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Theorem qustgpopn 23844

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 6069	. . . 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 7444	. . . . . . 7 ⊢ (𝐺 ~_QG 𝑌) ∈ V
8	7	a1i 11	. . . . . 6 ⊢ ((𝐺 ∈ TopGrp ∧ 𝑌 ∈ (NrmSGrp‘𝐺) ∧ 𝑆 ∈ 𝐽) → (𝐺 ~_QG 𝑌) ∈ V)
9		simp1 1134	. . . . . 6 ⊢ ((𝐺 ∈ TopGrp ∧ 𝑌 ∈ (NrmSGrp‘𝐺) ∧ 𝑆 ∈ 𝐽) → 𝐺 ∈ TopGrp)
10	3, 5, 6, 8, 9	quslem 17493	. . . . 5 ⊢ ((𝐺 ∈ TopGrp ∧ 𝑌 ∈ (NrmSGrp‘𝐺) ∧ 𝑆 ∈ 𝐽) → 𝐹:𝑋–onto→(𝑋 / (𝐺 ~_QG 𝑌)))
11		forn 6807	. . . . 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 4033	. . 3 ⊢ ((𝐺 ∈ TopGrp ∧ 𝑌 ∈ (NrmSGrp‘𝐺) ∧ 𝑆 ∈ 𝐽) → (𝐹 “ 𝑆) ⊆ (𝑋 / (𝐺 ~_QG 𝑌)))
14		eceq1 8743	. . . . . . . . . 10 ⊢ (𝑥 = 𝑦 → [𝑥](𝐺 ~_QG 𝑌) = [𝑦](𝐺 ~_QG 𝑌))
15	14	cbvmptv 5260	. . . . . . . . 9 ⊢ (𝑥 ∈ 𝑋 ↦ [𝑥](𝐺 ~_QG 𝑌)) = (𝑦 ∈ 𝑋 ↦ [𝑦](𝐺 ~_QG 𝑌))
16	6, 15	eqtri 2758	. . . . . . . 8 ⊢ 𝐹 = (𝑦 ∈ 𝑋 ↦ [𝑦](𝐺 ~_QG 𝑌))
17	16	mptpreima 6236	. . . . . . 7 ⊢ (^◡𝐹 “ (𝐹 “ 𝑆)) = {𝑦 ∈ 𝑋 ∣ [𝑦](𝐺 ~_QG 𝑌) ∈ (𝐹 “ 𝑆)}
18	17	reqabi 3452	. . . . . 6 ⊢ (𝑦 ∈ (^◡𝐹 “ (𝐹 “ 𝑆)) ↔ (𝑦 ∈ 𝑋 ∧ [𝑦](𝐺 ~_QG 𝑌) ∈ (𝐹 “ 𝑆)))
19	6	funmpt2 6586	. . . . . . . . 9 ⊢ Fun 𝐹
20		fvelima 6956	. . . . . . . . 9 ⊢ ((Fun 𝐹 ∧ [𝑦](𝐺 ~_QG 𝑌) ∈ (𝐹 “ 𝑆)) → ∃𝑧 ∈ 𝑆 (𝐹‘𝑧) = [𝑦](𝐺 ~_QG 𝑌))
21	19, 20	mpan 686	. . . . . . . 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 23806	. . . . . . . . . . . . . . . . . 18 ⊢ (𝐺 ∈ TopGrp → 𝐽 ∈ (TopOn‘𝑋))
24	9, 23	syl 17	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐺 ∈ TopGrp ∧ 𝑌 ∈ (NrmSGrp‘𝐺) ∧ 𝑆 ∈ 𝐽) → 𝐽 ∈ (TopOn‘𝑋))
25		simp3 1136	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐺 ∈ TopGrp ∧ 𝑌 ∈ (NrmSGrp‘𝐺) ∧ 𝑆 ∈ 𝐽) → 𝑆 ∈ 𝐽)
26		toponss 22649	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ 𝐽) → 𝑆 ⊆ 𝑋)
27	24, 25, 26	syl2anc 582	. . . . . . . . . . . . . . . 16 ⊢ ((𝐺 ∈ TopGrp ∧ 𝑌 ∈ (NrmSGrp‘𝐺) ∧ 𝑆 ∈ 𝐽) → 𝑆 ⊆ 𝑋)
28	27	adantr 479	. . . . . . . . . . . . . . 15 ⊢ (((𝐺 ∈ TopGrp ∧ 𝑌 ∈ (NrmSGrp‘𝐺) ∧ 𝑆 ∈ 𝐽) ∧ 𝑦 ∈ 𝑋) → 𝑆 ⊆ 𝑋)
29	28	sselda 3981	. . . . . . . . . . . . . 14 ⊢ ((((𝐺 ∈ TopGrp ∧ 𝑌 ∈ (NrmSGrp‘𝐺) ∧ 𝑆 ∈ 𝐽) ∧ 𝑦 ∈ 𝑋) ∧ 𝑧 ∈ 𝑆) → 𝑧 ∈ 𝑋)
30		eceq1 8743	. . . . . . . . . . . . . . 15 ⊢ (𝑥 = 𝑧 → [𝑥](𝐺 ~_QG 𝑌) = [𝑧](𝐺 ~_QG 𝑌))
31		ecexg 8709	. . . . . . . . . . . . . . . 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 6997	. . . . . . . . . . . . . 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 2732	. . . . . . . . . . . 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 2737	. . . . . . . . . . . 12 ⊢ ([𝑧](𝐺 ~_QG 𝑌) = [𝑦](𝐺 ~_QG 𝑌) ↔ [𝑦](𝐺 ~_QG 𝑌) = [𝑧](𝐺 ~_QG 𝑌))
37	35, 36	bitrdi 286	. . . . . . . . . . 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 19074	. . . . . . . . . . . . . . 15 ⊢ (𝑌 ∈ (NrmSGrp‘𝐺) → 𝑌 ∈ (SubGrp‘𝐺))
39	38	3ad2ant2 1132	. . . . . . . . . . . . . 14 ⊢ ((𝐺 ∈ TopGrp ∧ 𝑌 ∈ (NrmSGrp‘𝐺) ∧ 𝑆 ∈ 𝐽) → 𝑌 ∈ (SubGrp‘𝐺))
40	39	ad2antrr 722	. . . . . . . . . . . . 13 ⊢ ((((𝐺 ∈ TopGrp ∧ 𝑌 ∈ (NrmSGrp‘𝐺) ∧ 𝑆 ∈ 𝐽) ∧ 𝑦 ∈ 𝑋) ∧ 𝑧 ∈ 𝑆) → 𝑌 ∈ (SubGrp‘𝐺))
41		eqid 2730	. . . . . . . . . . . . . 14 ⊢ (𝐺 ~_QG 𝑌) = (𝐺 ~_QG 𝑌)
42	4, 41	eqger 19094	. . . . . . . . . . . . 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 765	. . . . . . . . . . . 12 ⊢ ((((𝐺 ∈ TopGrp ∧ 𝑌 ∈ (NrmSGrp‘𝐺) ∧ 𝑆 ∈ 𝐽) ∧ 𝑦 ∈ 𝑋) ∧ 𝑧 ∈ 𝑆) → 𝑦 ∈ 𝑋)
45	43, 44	erth 8754	. . . . . . . . . . 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 722	. . . . . . . . . . . 12 ⊢ ((((𝐺 ∈ TopGrp ∧ 𝑌 ∈ (NrmSGrp‘𝐺) ∧ 𝑆 ∈ 𝐽) ∧ 𝑦 ∈ 𝑋) ∧ 𝑧 ∈ 𝑆) → 𝐺 ∈ TopGrp)
47	4	subgss 19043	. . . . . . . . . . . . 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 2730	. . . . . . . . . . . . 13 ⊢ (inv_g‘𝐺) = (inv_g‘𝐺)
50		eqid 2730	. . . . . . . . . . . . 13 ⊢ (+_g‘𝐺) = (+_g‘𝐺)
51	4, 49, 50, 41	eqgval 19093	. . . . . . . . . . . 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 582	. . . . . . . . . . 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 306	. . . . . . . . . 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 2730	. . . . . . . . . . . . . . . . . 18 ⊢ (opp_g‘𝐺) = (opp_g‘𝐺)
55		eqid 2730	. . . . . . . . . . . . . . . . . 18 ⊢ (+_g‘(opp_g‘𝐺)) = (+_g‘(opp_g‘𝐺))
56	50, 54, 55	oppgplus 19254	. . . . . . . . . . . . . . . . 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 5252	. . . . . . . . . . . . . . . 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 479	. . . . . . . . . . . . . . . . . 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 23822	. . . . . . . . . . . . . . . . . 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 3981	. . . . . . . . . . . . . . . . 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 2730	. . . . . . . . . . . . . . . . . 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 19257	. . . . . . . . . . . . . . . . . 18 ⊢ 𝑋 = (Base‘(opp_g‘𝐺))
64	54, 22	oppgtopn 19261	. . . . . . . . . . . . . . . . . 18 ⊢ 𝐽 = (TopOpen‘(opp_g‘𝐺))
65	62, 63, 55, 64	tgplacthmeo 23827	. . . . . . . . . . . . . . . . 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 582	. . . . . . . . . . . . . . . 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 2836	. . . . . . . . . . . . . . 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 23484	. . . . . . . . . . . . . . 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 726	. . . . . . . . . . . . . 14 ⊢ (((((𝐺 ∈ TopGrp ∧ 𝑌 ∈ (NrmSGrp‘𝐺) ∧ 𝑆 ∈ 𝐽) ∧ 𝑦 ∈ 𝑋) ∧ 𝑧 ∈ 𝑆) ∧ (((inv_g‘𝐺)‘𝑦)(+_g‘𝐺)𝑧) ∈ 𝑌) → 𝑆 ∈ 𝐽)
71		cnima 22989	. . . . . . . . . . . . . 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 582	. . . . . . . . . . . . 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 479	. . . . . . . . . . . . . 14 ⊢ (((((𝐺 ∈ TopGrp ∧ 𝑌 ∈ (NrmSGrp‘𝐺) ∧ 𝑆 ∈ 𝐽) ∧ 𝑦 ∈ 𝑋) ∧ 𝑧 ∈ 𝑆) ∧ (((inv_g‘𝐺)‘𝑦)(+_g‘𝐺)𝑧) ∈ 𝑌) → 𝑦 ∈ 𝑋)
74		tgpgrp 23802	. . . . . . . . . . . . . . . . . . 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 2730	. . . . . . . . . . . . . . . . . . 19 ⊢ (0_g‘𝐺) = (0_g‘𝐺)
77	4, 50, 76, 49	grprinv 18911	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐺 ∈ Grp ∧ 𝑦 ∈ 𝑋) → (𝑦(+_g‘𝐺)((inv_g‘𝐺)‘𝑦)) = (0_g‘𝐺))
78	75, 73, 77	syl2anc 582	. . . . . . . . . . . . . . . . 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 7426	. . . . . . . . . . . . . . . 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 18908	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐺 ∈ Grp ∧ 𝑦 ∈ 𝑋) → ((inv_g‘𝐺)‘𝑦) ∈ 𝑋)
81	75, 73, 80	syl2anc 582	. . . . . . . . . . . . . . . . 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 479	. . . . . . . . . . . . . . . . 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 18864	. . . . . . . . . . . . . . . . 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 1370	. . . . . . . . . . . . . . . 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 18888	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐺 ∈ Grp ∧ 𝑧 ∈ 𝑋) → ((0_g‘𝐺)(+_g‘𝐺)𝑧) = 𝑧)
86	75, 82, 85	syl2anc 582	. . . . . . . . . . . . . . . 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 2778	. . . . . . . . . . . . . . 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 765	. . . . . . . . . . . . . . 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 2831	. . . . . . . . . . . . . 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 7418	. . . . . . . . . . . . . . . 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 2816	. . . . . . . . . . . . . . 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 2730	. . . . . . . . . . . . . . . 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 6236	. . . . . . . . . . . . . . 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 3685	. . . . . . . . . . . . . 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 581	. . . . . . . . . . . . 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 8709	. . . . . . . . . . . . . . . . . . 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 6692	. . . . . . . . . . . . . . . . 17 ⊢ 𝐹 Fn 𝑋
99	28	ad3antrrr 726	. . . . . . . . . . . . . . . . 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 7236	. . . . . . . . . . . . . . . . . 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 1119	. . . . . . . . . . . . . . . . 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 585	. . . . . . . . . . . . . . . 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 479	. . . . . . . . . . . . . . . . . . . 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 483	. . . . . . . . . . . . . . . . . . . 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 479	. . . . . . . . . . . . . . . . . . . 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 18863	. . . . . . . . . . . . . . . . . . . 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 1369	. . . . . . . . . . . . . . . . . . 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 8743	. . . . . . . . . . . . . . . . . . . 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 7002	. . . . . . . . . . . . . . . . . . 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 722	. . . . . . . . . . . . . . . . . . 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 18910	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝐺 ∈ Grp ∧ 𝑎 ∈ 𝑋) → (((inv_g‘𝐺)‘𝑎)(+_g‘𝐺)𝑎) = (0_g‘𝐺))
113	103, 104, 112	syl2anc 582	. . . . . . . . . . . . . . . . . . . . . . 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 7426	. . . . . . . . . . . . . . . . . . . . . 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 18908	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝐺 ∈ Grp ∧ 𝑎 ∈ 𝑋) → ((inv_g‘𝐺)‘𝑎) ∈ 𝑋)
116	103, 104, 115	syl2anc 582	. . . . . . . . . . . . . . . . . . . . . . 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 18864	. . . . . . . . . . . . . . . . . . . . . . 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 1370	. . . . . . . . . . . . . . . . . . . . . 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 18888	. . . . . . . . . . . . . . . . . . . . . . 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 582	. . . . . . . . . . . . . . . . . . . . . 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 2778	. . . . . . . . . . . . . . . . . . . . 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 765	. . . . . . . . . . . . . . . . . . . . 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 2831	. . . . . . . . . . . . . . . . . . . 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 722	. . . . . . . . . . . . . . . . . . . . 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 19093	. . . . . . . . . . . . . . . . . . . . 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 582	. . . . . . . . . . . . . . . . . . . 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 1340	. . . . . . . . . . . . . . . . . . 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 8756	. . . . . . . . . . . . . . . . . 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 2773	. . . . . . . . . . . . . . . . 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 2816	. . . . . . . . . . . . . . . 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 4072	. . . . . . . . . . . . . 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 8743	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 = 𝑎 → [𝑥](𝐺 ~_QG 𝑌) = [𝑎](𝐺 ~_QG 𝑌))
134	133	cbvmptv 5260	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 ∈ 𝑋 ↦ [𝑥](𝐺 ~_QG 𝑌)) = (𝑎 ∈ 𝑋 ↦ [𝑎](𝐺 ~_QG 𝑌))
135	6, 134	eqtri 2758	. . . . . . . . . . . . . . 15 ⊢ 𝐹 = (𝑎 ∈ 𝑋 ↦ [𝑎](𝐺 ~_QG 𝑌))
136	135	mptpreima 6236	. . . . . . . . . . . . . 14 ⊢ (^◡𝐹 “ (𝐹 “ 𝑆)) = {𝑎 ∈ 𝑋 ∣ [𝑎](𝐺 ~_QG 𝑌) ∈ (𝐹 “ 𝑆)}
137	132, 93, 136	3sstr4g 4026	. . . . . . . . . . . . 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 2820	. . . . . . . . . . . . . . 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 4006	. . . . . . . . . . . . . . 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 629	. . . . . . . . . . . . . 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 3611	. . . . . . . . . . . . 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 833	. . . . . . . . . . . 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 1188	. . . . . . . . . . 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 411	. . . . . . . . . 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 3153	. . . . . . . 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 452	. . . . . 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 3143	. . . 4 ⊢ ((𝐺 ∈ TopGrp ∧ 𝑌 ∈ (NrmSGrp‘𝐺) ∧ 𝑆 ∈ 𝐽) → ∀𝑦 ∈ (^◡𝐹 “ (𝐹 “ 𝑆))∃𝑢 ∈ 𝐽 (𝑦 ∈ 𝑢 ∧ 𝑢 ⊆ (^◡𝐹 “ (𝐹 “ 𝑆))))
151		topontop 22635	. . . . 5 ⊢ (𝐽 ∈ (TopOn‘𝑋) → 𝐽 ∈ Top)
152		eltop2 22698	. . . . 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 256	. . 3 ⊢ ((𝐺 ∈ TopGrp ∧ 𝑌 ∈ (NrmSGrp‘𝐺) ∧ 𝑆 ∈ 𝐽) → (^◡𝐹 “ (𝐹 “ 𝑆)) ∈ 𝐽)
155		elqtop3 23427	. . . 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 582	. . 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 709	. 2 ⊢ ((𝐺 ∈ TopGrp ∧ 𝑌 ∈ (NrmSGrp‘𝐺) ∧ 𝑆 ∈ 𝐽) → (𝐹 “ 𝑆) ∈ (𝐽 qTop 𝐹))
158	3, 5, 6, 8, 9	qusval 17492	. . 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 23444	. 2 ⊢ ((𝐺 ∈ TopGrp ∧ 𝑌 ∈ (NrmSGrp‘𝐺) ∧ 𝑆 ∈ 𝐽) → 𝐾 = (𝐽 qTop 𝐹))
161	157, 160	eleqtrrd 2834	1 ⊢ ((𝐺 ∈ TopGrp ∧ 𝑌 ∈ (NrmSGrp‘𝐺) ∧ 𝑆 ∈ 𝐽) → (𝐹 “ 𝑆) ∈ 𝐾)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 205 ∧ wa 394 ∧ w3a 1085 = wceq 1539 ∈ wcel 2104 ∀wral 3059 ∃wrex 3068 {crab 3430 Vcvv 3472 ⊆ wss 3947 class class class wbr 5147 ↦ cmpt 5230 ^◡ccnv 5674 ran crn 5676 “ cima 5678 Fun wfun 6536 Fn wfn 6537 –onto→wfo 6540 ‘cfv 6542 (class class class)co 7411 Er wer 8702 [cec 8703 / cqs 8704 Basecbs 17148 +_gcplusg 17201 TopOpenctopn 17371 0_gc0g 17389 qTop cqtop 17453 /_s cqus 17455 Grpcgrp 18855 inv_gcminusg 18856 SubGrpcsubg 19036 NrmSGrpcnsg 19037 ~_QG cqg 19038 opp_gcoppg 19250 Topctop 22615 TopOnctopon 22632 Cn ccn 22948 Homeochmeo 23477 TopGrpctgp 23795

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1911 ax-6 1969 ax-7 2009 ax-8 2106 ax-9 2114 ax-10 2135 ax-11 2152 ax-12 2169 ax-ext 2701 ax-rep 5284 ax-sep 5298 ax-nul 5305 ax-pow 5362 ax-pr 5426 ax-un 7727 ax-cnex 11168 ax-resscn 11169 ax-1cn 11170 ax-icn 11171 ax-addcl 11172 ax-addrcl 11173 ax-mulcl 11174 ax-mulrcl 11175 ax-mulcom 11176 ax-addass 11177 ax-mulass 11178 ax-distr 11179 ax-i2m1 11180 ax-1ne0 11181 ax-1rid 11182 ax-rnegex 11183 ax-rrecex 11184 ax-cnre 11185 ax-pre-lttri 11186 ax-pre-lttrn 11187 ax-pre-ltadd 11188 ax-pre-mulgt0 11189

This theorem depends on definitions: df-bi 206 df-an 395 df-or 844 df-3or 1086 df-3an 1087 df-tru 1542 df-fal 1552 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2532 df-eu 2561 df-clab 2708 df-cleq 2722 df-clel 2808 df-nfc 2883 df-ne 2939 df-nel 3045 df-ral 3060 df-rex 3069 df-rmo 3374 df-reu 3375 df-rab 3431 df-v 3474 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4322 df-if 4528 df-pw 4603 df-sn 4628 df-pr 4630 df-tp 4632 df-op 4634 df-uni 4908 df-iun 4998 df-br 5148 df-opab 5210 df-mpt 5231 df-tr 5265 df-id 5573 df-eprel 5579 df-po 5587 df-so 5588 df-fr 5630 df-we 5632 df-xp 5681 df-rel 5682 df-cnv 5683 df-co 5684 df-dm 5685 df-rn 5686 df-res 5687 df-ima 5688 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 7367 df-ov 7414 df-oprab 7415 df-mpo 7416 df-om 7858 df-1st 7977 df-2nd 7978 df-tpos 8213 df-frecs 8268 df-wrecs 8299 df-recs 8373 df-rdg 8412 df-1o 8468 df-er 8705 df-ec 8707 df-qs 8711 df-map 8824 df-en 8942 df-dom 8943 df-sdom 8944 df-fin 8945 df-sup 9439 df-inf 9440 df-pnf 11254 df-mnf 11255 df-xr 11256 df-ltxr 11257 df-le 11258 df-sub 11450 df-neg 11451 df-nn 12217 df-2 12279 df-3 12280 df-4 12281 df-5 12282 df-6 12283 df-7 12284 df-8 12285 df-9 12286 df-n0 12477 df-z 12563 df-dec 12682 df-uz 12827 df-fz 13489 df-struct 17084 df-sets 17101 df-slot 17119 df-ndx 17131 df-base 17149 df-ress 17178 df-plusg 17214 df-mulr 17215 df-sca 17217 df-vsca 17218 df-ip 17219 df-tset 17220 df-ple 17221 df-ds 17223 df-rest 17372 df-topn 17373 df-0g 17391 df-topgen 17393 df-qtop 17457 df-imas 17458 df-qus 17459 df-plusf 18564 df-mgm 18565 df-sgrp 18644 df-mnd 18660 df-grp 18858 df-minusg 18859 df-subg 19039 df-nsg 19040 df-eqg 19041 df-oppg 19251 df-top 22616 df-topon 22633 df-topsp 22655 df-bases 22669 df-cn 22951 df-cnp 22952 df-tx 23286 df-hmeo 23479 df-tmd 23796 df-tgp 23797

This theorem is referenced by: qustgplem 23845

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