Theorem qustgpopn 24234

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 6063	. . . 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 7433	. . . . . . 7 ⊢ (𝐺 ~QG 𝑌) ∈ V
8	7	a1i 11	. . . . . 6 ⊢ ((𝐺 ∈ TopGrp ∧ 𝑌 ∈ (NrmSGrp‘𝐺) ∧ 𝑆 ∈ 𝐽) → (𝐺 ~QG 𝑌) ∈ V)
9		simp1 1152	. . . . . 6 ⊢ ((𝐺 ∈ TopGrp ∧ 𝑌 ∈ (NrmSGrp‘𝐺) ∧ 𝑆 ∈ 𝐽) → 𝐺 ∈ TopGrp)
10	3, 5, 6, 8, 9	quslem 17585	. . . . 5 ⊢ ((𝐺 ∈ TopGrp ∧ 𝑌 ∈ (NrmSGrp‘𝐺) ∧ 𝑆 ∈ 𝐽) → 𝐹:𝑋–onto→(𝑋 / (𝐺 ~QG 𝑌)))
11		forn 6785	. . . . 5 ⊢ (𝐹:𝑋–onto→(𝑋 / (𝐺 ~QG 𝑌)) → ran 𝐹 = (𝑋 / (𝐺 ~QG 𝑌)))
12	10, 11	syl 18	. . . 4 ⊢ ((𝐺 ∈ TopGrp ∧ 𝑌 ∈ (NrmSGrp‘𝐺) ∧ 𝑆 ∈ 𝐽) → ran 𝐹 = (𝑋 / (𝐺 ~QG 𝑌)))
13	1, 12	sseqtrid 3981	. . 3 ⊢ ((𝐺 ∈ TopGrp ∧ 𝑌 ∈ (NrmSGrp‘𝐺) ∧ 𝑆 ∈ 𝐽) → (𝐹 “ 𝑆) ⊆ (𝑋 / (𝐺 ~QG 𝑌)))
14		eceq1 8722	. . . . . . . . . 10 ⊢ (𝑥 = 𝑦 → [𝑥](𝐺 ~QG 𝑌) = [𝑦](𝐺 ~QG 𝑌))
15	14	cbvmptv 5208	. . . . . . . . 9 ⊢ (𝑥 ∈ 𝑋 ↦ [𝑥](𝐺 ~QG 𝑌)) = (𝑦 ∈ 𝑋 ↦ [𝑦](𝐺 ~QG 𝑌))
16	6, 15	eqtri 2788	. . . . . . . 8 ⊢ 𝐹 = (𝑦 ∈ 𝑋 ↦ [𝑦](𝐺 ~QG 𝑌))
17	16	mptpreima 6228	. . . . . . 7 ⊢ (◡𝐹 “ (𝐹 “ 𝑆)) = {𝑦 ∈ 𝑋 ∣ [𝑦](𝐺 ~QG 𝑌) ∈ (𝐹 “ 𝑆)}
18	17	reqabi 3440	. . . . . 6 ⊢ (𝑦 ∈ (◡𝐹 “ (𝐹 “ 𝑆)) ↔ (𝑦 ∈ 𝑋 ∧ [𝑦](𝐺 ~QG 𝑌) ∈ (𝐹 “ 𝑆)))
19	6	funmpt2 6564	. . . . . . . . 9 ⊢ Fun 𝐹
20		fvelima 6936	. . . . . . . . 9 ⊢ ((Fun 𝐹 ∧ [𝑦](𝐺 ~QG 𝑌) ∈ (𝐹 “ 𝑆)) → ∃𝑧 ∈ 𝑆 (𝐹‘𝑧) = [𝑦](𝐺 ~QG 𝑌))
21	19, 20	mpan 702	. . . . . . . 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 24196	. . . . . . . . . . . . . . . . . 18 ⊢ (𝐺 ∈ TopGrp → 𝐽 ∈ (TopOn‘𝑋))
24	9, 23	syl 18	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐺 ∈ TopGrp ∧ 𝑌 ∈ (NrmSGrp‘𝐺) ∧ 𝑆 ∈ 𝐽) → 𝐽 ∈ (TopOn‘𝑋))
25		simp3 1154	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐺 ∈ TopGrp ∧ 𝑌 ∈ (NrmSGrp‘𝐺) ∧ 𝑆 ∈ 𝐽) → 𝑆 ∈ 𝐽)
26		toponss 23041	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ 𝐽) → 𝑆 ⊆ 𝑋)
27	24, 25, 26	syl2anc 595	. . . . . . . . . . . . . . . 16 ⊢ ((𝐺 ∈ TopGrp ∧ 𝑌 ∈ (NrmSGrp‘𝐺) ∧ 𝑆 ∈ 𝐽) → 𝑆 ⊆ 𝑋)
28	27	adantr 485	. . . . . . . . . . . . . . 15 ⊢ (((𝐺 ∈ TopGrp ∧ 𝑌 ∈ (NrmSGrp‘𝐺) ∧ 𝑆 ∈ 𝐽) ∧ 𝑦 ∈ 𝑋) → 𝑆 ⊆ 𝑋)
29	28	sselda 3939	. . . . . . . . . . . . . 14 ⊢ ((((𝐺 ∈ TopGrp ∧ 𝑌 ∈ (NrmSGrp‘𝐺) ∧ 𝑆 ∈ 𝐽) ∧ 𝑦 ∈ 𝑋) ∧ 𝑧 ∈ 𝑆) → 𝑧 ∈ 𝑋)
30		eceq1 8722	. . . . . . . . . . . . . . 15 ⊢ (𝑥 = 𝑧 → [𝑥](𝐺 ~QG 𝑌) = [𝑧](𝐺 ~QG 𝑌))
31		ecexg 8686	. . . . . . . . . . . . . . . 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 6979	. . . . . . . . . . . . . 14 ⊢ (𝑧 ∈ 𝑋 → (𝐹‘𝑧) = [𝑧](𝐺 ~QG 𝑌))
34	29, 33	syl 18	. . . . . . . . . . . . 13 ⊢ ((((𝐺 ∈ TopGrp ∧ 𝑌 ∈ (NrmSGrp‘𝐺) ∧ 𝑆 ∈ 𝐽) ∧ 𝑦 ∈ 𝑋) ∧ 𝑧 ∈ 𝑆) → (𝐹‘𝑧) = [𝑧](𝐺 ~QG 𝑌))
35	34	eqeq1d 2767	. . . . . . . . . . . 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 2772	. . . . . . . . . . . 12 ⊢ ([𝑧](𝐺 ~QG 𝑌) = [𝑦](𝐺 ~QG 𝑌) ↔ [𝑦](𝐺 ~QG 𝑌) = [𝑧](𝐺 ~QG 𝑌))
37	35, 36	bitrdi 290	. . . . . . . . . . 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 19212	. . . . . . . . . . . . . . 15 ⊢ (𝑌 ∈ (NrmSGrp‘𝐺) → 𝑌 ∈ (SubGrp‘𝐺))
39	38	3ad2ant2 1150	. . . . . . . . . . . . . 14 ⊢ ((𝐺 ∈ TopGrp ∧ 𝑌 ∈ (NrmSGrp‘𝐺) ∧ 𝑆 ∈ 𝐽) → 𝑌 ∈ (SubGrp‘𝐺))
40	39	ad2antrr 738	. . . . . . . . . . . . 13 ⊢ ((((𝐺 ∈ TopGrp ∧ 𝑌 ∈ (NrmSGrp‘𝐺) ∧ 𝑆 ∈ 𝐽) ∧ 𝑦 ∈ 𝑋) ∧ 𝑧 ∈ 𝑆) → 𝑌 ∈ (SubGrp‘𝐺))
41		eqid 2765	. . . . . . . . . . . . . 14 ⊢ (𝐺 ~QG 𝑌) = (𝐺 ~QG 𝑌)
42	4, 41	eqger 19234	. . . . . . . . . . . . 13 ⊢ (𝑌 ∈ (SubGrp‘𝐺) → (𝐺 ~QG 𝑌) Er 𝑋)
43	40, 42	syl 18	. . . . . . . . . . . 12 ⊢ ((((𝐺 ∈ TopGrp ∧ 𝑌 ∈ (NrmSGrp‘𝐺) ∧ 𝑆 ∈ 𝐽) ∧ 𝑦 ∈ 𝑋) ∧ 𝑧 ∈ 𝑆) → (𝐺 ~QG 𝑌) Er 𝑋)
44		simplr 780	. . . . . . . . . . . 12 ⊢ ((((𝐺 ∈ TopGrp ∧ 𝑌 ∈ (NrmSGrp‘𝐺) ∧ 𝑆 ∈ 𝐽) ∧ 𝑦 ∈ 𝑋) ∧ 𝑧 ∈ 𝑆) → 𝑦 ∈ 𝑋)
45	43, 44	erth 8737	. . . . . . . . . . 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 738	. . . . . . . . . . . 12 ⊢ ((((𝐺 ∈ TopGrp ∧ 𝑌 ∈ (NrmSGrp‘𝐺) ∧ 𝑆 ∈ 𝐽) ∧ 𝑦 ∈ 𝑋) ∧ 𝑧 ∈ 𝑆) → 𝐺 ∈ TopGrp)
47	4	subgss 19181	. . . . . . . . . . . . 13 ⊢ (𝑌 ∈ (SubGrp‘𝐺) → 𝑌 ⊆ 𝑋)
48	40, 47	syl 18	. . . . . . . . . . . 12 ⊢ ((((𝐺 ∈ TopGrp ∧ 𝑌 ∈ (NrmSGrp‘𝐺) ∧ 𝑆 ∈ 𝐽) ∧ 𝑦 ∈ 𝑋) ∧ 𝑧 ∈ 𝑆) → 𝑌 ⊆ 𝑋)
49		eqid 2765	. . . . . . . . . . . . 13 ⊢ (invg‘𝐺) = (invg‘𝐺)
50		eqid 2765	. . . . . . . . . . . . 13 ⊢ (+g‘𝐺) = (+g‘𝐺)
51	4, 49, 50, 41	eqgval 19233	. . . . . . . . . . . 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 595	. . . . . . . . . . 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 310	. . . . . . . . . 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 2765	. . . . . . . . . . . . . . . . . 18 ⊢ (oppg‘𝐺) = (oppg‘𝐺)
55		eqid 2765	. . . . . . . . . . . . . . . . . 18 ⊢ (+g‘(oppg‘𝐺)) = (+g‘(oppg‘𝐺))
56	50, 54, 55	oppgplus 19407	. . . . . . . . . . . . . . . . 17 ⊢ ((((invg‘𝐺)‘𝑦)(+g‘𝐺)𝑧)(+g‘(oppg‘𝐺))𝑎) = (𝑎(+g‘𝐺)(((invg‘𝐺)‘𝑦)(+g‘𝐺)𝑧))
57	56	mpteq2i 5200	. . . . . . . . . . . . . . . 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 485	. . . . . . . . . . . . . . . . . 18 ⊢ (((((𝐺 ∈ TopGrp ∧ 𝑌 ∈ (NrmSGrp‘𝐺) ∧ 𝑆 ∈ 𝐽) ∧ 𝑦 ∈ 𝑋) ∧ 𝑧 ∈ 𝑆) ∧ (((invg‘𝐺)‘𝑦)(+g‘𝐺)𝑧) ∈ 𝑌) → 𝐺 ∈ TopGrp)
59	54	oppgtgp 24212	. . . . . . . . . . . . . . . . . 18 ⊢ (𝐺 ∈ TopGrp → (oppg‘𝐺) ∈ TopGrp)
60	58, 59	syl 18	. . . . . . . . . . . . . . . . 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 3939	. . . . . . . . . . . . . . . . 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 2765	. . . . . . . . . . . . . . . . . 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 19409	. . . . . . . . . . . . . . . . . 18 ⊢ 𝑋 = (Base‘(oppg‘𝐺))
64	54, 22	oppgtopn 19411	. . . . . . . . . . . . . . . . . 18 ⊢ 𝐽 = (TopOpen‘(oppg‘𝐺))
65	62, 63, 55, 64	tgplacthmeo 24217	. . . . . . . . . . . . . . . . 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 595	. . . . . . . . . . . . . . . 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 2870	. . . . . . . . . . . . . . 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 23874	. . . . . . . . . . . . . . 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 18	. . . . . . . . . . . . . 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 742	. . . . . . . . . . . . . 14 ⊢ (((((𝐺 ∈ TopGrp ∧ 𝑌 ∈ (NrmSGrp‘𝐺) ∧ 𝑆 ∈ 𝐽) ∧ 𝑦 ∈ 𝑋) ∧ 𝑧 ∈ 𝑆) ∧ (((invg‘𝐺)‘𝑦)(+g‘𝐺)𝑧) ∈ 𝑌) → 𝑆 ∈ 𝐽)
71		cnima 23379	. . . . . . . . . . . . . 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 595	. . . . . . . . . . . . 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 485	. . . . . . . . . . . . . 14 ⊢ (((((𝐺 ∈ TopGrp ∧ 𝑌 ∈ (NrmSGrp‘𝐺) ∧ 𝑆 ∈ 𝐽) ∧ 𝑦 ∈ 𝑋) ∧ 𝑧 ∈ 𝑆) ∧ (((invg‘𝐺)‘𝑦)(+g‘𝐺)𝑧) ∈ 𝑌) → 𝑦 ∈ 𝑋)
74		tgpgrp 24192	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝐺 ∈ TopGrp → 𝐺 ∈ Grp)
75	58, 74	syl 18	. . . . . . . . . . . . . . . . . 18 ⊢ (((((𝐺 ∈ TopGrp ∧ 𝑌 ∈ (NrmSGrp‘𝐺) ∧ 𝑆 ∈ 𝐽) ∧ 𝑦 ∈ 𝑋) ∧ 𝑧 ∈ 𝑆) ∧ (((invg‘𝐺)‘𝑦)(+g‘𝐺)𝑧) ∈ 𝑌) → 𝐺 ∈ Grp)
76		eqid 2765	. . . . . . . . . . . . . . . . . . 19 ⊢ (0g‘𝐺) = (0g‘𝐺)
77	4, 50, 76, 49	grprinv 19045	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐺 ∈ Grp ∧ 𝑦 ∈ 𝑋) → (𝑦(+g‘𝐺)((invg‘𝐺)‘𝑦)) = (0g‘𝐺))
78	75, 73, 77	syl2anc 595	. . . . . . . . . . . . . . . . 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 7415	. . . . . . . . . . . . . . . 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 19042	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐺 ∈ Grp ∧ 𝑦 ∈ 𝑋) → ((invg‘𝐺)‘𝑦) ∈ 𝑋)
81	75, 73, 80	syl2anc 595	. . . . . . . . . . . . . . . . 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 485	. . . . . . . . . . . . . . . . 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 18997	. . . . . . . . . . . . . . . . 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 1395	. . . . . . . . . . . . . . . 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 19022	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐺 ∈ Grp ∧ 𝑧 ∈ 𝑋) → ((0g‘𝐺)(+g‘𝐺)𝑧) = 𝑧)
86	75, 82, 85	syl2anc 595	. . . . . . . . . . . . . . . 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 2808	. . . . . . . . . . . . . . 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 780	. . . . . . . . . . . . . . 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 2865	. . . . . . . . . . . . . 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 7407	. . . . . . . . . . . . . . . 16 ⊢ (𝑎 = 𝑦 → (𝑎(+g‘𝐺)(((invg‘𝐺)‘𝑦)(+g‘𝐺)𝑧)) = (𝑦(+g‘𝐺)(((invg‘𝐺)‘𝑦)(+g‘𝐺)𝑧)))
91	90	eleq1d 2850	. . . . . . . . . . . . . . 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 2765	. . . . . . . . . . . . . . . 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 6228	. . . . . . . . . . . . . . 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 3657	. . . . . . . . . . . . . 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 594	. . . . . . . . . . . . 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 8686	. . . . . . . . . . . . . . . . . . 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 6668	. . . . . . . . . . . . . . . . 17 ⊢ 𝐹 Fn 𝑋
99	28	ad3antrrr 742	. . . . . . . . . . . . . . . . 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 7221	. . . . . . . . . . . . . . . . . 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 1137	. . . . . . . . . . . . . . . . 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 598	. . . . . . . . . . . . . . . 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 485	. . . . . . . . . . . . . . . . . . . 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 489	. . . . . . . . . . . . . . . . . . . 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 485	. . . . . . . . . . . . . . . . . . . 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 18996	. . . . . . . . . . . . . . . . . . . 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 1394	. . . . . . . . . . . . . . . . . . 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 8722	. . . . . . . . . . . . . . . . . . . 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 6985	. . . . . . . . . . . . . . . . . . 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 18	. . . . . . . . . . . . . . . . . 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 738	. . . . . . . . . . . . . . . . . . 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 19044	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝐺 ∈ Grp ∧ 𝑎 ∈ 𝑋) → (((invg‘𝐺)‘𝑎)(+g‘𝐺)𝑎) = (0g‘𝐺))
113	103, 104, 112	syl2anc 595	. . . . . . . . . . . . . . . . . . . . . . 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 7415	. . . . . . . . . . . . . . . . . . . . . 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 19042	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝐺 ∈ Grp ∧ 𝑎 ∈ 𝑋) → ((invg‘𝐺)‘𝑎) ∈ 𝑋)
116	103, 104, 115	syl2anc 595	. . . . . . . . . . . . . . . . . . . . . . 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 18997	. . . . . . . . . . . . . . . . . . . . . . 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 1395	. . . . . . . . . . . . . . . . . . . . . 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 19022	. . . . . . . . . . . . . . . . . . . . . . 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 595	. . . . . . . . . . . . . . . . . . . . . 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 2808	. . . . . . . . . . . . . . . . . . . . 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 780	. . . . . . . . . . . . . . . . . . . . 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 2865	. . . . . . . . . . . . . . . . . . . 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 738	. . . . . . . . . . . . . . . . . . . . 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 19233	. . . . . . . . . . . . . . . . . . . . 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 595	. . . . . . . . . . . . . . . . . . . 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 1359	. . . . . . . . . . . . . . . . . . 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 8739	. . . . . . . . . . . . . . . . . 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 2803	. . . . . . . . . . . . . . . . 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 2850	. . . . . . . . . . . . . . . 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 242	. . . . . . . . . . . . . . 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 4031	. . . . . . . . . . . . . 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 8722	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 = 𝑎 → [𝑥](𝐺 ~QG 𝑌) = [𝑎](𝐺 ~QG 𝑌))
134	133	cbvmptv 5208	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 ∈ 𝑋 ↦ [𝑥](𝐺 ~QG 𝑌)) = (𝑎 ∈ 𝑋 ↦ [𝑎](𝐺 ~QG 𝑌))
135	6, 134	eqtri 2788	. . . . . . . . . . . . . . 15 ⊢ 𝐹 = (𝑎 ∈ 𝑋 ↦ [𝑎](𝐺 ~QG 𝑌))
136	135	mptpreima 6228	. . . . . . . . . . . . . 14 ⊢ (◡𝐹 “ (𝐹 “ 𝑆)) = {𝑎 ∈ 𝑋 ∣ [𝑎](𝐺 ~QG 𝑌) ∈ (𝐹 “ 𝑆)}
137	132, 93, 136	3sstr4g 3992	. . . . . . . . . . . . 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 2854	. . . . . . . . . . . . . . 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 3964	. . . . . . . . . . . . . . 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 643	. . . . . . . . . . . . . 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 3584	. . . . . . . . . . . . 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 849	. . . . . . . . . . . 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 1207	. . . . . . . . . . 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 417	. . . . . . . . . 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 243	. . . . . . . . 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 3166	. . . . . . . 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 35	. . . . . . 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 458	. . . . . 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 245	. . . . 5 ⊢ ((𝐺 ∈ TopGrp ∧ 𝑌 ∈ (NrmSGrp‘𝐺) ∧ 𝑆 ∈ 𝐽) → (𝑦 ∈ (◡𝐹 “ (𝐹 “ 𝑆)) → ∃𝑢 ∈ 𝐽 (𝑦 ∈ 𝑢 ∧ 𝑢 ⊆ (◡𝐹 “ (𝐹 “ 𝑆)))))
150	149	ralrimiv 3156	. . . 4 ⊢ ((𝐺 ∈ TopGrp ∧ 𝑌 ∈ (NrmSGrp‘𝐺) ∧ 𝑆 ∈ 𝐽) → ∀𝑦 ∈ (◡𝐹 “ (𝐹 “ 𝑆))∃𝑢 ∈ 𝐽 (𝑦 ∈ 𝑢 ∧ 𝑢 ⊆ (◡𝐹 “ (𝐹 “ 𝑆))))
151		topontop 23027	. . . . 5 ⊢ (𝐽 ∈ (TopOn‘𝑋) → 𝐽 ∈ Top)
152		eltop2 23089	. . . . 5 ⊢ (𝐽 ∈ Top → ((◡𝐹 “ (𝐹 “ 𝑆)) ∈ 𝐽 ↔ ∀𝑦 ∈ (◡𝐹 “ (𝐹 “ 𝑆))∃𝑢 ∈ 𝐽 (𝑦 ∈ 𝑢 ∧ 𝑢 ⊆ (◡𝐹 “ (𝐹 “ 𝑆)))))
153	24, 151, 152	3syl 19	. . . 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 260	. . 3 ⊢ ((𝐺 ∈ TopGrp ∧ 𝑌 ∈ (NrmSGrp‘𝐺) ∧ 𝑆 ∈ 𝐽) → (◡𝐹 “ (𝐹 “ 𝑆)) ∈ 𝐽)
155		elqtop3 23817	. . . 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 595	. . 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 725	. 2 ⊢ ((𝐺 ∈ TopGrp ∧ 𝑌 ∈ (NrmSGrp‘𝐺) ∧ 𝑆 ∈ 𝐽) → (𝐹 “ 𝑆) ∈ (𝐽 qTop 𝐹))
158	3, 5, 6, 8, 9	qusval 17584	. . 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 23834	. 2 ⊢ ((𝐺 ∈ TopGrp ∧ 𝑌 ∈ (NrmSGrp‘𝐺) ∧ 𝑆 ∈ 𝐽) → 𝐾 = (𝐽 qTop 𝐹))
161	157, 160	eleqtrrd 2868	1 ⊢ ((𝐺 ∈ TopGrp ∧ 𝑌 ∈ (NrmSGrp‘𝐺) ∧ 𝑆 ∈ 𝐽) → (𝐹 “ 𝑆) ∈ 𝐾)

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 400   ∧ w3a 1101   = wceq 1563   ∈ wcel 2145  ∀wral 3079  ∃wrex 3089  {crab 3417  Vcvv 3457   ⊆ wss 3907   class class class wbr 5104   ↦ cmpt 5185  ^◡ccnv 5650  ran crn 5652   “ cima 5654  Fun wfun 6519   Fn wfn 6520  –onto→wfo 6523  ‘cfv 6525  (class class class)co 7400   Er wer 8679  [cec 8680   / cqs 8681  Basecbs 17257  +_gcplusg 17298  TopOpenctopn 17462  0_gc0g 17480   qTop cqtop 17545   /_s cqus 17547  Grpcgrp 18988  inv_gcminusg 18989  SubGrpcsubg 19174  NrmSGrpcnsg 19175   ~_QG cqg 19176  opp_gcoppg 19403  Topctop 23007  TopOnctopon 23024   Cn ccn 23338  Homeochmeo 23867  TopGrpctgp 24185

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1818  ax-4 1832  ax-5 1933  ax-6 1990  ax-7 2031  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2215  ax-ext 2737  ax-rep 5231  ax-sep 5250  ax-nul 5260  ax-pow 5326  ax-pr 5394  ax-un 7722  ax-cnex 11144  ax-resscn 11145  ax-1cn 11146  ax-icn 11147  ax-addcl 11148  ax-addrcl 11149  ax-mulcl 11150  ax-mulrcl 11151  ax-mulcom 11152  ax-addass 11153  ax-mulass 11154  ax-distr 11155  ax-i2m1 11156  ax-1ne0 11157  ax-1rid 11158  ax-rnegex 11159  ax-rrecex 11160  ax-cnre 11161  ax-pre-lttri 11162  ax-pre-lttrn 11163  ax-pre-ltadd 11164  ax-pre-mulgt0 11165

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1102  df-3an 1103  df-tru 1566  df-fal 1576  df-ex 1803  df-nf 1807  df-sb 2094  df-mo 2569  df-eu 2599  df-clab 2744  df-cleq 2757  df-clel 2840  df-nfc 2914  df-ne 2961  df-nel 3065  df-ral 3080  df-rex 3090  df-rmo 3370  df-reu 3371  df-rab 3418  df-v 3459  df-sbc 3748  df-csb 3856  df-dif 3910  df-un 3912  df-in 3914  df-ss 3924  df-pss 3927  df-nul 4289  df-if 4484  df-pw 4560  df-sn 4586  df-pr 4588  df-tp 4590  df-op 4592  df-uni 4868  df-iun 4953  df-br 5105  df-opab 5167  df-mpt 5186  df-tr 5212  df-id 5546  df-eprel 5551  df-po 5559  df-so 5560  df-fr 5604  df-we 5606  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-rn 5662  df-res 5663  df-ima 5664  df-pred 6291  df-ord 6352  df-on 6353  df-lim 6354  df-suc 6355  df-iota 6481  df-fun 6527  df-fn 6528  df-f 6529  df-f1 6530  df-fo 6531  df-f1o 6532  df-fv 6533  df-riota 7357  df-ov 7403  df-oprab 7404  df-mpo 7405  df-om 7851  df-1st 7974  df-2nd 7975  df-tpos 8210  df-frecs 8266  df-wrecs 8297  df-recs 8346  df-rdg 8385  df-1o 8441  df-er 8682  df-ec 8684  df-qs 8688  df-map 8814  df-en 8932  df-dom 8933  df-sdom 8934  df-fin 8935  df-sup 9390  df-inf 9391  df-pnf 11233  df-mnf 11234  df-xr 11235  df-ltxr 11236  df-le 11237  df-sub 11431  df-neg 11432  df-nn 12222  df-2 12291  df-3 12292  df-4 12293  df-5 12294  df-6 12295  df-7 12296  df-8 12297  df-9 12298  df-n0 12493  df-z 12580  df-dec 12700  df-uz 12851  df-fz 13524  df-struct 17195  df-sets 17212  df-slot 17230  df-ndx 17242  df-base 17258  df-ress 17279  df-plusg 17311  df-mulr 17312  df-sca 17314  df-vsca 17315  df-ip 17316  df-tset 17317  df-ple 17318  df-ds 17320  df-rest 17463  df-topn 17464  df-0g 17482  df-topgen 17484  df-qtop 17549  df-imas 17550  df-qus 17551  df-plusf 18685  df-mgm 18686  df-sgrp 18765  df-mnd 18781  df-grp 18991  df-minusg 18992  df-subg 19177  df-nsg 19178  df-eqg 19179  df-oppg 19404  df-top 23008  df-topon 23025  df-topsp 23047  df-bases 23060  df-cn 23341  df-cnp 23342  df-tx 23676  df-hmeo 23869  df-tmd 24186  df-tgp 24187

This theorem is referenced by:  qustgplem  24235