Theorem qustgpopn 22720

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 5933	. . . 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 7181	. . . . . . 7 ⊢ (𝐺 ~QG 𝑌) ∈ V
8	7	a1i 11	. . . . . 6 ⊢ ((𝐺 ∈ TopGrp ∧ 𝑌 ∈ (NrmSGrp‘𝐺) ∧ 𝑆 ∈ 𝐽) → (𝐺 ~QG 𝑌) ∈ V)
9		simp1 1130	. . . . . 6 ⊢ ((𝐺 ∈ TopGrp ∧ 𝑌 ∈ (NrmSGrp‘𝐺) ∧ 𝑆 ∈ 𝐽) → 𝐺 ∈ TopGrp)
10	3, 5, 6, 8, 9	quslem 16808	. . . . 5 ⊢ ((𝐺 ∈ TopGrp ∧ 𝑌 ∈ (NrmSGrp‘𝐺) ∧ 𝑆 ∈ 𝐽) → 𝐹:𝑋–onto→(𝑋 / (𝐺 ~QG 𝑌)))
11		forn 6586	. . . . 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 4017	. . 3 ⊢ ((𝐺 ∈ TopGrp ∧ 𝑌 ∈ (NrmSGrp‘𝐺) ∧ 𝑆 ∈ 𝐽) → (𝐹 “ 𝑆) ⊆ (𝑋 / (𝐺 ~QG 𝑌)))
14		eceq1 8319	. . . . . . . . . 10 ⊢ (𝑥 = 𝑦 → [𝑥](𝐺 ~QG 𝑌) = [𝑦](𝐺 ~QG 𝑌))
15	14	cbvmptv 5160	. . . . . . . . 9 ⊢ (𝑥 ∈ 𝑋 ↦ [𝑥](𝐺 ~QG 𝑌)) = (𝑦 ∈ 𝑋 ↦ [𝑦](𝐺 ~QG 𝑌))
16	6, 15	eqtri 2842	. . . . . . . 8 ⊢ 𝐹 = (𝑦 ∈ 𝑋 ↦ [𝑦](𝐺 ~QG 𝑌))
17	16	mptpreima 6085	. . . . . . 7 ⊢ (◡𝐹 “ (𝐹 “ 𝑆)) = {𝑦 ∈ 𝑋 ∣ [𝑦](𝐺 ~QG 𝑌) ∈ (𝐹 “ 𝑆)}
18	17	rabeq2i 3486	. . . . . 6 ⊢ (𝑦 ∈ (◡𝐹 “ (𝐹 “ 𝑆)) ↔ (𝑦 ∈ 𝑋 ∧ [𝑦](𝐺 ~QG 𝑌) ∈ (𝐹 “ 𝑆)))
19	6	funmpt2 6387	. . . . . . . . 9 ⊢ Fun 𝐹
20		fvelima 6724	. . . . . . . . 9 ⊢ ((Fun 𝐹 ∧ [𝑦](𝐺 ~QG 𝑌) ∈ (𝐹 “ 𝑆)) → ∃𝑧 ∈ 𝑆 (𝐹‘𝑧) = [𝑦](𝐺 ~QG 𝑌))
21	19, 20	mpan 688	. . . . . . . 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 22682	. . . . . . . . . . . . . . . . . 18 ⊢ (𝐺 ∈ TopGrp → 𝐽 ∈ (TopOn‘𝑋))
24	9, 23	syl 17	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐺 ∈ TopGrp ∧ 𝑌 ∈ (NrmSGrp‘𝐺) ∧ 𝑆 ∈ 𝐽) → 𝐽 ∈ (TopOn‘𝑋))
25		simp3 1132	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐺 ∈ TopGrp ∧ 𝑌 ∈ (NrmSGrp‘𝐺) ∧ 𝑆 ∈ 𝐽) → 𝑆 ∈ 𝐽)
26		toponss 21527	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ 𝐽) → 𝑆 ⊆ 𝑋)
27	24, 25, 26	syl2anc 586	. . . . . . . . . . . . . . . 16 ⊢ ((𝐺 ∈ TopGrp ∧ 𝑌 ∈ (NrmSGrp‘𝐺) ∧ 𝑆 ∈ 𝐽) → 𝑆 ⊆ 𝑋)
28	27	adantr 483	. . . . . . . . . . . . . . 15 ⊢ (((𝐺 ∈ TopGrp ∧ 𝑌 ∈ (NrmSGrp‘𝐺) ∧ 𝑆 ∈ 𝐽) ∧ 𝑦 ∈ 𝑋) → 𝑆 ⊆ 𝑋)
29	28	sselda 3965	. . . . . . . . . . . . . 14 ⊢ ((((𝐺 ∈ TopGrp ∧ 𝑌 ∈ (NrmSGrp‘𝐺) ∧ 𝑆 ∈ 𝐽) ∧ 𝑦 ∈ 𝑋) ∧ 𝑧 ∈ 𝑆) → 𝑧 ∈ 𝑋)
30		eceq1 8319	. . . . . . . . . . . . . . 15 ⊢ (𝑥 = 𝑧 → [𝑥](𝐺 ~QG 𝑌) = [𝑧](𝐺 ~QG 𝑌))
31		ecexg 8285	. . . . . . . . . . . . . . . 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 6761	. . . . . . . . . . . . . 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 2821	. . . . . . . . . . . 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 2826	. . . . . . . . . . . 12 ⊢ ([𝑧](𝐺 ~QG 𝑌) = [𝑦](𝐺 ~QG 𝑌) ↔ [𝑦](𝐺 ~QG 𝑌) = [𝑧](𝐺 ~QG 𝑌))
37	35, 36	syl6bb 289	. . . . . . . . . . 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 18302	. . . . . . . . . . . . . . 15 ⊢ (𝑌 ∈ (NrmSGrp‘𝐺) → 𝑌 ∈ (SubGrp‘𝐺))
39	38	3ad2ant2 1128	. . . . . . . . . . . . . 14 ⊢ ((𝐺 ∈ TopGrp ∧ 𝑌 ∈ (NrmSGrp‘𝐺) ∧ 𝑆 ∈ 𝐽) → 𝑌 ∈ (SubGrp‘𝐺))
40	39	ad2antrr 724	. . . . . . . . . . . . 13 ⊢ ((((𝐺 ∈ TopGrp ∧ 𝑌 ∈ (NrmSGrp‘𝐺) ∧ 𝑆 ∈ 𝐽) ∧ 𝑦 ∈ 𝑋) ∧ 𝑧 ∈ 𝑆) → 𝑌 ∈ (SubGrp‘𝐺))
41		eqid 2819	. . . . . . . . . . . . . 14 ⊢ (𝐺 ~QG 𝑌) = (𝐺 ~QG 𝑌)
42	4, 41	eqger 18322	. . . . . . . . . . . . 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 767	. . . . . . . . . . . 12 ⊢ ((((𝐺 ∈ TopGrp ∧ 𝑌 ∈ (NrmSGrp‘𝐺) ∧ 𝑆 ∈ 𝐽) ∧ 𝑦 ∈ 𝑋) ∧ 𝑧 ∈ 𝑆) → 𝑦 ∈ 𝑋)
45	43, 44	erth 8330	. . . . . . . . . . 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 724	. . . . . . . . . . . 12 ⊢ ((((𝐺 ∈ TopGrp ∧ 𝑌 ∈ (NrmSGrp‘𝐺) ∧ 𝑆 ∈ 𝐽) ∧ 𝑦 ∈ 𝑋) ∧ 𝑧 ∈ 𝑆) → 𝐺 ∈ TopGrp)
47	4	subgss 18272	. . . . . . . . . . . . 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 2819	. . . . . . . . . . . . 13 ⊢ (invg‘𝐺) = (invg‘𝐺)
50		eqid 2819	. . . . . . . . . . . . 13 ⊢ (+g‘𝐺) = (+g‘𝐺)
51	4, 49, 50, 41	eqgval 18321	. . . . . . . . . . . 12 ⊢ ((𝐺 ∈ TopGrp ∧ 𝑌 ⊆ 𝑋) → (𝑦(𝐺 ~QG 𝑌)𝑧 ↔ (𝑦 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋 ∧ (((invg‘𝐺)‘𝑦)(+g‘𝐺)𝑧) ∈ 𝑌)))
52	46, 48, 51	syl2anc 586	. . . . . . . . . . 11 ⊢ ((((𝐺 ∈ TopGrp ∧ 𝑌 ∈ (NrmSGrp‘𝐺) ∧ 𝑆 ∈ 𝐽) ∧ 𝑦 ∈ 𝑋) ∧ 𝑧 ∈ 𝑆) → (𝑦(𝐺 ~QG 𝑌)𝑧 ↔ (𝑦 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋 ∧ (((invg‘𝐺)‘𝑦)(+g‘𝐺)𝑧) ∈ 𝑌)))
53	37, 45, 52	3bitr2d 309	. . . . . . . . . 10 ⊢ ((((𝐺 ∈ TopGrp ∧ 𝑌 ∈ (NrmSGrp‘𝐺) ∧ 𝑆 ∈ 𝐽) ∧ 𝑦 ∈ 𝑋) ∧ 𝑧 ∈ 𝑆) → ((𝐹‘𝑧) = [𝑦](𝐺 ~QG 𝑌) ↔ (𝑦 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋 ∧ (((invg‘𝐺)‘𝑦)(+g‘𝐺)𝑧) ∈ 𝑌)))
54		eqid 2819	. . . . . . . . . . . . . . . . . 18 ⊢ (oppg‘𝐺) = (oppg‘𝐺)
55		eqid 2819	. . . . . . . . . . . . . . . . . 18 ⊢ (+g‘(oppg‘𝐺)) = (+g‘(oppg‘𝐺))
56	50, 54, 55	oppgplus 18469	. . . . . . . . . . . . . . . . 17 ⊢ ((((invg‘𝐺)‘𝑦)(+g‘𝐺)𝑧)(+g‘(oppg‘𝐺))𝑎) = (𝑎(+g‘𝐺)(((invg‘𝐺)‘𝑦)(+g‘𝐺)𝑧))
57	56	mpteq2i 5149	. . . . . . . . . . . . . . . 16 ⊢ (𝑎 ∈ 𝑋 ↦ ((((invg‘𝐺)‘𝑦)(+g‘𝐺)𝑧)(+g‘(oppg‘𝐺))𝑎)) = (𝑎 ∈ 𝑋 ↦ (𝑎(+g‘𝐺)(((invg‘𝐺)‘𝑦)(+g‘𝐺)𝑧)))
58	46	adantr 483	. . . . . . . . . . . . . . . . . 18 ⊢ (((((𝐺 ∈ TopGrp ∧ 𝑌 ∈ (NrmSGrp‘𝐺) ∧ 𝑆 ∈ 𝐽) ∧ 𝑦 ∈ 𝑋) ∧ 𝑧 ∈ 𝑆) ∧ (((invg‘𝐺)‘𝑦)(+g‘𝐺)𝑧) ∈ 𝑌) → 𝐺 ∈ TopGrp)
59	54	oppgtgp 22698	. . . . . . . . . . . . . . . . . 18 ⊢ (𝐺 ∈ TopGrp → (oppg‘𝐺) ∈ TopGrp)
60	58, 59	syl 17	. . . . . . . . . . . . . . . . 17 ⊢ (((((𝐺 ∈ TopGrp ∧ 𝑌 ∈ (NrmSGrp‘𝐺) ∧ 𝑆 ∈ 𝐽) ∧ 𝑦 ∈ 𝑋) ∧ 𝑧 ∈ 𝑆) ∧ (((invg‘𝐺)‘𝑦)(+g‘𝐺)𝑧) ∈ 𝑌) → (oppg‘𝐺) ∈ TopGrp)
61	48	sselda 3965	. . . . . . . . . . . . . . . . 17 ⊢ (((((𝐺 ∈ TopGrp ∧ 𝑌 ∈ (NrmSGrp‘𝐺) ∧ 𝑆 ∈ 𝐽) ∧ 𝑦 ∈ 𝑋) ∧ 𝑧 ∈ 𝑆) ∧ (((invg‘𝐺)‘𝑦)(+g‘𝐺)𝑧) ∈ 𝑌) → (((invg‘𝐺)‘𝑦)(+g‘𝐺)𝑧) ∈ 𝑋)
62		eqid 2819	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑎 ∈ 𝑋 ↦ ((((invg‘𝐺)‘𝑦)(+g‘𝐺)𝑧)(+g‘(oppg‘𝐺))𝑎)) = (𝑎 ∈ 𝑋 ↦ ((((invg‘𝐺)‘𝑦)(+g‘𝐺)𝑧)(+g‘(oppg‘𝐺))𝑎))
63	54, 4	oppgbas 18471	. . . . . . . . . . . . . . . . . 18 ⊢ 𝑋 = (Base‘(oppg‘𝐺))
64	54, 22	oppgtopn 18473	. . . . . . . . . . . . . . . . . 18 ⊢ 𝐽 = (TopOpen‘(oppg‘𝐺))
65	62, 63, 55, 64	tgplacthmeo 22703	. . . . . . . . . . . . . . . . 17 ⊢ (((oppg‘𝐺) ∈ TopGrp ∧ (((invg‘𝐺)‘𝑦)(+g‘𝐺)𝑧) ∈ 𝑋) → (𝑎 ∈ 𝑋 ↦ ((((invg‘𝐺)‘𝑦)(+g‘𝐺)𝑧)(+g‘(oppg‘𝐺))𝑎)) ∈ (𝐽Homeo𝐽))
66	60, 61, 65	syl2anc 586	. . . . . . . . . . . . . . . 16 ⊢ (((((𝐺 ∈ TopGrp ∧ 𝑌 ∈ (NrmSGrp‘𝐺) ∧ 𝑆 ∈ 𝐽) ∧ 𝑦 ∈ 𝑋) ∧ 𝑧 ∈ 𝑆) ∧ (((invg‘𝐺)‘𝑦)(+g‘𝐺)𝑧) ∈ 𝑌) → (𝑎 ∈ 𝑋 ↦ ((((invg‘𝐺)‘𝑦)(+g‘𝐺)𝑧)(+g‘(oppg‘𝐺))𝑎)) ∈ (𝐽Homeo𝐽))
67	57, 66	eqeltrrid 2916	. . . . . . . . . . . . . . 15 ⊢ (((((𝐺 ∈ TopGrp ∧ 𝑌 ∈ (NrmSGrp‘𝐺) ∧ 𝑆 ∈ 𝐽) ∧ 𝑦 ∈ 𝑋) ∧ 𝑧 ∈ 𝑆) ∧ (((invg‘𝐺)‘𝑦)(+g‘𝐺)𝑧) ∈ 𝑌) → (𝑎 ∈ 𝑋 ↦ (𝑎(+g‘𝐺)(((invg‘𝐺)‘𝑦)(+g‘𝐺)𝑧))) ∈ (𝐽Homeo𝐽))
68		hmeocn 22360	. . . . . . . . . . . . . . 15 ⊢ ((𝑎 ∈ 𝑋 ↦ (𝑎(+g‘𝐺)(((invg‘𝐺)‘𝑦)(+g‘𝐺)𝑧))) ∈ (𝐽Homeo𝐽) → (𝑎 ∈ 𝑋 ↦ (𝑎(+g‘𝐺)(((invg‘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) ∧ (((invg‘𝐺)‘𝑦)(+g‘𝐺)𝑧) ∈ 𝑌) → (𝑎 ∈ 𝑋 ↦ (𝑎(+g‘𝐺)(((invg‘𝐺)‘𝑦)(+g‘𝐺)𝑧))) ∈ (𝐽 Cn 𝐽))
70	25	ad3antrrr 728	. . . . . . . . . . . . . 14 ⊢ (((((𝐺 ∈ TopGrp ∧ 𝑌 ∈ (NrmSGrp‘𝐺) ∧ 𝑆 ∈ 𝐽) ∧ 𝑦 ∈ 𝑋) ∧ 𝑧 ∈ 𝑆) ∧ (((invg‘𝐺)‘𝑦)(+g‘𝐺)𝑧) ∈ 𝑌) → 𝑆 ∈ 𝐽)
71		cnima 21865	. . . . . . . . . . . . . 14 ⊢ (((𝑎 ∈ 𝑋 ↦ (𝑎(+g‘𝐺)(((invg‘𝐺)‘𝑦)(+g‘𝐺)𝑧))) ∈ (𝐽 Cn 𝐽) ∧ 𝑆 ∈ 𝐽) → (◡(𝑎 ∈ 𝑋 ↦ (𝑎(+g‘𝐺)(((invg‘𝐺)‘𝑦)(+g‘𝐺)𝑧))) “ 𝑆) ∈ 𝐽)
72	69, 70, 71	syl2anc 586	. . . . . . . . . . . . 13 ⊢ (((((𝐺 ∈ TopGrp ∧ 𝑌 ∈ (NrmSGrp‘𝐺) ∧ 𝑆 ∈ 𝐽) ∧ 𝑦 ∈ 𝑋) ∧ 𝑧 ∈ 𝑆) ∧ (((invg‘𝐺)‘𝑦)(+g‘𝐺)𝑧) ∈ 𝑌) → (◡(𝑎 ∈ 𝑋 ↦ (𝑎(+g‘𝐺)(((invg‘𝐺)‘𝑦)(+g‘𝐺)𝑧))) “ 𝑆) ∈ 𝐽)
73	44	adantr 483	. . . . . . . . . . . . . 14 ⊢ (((((𝐺 ∈ TopGrp ∧ 𝑌 ∈ (NrmSGrp‘𝐺) ∧ 𝑆 ∈ 𝐽) ∧ 𝑦 ∈ 𝑋) ∧ 𝑧 ∈ 𝑆) ∧ (((invg‘𝐺)‘𝑦)(+g‘𝐺)𝑧) ∈ 𝑌) → 𝑦 ∈ 𝑋)
74		tgpgrp 22678	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝐺 ∈ TopGrp → 𝐺 ∈ Grp)
75	58, 74	syl 17	. . . . . . . . . . . . . . . . . 18 ⊢ (((((𝐺 ∈ TopGrp ∧ 𝑌 ∈ (NrmSGrp‘𝐺) ∧ 𝑆 ∈ 𝐽) ∧ 𝑦 ∈ 𝑋) ∧ 𝑧 ∈ 𝑆) ∧ (((invg‘𝐺)‘𝑦)(+g‘𝐺)𝑧) ∈ 𝑌) → 𝐺 ∈ Grp)
76		eqid 2819	. . . . . . . . . . . . . . . . . . 19 ⊢ (0g‘𝐺) = (0g‘𝐺)
77	4, 50, 76, 49	grprinv 18145	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐺 ∈ Grp ∧ 𝑦 ∈ 𝑋) → (𝑦(+g‘𝐺)((invg‘𝐺)‘𝑦)) = (0g‘𝐺))
78	75, 73, 77	syl2anc 586	. . . . . . . . . . . . . . . . 17 ⊢ (((((𝐺 ∈ TopGrp ∧ 𝑌 ∈ (NrmSGrp‘𝐺) ∧ 𝑆 ∈ 𝐽) ∧ 𝑦 ∈ 𝑋) ∧ 𝑧 ∈ 𝑆) ∧ (((invg‘𝐺)‘𝑦)(+g‘𝐺)𝑧) ∈ 𝑌) → (𝑦(+g‘𝐺)((invg‘𝐺)‘𝑦)) = (0g‘𝐺))
79	78	oveq1d 7163	. . . . . . . . . . . . . . . 16 ⊢ (((((𝐺 ∈ TopGrp ∧ 𝑌 ∈ (NrmSGrp‘𝐺) ∧ 𝑆 ∈ 𝐽) ∧ 𝑦 ∈ 𝑋) ∧ 𝑧 ∈ 𝑆) ∧ (((invg‘𝐺)‘𝑦)(+g‘𝐺)𝑧) ∈ 𝑌) → ((𝑦(+g‘𝐺)((invg‘𝐺)‘𝑦))(+g‘𝐺)𝑧) = ((0g‘𝐺)(+g‘𝐺)𝑧))
80	4, 49	grpinvcl 18143	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐺 ∈ Grp ∧ 𝑦 ∈ 𝑋) → ((invg‘𝐺)‘𝑦) ∈ 𝑋)
81	75, 73, 80	syl2anc 586	. . . . . . . . . . . . . . . . 17 ⊢ (((((𝐺 ∈ TopGrp ∧ 𝑌 ∈ (NrmSGrp‘𝐺) ∧ 𝑆 ∈ 𝐽) ∧ 𝑦 ∈ 𝑋) ∧ 𝑧 ∈ 𝑆) ∧ (((invg‘𝐺)‘𝑦)(+g‘𝐺)𝑧) ∈ 𝑌) → ((invg‘𝐺)‘𝑦) ∈ 𝑋)
82	29	adantr 483	. . . . . . . . . . . . . . . . 17 ⊢ (((((𝐺 ∈ TopGrp ∧ 𝑌 ∈ (NrmSGrp‘𝐺) ∧ 𝑆 ∈ 𝐽) ∧ 𝑦 ∈ 𝑋) ∧ 𝑧 ∈ 𝑆) ∧ (((invg‘𝐺)‘𝑦)(+g‘𝐺)𝑧) ∈ 𝑌) → 𝑧 ∈ 𝑋)
83	4, 50	grpass 18104	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐺 ∈ Grp ∧ (𝑦 ∈ 𝑋 ∧ ((invg‘𝐺)‘𝑦) ∈ 𝑋 ∧ 𝑧 ∈ 𝑋)) → ((𝑦(+g‘𝐺)((invg‘𝐺)‘𝑦))(+g‘𝐺)𝑧) = (𝑦(+g‘𝐺)(((invg‘𝐺)‘𝑦)(+g‘𝐺)𝑧)))
84	75, 73, 81, 82, 83	syl13anc 1366	. . . . . . . . . . . . . . . 16 ⊢ (((((𝐺 ∈ TopGrp ∧ 𝑌 ∈ (NrmSGrp‘𝐺) ∧ 𝑆 ∈ 𝐽) ∧ 𝑦 ∈ 𝑋) ∧ 𝑧 ∈ 𝑆) ∧ (((invg‘𝐺)‘𝑦)(+g‘𝐺)𝑧) ∈ 𝑌) → ((𝑦(+g‘𝐺)((invg‘𝐺)‘𝑦))(+g‘𝐺)𝑧) = (𝑦(+g‘𝐺)(((invg‘𝐺)‘𝑦)(+g‘𝐺)𝑧)))
85	4, 50, 76	grplid 18125	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐺 ∈ Grp ∧ 𝑧 ∈ 𝑋) → ((0g‘𝐺)(+g‘𝐺)𝑧) = 𝑧)
86	75, 82, 85	syl2anc 586	. . . . . . . . . . . . . . . 16 ⊢ (((((𝐺 ∈ TopGrp ∧ 𝑌 ∈ (NrmSGrp‘𝐺) ∧ 𝑆 ∈ 𝐽) ∧ 𝑦 ∈ 𝑋) ∧ 𝑧 ∈ 𝑆) ∧ (((invg‘𝐺)‘𝑦)(+g‘𝐺)𝑧) ∈ 𝑌) → ((0g‘𝐺)(+g‘𝐺)𝑧) = 𝑧)
87	79, 84, 86	3eqtr3d 2862	. . . . . . . . . . . . . . 15 ⊢ (((((𝐺 ∈ TopGrp ∧ 𝑌 ∈ (NrmSGrp‘𝐺) ∧ 𝑆 ∈ 𝐽) ∧ 𝑦 ∈ 𝑋) ∧ 𝑧 ∈ 𝑆) ∧ (((invg‘𝐺)‘𝑦)(+g‘𝐺)𝑧) ∈ 𝑌) → (𝑦(+g‘𝐺)(((invg‘𝐺)‘𝑦)(+g‘𝐺)𝑧)) = 𝑧)
88		simplr 767	. . . . . . . . . . . . . . 15 ⊢ (((((𝐺 ∈ TopGrp ∧ 𝑌 ∈ (NrmSGrp‘𝐺) ∧ 𝑆 ∈ 𝐽) ∧ 𝑦 ∈ 𝑋) ∧ 𝑧 ∈ 𝑆) ∧ (((invg‘𝐺)‘𝑦)(+g‘𝐺)𝑧) ∈ 𝑌) → 𝑧 ∈ 𝑆)
89	87, 88	eqeltrd 2911	. . . . . . . . . . . . . 14 ⊢ (((((𝐺 ∈ TopGrp ∧ 𝑌 ∈ (NrmSGrp‘𝐺) ∧ 𝑆 ∈ 𝐽) ∧ 𝑦 ∈ 𝑋) ∧ 𝑧 ∈ 𝑆) ∧ (((invg‘𝐺)‘𝑦)(+g‘𝐺)𝑧) ∈ 𝑌) → (𝑦(+g‘𝐺)(((invg‘𝐺)‘𝑦)(+g‘𝐺)𝑧)) ∈ 𝑆)
90		oveq1 7155	. . . . . . . . . . . . . . . 16 ⊢ (𝑎 = 𝑦 → (𝑎(+g‘𝐺)(((invg‘𝐺)‘𝑦)(+g‘𝐺)𝑧)) = (𝑦(+g‘𝐺)(((invg‘𝐺)‘𝑦)(+g‘𝐺)𝑧)))
91	90	eleq1d 2895	. . . . . . . . . . . . . . 15 ⊢ (𝑎 = 𝑦 → ((𝑎(+g‘𝐺)(((invg‘𝐺)‘𝑦)(+g‘𝐺)𝑧)) ∈ 𝑆 ↔ (𝑦(+g‘𝐺)(((invg‘𝐺)‘𝑦)(+g‘𝐺)𝑧)) ∈ 𝑆))
92		eqid 2819	. . . . . . . . . . . . . . . 16 ⊢ (𝑎 ∈ 𝑋 ↦ (𝑎(+g‘𝐺)(((invg‘𝐺)‘𝑦)(+g‘𝐺)𝑧))) = (𝑎 ∈ 𝑋 ↦ (𝑎(+g‘𝐺)(((invg‘𝐺)‘𝑦)(+g‘𝐺)𝑧)))
93	92	mptpreima 6085	. . . . . . . . . . . . . . 15 ⊢ (◡(𝑎 ∈ 𝑋 ↦ (𝑎(+g‘𝐺)(((invg‘𝐺)‘𝑦)(+g‘𝐺)𝑧))) “ 𝑆) = {𝑎 ∈ 𝑋 ∣ (𝑎(+g‘𝐺)(((invg‘𝐺)‘𝑦)(+g‘𝐺)𝑧)) ∈ 𝑆}
94	91, 93	elrab2 3681	. . . . . . . . . . . . . 14 ⊢ (𝑦 ∈ (◡(𝑎 ∈ 𝑋 ↦ (𝑎(+g‘𝐺)(((invg‘𝐺)‘𝑦)(+g‘𝐺)𝑧))) “ 𝑆) ↔ (𝑦 ∈ 𝑋 ∧ (𝑦(+g‘𝐺)(((invg‘𝐺)‘𝑦)(+g‘𝐺)𝑧)) ∈ 𝑆))
95	73, 89, 94	sylanbrc 585	. . . . . . . . . . . . 13 ⊢ (((((𝐺 ∈ TopGrp ∧ 𝑌 ∈ (NrmSGrp‘𝐺) ∧ 𝑆 ∈ 𝐽) ∧ 𝑦 ∈ 𝑋) ∧ 𝑧 ∈ 𝑆) ∧ (((invg‘𝐺)‘𝑦)(+g‘𝐺)𝑧) ∈ 𝑌) → 𝑦 ∈ (◡(𝑎 ∈ 𝑋 ↦ (𝑎(+g‘𝐺)(((invg‘𝐺)‘𝑦)(+g‘𝐺)𝑧))) “ 𝑆))
96		ecexg 8285	. . . . . . . . . . . . . . . . . . 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 6484	. . . . . . . . . . . . . . . . 17 ⊢ 𝐹 Fn 𝑋
99	28	ad3antrrr 728	. . . . . . . . . . . . . . . . 17 ⊢ ((((((𝐺 ∈ TopGrp ∧ 𝑌 ∈ (NrmSGrp‘𝐺) ∧ 𝑆 ∈ 𝐽) ∧ 𝑦 ∈ 𝑋) ∧ 𝑧 ∈ 𝑆) ∧ (((invg‘𝐺)‘𝑦)(+g‘𝐺)𝑧) ∈ 𝑌) ∧ 𝑎 ∈ 𝑋) → 𝑆 ⊆ 𝑋)
100		fnfvima 6987	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐹 Fn 𝑋 ∧ 𝑆 ⊆ 𝑋 ∧ (𝑎(+g‘𝐺)(((invg‘𝐺)‘𝑦)(+g‘𝐺)𝑧)) ∈ 𝑆) → (𝐹‘(𝑎(+g‘𝐺)(((invg‘𝐺)‘𝑦)(+g‘𝐺)𝑧))) ∈ (𝐹 “ 𝑆))
101	100	3expia 1115	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐹 Fn 𝑋 ∧ 𝑆 ⊆ 𝑋) → ((𝑎(+g‘𝐺)(((invg‘𝐺)‘𝑦)(+g‘𝐺)𝑧)) ∈ 𝑆 → (𝐹‘(𝑎(+g‘𝐺)(((invg‘𝐺)‘𝑦)(+g‘𝐺)𝑧))) ∈ (𝐹 “ 𝑆)))
102	98, 99, 101	sylancr 589	. . . . . . . . . . . . . . . 16 ⊢ ((((((𝐺 ∈ TopGrp ∧ 𝑌 ∈ (NrmSGrp‘𝐺) ∧ 𝑆 ∈ 𝐽) ∧ 𝑦 ∈ 𝑋) ∧ 𝑧 ∈ 𝑆) ∧ (((invg‘𝐺)‘𝑦)(+g‘𝐺)𝑧) ∈ 𝑌) ∧ 𝑎 ∈ 𝑋) → ((𝑎(+g‘𝐺)(((invg‘𝐺)‘𝑦)(+g‘𝐺)𝑧)) ∈ 𝑆 → (𝐹‘(𝑎(+g‘𝐺)(((invg‘𝐺)‘𝑦)(+g‘𝐺)𝑧))) ∈ (𝐹 “ 𝑆)))
103	75	adantr 483	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((((𝐺 ∈ TopGrp ∧ 𝑌 ∈ (NrmSGrp‘𝐺) ∧ 𝑆 ∈ 𝐽) ∧ 𝑦 ∈ 𝑋) ∧ 𝑧 ∈ 𝑆) ∧ (((invg‘𝐺)‘𝑦)(+g‘𝐺)𝑧) ∈ 𝑌) ∧ 𝑎 ∈ 𝑋) → 𝐺 ∈ Grp)
104		simpr 487	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((((𝐺 ∈ TopGrp ∧ 𝑌 ∈ (NrmSGrp‘𝐺) ∧ 𝑆 ∈ 𝐽) ∧ 𝑦 ∈ 𝑋) ∧ 𝑧 ∈ 𝑆) ∧ (((invg‘𝐺)‘𝑦)(+g‘𝐺)𝑧) ∈ 𝑌) ∧ 𝑎 ∈ 𝑋) → 𝑎 ∈ 𝑋)
105	61	adantr 483	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((((𝐺 ∈ TopGrp ∧ 𝑌 ∈ (NrmSGrp‘𝐺) ∧ 𝑆 ∈ 𝐽) ∧ 𝑦 ∈ 𝑋) ∧ 𝑧 ∈ 𝑆) ∧ (((invg‘𝐺)‘𝑦)(+g‘𝐺)𝑧) ∈ 𝑌) ∧ 𝑎 ∈ 𝑋) → (((invg‘𝐺)‘𝑦)(+g‘𝐺)𝑧) ∈ 𝑋)
106	4, 50	grpcl 18103	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝐺 ∈ Grp ∧ 𝑎 ∈ 𝑋 ∧ (((invg‘𝐺)‘𝑦)(+g‘𝐺)𝑧) ∈ 𝑋) → (𝑎(+g‘𝐺)(((invg‘𝐺)‘𝑦)(+g‘𝐺)𝑧)) ∈ 𝑋)
107	103, 104, 105, 106	syl3anc 1365	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((((𝐺 ∈ TopGrp ∧ 𝑌 ∈ (NrmSGrp‘𝐺) ∧ 𝑆 ∈ 𝐽) ∧ 𝑦 ∈ 𝑋) ∧ 𝑧 ∈ 𝑆) ∧ (((invg‘𝐺)‘𝑦)(+g‘𝐺)𝑧) ∈ 𝑌) ∧ 𝑎 ∈ 𝑋) → (𝑎(+g‘𝐺)(((invg‘𝐺)‘𝑦)(+g‘𝐺)𝑧)) ∈ 𝑋)
108		eceq1 8319	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑥 = (𝑎(+g‘𝐺)(((invg‘𝐺)‘𝑦)(+g‘𝐺)𝑧)) → [𝑥](𝐺 ~QG 𝑌) = [(𝑎(+g‘𝐺)(((invg‘𝐺)‘𝑦)(+g‘𝐺)𝑧))](𝐺 ~QG 𝑌))
109	108, 6, 97	fvmpt3i 6766	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑎(+g‘𝐺)(((invg‘𝐺)‘𝑦)(+g‘𝐺)𝑧)) ∈ 𝑋 → (𝐹‘(𝑎(+g‘𝐺)(((invg‘𝐺)‘𝑦)(+g‘𝐺)𝑧))) = [(𝑎(+g‘𝐺)(((invg‘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) ∧ (((invg‘𝐺)‘𝑦)(+g‘𝐺)𝑧) ∈ 𝑌) ∧ 𝑎 ∈ 𝑋) → (𝐹‘(𝑎(+g‘𝐺)(((invg‘𝐺)‘𝑦)(+g‘𝐺)𝑧))) = [(𝑎(+g‘𝐺)(((invg‘𝐺)‘𝑦)(+g‘𝐺)𝑧))](𝐺 ~QG 𝑌))
111	43	ad2antrr 724	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((((𝐺 ∈ TopGrp ∧ 𝑌 ∈ (NrmSGrp‘𝐺) ∧ 𝑆 ∈ 𝐽) ∧ 𝑦 ∈ 𝑋) ∧ 𝑧 ∈ 𝑆) ∧ (((invg‘𝐺)‘𝑦)(+g‘𝐺)𝑧) ∈ 𝑌) ∧ 𝑎 ∈ 𝑋) → (𝐺 ~QG 𝑌) Er 𝑋)
112	4, 50, 76, 49	grplinv 18144	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝐺 ∈ Grp ∧ 𝑎 ∈ 𝑋) → (((invg‘𝐺)‘𝑎)(+g‘𝐺)𝑎) = (0g‘𝐺))
113	103, 104, 112	syl2anc 586	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((((((𝐺 ∈ TopGrp ∧ 𝑌 ∈ (NrmSGrp‘𝐺) ∧ 𝑆 ∈ 𝐽) ∧ 𝑦 ∈ 𝑋) ∧ 𝑧 ∈ 𝑆) ∧ (((invg‘𝐺)‘𝑦)(+g‘𝐺)𝑧) ∈ 𝑌) ∧ 𝑎 ∈ 𝑋) → (((invg‘𝐺)‘𝑎)(+g‘𝐺)𝑎) = (0g‘𝐺))
114	113	oveq1d 7163	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((((((𝐺 ∈ TopGrp ∧ 𝑌 ∈ (NrmSGrp‘𝐺) ∧ 𝑆 ∈ 𝐽) ∧ 𝑦 ∈ 𝑋) ∧ 𝑧 ∈ 𝑆) ∧ (((invg‘𝐺)‘𝑦)(+g‘𝐺)𝑧) ∈ 𝑌) ∧ 𝑎 ∈ 𝑋) → ((((invg‘𝐺)‘𝑎)(+g‘𝐺)𝑎)(+g‘𝐺)(((invg‘𝐺)‘𝑦)(+g‘𝐺)𝑧)) = ((0g‘𝐺)(+g‘𝐺)(((invg‘𝐺)‘𝑦)(+g‘𝐺)𝑧)))
115	4, 49	grpinvcl 18143	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝐺 ∈ Grp ∧ 𝑎 ∈ 𝑋) → ((invg‘𝐺)‘𝑎) ∈ 𝑋)
116	103, 104, 115	syl2anc 586	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((((((𝐺 ∈ TopGrp ∧ 𝑌 ∈ (NrmSGrp‘𝐺) ∧ 𝑆 ∈ 𝐽) ∧ 𝑦 ∈ 𝑋) ∧ 𝑧 ∈ 𝑆) ∧ (((invg‘𝐺)‘𝑦)(+g‘𝐺)𝑧) ∈ 𝑌) ∧ 𝑎 ∈ 𝑋) → ((invg‘𝐺)‘𝑎) ∈ 𝑋)
117	4, 50	grpass 18104	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝐺 ∈ Grp ∧ (((invg‘𝐺)‘𝑎) ∈ 𝑋 ∧ 𝑎 ∈ 𝑋 ∧ (((invg‘𝐺)‘𝑦)(+g‘𝐺)𝑧) ∈ 𝑋)) → ((((invg‘𝐺)‘𝑎)(+g‘𝐺)𝑎)(+g‘𝐺)(((invg‘𝐺)‘𝑦)(+g‘𝐺)𝑧)) = (((invg‘𝐺)‘𝑎)(+g‘𝐺)(𝑎(+g‘𝐺)(((invg‘𝐺)‘𝑦)(+g‘𝐺)𝑧))))
118	103, 116, 104, 105, 117	syl13anc 1366	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((((((𝐺 ∈ TopGrp ∧ 𝑌 ∈ (NrmSGrp‘𝐺) ∧ 𝑆 ∈ 𝐽) ∧ 𝑦 ∈ 𝑋) ∧ 𝑧 ∈ 𝑆) ∧ (((invg‘𝐺)‘𝑦)(+g‘𝐺)𝑧) ∈ 𝑌) ∧ 𝑎 ∈ 𝑋) → ((((invg‘𝐺)‘𝑎)(+g‘𝐺)𝑎)(+g‘𝐺)(((invg‘𝐺)‘𝑦)(+g‘𝐺)𝑧)) = (((invg‘𝐺)‘𝑎)(+g‘𝐺)(𝑎(+g‘𝐺)(((invg‘𝐺)‘𝑦)(+g‘𝐺)𝑧))))
119	4, 50, 76	grplid 18125	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝐺 ∈ Grp ∧ (((invg‘𝐺)‘𝑦)(+g‘𝐺)𝑧) ∈ 𝑋) → ((0g‘𝐺)(+g‘𝐺)(((invg‘𝐺)‘𝑦)(+g‘𝐺)𝑧)) = (((invg‘𝐺)‘𝑦)(+g‘𝐺)𝑧))
120	103, 105, 119	syl2anc 586	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((((((𝐺 ∈ TopGrp ∧ 𝑌 ∈ (NrmSGrp‘𝐺) ∧ 𝑆 ∈ 𝐽) ∧ 𝑦 ∈ 𝑋) ∧ 𝑧 ∈ 𝑆) ∧ (((invg‘𝐺)‘𝑦)(+g‘𝐺)𝑧) ∈ 𝑌) ∧ 𝑎 ∈ 𝑋) → ((0g‘𝐺)(+g‘𝐺)(((invg‘𝐺)‘𝑦)(+g‘𝐺)𝑧)) = (((invg‘𝐺)‘𝑦)(+g‘𝐺)𝑧))
121	114, 118, 120	3eqtr3d 2862	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((((((𝐺 ∈ TopGrp ∧ 𝑌 ∈ (NrmSGrp‘𝐺) ∧ 𝑆 ∈ 𝐽) ∧ 𝑦 ∈ 𝑋) ∧ 𝑧 ∈ 𝑆) ∧ (((invg‘𝐺)‘𝑦)(+g‘𝐺)𝑧) ∈ 𝑌) ∧ 𝑎 ∈ 𝑋) → (((invg‘𝐺)‘𝑎)(+g‘𝐺)(𝑎(+g‘𝐺)(((invg‘𝐺)‘𝑦)(+g‘𝐺)𝑧))) = (((invg‘𝐺)‘𝑦)(+g‘𝐺)𝑧))
122		simplr 767	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((((((𝐺 ∈ TopGrp ∧ 𝑌 ∈ (NrmSGrp‘𝐺) ∧ 𝑆 ∈ 𝐽) ∧ 𝑦 ∈ 𝑋) ∧ 𝑧 ∈ 𝑆) ∧ (((invg‘𝐺)‘𝑦)(+g‘𝐺)𝑧) ∈ 𝑌) ∧ 𝑎 ∈ 𝑋) → (((invg‘𝐺)‘𝑦)(+g‘𝐺)𝑧) ∈ 𝑌)
123	121, 122	eqeltrd 2911	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((((𝐺 ∈ TopGrp ∧ 𝑌 ∈ (NrmSGrp‘𝐺) ∧ 𝑆 ∈ 𝐽) ∧ 𝑦 ∈ 𝑋) ∧ 𝑧 ∈ 𝑆) ∧ (((invg‘𝐺)‘𝑦)(+g‘𝐺)𝑧) ∈ 𝑌) ∧ 𝑎 ∈ 𝑋) → (((invg‘𝐺)‘𝑎)(+g‘𝐺)(𝑎(+g‘𝐺)(((invg‘𝐺)‘𝑦)(+g‘𝐺)𝑧))) ∈ 𝑌)
124	48	ad2antrr 724	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((((((𝐺 ∈ TopGrp ∧ 𝑌 ∈ (NrmSGrp‘𝐺) ∧ 𝑆 ∈ 𝐽) ∧ 𝑦 ∈ 𝑋) ∧ 𝑧 ∈ 𝑆) ∧ (((invg‘𝐺)‘𝑦)(+g‘𝐺)𝑧) ∈ 𝑌) ∧ 𝑎 ∈ 𝑋) → 𝑌 ⊆ 𝑋)
125	4, 49, 50, 41	eqgval 18321	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝐺 ∈ Grp ∧ 𝑌 ⊆ 𝑋) → (𝑎(𝐺 ~QG 𝑌)(𝑎(+g‘𝐺)(((invg‘𝐺)‘𝑦)(+g‘𝐺)𝑧)) ↔ (𝑎 ∈ 𝑋 ∧ (𝑎(+g‘𝐺)(((invg‘𝐺)‘𝑦)(+g‘𝐺)𝑧)) ∈ 𝑋 ∧ (((invg‘𝐺)‘𝑎)(+g‘𝐺)(𝑎(+g‘𝐺)(((invg‘𝐺)‘𝑦)(+g‘𝐺)𝑧))) ∈ 𝑌)))
126	103, 124, 125	syl2anc 586	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((((𝐺 ∈ TopGrp ∧ 𝑌 ∈ (NrmSGrp‘𝐺) ∧ 𝑆 ∈ 𝐽) ∧ 𝑦 ∈ 𝑋) ∧ 𝑧 ∈ 𝑆) ∧ (((invg‘𝐺)‘𝑦)(+g‘𝐺)𝑧) ∈ 𝑌) ∧ 𝑎 ∈ 𝑋) → (𝑎(𝐺 ~QG 𝑌)(𝑎(+g‘𝐺)(((invg‘𝐺)‘𝑦)(+g‘𝐺)𝑧)) ↔ (𝑎 ∈ 𝑋 ∧ (𝑎(+g‘𝐺)(((invg‘𝐺)‘𝑦)(+g‘𝐺)𝑧)) ∈ 𝑋 ∧ (((invg‘𝐺)‘𝑎)(+g‘𝐺)(𝑎(+g‘𝐺)(((invg‘𝐺)‘𝑦)(+g‘𝐺)𝑧))) ∈ 𝑌)))
127	104, 107, 123, 126	mpbir3and 1336	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((((𝐺 ∈ TopGrp ∧ 𝑌 ∈ (NrmSGrp‘𝐺) ∧ 𝑆 ∈ 𝐽) ∧ 𝑦 ∈ 𝑋) ∧ 𝑧 ∈ 𝑆) ∧ (((invg‘𝐺)‘𝑦)(+g‘𝐺)𝑧) ∈ 𝑌) ∧ 𝑎 ∈ 𝑋) → 𝑎(𝐺 ~QG 𝑌)(𝑎(+g‘𝐺)(((invg‘𝐺)‘𝑦)(+g‘𝐺)𝑧)))
128	111, 127	erthi 8332	. . . . . . . . . . . . . . . . . 18 ⊢ ((((((𝐺 ∈ TopGrp ∧ 𝑌 ∈ (NrmSGrp‘𝐺) ∧ 𝑆 ∈ 𝐽) ∧ 𝑦 ∈ 𝑋) ∧ 𝑧 ∈ 𝑆) ∧ (((invg‘𝐺)‘𝑦)(+g‘𝐺)𝑧) ∈ 𝑌) ∧ 𝑎 ∈ 𝑋) → [𝑎](𝐺 ~QG 𝑌) = [(𝑎(+g‘𝐺)(((invg‘𝐺)‘𝑦)(+g‘𝐺)𝑧))](𝐺 ~QG 𝑌))
129	110, 128	eqtr4d 2857	. . . . . . . . . . . . . . . . 17 ⊢ ((((((𝐺 ∈ TopGrp ∧ 𝑌 ∈ (NrmSGrp‘𝐺) ∧ 𝑆 ∈ 𝐽) ∧ 𝑦 ∈ 𝑋) ∧ 𝑧 ∈ 𝑆) ∧ (((invg‘𝐺)‘𝑦)(+g‘𝐺)𝑧) ∈ 𝑌) ∧ 𝑎 ∈ 𝑋) → (𝐹‘(𝑎(+g‘𝐺)(((invg‘𝐺)‘𝑦)(+g‘𝐺)𝑧))) = [𝑎](𝐺 ~QG 𝑌))
130	129	eleq1d 2895	. . . . . . . . . . . . . . . 16 ⊢ ((((((𝐺 ∈ TopGrp ∧ 𝑌 ∈ (NrmSGrp‘𝐺) ∧ 𝑆 ∈ 𝐽) ∧ 𝑦 ∈ 𝑋) ∧ 𝑧 ∈ 𝑆) ∧ (((invg‘𝐺)‘𝑦)(+g‘𝐺)𝑧) ∈ 𝑌) ∧ 𝑎 ∈ 𝑋) → ((𝐹‘(𝑎(+g‘𝐺)(((invg‘𝐺)‘𝑦)(+g‘𝐺)𝑧))) ∈ (𝐹 “ 𝑆) ↔ [𝑎](𝐺 ~QG 𝑌) ∈ (𝐹 “ 𝑆)))
131	102, 130	sylibd 241	. . . . . . . . . . . . . . 15 ⊢ ((((((𝐺 ∈ TopGrp ∧ 𝑌 ∈ (NrmSGrp‘𝐺) ∧ 𝑆 ∈ 𝐽) ∧ 𝑦 ∈ 𝑋) ∧ 𝑧 ∈ 𝑆) ∧ (((invg‘𝐺)‘𝑦)(+g‘𝐺)𝑧) ∈ 𝑌) ∧ 𝑎 ∈ 𝑋) → ((𝑎(+g‘𝐺)(((invg‘𝐺)‘𝑦)(+g‘𝐺)𝑧)) ∈ 𝑆 → [𝑎](𝐺 ~QG 𝑌) ∈ (𝐹 “ 𝑆)))
132	131	ss2rabdv 4050	. . . . . . . . . . . . . 14 ⊢ (((((𝐺 ∈ TopGrp ∧ 𝑌 ∈ (NrmSGrp‘𝐺) ∧ 𝑆 ∈ 𝐽) ∧ 𝑦 ∈ 𝑋) ∧ 𝑧 ∈ 𝑆) ∧ (((invg‘𝐺)‘𝑦)(+g‘𝐺)𝑧) ∈ 𝑌) → {𝑎 ∈ 𝑋 ∣ (𝑎(+g‘𝐺)(((invg‘𝐺)‘𝑦)(+g‘𝐺)𝑧)) ∈ 𝑆} ⊆ {𝑎 ∈ 𝑋 ∣ [𝑎](𝐺 ~QG 𝑌) ∈ (𝐹 “ 𝑆)})
133		eceq1 8319	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 = 𝑎 → [𝑥](𝐺 ~QG 𝑌) = [𝑎](𝐺 ~QG 𝑌))
134	133	cbvmptv 5160	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 ∈ 𝑋 ↦ [𝑥](𝐺 ~QG 𝑌)) = (𝑎 ∈ 𝑋 ↦ [𝑎](𝐺 ~QG 𝑌))
135	6, 134	eqtri 2842	. . . . . . . . . . . . . . 15 ⊢ 𝐹 = (𝑎 ∈ 𝑋 ↦ [𝑎](𝐺 ~QG 𝑌))
136	135	mptpreima 6085	. . . . . . . . . . . . . 14 ⊢ (◡𝐹 “ (𝐹 “ 𝑆)) = {𝑎 ∈ 𝑋 ∣ [𝑎](𝐺 ~QG 𝑌) ∈ (𝐹 “ 𝑆)}
137	132, 93, 136	3sstr4g 4010	. . . . . . . . . . . . 13 ⊢ (((((𝐺 ∈ TopGrp ∧ 𝑌 ∈ (NrmSGrp‘𝐺) ∧ 𝑆 ∈ 𝐽) ∧ 𝑦 ∈ 𝑋) ∧ 𝑧 ∈ 𝑆) ∧ (((invg‘𝐺)‘𝑦)(+g‘𝐺)𝑧) ∈ 𝑌) → (◡(𝑎 ∈ 𝑋 ↦ (𝑎(+g‘𝐺)(((invg‘𝐺)‘𝑦)(+g‘𝐺)𝑧))) “ 𝑆) ⊆ (◡𝐹 “ (𝐹 “ 𝑆)))
138		eleq2 2899	. . . . . . . . . . . . . . 15 ⊢ (𝑢 = (◡(𝑎 ∈ 𝑋 ↦ (𝑎(+g‘𝐺)(((invg‘𝐺)‘𝑦)(+g‘𝐺)𝑧))) “ 𝑆) → (𝑦 ∈ 𝑢 ↔ 𝑦 ∈ (◡(𝑎 ∈ 𝑋 ↦ (𝑎(+g‘𝐺)(((invg‘𝐺)‘𝑦)(+g‘𝐺)𝑧))) “ 𝑆)))
139		sseq1 3990	. . . . . . . . . . . . . . 15 ⊢ (𝑢 = (◡(𝑎 ∈ 𝑋 ↦ (𝑎(+g‘𝐺)(((invg‘𝐺)‘𝑦)(+g‘𝐺)𝑧))) “ 𝑆) → (𝑢 ⊆ (◡𝐹 “ (𝐹 “ 𝑆)) ↔ (◡(𝑎 ∈ 𝑋 ↦ (𝑎(+g‘𝐺)(((invg‘𝐺)‘𝑦)(+g‘𝐺)𝑧))) “ 𝑆) ⊆ (◡𝐹 “ (𝐹 “ 𝑆))))
140	138, 139	anbi12d 632	. . . . . . . . . . . . . 14 ⊢ (𝑢 = (◡(𝑎 ∈ 𝑋 ↦ (𝑎(+g‘𝐺)(((invg‘𝐺)‘𝑦)(+g‘𝐺)𝑧))) “ 𝑆) → ((𝑦 ∈ 𝑢 ∧ 𝑢 ⊆ (◡𝐹 “ (𝐹 “ 𝑆))) ↔ (𝑦 ∈ (◡(𝑎 ∈ 𝑋 ↦ (𝑎(+g‘𝐺)(((invg‘𝐺)‘𝑦)(+g‘𝐺)𝑧))) “ 𝑆) ∧ (◡(𝑎 ∈ 𝑋 ↦ (𝑎(+g‘𝐺)(((invg‘𝐺)‘𝑦)(+g‘𝐺)𝑧))) “ 𝑆) ⊆ (◡𝐹 “ (𝐹 “ 𝑆)))))
141	140	rspcev 3621	. . . . . . . . . . . . 13 ⊢ (((◡(𝑎 ∈ 𝑋 ↦ (𝑎(+g‘𝐺)(((invg‘𝐺)‘𝑦)(+g‘𝐺)𝑧))) “ 𝑆) ∈ 𝐽 ∧ (𝑦 ∈ (◡(𝑎 ∈ 𝑋 ↦ (𝑎(+g‘𝐺)(((invg‘𝐺)‘𝑦)(+g‘𝐺)𝑧))) “ 𝑆) ∧ (◡(𝑎 ∈ 𝑋 ↦ (𝑎(+g‘𝐺)(((invg‘𝐺)‘𝑦)(+g‘𝐺)𝑧))) “ 𝑆) ⊆ (◡𝐹 “ (𝐹 “ 𝑆)))) → ∃𝑢 ∈ 𝐽 (𝑦 ∈ 𝑢 ∧ 𝑢 ⊆ (◡𝐹 “ (𝐹 “ 𝑆))))
142	72, 95, 137, 141	syl12anc 834	. . . . . . . . . . . 12 ⊢ (((((𝐺 ∈ TopGrp ∧ 𝑌 ∈ (NrmSGrp‘𝐺) ∧ 𝑆 ∈ 𝐽) ∧ 𝑦 ∈ 𝑋) ∧ 𝑧 ∈ 𝑆) ∧ (((invg‘𝐺)‘𝑦)(+g‘𝐺)𝑧) ∈ 𝑌) → ∃𝑢 ∈ 𝐽 (𝑦 ∈ 𝑢 ∧ 𝑢 ⊆ (◡𝐹 “ (𝐹 “ 𝑆))))
143	142	3ad2antr3 1184	. . . . . . . . . . 11 ⊢ (((((𝐺 ∈ TopGrp ∧ 𝑌 ∈ (NrmSGrp‘𝐺) ∧ 𝑆 ∈ 𝐽) ∧ 𝑦 ∈ 𝑋) ∧ 𝑧 ∈ 𝑆) ∧ (𝑦 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋 ∧ (((invg‘𝐺)‘𝑦)(+g‘𝐺)𝑧) ∈ 𝑌)) → ∃𝑢 ∈ 𝐽 (𝑦 ∈ 𝑢 ∧ 𝑢 ⊆ (◡𝐹 “ (𝐹 “ 𝑆))))
144	143	ex 415	. . . . . . . . . 10 ⊢ ((((𝐺 ∈ TopGrp ∧ 𝑌 ∈ (NrmSGrp‘𝐺) ∧ 𝑆 ∈ 𝐽) ∧ 𝑦 ∈ 𝑋) ∧ 𝑧 ∈ 𝑆) → ((𝑦 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋 ∧ (((invg‘𝐺)‘𝑦)(+g‘𝐺)𝑧) ∈ 𝑌) → ∃𝑢 ∈ 𝐽 (𝑦 ∈ 𝑢 ∧ 𝑢 ⊆ (◡𝐹 “ (𝐹 “ 𝑆)))))
145	53, 144	sylbid 242	. . . . . . . . 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 3282	. . . . . . . 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 456	. . . . . 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	syl5bi 244	. . . . 5 ⊢ ((𝐺 ∈ TopGrp ∧ 𝑌 ∈ (NrmSGrp‘𝐺) ∧ 𝑆 ∈ 𝐽) → (𝑦 ∈ (◡𝐹 “ (𝐹 “ 𝑆)) → ∃𝑢 ∈ 𝐽 (𝑦 ∈ 𝑢 ∧ 𝑢 ⊆ (◡𝐹 “ (𝐹 “ 𝑆)))))
150	149	ralrimiv 3179	. . . 4 ⊢ ((𝐺 ∈ TopGrp ∧ 𝑌 ∈ (NrmSGrp‘𝐺) ∧ 𝑆 ∈ 𝐽) → ∀𝑦 ∈ (◡𝐹 “ (𝐹 “ 𝑆))∃𝑢 ∈ 𝐽 (𝑦 ∈ 𝑢 ∧ 𝑢 ⊆ (◡𝐹 “ (𝐹 “ 𝑆))))
151		topontop 21513	. . . . 5 ⊢ (𝐽 ∈ (TopOn‘𝑋) → 𝐽 ∈ Top)
152		eltop2 21575	. . . . 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 259	. . 3 ⊢ ((𝐺 ∈ TopGrp ∧ 𝑌 ∈ (NrmSGrp‘𝐺) ∧ 𝑆 ∈ 𝐽) → (◡𝐹 “ (𝐹 “ 𝑆)) ∈ 𝐽)
155		elqtop3 22303	. . . 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 586	. . 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 711	. 2 ⊢ ((𝐺 ∈ TopGrp ∧ 𝑌 ∈ (NrmSGrp‘𝐺) ∧ 𝑆 ∈ 𝐽) → (𝐹 “ 𝑆) ∈ (𝐽 qTop 𝐹))
158	3, 5, 6, 8, 9	qusval 16807	. . 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 22320	. 2 ⊢ ((𝐺 ∈ TopGrp ∧ 𝑌 ∈ (NrmSGrp‘𝐺) ∧ 𝑆 ∈ 𝐽) → 𝐾 = (𝐽 qTop 𝐹))
161	157, 160	eleqtrrd 2914	1 ⊢ ((𝐺 ∈ TopGrp ∧ 𝑌 ∈ (NrmSGrp‘𝐺) ∧ 𝑆 ∈ 𝐽) → (𝐹 “ 𝑆) ∈ 𝐾)

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 398   ∧ w3a 1081   = wceq 1530   ∈ wcel 2107  ∀wral 3136  ∃wrex 3137  {crab 3140  Vcvv 3493   ⊆ wss 3934   class class class wbr 5057   ↦ cmpt 5137  ^◡ccnv 5547  ran crn 5549   “ cima 5551  Fun wfun 6342   Fn wfn 6343  –onto→wfo 6346  ‘cfv 6348  (class class class)co 7148   Er wer 8278  [cec 8279   / cqs 8280  Basecbs 16475  +_gcplusg 16557  TopOpenctopn 16687  0_gc0g 16705   qTop cqtop 16768   /_s cqus 16770  Grpcgrp 18095  inv_gcminusg 18096  SubGrpcsubg 18265  NrmSGrpcnsg 18266   ~_QG cqg 18267  opp_gcoppg 18465  Topctop 21493  TopOnctopon 21510   Cn ccn 21824  Homeochmeo 22353  TopGrpctgp 22671

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2791  ax-rep 5181  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7453  ax-cnex 10585  ax-resscn 10586  ax-1cn 10587  ax-icn 10588  ax-addcl 10589  ax-addrcl 10590  ax-mulcl 10591  ax-mulrcl 10592  ax-mulcom 10593  ax-addass 10594  ax-mulass 10595  ax-distr 10596  ax-i2m1 10597  ax-1ne0 10598  ax-1rid 10599  ax-rnegex 10600  ax-rrecex 10601  ax-cnre 10602  ax-pre-lttri 10603  ax-pre-lttrn 10604  ax-pre-ltadd 10605  ax-pre-mulgt0 10606

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1082  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2616  df-eu 2648  df-clab 2798  df-cleq 2812  df-clel 2891  df-nfc 2961  df-ne 3015  df-nel 3122  df-ral 3141  df-rex 3142  df-reu 3143  df-rmo 3144  df-rab 3145  df-v 3495  df-sbc 3771  df-csb 3882  df-dif 3937  df-un 3939  df-in 3941  df-ss 3950  df-pss 3952  df-nul 4290  df-if 4466  df-pw 4539  df-sn 4560  df-pr 4562  df-tp 4564  df-op 4566  df-uni 4831  df-int 4868  df-iun 4912  df-br 5058  df-opab 5120  df-mpt 5138  df-tr 5164  df-id 5453  df-eprel 5458  df-po 5467  df-so 5468  df-fr 5507  df-we 5509  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-pred 6141  df-ord 6187  df-on 6188  df-lim 6189  df-suc 6190  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-f1 6353  df-fo 6354  df-f1o 6355  df-fv 6356  df-riota 7106  df-ov 7151  df-oprab 7152  df-mpo 7153  df-om 7573  df-1st 7681  df-2nd 7682  df-tpos 7884  df-wrecs 7939  df-recs 8000  df-rdg 8038  df-1o 8094  df-oadd 8098  df-er 8281  df-ec 8283  df-qs 8287  df-map 8400  df-en 8502  df-dom 8503  df-sdom 8504  df-fin 8505  df-sup 8898  df-inf 8899  df-pnf 10669  df-mnf 10670  df-xr 10671  df-ltxr 10672  df-le 10673  df-sub 10864  df-neg 10865  df-nn 11631  df-2 11692  df-3 11693  df-4 11694  df-5 11695  df-6 11696  df-7 11697  df-8 11698  df-9 11699  df-n0 11890  df-z 11974  df-dec 12091  df-uz 12236  df-fz 12885  df-struct 16477  df-ndx 16478  df-slot 16479  df-base 16481  df-sets 16482  df-ress 16483  df-plusg 16570  df-mulr 16571  df-sca 16573  df-vsca 16574  df-ip 16575  df-tset 16576  df-ple 16577  df-ds 16579  df-rest 16688  df-topn 16689  df-0g 16707  df-topgen 16709  df-qtop 16772  df-imas 16773  df-qus 16774  df-plusf 17843  df-mgm 17844  df-sgrp 17893  df-mnd 17904  df-grp 18098  df-minusg 18099  df-subg 18268  df-nsg 18269  df-eqg 18270  df-oppg 18466  df-top 21494  df-topon 21511  df-topsp 21533  df-bases 21546  df-cn 21827  df-cnp 21828  df-tx 22162  df-hmeo 22355  df-tmd 22672  df-tgp 22673

This theorem is referenced by:  qustgplem  22721