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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  –onto→wfo 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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