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Theorem qustgpopn 21733
 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 5396 . . . 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 6577 . . . . . . 7 (𝐺 ~QG 𝑌) ∈ V
87a1i 11 . . . . . 6 ((𝐺 ∈ TopGrp ∧ 𝑌 ∈ (NrmSGrp‘𝐺) ∧ 𝑆𝐽) → (𝐺 ~QG 𝑌) ∈ V)
9 simp1 1054 . . . . . 6 ((𝐺 ∈ TopGrp ∧ 𝑌 ∈ (NrmSGrp‘𝐺) ∧ 𝑆𝐽) → 𝐺 ∈ TopGrp)
103, 5, 6, 8, 9quslem 16026 . . . . 5 ((𝐺 ∈ TopGrp ∧ 𝑌 ∈ (NrmSGrp‘𝐺) ∧ 𝑆𝐽) → 𝐹:𝑋onto→(𝑋 / (𝐺 ~QG 𝑌)))
11 forn 6031 . . . . 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, 12syl5sseq 3616 . . 3 ((𝐺 ∈ TopGrp ∧ 𝑌 ∈ (NrmSGrp‘𝐺) ∧ 𝑆𝐽) → (𝐹𝑆) ⊆ (𝑋 / (𝐺 ~QG 𝑌)))
14 eceq1 7669 . . . . . . . . . 10 (𝑥 = 𝑦 → [𝑥](𝐺 ~QG 𝑌) = [𝑦](𝐺 ~QG 𝑌))
1514cbvmptv 4678 . . . . . . . . 9 (𝑥𝑋 ↦ [𝑥](𝐺 ~QG 𝑌)) = (𝑦𝑋 ↦ [𝑦](𝐺 ~QG 𝑌))
166, 15eqtri 2632 . . . . . . . 8 𝐹 = (𝑦𝑋 ↦ [𝑦](𝐺 ~QG 𝑌))
1716mptpreima 5545 . . . . . . 7 (𝐹 “ (𝐹𝑆)) = {𝑦𝑋 ∣ [𝑦](𝐺 ~QG 𝑌) ∈ (𝐹𝑆)}
1817rabeq2i 3170 . . . . . 6 (𝑦 ∈ (𝐹 “ (𝐹𝑆)) ↔ (𝑦𝑋 ∧ [𝑦](𝐺 ~QG 𝑌) ∈ (𝐹𝑆)))
196funmpt2 5841 . . . . . . . . 9 Fun 𝐹
20 fvelima 6158 . . . . . . . . 9 ((Fun 𝐹 ∧ [𝑦](𝐺 ~QG 𝑌) ∈ (𝐹𝑆)) → ∃𝑧𝑆 (𝐹𝑧) = [𝑦](𝐺 ~QG 𝑌))
2119, 20mpan 702 . . . . . . . 8 ([𝑦](𝐺 ~QG 𝑌) ∈ (𝐹𝑆) → ∃𝑧𝑆 (𝐹𝑧) = [𝑦](𝐺 ~QG 𝑌))
22 qustgpopn.j . . . . . . . . . . . . . . . . . . 19 𝐽 = (TopOpen‘𝐺)
2322, 4tgptopon 21696 . . . . . . . . . . . . . . . . . 18 (𝐺 ∈ TopGrp → 𝐽 ∈ (TopOn‘𝑋))
249, 23syl 17 . . . . . . . . . . . . . . . . 17 ((𝐺 ∈ TopGrp ∧ 𝑌 ∈ (NrmSGrp‘𝐺) ∧ 𝑆𝐽) → 𝐽 ∈ (TopOn‘𝑋))
25 simp3 1056 . . . . . . . . . . . . . . . . 17 ((𝐺 ∈ TopGrp ∧ 𝑌 ∈ (NrmSGrp‘𝐺) ∧ 𝑆𝐽) → 𝑆𝐽)
26 toponss 20544 . . . . . . . . . . . . . . . . 17 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑆𝐽) → 𝑆𝑋)
2724, 25, 26syl2anc 691 . . . . . . . . . . . . . . . 16 ((𝐺 ∈ TopGrp ∧ 𝑌 ∈ (NrmSGrp‘𝐺) ∧ 𝑆𝐽) → 𝑆𝑋)
2827adantr 480 . . . . . . . . . . . . . . 15 (((𝐺 ∈ TopGrp ∧ 𝑌 ∈ (NrmSGrp‘𝐺) ∧ 𝑆𝐽) ∧ 𝑦𝑋) → 𝑆𝑋)
2928sselda 3568 . . . . . . . . . . . . . 14 ((((𝐺 ∈ TopGrp ∧ 𝑌 ∈ (NrmSGrp‘𝐺) ∧ 𝑆𝐽) ∧ 𝑦𝑋) ∧ 𝑧𝑆) → 𝑧𝑋)
30 eceq1 7669 . . . . . . . . . . . . . . 15 (𝑥 = 𝑧 → [𝑥](𝐺 ~QG 𝑌) = [𝑧](𝐺 ~QG 𝑌))
31 ecexg 7633 . . . . . . . . . . . . . . . 16 ((𝐺 ~QG 𝑌) ∈ V → [𝑧](𝐺 ~QG 𝑌) ∈ V)
327, 31ax-mp 5 . . . . . . . . . . . . . . 15 [𝑧](𝐺 ~QG 𝑌) ∈ V
3330, 6, 32fvmpt 6191 . . . . . . . . . . . . . 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 2612 . . . . . . . . . . . 12 ((((𝐺 ∈ TopGrp ∧ 𝑌 ∈ (NrmSGrp‘𝐺) ∧ 𝑆𝐽) ∧ 𝑦𝑋) ∧ 𝑧𝑆) → ((𝐹𝑧) = [𝑦](𝐺 ~QG 𝑌) ↔ [𝑧](𝐺 ~QG 𝑌) = [𝑦](𝐺 ~QG 𝑌)))
36 eqcom 2617 . . . . . . . . . . . 12 ([𝑧](𝐺 ~QG 𝑌) = [𝑦](𝐺 ~QG 𝑌) ↔ [𝑦](𝐺 ~QG 𝑌) = [𝑧](𝐺 ~QG 𝑌))
3735, 36syl6bb 275 . . . . . . . . . . 11 ((((𝐺 ∈ TopGrp ∧ 𝑌 ∈ (NrmSGrp‘𝐺) ∧ 𝑆𝐽) ∧ 𝑦𝑋) ∧ 𝑧𝑆) → ((𝐹𝑧) = [𝑦](𝐺 ~QG 𝑌) ↔ [𝑦](𝐺 ~QG 𝑌) = [𝑧](𝐺 ~QG 𝑌)))
38 nsgsubg 17449 . . . . . . . . . . . . . . 15 (𝑌 ∈ (NrmSGrp‘𝐺) → 𝑌 ∈ (SubGrp‘𝐺))
39383ad2ant2 1076 . . . . . . . . . . . . . 14 ((𝐺 ∈ TopGrp ∧ 𝑌 ∈ (NrmSGrp‘𝐺) ∧ 𝑆𝐽) → 𝑌 ∈ (SubGrp‘𝐺))
4039ad2antrr 758 . . . . . . . . . . . . 13 ((((𝐺 ∈ TopGrp ∧ 𝑌 ∈ (NrmSGrp‘𝐺) ∧ 𝑆𝐽) ∧ 𝑦𝑋) ∧ 𝑧𝑆) → 𝑌 ∈ (SubGrp‘𝐺))
41 eqid 2610 . . . . . . . . . . . . . 14 (𝐺 ~QG 𝑌) = (𝐺 ~QG 𝑌)
424, 41eqger 17467 . . . . . . . . . . . . 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 788 . . . . . . . . . . . 12 ((((𝐺 ∈ TopGrp ∧ 𝑌 ∈ (NrmSGrp‘𝐺) ∧ 𝑆𝐽) ∧ 𝑦𝑋) ∧ 𝑧𝑆) → 𝑦𝑋)
4543, 44erth 7678 . . . . . . . . . . 11 ((((𝐺 ∈ TopGrp ∧ 𝑌 ∈ (NrmSGrp‘𝐺) ∧ 𝑆𝐽) ∧ 𝑦𝑋) ∧ 𝑧𝑆) → (𝑦(𝐺 ~QG 𝑌)𝑧 ↔ [𝑦](𝐺 ~QG 𝑌) = [𝑧](𝐺 ~QG 𝑌)))
469ad2antrr 758 . . . . . . . . . . . 12 ((((𝐺 ∈ TopGrp ∧ 𝑌 ∈ (NrmSGrp‘𝐺) ∧ 𝑆𝐽) ∧ 𝑦𝑋) ∧ 𝑧𝑆) → 𝐺 ∈ TopGrp)
474subgss 17418 . . . . . . . . . . . . 13 (𝑌 ∈ (SubGrp‘𝐺) → 𝑌𝑋)
4840, 47syl 17 . . . . . . . . . . . 12 ((((𝐺 ∈ TopGrp ∧ 𝑌 ∈ (NrmSGrp‘𝐺) ∧ 𝑆𝐽) ∧ 𝑦𝑋) ∧ 𝑧𝑆) → 𝑌𝑋)
49 eqid 2610 . . . . . . . . . . . . 13 (invg𝐺) = (invg𝐺)
50 eqid 2610 . . . . . . . . . . . . 13 (+g𝐺) = (+g𝐺)
514, 49, 50, 41eqgval 17466 . . . . . . . . . . . 12 ((𝐺 ∈ TopGrp ∧ 𝑌𝑋) → (𝑦(𝐺 ~QG 𝑌)𝑧 ↔ (𝑦𝑋𝑧𝑋 ∧ (((invg𝐺)‘𝑦)(+g𝐺)𝑧) ∈ 𝑌)))
5246, 48, 51syl2anc 691 . . . . . . . . . . 11 ((((𝐺 ∈ TopGrp ∧ 𝑌 ∈ (NrmSGrp‘𝐺) ∧ 𝑆𝐽) ∧ 𝑦𝑋) ∧ 𝑧𝑆) → (𝑦(𝐺 ~QG 𝑌)𝑧 ↔ (𝑦𝑋𝑧𝑋 ∧ (((invg𝐺)‘𝑦)(+g𝐺)𝑧) ∈ 𝑌)))
5337, 45, 523bitr2d 295 . . . . . . . . . 10 ((((𝐺 ∈ TopGrp ∧ 𝑌 ∈ (NrmSGrp‘𝐺) ∧ 𝑆𝐽) ∧ 𝑦𝑋) ∧ 𝑧𝑆) → ((𝐹𝑧) = [𝑦](𝐺 ~QG 𝑌) ↔ (𝑦𝑋𝑧𝑋 ∧ (((invg𝐺)‘𝑦)(+g𝐺)𝑧) ∈ 𝑌)))
54 eqid 2610 . . . . . . . . . . . . . . . . . 18 (oppg𝐺) = (oppg𝐺)
55 eqid 2610 . . . . . . . . . . . . . . . . . 18 (+g‘(oppg𝐺)) = (+g‘(oppg𝐺))
5650, 54, 55oppgplus 17602 . . . . . . . . . . . . . . . . 17 ((((invg𝐺)‘𝑦)(+g𝐺)𝑧)(+g‘(oppg𝐺))𝑎) = (𝑎(+g𝐺)(((invg𝐺)‘𝑦)(+g𝐺)𝑧))
5756mpteq2i 4669 . . . . . . . . . . . . . . . 16 (𝑎𝑋 ↦ ((((invg𝐺)‘𝑦)(+g𝐺)𝑧)(+g‘(oppg𝐺))𝑎)) = (𝑎𝑋 ↦ (𝑎(+g𝐺)(((invg𝐺)‘𝑦)(+g𝐺)𝑧)))
5846adantr 480 . . . . . . . . . . . . . . . . . 18 (((((𝐺 ∈ TopGrp ∧ 𝑌 ∈ (NrmSGrp‘𝐺) ∧ 𝑆𝐽) ∧ 𝑦𝑋) ∧ 𝑧𝑆) ∧ (((invg𝐺)‘𝑦)(+g𝐺)𝑧) ∈ 𝑌) → 𝐺 ∈ TopGrp)
5954oppgtgp 21712 . . . . . . . . . . . . . . . . . 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 3568 . . . . . . . . . . . . . . . . 17 (((((𝐺 ∈ TopGrp ∧ 𝑌 ∈ (NrmSGrp‘𝐺) ∧ 𝑆𝐽) ∧ 𝑦𝑋) ∧ 𝑧𝑆) ∧ (((invg𝐺)‘𝑦)(+g𝐺)𝑧) ∈ 𝑌) → (((invg𝐺)‘𝑦)(+g𝐺)𝑧) ∈ 𝑋)
62 eqid 2610 . . . . . . . . . . . . . . . . . 18 (𝑎𝑋 ↦ ((((invg𝐺)‘𝑦)(+g𝐺)𝑧)(+g‘(oppg𝐺))𝑎)) = (𝑎𝑋 ↦ ((((invg𝐺)‘𝑦)(+g𝐺)𝑧)(+g‘(oppg𝐺))𝑎))
6354, 4oppgbas 17604 . . . . . . . . . . . . . . . . . 18 𝑋 = (Base‘(oppg𝐺))
6454, 22oppgtopn 17606 . . . . . . . . . . . . . . . . . 18 𝐽 = (TopOpen‘(oppg𝐺))
6562, 63, 55, 64tgplacthmeo 21717 . . . . . . . . . . . . . . . . 17 (((oppg𝐺) ∈ TopGrp ∧ (((invg𝐺)‘𝑦)(+g𝐺)𝑧) ∈ 𝑋) → (𝑎𝑋 ↦ ((((invg𝐺)‘𝑦)(+g𝐺)𝑧)(+g‘(oppg𝐺))𝑎)) ∈ (𝐽Homeo𝐽))
6660, 61, 65syl2anc 691 . . . . . . . . . . . . . . . 16 (((((𝐺 ∈ TopGrp ∧ 𝑌 ∈ (NrmSGrp‘𝐺) ∧ 𝑆𝐽) ∧ 𝑦𝑋) ∧ 𝑧𝑆) ∧ (((invg𝐺)‘𝑦)(+g𝐺)𝑧) ∈ 𝑌) → (𝑎𝑋 ↦ ((((invg𝐺)‘𝑦)(+g𝐺)𝑧)(+g‘(oppg𝐺))𝑎)) ∈ (𝐽Homeo𝐽))
6757, 66syl5eqelr 2693 . . . . . . . . . . . . . . 15 (((((𝐺 ∈ TopGrp ∧ 𝑌 ∈ (NrmSGrp‘𝐺) ∧ 𝑆𝐽) ∧ 𝑦𝑋) ∧ 𝑧𝑆) ∧ (((invg𝐺)‘𝑦)(+g𝐺)𝑧) ∈ 𝑌) → (𝑎𝑋 ↦ (𝑎(+g𝐺)(((invg𝐺)‘𝑦)(+g𝐺)𝑧))) ∈ (𝐽Homeo𝐽))
68 hmeocn 21373 . . . . . . . . . . . . . . 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 762 . . . . . . . . . . . . . 14 (((((𝐺 ∈ TopGrp ∧ 𝑌 ∈ (NrmSGrp‘𝐺) ∧ 𝑆𝐽) ∧ 𝑦𝑋) ∧ 𝑧𝑆) ∧ (((invg𝐺)‘𝑦)(+g𝐺)𝑧) ∈ 𝑌) → 𝑆𝐽)
71 cnima 20879 . . . . . . . . . . . . . 14 (((𝑎𝑋 ↦ (𝑎(+g𝐺)(((invg𝐺)‘𝑦)(+g𝐺)𝑧))) ∈ (𝐽 Cn 𝐽) ∧ 𝑆𝐽) → ((𝑎𝑋 ↦ (𝑎(+g𝐺)(((invg𝐺)‘𝑦)(+g𝐺)𝑧))) “ 𝑆) ∈ 𝐽)
7269, 70, 71syl2anc 691 . . . . . . . . . . . . 13 (((((𝐺 ∈ TopGrp ∧ 𝑌 ∈ (NrmSGrp‘𝐺) ∧ 𝑆𝐽) ∧ 𝑦𝑋) ∧ 𝑧𝑆) ∧ (((invg𝐺)‘𝑦)(+g𝐺)𝑧) ∈ 𝑌) → ((𝑎𝑋 ↦ (𝑎(+g𝐺)(((invg𝐺)‘𝑦)(+g𝐺)𝑧))) “ 𝑆) ∈ 𝐽)
7344adantr 480 . . . . . . . . . . . . . 14 (((((𝐺 ∈ TopGrp ∧ 𝑌 ∈ (NrmSGrp‘𝐺) ∧ 𝑆𝐽) ∧ 𝑦𝑋) ∧ 𝑧𝑆) ∧ (((invg𝐺)‘𝑦)(+g𝐺)𝑧) ∈ 𝑌) → 𝑦𝑋)
74 tgpgrp 21692 . . . . . . . . . . . . . . . . . . 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 2610 . . . . . . . . . . . . . . . . . . 19 (0g𝐺) = (0g𝐺)
774, 50, 76, 49grprinv 17292 . . . . . . . . . . . . . . . . . 18 ((𝐺 ∈ Grp ∧ 𝑦𝑋) → (𝑦(+g𝐺)((invg𝐺)‘𝑦)) = (0g𝐺))
7875, 73, 77syl2anc 691 . . . . . . . . . . . . . . . . 17 (((((𝐺 ∈ TopGrp ∧ 𝑌 ∈ (NrmSGrp‘𝐺) ∧ 𝑆𝐽) ∧ 𝑦𝑋) ∧ 𝑧𝑆) ∧ (((invg𝐺)‘𝑦)(+g𝐺)𝑧) ∈ 𝑌) → (𝑦(+g𝐺)((invg𝐺)‘𝑦)) = (0g𝐺))
7978oveq1d 6564 . . . . . . . . . . . . . . . 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 17290 . . . . . . . . . . . . . . . . . 18 ((𝐺 ∈ Grp ∧ 𝑦𝑋) → ((invg𝐺)‘𝑦) ∈ 𝑋)
8175, 73, 80syl2anc 691 . . . . . . . . . . . . . . . . 17 (((((𝐺 ∈ TopGrp ∧ 𝑌 ∈ (NrmSGrp‘𝐺) ∧ 𝑆𝐽) ∧ 𝑦𝑋) ∧ 𝑧𝑆) ∧ (((invg𝐺)‘𝑦)(+g𝐺)𝑧) ∈ 𝑌) → ((invg𝐺)‘𝑦) ∈ 𝑋)
8229adantr 480 . . . . . . . . . . . . . . . . 17 (((((𝐺 ∈ TopGrp ∧ 𝑌 ∈ (NrmSGrp‘𝐺) ∧ 𝑆𝐽) ∧ 𝑦𝑋) ∧ 𝑧𝑆) ∧ (((invg𝐺)‘𝑦)(+g𝐺)𝑧) ∈ 𝑌) → 𝑧𝑋)
834, 50grpass 17254 . . . . . . . . . . . . . . . . 17 ((𝐺 ∈ Grp ∧ (𝑦𝑋 ∧ ((invg𝐺)‘𝑦) ∈ 𝑋𝑧𝑋)) → ((𝑦(+g𝐺)((invg𝐺)‘𝑦))(+g𝐺)𝑧) = (𝑦(+g𝐺)(((invg𝐺)‘𝑦)(+g𝐺)𝑧)))
8475, 73, 81, 82, 83syl13anc 1320 . . . . . . . . . . . . . . . 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 17275 . . . . . . . . . . . . . . . . 17 ((𝐺 ∈ Grp ∧ 𝑧𝑋) → ((0g𝐺)(+g𝐺)𝑧) = 𝑧)
8675, 82, 85syl2anc 691 . . . . . . . . . . . . . . . 16 (((((𝐺 ∈ TopGrp ∧ 𝑌 ∈ (NrmSGrp‘𝐺) ∧ 𝑆𝐽) ∧ 𝑦𝑋) ∧ 𝑧𝑆) ∧ (((invg𝐺)‘𝑦)(+g𝐺)𝑧) ∈ 𝑌) → ((0g𝐺)(+g𝐺)𝑧) = 𝑧)
8779, 84, 863eqtr3d 2652 . . . . . . . . . . . . . . 15 (((((𝐺 ∈ TopGrp ∧ 𝑌 ∈ (NrmSGrp‘𝐺) ∧ 𝑆𝐽) ∧ 𝑦𝑋) ∧ 𝑧𝑆) ∧ (((invg𝐺)‘𝑦)(+g𝐺)𝑧) ∈ 𝑌) → (𝑦(+g𝐺)(((invg𝐺)‘𝑦)(+g𝐺)𝑧)) = 𝑧)
88 simplr 788 . . . . . . . . . . . . . . 15 (((((𝐺 ∈ TopGrp ∧ 𝑌 ∈ (NrmSGrp‘𝐺) ∧ 𝑆𝐽) ∧ 𝑦𝑋) ∧ 𝑧𝑆) ∧ (((invg𝐺)‘𝑦)(+g𝐺)𝑧) ∈ 𝑌) → 𝑧𝑆)
8987, 88eqeltrd 2688 . . . . . . . . . . . . . 14 (((((𝐺 ∈ TopGrp ∧ 𝑌 ∈ (NrmSGrp‘𝐺) ∧ 𝑆𝐽) ∧ 𝑦𝑋) ∧ 𝑧𝑆) ∧ (((invg𝐺)‘𝑦)(+g𝐺)𝑧) ∈ 𝑌) → (𝑦(+g𝐺)(((invg𝐺)‘𝑦)(+g𝐺)𝑧)) ∈ 𝑆)
90 oveq1 6556 . . . . . . . . . . . . . . . 16 (𝑎 = 𝑦 → (𝑎(+g𝐺)(((invg𝐺)‘𝑦)(+g𝐺)𝑧)) = (𝑦(+g𝐺)(((invg𝐺)‘𝑦)(+g𝐺)𝑧)))
9190eleq1d 2672 . . . . . . . . . . . . . . 15 (𝑎 = 𝑦 → ((𝑎(+g𝐺)(((invg𝐺)‘𝑦)(+g𝐺)𝑧)) ∈ 𝑆 ↔ (𝑦(+g𝐺)(((invg𝐺)‘𝑦)(+g𝐺)𝑧)) ∈ 𝑆))
92 eqid 2610 . . . . . . . . . . . . . . . 16 (𝑎𝑋 ↦ (𝑎(+g𝐺)(((invg𝐺)‘𝑦)(+g𝐺)𝑧))) = (𝑎𝑋 ↦ (𝑎(+g𝐺)(((invg𝐺)‘𝑦)(+g𝐺)𝑧)))
9392mptpreima 5545 . . . . . . . . . . . . . . 15 ((𝑎𝑋 ↦ (𝑎(+g𝐺)(((invg𝐺)‘𝑦)(+g𝐺)𝑧))) “ 𝑆) = {𝑎𝑋 ∣ (𝑎(+g𝐺)(((invg𝐺)‘𝑦)(+g𝐺)𝑧)) ∈ 𝑆}
9491, 93elrab2 3333 . . . . . . . . . . . . . 14 (𝑦 ∈ ((𝑎𝑋 ↦ (𝑎(+g𝐺)(((invg𝐺)‘𝑦)(+g𝐺)𝑧))) “ 𝑆) ↔ (𝑦𝑋 ∧ (𝑦(+g𝐺)(((invg𝐺)‘𝑦)(+g𝐺)𝑧)) ∈ 𝑆))
9573, 89, 94sylanbrc 695 . . . . . . . . . . . . 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 7633 . . . . . . . . . . . . . . . . . . 19 ((𝐺 ~QG 𝑌) ∈ V → [𝑥](𝐺 ~QG 𝑌) ∈ V)
977, 96ax-mp 5 . . . . . . . . . . . . . . . . . 18 [𝑥](𝐺 ~QG 𝑌) ∈ V
9897, 6fnmpti 5935 . . . . . . . . . . . . . . . . 17 𝐹 Fn 𝑋
9928ad3antrrr 762 . . . . . . . . . . . . . . . . 17 ((((((𝐺 ∈ TopGrp ∧ 𝑌 ∈ (NrmSGrp‘𝐺) ∧ 𝑆𝐽) ∧ 𝑦𝑋) ∧ 𝑧𝑆) ∧ (((invg𝐺)‘𝑦)(+g𝐺)𝑧) ∈ 𝑌) ∧ 𝑎𝑋) → 𝑆𝑋)
100 fnfvima 6400 . . . . . . . . . . . . . . . . . 18 ((𝐹 Fn 𝑋𝑆𝑋 ∧ (𝑎(+g𝐺)(((invg𝐺)‘𝑦)(+g𝐺)𝑧)) ∈ 𝑆) → (𝐹‘(𝑎(+g𝐺)(((invg𝐺)‘𝑦)(+g𝐺)𝑧))) ∈ (𝐹𝑆))
1011003expia 1259 . . . . . . . . . . . . . . . . 17 ((𝐹 Fn 𝑋𝑆𝑋) → ((𝑎(+g𝐺)(((invg𝐺)‘𝑦)(+g𝐺)𝑧)) ∈ 𝑆 → (𝐹‘(𝑎(+g𝐺)(((invg𝐺)‘𝑦)(+g𝐺)𝑧))) ∈ (𝐹𝑆)))
10298, 99, 101sylancr 694 . . . . . . . . . . . . . . . 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 480 . . . . . . . . . . . . . . . . . . . 20 ((((((𝐺 ∈ TopGrp ∧ 𝑌 ∈ (NrmSGrp‘𝐺) ∧ 𝑆𝐽) ∧ 𝑦𝑋) ∧ 𝑧𝑆) ∧ (((invg𝐺)‘𝑦)(+g𝐺)𝑧) ∈ 𝑌) ∧ 𝑎𝑋) → 𝐺 ∈ Grp)
104 simpr 476 . . . . . . . . . . . . . . . . . . . 20 ((((((𝐺 ∈ TopGrp ∧ 𝑌 ∈ (NrmSGrp‘𝐺) ∧ 𝑆𝐽) ∧ 𝑦𝑋) ∧ 𝑧𝑆) ∧ (((invg𝐺)‘𝑦)(+g𝐺)𝑧) ∈ 𝑌) ∧ 𝑎𝑋) → 𝑎𝑋)
10561adantr 480 . . . . . . . . . . . . . . . . . . . 20 ((((((𝐺 ∈ TopGrp ∧ 𝑌 ∈ (NrmSGrp‘𝐺) ∧ 𝑆𝐽) ∧ 𝑦𝑋) ∧ 𝑧𝑆) ∧ (((invg𝐺)‘𝑦)(+g𝐺)𝑧) ∈ 𝑌) ∧ 𝑎𝑋) → (((invg𝐺)‘𝑦)(+g𝐺)𝑧) ∈ 𝑋)
1064, 50grpcl 17253 . . . . . . . . . . . . . . . . . . . 20 ((𝐺 ∈ Grp ∧ 𝑎𝑋 ∧ (((invg𝐺)‘𝑦)(+g𝐺)𝑧) ∈ 𝑋) → (𝑎(+g𝐺)(((invg𝐺)‘𝑦)(+g𝐺)𝑧)) ∈ 𝑋)
107103, 104, 105, 106syl3anc 1318 . . . . . . . . . . . . . . . . . . 19 ((((((𝐺 ∈ TopGrp ∧ 𝑌 ∈ (NrmSGrp‘𝐺) ∧ 𝑆𝐽) ∧ 𝑦𝑋) ∧ 𝑧𝑆) ∧ (((invg𝐺)‘𝑦)(+g𝐺)𝑧) ∈ 𝑌) ∧ 𝑎𝑋) → (𝑎(+g𝐺)(((invg𝐺)‘𝑦)(+g𝐺)𝑧)) ∈ 𝑋)
108 eceq1 7669 . . . . . . . . . . . . . . . . . . . 20 (𝑥 = (𝑎(+g𝐺)(((invg𝐺)‘𝑦)(+g𝐺)𝑧)) → [𝑥](𝐺 ~QG 𝑌) = [(𝑎(+g𝐺)(((invg𝐺)‘𝑦)(+g𝐺)𝑧))](𝐺 ~QG 𝑌))
109108, 6, 97fvmpt3i 6196 . . . . . . . . . . . . . . . . . . 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 758 . . . . . . . . . . . . . . . . . . 19 ((((((𝐺 ∈ TopGrp ∧ 𝑌 ∈ (NrmSGrp‘𝐺) ∧ 𝑆𝐽) ∧ 𝑦𝑋) ∧ 𝑧𝑆) ∧ (((invg𝐺)‘𝑦)(+g𝐺)𝑧) ∈ 𝑌) ∧ 𝑎𝑋) → (𝐺 ~QG 𝑌) Er 𝑋)
1124, 50, 76, 49grplinv 17291 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝐺 ∈ Grp ∧ 𝑎𝑋) → (((invg𝐺)‘𝑎)(+g𝐺)𝑎) = (0g𝐺))
113103, 104, 112syl2anc 691 . . . . . . . . . . . . . . . . . . . . . . 23 ((((((𝐺 ∈ TopGrp ∧ 𝑌 ∈ (NrmSGrp‘𝐺) ∧ 𝑆𝐽) ∧ 𝑦𝑋) ∧ 𝑧𝑆) ∧ (((invg𝐺)‘𝑦)(+g𝐺)𝑧) ∈ 𝑌) ∧ 𝑎𝑋) → (((invg𝐺)‘𝑎)(+g𝐺)𝑎) = (0g𝐺))
114113oveq1d 6564 . . . . . . . . . . . . . . . . . . . . . 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 17290 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝐺 ∈ Grp ∧ 𝑎𝑋) → ((invg𝐺)‘𝑎) ∈ 𝑋)
116103, 104, 115syl2anc 691 . . . . . . . . . . . . . . . . . . . . . . 23 ((((((𝐺 ∈ TopGrp ∧ 𝑌 ∈ (NrmSGrp‘𝐺) ∧ 𝑆𝐽) ∧ 𝑦𝑋) ∧ 𝑧𝑆) ∧ (((invg𝐺)‘𝑦)(+g𝐺)𝑧) ∈ 𝑌) ∧ 𝑎𝑋) → ((invg𝐺)‘𝑎) ∈ 𝑋)
1174, 50grpass 17254 . . . . . . . . . . . . . . . . . . . . . . 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 1320 . . . . . . . . . . . . . . . . . . . . . 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 17275 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝐺 ∈ Grp ∧ (((invg𝐺)‘𝑦)(+g𝐺)𝑧) ∈ 𝑋) → ((0g𝐺)(+g𝐺)(((invg𝐺)‘𝑦)(+g𝐺)𝑧)) = (((invg𝐺)‘𝑦)(+g𝐺)𝑧))
120103, 105, 119syl2anc 691 . . . . . . . . . . . . . . . . . . . . . 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 2652 . . . . . . . . . . . . . . . . . . . . 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 788 . . . . . . . . . . . . . . . . . . . . 21 ((((((𝐺 ∈ TopGrp ∧ 𝑌 ∈ (NrmSGrp‘𝐺) ∧ 𝑆𝐽) ∧ 𝑦𝑋) ∧ 𝑧𝑆) ∧ (((invg𝐺)‘𝑦)(+g𝐺)𝑧) ∈ 𝑌) ∧ 𝑎𝑋) → (((invg𝐺)‘𝑦)(+g𝐺)𝑧) ∈ 𝑌)
123121, 122eqeltrd 2688 . . . . . . . . . . . . . . . . . . . 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 758 . . . . . . . . . . . . . . . . . . . . 21 ((((((𝐺 ∈ TopGrp ∧ 𝑌 ∈ (NrmSGrp‘𝐺) ∧ 𝑆𝐽) ∧ 𝑦𝑋) ∧ 𝑧𝑆) ∧ (((invg𝐺)‘𝑦)(+g𝐺)𝑧) ∈ 𝑌) ∧ 𝑎𝑋) → 𝑌𝑋)
1254, 49, 50, 41eqgval 17466 . . . . . . . . . . . . . . . . . . . . 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 691 . . . . . . . . . . . . . . . . . . . 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 1238 . . . . . . . . . . . . . . . . . . 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 7680 . . . . . . . . . . . . . . . . . 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 2647 . . . . . . . . . . . . . . . . 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 2672 . . . . . . . . . . . . . . . 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 228 . . . . . . . . . . . . . . 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 3646 . . . . . . . . . . . . . 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 7669 . . . . . . . . . . . . . . . . 17 (𝑥 = 𝑎 → [𝑥](𝐺 ~QG 𝑌) = [𝑎](𝐺 ~QG 𝑌))
134133cbvmptv 4678 . . . . . . . . . . . . . . . 16 (𝑥𝑋 ↦ [𝑥](𝐺 ~QG 𝑌)) = (𝑎𝑋 ↦ [𝑎](𝐺 ~QG 𝑌))
1356, 134eqtri 2632 . . . . . . . . . . . . . . 15 𝐹 = (𝑎𝑋 ↦ [𝑎](𝐺 ~QG 𝑌))
136135mptpreima 5545 . . . . . . . . . . . . . 14 (𝐹 “ (𝐹𝑆)) = {𝑎𝑋 ∣ [𝑎](𝐺 ~QG 𝑌) ∈ (𝐹𝑆)}
137132, 93, 1363sstr4g 3609 . . . . . . . . . . . . 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 2677 . . . . . . . . . . . . . . 15 (𝑢 = ((𝑎𝑋 ↦ (𝑎(+g𝐺)(((invg𝐺)‘𝑦)(+g𝐺)𝑧))) “ 𝑆) → (𝑦𝑢𝑦 ∈ ((𝑎𝑋 ↦ (𝑎(+g𝐺)(((invg𝐺)‘𝑦)(+g𝐺)𝑧))) “ 𝑆)))
139 sseq1 3589 . . . . . . . . . . . . . . 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 743 . . . . . . . . . . . . . 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 3282 . . . . . . . . . . . . 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 1316 . . . . . . . . . . . 12 (((((𝐺 ∈ TopGrp ∧ 𝑌 ∈ (NrmSGrp‘𝐺) ∧ 𝑆𝐽) ∧ 𝑦𝑋) ∧ 𝑧𝑆) ∧ (((invg𝐺)‘𝑦)(+g𝐺)𝑧) ∈ 𝑌) → ∃𝑢𝐽 (𝑦𝑢𝑢 ⊆ (𝐹 “ (𝐹𝑆))))
1431423ad2antr3 1221 . . . . . . . . . . 11 (((((𝐺 ∈ TopGrp ∧ 𝑌 ∈ (NrmSGrp‘𝐺) ∧ 𝑆𝐽) ∧ 𝑦𝑋) ∧ 𝑧𝑆) ∧ (𝑦𝑋𝑧𝑋 ∧ (((invg𝐺)‘𝑦)(+g𝐺)𝑧) ∈ 𝑌)) → ∃𝑢𝐽 (𝑦𝑢𝑢 ⊆ (𝐹 “ (𝐹𝑆))))
144143ex 449 . . . . . . . . . 10 ((((𝐺 ∈ TopGrp ∧ 𝑌 ∈ (NrmSGrp‘𝐺) ∧ 𝑆𝐽) ∧ 𝑦𝑋) ∧ 𝑧𝑆) → ((𝑦𝑋𝑧𝑋 ∧ (((invg𝐺)‘𝑦)(+g𝐺)𝑧) ∈ 𝑌) → ∃𝑢𝐽 (𝑦𝑢𝑢 ⊆ (𝐹 “ (𝐹𝑆)))))
14553, 144sylbid 229 . . . . . . . . 9 ((((𝐺 ∈ TopGrp ∧ 𝑌 ∈ (NrmSGrp‘𝐺) ∧ 𝑆𝐽) ∧ 𝑦𝑋) ∧ 𝑧𝑆) → ((𝐹𝑧) = [𝑦](𝐺 ~QG 𝑌) → ∃𝑢𝐽 (𝑦𝑢𝑢 ⊆ (𝐹 “ (𝐹𝑆)))))
146145rexlimdva 3013 . . . . . . . 8 (((𝐺 ∈ TopGrp ∧ 𝑌 ∈ (NrmSGrp‘𝐺) ∧ 𝑆𝐽) ∧ 𝑦𝑋) → (∃𝑧𝑆 (𝐹𝑧) = [𝑦](𝐺 ~QG 𝑌) → ∃𝑢𝐽 (𝑦𝑢𝑢 ⊆ (𝐹 “ (𝐹𝑆)))))
14721, 146syl5 33 . . . . . . 7 (((𝐺 ∈ TopGrp ∧ 𝑌 ∈ (NrmSGrp‘𝐺) ∧ 𝑆𝐽) ∧ 𝑦𝑋) → ([𝑦](𝐺 ~QG 𝑌) ∈ (𝐹𝑆) → ∃𝑢𝐽 (𝑦𝑢𝑢 ⊆ (𝐹 “ (𝐹𝑆)))))
148147expimpd 627 . . . . . 6 ((𝐺 ∈ TopGrp ∧ 𝑌 ∈ (NrmSGrp‘𝐺) ∧ 𝑆𝐽) → ((𝑦𝑋 ∧ [𝑦](𝐺 ~QG 𝑌) ∈ (𝐹𝑆)) → ∃𝑢𝐽 (𝑦𝑢𝑢 ⊆ (𝐹 “ (𝐹𝑆)))))
14918, 148syl5bi 231 . . . . 5 ((𝐺 ∈ TopGrp ∧ 𝑌 ∈ (NrmSGrp‘𝐺) ∧ 𝑆𝐽) → (𝑦 ∈ (𝐹 “ (𝐹𝑆)) → ∃𝑢𝐽 (𝑦𝑢𝑢 ⊆ (𝐹 “ (𝐹𝑆)))))
150149ralrimiv 2948 . . . 4 ((𝐺 ∈ TopGrp ∧ 𝑌 ∈ (NrmSGrp‘𝐺) ∧ 𝑆𝐽) → ∀𝑦 ∈ (𝐹 “ (𝐹𝑆))∃𝑢𝐽 (𝑦𝑢𝑢 ⊆ (𝐹 “ (𝐹𝑆))))
151 topontop 20541 . . . . 5 (𝐽 ∈ (TopOn‘𝑋) → 𝐽 ∈ Top)
152 eltop2 20590 . . . . 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 246 . . 3 ((𝐺 ∈ TopGrp ∧ 𝑌 ∈ (NrmSGrp‘𝐺) ∧ 𝑆𝐽) → (𝐹 “ (𝐹𝑆)) ∈ 𝐽)
155 elqtop3 21316 . . . 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 691 . . 3 ((𝐺 ∈ TopGrp ∧ 𝑌 ∈ (NrmSGrp‘𝐺) ∧ 𝑆𝐽) → ((𝐹𝑆) ∈ (𝐽 qTop 𝐹) ↔ ((𝐹𝑆) ⊆ (𝑋 / (𝐺 ~QG 𝑌)) ∧ (𝐹 “ (𝐹𝑆)) ∈ 𝐽)))
15713, 154, 156mpbir2and 959 . 2 ((𝐺 ∈ TopGrp ∧ 𝑌 ∈ (NrmSGrp‘𝐺) ∧ 𝑆𝐽) → (𝐹𝑆) ∈ (𝐽 qTop 𝐹))
1583, 5, 6, 8, 9qusval 16025 . . 3 ((𝐺 ∈ TopGrp ∧ 𝑌 ∈ (NrmSGrp‘𝐺) ∧ 𝑆𝐽) → 𝐻 = (𝐹s 𝐺))
159 qustgpopn.k . . 3 𝐾 = (TopOpen‘𝐻)
160158, 5, 10, 9, 22, 159imastopn 21333 . 2 ((𝐺 ∈ TopGrp ∧ 𝑌 ∈ (NrmSGrp‘𝐺) ∧ 𝑆𝐽) → 𝐾 = (𝐽 qTop 𝐹))
161157, 160eleqtrrd 2691 1 ((𝐺 ∈ TopGrp ∧ 𝑌 ∈ (NrmSGrp‘𝐺) ∧ 𝑆𝐽) → (𝐹𝑆) ∈ 𝐾)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383   ∧ w3a 1031   = wceq 1475   ∈ wcel 1977  ∀wral 2896  ∃wrex 2897  {crab 2900  Vcvv 3173   ⊆ wss 3540   class class class wbr 4583   ↦ cmpt 4643  ◡ccnv 5037  ran crn 5039   “ cima 5041  Fun wfun 5798   Fn wfn 5799  –onto→wfo 5802  ‘cfv 5804  (class class class)co 6549   Er wer 7626  [cec 7627   / cqs 7628  Basecbs 15695  +gcplusg 15768  TopOpenctopn 15905  0gc0g 15923   qTop cqtop 15986   /s cqus 15988  Grpcgrp 17245  invgcminusg 17246  SubGrpcsubg 17411  NrmSGrpcnsg 17412   ~QG cqg 17413  oppgcoppg 17598  Topctop 20517  TopOnctopon 20518   Cn ccn 20838  Homeochmeo 21366  TopGrpctgp 21685 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-cnex 9871  ax-resscn 9872  ax-1cn 9873  ax-icn 9874  ax-addcl 9875  ax-addrcl 9876  ax-mulcl 9877  ax-mulrcl 9878  ax-mulcom 9879  ax-addass 9880  ax-mulass 9881  ax-distr 9882  ax-i2m1 9883  ax-1ne0 9884  ax-1rid 9885  ax-rnegex 9886  ax-rrecex 9887  ax-cnre 9888  ax-pre-lttri 9889  ax-pre-lttrn 9890  ax-pre-ltadd 9891  ax-pre-mulgt0 9892 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-nel 2783  df-ral 2901  df-rex 2902  df-reu 2903  df-rmo 2904  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-int 4411  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-riota 6511  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-om 6958  df-1st 7059  df-2nd 7060  df-tpos 7239  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-1o 7447  df-oadd 7451  df-er 7629  df-ec 7631  df-qs 7635  df-map 7746  df-en 7842  df-dom 7843  df-sdom 7844  df-fin 7845  df-sup 8231  df-inf 8232  df-pnf 9955  df-mnf 9956  df-xr 9957  df-ltxr 9958  df-le 9959  df-sub 10147  df-neg 10148  df-nn 10898  df-2 10956  df-3 10957  df-4 10958  df-5 10959  df-6 10960  df-7 10961  df-8 10962  df-9 10963  df-n0 11170  df-z 11255  df-dec 11370  df-uz 11564  df-fz 12198  df-struct 15697  df-ndx 15698  df-slot 15699  df-base 15700  df-sets 15701  df-ress 15702  df-plusg 15781  df-mulr 15782  df-sca 15784  df-vsca 15785  df-ip 15786  df-tset 15787  df-ple 15788  df-ds 15791  df-rest 15906  df-topn 15907  df-0g 15925  df-topgen 15927  df-qtop 15990  df-imas 15991  df-qus 15992  df-plusf 17064  df-mgm 17065  df-sgrp 17107  df-mnd 17118  df-grp 17248  df-minusg 17249  df-subg 17414  df-nsg 17415  df-eqg 17416  df-oppg 17599  df-top 20521  df-bases 20522  df-topon 20523  df-topsp 20524  df-cn 20841  df-cnp 20842  df-tx 21175  df-hmeo 21368  df-tmd 21686  df-tgp 21687 This theorem is referenced by:  qustgplem  21734
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