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Theorem qustgpopn 22725
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 5907 . . . 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 7168 . . . . . . 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 16808 . . . . 5 ((𝐺 ∈ TopGrp ∧ 𝑌 ∈ (NrmSGrp‘𝐺) ∧ 𝑆𝐽) → 𝐹:𝑋onto→(𝑋 / (𝐺 ~QG 𝑌)))
11 forn 6568 . . . . 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 3967 . . 3 ((𝐺 ∈ TopGrp ∧ 𝑌 ∈ (NrmSGrp‘𝐺) ∧ 𝑆𝐽) → (𝐹𝑆) ⊆ (𝑋 / (𝐺 ~QG 𝑌)))
14 eceq1 8310 . . . . . . . . . 10 (𝑥 = 𝑦 → [𝑥](𝐺 ~QG 𝑌) = [𝑦](𝐺 ~QG 𝑌))
1514cbvmptv 5133 . . . . . . . . 9 (𝑥𝑋 ↦ [𝑥](𝐺 ~QG 𝑌)) = (𝑦𝑋 ↦ [𝑦](𝐺 ~QG 𝑌))
166, 15eqtri 2821 . . . . . . . 8 𝐹 = (𝑦𝑋 ↦ [𝑦](𝐺 ~QG 𝑌))
1716mptpreima 6059 . . . . . . 7 (𝐹 “ (𝐹𝑆)) = {𝑦𝑋 ∣ [𝑦](𝐺 ~QG 𝑌) ∈ (𝐹𝑆)}
1817rabeq2i 3435 . . . . . 6 (𝑦 ∈ (𝐹 “ (𝐹𝑆)) ↔ (𝑦𝑋 ∧ [𝑦](𝐺 ~QG 𝑌) ∈ (𝐹𝑆)))
196funmpt2 6363 . . . . . . . . 9 Fun 𝐹
20 fvelima 6706 . . . . . . . . 9 ((Fun 𝐹 ∧ [𝑦](𝐺 ~QG 𝑌) ∈ (𝐹𝑆)) → ∃𝑧𝑆 (𝐹𝑧) = [𝑦](𝐺 ~QG 𝑌))
2119, 20mpan 689 . . . . . . . 8 ([𝑦](𝐺 ~QG 𝑌) ∈ (𝐹𝑆) → ∃𝑧𝑆 (𝐹𝑧) = [𝑦](𝐺 ~QG 𝑌))
22 qustgpopn.j . . . . . . . . . . . . . . . . . . 19 𝐽 = (TopOpen‘𝐺)
2322, 4tgptopon 22687 . . . . . . . . . . . . . . . . . 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 21532 . . . . . . . . . . . . . . . . 17 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑆𝐽) → 𝑆𝑋)
2724, 25, 26syl2anc 587 . . . . . . . . . . . . . . . 16 ((𝐺 ∈ TopGrp ∧ 𝑌 ∈ (NrmSGrp‘𝐺) ∧ 𝑆𝐽) → 𝑆𝑋)
2827adantr 484 . . . . . . . . . . . . . . 15 (((𝐺 ∈ TopGrp ∧ 𝑌 ∈ (NrmSGrp‘𝐺) ∧ 𝑆𝐽) ∧ 𝑦𝑋) → 𝑆𝑋)
2928sselda 3915 . . . . . . . . . . . . . 14 ((((𝐺 ∈ TopGrp ∧ 𝑌 ∈ (NrmSGrp‘𝐺) ∧ 𝑆𝐽) ∧ 𝑦𝑋) ∧ 𝑧𝑆) → 𝑧𝑋)
30 eceq1 8310 . . . . . . . . . . . . . . 15 (𝑥 = 𝑧 → [𝑥](𝐺 ~QG 𝑌) = [𝑧](𝐺 ~QG 𝑌))
31 ecexg 8276 . . . . . . . . . . . . . . . 16 ((𝐺 ~QG 𝑌) ∈ V → [𝑧](𝐺 ~QG 𝑌) ∈ V)
327, 31ax-mp 5 . . . . . . . . . . . . . . 15 [𝑧](𝐺 ~QG 𝑌) ∈ V
3330, 6, 32fvmpt 6745 . . . . . . . . . . . . . 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 2800 . . . . . . . . . . . 12 ((((𝐺 ∈ TopGrp ∧ 𝑌 ∈ (NrmSGrp‘𝐺) ∧ 𝑆𝐽) ∧ 𝑦𝑋) ∧ 𝑧𝑆) → ((𝐹𝑧) = [𝑦](𝐺 ~QG 𝑌) ↔ [𝑧](𝐺 ~QG 𝑌) = [𝑦](𝐺 ~QG 𝑌)))
36 eqcom 2805 . . . . . . . . . . . 12 ([𝑧](𝐺 ~QG 𝑌) = [𝑦](𝐺 ~QG 𝑌) ↔ [𝑦](𝐺 ~QG 𝑌) = [𝑧](𝐺 ~QG 𝑌))
3735, 36syl6bb 290 . . . . . . . . . . 11 ((((𝐺 ∈ TopGrp ∧ 𝑌 ∈ (NrmSGrp‘𝐺) ∧ 𝑆𝐽) ∧ 𝑦𝑋) ∧ 𝑧𝑆) → ((𝐹𝑧) = [𝑦](𝐺 ~QG 𝑌) ↔ [𝑦](𝐺 ~QG 𝑌) = [𝑧](𝐺 ~QG 𝑌)))
38 nsgsubg 18302 . . . . . . . . . . . . . . 15 (𝑌 ∈ (NrmSGrp‘𝐺) → 𝑌 ∈ (SubGrp‘𝐺))
39383ad2ant2 1131 . . . . . . . . . . . . . 14 ((𝐺 ∈ TopGrp ∧ 𝑌 ∈ (NrmSGrp‘𝐺) ∧ 𝑆𝐽) → 𝑌 ∈ (SubGrp‘𝐺))
4039ad2antrr 725 . . . . . . . . . . . . 13 ((((𝐺 ∈ TopGrp ∧ 𝑌 ∈ (NrmSGrp‘𝐺) ∧ 𝑆𝐽) ∧ 𝑦𝑋) ∧ 𝑧𝑆) → 𝑌 ∈ (SubGrp‘𝐺))
41 eqid 2798 . . . . . . . . . . . . . 14 (𝐺 ~QG 𝑌) = (𝐺 ~QG 𝑌)
424, 41eqger 18322 . . . . . . . . . . . . 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 768 . . . . . . . . . . . 12 ((((𝐺 ∈ TopGrp ∧ 𝑌 ∈ (NrmSGrp‘𝐺) ∧ 𝑆𝐽) ∧ 𝑦𝑋) ∧ 𝑧𝑆) → 𝑦𝑋)
4543, 44erth 8321 . . . . . . . . . . 11 ((((𝐺 ∈ TopGrp ∧ 𝑌 ∈ (NrmSGrp‘𝐺) ∧ 𝑆𝐽) ∧ 𝑦𝑋) ∧ 𝑧𝑆) → (𝑦(𝐺 ~QG 𝑌)𝑧 ↔ [𝑦](𝐺 ~QG 𝑌) = [𝑧](𝐺 ~QG 𝑌)))
469ad2antrr 725 . . . . . . . . . . . 12 ((((𝐺 ∈ TopGrp ∧ 𝑌 ∈ (NrmSGrp‘𝐺) ∧ 𝑆𝐽) ∧ 𝑦𝑋) ∧ 𝑧𝑆) → 𝐺 ∈ TopGrp)
474subgss 18272 . . . . . . . . . . . . 13 (𝑌 ∈ (SubGrp‘𝐺) → 𝑌𝑋)
4840, 47syl 17 . . . . . . . . . . . 12 ((((𝐺 ∈ TopGrp ∧ 𝑌 ∈ (NrmSGrp‘𝐺) ∧ 𝑆𝐽) ∧ 𝑦𝑋) ∧ 𝑧𝑆) → 𝑌𝑋)
49 eqid 2798 . . . . . . . . . . . . 13 (invg𝐺) = (invg𝐺)
50 eqid 2798 . . . . . . . . . . . . 13 (+g𝐺) = (+g𝐺)
514, 49, 50, 41eqgval 18321 . . . . . . . . . . . 12 ((𝐺 ∈ TopGrp ∧ 𝑌𝑋) → (𝑦(𝐺 ~QG 𝑌)𝑧 ↔ (𝑦𝑋𝑧𝑋 ∧ (((invg𝐺)‘𝑦)(+g𝐺)𝑧) ∈ 𝑌)))
5246, 48, 51syl2anc 587 . . . . . . . . . . 11 ((((𝐺 ∈ TopGrp ∧ 𝑌 ∈ (NrmSGrp‘𝐺) ∧ 𝑆𝐽) ∧ 𝑦𝑋) ∧ 𝑧𝑆) → (𝑦(𝐺 ~QG 𝑌)𝑧 ↔ (𝑦𝑋𝑧𝑋 ∧ (((invg𝐺)‘𝑦)(+g𝐺)𝑧) ∈ 𝑌)))
5337, 45, 523bitr2d 310 . . . . . . . . . 10 ((((𝐺 ∈ TopGrp ∧ 𝑌 ∈ (NrmSGrp‘𝐺) ∧ 𝑆𝐽) ∧ 𝑦𝑋) ∧ 𝑧𝑆) → ((𝐹𝑧) = [𝑦](𝐺 ~QG 𝑌) ↔ (𝑦𝑋𝑧𝑋 ∧ (((invg𝐺)‘𝑦)(+g𝐺)𝑧) ∈ 𝑌)))
54 eqid 2798 . . . . . . . . . . . . . . . . . 18 (oppg𝐺) = (oppg𝐺)
55 eqid 2798 . . . . . . . . . . . . . . . . . 18 (+g‘(oppg𝐺)) = (+g‘(oppg𝐺))
5650, 54, 55oppgplus 18469 . . . . . . . . . . . . . . . . 17 ((((invg𝐺)‘𝑦)(+g𝐺)𝑧)(+g‘(oppg𝐺))𝑎) = (𝑎(+g𝐺)(((invg𝐺)‘𝑦)(+g𝐺)𝑧))
5756mpteq2i 5122 . . . . . . . . . . . . . . . 16 (𝑎𝑋 ↦ ((((invg𝐺)‘𝑦)(+g𝐺)𝑧)(+g‘(oppg𝐺))𝑎)) = (𝑎𝑋 ↦ (𝑎(+g𝐺)(((invg𝐺)‘𝑦)(+g𝐺)𝑧)))
5846adantr 484 . . . . . . . . . . . . . . . . . 18 (((((𝐺 ∈ TopGrp ∧ 𝑌 ∈ (NrmSGrp‘𝐺) ∧ 𝑆𝐽) ∧ 𝑦𝑋) ∧ 𝑧𝑆) ∧ (((invg𝐺)‘𝑦)(+g𝐺)𝑧) ∈ 𝑌) → 𝐺 ∈ TopGrp)
5954oppgtgp 22703 . . . . . . . . . . . . . . . . . 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 3915 . . . . . . . . . . . . . . . . 17 (((((𝐺 ∈ TopGrp ∧ 𝑌 ∈ (NrmSGrp‘𝐺) ∧ 𝑆𝐽) ∧ 𝑦𝑋) ∧ 𝑧𝑆) ∧ (((invg𝐺)‘𝑦)(+g𝐺)𝑧) ∈ 𝑌) → (((invg𝐺)‘𝑦)(+g𝐺)𝑧) ∈ 𝑋)
62 eqid 2798 . . . . . . . . . . . . . . . . . 18 (𝑎𝑋 ↦ ((((invg𝐺)‘𝑦)(+g𝐺)𝑧)(+g‘(oppg𝐺))𝑎)) = (𝑎𝑋 ↦ ((((invg𝐺)‘𝑦)(+g𝐺)𝑧)(+g‘(oppg𝐺))𝑎))
6354, 4oppgbas 18471 . . . . . . . . . . . . . . . . . 18 𝑋 = (Base‘(oppg𝐺))
6454, 22oppgtopn 18473 . . . . . . . . . . . . . . . . . 18 𝐽 = (TopOpen‘(oppg𝐺))
6562, 63, 55, 64tgplacthmeo 22708 . . . . . . . . . . . . . . . . 17 (((oppg𝐺) ∈ TopGrp ∧ (((invg𝐺)‘𝑦)(+g𝐺)𝑧) ∈ 𝑋) → (𝑎𝑋 ↦ ((((invg𝐺)‘𝑦)(+g𝐺)𝑧)(+g‘(oppg𝐺))𝑎)) ∈ (𝐽Homeo𝐽))
6660, 61, 65syl2anc 587 . . . . . . . . . . . . . . . 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 2895 . . . . . . . . . . . . . . 15 (((((𝐺 ∈ TopGrp ∧ 𝑌 ∈ (NrmSGrp‘𝐺) ∧ 𝑆𝐽) ∧ 𝑦𝑋) ∧ 𝑧𝑆) ∧ (((invg𝐺)‘𝑦)(+g𝐺)𝑧) ∈ 𝑌) → (𝑎𝑋 ↦ (𝑎(+g𝐺)(((invg𝐺)‘𝑦)(+g𝐺)𝑧))) ∈ (𝐽Homeo𝐽))
68 hmeocn 22365 . . . . . . . . . . . . . . 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 729 . . . . . . . . . . . . . 14 (((((𝐺 ∈ TopGrp ∧ 𝑌 ∈ (NrmSGrp‘𝐺) ∧ 𝑆𝐽) ∧ 𝑦𝑋) ∧ 𝑧𝑆) ∧ (((invg𝐺)‘𝑦)(+g𝐺)𝑧) ∈ 𝑌) → 𝑆𝐽)
71 cnima 21870 . . . . . . . . . . . . . 14 (((𝑎𝑋 ↦ (𝑎(+g𝐺)(((invg𝐺)‘𝑦)(+g𝐺)𝑧))) ∈ (𝐽 Cn 𝐽) ∧ 𝑆𝐽) → ((𝑎𝑋 ↦ (𝑎(+g𝐺)(((invg𝐺)‘𝑦)(+g𝐺)𝑧))) “ 𝑆) ∈ 𝐽)
7269, 70, 71syl2anc 587 . . . . . . . . . . . . 13 (((((𝐺 ∈ TopGrp ∧ 𝑌 ∈ (NrmSGrp‘𝐺) ∧ 𝑆𝐽) ∧ 𝑦𝑋) ∧ 𝑧𝑆) ∧ (((invg𝐺)‘𝑦)(+g𝐺)𝑧) ∈ 𝑌) → ((𝑎𝑋 ↦ (𝑎(+g𝐺)(((invg𝐺)‘𝑦)(+g𝐺)𝑧))) “ 𝑆) ∈ 𝐽)
7344adantr 484 . . . . . . . . . . . . . 14 (((((𝐺 ∈ TopGrp ∧ 𝑌 ∈ (NrmSGrp‘𝐺) ∧ 𝑆𝐽) ∧ 𝑦𝑋) ∧ 𝑧𝑆) ∧ (((invg𝐺)‘𝑦)(+g𝐺)𝑧) ∈ 𝑌) → 𝑦𝑋)
74 tgpgrp 22683 . . . . . . . . . . . . . . . . . . 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 2798 . . . . . . . . . . . . . . . . . . 19 (0g𝐺) = (0g𝐺)
774, 50, 76, 49grprinv 18145 . . . . . . . . . . . . . . . . . 18 ((𝐺 ∈ Grp ∧ 𝑦𝑋) → (𝑦(+g𝐺)((invg𝐺)‘𝑦)) = (0g𝐺))
7875, 73, 77syl2anc 587 . . . . . . . . . . . . . . . . 17 (((((𝐺 ∈ TopGrp ∧ 𝑌 ∈ (NrmSGrp‘𝐺) ∧ 𝑆𝐽) ∧ 𝑦𝑋) ∧ 𝑧𝑆) ∧ (((invg𝐺)‘𝑦)(+g𝐺)𝑧) ∈ 𝑌) → (𝑦(+g𝐺)((invg𝐺)‘𝑦)) = (0g𝐺))
7978oveq1d 7150 . . . . . . . . . . . . . . . 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 18143 . . . . . . . . . . . . . . . . . 18 ((𝐺 ∈ Grp ∧ 𝑦𝑋) → ((invg𝐺)‘𝑦) ∈ 𝑋)
8175, 73, 80syl2anc 587 . . . . . . . . . . . . . . . . 17 (((((𝐺 ∈ TopGrp ∧ 𝑌 ∈ (NrmSGrp‘𝐺) ∧ 𝑆𝐽) ∧ 𝑦𝑋) ∧ 𝑧𝑆) ∧ (((invg𝐺)‘𝑦)(+g𝐺)𝑧) ∈ 𝑌) → ((invg𝐺)‘𝑦) ∈ 𝑋)
8229adantr 484 . . . . . . . . . . . . . . . . 17 (((((𝐺 ∈ TopGrp ∧ 𝑌 ∈ (NrmSGrp‘𝐺) ∧ 𝑆𝐽) ∧ 𝑦𝑋) ∧ 𝑧𝑆) ∧ (((invg𝐺)‘𝑦)(+g𝐺)𝑧) ∈ 𝑌) → 𝑧𝑋)
834, 50grpass 18104 . . . . . . . . . . . . . . . . 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 18125 . . . . . . . . . . . . . . . . 17 ((𝐺 ∈ Grp ∧ 𝑧𝑋) → ((0g𝐺)(+g𝐺)𝑧) = 𝑧)
8675, 82, 85syl2anc 587 . . . . . . . . . . . . . . . 16 (((((𝐺 ∈ TopGrp ∧ 𝑌 ∈ (NrmSGrp‘𝐺) ∧ 𝑆𝐽) ∧ 𝑦𝑋) ∧ 𝑧𝑆) ∧ (((invg𝐺)‘𝑦)(+g𝐺)𝑧) ∈ 𝑌) → ((0g𝐺)(+g𝐺)𝑧) = 𝑧)
8779, 84, 863eqtr3d 2841 . . . . . . . . . . . . . . 15 (((((𝐺 ∈ TopGrp ∧ 𝑌 ∈ (NrmSGrp‘𝐺) ∧ 𝑆𝐽) ∧ 𝑦𝑋) ∧ 𝑧𝑆) ∧ (((invg𝐺)‘𝑦)(+g𝐺)𝑧) ∈ 𝑌) → (𝑦(+g𝐺)(((invg𝐺)‘𝑦)(+g𝐺)𝑧)) = 𝑧)
88 simplr 768 . . . . . . . . . . . . . . 15 (((((𝐺 ∈ TopGrp ∧ 𝑌 ∈ (NrmSGrp‘𝐺) ∧ 𝑆𝐽) ∧ 𝑦𝑋) ∧ 𝑧𝑆) ∧ (((invg𝐺)‘𝑦)(+g𝐺)𝑧) ∈ 𝑌) → 𝑧𝑆)
8987, 88eqeltrd 2890 . . . . . . . . . . . . . 14 (((((𝐺 ∈ TopGrp ∧ 𝑌 ∈ (NrmSGrp‘𝐺) ∧ 𝑆𝐽) ∧ 𝑦𝑋) ∧ 𝑧𝑆) ∧ (((invg𝐺)‘𝑦)(+g𝐺)𝑧) ∈ 𝑌) → (𝑦(+g𝐺)(((invg𝐺)‘𝑦)(+g𝐺)𝑧)) ∈ 𝑆)
90 oveq1 7142 . . . . . . . . . . . . . . . 16 (𝑎 = 𝑦 → (𝑎(+g𝐺)(((invg𝐺)‘𝑦)(+g𝐺)𝑧)) = (𝑦(+g𝐺)(((invg𝐺)‘𝑦)(+g𝐺)𝑧)))
9190eleq1d 2874 . . . . . . . . . . . . . . 15 (𝑎 = 𝑦 → ((𝑎(+g𝐺)(((invg𝐺)‘𝑦)(+g𝐺)𝑧)) ∈ 𝑆 ↔ (𝑦(+g𝐺)(((invg𝐺)‘𝑦)(+g𝐺)𝑧)) ∈ 𝑆))
92 eqid 2798 . . . . . . . . . . . . . . . 16 (𝑎𝑋 ↦ (𝑎(+g𝐺)(((invg𝐺)‘𝑦)(+g𝐺)𝑧))) = (𝑎𝑋 ↦ (𝑎(+g𝐺)(((invg𝐺)‘𝑦)(+g𝐺)𝑧)))
9392mptpreima 6059 . . . . . . . . . . . . . . 15 ((𝑎𝑋 ↦ (𝑎(+g𝐺)(((invg𝐺)‘𝑦)(+g𝐺)𝑧))) “ 𝑆) = {𝑎𝑋 ∣ (𝑎(+g𝐺)(((invg𝐺)‘𝑦)(+g𝐺)𝑧)) ∈ 𝑆}
9491, 93elrab2 3631 . . . . . . . . . . . . . 14 (𝑦 ∈ ((𝑎𝑋 ↦ (𝑎(+g𝐺)(((invg𝐺)‘𝑦)(+g𝐺)𝑧))) “ 𝑆) ↔ (𝑦𝑋 ∧ (𝑦(+g𝐺)(((invg𝐺)‘𝑦)(+g𝐺)𝑧)) ∈ 𝑆))
9573, 89, 94sylanbrc 586 . . . . . . . . . . . . 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 8276 . . . . . . . . . . . . . . . . . . 19 ((𝐺 ~QG 𝑌) ∈ V → [𝑥](𝐺 ~QG 𝑌) ∈ V)
977, 96ax-mp 5 . . . . . . . . . . . . . . . . . 18 [𝑥](𝐺 ~QG 𝑌) ∈ V
9897, 6fnmpti 6463 . . . . . . . . . . . . . . . . 17 𝐹 Fn 𝑋
9928ad3antrrr 729 . . . . . . . . . . . . . . . . 17 ((((((𝐺 ∈ TopGrp ∧ 𝑌 ∈ (NrmSGrp‘𝐺) ∧ 𝑆𝐽) ∧ 𝑦𝑋) ∧ 𝑧𝑆) ∧ (((invg𝐺)‘𝑦)(+g𝐺)𝑧) ∈ 𝑌) ∧ 𝑎𝑋) → 𝑆𝑋)
100 fnfvima 6973 . . . . . . . . . . . . . . . . . 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 590 . . . . . . . . . . . . . . . 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 484 . . . . . . . . . . . . . . . . . . . 20 ((((((𝐺 ∈ TopGrp ∧ 𝑌 ∈ (NrmSGrp‘𝐺) ∧ 𝑆𝐽) ∧ 𝑦𝑋) ∧ 𝑧𝑆) ∧ (((invg𝐺)‘𝑦)(+g𝐺)𝑧) ∈ 𝑌) ∧ 𝑎𝑋) → 𝐺 ∈ Grp)
104 simpr 488 . . . . . . . . . . . . . . . . . . . 20 ((((((𝐺 ∈ TopGrp ∧ 𝑌 ∈ (NrmSGrp‘𝐺) ∧ 𝑆𝐽) ∧ 𝑦𝑋) ∧ 𝑧𝑆) ∧ (((invg𝐺)‘𝑦)(+g𝐺)𝑧) ∈ 𝑌) ∧ 𝑎𝑋) → 𝑎𝑋)
10561adantr 484 . . . . . . . . . . . . . . . . . . . 20 ((((((𝐺 ∈ TopGrp ∧ 𝑌 ∈ (NrmSGrp‘𝐺) ∧ 𝑆𝐽) ∧ 𝑦𝑋) ∧ 𝑧𝑆) ∧ (((invg𝐺)‘𝑦)(+g𝐺)𝑧) ∈ 𝑌) ∧ 𝑎𝑋) → (((invg𝐺)‘𝑦)(+g𝐺)𝑧) ∈ 𝑋)
1064, 50grpcl 18103 . . . . . . . . . . . . . . . . . . . 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 8310 . . . . . . . . . . . . . . . . . . . 20 (𝑥 = (𝑎(+g𝐺)(((invg𝐺)‘𝑦)(+g𝐺)𝑧)) → [𝑥](𝐺 ~QG 𝑌) = [(𝑎(+g𝐺)(((invg𝐺)‘𝑦)(+g𝐺)𝑧))](𝐺 ~QG 𝑌))
109108, 6, 97fvmpt3i 6750 . . . . . . . . . . . . . . . . . . 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 725 . . . . . . . . . . . . . . . . . . 19 ((((((𝐺 ∈ TopGrp ∧ 𝑌 ∈ (NrmSGrp‘𝐺) ∧ 𝑆𝐽) ∧ 𝑦𝑋) ∧ 𝑧𝑆) ∧ (((invg𝐺)‘𝑦)(+g𝐺)𝑧) ∈ 𝑌) ∧ 𝑎𝑋) → (𝐺 ~QG 𝑌) Er 𝑋)
1124, 50, 76, 49grplinv 18144 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝐺 ∈ Grp ∧ 𝑎𝑋) → (((invg𝐺)‘𝑎)(+g𝐺)𝑎) = (0g𝐺))
113103, 104, 112syl2anc 587 . . . . . . . . . . . . . . . . . . . . . . 23 ((((((𝐺 ∈ TopGrp ∧ 𝑌 ∈ (NrmSGrp‘𝐺) ∧ 𝑆𝐽) ∧ 𝑦𝑋) ∧ 𝑧𝑆) ∧ (((invg𝐺)‘𝑦)(+g𝐺)𝑧) ∈ 𝑌) ∧ 𝑎𝑋) → (((invg𝐺)‘𝑎)(+g𝐺)𝑎) = (0g𝐺))
114113oveq1d 7150 . . . . . . . . . . . . . . . . . . . . . 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 18143 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝐺 ∈ Grp ∧ 𝑎𝑋) → ((invg𝐺)‘𝑎) ∈ 𝑋)
116103, 104, 115syl2anc 587 . . . . . . . . . . . . . . . . . . . . . . 23 ((((((𝐺 ∈ TopGrp ∧ 𝑌 ∈ (NrmSGrp‘𝐺) ∧ 𝑆𝐽) ∧ 𝑦𝑋) ∧ 𝑧𝑆) ∧ (((invg𝐺)‘𝑦)(+g𝐺)𝑧) ∈ 𝑌) ∧ 𝑎𝑋) → ((invg𝐺)‘𝑎) ∈ 𝑋)
1174, 50grpass 18104 . . . . . . . . . . . . . . . . . . . . . . 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 18125 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝐺 ∈ Grp ∧ (((invg𝐺)‘𝑦)(+g𝐺)𝑧) ∈ 𝑋) → ((0g𝐺)(+g𝐺)(((invg𝐺)‘𝑦)(+g𝐺)𝑧)) = (((invg𝐺)‘𝑦)(+g𝐺)𝑧))
120103, 105, 119syl2anc 587 . . . . . . . . . . . . . . . . . . . . . 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 2841 . . . . . . . . . . . . . . . . . . . . 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 768 . . . . . . . . . . . . . . . . . . . . 21 ((((((𝐺 ∈ TopGrp ∧ 𝑌 ∈ (NrmSGrp‘𝐺) ∧ 𝑆𝐽) ∧ 𝑦𝑋) ∧ 𝑧𝑆) ∧ (((invg𝐺)‘𝑦)(+g𝐺)𝑧) ∈ 𝑌) ∧ 𝑎𝑋) → (((invg𝐺)‘𝑦)(+g𝐺)𝑧) ∈ 𝑌)
123121, 122eqeltrd 2890 . . . . . . . . . . . . . . . . . . . 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 725 . . . . . . . . . . . . . . . . . . . . 21 ((((((𝐺 ∈ TopGrp ∧ 𝑌 ∈ (NrmSGrp‘𝐺) ∧ 𝑆𝐽) ∧ 𝑦𝑋) ∧ 𝑧𝑆) ∧ (((invg𝐺)‘𝑦)(+g𝐺)𝑧) ∈ 𝑌) ∧ 𝑎𝑋) → 𝑌𝑋)
1254, 49, 50, 41eqgval 18321 . . . . . . . . . . . . . . . . . . . . 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 587 . . . . . . . . . . . . . . . . . . . 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 8323 . . . . . . . . . . . . . . . . . 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 2836 . . . . . . . . . . . . . . . . 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 2874 . . . . . . . . . . . . . . . 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 242 . . . . . . . . . . . . . . 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 4003 . . . . . . . . . . . . . 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 8310 . . . . . . . . . . . . . . . . 17 (𝑥 = 𝑎 → [𝑥](𝐺 ~QG 𝑌) = [𝑎](𝐺 ~QG 𝑌))
134133cbvmptv 5133 . . . . . . . . . . . . . . . 16 (𝑥𝑋 ↦ [𝑥](𝐺 ~QG 𝑌)) = (𝑎𝑋 ↦ [𝑎](𝐺 ~QG 𝑌))
1356, 134eqtri 2821 . . . . . . . . . . . . . . 15 𝐹 = (𝑎𝑋 ↦ [𝑎](𝐺 ~QG 𝑌))
136135mptpreima 6059 . . . . . . . . . . . . . 14 (𝐹 “ (𝐹𝑆)) = {𝑎𝑋 ∣ [𝑎](𝐺 ~QG 𝑌) ∈ (𝐹𝑆)}
137132, 93, 1363sstr4g 3960 . . . . . . . . . . . . 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 2878 . . . . . . . . . . . . . . 15 (𝑢 = ((𝑎𝑋 ↦ (𝑎(+g𝐺)(((invg𝐺)‘𝑦)(+g𝐺)𝑧))) “ 𝑆) → (𝑦𝑢𝑦 ∈ ((𝑎𝑋 ↦ (𝑎(+g𝐺)(((invg𝐺)‘𝑦)(+g𝐺)𝑧))) “ 𝑆)))
139 sseq1 3940 . . . . . . . . . . . . . . 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 633 . . . . . . . . . . . . . 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 3571 . . . . . . . . . . . . 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 416 . . . . . . . . . 10 ((((𝐺 ∈ TopGrp ∧ 𝑌 ∈ (NrmSGrp‘𝐺) ∧ 𝑆𝐽) ∧ 𝑦𝑋) ∧ 𝑧𝑆) → ((𝑦𝑋𝑧𝑋 ∧ (((invg𝐺)‘𝑦)(+g𝐺)𝑧) ∈ 𝑌) → ∃𝑢𝐽 (𝑦𝑢𝑢 ⊆ (𝐹 “ (𝐹𝑆)))))
14553, 144sylbid 243 . . . . . . . . 9 ((((𝐺 ∈ TopGrp ∧ 𝑌 ∈ (NrmSGrp‘𝐺) ∧ 𝑆𝐽) ∧ 𝑦𝑋) ∧ 𝑧𝑆) → ((𝐹𝑧) = [𝑦](𝐺 ~QG 𝑌) → ∃𝑢𝐽 (𝑦𝑢𝑢 ⊆ (𝐹 “ (𝐹𝑆)))))
146145rexlimdva 3243 . . . . . . . 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 457 . . . . . 6 ((𝐺 ∈ TopGrp ∧ 𝑌 ∈ (NrmSGrp‘𝐺) ∧ 𝑆𝐽) → ((𝑦𝑋 ∧ [𝑦](𝐺 ~QG 𝑌) ∈ (𝐹𝑆)) → ∃𝑢𝐽 (𝑦𝑢𝑢 ⊆ (𝐹 “ (𝐹𝑆)))))
14918, 148syl5bi 245 . . . . 5 ((𝐺 ∈ TopGrp ∧ 𝑌 ∈ (NrmSGrp‘𝐺) ∧ 𝑆𝐽) → (𝑦 ∈ (𝐹 “ (𝐹𝑆)) → ∃𝑢𝐽 (𝑦𝑢𝑢 ⊆ (𝐹 “ (𝐹𝑆)))))
150149ralrimiv 3148 . . . 4 ((𝐺 ∈ TopGrp ∧ 𝑌 ∈ (NrmSGrp‘𝐺) ∧ 𝑆𝐽) → ∀𝑦 ∈ (𝐹 “ (𝐹𝑆))∃𝑢𝐽 (𝑦𝑢𝑢 ⊆ (𝐹 “ (𝐹𝑆))))
151 topontop 21518 . . . . 5 (𝐽 ∈ (TopOn‘𝑋) → 𝐽 ∈ Top)
152 eltop2 21580 . . . . 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 260 . . 3 ((𝐺 ∈ TopGrp ∧ 𝑌 ∈ (NrmSGrp‘𝐺) ∧ 𝑆𝐽) → (𝐹 “ (𝐹𝑆)) ∈ 𝐽)
155 elqtop3 22308 . . . 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 587 . . 3 ((𝐺 ∈ TopGrp ∧ 𝑌 ∈ (NrmSGrp‘𝐺) ∧ 𝑆𝐽) → ((𝐹𝑆) ∈ (𝐽 qTop 𝐹) ↔ ((𝐹𝑆) ⊆ (𝑋 / (𝐺 ~QG 𝑌)) ∧ (𝐹 “ (𝐹𝑆)) ∈ 𝐽)))
15713, 154, 156mpbir2and 712 . 2 ((𝐺 ∈ TopGrp ∧ 𝑌 ∈ (NrmSGrp‘𝐺) ∧ 𝑆𝐽) → (𝐹𝑆) ∈ (𝐽 qTop 𝐹))
1583, 5, 6, 8, 9qusval 16807 . . 3 ((𝐺 ∈ TopGrp ∧ 𝑌 ∈ (NrmSGrp‘𝐺) ∧ 𝑆𝐽) → 𝐻 = (𝐹s 𝐺))
159 qustgpopn.k . . 3 𝐾 = (TopOpen‘𝐻)
160158, 5, 10, 9, 22, 159imastopn 22325 . 2 ((𝐺 ∈ TopGrp ∧ 𝑌 ∈ (NrmSGrp‘𝐺) ∧ 𝑆𝐽) → 𝐾 = (𝐽 qTop 𝐹))
161157, 160eleqtrrd 2893 1 ((𝐺 ∈ TopGrp ∧ 𝑌 ∈ (NrmSGrp‘𝐺) ∧ 𝑆𝐽) → (𝐹𝑆) ∈ 𝐾)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wa 399  w3a 1084   = wceq 1538  wcel 2111  wral 3106  wrex 3107  {crab 3110  Vcvv 3441  wss 3881   class class class wbr 5030  cmpt 5110  ccnv 5518  ran crn 5520  cima 5522  Fun wfun 6318   Fn wfn 6319  ontowfo 6322  cfv 6324  (class class class)co 7135   Er wer 8269  [cec 8270   / cqs 8271  Basecbs 16475  +gcplusg 16557  TopOpenctopn 16687  0gc0g 16705   qTop cqtop 16768   /s cqus 16770  Grpcgrp 18095  invgcminusg 18096  SubGrpcsubg 18265  NrmSGrpcnsg 18266   ~QG cqg 18267  oppgcoppg 18465  Topctop 21498  TopOnctopon 21515   Cn ccn 21829  Homeochmeo 22358  TopGrpctgp 22676
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 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-rep 5154  ax-sep 5167  ax-nul 5174  ax-pow 5231  ax-pr 5295  ax-un 7441  ax-cnex 10582  ax-resscn 10583  ax-1cn 10584  ax-icn 10585  ax-addcl 10586  ax-addrcl 10587  ax-mulcl 10588  ax-mulrcl 10589  ax-mulcom 10590  ax-addass 10591  ax-mulass 10592  ax-distr 10593  ax-i2m1 10594  ax-1ne0 10595  ax-1rid 10596  ax-rnegex 10597  ax-rrecex 10598  ax-cnre 10599  ax-pre-lttri 10600  ax-pre-lttrn 10601  ax-pre-ltadd 10602  ax-pre-mulgt0 10603
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-nel 3092  df-ral 3111  df-rex 3112  df-reu 3113  df-rmo 3114  df-rab 3115  df-v 3443  df-sbc 3721  df-csb 3829  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-pss 3900  df-nul 4244  df-if 4426  df-pw 4499  df-sn 4526  df-pr 4528  df-tp 4530  df-op 4532  df-uni 4801  df-int 4839  df-iun 4883  df-br 5031  df-opab 5093  df-mpt 5111  df-tr 5137  df-id 5425  df-eprel 5430  df-po 5438  df-so 5439  df-fr 5478  df-we 5480  df-xp 5525  df-rel 5526  df-cnv 5527  df-co 5528  df-dm 5529  df-rn 5530  df-res 5531  df-ima 5532  df-pred 6116  df-ord 6162  df-on 6163  df-lim 6164  df-suc 6165  df-iota 6283  df-fun 6326  df-fn 6327  df-f 6328  df-f1 6329  df-fo 6330  df-f1o 6331  df-fv 6332  df-riota 7093  df-ov 7138  df-oprab 7139  df-mpo 7140  df-om 7561  df-1st 7671  df-2nd 7672  df-tpos 7875  df-wrecs 7930  df-recs 7991  df-rdg 8029  df-1o 8085  df-oadd 8089  df-er 8272  df-ec 8274  df-qs 8278  df-map 8391  df-en 8493  df-dom 8494  df-sdom 8495  df-fin 8496  df-sup 8890  df-inf 8891  df-pnf 10666  df-mnf 10667  df-xr 10668  df-ltxr 10669  df-le 10670  df-sub 10861  df-neg 10862  df-nn 11626  df-2 11688  df-3 11689  df-4 11690  df-5 11691  df-6 11692  df-7 11693  df-8 11694  df-9 11695  df-n0 11886  df-z 11970  df-dec 12087  df-uz 12232  df-fz 12886  df-struct 16477  df-ndx 16478  df-slot 16479  df-base 16481  df-sets 16482  df-ress 16483  df-plusg 16570  df-mulr 16571  df-sca 16573  df-vsca 16574  df-ip 16575  df-tset 16576  df-ple 16577  df-ds 16579  df-rest 16688  df-topn 16689  df-0g 16707  df-topgen 16709  df-qtop 16772  df-imas 16773  df-qus 16774  df-plusf 17843  df-mgm 17844  df-sgrp 17893  df-mnd 17904  df-grp 18098  df-minusg 18099  df-subg 18268  df-nsg 18269  df-eqg 18270  df-oppg 18466  df-top 21499  df-topon 21516  df-topsp 21538  df-bases 21551  df-cn 21832  df-cnp 21833  df-tx 22167  df-hmeo 22360  df-tmd 22677  df-tgp 22678
This theorem is referenced by:  qustgplem  22726
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