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Theorem qustgpopn 23376
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 6014 . . . 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 7374 . . . . . . 7 (𝐺 ~QG 𝑌) ∈ V
87a1i 11 . . . . . 6 ((𝐺 ∈ TopGrp ∧ 𝑌 ∈ (NrmSGrp‘𝐺) ∧ 𝑆𝐽) → (𝐺 ~QG 𝑌) ∈ V)
9 simp1 1136 . . . . . 6 ((𝐺 ∈ TopGrp ∧ 𝑌 ∈ (NrmSGrp‘𝐺) ∧ 𝑆𝐽) → 𝐺 ∈ TopGrp)
103, 5, 6, 8, 9quslem 17351 . . . . 5 ((𝐺 ∈ TopGrp ∧ 𝑌 ∈ (NrmSGrp‘𝐺) ∧ 𝑆𝐽) → 𝐹:𝑋onto→(𝑋 / (𝐺 ~QG 𝑌)))
11 forn 6746 . . . . 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 3987 . . 3 ((𝐺 ∈ TopGrp ∧ 𝑌 ∈ (NrmSGrp‘𝐺) ∧ 𝑆𝐽) → (𝐹𝑆) ⊆ (𝑋 / (𝐺 ~QG 𝑌)))
14 eceq1 8611 . . . . . . . . . 10 (𝑥 = 𝑦 → [𝑥](𝐺 ~QG 𝑌) = [𝑦](𝐺 ~QG 𝑌))
1514cbvmptv 5209 . . . . . . . . 9 (𝑥𝑋 ↦ [𝑥](𝐺 ~QG 𝑌)) = (𝑦𝑋 ↦ [𝑦](𝐺 ~QG 𝑌))
166, 15eqtri 2765 . . . . . . . 8 𝐹 = (𝑦𝑋 ↦ [𝑦](𝐺 ~QG 𝑌))
1716mptpreima 6180 . . . . . . 7 (𝐹 “ (𝐹𝑆)) = {𝑦𝑋 ∣ [𝑦](𝐺 ~QG 𝑌) ∈ (𝐹𝑆)}
1817rabeq2i 3426 . . . . . 6 (𝑦 ∈ (𝐹 “ (𝐹𝑆)) ↔ (𝑦𝑋 ∧ [𝑦](𝐺 ~QG 𝑌) ∈ (𝐹𝑆)))
196funmpt2 6527 . . . . . . . . 9 Fun 𝐹
20 fvelima 6895 . . . . . . . . 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 23338 . . . . . . . . . . . . . . . . . 18 (𝐺 ∈ TopGrp → 𝐽 ∈ (TopOn‘𝑋))
249, 23syl 17 . . . . . . . . . . . . . . . . 17 ((𝐺 ∈ TopGrp ∧ 𝑌 ∈ (NrmSGrp‘𝐺) ∧ 𝑆𝐽) → 𝐽 ∈ (TopOn‘𝑋))
25 simp3 1138 . . . . . . . . . . . . . . . . 17 ((𝐺 ∈ TopGrp ∧ 𝑌 ∈ (NrmSGrp‘𝐺) ∧ 𝑆𝐽) → 𝑆𝐽)
26 toponss 22181 . . . . . . . . . . . . . . . . 17 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑆𝐽) → 𝑆𝑋)
2724, 25, 26syl2anc 585 . . . . . . . . . . . . . . . 16 ((𝐺 ∈ TopGrp ∧ 𝑌 ∈ (NrmSGrp‘𝐺) ∧ 𝑆𝐽) → 𝑆𝑋)
2827adantr 482 . . . . . . . . . . . . . . 15 (((𝐺 ∈ TopGrp ∧ 𝑌 ∈ (NrmSGrp‘𝐺) ∧ 𝑆𝐽) ∧ 𝑦𝑋) → 𝑆𝑋)
2928sselda 3935 . . . . . . . . . . . . . 14 ((((𝐺 ∈ TopGrp ∧ 𝑌 ∈ (NrmSGrp‘𝐺) ∧ 𝑆𝐽) ∧ 𝑦𝑋) ∧ 𝑧𝑆) → 𝑧𝑋)
30 eceq1 8611 . . . . . . . . . . . . . . 15 (𝑥 = 𝑧 → [𝑥](𝐺 ~QG 𝑌) = [𝑧](𝐺 ~QG 𝑌))
31 ecexg 8577 . . . . . . . . . . . . . . . 16 ((𝐺 ~QG 𝑌) ∈ V → [𝑧](𝐺 ~QG 𝑌) ∈ V)
327, 31ax-mp 5 . . . . . . . . . . . . . . 15 [𝑧](𝐺 ~QG 𝑌) ∈ V
3330, 6, 32fvmpt 6935 . . . . . . . . . . . . . 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 2739 . . . . . . . . . . . 12 ((((𝐺 ∈ TopGrp ∧ 𝑌 ∈ (NrmSGrp‘𝐺) ∧ 𝑆𝐽) ∧ 𝑦𝑋) ∧ 𝑧𝑆) → ((𝐹𝑧) = [𝑦](𝐺 ~QG 𝑌) ↔ [𝑧](𝐺 ~QG 𝑌) = [𝑦](𝐺 ~QG 𝑌)))
36 eqcom 2744 . . . . . . . . . . . 12 ([𝑧](𝐺 ~QG 𝑌) = [𝑦](𝐺 ~QG 𝑌) ↔ [𝑦](𝐺 ~QG 𝑌) = [𝑧](𝐺 ~QG 𝑌))
3735, 36bitrdi 287 . . . . . . . . . . 11 ((((𝐺 ∈ TopGrp ∧ 𝑌 ∈ (NrmSGrp‘𝐺) ∧ 𝑆𝐽) ∧ 𝑦𝑋) ∧ 𝑧𝑆) → ((𝐹𝑧) = [𝑦](𝐺 ~QG 𝑌) ↔ [𝑦](𝐺 ~QG 𝑌) = [𝑧](𝐺 ~QG 𝑌)))
38 nsgsubg 18882 . . . . . . . . . . . . . . 15 (𝑌 ∈ (NrmSGrp‘𝐺) → 𝑌 ∈ (SubGrp‘𝐺))
39383ad2ant2 1134 . . . . . . . . . . . . . 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 2737 . . . . . . . . . . . . . 14 (𝐺 ~QG 𝑌) = (𝐺 ~QG 𝑌)
424, 41eqger 18902 . . . . . . . . . . . . 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 8622 . . . . . . . . . . 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 18852 . . . . . . . . . . . . 13 (𝑌 ∈ (SubGrp‘𝐺) → 𝑌𝑋)
4840, 47syl 17 . . . . . . . . . . . 12 ((((𝐺 ∈ TopGrp ∧ 𝑌 ∈ (NrmSGrp‘𝐺) ∧ 𝑆𝐽) ∧ 𝑦𝑋) ∧ 𝑧𝑆) → 𝑌𝑋)
49 eqid 2737 . . . . . . . . . . . . 13 (invg𝐺) = (invg𝐺)
50 eqid 2737 . . . . . . . . . . . . 13 (+g𝐺) = (+g𝐺)
514, 49, 50, 41eqgval 18901 . . . . . . . . . . . 12 ((𝐺 ∈ TopGrp ∧ 𝑌𝑋) → (𝑦(𝐺 ~QG 𝑌)𝑧 ↔ (𝑦𝑋𝑧𝑋 ∧ (((invg𝐺)‘𝑦)(+g𝐺)𝑧) ∈ 𝑌)))
5246, 48, 51syl2anc 585 . . . . . . . . . . 11 ((((𝐺 ∈ TopGrp ∧ 𝑌 ∈ (NrmSGrp‘𝐺) ∧ 𝑆𝐽) ∧ 𝑦𝑋) ∧ 𝑧𝑆) → (𝑦(𝐺 ~QG 𝑌)𝑧 ↔ (𝑦𝑋𝑧𝑋 ∧ (((invg𝐺)‘𝑦)(+g𝐺)𝑧) ∈ 𝑌)))
5337, 45, 523bitr2d 307 . . . . . . . . . 10 ((((𝐺 ∈ TopGrp ∧ 𝑌 ∈ (NrmSGrp‘𝐺) ∧ 𝑆𝐽) ∧ 𝑦𝑋) ∧ 𝑧𝑆) → ((𝐹𝑧) = [𝑦](𝐺 ~QG 𝑌) ↔ (𝑦𝑋𝑧𝑋 ∧ (((invg𝐺)‘𝑦)(+g𝐺)𝑧) ∈ 𝑌)))
54 eqid 2737 . . . . . . . . . . . . . . . . . 18 (oppg𝐺) = (oppg𝐺)
55 eqid 2737 . . . . . . . . . . . . . . . . . 18 (+g‘(oppg𝐺)) = (+g‘(oppg𝐺))
5650, 54, 55oppgplus 19049 . . . . . . . . . . . . . . . . 17 ((((invg𝐺)‘𝑦)(+g𝐺)𝑧)(+g‘(oppg𝐺))𝑎) = (𝑎(+g𝐺)(((invg𝐺)‘𝑦)(+g𝐺)𝑧))
5756mpteq2i 5201 . . . . . . . . . . . . . . . 16 (𝑎𝑋 ↦ ((((invg𝐺)‘𝑦)(+g𝐺)𝑧)(+g‘(oppg𝐺))𝑎)) = (𝑎𝑋 ↦ (𝑎(+g𝐺)(((invg𝐺)‘𝑦)(+g𝐺)𝑧)))
5846adantr 482 . . . . . . . . . . . . . . . . . 18 (((((𝐺 ∈ TopGrp ∧ 𝑌 ∈ (NrmSGrp‘𝐺) ∧ 𝑆𝐽) ∧ 𝑦𝑋) ∧ 𝑧𝑆) ∧ (((invg𝐺)‘𝑦)(+g𝐺)𝑧) ∈ 𝑌) → 𝐺 ∈ TopGrp)
5954oppgtgp 23354 . . . . . . . . . . . . . . . . . 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 3935 . . . . . . . . . . . . . . . . 17 (((((𝐺 ∈ TopGrp ∧ 𝑌 ∈ (NrmSGrp‘𝐺) ∧ 𝑆𝐽) ∧ 𝑦𝑋) ∧ 𝑧𝑆) ∧ (((invg𝐺)‘𝑦)(+g𝐺)𝑧) ∈ 𝑌) → (((invg𝐺)‘𝑦)(+g𝐺)𝑧) ∈ 𝑋)
62 eqid 2737 . . . . . . . . . . . . . . . . . 18 (𝑎𝑋 ↦ ((((invg𝐺)‘𝑦)(+g𝐺)𝑧)(+g‘(oppg𝐺))𝑎)) = (𝑎𝑋 ↦ ((((invg𝐺)‘𝑦)(+g𝐺)𝑧)(+g‘(oppg𝐺))𝑎))
6354, 4oppgbas 19052 . . . . . . . . . . . . . . . . . 18 𝑋 = (Base‘(oppg𝐺))
6454, 22oppgtopn 19056 . . . . . . . . . . . . . . . . . 18 𝐽 = (TopOpen‘(oppg𝐺))
6562, 63, 55, 64tgplacthmeo 23359 . . . . . . . . . . . . . . . . 17 (((oppg𝐺) ∈ TopGrp ∧ (((invg𝐺)‘𝑦)(+g𝐺)𝑧) ∈ 𝑋) → (𝑎𝑋 ↦ ((((invg𝐺)‘𝑦)(+g𝐺)𝑧)(+g‘(oppg𝐺))𝑎)) ∈ (𝐽Homeo𝐽))
6660, 61, 65syl2anc 585 . . . . . . . . . . . . . . . 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 2843 . . . . . . . . . . . . . . 15 (((((𝐺 ∈ TopGrp ∧ 𝑌 ∈ (NrmSGrp‘𝐺) ∧ 𝑆𝐽) ∧ 𝑦𝑋) ∧ 𝑧𝑆) ∧ (((invg𝐺)‘𝑦)(+g𝐺)𝑧) ∈ 𝑌) → (𝑎𝑋 ↦ (𝑎(+g𝐺)(((invg𝐺)‘𝑦)(+g𝐺)𝑧))) ∈ (𝐽Homeo𝐽))
68 hmeocn 23016 . . . . . . . . . . . . . . 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 22521 . . . . . . . . . . . . . 14 (((𝑎𝑋 ↦ (𝑎(+g𝐺)(((invg𝐺)‘𝑦)(+g𝐺)𝑧))) ∈ (𝐽 Cn 𝐽) ∧ 𝑆𝐽) → ((𝑎𝑋 ↦ (𝑎(+g𝐺)(((invg𝐺)‘𝑦)(+g𝐺)𝑧))) “ 𝑆) ∈ 𝐽)
7269, 70, 71syl2anc 585 . . . . . . . . . . . . 13 (((((𝐺 ∈ TopGrp ∧ 𝑌 ∈ (NrmSGrp‘𝐺) ∧ 𝑆𝐽) ∧ 𝑦𝑋) ∧ 𝑧𝑆) ∧ (((invg𝐺)‘𝑦)(+g𝐺)𝑧) ∈ 𝑌) → ((𝑎𝑋 ↦ (𝑎(+g𝐺)(((invg𝐺)‘𝑦)(+g𝐺)𝑧))) “ 𝑆) ∈ 𝐽)
7344adantr 482 . . . . . . . . . . . . . 14 (((((𝐺 ∈ TopGrp ∧ 𝑌 ∈ (NrmSGrp‘𝐺) ∧ 𝑆𝐽) ∧ 𝑦𝑋) ∧ 𝑧𝑆) ∧ (((invg𝐺)‘𝑦)(+g𝐺)𝑧) ∈ 𝑌) → 𝑦𝑋)
74 tgpgrp 23334 . . . . . . . . . . . . . . . . . . 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 2737 . . . . . . . . . . . . . . . . . . 19 (0g𝐺) = (0g𝐺)
774, 50, 76, 49grprinv 18725 . . . . . . . . . . . . . . . . . 18 ((𝐺 ∈ Grp ∧ 𝑦𝑋) → (𝑦(+g𝐺)((invg𝐺)‘𝑦)) = (0g𝐺))
7875, 73, 77syl2anc 585 . . . . . . . . . . . . . . . . 17 (((((𝐺 ∈ TopGrp ∧ 𝑌 ∈ (NrmSGrp‘𝐺) ∧ 𝑆𝐽) ∧ 𝑦𝑋) ∧ 𝑧𝑆) ∧ (((invg𝐺)‘𝑦)(+g𝐺)𝑧) ∈ 𝑌) → (𝑦(+g𝐺)((invg𝐺)‘𝑦)) = (0g𝐺))
7978oveq1d 7356 . . . . . . . . . . . . . . . 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 18723 . . . . . . . . . . . . . . . . . 18 ((𝐺 ∈ Grp ∧ 𝑦𝑋) → ((invg𝐺)‘𝑦) ∈ 𝑋)
8175, 73, 80syl2anc 585 . . . . . . . . . . . . . . . . 17 (((((𝐺 ∈ TopGrp ∧ 𝑌 ∈ (NrmSGrp‘𝐺) ∧ 𝑆𝐽) ∧ 𝑦𝑋) ∧ 𝑧𝑆) ∧ (((invg𝐺)‘𝑦)(+g𝐺)𝑧) ∈ 𝑌) → ((invg𝐺)‘𝑦) ∈ 𝑋)
8229adantr 482 . . . . . . . . . . . . . . . . 17 (((((𝐺 ∈ TopGrp ∧ 𝑌 ∈ (NrmSGrp‘𝐺) ∧ 𝑆𝐽) ∧ 𝑦𝑋) ∧ 𝑧𝑆) ∧ (((invg𝐺)‘𝑦)(+g𝐺)𝑧) ∈ 𝑌) → 𝑧𝑋)
834, 50grpass 18682 . . . . . . . . . . . . . . . . 17 ((𝐺 ∈ Grp ∧ (𝑦𝑋 ∧ ((invg𝐺)‘𝑦) ∈ 𝑋𝑧𝑋)) → ((𝑦(+g𝐺)((invg𝐺)‘𝑦))(+g𝐺)𝑧) = (𝑦(+g𝐺)(((invg𝐺)‘𝑦)(+g𝐺)𝑧)))
8475, 73, 81, 82, 83syl13anc 1372 . . . . . . . . . . . . . . . 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 18705 . . . . . . . . . . . . . . . . 17 ((𝐺 ∈ Grp ∧ 𝑧𝑋) → ((0g𝐺)(+g𝐺)𝑧) = 𝑧)
8675, 82, 85syl2anc 585 . . . . . . . . . . . . . . . 16 (((((𝐺 ∈ TopGrp ∧ 𝑌 ∈ (NrmSGrp‘𝐺) ∧ 𝑆𝐽) ∧ 𝑦𝑋) ∧ 𝑧𝑆) ∧ (((invg𝐺)‘𝑦)(+g𝐺)𝑧) ∈ 𝑌) → ((0g𝐺)(+g𝐺)𝑧) = 𝑧)
8779, 84, 863eqtr3d 2785 . . . . . . . . . . . . . . 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 2838 . . . . . . . . . . . . . 14 (((((𝐺 ∈ TopGrp ∧ 𝑌 ∈ (NrmSGrp‘𝐺) ∧ 𝑆𝐽) ∧ 𝑦𝑋) ∧ 𝑧𝑆) ∧ (((invg𝐺)‘𝑦)(+g𝐺)𝑧) ∈ 𝑌) → (𝑦(+g𝐺)(((invg𝐺)‘𝑦)(+g𝐺)𝑧)) ∈ 𝑆)
90 oveq1 7348 . . . . . . . . . . . . . . . 16 (𝑎 = 𝑦 → (𝑎(+g𝐺)(((invg𝐺)‘𝑦)(+g𝐺)𝑧)) = (𝑦(+g𝐺)(((invg𝐺)‘𝑦)(+g𝐺)𝑧)))
9190eleq1d 2822 . . . . . . . . . . . . . . 15 (𝑎 = 𝑦 → ((𝑎(+g𝐺)(((invg𝐺)‘𝑦)(+g𝐺)𝑧)) ∈ 𝑆 ↔ (𝑦(+g𝐺)(((invg𝐺)‘𝑦)(+g𝐺)𝑧)) ∈ 𝑆))
92 eqid 2737 . . . . . . . . . . . . . . . 16 (𝑎𝑋 ↦ (𝑎(+g𝐺)(((invg𝐺)‘𝑦)(+g𝐺)𝑧))) = (𝑎𝑋 ↦ (𝑎(+g𝐺)(((invg𝐺)‘𝑦)(+g𝐺)𝑧)))
9392mptpreima 6180 . . . . . . . . . . . . . . 15 ((𝑎𝑋 ↦ (𝑎(+g𝐺)(((invg𝐺)‘𝑦)(+g𝐺)𝑧))) “ 𝑆) = {𝑎𝑋 ∣ (𝑎(+g𝐺)(((invg𝐺)‘𝑦)(+g𝐺)𝑧)) ∈ 𝑆}
9491, 93elrab2 3640 . . . . . . . . . . . . . 14 (𝑦 ∈ ((𝑎𝑋 ↦ (𝑎(+g𝐺)(((invg𝐺)‘𝑦)(+g𝐺)𝑧))) “ 𝑆) ↔ (𝑦𝑋 ∧ (𝑦(+g𝐺)(((invg𝐺)‘𝑦)(+g𝐺)𝑧)) ∈ 𝑆))
9573, 89, 94sylanbrc 584 . . . . . . . . . . . . 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 8577 . . . . . . . . . . . . . . . . . . 19 ((𝐺 ~QG 𝑌) ∈ V → [𝑥](𝐺 ~QG 𝑌) ∈ V)
977, 96ax-mp 5 . . . . . . . . . . . . . . . . . 18 [𝑥](𝐺 ~QG 𝑌) ∈ V
9897, 6fnmpti 6631 . . . . . . . . . . . . . . . . 17 𝐹 Fn 𝑋
9928ad3antrrr 728 . . . . . . . . . . . . . . . . 17 ((((((𝐺 ∈ TopGrp ∧ 𝑌 ∈ (NrmSGrp‘𝐺) ∧ 𝑆𝐽) ∧ 𝑦𝑋) ∧ 𝑧𝑆) ∧ (((invg𝐺)‘𝑦)(+g𝐺)𝑧) ∈ 𝑌) ∧ 𝑎𝑋) → 𝑆𝑋)
100 fnfvima 7169 . . . . . . . . . . . . . . . . . 18 ((𝐹 Fn 𝑋𝑆𝑋 ∧ (𝑎(+g𝐺)(((invg𝐺)‘𝑦)(+g𝐺)𝑧)) ∈ 𝑆) → (𝐹‘(𝑎(+g𝐺)(((invg𝐺)‘𝑦)(+g𝐺)𝑧))) ∈ (𝐹𝑆))
1011003expia 1121 . . . . . . . . . . . . . . . . 17 ((𝐹 Fn 𝑋𝑆𝑋) → ((𝑎(+g𝐺)(((invg𝐺)‘𝑦)(+g𝐺)𝑧)) ∈ 𝑆 → (𝐹‘(𝑎(+g𝐺)(((invg𝐺)‘𝑦)(+g𝐺)𝑧))) ∈ (𝐹𝑆)))
10298, 99, 101sylancr 588 . . . . . . . . . . . . . . . 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 482 . . . . . . . . . . . . . . . . . . . 20 ((((((𝐺 ∈ TopGrp ∧ 𝑌 ∈ (NrmSGrp‘𝐺) ∧ 𝑆𝐽) ∧ 𝑦𝑋) ∧ 𝑧𝑆) ∧ (((invg𝐺)‘𝑦)(+g𝐺)𝑧) ∈ 𝑌) ∧ 𝑎𝑋) → 𝐺 ∈ Grp)
104 simpr 486 . . . . . . . . . . . . . . . . . . . 20 ((((((𝐺 ∈ TopGrp ∧ 𝑌 ∈ (NrmSGrp‘𝐺) ∧ 𝑆𝐽) ∧ 𝑦𝑋) ∧ 𝑧𝑆) ∧ (((invg𝐺)‘𝑦)(+g𝐺)𝑧) ∈ 𝑌) ∧ 𝑎𝑋) → 𝑎𝑋)
10561adantr 482 . . . . . . . . . . . . . . . . . . . 20 ((((((𝐺 ∈ TopGrp ∧ 𝑌 ∈ (NrmSGrp‘𝐺) ∧ 𝑆𝐽) ∧ 𝑦𝑋) ∧ 𝑧𝑆) ∧ (((invg𝐺)‘𝑦)(+g𝐺)𝑧) ∈ 𝑌) ∧ 𝑎𝑋) → (((invg𝐺)‘𝑦)(+g𝐺)𝑧) ∈ 𝑋)
1064, 50grpcl 18681 . . . . . . . . . . . . . . . . . . . 20 ((𝐺 ∈ Grp ∧ 𝑎𝑋 ∧ (((invg𝐺)‘𝑦)(+g𝐺)𝑧) ∈ 𝑋) → (𝑎(+g𝐺)(((invg𝐺)‘𝑦)(+g𝐺)𝑧)) ∈ 𝑋)
107103, 104, 105, 106syl3anc 1371 . . . . . . . . . . . . . . . . . . 19 ((((((𝐺 ∈ TopGrp ∧ 𝑌 ∈ (NrmSGrp‘𝐺) ∧ 𝑆𝐽) ∧ 𝑦𝑋) ∧ 𝑧𝑆) ∧ (((invg𝐺)‘𝑦)(+g𝐺)𝑧) ∈ 𝑌) ∧ 𝑎𝑋) → (𝑎(+g𝐺)(((invg𝐺)‘𝑦)(+g𝐺)𝑧)) ∈ 𝑋)
108 eceq1 8611 . . . . . . . . . . . . . . . . . . . 20 (𝑥 = (𝑎(+g𝐺)(((invg𝐺)‘𝑦)(+g𝐺)𝑧)) → [𝑥](𝐺 ~QG 𝑌) = [(𝑎(+g𝐺)(((invg𝐺)‘𝑦)(+g𝐺)𝑧))](𝐺 ~QG 𝑌))
109108, 6, 97fvmpt3i 6940 . . . . . . . . . . . . . . . . . . 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 18724 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝐺 ∈ Grp ∧ 𝑎𝑋) → (((invg𝐺)‘𝑎)(+g𝐺)𝑎) = (0g𝐺))
113103, 104, 112syl2anc 585 . . . . . . . . . . . . . . . . . . . . . . 23 ((((((𝐺 ∈ TopGrp ∧ 𝑌 ∈ (NrmSGrp‘𝐺) ∧ 𝑆𝐽) ∧ 𝑦𝑋) ∧ 𝑧𝑆) ∧ (((invg𝐺)‘𝑦)(+g𝐺)𝑧) ∈ 𝑌) ∧ 𝑎𝑋) → (((invg𝐺)‘𝑎)(+g𝐺)𝑎) = (0g𝐺))
114113oveq1d 7356 . . . . . . . . . . . . . . . . . . . . . 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 18723 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝐺 ∈ Grp ∧ 𝑎𝑋) → ((invg𝐺)‘𝑎) ∈ 𝑋)
116103, 104, 115syl2anc 585 . . . . . . . . . . . . . . . . . . . . . . 23 ((((((𝐺 ∈ TopGrp ∧ 𝑌 ∈ (NrmSGrp‘𝐺) ∧ 𝑆𝐽) ∧ 𝑦𝑋) ∧ 𝑧𝑆) ∧ (((invg𝐺)‘𝑦)(+g𝐺)𝑧) ∈ 𝑌) ∧ 𝑎𝑋) → ((invg𝐺)‘𝑎) ∈ 𝑋)
1174, 50grpass 18682 . . . . . . . . . . . . . . . . . . . . . . 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 1372 . . . . . . . . . . . . . . . . . . . . . 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 18705 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝐺 ∈ Grp ∧ (((invg𝐺)‘𝑦)(+g𝐺)𝑧) ∈ 𝑋) → ((0g𝐺)(+g𝐺)(((invg𝐺)‘𝑦)(+g𝐺)𝑧)) = (((invg𝐺)‘𝑦)(+g𝐺)𝑧))
120103, 105, 119syl2anc 585 . . . . . . . . . . . . . . . . . . . . . 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 2785 . . . . . . . . . . . . . . . . . . . . 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 2838 . . . . . . . . . . . . . . . . . . . 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 18901 . . . . . . . . . . . . . . . . . . . . 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 585 . . . . . . . . . . . . . . . . . . . 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 1342 . . . . . . . . . . . . . . . . . . 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 8624 . . . . . . . . . . . . . . . . . 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 2780 . . . . . . . . . . . . . . . . 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 2822 . . . . . . . . . . . . . . . 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 4024 . . . . . . . . . . . . . 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 8611 . . . . . . . . . . . . . . . . 17 (𝑥 = 𝑎 → [𝑥](𝐺 ~QG 𝑌) = [𝑎](𝐺 ~QG 𝑌))
134133cbvmptv 5209 . . . . . . . . . . . . . . . 16 (𝑥𝑋 ↦ [𝑥](𝐺 ~QG 𝑌)) = (𝑎𝑋 ↦ [𝑎](𝐺 ~QG 𝑌))
1356, 134eqtri 2765 . . . . . . . . . . . . . . 15 𝐹 = (𝑎𝑋 ↦ [𝑎](𝐺 ~QG 𝑌))
136135mptpreima 6180 . . . . . . . . . . . . . 14 (𝐹 “ (𝐹𝑆)) = {𝑎𝑋 ∣ [𝑎](𝐺 ~QG 𝑌) ∈ (𝐹𝑆)}
137132, 93, 1363sstr4g 3980 . . . . . . . . . . . . 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 2826 . . . . . . . . . . . . . . 15 (𝑢 = ((𝑎𝑋 ↦ (𝑎(+g𝐺)(((invg𝐺)‘𝑦)(+g𝐺)𝑧))) “ 𝑆) → (𝑦𝑢𝑦 ∈ ((𝑎𝑋 ↦ (𝑎(+g𝐺)(((invg𝐺)‘𝑦)(+g𝐺)𝑧))) “ 𝑆)))
139 sseq1 3960 . . . . . . . . . . . . . . 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 632 . . . . . . . . . . . . . 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 3573 . . . . . . . . . . . . 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 1190 . . . . . . . . . . 11 (((((𝐺 ∈ TopGrp ∧ 𝑌 ∈ (NrmSGrp‘𝐺) ∧ 𝑆𝐽) ∧ 𝑦𝑋) ∧ 𝑧𝑆) ∧ (𝑦𝑋𝑧𝑋 ∧ (((invg𝐺)‘𝑦)(+g𝐺)𝑧) ∈ 𝑌)) → ∃𝑢𝐽 (𝑦𝑢𝑢 ⊆ (𝐹 “ (𝐹𝑆))))
144143ex 414 . . . . . . . . . 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 3149 . . . . . . . 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 455 . . . . . 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 3139 . . . 4 ((𝐺 ∈ TopGrp ∧ 𝑌 ∈ (NrmSGrp‘𝐺) ∧ 𝑆𝐽) → ∀𝑦 ∈ (𝐹 “ (𝐹𝑆))∃𝑢𝐽 (𝑦𝑢𝑢 ⊆ (𝐹 “ (𝐹𝑆))))
151 topontop 22167 . . . . 5 (𝐽 ∈ (TopOn‘𝑋) → 𝐽 ∈ Top)
152 eltop2 22230 . . . . 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 257 . . 3 ((𝐺 ∈ TopGrp ∧ 𝑌 ∈ (NrmSGrp‘𝐺) ∧ 𝑆𝐽) → (𝐹 “ (𝐹𝑆)) ∈ 𝐽)
155 elqtop3 22959 . . . 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 585 . . 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 17350 . . 3 ((𝐺 ∈ TopGrp ∧ 𝑌 ∈ (NrmSGrp‘𝐺) ∧ 𝑆𝐽) → 𝐻 = (𝐹s 𝐺))
159 qustgpopn.k . . 3 𝐾 = (TopOpen‘𝐻)
160158, 5, 10, 9, 22, 159imastopn 22976 . 2 ((𝐺 ∈ TopGrp ∧ 𝑌 ∈ (NrmSGrp‘𝐺) ∧ 𝑆𝐽) → 𝐾 = (𝐽 qTop 𝐹))
161157, 160eleqtrrd 2841 1 ((𝐺 ∈ TopGrp ∧ 𝑌 ∈ (NrmSGrp‘𝐺) ∧ 𝑆𝐽) → (𝐹𝑆) ∈ 𝐾)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 397  w3a 1087   = wceq 1541  wcel 2106  wral 3062  wrex 3071  {crab 3404  Vcvv 3442  wss 3901   class class class wbr 5096  cmpt 5179  ccnv 5623  ran crn 5625  cima 5627  Fun wfun 6477   Fn wfn 6478  ontowfo 6481  cfv 6483  (class class class)co 7341   Er wer 8570  [cec 8571   / cqs 8572  Basecbs 17009  +gcplusg 17059  TopOpenctopn 17229  0gc0g 17247   qTop cqtop 17311   /s cqus 17313  Grpcgrp 18673  invgcminusg 18674  SubGrpcsubg 18845  NrmSGrpcnsg 18846   ~QG cqg 18847  oppgcoppg 19045  Topctop 22147  TopOnctopon 22164   Cn ccn 22480  Homeochmeo 23009  TopGrpctgp 23327
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 2708  ax-rep 5233  ax-sep 5247  ax-nul 5254  ax-pow 5312  ax-pr 5376  ax-un 7654  ax-cnex 11032  ax-resscn 11033  ax-1cn 11034  ax-icn 11035  ax-addcl 11036  ax-addrcl 11037  ax-mulcl 11038  ax-mulrcl 11039  ax-mulcom 11040  ax-addass 11041  ax-mulass 11042  ax-distr 11043  ax-i2m1 11044  ax-1ne0 11045  ax-1rid 11046  ax-rnegex 11047  ax-rrecex 11048  ax-cnre 11049  ax-pre-lttri 11050  ax-pre-lttrn 11051  ax-pre-ltadd 11052  ax-pre-mulgt0 11053
This theorem depends on definitions:  df-bi 206  df-an 398  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 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2887  df-ne 2942  df-nel 3048  df-ral 3063  df-rex 3072  df-rmo 3350  df-reu 3351  df-rab 3405  df-v 3444  df-sbc 3731  df-csb 3847  df-dif 3904  df-un 3906  df-in 3908  df-ss 3918  df-pss 3920  df-nul 4274  df-if 4478  df-pw 4553  df-sn 4578  df-pr 4580  df-tp 4582  df-op 4584  df-uni 4857  df-iun 4947  df-br 5097  df-opab 5159  df-mpt 5180  df-tr 5214  df-id 5522  df-eprel 5528  df-po 5536  df-so 5537  df-fr 5579  df-we 5581  df-xp 5630  df-rel 5631  df-cnv 5632  df-co 5633  df-dm 5634  df-rn 5635  df-res 5636  df-ima 5637  df-pred 6242  df-ord 6309  df-on 6310  df-lim 6311  df-suc 6312  df-iota 6435  df-fun 6485  df-fn 6486  df-f 6487  df-f1 6488  df-fo 6489  df-f1o 6490  df-fv 6491  df-riota 7297  df-ov 7344  df-oprab 7345  df-mpo 7346  df-om 7785  df-1st 7903  df-2nd 7904  df-tpos 8116  df-frecs 8171  df-wrecs 8202  df-recs 8276  df-rdg 8315  df-1o 8371  df-er 8573  df-ec 8575  df-qs 8579  df-map 8692  df-en 8809  df-dom 8810  df-sdom 8811  df-fin 8812  df-sup 9303  df-inf 9304  df-pnf 11116  df-mnf 11117  df-xr 11118  df-ltxr 11119  df-le 11120  df-sub 11312  df-neg 11313  df-nn 12079  df-2 12141  df-3 12142  df-4 12143  df-5 12144  df-6 12145  df-7 12146  df-8 12147  df-9 12148  df-n0 12339  df-z 12425  df-dec 12543  df-uz 12688  df-fz 13345  df-struct 16945  df-sets 16962  df-slot 16980  df-ndx 16992  df-base 17010  df-ress 17039  df-plusg 17072  df-mulr 17073  df-sca 17075  df-vsca 17076  df-ip 17077  df-tset 17078  df-ple 17079  df-ds 17081  df-rest 17230  df-topn 17231  df-0g 17249  df-topgen 17251  df-qtop 17315  df-imas 17316  df-qus 17317  df-plusf 18422  df-mgm 18423  df-sgrp 18472  df-mnd 18483  df-grp 18676  df-minusg 18677  df-subg 18848  df-nsg 18849  df-eqg 18850  df-oppg 19046  df-top 22148  df-topon 22165  df-topsp 22187  df-bases 22201  df-cn 22483  df-cnp 22484  df-tx 22818  df-hmeo 23011  df-tmd 23328  df-tgp 23329
This theorem is referenced by:  qustgplem  23377
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