MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  qustgpopn Structured version   Visualization version   GIF version

Theorem qustgpopn 24115
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 = (𝑥𝑋 ↦ [𝑥](𝐺 ~QG 𝑌))
Assertion
Ref Expression
qustgpopn ((𝐺 ∈ TopGrp ∧ 𝑌 ∈ (NrmSGrp‘𝐺) ∧ 𝑆𝐽) → (𝐹𝑆) ∈ 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.
StepHypRef Expression
1 imassrn 6080 . . . 4 (𝐹𝑆) ⊆ ran 𝐹
2 qustgp.h . . . . . . 7 𝐻 = (𝐺 /s (𝐺 ~QG 𝑌))
32a1i 11 . . . . . 6 ((𝐺 ∈ TopGrp ∧ 𝑌 ∈ (NrmSGrp‘𝐺) ∧ 𝑆𝐽) → 𝐻 = (𝐺 /s (𝐺 ~QG 𝑌)))
4 qustgpopn.x . . . . . . 7 𝑋 = (Base‘𝐺)
54a1i 11 . . . . . 6 ((𝐺 ∈ TopGrp ∧ 𝑌 ∈ (NrmSGrp‘𝐺) ∧ 𝑆𝐽) → 𝑋 = (Base‘𝐺))
6 qustgpopn.f . . . . . 6 𝐹 = (𝑥𝑋 ↦ [𝑥](𝐺 ~QG 𝑌))
7 ovex 7457 . . . . . . 7 (𝐺 ~QG 𝑌) ∈ V
87a1i 11 . . . . . 6 ((𝐺 ∈ TopGrp ∧ 𝑌 ∈ (NrmSGrp‘𝐺) ∧ 𝑆𝐽) → (𝐺 ~QG 𝑌) ∈ V)
9 simp1 1133 . . . . . 6 ((𝐺 ∈ TopGrp ∧ 𝑌 ∈ (NrmSGrp‘𝐺) ∧ 𝑆𝐽) → 𝐺 ∈ TopGrp)
103, 5, 6, 8, 9quslem 17558 . . . . 5 ((𝐺 ∈ TopGrp ∧ 𝑌 ∈ (NrmSGrp‘𝐺) ∧ 𝑆𝐽) → 𝐹:𝑋onto→(𝑋 / (𝐺 ~QG 𝑌)))
11 forn 6818 . . . . 5 (𝐹:𝑋onto→(𝑋 / (𝐺 ~QG 𝑌)) → ran 𝐹 = (𝑋 / (𝐺 ~QG 𝑌)))
1210, 11syl 17 . . . 4 ((𝐺 ∈ TopGrp ∧ 𝑌 ∈ (NrmSGrp‘𝐺) ∧ 𝑆𝐽) → ran 𝐹 = (𝑋 / (𝐺 ~QG 𝑌)))
131, 12sseqtrid 4032 . . 3 ((𝐺 ∈ TopGrp ∧ 𝑌 ∈ (NrmSGrp‘𝐺) ∧ 𝑆𝐽) → (𝐹𝑆) ⊆ (𝑋 / (𝐺 ~QG 𝑌)))
14 eceq1 8773 . . . . . . . . . 10 (𝑥 = 𝑦 → [𝑥](𝐺 ~QG 𝑌) = [𝑦](𝐺 ~QG 𝑌))
1514cbvmptv 5266 . . . . . . . . 9 (𝑥𝑋 ↦ [𝑥](𝐺 ~QG 𝑌)) = (𝑦𝑋 ↦ [𝑦](𝐺 ~QG 𝑌))
166, 15eqtri 2754 . . . . . . . 8 𝐹 = (𝑦𝑋 ↦ [𝑦](𝐺 ~QG 𝑌))
1716mptpreima 6249 . . . . . . 7 (𝐹 “ (𝐹𝑆)) = {𝑦𝑋 ∣ [𝑦](𝐺 ~QG 𝑌) ∈ (𝐹𝑆)}
1817reqabi 3442 . . . . . 6 (𝑦 ∈ (𝐹 “ (𝐹𝑆)) ↔ (𝑦𝑋 ∧ [𝑦](𝐺 ~QG 𝑌) ∈ (𝐹𝑆)))
196funmpt2 6598 . . . . . . . . 9 Fun 𝐹
20 fvelima 6968 . . . . . . . . 9 ((Fun 𝐹 ∧ [𝑦](𝐺 ~QG 𝑌) ∈ (𝐹𝑆)) → ∃𝑧𝑆 (𝐹𝑧) = [𝑦](𝐺 ~QG 𝑌))
2119, 20mpan 688 . . . . . . . 8 ([𝑦](𝐺 ~QG 𝑌) ∈ (𝐹𝑆) → ∃𝑧𝑆 (𝐹𝑧) = [𝑦](𝐺 ~QG 𝑌))
22 qustgpopn.j . . . . . . . . . . . . . . . . . . 19 𝐽 = (TopOpen‘𝐺)
2322, 4tgptopon 24077 . . . . . . . . . . . . . . . . . 18 (𝐺 ∈ TopGrp → 𝐽 ∈ (TopOn‘𝑋))
249, 23syl 17 . . . . . . . . . . . . . . . . 17 ((𝐺 ∈ TopGrp ∧ 𝑌 ∈ (NrmSGrp‘𝐺) ∧ 𝑆𝐽) → 𝐽 ∈ (TopOn‘𝑋))
25 simp3 1135 . . . . . . . . . . . . . . . . 17 ((𝐺 ∈ TopGrp ∧ 𝑌 ∈ (NrmSGrp‘𝐺) ∧ 𝑆𝐽) → 𝑆𝐽)
26 toponss 22920 . . . . . . . . . . . . . . . . 17 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑆𝐽) → 𝑆𝑋)
2724, 25, 26syl2anc 582 . . . . . . . . . . . . . . . 16 ((𝐺 ∈ TopGrp ∧ 𝑌 ∈ (NrmSGrp‘𝐺) ∧ 𝑆𝐽) → 𝑆𝑋)
2827adantr 479 . . . . . . . . . . . . . . 15 (((𝐺 ∈ TopGrp ∧ 𝑌 ∈ (NrmSGrp‘𝐺) ∧ 𝑆𝐽) ∧ 𝑦𝑋) → 𝑆𝑋)
2928sselda 3979 . . . . . . . . . . . . . 14 ((((𝐺 ∈ TopGrp ∧ 𝑌 ∈ (NrmSGrp‘𝐺) ∧ 𝑆𝐽) ∧ 𝑦𝑋) ∧ 𝑧𝑆) → 𝑧𝑋)
30 eceq1 8773 . . . . . . . . . . . . . . 15 (𝑥 = 𝑧 → [𝑥](𝐺 ~QG 𝑌) = [𝑧](𝐺 ~QG 𝑌))
31 ecexg 8738 . . . . . . . . . . . . . . . 16 ((𝐺 ~QG 𝑌) ∈ V → [𝑧](𝐺 ~QG 𝑌) ∈ V)
327, 31ax-mp 5 . . . . . . . . . . . . . . 15 [𝑧](𝐺 ~QG 𝑌) ∈ V
3330, 6, 32fvmpt 7009 . . . . . . . . . . . . . 14 (𝑧𝑋 → (𝐹𝑧) = [𝑧](𝐺 ~QG 𝑌))
3429, 33syl 17 . . . . . . . . . . . . 13 ((((𝐺 ∈ TopGrp ∧ 𝑌 ∈ (NrmSGrp‘𝐺) ∧ 𝑆𝐽) ∧ 𝑦𝑋) ∧ 𝑧𝑆) → (𝐹𝑧) = [𝑧](𝐺 ~QG 𝑌))
3534eqeq1d 2728 . . . . . . . . . . . 12 ((((𝐺 ∈ TopGrp ∧ 𝑌 ∈ (NrmSGrp‘𝐺) ∧ 𝑆𝐽) ∧ 𝑦𝑋) ∧ 𝑧𝑆) → ((𝐹𝑧) = [𝑦](𝐺 ~QG 𝑌) ↔ [𝑧](𝐺 ~QG 𝑌) = [𝑦](𝐺 ~QG 𝑌)))
36 eqcom 2733 . . . . . . . . . . . 12 ([𝑧](𝐺 ~QG 𝑌) = [𝑦](𝐺 ~QG 𝑌) ↔ [𝑦](𝐺 ~QG 𝑌) = [𝑧](𝐺 ~QG 𝑌))
3735, 36bitrdi 286 . . . . . . . . . . 11 ((((𝐺 ∈ TopGrp ∧ 𝑌 ∈ (NrmSGrp‘𝐺) ∧ 𝑆𝐽) ∧ 𝑦𝑋) ∧ 𝑧𝑆) → ((𝐹𝑧) = [𝑦](𝐺 ~QG 𝑌) ↔ [𝑦](𝐺 ~QG 𝑌) = [𝑧](𝐺 ~QG 𝑌)))
38 nsgsubg 19152 . . . . . . . . . . . . . . 15 (𝑌 ∈ (NrmSGrp‘𝐺) → 𝑌 ∈ (SubGrp‘𝐺))
39383ad2ant2 1131 . . . . . . . . . . . . . 14 ((𝐺 ∈ TopGrp ∧ 𝑌 ∈ (NrmSGrp‘𝐺) ∧ 𝑆𝐽) → 𝑌 ∈ (SubGrp‘𝐺))
4039ad2antrr 724 . . . . . . . . . . . . 13 ((((𝐺 ∈ TopGrp ∧ 𝑌 ∈ (NrmSGrp‘𝐺) ∧ 𝑆𝐽) ∧ 𝑦𝑋) ∧ 𝑧𝑆) → 𝑌 ∈ (SubGrp‘𝐺))
41 eqid 2726 . . . . . . . . . . . . . 14 (𝐺 ~QG 𝑌) = (𝐺 ~QG 𝑌)
424, 41eqger 19172 . . . . . . . . . . . . 13 (𝑌 ∈ (SubGrp‘𝐺) → (𝐺 ~QG 𝑌) Er 𝑋)
4340, 42syl 17 . . . . . . . . . . . 12 ((((𝐺 ∈ TopGrp ∧ 𝑌 ∈ (NrmSGrp‘𝐺) ∧ 𝑆𝐽) ∧ 𝑦𝑋) ∧ 𝑧𝑆) → (𝐺 ~QG 𝑌) Er 𝑋)
44 simplr 767 . . . . . . . . . . . 12 ((((𝐺 ∈ TopGrp ∧ 𝑌 ∈ (NrmSGrp‘𝐺) ∧ 𝑆𝐽) ∧ 𝑦𝑋) ∧ 𝑧𝑆) → 𝑦𝑋)
4543, 44erth 8785 . . . . . . . . . . 11 ((((𝐺 ∈ TopGrp ∧ 𝑌 ∈ (NrmSGrp‘𝐺) ∧ 𝑆𝐽) ∧ 𝑦𝑋) ∧ 𝑧𝑆) → (𝑦(𝐺 ~QG 𝑌)𝑧 ↔ [𝑦](𝐺 ~QG 𝑌) = [𝑧](𝐺 ~QG 𝑌)))
469ad2antrr 724 . . . . . . . . . . . 12 ((((𝐺 ∈ TopGrp ∧ 𝑌 ∈ (NrmSGrp‘𝐺) ∧ 𝑆𝐽) ∧ 𝑦𝑋) ∧ 𝑧𝑆) → 𝐺 ∈ TopGrp)
474subgss 19121 . . . . . . . . . . . . 13 (𝑌 ∈ (SubGrp‘𝐺) → 𝑌𝑋)
4840, 47syl 17 . . . . . . . . . . . 12 ((((𝐺 ∈ TopGrp ∧ 𝑌 ∈ (NrmSGrp‘𝐺) ∧ 𝑆𝐽) ∧ 𝑦𝑋) ∧ 𝑧𝑆) → 𝑌𝑋)
49 eqid 2726 . . . . . . . . . . . . 13 (invg𝐺) = (invg𝐺)
50 eqid 2726 . . . . . . . . . . . . 13 (+g𝐺) = (+g𝐺)
514, 49, 50, 41eqgval 19171 . . . . . . . . . . . 12 ((𝐺 ∈ TopGrp ∧ 𝑌𝑋) → (𝑦(𝐺 ~QG 𝑌)𝑧 ↔ (𝑦𝑋𝑧𝑋 ∧ (((invg𝐺)‘𝑦)(+g𝐺)𝑧) ∈ 𝑌)))
5246, 48, 51syl2anc 582 . . . . . . . . . . 11 ((((𝐺 ∈ TopGrp ∧ 𝑌 ∈ (NrmSGrp‘𝐺) ∧ 𝑆𝐽) ∧ 𝑦𝑋) ∧ 𝑧𝑆) → (𝑦(𝐺 ~QG 𝑌)𝑧 ↔ (𝑦𝑋𝑧𝑋 ∧ (((invg𝐺)‘𝑦)(+g𝐺)𝑧) ∈ 𝑌)))
5337, 45, 523bitr2d 306 . . . . . . . . . 10 ((((𝐺 ∈ TopGrp ∧ 𝑌 ∈ (NrmSGrp‘𝐺) ∧ 𝑆𝐽) ∧ 𝑦𝑋) ∧ 𝑧𝑆) → ((𝐹𝑧) = [𝑦](𝐺 ~QG 𝑌) ↔ (𝑦𝑋𝑧𝑋 ∧ (((invg𝐺)‘𝑦)(+g𝐺)𝑧) ∈ 𝑌)))
54 eqid 2726 . . . . . . . . . . . . . . . . . 18 (oppg𝐺) = (oppg𝐺)
55 eqid 2726 . . . . . . . . . . . . . . . . . 18 (+g‘(oppg𝐺)) = (+g‘(oppg𝐺))
5650, 54, 55oppgplus 19343 . . . . . . . . . . . . . . . . 17 ((((invg𝐺)‘𝑦)(+g𝐺)𝑧)(+g‘(oppg𝐺))𝑎) = (𝑎(+g𝐺)(((invg𝐺)‘𝑦)(+g𝐺)𝑧))
5756mpteq2i 5258 . . . . . . . . . . . . . . . 16 (𝑎𝑋 ↦ ((((invg𝐺)‘𝑦)(+g𝐺)𝑧)(+g‘(oppg𝐺))𝑎)) = (𝑎𝑋 ↦ (𝑎(+g𝐺)(((invg𝐺)‘𝑦)(+g𝐺)𝑧)))
5846adantr 479 . . . . . . . . . . . . . . . . . 18 (((((𝐺 ∈ TopGrp ∧ 𝑌 ∈ (NrmSGrp‘𝐺) ∧ 𝑆𝐽) ∧ 𝑦𝑋) ∧ 𝑧𝑆) ∧ (((invg𝐺)‘𝑦)(+g𝐺)𝑧) ∈ 𝑌) → 𝐺 ∈ TopGrp)
5954oppgtgp 24093 . . . . . . . . . . . . . . . . . 18 (𝐺 ∈ TopGrp → (oppg𝐺) ∈ TopGrp)
6058, 59syl 17 . . . . . . . . . . . . . . . . 17 (((((𝐺 ∈ TopGrp ∧ 𝑌 ∈ (NrmSGrp‘𝐺) ∧ 𝑆𝐽) ∧ 𝑦𝑋) ∧ 𝑧𝑆) ∧ (((invg𝐺)‘𝑦)(+g𝐺)𝑧) ∈ 𝑌) → (oppg𝐺) ∈ TopGrp)
6148sselda 3979 . . . . . . . . . . . . . . . . 17 (((((𝐺 ∈ TopGrp ∧ 𝑌 ∈ (NrmSGrp‘𝐺) ∧ 𝑆𝐽) ∧ 𝑦𝑋) ∧ 𝑧𝑆) ∧ (((invg𝐺)‘𝑦)(+g𝐺)𝑧) ∈ 𝑌) → (((invg𝐺)‘𝑦)(+g𝐺)𝑧) ∈ 𝑋)
62 eqid 2726 . . . . . . . . . . . . . . . . . 18 (𝑎𝑋 ↦ ((((invg𝐺)‘𝑦)(+g𝐺)𝑧)(+g‘(oppg𝐺))𝑎)) = (𝑎𝑋 ↦ ((((invg𝐺)‘𝑦)(+g𝐺)𝑧)(+g‘(oppg𝐺))𝑎))
6354, 4oppgbas 19346 . . . . . . . . . . . . . . . . . 18 𝑋 = (Base‘(oppg𝐺))
6454, 22oppgtopn 19350 . . . . . . . . . . . . . . . . . 18 𝐽 = (TopOpen‘(oppg𝐺))
6562, 63, 55, 64tgplacthmeo 24098 . . . . . . . . . . . . . . . . 17 (((oppg𝐺) ∈ TopGrp ∧ (((invg𝐺)‘𝑦)(+g𝐺)𝑧) ∈ 𝑋) → (𝑎𝑋 ↦ ((((invg𝐺)‘𝑦)(+g𝐺)𝑧)(+g‘(oppg𝐺))𝑎)) ∈ (𝐽Homeo𝐽))
6660, 61, 65syl2anc 582 . . . . . . . . . . . . . . . 16 (((((𝐺 ∈ TopGrp ∧ 𝑌 ∈ (NrmSGrp‘𝐺) ∧ 𝑆𝐽) ∧ 𝑦𝑋) ∧ 𝑧𝑆) ∧ (((invg𝐺)‘𝑦)(+g𝐺)𝑧) ∈ 𝑌) → (𝑎𝑋 ↦ ((((invg𝐺)‘𝑦)(+g𝐺)𝑧)(+g‘(oppg𝐺))𝑎)) ∈ (𝐽Homeo𝐽))
6757, 66eqeltrrid 2831 . . . . . . . . . . . . . . 15 (((((𝐺 ∈ TopGrp ∧ 𝑌 ∈ (NrmSGrp‘𝐺) ∧ 𝑆𝐽) ∧ 𝑦𝑋) ∧ 𝑧𝑆) ∧ (((invg𝐺)‘𝑦)(+g𝐺)𝑧) ∈ 𝑌) → (𝑎𝑋 ↦ (𝑎(+g𝐺)(((invg𝐺)‘𝑦)(+g𝐺)𝑧))) ∈ (𝐽Homeo𝐽))
68 hmeocn 23755 . . . . . . . . . . . . . . 15 ((𝑎𝑋 ↦ (𝑎(+g𝐺)(((invg𝐺)‘𝑦)(+g𝐺)𝑧))) ∈ (𝐽Homeo𝐽) → (𝑎𝑋 ↦ (𝑎(+g𝐺)(((invg𝐺)‘𝑦)(+g𝐺)𝑧))) ∈ (𝐽 Cn 𝐽))
6967, 68syl 17 . . . . . . . . . . . . . 14 (((((𝐺 ∈ TopGrp ∧ 𝑌 ∈ (NrmSGrp‘𝐺) ∧ 𝑆𝐽) ∧ 𝑦𝑋) ∧ 𝑧𝑆) ∧ (((invg𝐺)‘𝑦)(+g𝐺)𝑧) ∈ 𝑌) → (𝑎𝑋 ↦ (𝑎(+g𝐺)(((invg𝐺)‘𝑦)(+g𝐺)𝑧))) ∈ (𝐽 Cn 𝐽))
7025ad3antrrr 728 . . . . . . . . . . . . . 14 (((((𝐺 ∈ TopGrp ∧ 𝑌 ∈ (NrmSGrp‘𝐺) ∧ 𝑆𝐽) ∧ 𝑦𝑋) ∧ 𝑧𝑆) ∧ (((invg𝐺)‘𝑦)(+g𝐺)𝑧) ∈ 𝑌) → 𝑆𝐽)
71 cnima 23260 . . . . . . . . . . . . . 14 (((𝑎𝑋 ↦ (𝑎(+g𝐺)(((invg𝐺)‘𝑦)(+g𝐺)𝑧))) ∈ (𝐽 Cn 𝐽) ∧ 𝑆𝐽) → ((𝑎𝑋 ↦ (𝑎(+g𝐺)(((invg𝐺)‘𝑦)(+g𝐺)𝑧))) “ 𝑆) ∈ 𝐽)
7269, 70, 71syl2anc 582 . . . . . . . . . . . . 13 (((((𝐺 ∈ TopGrp ∧ 𝑌 ∈ (NrmSGrp‘𝐺) ∧ 𝑆𝐽) ∧ 𝑦𝑋) ∧ 𝑧𝑆) ∧ (((invg𝐺)‘𝑦)(+g𝐺)𝑧) ∈ 𝑌) → ((𝑎𝑋 ↦ (𝑎(+g𝐺)(((invg𝐺)‘𝑦)(+g𝐺)𝑧))) “ 𝑆) ∈ 𝐽)
7344adantr 479 . . . . . . . . . . . . . 14 (((((𝐺 ∈ TopGrp ∧ 𝑌 ∈ (NrmSGrp‘𝐺) ∧ 𝑆𝐽) ∧ 𝑦𝑋) ∧ 𝑧𝑆) ∧ (((invg𝐺)‘𝑦)(+g𝐺)𝑧) ∈ 𝑌) → 𝑦𝑋)
74 tgpgrp 24073 . . . . . . . . . . . . . . . . . . 19 (𝐺 ∈ TopGrp → 𝐺 ∈ Grp)
7558, 74syl 17 . . . . . . . . . . . . . . . . . 18 (((((𝐺 ∈ TopGrp ∧ 𝑌 ∈ (NrmSGrp‘𝐺) ∧ 𝑆𝐽) ∧ 𝑦𝑋) ∧ 𝑧𝑆) ∧ (((invg𝐺)‘𝑦)(+g𝐺)𝑧) ∈ 𝑌) → 𝐺 ∈ Grp)
76 eqid 2726 . . . . . . . . . . . . . . . . . . 19 (0g𝐺) = (0g𝐺)
774, 50, 76, 49grprinv 18985 . . . . . . . . . . . . . . . . . 18 ((𝐺 ∈ Grp ∧ 𝑦𝑋) → (𝑦(+g𝐺)((invg𝐺)‘𝑦)) = (0g𝐺))
7875, 73, 77syl2anc 582 . . . . . . . . . . . . . . . . 17 (((((𝐺 ∈ TopGrp ∧ 𝑌 ∈ (NrmSGrp‘𝐺) ∧ 𝑆𝐽) ∧ 𝑦𝑋) ∧ 𝑧𝑆) ∧ (((invg𝐺)‘𝑦)(+g𝐺)𝑧) ∈ 𝑌) → (𝑦(+g𝐺)((invg𝐺)‘𝑦)) = (0g𝐺))
7978oveq1d 7439 . . . . . . . . . . . . . . . 16 (((((𝐺 ∈ TopGrp ∧ 𝑌 ∈ (NrmSGrp‘𝐺) ∧ 𝑆𝐽) ∧ 𝑦𝑋) ∧ 𝑧𝑆) ∧ (((invg𝐺)‘𝑦)(+g𝐺)𝑧) ∈ 𝑌) → ((𝑦(+g𝐺)((invg𝐺)‘𝑦))(+g𝐺)𝑧) = ((0g𝐺)(+g𝐺)𝑧))
804, 49grpinvcl 18982 . . . . . . . . . . . . . . . . . 18 ((𝐺 ∈ Grp ∧ 𝑦𝑋) → ((invg𝐺)‘𝑦) ∈ 𝑋)
8175, 73, 80syl2anc 582 . . . . . . . . . . . . . . . . 17 (((((𝐺 ∈ TopGrp ∧ 𝑌 ∈ (NrmSGrp‘𝐺) ∧ 𝑆𝐽) ∧ 𝑦𝑋) ∧ 𝑧𝑆) ∧ (((invg𝐺)‘𝑦)(+g𝐺)𝑧) ∈ 𝑌) → ((invg𝐺)‘𝑦) ∈ 𝑋)
8229adantr 479 . . . . . . . . . . . . . . . . 17 (((((𝐺 ∈ TopGrp ∧ 𝑌 ∈ (NrmSGrp‘𝐺) ∧ 𝑆𝐽) ∧ 𝑦𝑋) ∧ 𝑧𝑆) ∧ (((invg𝐺)‘𝑦)(+g𝐺)𝑧) ∈ 𝑌) → 𝑧𝑋)
834, 50grpass 18937 . . . . . . . . . . . . . . . . 17 ((𝐺 ∈ Grp ∧ (𝑦𝑋 ∧ ((invg𝐺)‘𝑦) ∈ 𝑋𝑧𝑋)) → ((𝑦(+g𝐺)((invg𝐺)‘𝑦))(+g𝐺)𝑧) = (𝑦(+g𝐺)(((invg𝐺)‘𝑦)(+g𝐺)𝑧)))
8475, 73, 81, 82, 83syl13anc 1369 . . . . . . . . . . . . . . . 16 (((((𝐺 ∈ TopGrp ∧ 𝑌 ∈ (NrmSGrp‘𝐺) ∧ 𝑆𝐽) ∧ 𝑦𝑋) ∧ 𝑧𝑆) ∧ (((invg𝐺)‘𝑦)(+g𝐺)𝑧) ∈ 𝑌) → ((𝑦(+g𝐺)((invg𝐺)‘𝑦))(+g𝐺)𝑧) = (𝑦(+g𝐺)(((invg𝐺)‘𝑦)(+g𝐺)𝑧)))
854, 50, 76grplid 18962 . . . . . . . . . . . . . . . . 17 ((𝐺 ∈ Grp ∧ 𝑧𝑋) → ((0g𝐺)(+g𝐺)𝑧) = 𝑧)
8675, 82, 85syl2anc 582 . . . . . . . . . . . . . . . 16 (((((𝐺 ∈ TopGrp ∧ 𝑌 ∈ (NrmSGrp‘𝐺) ∧ 𝑆𝐽) ∧ 𝑦𝑋) ∧ 𝑧𝑆) ∧ (((invg𝐺)‘𝑦)(+g𝐺)𝑧) ∈ 𝑌) → ((0g𝐺)(+g𝐺)𝑧) = 𝑧)
8779, 84, 863eqtr3d 2774 . . . . . . . . . . . . . . 15 (((((𝐺 ∈ TopGrp ∧ 𝑌 ∈ (NrmSGrp‘𝐺) ∧ 𝑆𝐽) ∧ 𝑦𝑋) ∧ 𝑧𝑆) ∧ (((invg𝐺)‘𝑦)(+g𝐺)𝑧) ∈ 𝑌) → (𝑦(+g𝐺)(((invg𝐺)‘𝑦)(+g𝐺)𝑧)) = 𝑧)
88 simplr 767 . . . . . . . . . . . . . . 15 (((((𝐺 ∈ TopGrp ∧ 𝑌 ∈ (NrmSGrp‘𝐺) ∧ 𝑆𝐽) ∧ 𝑦𝑋) ∧ 𝑧𝑆) ∧ (((invg𝐺)‘𝑦)(+g𝐺)𝑧) ∈ 𝑌) → 𝑧𝑆)
8987, 88eqeltrd 2826 . . . . . . . . . . . . . 14 (((((𝐺 ∈ TopGrp ∧ 𝑌 ∈ (NrmSGrp‘𝐺) ∧ 𝑆𝐽) ∧ 𝑦𝑋) ∧ 𝑧𝑆) ∧ (((invg𝐺)‘𝑦)(+g𝐺)𝑧) ∈ 𝑌) → (𝑦(+g𝐺)(((invg𝐺)‘𝑦)(+g𝐺)𝑧)) ∈ 𝑆)
90 oveq1 7431 . . . . . . . . . . . . . . . 16 (𝑎 = 𝑦 → (𝑎(+g𝐺)(((invg𝐺)‘𝑦)(+g𝐺)𝑧)) = (𝑦(+g𝐺)(((invg𝐺)‘𝑦)(+g𝐺)𝑧)))
9190eleq1d 2811 . . . . . . . . . . . . . . 15 (𝑎 = 𝑦 → ((𝑎(+g𝐺)(((invg𝐺)‘𝑦)(+g𝐺)𝑧)) ∈ 𝑆 ↔ (𝑦(+g𝐺)(((invg𝐺)‘𝑦)(+g𝐺)𝑧)) ∈ 𝑆))
92 eqid 2726 . . . . . . . . . . . . . . . 16 (𝑎𝑋 ↦ (𝑎(+g𝐺)(((invg𝐺)‘𝑦)(+g𝐺)𝑧))) = (𝑎𝑋 ↦ (𝑎(+g𝐺)(((invg𝐺)‘𝑦)(+g𝐺)𝑧)))
9392mptpreima 6249 . . . . . . . . . . . . . . 15 ((𝑎𝑋 ↦ (𝑎(+g𝐺)(((invg𝐺)‘𝑦)(+g𝐺)𝑧))) “ 𝑆) = {𝑎𝑋 ∣ (𝑎(+g𝐺)(((invg𝐺)‘𝑦)(+g𝐺)𝑧)) ∈ 𝑆}
9491, 93elrab2 3684 . . . . . . . . . . . . . 14 (𝑦 ∈ ((𝑎𝑋 ↦ (𝑎(+g𝐺)(((invg𝐺)‘𝑦)(+g𝐺)𝑧))) “ 𝑆) ↔ (𝑦𝑋 ∧ (𝑦(+g𝐺)(((invg𝐺)‘𝑦)(+g𝐺)𝑧)) ∈ 𝑆))
9573, 89, 94sylanbrc 581 . . . . . . . . . . . . 13 (((((𝐺 ∈ TopGrp ∧ 𝑌 ∈ (NrmSGrp‘𝐺) ∧ 𝑆𝐽) ∧ 𝑦𝑋) ∧ 𝑧𝑆) ∧ (((invg𝐺)‘𝑦)(+g𝐺)𝑧) ∈ 𝑌) → 𝑦 ∈ ((𝑎𝑋 ↦ (𝑎(+g𝐺)(((invg𝐺)‘𝑦)(+g𝐺)𝑧))) “ 𝑆))
96 ecexg 8738 . . . . . . . . . . . . . . . . . . 19 ((𝐺 ~QG 𝑌) ∈ V → [𝑥](𝐺 ~QG 𝑌) ∈ V)
977, 96ax-mp 5 . . . . . . . . . . . . . . . . . 18 [𝑥](𝐺 ~QG 𝑌) ∈ V
9897, 6fnmpti 6704 . . . . . . . . . . . . . . . . 17 𝐹 Fn 𝑋
9928ad3antrrr 728 . . . . . . . . . . . . . . . . 17 ((((((𝐺 ∈ TopGrp ∧ 𝑌 ∈ (NrmSGrp‘𝐺) ∧ 𝑆𝐽) ∧ 𝑦𝑋) ∧ 𝑧𝑆) ∧ (((invg𝐺)‘𝑦)(+g𝐺)𝑧) ∈ 𝑌) ∧ 𝑎𝑋) → 𝑆𝑋)
100 fnfvima 7250 . . . . . . . . . . . . . . . . . 18 ((𝐹 Fn 𝑋𝑆𝑋 ∧ (𝑎(+g𝐺)(((invg𝐺)‘𝑦)(+g𝐺)𝑧)) ∈ 𝑆) → (𝐹‘(𝑎(+g𝐺)(((invg𝐺)‘𝑦)(+g𝐺)𝑧))) ∈ (𝐹𝑆))
1011003expia 1118 . . . . . . . . . . . . . . . . 17 ((𝐹 Fn 𝑋𝑆𝑋) → ((𝑎(+g𝐺)(((invg𝐺)‘𝑦)(+g𝐺)𝑧)) ∈ 𝑆 → (𝐹‘(𝑎(+g𝐺)(((invg𝐺)‘𝑦)(+g𝐺)𝑧))) ∈ (𝐹𝑆)))
10298, 99, 101sylancr 585 . . . . . . . . . . . . . . . 16 ((((((𝐺 ∈ TopGrp ∧ 𝑌 ∈ (NrmSGrp‘𝐺) ∧ 𝑆𝐽) ∧ 𝑦𝑋) ∧ 𝑧𝑆) ∧ (((invg𝐺)‘𝑦)(+g𝐺)𝑧) ∈ 𝑌) ∧ 𝑎𝑋) → ((𝑎(+g𝐺)(((invg𝐺)‘𝑦)(+g𝐺)𝑧)) ∈ 𝑆 → (𝐹‘(𝑎(+g𝐺)(((invg𝐺)‘𝑦)(+g𝐺)𝑧))) ∈ (𝐹𝑆)))
10375adantr 479 . . . . . . . . . . . . . . . . . . . 20 ((((((𝐺 ∈ TopGrp ∧ 𝑌 ∈ (NrmSGrp‘𝐺) ∧ 𝑆𝐽) ∧ 𝑦𝑋) ∧ 𝑧𝑆) ∧ (((invg𝐺)‘𝑦)(+g𝐺)𝑧) ∈ 𝑌) ∧ 𝑎𝑋) → 𝐺 ∈ Grp)
104 simpr 483 . . . . . . . . . . . . . . . . . . . 20 ((((((𝐺 ∈ TopGrp ∧ 𝑌 ∈ (NrmSGrp‘𝐺) ∧ 𝑆𝐽) ∧ 𝑦𝑋) ∧ 𝑧𝑆) ∧ (((invg𝐺)‘𝑦)(+g𝐺)𝑧) ∈ 𝑌) ∧ 𝑎𝑋) → 𝑎𝑋)
10561adantr 479 . . . . . . . . . . . . . . . . . . . 20 ((((((𝐺 ∈ TopGrp ∧ 𝑌 ∈ (NrmSGrp‘𝐺) ∧ 𝑆𝐽) ∧ 𝑦𝑋) ∧ 𝑧𝑆) ∧ (((invg𝐺)‘𝑦)(+g𝐺)𝑧) ∈ 𝑌) ∧ 𝑎𝑋) → (((invg𝐺)‘𝑦)(+g𝐺)𝑧) ∈ 𝑋)
1064, 50grpcl 18936 . . . . . . . . . . . . . . . . . . . 20 ((𝐺 ∈ Grp ∧ 𝑎𝑋 ∧ (((invg𝐺)‘𝑦)(+g𝐺)𝑧) ∈ 𝑋) → (𝑎(+g𝐺)(((invg𝐺)‘𝑦)(+g𝐺)𝑧)) ∈ 𝑋)
107103, 104, 105, 106syl3anc 1368 . . . . . . . . . . . . . . . . . . 19 ((((((𝐺 ∈ TopGrp ∧ 𝑌 ∈ (NrmSGrp‘𝐺) ∧ 𝑆𝐽) ∧ 𝑦𝑋) ∧ 𝑧𝑆) ∧ (((invg𝐺)‘𝑦)(+g𝐺)𝑧) ∈ 𝑌) ∧ 𝑎𝑋) → (𝑎(+g𝐺)(((invg𝐺)‘𝑦)(+g𝐺)𝑧)) ∈ 𝑋)
108 eceq1 8773 . . . . . . . . . . . . . . . . . . . 20 (𝑥 = (𝑎(+g𝐺)(((invg𝐺)‘𝑦)(+g𝐺)𝑧)) → [𝑥](𝐺 ~QG 𝑌) = [(𝑎(+g𝐺)(((invg𝐺)‘𝑦)(+g𝐺)𝑧))](𝐺 ~QG 𝑌))
109108, 6, 97fvmpt3i 7014 . . . . . . . . . . . . . . . . . . 19 ((𝑎(+g𝐺)(((invg𝐺)‘𝑦)(+g𝐺)𝑧)) ∈ 𝑋 → (𝐹‘(𝑎(+g𝐺)(((invg𝐺)‘𝑦)(+g𝐺)𝑧))) = [(𝑎(+g𝐺)(((invg𝐺)‘𝑦)(+g𝐺)𝑧))](𝐺 ~QG 𝑌))
110107, 109syl 17 . . . . . . . . . . . . . . . . . 18 ((((((𝐺 ∈ TopGrp ∧ 𝑌 ∈ (NrmSGrp‘𝐺) ∧ 𝑆𝐽) ∧ 𝑦𝑋) ∧ 𝑧𝑆) ∧ (((invg𝐺)‘𝑦)(+g𝐺)𝑧) ∈ 𝑌) ∧ 𝑎𝑋) → (𝐹‘(𝑎(+g𝐺)(((invg𝐺)‘𝑦)(+g𝐺)𝑧))) = [(𝑎(+g𝐺)(((invg𝐺)‘𝑦)(+g𝐺)𝑧))](𝐺 ~QG 𝑌))
11143ad2antrr 724 . . . . . . . . . . . . . . . . . . 19 ((((((𝐺 ∈ TopGrp ∧ 𝑌 ∈ (NrmSGrp‘𝐺) ∧ 𝑆𝐽) ∧ 𝑦𝑋) ∧ 𝑧𝑆) ∧ (((invg𝐺)‘𝑦)(+g𝐺)𝑧) ∈ 𝑌) ∧ 𝑎𝑋) → (𝐺 ~QG 𝑌) Er 𝑋)
1124, 50, 76, 49grplinv 18984 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝐺 ∈ Grp ∧ 𝑎𝑋) → (((invg𝐺)‘𝑎)(+g𝐺)𝑎) = (0g𝐺))
113103, 104, 112syl2anc 582 . . . . . . . . . . . . . . . . . . . . . . 23 ((((((𝐺 ∈ TopGrp ∧ 𝑌 ∈ (NrmSGrp‘𝐺) ∧ 𝑆𝐽) ∧ 𝑦𝑋) ∧ 𝑧𝑆) ∧ (((invg𝐺)‘𝑦)(+g𝐺)𝑧) ∈ 𝑌) ∧ 𝑎𝑋) → (((invg𝐺)‘𝑎)(+g𝐺)𝑎) = (0g𝐺))
114113oveq1d 7439 . . . . . . . . . . . . . . . . . . . . . 22 ((((((𝐺 ∈ TopGrp ∧ 𝑌 ∈ (NrmSGrp‘𝐺) ∧ 𝑆𝐽) ∧ 𝑦𝑋) ∧ 𝑧𝑆) ∧ (((invg𝐺)‘𝑦)(+g𝐺)𝑧) ∈ 𝑌) ∧ 𝑎𝑋) → ((((invg𝐺)‘𝑎)(+g𝐺)𝑎)(+g𝐺)(((invg𝐺)‘𝑦)(+g𝐺)𝑧)) = ((0g𝐺)(+g𝐺)(((invg𝐺)‘𝑦)(+g𝐺)𝑧)))
1154, 49grpinvcl 18982 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝐺 ∈ Grp ∧ 𝑎𝑋) → ((invg𝐺)‘𝑎) ∈ 𝑋)
116103, 104, 115syl2anc 582 . . . . . . . . . . . . . . . . . . . . . . 23 ((((((𝐺 ∈ TopGrp ∧ 𝑌 ∈ (NrmSGrp‘𝐺) ∧ 𝑆𝐽) ∧ 𝑦𝑋) ∧ 𝑧𝑆) ∧ (((invg𝐺)‘𝑦)(+g𝐺)𝑧) ∈ 𝑌) ∧ 𝑎𝑋) → ((invg𝐺)‘𝑎) ∈ 𝑋)
1174, 50grpass 18937 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝐺 ∈ Grp ∧ (((invg𝐺)‘𝑎) ∈ 𝑋𝑎𝑋 ∧ (((invg𝐺)‘𝑦)(+g𝐺)𝑧) ∈ 𝑋)) → ((((invg𝐺)‘𝑎)(+g𝐺)𝑎)(+g𝐺)(((invg𝐺)‘𝑦)(+g𝐺)𝑧)) = (((invg𝐺)‘𝑎)(+g𝐺)(𝑎(+g𝐺)(((invg𝐺)‘𝑦)(+g𝐺)𝑧))))
118103, 116, 104, 105, 117syl13anc 1369 . . . . . . . . . . . . . . . . . . . . . 22 ((((((𝐺 ∈ TopGrp ∧ 𝑌 ∈ (NrmSGrp‘𝐺) ∧ 𝑆𝐽) ∧ 𝑦𝑋) ∧ 𝑧𝑆) ∧ (((invg𝐺)‘𝑦)(+g𝐺)𝑧) ∈ 𝑌) ∧ 𝑎𝑋) → ((((invg𝐺)‘𝑎)(+g𝐺)𝑎)(+g𝐺)(((invg𝐺)‘𝑦)(+g𝐺)𝑧)) = (((invg𝐺)‘𝑎)(+g𝐺)(𝑎(+g𝐺)(((invg𝐺)‘𝑦)(+g𝐺)𝑧))))
1194, 50, 76grplid 18962 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝐺 ∈ Grp ∧ (((invg𝐺)‘𝑦)(+g𝐺)𝑧) ∈ 𝑋) → ((0g𝐺)(+g𝐺)(((invg𝐺)‘𝑦)(+g𝐺)𝑧)) = (((invg𝐺)‘𝑦)(+g𝐺)𝑧))
120103, 105, 119syl2anc 582 . . . . . . . . . . . . . . . . . . . . . 22 ((((((𝐺 ∈ TopGrp ∧ 𝑌 ∈ (NrmSGrp‘𝐺) ∧ 𝑆𝐽) ∧ 𝑦𝑋) ∧ 𝑧𝑆) ∧ (((invg𝐺)‘𝑦)(+g𝐺)𝑧) ∈ 𝑌) ∧ 𝑎𝑋) → ((0g𝐺)(+g𝐺)(((invg𝐺)‘𝑦)(+g𝐺)𝑧)) = (((invg𝐺)‘𝑦)(+g𝐺)𝑧))
121114, 118, 1203eqtr3d 2774 . . . . . . . . . . . . . . . . . . . . 21 ((((((𝐺 ∈ TopGrp ∧ 𝑌 ∈ (NrmSGrp‘𝐺) ∧ 𝑆𝐽) ∧ 𝑦𝑋) ∧ 𝑧𝑆) ∧ (((invg𝐺)‘𝑦)(+g𝐺)𝑧) ∈ 𝑌) ∧ 𝑎𝑋) → (((invg𝐺)‘𝑎)(+g𝐺)(𝑎(+g𝐺)(((invg𝐺)‘𝑦)(+g𝐺)𝑧))) = (((invg𝐺)‘𝑦)(+g𝐺)𝑧))
122 simplr 767 . . . . . . . . . . . . . . . . . . . . 21 ((((((𝐺 ∈ TopGrp ∧ 𝑌 ∈ (NrmSGrp‘𝐺) ∧ 𝑆𝐽) ∧ 𝑦𝑋) ∧ 𝑧𝑆) ∧ (((invg𝐺)‘𝑦)(+g𝐺)𝑧) ∈ 𝑌) ∧ 𝑎𝑋) → (((invg𝐺)‘𝑦)(+g𝐺)𝑧) ∈ 𝑌)
123121, 122eqeltrd 2826 . . . . . . . . . . . . . . . . . . . 20 ((((((𝐺 ∈ TopGrp ∧ 𝑌 ∈ (NrmSGrp‘𝐺) ∧ 𝑆𝐽) ∧ 𝑦𝑋) ∧ 𝑧𝑆) ∧ (((invg𝐺)‘𝑦)(+g𝐺)𝑧) ∈ 𝑌) ∧ 𝑎𝑋) → (((invg𝐺)‘𝑎)(+g𝐺)(𝑎(+g𝐺)(((invg𝐺)‘𝑦)(+g𝐺)𝑧))) ∈ 𝑌)
12448ad2antrr 724 . . . . . . . . . . . . . . . . . . . . 21 ((((((𝐺 ∈ TopGrp ∧ 𝑌 ∈ (NrmSGrp‘𝐺) ∧ 𝑆𝐽) ∧ 𝑦𝑋) ∧ 𝑧𝑆) ∧ (((invg𝐺)‘𝑦)(+g𝐺)𝑧) ∈ 𝑌) ∧ 𝑎𝑋) → 𝑌𝑋)
1254, 49, 50, 41eqgval 19171 . . . . . . . . . . . . . . . . . . . . 21 ((𝐺 ∈ Grp ∧ 𝑌𝑋) → (𝑎(𝐺 ~QG 𝑌)(𝑎(+g𝐺)(((invg𝐺)‘𝑦)(+g𝐺)𝑧)) ↔ (𝑎𝑋 ∧ (𝑎(+g𝐺)(((invg𝐺)‘𝑦)(+g𝐺)𝑧)) ∈ 𝑋 ∧ (((invg𝐺)‘𝑎)(+g𝐺)(𝑎(+g𝐺)(((invg𝐺)‘𝑦)(+g𝐺)𝑧))) ∈ 𝑌)))
126103, 124, 125syl2anc 582 . . . . . . . . . . . . . . . . . . . 20 ((((((𝐺 ∈ TopGrp ∧ 𝑌 ∈ (NrmSGrp‘𝐺) ∧ 𝑆𝐽) ∧ 𝑦𝑋) ∧ 𝑧𝑆) ∧ (((invg𝐺)‘𝑦)(+g𝐺)𝑧) ∈ 𝑌) ∧ 𝑎𝑋) → (𝑎(𝐺 ~QG 𝑌)(𝑎(+g𝐺)(((invg𝐺)‘𝑦)(+g𝐺)𝑧)) ↔ (𝑎𝑋 ∧ (𝑎(+g𝐺)(((invg𝐺)‘𝑦)(+g𝐺)𝑧)) ∈ 𝑋 ∧ (((invg𝐺)‘𝑎)(+g𝐺)(𝑎(+g𝐺)(((invg𝐺)‘𝑦)(+g𝐺)𝑧))) ∈ 𝑌)))
127104, 107, 123, 126mpbir3and 1339 . . . . . . . . . . . . . . . . . . 19 ((((((𝐺 ∈ TopGrp ∧ 𝑌 ∈ (NrmSGrp‘𝐺) ∧ 𝑆𝐽) ∧ 𝑦𝑋) ∧ 𝑧𝑆) ∧ (((invg𝐺)‘𝑦)(+g𝐺)𝑧) ∈ 𝑌) ∧ 𝑎𝑋) → 𝑎(𝐺 ~QG 𝑌)(𝑎(+g𝐺)(((invg𝐺)‘𝑦)(+g𝐺)𝑧)))
128111, 127erthi 8787 . . . . . . . . . . . . . . . . . 18 ((((((𝐺 ∈ TopGrp ∧ 𝑌 ∈ (NrmSGrp‘𝐺) ∧ 𝑆𝐽) ∧ 𝑦𝑋) ∧ 𝑧𝑆) ∧ (((invg𝐺)‘𝑦)(+g𝐺)𝑧) ∈ 𝑌) ∧ 𝑎𝑋) → [𝑎](𝐺 ~QG 𝑌) = [(𝑎(+g𝐺)(((invg𝐺)‘𝑦)(+g𝐺)𝑧))](𝐺 ~QG 𝑌))
129110, 128eqtr4d 2769 . . . . . . . . . . . . . . . . 17 ((((((𝐺 ∈ TopGrp ∧ 𝑌 ∈ (NrmSGrp‘𝐺) ∧ 𝑆𝐽) ∧ 𝑦𝑋) ∧ 𝑧𝑆) ∧ (((invg𝐺)‘𝑦)(+g𝐺)𝑧) ∈ 𝑌) ∧ 𝑎𝑋) → (𝐹‘(𝑎(+g𝐺)(((invg𝐺)‘𝑦)(+g𝐺)𝑧))) = [𝑎](𝐺 ~QG 𝑌))
130129eleq1d 2811 . . . . . . . . . . . . . . . 16 ((((((𝐺 ∈ TopGrp ∧ 𝑌 ∈ (NrmSGrp‘𝐺) ∧ 𝑆𝐽) ∧ 𝑦𝑋) ∧ 𝑧𝑆) ∧ (((invg𝐺)‘𝑦)(+g𝐺)𝑧) ∈ 𝑌) ∧ 𝑎𝑋) → ((𝐹‘(𝑎(+g𝐺)(((invg𝐺)‘𝑦)(+g𝐺)𝑧))) ∈ (𝐹𝑆) ↔ [𝑎](𝐺 ~QG 𝑌) ∈ (𝐹𝑆)))
131102, 130sylibd 238 . . . . . . . . . . . . . . 15 ((((((𝐺 ∈ TopGrp ∧ 𝑌 ∈ (NrmSGrp‘𝐺) ∧ 𝑆𝐽) ∧ 𝑦𝑋) ∧ 𝑧𝑆) ∧ (((invg𝐺)‘𝑦)(+g𝐺)𝑧) ∈ 𝑌) ∧ 𝑎𝑋) → ((𝑎(+g𝐺)(((invg𝐺)‘𝑦)(+g𝐺)𝑧)) ∈ 𝑆 → [𝑎](𝐺 ~QG 𝑌) ∈ (𝐹𝑆)))
132131ss2rabdv 4072 . . . . . . . . . . . . . 14 (((((𝐺 ∈ TopGrp ∧ 𝑌 ∈ (NrmSGrp‘𝐺) ∧ 𝑆𝐽) ∧ 𝑦𝑋) ∧ 𝑧𝑆) ∧ (((invg𝐺)‘𝑦)(+g𝐺)𝑧) ∈ 𝑌) → {𝑎𝑋 ∣ (𝑎(+g𝐺)(((invg𝐺)‘𝑦)(+g𝐺)𝑧)) ∈ 𝑆} ⊆ {𝑎𝑋 ∣ [𝑎](𝐺 ~QG 𝑌) ∈ (𝐹𝑆)})
133 eceq1 8773 . . . . . . . . . . . . . . . . 17 (𝑥 = 𝑎 → [𝑥](𝐺 ~QG 𝑌) = [𝑎](𝐺 ~QG 𝑌))
134133cbvmptv 5266 . . . . . . . . . . . . . . . 16 (𝑥𝑋 ↦ [𝑥](𝐺 ~QG 𝑌)) = (𝑎𝑋 ↦ [𝑎](𝐺 ~QG 𝑌))
1356, 134eqtri 2754 . . . . . . . . . . . . . . 15 𝐹 = (𝑎𝑋 ↦ [𝑎](𝐺 ~QG 𝑌))
136135mptpreima 6249 . . . . . . . . . . . . . 14 (𝐹 “ (𝐹𝑆)) = {𝑎𝑋 ∣ [𝑎](𝐺 ~QG 𝑌) ∈ (𝐹𝑆)}
137132, 93, 1363sstr4g 4025 . . . . . . . . . . . . 13 (((((𝐺 ∈ TopGrp ∧ 𝑌 ∈ (NrmSGrp‘𝐺) ∧ 𝑆𝐽) ∧ 𝑦𝑋) ∧ 𝑧𝑆) ∧ (((invg𝐺)‘𝑦)(+g𝐺)𝑧) ∈ 𝑌) → ((𝑎𝑋 ↦ (𝑎(+g𝐺)(((invg𝐺)‘𝑦)(+g𝐺)𝑧))) “ 𝑆) ⊆ (𝐹 “ (𝐹𝑆)))
138 eleq2 2815 . . . . . . . . . . . . . . 15 (𝑢 = ((𝑎𝑋 ↦ (𝑎(+g𝐺)(((invg𝐺)‘𝑦)(+g𝐺)𝑧))) “ 𝑆) → (𝑦𝑢𝑦 ∈ ((𝑎𝑋 ↦ (𝑎(+g𝐺)(((invg𝐺)‘𝑦)(+g𝐺)𝑧))) “ 𝑆)))
139 sseq1 4005 . . . . . . . . . . . . . . 15 (𝑢 = ((𝑎𝑋 ↦ (𝑎(+g𝐺)(((invg𝐺)‘𝑦)(+g𝐺)𝑧))) “ 𝑆) → (𝑢 ⊆ (𝐹 “ (𝐹𝑆)) ↔ ((𝑎𝑋 ↦ (𝑎(+g𝐺)(((invg𝐺)‘𝑦)(+g𝐺)𝑧))) “ 𝑆) ⊆ (𝐹 “ (𝐹𝑆))))
140138, 139anbi12d 630 . . . . . . . . . . . . . 14 (𝑢 = ((𝑎𝑋 ↦ (𝑎(+g𝐺)(((invg𝐺)‘𝑦)(+g𝐺)𝑧))) “ 𝑆) → ((𝑦𝑢𝑢 ⊆ (𝐹 “ (𝐹𝑆))) ↔ (𝑦 ∈ ((𝑎𝑋 ↦ (𝑎(+g𝐺)(((invg𝐺)‘𝑦)(+g𝐺)𝑧))) “ 𝑆) ∧ ((𝑎𝑋 ↦ (𝑎(+g𝐺)(((invg𝐺)‘𝑦)(+g𝐺)𝑧))) “ 𝑆) ⊆ (𝐹 “ (𝐹𝑆)))))
141140rspcev 3608 . . . . . . . . . . . . 13 ((((𝑎𝑋 ↦ (𝑎(+g𝐺)(((invg𝐺)‘𝑦)(+g𝐺)𝑧))) “ 𝑆) ∈ 𝐽 ∧ (𝑦 ∈ ((𝑎𝑋 ↦ (𝑎(+g𝐺)(((invg𝐺)‘𝑦)(+g𝐺)𝑧))) “ 𝑆) ∧ ((𝑎𝑋 ↦ (𝑎(+g𝐺)(((invg𝐺)‘𝑦)(+g𝐺)𝑧))) “ 𝑆) ⊆ (𝐹 “ (𝐹𝑆)))) → ∃𝑢𝐽 (𝑦𝑢𝑢 ⊆ (𝐹 “ (𝐹𝑆))))
14272, 95, 137, 141syl12anc 835 . . . . . . . . . . . 12 (((((𝐺 ∈ TopGrp ∧ 𝑌 ∈ (NrmSGrp‘𝐺) ∧ 𝑆𝐽) ∧ 𝑦𝑋) ∧ 𝑧𝑆) ∧ (((invg𝐺)‘𝑦)(+g𝐺)𝑧) ∈ 𝑌) → ∃𝑢𝐽 (𝑦𝑢𝑢 ⊆ (𝐹 “ (𝐹𝑆))))
1431423ad2antr3 1187 . . . . . . . . . . 11 (((((𝐺 ∈ TopGrp ∧ 𝑌 ∈ (NrmSGrp‘𝐺) ∧ 𝑆𝐽) ∧ 𝑦𝑋) ∧ 𝑧𝑆) ∧ (𝑦𝑋𝑧𝑋 ∧ (((invg𝐺)‘𝑦)(+g𝐺)𝑧) ∈ 𝑌)) → ∃𝑢𝐽 (𝑦𝑢𝑢 ⊆ (𝐹 “ (𝐹𝑆))))
144143ex 411 . . . . . . . . . 10 ((((𝐺 ∈ TopGrp ∧ 𝑌 ∈ (NrmSGrp‘𝐺) ∧ 𝑆𝐽) ∧ 𝑦𝑋) ∧ 𝑧𝑆) → ((𝑦𝑋𝑧𝑋 ∧ (((invg𝐺)‘𝑦)(+g𝐺)𝑧) ∈ 𝑌) → ∃𝑢𝐽 (𝑦𝑢𝑢 ⊆ (𝐹 “ (𝐹𝑆)))))
14553, 144sylbid 239 . . . . . . . . 9 ((((𝐺 ∈ TopGrp ∧ 𝑌 ∈ (NrmSGrp‘𝐺) ∧ 𝑆𝐽) ∧ 𝑦𝑋) ∧ 𝑧𝑆) → ((𝐹𝑧) = [𝑦](𝐺 ~QG 𝑌) → ∃𝑢𝐽 (𝑦𝑢𝑢 ⊆ (𝐹 “ (𝐹𝑆)))))
146145rexlimdva 3145 . . . . . . . 8 (((𝐺 ∈ TopGrp ∧ 𝑌 ∈ (NrmSGrp‘𝐺) ∧ 𝑆𝐽) ∧ 𝑦𝑋) → (∃𝑧𝑆 (𝐹𝑧) = [𝑦](𝐺 ~QG 𝑌) → ∃𝑢𝐽 (𝑦𝑢𝑢 ⊆ (𝐹 “ (𝐹𝑆)))))
14721, 146syl5 34 . . . . . . 7 (((𝐺 ∈ TopGrp ∧ 𝑌 ∈ (NrmSGrp‘𝐺) ∧ 𝑆𝐽) ∧ 𝑦𝑋) → ([𝑦](𝐺 ~QG 𝑌) ∈ (𝐹𝑆) → ∃𝑢𝐽 (𝑦𝑢𝑢 ⊆ (𝐹 “ (𝐹𝑆)))))
148147expimpd 452 . . . . . 6 ((𝐺 ∈ TopGrp ∧ 𝑌 ∈ (NrmSGrp‘𝐺) ∧ 𝑆𝐽) → ((𝑦𝑋 ∧ [𝑦](𝐺 ~QG 𝑌) ∈ (𝐹𝑆)) → ∃𝑢𝐽 (𝑦𝑢𝑢 ⊆ (𝐹 “ (𝐹𝑆)))))
14918, 148biimtrid 241 . . . . 5 ((𝐺 ∈ TopGrp ∧ 𝑌 ∈ (NrmSGrp‘𝐺) ∧ 𝑆𝐽) → (𝑦 ∈ (𝐹 “ (𝐹𝑆)) → ∃𝑢𝐽 (𝑦𝑢𝑢 ⊆ (𝐹 “ (𝐹𝑆)))))
150149ralrimiv 3135 . . . 4 ((𝐺 ∈ TopGrp ∧ 𝑌 ∈ (NrmSGrp‘𝐺) ∧ 𝑆𝐽) → ∀𝑦 ∈ (𝐹 “ (𝐹𝑆))∃𝑢𝐽 (𝑦𝑢𝑢 ⊆ (𝐹 “ (𝐹𝑆))))
151 topontop 22906 . . . . 5 (𝐽 ∈ (TopOn‘𝑋) → 𝐽 ∈ Top)
152 eltop2 22969 . . . . 5 (𝐽 ∈ Top → ((𝐹 “ (𝐹𝑆)) ∈ 𝐽 ↔ ∀𝑦 ∈ (𝐹 “ (𝐹𝑆))∃𝑢𝐽 (𝑦𝑢𝑢 ⊆ (𝐹 “ (𝐹𝑆)))))
15324, 151, 1523syl 18 . . . 4 ((𝐺 ∈ TopGrp ∧ 𝑌 ∈ (NrmSGrp‘𝐺) ∧ 𝑆𝐽) → ((𝐹 “ (𝐹𝑆)) ∈ 𝐽 ↔ ∀𝑦 ∈ (𝐹 “ (𝐹𝑆))∃𝑢𝐽 (𝑦𝑢𝑢 ⊆ (𝐹 “ (𝐹𝑆)))))
154150, 153mpbird 256 . . 3 ((𝐺 ∈ TopGrp ∧ 𝑌 ∈ (NrmSGrp‘𝐺) ∧ 𝑆𝐽) → (𝐹 “ (𝐹𝑆)) ∈ 𝐽)
155 elqtop3 23698 . . . 4 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹:𝑋onto→(𝑋 / (𝐺 ~QG 𝑌))) → ((𝐹𝑆) ∈ (𝐽 qTop 𝐹) ↔ ((𝐹𝑆) ⊆ (𝑋 / (𝐺 ~QG 𝑌)) ∧ (𝐹 “ (𝐹𝑆)) ∈ 𝐽)))
15624, 10, 155syl2anc 582 . . 3 ((𝐺 ∈ TopGrp ∧ 𝑌 ∈ (NrmSGrp‘𝐺) ∧ 𝑆𝐽) → ((𝐹𝑆) ∈ (𝐽 qTop 𝐹) ↔ ((𝐹𝑆) ⊆ (𝑋 / (𝐺 ~QG 𝑌)) ∧ (𝐹 “ (𝐹𝑆)) ∈ 𝐽)))
15713, 154, 156mpbir2and 711 . 2 ((𝐺 ∈ TopGrp ∧ 𝑌 ∈ (NrmSGrp‘𝐺) ∧ 𝑆𝐽) → (𝐹𝑆) ∈ (𝐽 qTop 𝐹))
1583, 5, 6, 8, 9qusval 17557 . . 3 ((𝐺 ∈ TopGrp ∧ 𝑌 ∈ (NrmSGrp‘𝐺) ∧ 𝑆𝐽) → 𝐻 = (𝐹s 𝐺))
159 qustgpopn.k . . 3 𝐾 = (TopOpen‘𝐻)
160158, 5, 10, 9, 22, 159imastopn 23715 . 2 ((𝐺 ∈ TopGrp ∧ 𝑌 ∈ (NrmSGrp‘𝐺) ∧ 𝑆𝐽) → 𝐾 = (𝐽 qTop 𝐹))
161157, 160eleqtrrd 2829 1 ((𝐺 ∈ TopGrp ∧ 𝑌 ∈ (NrmSGrp‘𝐺) ∧ 𝑆𝐽) → (𝐹𝑆) ∈ 𝐾)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 394  w3a 1084   = wceq 1534  wcel 2099  wral 3051  wrex 3060  {crab 3419  Vcvv 3462  wss 3947   class class class wbr 5153  cmpt 5236  ccnv 5681  ran crn 5683  cima 5685  Fun wfun 6548   Fn wfn 6549  ontowfo 6552  cfv 6554  (class class class)co 7424   Er wer 8731  [cec 8732   / cqs 8733  Basecbs 17213  +gcplusg 17266  TopOpenctopn 17436  0gc0g 17454   qTop cqtop 17518   /s cqus 17520  Grpcgrp 18928  invgcminusg 18929  SubGrpcsubg 19114  NrmSGrpcnsg 19115   ~QG cqg 19116  oppgcoppg 19339  Topctop 22886  TopOnctopon 22903   Cn ccn 23219  Homeochmeo 23748  TopGrpctgp 24066
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2697  ax-rep 5290  ax-sep 5304  ax-nul 5311  ax-pow 5369  ax-pr 5433  ax-un 7746  ax-cnex 11214  ax-resscn 11215  ax-1cn 11216  ax-icn 11217  ax-addcl 11218  ax-addrcl 11219  ax-mulcl 11220  ax-mulrcl 11221  ax-mulcom 11222  ax-addass 11223  ax-mulass 11224  ax-distr 11225  ax-i2m1 11226  ax-1ne0 11227  ax-1rid 11228  ax-rnegex 11229  ax-rrecex 11230  ax-cnre 11231  ax-pre-lttri 11232  ax-pre-lttrn 11233  ax-pre-ltadd 11234  ax-pre-mulgt0 11235
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2704  df-cleq 2718  df-clel 2803  df-nfc 2878  df-ne 2931  df-nel 3037  df-ral 3052  df-rex 3061  df-rmo 3364  df-reu 3365  df-rab 3420  df-v 3464  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-pss 3967  df-nul 4326  df-if 4534  df-pw 4609  df-sn 4634  df-pr 4636  df-tp 4638  df-op 4640  df-uni 4914  df-iun 5003  df-br 5154  df-opab 5216  df-mpt 5237  df-tr 5271  df-id 5580  df-eprel 5586  df-po 5594  df-so 5595  df-fr 5637  df-we 5639  df-xp 5688  df-rel 5689  df-cnv 5690  df-co 5691  df-dm 5692  df-rn 5693  df-res 5694  df-ima 5695  df-pred 6312  df-ord 6379  df-on 6380  df-lim 6381  df-suc 6382  df-iota 6506  df-fun 6556  df-fn 6557  df-f 6558  df-f1 6559  df-fo 6560  df-f1o 6561  df-fv 6562  df-riota 7380  df-ov 7427  df-oprab 7428  df-mpo 7429  df-om 7877  df-1st 8003  df-2nd 8004  df-tpos 8241  df-frecs 8296  df-wrecs 8327  df-recs 8401  df-rdg 8440  df-1o 8496  df-er 8734  df-ec 8736  df-qs 8740  df-map 8857  df-en 8975  df-dom 8976  df-sdom 8977  df-fin 8978  df-sup 9485  df-inf 9486  df-pnf 11300  df-mnf 11301  df-xr 11302  df-ltxr 11303  df-le 11304  df-sub 11496  df-neg 11497  df-nn 12265  df-2 12327  df-3 12328  df-4 12329  df-5 12330  df-6 12331  df-7 12332  df-8 12333  df-9 12334  df-n0 12525  df-z 12611  df-dec 12730  df-uz 12875  df-fz 13539  df-struct 17149  df-sets 17166  df-slot 17184  df-ndx 17196  df-base 17214  df-ress 17243  df-plusg 17279  df-mulr 17280  df-sca 17282  df-vsca 17283  df-ip 17284  df-tset 17285  df-ple 17286  df-ds 17288  df-rest 17437  df-topn 17438  df-0g 17456  df-topgen 17458  df-qtop 17522  df-imas 17523  df-qus 17524  df-plusf 18632  df-mgm 18633  df-sgrp 18712  df-mnd 18728  df-grp 18931  df-minusg 18932  df-subg 19117  df-nsg 19118  df-eqg 19119  df-oppg 19340  df-top 22887  df-topon 22904  df-topsp 22926  df-bases 22940  df-cn 23222  df-cnp 23223  df-tx 23557  df-hmeo 23750  df-tmd 24067  df-tgp 24068
This theorem is referenced by:  qustgplem  24116
  Copyright terms: Public domain W3C validator