Theorem qustgpopn 23471

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 6024	. . . 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 7390	. . . . . . 7 ⊢ (𝐺 ~QG 𝑌) ∈ V
8	7	a1i 11	. . . . . 6 ⊢ ((𝐺 ∈ TopGrp ∧ 𝑌 ∈ (NrmSGrp‘𝐺) ∧ 𝑆 ∈ 𝐽) → (𝐺 ~QG 𝑌) ∈ V)
9		simp1 1136	. . . . . 6 ⊢ ((𝐺 ∈ TopGrp ∧ 𝑌 ∈ (NrmSGrp‘𝐺) ∧ 𝑆 ∈ 𝐽) → 𝐺 ∈ TopGrp)
10	3, 5, 6, 8, 9	quslem 17425	. . . . 5 ⊢ ((𝐺 ∈ TopGrp ∧ 𝑌 ∈ (NrmSGrp‘𝐺) ∧ 𝑆 ∈ 𝐽) → 𝐹:𝑋–onto→(𝑋 / (𝐺 ~QG 𝑌)))
11		forn 6759	. . . . 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 3996	. . 3 ⊢ ((𝐺 ∈ TopGrp ∧ 𝑌 ∈ (NrmSGrp‘𝐺) ∧ 𝑆 ∈ 𝐽) → (𝐹 “ 𝑆) ⊆ (𝑋 / (𝐺 ~QG 𝑌)))
14		eceq1 8686	. . . . . . . . . 10 ⊢ (𝑥 = 𝑦 → [𝑥](𝐺 ~QG 𝑌) = [𝑦](𝐺 ~QG 𝑌))
15	14	cbvmptv 5218	. . . . . . . . 9 ⊢ (𝑥 ∈ 𝑋 ↦ [𝑥](𝐺 ~QG 𝑌)) = (𝑦 ∈ 𝑋 ↦ [𝑦](𝐺 ~QG 𝑌))
16	6, 15	eqtri 2764	. . . . . . . 8 ⊢ 𝐹 = (𝑦 ∈ 𝑋 ↦ [𝑦](𝐺 ~QG 𝑌))
17	16	mptpreima 6190	. . . . . . 7 ⊢ (◡𝐹 “ (𝐹 “ 𝑆)) = {𝑦 ∈ 𝑋 ∣ [𝑦](𝐺 ~QG 𝑌) ∈ (𝐹 “ 𝑆)}
18	17	reqabi 3429	. . . . . 6 ⊢ (𝑦 ∈ (◡𝐹 “ (𝐹 “ 𝑆)) ↔ (𝑦 ∈ 𝑋 ∧ [𝑦](𝐺 ~QG 𝑌) ∈ (𝐹 “ 𝑆)))
19	6	funmpt2 6540	. . . . . . . . 9 ⊢ Fun 𝐹
20		fvelima 6908	. . . . . . . . 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 23433	. . . . . . . . . . . . . . . . . 18 ⊢ (𝐺 ∈ TopGrp → 𝐽 ∈ (TopOn‘𝑋))
24	9, 23	syl 17	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐺 ∈ TopGrp ∧ 𝑌 ∈ (NrmSGrp‘𝐺) ∧ 𝑆 ∈ 𝐽) → 𝐽 ∈ (TopOn‘𝑋))
25		simp3 1138	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐺 ∈ TopGrp ∧ 𝑌 ∈ (NrmSGrp‘𝐺) ∧ 𝑆 ∈ 𝐽) → 𝑆 ∈ 𝐽)
26		toponss 22276	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ 𝐽) → 𝑆 ⊆ 𝑋)
27	24, 25, 26	syl2anc 584	. . . . . . . . . . . . . . . 16 ⊢ ((𝐺 ∈ TopGrp ∧ 𝑌 ∈ (NrmSGrp‘𝐺) ∧ 𝑆 ∈ 𝐽) → 𝑆 ⊆ 𝑋)
28	27	adantr 481	. . . . . . . . . . . . . . 15 ⊢ (((𝐺 ∈ TopGrp ∧ 𝑌 ∈ (NrmSGrp‘𝐺) ∧ 𝑆 ∈ 𝐽) ∧ 𝑦 ∈ 𝑋) → 𝑆 ⊆ 𝑋)
29	28	sselda 3944	. . . . . . . . . . . . . 14 ⊢ ((((𝐺 ∈ TopGrp ∧ 𝑌 ∈ (NrmSGrp‘𝐺) ∧ 𝑆 ∈ 𝐽) ∧ 𝑦 ∈ 𝑋) ∧ 𝑧 ∈ 𝑆) → 𝑧 ∈ 𝑋)
30		eceq1 8686	. . . . . . . . . . . . . . 15 ⊢ (𝑥 = 𝑧 → [𝑥](𝐺 ~QG 𝑌) = [𝑧](𝐺 ~QG 𝑌))
31		ecexg 8652	. . . . . . . . . . . . . . . 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 6948	. . . . . . . . . . . . . 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 2738	. . . . . . . . . . . 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 2743	. . . . . . . . . . . 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 18960	. . . . . . . . . . . . . . 15 ⊢ (𝑌 ∈ (NrmSGrp‘𝐺) → 𝑌 ∈ (SubGrp‘𝐺))
39	38	3ad2ant2 1134	. . . . . . . . . . . . . 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 2736	. . . . . . . . . . . . . 14 ⊢ (𝐺 ~QG 𝑌) = (𝐺 ~QG 𝑌)
42	4, 41	eqger 18980	. . . . . . . . . . . . 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 8697	. . . . . . . . . . 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 18929	. . . . . . . . . . . . 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 2736	. . . . . . . . . . . . 13 ⊢ (invg‘𝐺) = (invg‘𝐺)
50		eqid 2736	. . . . . . . . . . . . 13 ⊢ (+g‘𝐺) = (+g‘𝐺)
51	4, 49, 50, 41	eqgval 18979	. . . . . . . . . . . 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 584	. . . . . . . . . . 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 306	. . . . . . . . . 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 2736	. . . . . . . . . . . . . . . . . 18 ⊢ (oppg‘𝐺) = (oppg‘𝐺)
55		eqid 2736	. . . . . . . . . . . . . . . . . 18 ⊢ (+g‘(oppg‘𝐺)) = (+g‘(oppg‘𝐺))
56	50, 54, 55	oppgplus 19127	. . . . . . . . . . . . . . . . 17 ⊢ ((((invg‘𝐺)‘𝑦)(+g‘𝐺)𝑧)(+g‘(oppg‘𝐺))𝑎) = (𝑎(+g‘𝐺)(((invg‘𝐺)‘𝑦)(+g‘𝐺)𝑧))
57	56	mpteq2i 5210	. . . . . . . . . . . . . . . 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 481	. . . . . . . . . . . . . . . . . 18 ⊢ (((((𝐺 ∈ TopGrp ∧ 𝑌 ∈ (NrmSGrp‘𝐺) ∧ 𝑆 ∈ 𝐽) ∧ 𝑦 ∈ 𝑋) ∧ 𝑧 ∈ 𝑆) ∧ (((invg‘𝐺)‘𝑦)(+g‘𝐺)𝑧) ∈ 𝑌) → 𝐺 ∈ TopGrp)
59	54	oppgtgp 23449	. . . . . . . . . . . . . . . . . 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 3944	. . . . . . . . . . . . . . . . 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 2736	. . . . . . . . . . . . . . . . . 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 19130	. . . . . . . . . . . . . . . . . 18 ⊢ 𝑋 = (Base‘(oppg‘𝐺))
64	54, 22	oppgtopn 19134	. . . . . . . . . . . . . . . . . 18 ⊢ 𝐽 = (TopOpen‘(oppg‘𝐺))
65	62, 63, 55, 64	tgplacthmeo 23454	. . . . . . . . . . . . . . . . 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 584	. . . . . . . . . . . . . . . 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 2843	. . . . . . . . . . . . . . 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 23111	. . . . . . . . . . . . . . 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 22616	. . . . . . . . . . . . . 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 584	. . . . . . . . . . . . 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 481	. . . . . . . . . . . . . 14 ⊢ (((((𝐺 ∈ TopGrp ∧ 𝑌 ∈ (NrmSGrp‘𝐺) ∧ 𝑆 ∈ 𝐽) ∧ 𝑦 ∈ 𝑋) ∧ 𝑧 ∈ 𝑆) ∧ (((invg‘𝐺)‘𝑦)(+g‘𝐺)𝑧) ∈ 𝑌) → 𝑦 ∈ 𝑋)
74		tgpgrp 23429	. . . . . . . . . . . . . . . . . . 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 2736	. . . . . . . . . . . . . . . . . . 19 ⊢ (0g‘𝐺) = (0g‘𝐺)
77	4, 50, 76, 49	grprinv 18801	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐺 ∈ Grp ∧ 𝑦 ∈ 𝑋) → (𝑦(+g‘𝐺)((invg‘𝐺)‘𝑦)) = (0g‘𝐺))
78	75, 73, 77	syl2anc 584	. . . . . . . . . . . . . . . . 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 7372	. . . . . . . . . . . . . . . 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 18798	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐺 ∈ Grp ∧ 𝑦 ∈ 𝑋) → ((invg‘𝐺)‘𝑦) ∈ 𝑋)
81	75, 73, 80	syl2anc 584	. . . . . . . . . . . . . . . . 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 481	. . . . . . . . . . . . . . . . 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 18757	. . . . . . . . . . . . . . . . 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 1372	. . . . . . . . . . . . . . . 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 18780	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐺 ∈ Grp ∧ 𝑧 ∈ 𝑋) → ((0g‘𝐺)(+g‘𝐺)𝑧) = 𝑧)
86	75, 82, 85	syl2anc 584	. . . . . . . . . . . . . . . 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 2784	. . . . . . . . . . . . . . 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 2838	. . . . . . . . . . . . . 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 7364	. . . . . . . . . . . . . . . 16 ⊢ (𝑎 = 𝑦 → (𝑎(+g‘𝐺)(((invg‘𝐺)‘𝑦)(+g‘𝐺)𝑧)) = (𝑦(+g‘𝐺)(((invg‘𝐺)‘𝑦)(+g‘𝐺)𝑧)))
91	90	eleq1d 2822	. . . . . . . . . . . . . . 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 2736	. . . . . . . . . . . . . . . 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 6190	. . . . . . . . . . . . . . 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 3648	. . . . . . . . . . . . . 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 583	. . . . . . . . . . . . 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 8652	. . . . . . . . . . . . . . . . . . 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 6644	. . . . . . . . . . . . . . . . 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 7183	. . . . . . . . . . . . . . . . . 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 1121	. . . . . . . . . . . . . . . . 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 587	. . . . . . . . . . . . . . . 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 481	. . . . . . . . . . . . . . . . . . . 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 485	. . . . . . . . . . . . . . . . . . . 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 481	. . . . . . . . . . . . . . . . . . . 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 18756	. . . . . . . . . . . . . . . . . . . 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 1371	. . . . . . . . . . . . . . . . . . 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 8686	. . . . . . . . . . . . . . . . . . . 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 6953	. . . . . . . . . . . . . . . . . . 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 18800	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝐺 ∈ Grp ∧ 𝑎 ∈ 𝑋) → (((invg‘𝐺)‘𝑎)(+g‘𝐺)𝑎) = (0g‘𝐺))
113	103, 104, 112	syl2anc 584	. . . . . . . . . . . . . . . . . . . . . . 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 7372	. . . . . . . . . . . . . . . . . . . . . 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 18798	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝐺 ∈ Grp ∧ 𝑎 ∈ 𝑋) → ((invg‘𝐺)‘𝑎) ∈ 𝑋)
116	103, 104, 115	syl2anc 584	. . . . . . . . . . . . . . . . . . . . . . 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 18757	. . . . . . . . . . . . . . . . . . . . . . 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 1372	. . . . . . . . . . . . . . . . . . . . . 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 18780	. . . . . . . . . . . . . . . . . . . . . . 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 584	. . . . . . . . . . . . . . . . . . . . . 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 2784	. . . . . . . . . . . . . . . . . . . . 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 2838	. . . . . . . . . . . . . . . . . . . 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 18979	. . . . . . . . . . . . . . . . . . . . 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 584	. . . . . . . . . . . . . . . . . . . 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 1342	. . . . . . . . . . . . . . . . . . 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 8699	. . . . . . . . . . . . . . . . . 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 2779	. . . . . . . . . . . . . . . . 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 2822	. . . . . . . . . . . . . . . 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 238	. . . . . . . . . . . . . . 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 4033	. . . . . . . . . . . . . 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 8686	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 = 𝑎 → [𝑥](𝐺 ~QG 𝑌) = [𝑎](𝐺 ~QG 𝑌))
134	133	cbvmptv 5218	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 ∈ 𝑋 ↦ [𝑥](𝐺 ~QG 𝑌)) = (𝑎 ∈ 𝑋 ↦ [𝑎](𝐺 ~QG 𝑌))
135	6, 134	eqtri 2764	. . . . . . . . . . . . . . 15 ⊢ 𝐹 = (𝑎 ∈ 𝑋 ↦ [𝑎](𝐺 ~QG 𝑌))
136	135	mptpreima 6190	. . . . . . . . . . . . . 14 ⊢ (◡𝐹 “ (𝐹 “ 𝑆)) = {𝑎 ∈ 𝑋 ∣ [𝑎](𝐺 ~QG 𝑌) ∈ (𝐹 “ 𝑆)}
137	132, 93, 136	3sstr4g 3989	. . . . . . . . . . . . 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 2826	. . . . . . . . . . . . . . 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 3969	. . . . . . . . . . . . . . 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 631	. . . . . . . . . . . . . 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 3581	. . . . . . . . . . . . 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 835	. . . . . . . . . . . 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 1190	. . . . . . . . . . 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 413	. . . . . . . . . 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 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 3152	. . . . . . . 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 454	. . . . . 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 3142	. . . 4 ⊢ ((𝐺 ∈ TopGrp ∧ 𝑌 ∈ (NrmSGrp‘𝐺) ∧ 𝑆 ∈ 𝐽) → ∀𝑦 ∈ (◡𝐹 “ (𝐹 “ 𝑆))∃𝑢 ∈ 𝐽 (𝑦 ∈ 𝑢 ∧ 𝑢 ⊆ (◡𝐹 “ (𝐹 “ 𝑆))))
151		topontop 22262	. . . . 5 ⊢ (𝐽 ∈ (TopOn‘𝑋) → 𝐽 ∈ Top)
152		eltop2 22325	. . . . 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 23054	. . . 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 584	. . 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 17424	. . 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 23071	. 2 ⊢ ((𝐺 ∈ TopGrp ∧ 𝑌 ∈ (NrmSGrp‘𝐺) ∧ 𝑆 ∈ 𝐽) → 𝐾 = (𝐽 qTop 𝐹))
161	157, 160	eleqtrrd 2841	1 ⊢ ((𝐺 ∈ TopGrp ∧ 𝑌 ∈ (NrmSGrp‘𝐺) ∧ 𝑆 ∈ 𝐽) → (𝐹 “ 𝑆) ∈ 𝐾)

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 396   ∧ w3a 1087   = wceq 1541   ∈ wcel 2106  ∀wral 3064  ∃wrex 3073  {crab 3407  Vcvv 3445   ⊆ wss 3910   class class class wbr 5105   ↦ cmpt 5188  ^◡ccnv 5632  ran crn 5634   “ cima 5636  Fun wfun 6490   Fn wfn 6491  –onto→wfo 6494  ‘cfv 6496  (class class class)co 7357   Er wer 8645  [cec 8646   / cqs 8647  Basecbs 17083  +_gcplusg 17133  TopOpenctopn 17303  0_gc0g 17321   qTop cqtop 17385   /_s cqus 17387  Grpcgrp 18748  inv_gcminusg 18749  SubGrpcsubg 18922  NrmSGrpcnsg 18923   ~_QG cqg 18924  opp_gcoppg 19123  Topctop 22242  TopOnctopon 22259   Cn ccn 22575  Homeochmeo 23104  TopGrpctgp 23422

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-rep 5242  ax-sep 5256  ax-nul 5263  ax-pow 5320  ax-pr 5384  ax-un 7672  ax-cnex 11107  ax-resscn 11108  ax-1cn 11109  ax-icn 11110  ax-addcl 11111  ax-addrcl 11112  ax-mulcl 11113  ax-mulrcl 11114  ax-mulcom 11115  ax-addass 11116  ax-mulass 11117  ax-distr 11118  ax-i2m1 11119  ax-1ne0 11120  ax-1rid 11121  ax-rnegex 11122  ax-rrecex 11123  ax-cnre 11124  ax-pre-lttri 11125  ax-pre-lttrn 11126  ax-pre-ltadd 11127  ax-pre-mulgt0 11128

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-nel 3050  df-ral 3065  df-rex 3074  df-rmo 3353  df-reu 3354  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-pss 3929  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-tp 4591  df-op 4593  df-uni 4866  df-iun 4956  df-br 5106  df-opab 5168  df-mpt 5189  df-tr 5223  df-id 5531  df-eprel 5537  df-po 5545  df-so 5546  df-fr 5588  df-we 5590  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-pred 6253  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-f1 6501  df-fo 6502  df-f1o 6503  df-fv 6504  df-riota 7313  df-ov 7360  df-oprab 7361  df-mpo 7362  df-om 7803  df-1st 7921  df-2nd 7922  df-tpos 8157  df-frecs 8212  df-wrecs 8243  df-recs 8317  df-rdg 8356  df-1o 8412  df-er 8648  df-ec 8650  df-qs 8654  df-map 8767  df-en 8884  df-dom 8885  df-sdom 8886  df-fin 8887  df-sup 9378  df-inf 9379  df-pnf 11191  df-mnf 11192  df-xr 11193  df-ltxr 11194  df-le 11195  df-sub 11387  df-neg 11388  df-nn 12154  df-2 12216  df-3 12217  df-4 12218  df-5 12219  df-6 12220  df-7 12221  df-8 12222  df-9 12223  df-n0 12414  df-z 12500  df-dec 12619  df-uz 12764  df-fz 13425  df-struct 17019  df-sets 17036  df-slot 17054  df-ndx 17066  df-base 17084  df-ress 17113  df-plusg 17146  df-mulr 17147  df-sca 17149  df-vsca 17150  df-ip 17151  df-tset 17152  df-ple 17153  df-ds 17155  df-rest 17304  df-topn 17305  df-0g 17323  df-topgen 17325  df-qtop 17389  df-imas 17390  df-qus 17391  df-plusf 18496  df-mgm 18497  df-sgrp 18546  df-mnd 18557  df-grp 18751  df-minusg 18752  df-subg 18925  df-nsg 18926  df-eqg 18927  df-oppg 19124  df-top 22243  df-topon 22260  df-topsp 22282  df-bases 22296  df-cn 22578  df-cnp 22579  df-tx 22913  df-hmeo 23106  df-tmd 23423  df-tgp 23424

This theorem is referenced by:  qustgplem  23472