (^◡𝐹 “ (𝐹 “ 𝑆)) ∈ 𝐽)
155		elqtop3 23207	. . . 4 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹:𝑋–onto→(𝑋 / (𝐺 ~_QG 𝑌))) → ((𝐹 “ 𝑆) ∈ (𝐽 qTop 𝐹) ↔ ((𝐹 “ 𝑆) ⊆ (𝑋 / (𝐺 ~_QG 𝑌)) ∧ (^◡𝐹 “ (𝐹 “ 𝑆)) ∈ 𝐽)))
156	24, 10, 155	syl2anc 585	. . 3 ⊢ ((𝐺 ∈ TopGrp ∧ 𝑌 ∈ (NrmSGrp‘𝐺) ∧ 𝑆 ∈ 𝐽) → ((𝐹 “ 𝑆) ∈ (𝐽 qTop 𝐹) ↔ ((𝐹 “ 𝑆) ⊆ (𝑋 / (𝐺 ~_QG 𝑌)) ∧ (^◡𝐹 “ (𝐹 “ 𝑆)) ∈ 𝐽)))
157	13, 154, 156	mpbir2and 712	. 2 ⊢ ((𝐺 ∈ TopGrp ∧ 𝑌 ∈ (NrmSGrp‘𝐺) ∧ 𝑆 ∈ 𝐽) → (𝐹 “ 𝑆) ∈ (𝐽 qTop 𝐹))
158	3, 5, 6, 8, 9	qusval 17488	. . 3 ⊢ ((𝐺 ∈ TopGrp ∧ 𝑌 ∈ (NrmSGrp‘𝐺) ∧ 𝑆 ∈ 𝐽) → 𝐻 = (𝐹 “_s 𝐺))
159		qustgpopn.k	. . 3 ⊢ 𝐾 = (TopOpen‘𝐻)
160	158, 5, 10, 9, 22, 159	imastopn 23224	. 2 ⊢ ((𝐺 ∈ TopGrp ∧ 𝑌 ∈ (NrmSGrp‘𝐺) ∧ 𝑆 ∈ 𝐽) → 𝐾 = (𝐽 qTop 𝐹))
161	157, 160	eleqtrrd 2837	1 ⊢ ((𝐺 ∈ TopGrp ∧ 𝑌 ∈ (NrmSGrp‘𝐺) ∧ 𝑆 ∈ 𝐽) → (𝐹 “ 𝑆) ∈ 𝐾)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 205 ∧ wa 397 ∧ w3a 1088 = wceq 1542 ∈ wcel 2107 ∀wral 3062 ∃wrex 3071 {crab 3433 Vcvv 3475 ⊆ wss 3949 class class class wbr 5149 ↦ cmpt 5232 ^◡ccnv 5676 ran crn 5678 “ cima 5680 Fun wfun 6538 Fn wfn 6539 –onto→wfo 6542 ‘cfv 6544 (class class class)co 7409 Er wer 8700 [cec 8701 / cqs 8702 Basecbs 17144 +_gcplusg 17197 TopOpenctopn 17367 0_gc0g 17385 qTop cqtop 17449 /_s cqus 17451 Grpcgrp 18819 inv_gcminusg 18820 SubGrpcsubg 19000 NrmSGrpcnsg 19001 ~_QG cqg 19002 opp_gcoppg 19209 Topctop 22395 TopOnctopon 22412 Cn ccn 22728 Homeochmeo 23257 TopGrpctgp 23575

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-rep 5286 ax-sep 5300 ax-nul 5307 ax-pow 5364 ax-pr 5428 ax-un 7725 ax-cnex 11166 ax-resscn 11167 ax-1cn 11168 ax-icn 11169 ax-addcl 11170 ax-addrcl 11171 ax-mulcl 11172 ax-mulrcl 11173 ax-mulcom 11174 ax-addass 11175 ax-mulass 11176 ax-distr 11177 ax-i2m1 11178 ax-1ne0 11179 ax-1rid 11180 ax-rnegex 11181 ax-rrecex 11182 ax-cnre 11183 ax-pre-lttri 11184 ax-pre-lttrn 11185 ax-pre-ltadd 11186 ax-pre-mulgt0 11187

This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-rmo 3377 df-reu 3378 df-rab 3434 df-v 3477 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-pss 3968 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-tp 4634 df-op 4636 df-uni 4910 df-iun 5000 df-br 5150 df-opab 5212 df-mpt 5233 df-tr 5267 df-id 5575 df-eprel 5581 df-po 5589 df-so 5590 df-fr 5632 df-we 5634 df-xp 5683 df-rel 5684 df-cnv 5685 df-co 5686 df-dm 5687 df-rn 5688 df-res 5689 df-ima 5690 df-pred 6301 df-ord 6368 df-on 6369 df-lim 6370 df-suc 6371 df-iota 6496 df-fun 6546 df-fn 6547 df-f 6548 df-f1 6549 df-fo 6550 df-f1o 6551 df-fv 6552 df-riota 7365 df-ov 7412 df-oprab 7413 df-mpo 7414 df-om 7856 df-1st 7975 df-2nd 7976 df-tpos 8211 df-frecs 8266 df-wrecs 8297 df-recs 8371 df-rdg 8410 df-1o 8466 df-er 8703 df-ec 8705 df-qs 8709 df-map 8822 df-en 8940 df-dom 8941 df-sdom 8942 df-fin 8943 df-sup 9437 df-inf 9438 df-pnf 11250 df-mnf 11251 df-xr 11252 df-ltxr 11253 df-le 11254 df-sub 11446 df-neg 11447 df-nn 12213 df-2 12275 df-3 12276 df-4 12277 df-5 12278 df-6 12279 df-7 12280 df-8 12281 df-9 12282 df-n0 12473 df-z 12559 df-dec 12678 df-uz 12823 df-fz 13485 df-struct 17080 df-sets 17097 df-slot 17115 df-ndx 17127 df-base 17145 df-ress 17174 df-plusg 17210 df-mulr 17211 df-sca 17213 df-vsca 17214 df-ip 17215 df-tset 17216 df-ple 17217 df-ds 17219 df-rest 17368 df-topn 17369 df-0g 17387 df-topgen 17389 df-qtop 17453 df-imas 17454 df-qus 17455 df-plusf 18560 df-mgm 18561 df-sgrp 18610 df-mnd 18626 df-grp 18822 df-minusg 18823 df-subg 19003 df-nsg 19004 df-eqg 19005 df-oppg 19210 df-top 22396 df-topon 22413 df-topsp 22435 df-bases 22449 df-cn 22731 df-cnp 22732 df-tx 23066 df-hmeo 23259 df-tmd 23576 df-tgp 23577

This theorem is referenced by: qustgplem 23625

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