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Theorem pm2mpmhmlem2 21419
Description: Lemma 2 for pm2mpmhm 21420. (Contributed by AV, 22-Oct-2019.) (Revised by AV, 6-Dec-2019.)
Hypotheses
Ref Expression
pm2mpmhm.p 𝑃 = (Poly1𝑅)
pm2mpmhm.c 𝐶 = (𝑁 Mat 𝑃)
pm2mpmhm.a 𝐴 = (𝑁 Mat 𝑅)
pm2mpmhm.q 𝑄 = (Poly1𝐴)
pm2mpmhm.t 𝑇 = (𝑁 pMatToMatPoly 𝑅)
pm2mpmhm.b 𝐵 = (Base‘𝐶)
Assertion
Ref Expression
pm2mpmhmlem2 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → ∀𝑥𝐵𝑦𝐵 (𝑇‘(𝑥(.r𝐶)𝑦)) = ((𝑇𝑥)(.r𝑄)(𝑇𝑦)))
Distinct variable groups:   𝑥,𝐵,𝑦   𝑥,𝑁,𝑦   𝑥,𝑅,𝑦
Allowed substitution hints:   𝐴(𝑥,𝑦)   𝐶(𝑥,𝑦)   𝑃(𝑥,𝑦)   𝑄(𝑥,𝑦)   𝑇(𝑥,𝑦)

Proof of Theorem pm2mpmhmlem2
Dummy variables 𝑘 𝑙 𝑛 𝑟 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 simpll 765 . . . . 5 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥𝐵𝑦𝐵)) → 𝑁 ∈ Fin)
2 simplr 767 . . . . 5 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥𝐵𝑦𝐵)) → 𝑅 ∈ Ring)
3 pm2mpmhm.p . . . . . . . 8 𝑃 = (Poly1𝑅)
4 pm2mpmhm.c . . . . . . . 8 𝐶 = (𝑁 Mat 𝑃)
53, 4pmatring 21293 . . . . . . 7 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝐶 ∈ Ring)
65adantr 483 . . . . . 6 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥𝐵𝑦𝐵)) → 𝐶 ∈ Ring)
7 simpl 485 . . . . . . 7 ((𝑥𝐵𝑦𝐵) → 𝑥𝐵)
87adantl 484 . . . . . 6 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥𝐵𝑦𝐵)) → 𝑥𝐵)
9 simpr 487 . . . . . . 7 ((𝑥𝐵𝑦𝐵) → 𝑦𝐵)
109adantl 484 . . . . . 6 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥𝐵𝑦𝐵)) → 𝑦𝐵)
11 pm2mpmhm.b . . . . . . 7 𝐵 = (Base‘𝐶)
12 eqid 2819 . . . . . . 7 (.r𝐶) = (.r𝐶)
1311, 12ringcl 19303 . . . . . 6 ((𝐶 ∈ Ring ∧ 𝑥𝐵𝑦𝐵) → (𝑥(.r𝐶)𝑦) ∈ 𝐵)
146, 8, 10, 13syl3anc 1365 . . . . 5 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥𝐵𝑦𝐵)) → (𝑥(.r𝐶)𝑦) ∈ 𝐵)
15 eqid 2819 . . . . . 6 ( ·𝑠𝑄) = ( ·𝑠𝑄)
16 eqid 2819 . . . . . 6 (.g‘(mulGrp‘𝑄)) = (.g‘(mulGrp‘𝑄))
17 eqid 2819 . . . . . 6 (var1𝐴) = (var1𝐴)
18 pm2mpmhm.a . . . . . 6 𝐴 = (𝑁 Mat 𝑅)
19 pm2mpmhm.q . . . . . 6 𝑄 = (Poly1𝐴)
20 pm2mpmhm.t . . . . . 6 𝑇 = (𝑁 pMatToMatPoly 𝑅)
213, 4, 11, 15, 16, 17, 18, 19, 20pm2mpfval 21396 . . . . 5 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ (𝑥(.r𝐶)𝑦) ∈ 𝐵) → (𝑇‘(𝑥(.r𝐶)𝑦)) = (𝑄 Σg (𝑘 ∈ ℕ0 ↦ (((𝑥(.r𝐶)𝑦) decompPMat 𝑘)( ·𝑠𝑄)(𝑘(.g‘(mulGrp‘𝑄))(var1𝐴))))))
221, 2, 14, 21syl3anc 1365 . . . 4 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥𝐵𝑦𝐵)) → (𝑇‘(𝑥(.r𝐶)𝑦)) = (𝑄 Σg (𝑘 ∈ ℕ0 ↦ (((𝑥(.r𝐶)𝑦) decompPMat 𝑘)( ·𝑠𝑄)(𝑘(.g‘(mulGrp‘𝑄))(var1𝐴))))))
233, 4, 11, 18decpmatmul 21372 . . . . . . . 8 ((𝑅 ∈ Ring ∧ (𝑥𝐵𝑦𝐵) ∧ 𝑘 ∈ ℕ0) → ((𝑥(.r𝐶)𝑦) decompPMat 𝑘) = (𝐴 Σg (𝑧 ∈ (0...𝑘) ↦ ((𝑥 decompPMat 𝑧)(.r𝐴)(𝑦 decompPMat (𝑘𝑧))))))
2423ad4ant234 1169 . . . . . . 7 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥𝐵𝑦𝐵)) ∧ 𝑘 ∈ ℕ0) → ((𝑥(.r𝐶)𝑦) decompPMat 𝑘) = (𝐴 Σg (𝑧 ∈ (0...𝑘) ↦ ((𝑥 decompPMat 𝑧)(.r𝐴)(𝑦 decompPMat (𝑘𝑧))))))
2524oveq1d 7163 . . . . . 6 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥𝐵𝑦𝐵)) ∧ 𝑘 ∈ ℕ0) → (((𝑥(.r𝐶)𝑦) decompPMat 𝑘)( ·𝑠𝑄)(𝑘(.g‘(mulGrp‘𝑄))(var1𝐴))) = ((𝐴 Σg (𝑧 ∈ (0...𝑘) ↦ ((𝑥 decompPMat 𝑧)(.r𝐴)(𝑦 decompPMat (𝑘𝑧)))))( ·𝑠𝑄)(𝑘(.g‘(mulGrp‘𝑄))(var1𝐴))))
2625mpteq2dva 5152 . . . . 5 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥𝐵𝑦𝐵)) → (𝑘 ∈ ℕ0 ↦ (((𝑥(.r𝐶)𝑦) decompPMat 𝑘)( ·𝑠𝑄)(𝑘(.g‘(mulGrp‘𝑄))(var1𝐴)))) = (𝑘 ∈ ℕ0 ↦ ((𝐴 Σg (𝑧 ∈ (0...𝑘) ↦ ((𝑥 decompPMat 𝑧)(.r𝐴)(𝑦 decompPMat (𝑘𝑧)))))( ·𝑠𝑄)(𝑘(.g‘(mulGrp‘𝑄))(var1𝐴)))))
2726oveq2d 7164 . . . 4 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥𝐵𝑦𝐵)) → (𝑄 Σg (𝑘 ∈ ℕ0 ↦ (((𝑥(.r𝐶)𝑦) decompPMat 𝑘)( ·𝑠𝑄)(𝑘(.g‘(mulGrp‘𝑄))(var1𝐴))))) = (𝑄 Σg (𝑘 ∈ ℕ0 ↦ ((𝐴 Σg (𝑧 ∈ (0...𝑘) ↦ ((𝑥 decompPMat 𝑧)(.r𝐴)(𝑦 decompPMat (𝑘𝑧)))))( ·𝑠𝑄)(𝑘(.g‘(mulGrp‘𝑄))(var1𝐴))))))
28 eqid 2819 . . . . . . . 8 (Base‘𝑄) = (Base‘𝑄)
2918matring 21044 . . . . . . . . 9 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝐴 ∈ Ring)
3029ad2antrr 724 . . . . . . . 8 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥𝐵𝑦𝐵)) ∧ 𝑛 ∈ ℕ0) → 𝐴 ∈ Ring)
31 eqid 2819 . . . . . . . 8 (Base‘𝐴) = (Base‘𝐴)
32 eqid 2819 . . . . . . . 8 (0g𝐴) = (0g𝐴)
33 ringcmn 19323 . . . . . . . . . . . 12 (𝐴 ∈ Ring → 𝐴 ∈ CMnd)
3429, 33syl 17 . . . . . . . . . . 11 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝐴 ∈ CMnd)
3534ad3antrrr 728 . . . . . . . . . 10 (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥𝐵𝑦𝐵)) ∧ 𝑛 ∈ ℕ0) ∧ 𝑘 ∈ ℕ0) → 𝐴 ∈ CMnd)
36 fzfid 13333 . . . . . . . . . 10 (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥𝐵𝑦𝐵)) ∧ 𝑛 ∈ ℕ0) ∧ 𝑘 ∈ ℕ0) → (0...𝑘) ∈ Fin)
3730ad2antrr 724 . . . . . . . . . . . 12 ((((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥𝐵𝑦𝐵)) ∧ 𝑛 ∈ ℕ0) ∧ 𝑘 ∈ ℕ0) ∧ 𝑧 ∈ (0...𝑘)) → 𝐴 ∈ Ring)
38 simp-5r 784 . . . . . . . . . . . . 13 ((((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥𝐵𝑦𝐵)) ∧ 𝑛 ∈ ℕ0) ∧ 𝑘 ∈ ℕ0) ∧ 𝑧 ∈ (0...𝑘)) → 𝑅 ∈ Ring)
398ad3antrrr 728 . . . . . . . . . . . . 13 ((((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥𝐵𝑦𝐵)) ∧ 𝑛 ∈ ℕ0) ∧ 𝑘 ∈ ℕ0) ∧ 𝑧 ∈ (0...𝑘)) → 𝑥𝐵)
40 elfznn0 12992 . . . . . . . . . . . . . 14 (𝑧 ∈ (0...𝑘) → 𝑧 ∈ ℕ0)
4140adantl 484 . . . . . . . . . . . . 13 ((((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥𝐵𝑦𝐵)) ∧ 𝑛 ∈ ℕ0) ∧ 𝑘 ∈ ℕ0) ∧ 𝑧 ∈ (0...𝑘)) → 𝑧 ∈ ℕ0)
423, 4, 11, 18, 31decpmatcl 21367 . . . . . . . . . . . . 13 ((𝑅 ∈ Ring ∧ 𝑥𝐵𝑧 ∈ ℕ0) → (𝑥 decompPMat 𝑧) ∈ (Base‘𝐴))
4338, 39, 41, 42syl3anc 1365 . . . . . . . . . . . 12 ((((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥𝐵𝑦𝐵)) ∧ 𝑛 ∈ ℕ0) ∧ 𝑘 ∈ ℕ0) ∧ 𝑧 ∈ (0...𝑘)) → (𝑥 decompPMat 𝑧) ∈ (Base‘𝐴))
4410ad3antrrr 728 . . . . . . . . . . . . 13 ((((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥𝐵𝑦𝐵)) ∧ 𝑛 ∈ ℕ0) ∧ 𝑘 ∈ ℕ0) ∧ 𝑧 ∈ (0...𝑘)) → 𝑦𝐵)
45 fznn0sub 12931 . . . . . . . . . . . . . 14 (𝑧 ∈ (0...𝑘) → (𝑘𝑧) ∈ ℕ0)
4645adantl 484 . . . . . . . . . . . . 13 ((((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥𝐵𝑦𝐵)) ∧ 𝑛 ∈ ℕ0) ∧ 𝑘 ∈ ℕ0) ∧ 𝑧 ∈ (0...𝑘)) → (𝑘𝑧) ∈ ℕ0)
473, 4, 11, 18, 31decpmatcl 21367 . . . . . . . . . . . . 13 ((𝑅 ∈ Ring ∧ 𝑦𝐵 ∧ (𝑘𝑧) ∈ ℕ0) → (𝑦 decompPMat (𝑘𝑧)) ∈ (Base‘𝐴))
4838, 44, 46, 47syl3anc 1365 . . . . . . . . . . . 12 ((((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥𝐵𝑦𝐵)) ∧ 𝑛 ∈ ℕ0) ∧ 𝑘 ∈ ℕ0) ∧ 𝑧 ∈ (0...𝑘)) → (𝑦 decompPMat (𝑘𝑧)) ∈ (Base‘𝐴))
49 eqid 2819 . . . . . . . . . . . . 13 (.r𝐴) = (.r𝐴)
5031, 49ringcl 19303 . . . . . . . . . . . 12 ((𝐴 ∈ Ring ∧ (𝑥 decompPMat 𝑧) ∈ (Base‘𝐴) ∧ (𝑦 decompPMat (𝑘𝑧)) ∈ (Base‘𝐴)) → ((𝑥 decompPMat 𝑧)(.r𝐴)(𝑦 decompPMat (𝑘𝑧))) ∈ (Base‘𝐴))
5137, 43, 48, 50syl3anc 1365 . . . . . . . . . . 11 ((((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥𝐵𝑦𝐵)) ∧ 𝑛 ∈ ℕ0) ∧ 𝑘 ∈ ℕ0) ∧ 𝑧 ∈ (0...𝑘)) → ((𝑥 decompPMat 𝑧)(.r𝐴)(𝑦 decompPMat (𝑘𝑧))) ∈ (Base‘𝐴))
5251ralrimiva 3180 . . . . . . . . . 10 (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥𝐵𝑦𝐵)) ∧ 𝑛 ∈ ℕ0) ∧ 𝑘 ∈ ℕ0) → ∀𝑧 ∈ (0...𝑘)((𝑥 decompPMat 𝑧)(.r𝐴)(𝑦 decompPMat (𝑘𝑧))) ∈ (Base‘𝐴))
5331, 35, 36, 52gsummptcl 19079 . . . . . . . . 9 (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥𝐵𝑦𝐵)) ∧ 𝑛 ∈ ℕ0) ∧ 𝑘 ∈ ℕ0) → (𝐴 Σg (𝑧 ∈ (0...𝑘) ↦ ((𝑥 decompPMat 𝑧)(.r𝐴)(𝑦 decompPMat (𝑘𝑧))))) ∈ (Base‘𝐴))
5453ralrimiva 3180 . . . . . . . 8 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥𝐵𝑦𝐵)) ∧ 𝑛 ∈ ℕ0) → ∀𝑘 ∈ ℕ0 (𝐴 Σg (𝑧 ∈ (0...𝑘) ↦ ((𝑥 decompPMat 𝑧)(.r𝐴)(𝑦 decompPMat (𝑘𝑧))))) ∈ (Base‘𝐴))
553, 4, 11, 18, 49, 32decpmatmulsumfsupp 21373 . . . . . . . . 9 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥𝐵𝑦𝐵)) → (𝑘 ∈ ℕ0 ↦ (𝐴 Σg (𝑧 ∈ (0...𝑘) ↦ ((𝑥 decompPMat 𝑧)(.r𝐴)(𝑦 decompPMat (𝑘𝑧)))))) finSupp (0g𝐴))
5655adantr 483 . . . . . . . 8 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥𝐵𝑦𝐵)) ∧ 𝑛 ∈ ℕ0) → (𝑘 ∈ ℕ0 ↦ (𝐴 Σg (𝑧 ∈ (0...𝑘) ↦ ((𝑥 decompPMat 𝑧)(.r𝐴)(𝑦 decompPMat (𝑘𝑧)))))) finSupp (0g𝐴))
57 simpr 487 . . . . . . . 8 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥𝐵𝑦𝐵)) ∧ 𝑛 ∈ ℕ0) → 𝑛 ∈ ℕ0)
5819, 28, 17, 16, 30, 31, 15, 32, 54, 56, 57gsummoncoe1 20464 . . . . . . 7 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥𝐵𝑦𝐵)) ∧ 𝑛 ∈ ℕ0) → ((coe1‘(𝑄 Σg (𝑘 ∈ ℕ0 ↦ ((𝐴 Σg (𝑧 ∈ (0...𝑘) ↦ ((𝑥 decompPMat 𝑧)(.r𝐴)(𝑦 decompPMat (𝑘𝑧)))))( ·𝑠𝑄)(𝑘(.g‘(mulGrp‘𝑄))(var1𝐴))))))‘𝑛) = 𝑛 / 𝑘(𝐴 Σg (𝑧 ∈ (0...𝑘) ↦ ((𝑥 decompPMat 𝑧)(.r𝐴)(𝑦 decompPMat (𝑘𝑧))))))
59 csbov2g 7194 . . . . . . . . 9 (𝑛 ∈ ℕ0𝑛 / 𝑘(𝐴 Σg (𝑧 ∈ (0...𝑘) ↦ ((𝑥 decompPMat 𝑧)(.r𝐴)(𝑦 decompPMat (𝑘𝑧))))) = (𝐴 Σg 𝑛 / 𝑘(𝑧 ∈ (0...𝑘) ↦ ((𝑥 decompPMat 𝑧)(.r𝐴)(𝑦 decompPMat (𝑘𝑧))))))
60 id 22 . . . . . . . . . . 11 (𝑛 ∈ ℕ0𝑛 ∈ ℕ0)
61 oveq2 7156 . . . . . . . . . . . . 13 (𝑘 = 𝑛 → (0...𝑘) = (0...𝑛))
62 oveq1 7155 . . . . . . . . . . . . . . 15 (𝑘 = 𝑛 → (𝑘𝑧) = (𝑛𝑧))
6362oveq2d 7164 . . . . . . . . . . . . . 14 (𝑘 = 𝑛 → (𝑦 decompPMat (𝑘𝑧)) = (𝑦 decompPMat (𝑛𝑧)))
6463oveq2d 7164 . . . . . . . . . . . . 13 (𝑘 = 𝑛 → ((𝑥 decompPMat 𝑧)(.r𝐴)(𝑦 decompPMat (𝑘𝑧))) = ((𝑥 decompPMat 𝑧)(.r𝐴)(𝑦 decompPMat (𝑛𝑧))))
6561, 64mpteq12dv 5142 . . . . . . . . . . . 12 (𝑘 = 𝑛 → (𝑧 ∈ (0...𝑘) ↦ ((𝑥 decompPMat 𝑧)(.r𝐴)(𝑦 decompPMat (𝑘𝑧)))) = (𝑧 ∈ (0...𝑛) ↦ ((𝑥 decompPMat 𝑧)(.r𝐴)(𝑦 decompPMat (𝑛𝑧)))))
6665adantl 484 . . . . . . . . . . 11 ((𝑛 ∈ ℕ0𝑘 = 𝑛) → (𝑧 ∈ (0...𝑘) ↦ ((𝑥 decompPMat 𝑧)(.r𝐴)(𝑦 decompPMat (𝑘𝑧)))) = (𝑧 ∈ (0...𝑛) ↦ ((𝑥 decompPMat 𝑧)(.r𝐴)(𝑦 decompPMat (𝑛𝑧)))))
6760, 66csbied 3917 . . . . . . . . . 10 (𝑛 ∈ ℕ0𝑛 / 𝑘(𝑧 ∈ (0...𝑘) ↦ ((𝑥 decompPMat 𝑧)(.r𝐴)(𝑦 decompPMat (𝑘𝑧)))) = (𝑧 ∈ (0...𝑛) ↦ ((𝑥 decompPMat 𝑧)(.r𝐴)(𝑦 decompPMat (𝑛𝑧)))))
6867oveq2d 7164 . . . . . . . . 9 (𝑛 ∈ ℕ0 → (𝐴 Σg 𝑛 / 𝑘(𝑧 ∈ (0...𝑘) ↦ ((𝑥 decompPMat 𝑧)(.r𝐴)(𝑦 decompPMat (𝑘𝑧))))) = (𝐴 Σg (𝑧 ∈ (0...𝑛) ↦ ((𝑥 decompPMat 𝑧)(.r𝐴)(𝑦 decompPMat (𝑛𝑧))))))
6959, 68eqtrd 2854 . . . . . . . 8 (𝑛 ∈ ℕ0𝑛 / 𝑘(𝐴 Σg (𝑧 ∈ (0...𝑘) ↦ ((𝑥 decompPMat 𝑧)(.r𝐴)(𝑦 decompPMat (𝑘𝑧))))) = (𝐴 Σg (𝑧 ∈ (0...𝑛) ↦ ((𝑥 decompPMat 𝑧)(.r𝐴)(𝑦 decompPMat (𝑛𝑧))))))
7069adantl 484 . . . . . . 7 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥𝐵𝑦𝐵)) ∧ 𝑛 ∈ ℕ0) → 𝑛 / 𝑘(𝐴 Σg (𝑧 ∈ (0...𝑘) ↦ ((𝑥 decompPMat 𝑧)(.r𝐴)(𝑦 decompPMat (𝑘𝑧))))) = (𝐴 Σg (𝑧 ∈ (0...𝑛) ↦ ((𝑥 decompPMat 𝑧)(.r𝐴)(𝑦 decompPMat (𝑛𝑧))))))
71 eqidd 2820 . . . . . . . . 9 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥𝐵𝑦𝐵)) ∧ 𝑛 ∈ ℕ0) → (𝑟 ∈ ℕ0 ↦ (𝐴 Σg (𝑙 ∈ (0...𝑟) ↦ (((coe1‘(𝑄 Σg (𝑘 ∈ ℕ0 ↦ ((𝑥 decompPMat 𝑘)( ·𝑠𝑄)(𝑘(.g‘(mulGrp‘𝑄))(var1𝐴))))))‘𝑙)(.r𝐴)((coe1‘(𝑄 Σg (𝑘 ∈ ℕ0 ↦ ((𝑦 decompPMat 𝑘)( ·𝑠𝑄)(𝑘(.g‘(mulGrp‘𝑄))(var1𝐴))))))‘(𝑟𝑙)))))) = (𝑟 ∈ ℕ0 ↦ (𝐴 Σg (𝑙 ∈ (0...𝑟) ↦ (((coe1‘(𝑄 Σg (𝑘 ∈ ℕ0 ↦ ((𝑥 decompPMat 𝑘)( ·𝑠𝑄)(𝑘(.g‘(mulGrp‘𝑄))(var1𝐴))))))‘𝑙)(.r𝐴)((coe1‘(𝑄 Σg (𝑘 ∈ ℕ0 ↦ ((𝑦 decompPMat 𝑘)( ·𝑠𝑄)(𝑘(.g‘(mulGrp‘𝑄))(var1𝐴))))))‘(𝑟𝑙)))))))
72 oveq2 7156 . . . . . . . . . . . 12 (𝑟 = 𝑛 → (0...𝑟) = (0...𝑛))
73 fvoveq1 7171 . . . . . . . . . . . . 13 (𝑟 = 𝑛 → ((coe1‘(𝑄 Σg (𝑘 ∈ ℕ0 ↦ ((𝑦 decompPMat 𝑘)( ·𝑠𝑄)(𝑘(.g‘(mulGrp‘𝑄))(var1𝐴))))))‘(𝑟𝑙)) = ((coe1‘(𝑄 Σg (𝑘 ∈ ℕ0 ↦ ((𝑦 decompPMat 𝑘)( ·𝑠𝑄)(𝑘(.g‘(mulGrp‘𝑄))(var1𝐴))))))‘(𝑛𝑙)))
7473oveq2d 7164 . . . . . . . . . . . 12 (𝑟 = 𝑛 → (((coe1‘(𝑄 Σg (𝑘 ∈ ℕ0 ↦ ((𝑥 decompPMat 𝑘)( ·𝑠𝑄)(𝑘(.g‘(mulGrp‘𝑄))(var1𝐴))))))‘𝑙)(.r𝐴)((coe1‘(𝑄 Σg (𝑘 ∈ ℕ0 ↦ ((𝑦 decompPMat 𝑘)( ·𝑠𝑄)(𝑘(.g‘(mulGrp‘𝑄))(var1𝐴))))))‘(𝑟𝑙))) = (((coe1‘(𝑄 Σg (𝑘 ∈ ℕ0 ↦ ((𝑥 decompPMat 𝑘)( ·𝑠𝑄)(𝑘(.g‘(mulGrp‘𝑄))(var1𝐴))))))‘𝑙)(.r𝐴)((coe1‘(𝑄 Σg (𝑘 ∈ ℕ0 ↦ ((𝑦 decompPMat 𝑘)( ·𝑠𝑄)(𝑘(.g‘(mulGrp‘𝑄))(var1𝐴))))))‘(𝑛𝑙))))
7572, 74mpteq12dv 5142 . . . . . . . . . . 11 (𝑟 = 𝑛 → (𝑙 ∈ (0...𝑟) ↦ (((coe1‘(𝑄 Σg (𝑘 ∈ ℕ0 ↦ ((𝑥 decompPMat 𝑘)( ·𝑠𝑄)(𝑘(.g‘(mulGrp‘𝑄))(var1𝐴))))))‘𝑙)(.r𝐴)((coe1‘(𝑄 Σg (𝑘 ∈ ℕ0 ↦ ((𝑦 decompPMat 𝑘)( ·𝑠𝑄)(𝑘(.g‘(mulGrp‘𝑄))(var1𝐴))))))‘(𝑟𝑙)))) = (𝑙 ∈ (0...𝑛) ↦ (((coe1‘(𝑄 Σg (𝑘 ∈ ℕ0 ↦ ((𝑥 decompPMat 𝑘)( ·𝑠𝑄)(𝑘(.g‘(mulGrp‘𝑄))(var1𝐴))))))‘𝑙)(.r𝐴)((coe1‘(𝑄 Σg (𝑘 ∈ ℕ0 ↦ ((𝑦 decompPMat 𝑘)( ·𝑠𝑄)(𝑘(.g‘(mulGrp‘𝑄))(var1𝐴))))))‘(𝑛𝑙)))))
7675oveq2d 7164 . . . . . . . . . 10 (𝑟 = 𝑛 → (𝐴 Σg (𝑙 ∈ (0...𝑟) ↦ (((coe1‘(𝑄 Σg (𝑘 ∈ ℕ0 ↦ ((𝑥 decompPMat 𝑘)( ·𝑠𝑄)(𝑘(.g‘(mulGrp‘𝑄))(var1𝐴))))))‘𝑙)(.r𝐴)((coe1‘(𝑄 Σg (𝑘 ∈ ℕ0 ↦ ((𝑦 decompPMat 𝑘)( ·𝑠𝑄)(𝑘(.g‘(mulGrp‘𝑄))(var1𝐴))))))‘(𝑟𝑙))))) = (𝐴 Σg (𝑙 ∈ (0...𝑛) ↦ (((coe1‘(𝑄 Σg (𝑘 ∈ ℕ0 ↦ ((𝑥 decompPMat 𝑘)( ·𝑠𝑄)(𝑘(.g‘(mulGrp‘𝑄))(var1𝐴))))))‘𝑙)(.r𝐴)((coe1‘(𝑄 Σg (𝑘 ∈ ℕ0 ↦ ((𝑦 decompPMat 𝑘)( ·𝑠𝑄)(𝑘(.g‘(mulGrp‘𝑄))(var1𝐴))))))‘(𝑛𝑙))))))
7776adantl 484 . . . . . . . . 9 (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥𝐵𝑦𝐵)) ∧ 𝑛 ∈ ℕ0) ∧ 𝑟 = 𝑛) → (𝐴 Σg (𝑙 ∈ (0...𝑟) ↦ (((coe1‘(𝑄 Σg (𝑘 ∈ ℕ0 ↦ ((𝑥 decompPMat 𝑘)( ·𝑠𝑄)(𝑘(.g‘(mulGrp‘𝑄))(var1𝐴))))))‘𝑙)(.r𝐴)((coe1‘(𝑄 Σg (𝑘 ∈ ℕ0 ↦ ((𝑦 decompPMat 𝑘)( ·𝑠𝑄)(𝑘(.g‘(mulGrp‘𝑄))(var1𝐴))))))‘(𝑟𝑙))))) = (𝐴 Σg (𝑙 ∈ (0...𝑛) ↦ (((coe1‘(𝑄 Σg (𝑘 ∈ ℕ0 ↦ ((𝑥 decompPMat 𝑘)( ·𝑠𝑄)(𝑘(.g‘(mulGrp‘𝑄))(var1𝐴))))))‘𝑙)(.r𝐴)((coe1‘(𝑄 Σg (𝑘 ∈ ℕ0 ↦ ((𝑦 decompPMat 𝑘)( ·𝑠𝑄)(𝑘(.g‘(mulGrp‘𝑄))(var1𝐴))))))‘(𝑛𝑙))))))
78 ovexd 7183 . . . . . . . . 9 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥𝐵𝑦𝐵)) ∧ 𝑛 ∈ ℕ0) → (𝐴 Σg (𝑙 ∈ (0...𝑛) ↦ (((coe1‘(𝑄 Σg (𝑘 ∈ ℕ0 ↦ ((𝑥 decompPMat 𝑘)( ·𝑠𝑄)(𝑘(.g‘(mulGrp‘𝑄))(var1𝐴))))))‘𝑙)(.r𝐴)((coe1‘(𝑄 Σg (𝑘 ∈ ℕ0 ↦ ((𝑦 decompPMat 𝑘)( ·𝑠𝑄)(𝑘(.g‘(mulGrp‘𝑄))(var1𝐴))))))‘(𝑛𝑙))))) ∈ V)
7971, 77, 57, 78fvmptd 6768 . . . . . . . 8 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥𝐵𝑦𝐵)) ∧ 𝑛 ∈ ℕ0) → ((𝑟 ∈ ℕ0 ↦ (𝐴 Σg (𝑙 ∈ (0...𝑟) ↦ (((coe1‘(𝑄 Σg (𝑘 ∈ ℕ0 ↦ ((𝑥 decompPMat 𝑘)( ·𝑠𝑄)(𝑘(.g‘(mulGrp‘𝑄))(var1𝐴))))))‘𝑙)(.r𝐴)((coe1‘(𝑄 Σg (𝑘 ∈ ℕ0 ↦ ((𝑦 decompPMat 𝑘)( ·𝑠𝑄)(𝑘(.g‘(mulGrp‘𝑄))(var1𝐴))))))‘(𝑟𝑙))))))‘𝑛) = (𝐴 Σg (𝑙 ∈ (0...𝑛) ↦ (((coe1‘(𝑄 Σg (𝑘 ∈ ℕ0 ↦ ((𝑥 decompPMat 𝑘)( ·𝑠𝑄)(𝑘(.g‘(mulGrp‘𝑄))(var1𝐴))))))‘𝑙)(.r𝐴)((coe1‘(𝑄 Σg (𝑘 ∈ ℕ0 ↦ ((𝑦 decompPMat 𝑘)( ·𝑠𝑄)(𝑘(.g‘(mulGrp‘𝑄))(var1𝐴))))))‘(𝑛𝑙))))))
80 eqid 2819 . . . . . . . . . 10 (0g𝑄) = (0g𝑄)
8119ply1ring 20408 . . . . . . . . . . . . 13 (𝐴 ∈ Ring → 𝑄 ∈ Ring)
8229, 81syl 17 . . . . . . . . . . . 12 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝑄 ∈ Ring)
83 ringcmn 19323 . . . . . . . . . . . 12 (𝑄 ∈ Ring → 𝑄 ∈ CMnd)
8482, 83syl 17 . . . . . . . . . . 11 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝑄 ∈ CMnd)
8584ad2antrr 724 . . . . . . . . . 10 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥𝐵𝑦𝐵)) ∧ 𝑛 ∈ ℕ0) → 𝑄 ∈ CMnd)
86 nn0ex 11895 . . . . . . . . . . 11 0 ∈ V
8786a1i 11 . . . . . . . . . 10 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥𝐵𝑦𝐵)) ∧ 𝑛 ∈ ℕ0) → ℕ0 ∈ V)
887anim2i 618 . . . . . . . . . . . . . 14 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥𝐵𝑦𝐵)) → ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝑥𝐵))
89 df-3an 1083 . . . . . . . . . . . . . 14 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑥𝐵) ↔ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝑥𝐵))
9088, 89sylibr 236 . . . . . . . . . . . . 13 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥𝐵𝑦𝐵)) → (𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑥𝐵))
9190adantr 483 . . . . . . . . . . . 12 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥𝐵𝑦𝐵)) ∧ 𝑛 ∈ ℕ0) → (𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑥𝐵))
923, 4, 11, 15, 16, 17, 18, 19, 28pm2mpghmlem1 21413 . . . . . . . . . . . 12 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑥𝐵) ∧ 𝑘 ∈ ℕ0) → ((𝑥 decompPMat 𝑘)( ·𝑠𝑄)(𝑘(.g‘(mulGrp‘𝑄))(var1𝐴))) ∈ (Base‘𝑄))
9391, 92sylan 582 . . . . . . . . . . 11 (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥𝐵𝑦𝐵)) ∧ 𝑛 ∈ ℕ0) ∧ 𝑘 ∈ ℕ0) → ((𝑥 decompPMat 𝑘)( ·𝑠𝑄)(𝑘(.g‘(mulGrp‘𝑄))(var1𝐴))) ∈ (Base‘𝑄))
9493fmpttd 6872 . . . . . . . . . 10 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥𝐵𝑦𝐵)) ∧ 𝑛 ∈ ℕ0) → (𝑘 ∈ ℕ0 ↦ ((𝑥 decompPMat 𝑘)( ·𝑠𝑄)(𝑘(.g‘(mulGrp‘𝑄))(var1𝐴)))):ℕ0⟶(Base‘𝑄))
953, 4, 11, 15, 16, 17, 18, 19pm2mpghmlem2 21412 . . . . . . . . . . 11 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑥𝐵) → (𝑘 ∈ ℕ0 ↦ ((𝑥 decompPMat 𝑘)( ·𝑠𝑄)(𝑘(.g‘(mulGrp‘𝑄))(var1𝐴)))) finSupp (0g𝑄))
9691, 95syl 17 . . . . . . . . . 10 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥𝐵𝑦𝐵)) ∧ 𝑛 ∈ ℕ0) → (𝑘 ∈ ℕ0 ↦ ((𝑥 decompPMat 𝑘)( ·𝑠𝑄)(𝑘(.g‘(mulGrp‘𝑄))(var1𝐴)))) finSupp (0g𝑄))
9728, 80, 85, 87, 94, 96gsumcl 19027 . . . . . . . . 9 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥𝐵𝑦𝐵)) ∧ 𝑛 ∈ ℕ0) → (𝑄 Σg (𝑘 ∈ ℕ0 ↦ ((𝑥 decompPMat 𝑘)( ·𝑠𝑄)(𝑘(.g‘(mulGrp‘𝑄))(var1𝐴))))) ∈ (Base‘𝑄))
989anim2i 618 . . . . . . . . . . . . . 14 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥𝐵𝑦𝐵)) → ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝑦𝐵))
99 df-3an 1083 . . . . . . . . . . . . . 14 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑦𝐵) ↔ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝑦𝐵))
10098, 99sylibr 236 . . . . . . . . . . . . 13 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥𝐵𝑦𝐵)) → (𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑦𝐵))
101100adantr 483 . . . . . . . . . . . 12 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥𝐵𝑦𝐵)) ∧ 𝑛 ∈ ℕ0) → (𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑦𝐵))
1023, 4, 11, 15, 16, 17, 18, 19, 28pm2mpghmlem1 21413 . . . . . . . . . . . 12 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑦𝐵) ∧ 𝑘 ∈ ℕ0) → ((𝑦 decompPMat 𝑘)( ·𝑠𝑄)(𝑘(.g‘(mulGrp‘𝑄))(var1𝐴))) ∈ (Base‘𝑄))
103101, 102sylan 582 . . . . . . . . . . 11 (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥𝐵𝑦𝐵)) ∧ 𝑛 ∈ ℕ0) ∧ 𝑘 ∈ ℕ0) → ((𝑦 decompPMat 𝑘)( ·𝑠𝑄)(𝑘(.g‘(mulGrp‘𝑄))(var1𝐴))) ∈ (Base‘𝑄))
104103fmpttd 6872 . . . . . . . . . 10 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥𝐵𝑦𝐵)) ∧ 𝑛 ∈ ℕ0) → (𝑘 ∈ ℕ0 ↦ ((𝑦 decompPMat 𝑘)( ·𝑠𝑄)(𝑘(.g‘(mulGrp‘𝑄))(var1𝐴)))):ℕ0⟶(Base‘𝑄))
1051, 2, 103jca 1122 . . . . . . . . . . . 12 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥𝐵𝑦𝐵)) → (𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑦𝐵))
106105adantr 483 . . . . . . . . . . 11 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥𝐵𝑦𝐵)) ∧ 𝑛 ∈ ℕ0) → (𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑦𝐵))
1073, 4, 11, 15, 16, 17, 18, 19pm2mpghmlem2 21412 . . . . . . . . . . 11 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑦𝐵) → (𝑘 ∈ ℕ0 ↦ ((𝑦 decompPMat 𝑘)( ·𝑠𝑄)(𝑘(.g‘(mulGrp‘𝑄))(var1𝐴)))) finSupp (0g𝑄))
108106, 107syl 17 . . . . . . . . . 10 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥𝐵𝑦𝐵)) ∧ 𝑛 ∈ ℕ0) → (𝑘 ∈ ℕ0 ↦ ((𝑦 decompPMat 𝑘)( ·𝑠𝑄)(𝑘(.g‘(mulGrp‘𝑄))(var1𝐴)))) finSupp (0g𝑄))
10928, 80, 85, 87, 104, 108gsumcl 19027 . . . . . . . . 9 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥𝐵𝑦𝐵)) ∧ 𝑛 ∈ ℕ0) → (𝑄 Σg (𝑘 ∈ ℕ0 ↦ ((𝑦 decompPMat 𝑘)( ·𝑠𝑄)(𝑘(.g‘(mulGrp‘𝑄))(var1𝐴))))) ∈ (Base‘𝑄))
110 eqid 2819 . . . . . . . . . . 11 (.r𝑄) = (.r𝑄)
11119, 110, 49, 28coe1mul 20430 . . . . . . . . . 10 ((𝐴 ∈ Ring ∧ (𝑄 Σg (𝑘 ∈ ℕ0 ↦ ((𝑥 decompPMat 𝑘)( ·𝑠𝑄)(𝑘(.g‘(mulGrp‘𝑄))(var1𝐴))))) ∈ (Base‘𝑄) ∧ (𝑄 Σg (𝑘 ∈ ℕ0 ↦ ((𝑦 decompPMat 𝑘)( ·𝑠𝑄)(𝑘(.g‘(mulGrp‘𝑄))(var1𝐴))))) ∈ (Base‘𝑄)) → (coe1‘((𝑄 Σg (𝑘 ∈ ℕ0 ↦ ((𝑥 decompPMat 𝑘)( ·𝑠𝑄)(𝑘(.g‘(mulGrp‘𝑄))(var1𝐴)))))(.r𝑄)(𝑄 Σg (𝑘 ∈ ℕ0 ↦ ((𝑦 decompPMat 𝑘)( ·𝑠𝑄)(𝑘(.g‘(mulGrp‘𝑄))(var1𝐴))))))) = (𝑟 ∈ ℕ0 ↦ (𝐴 Σg (𝑙 ∈ (0...𝑟) ↦ (((coe1‘(𝑄 Σg (𝑘 ∈ ℕ0 ↦ ((𝑥 decompPMat 𝑘)( ·𝑠𝑄)(𝑘(.g‘(mulGrp‘𝑄))(var1𝐴))))))‘𝑙)(.r𝐴)((coe1‘(𝑄 Σg (𝑘 ∈ ℕ0 ↦ ((𝑦 decompPMat 𝑘)( ·𝑠𝑄)(𝑘(.g‘(mulGrp‘𝑄))(var1𝐴))))))‘(𝑟𝑙)))))))
112111fveq1d 6665 . . . . . . . . 9 ((𝐴 ∈ Ring ∧ (𝑄 Σg (𝑘 ∈ ℕ0 ↦ ((𝑥 decompPMat 𝑘)( ·𝑠𝑄)(𝑘(.g‘(mulGrp‘𝑄))(var1𝐴))))) ∈ (Base‘𝑄) ∧ (𝑄 Σg (𝑘 ∈ ℕ0 ↦ ((𝑦 decompPMat 𝑘)( ·𝑠𝑄)(𝑘(.g‘(mulGrp‘𝑄))(var1𝐴))))) ∈ (Base‘𝑄)) → ((coe1‘((𝑄 Σg (𝑘 ∈ ℕ0 ↦ ((𝑥 decompPMat 𝑘)( ·𝑠𝑄)(𝑘(.g‘(mulGrp‘𝑄))(var1𝐴)))))(.r𝑄)(𝑄 Σg (𝑘 ∈ ℕ0 ↦ ((𝑦 decompPMat 𝑘)( ·𝑠𝑄)(𝑘(.g‘(mulGrp‘𝑄))(var1𝐴)))))))‘𝑛) = ((𝑟 ∈ ℕ0 ↦ (𝐴 Σg (𝑙 ∈ (0...𝑟) ↦ (((coe1‘(𝑄 Σg (𝑘 ∈ ℕ0 ↦ ((𝑥 decompPMat 𝑘)( ·𝑠𝑄)(𝑘(.g‘(mulGrp‘𝑄))(var1𝐴))))))‘𝑙)(.r𝐴)((coe1‘(𝑄 Σg (𝑘 ∈ ℕ0 ↦ ((𝑦 decompPMat 𝑘)( ·𝑠𝑄)(𝑘(.g‘(mulGrp‘𝑄))(var1𝐴))))))‘(𝑟𝑙))))))‘𝑛))
11330, 97, 109, 112syl3anc 1365 . . . . . . . 8 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥𝐵𝑦𝐵)) ∧ 𝑛 ∈ ℕ0) → ((coe1‘((𝑄 Σg (𝑘 ∈ ℕ0 ↦ ((𝑥 decompPMat 𝑘)( ·𝑠𝑄)(𝑘(.g‘(mulGrp‘𝑄))(var1𝐴)))))(.r𝑄)(𝑄 Σg (𝑘 ∈ ℕ0 ↦ ((𝑦 decompPMat 𝑘)( ·𝑠𝑄)(𝑘(.g‘(mulGrp‘𝑄))(var1𝐴)))))))‘𝑛) = ((𝑟 ∈ ℕ0 ↦ (𝐴 Σg (𝑙 ∈ (0...𝑟) ↦ (((coe1‘(𝑄 Σg (𝑘 ∈ ℕ0 ↦ ((𝑥 decompPMat 𝑘)( ·𝑠𝑄)(𝑘(.g‘(mulGrp‘𝑄))(var1𝐴))))))‘𝑙)(.r𝐴)((coe1‘(𝑄 Σg (𝑘 ∈ ℕ0 ↦ ((𝑦 decompPMat 𝑘)( ·𝑠𝑄)(𝑘(.g‘(mulGrp‘𝑄))(var1𝐴))))))‘(𝑟𝑙))))))‘𝑛))
114 oveq2 7156 . . . . . . . . . . . 12 (𝑧 = 𝑙 → (𝑥 decompPMat 𝑧) = (𝑥 decompPMat 𝑙))
115 oveq2 7156 . . . . . . . . . . . . 13 (𝑧 = 𝑙 → (𝑛𝑧) = (𝑛𝑙))
116115oveq2d 7164 . . . . . . . . . . . 12 (𝑧 = 𝑙 → (𝑦 decompPMat (𝑛𝑧)) = (𝑦 decompPMat (𝑛𝑙)))
117114, 116oveq12d 7166 . . . . . . . . . . 11 (𝑧 = 𝑙 → ((𝑥 decompPMat 𝑧)(.r𝐴)(𝑦 decompPMat (𝑛𝑧))) = ((𝑥 decompPMat 𝑙)(.r𝐴)(𝑦 decompPMat (𝑛𝑙))))
118117cbvmptv 5160 . . . . . . . . . 10 (𝑧 ∈ (0...𝑛) ↦ ((𝑥 decompPMat 𝑧)(.r𝐴)(𝑦 decompPMat (𝑛𝑧)))) = (𝑙 ∈ (0...𝑛) ↦ ((𝑥 decompPMat 𝑙)(.r𝐴)(𝑦 decompPMat (𝑛𝑙))))
11929ad3antrrr 728 . . . . . . . . . . . . . 14 (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥𝐵𝑦𝐵)) ∧ 𝑛 ∈ ℕ0) ∧ 𝑙 ∈ (0...𝑛)) → 𝐴 ∈ Ring)
120 simp-5r 784 . . . . . . . . . . . . . . . 16 ((((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥𝐵𝑦𝐵)) ∧ 𝑛 ∈ ℕ0) ∧ 𝑙 ∈ (0...𝑛)) ∧ 𝑘 ∈ ℕ0) → 𝑅 ∈ Ring)
1218ad3antrrr 728 . . . . . . . . . . . . . . . 16 ((((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥𝐵𝑦𝐵)) ∧ 𝑛 ∈ ℕ0) ∧ 𝑙 ∈ (0...𝑛)) ∧ 𝑘 ∈ ℕ0) → 𝑥𝐵)
122 simpr 487 . . . . . . . . . . . . . . . 16 ((((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥𝐵𝑦𝐵)) ∧ 𝑛 ∈ ℕ0) ∧ 𝑙 ∈ (0...𝑛)) ∧ 𝑘 ∈ ℕ0) → 𝑘 ∈ ℕ0)
1233, 4, 11, 18, 31decpmatcl 21367 . . . . . . . . . . . . . . . 16 ((𝑅 ∈ Ring ∧ 𝑥𝐵𝑘 ∈ ℕ0) → (𝑥 decompPMat 𝑘) ∈ (Base‘𝐴))
124120, 121, 122, 123syl3anc 1365 . . . . . . . . . . . . . . 15 ((((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥𝐵𝑦𝐵)) ∧ 𝑛 ∈ ℕ0) ∧ 𝑙 ∈ (0...𝑛)) ∧ 𝑘 ∈ ℕ0) → (𝑥 decompPMat 𝑘) ∈ (Base‘𝐴))
125124ralrimiva 3180 . . . . . . . . . . . . . 14 (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥𝐵𝑦𝐵)) ∧ 𝑛 ∈ ℕ0) ∧ 𝑙 ∈ (0...𝑛)) → ∀𝑘 ∈ ℕ0 (𝑥 decompPMat 𝑘) ∈ (Base‘𝐴))
1262, 8jca 514 . . . . . . . . . . . . . . . 16 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥𝐵𝑦𝐵)) → (𝑅 ∈ Ring ∧ 𝑥𝐵))
127126ad2antrr 724 . . . . . . . . . . . . . . 15 (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥𝐵𝑦𝐵)) ∧ 𝑛 ∈ ℕ0) ∧ 𝑙 ∈ (0...𝑛)) → (𝑅 ∈ Ring ∧ 𝑥𝐵))
1283, 4, 11, 18, 32decpmatfsupp 21369 . . . . . . . . . . . . . . 15 ((𝑅 ∈ Ring ∧ 𝑥𝐵) → (𝑘 ∈ ℕ0 ↦ (𝑥 decompPMat 𝑘)) finSupp (0g𝐴))
129127, 128syl 17 . . . . . . . . . . . . . 14 (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥𝐵𝑦𝐵)) ∧ 𝑛 ∈ ℕ0) ∧ 𝑙 ∈ (0...𝑛)) → (𝑘 ∈ ℕ0 ↦ (𝑥 decompPMat 𝑘)) finSupp (0g𝐴))
130 elfznn0 12992 . . . . . . . . . . . . . . 15 (𝑙 ∈ (0...𝑛) → 𝑙 ∈ ℕ0)
131130adantl 484 . . . . . . . . . . . . . 14 (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥𝐵𝑦𝐵)) ∧ 𝑛 ∈ ℕ0) ∧ 𝑙 ∈ (0...𝑛)) → 𝑙 ∈ ℕ0)
13219, 28, 17, 16, 119, 31, 15, 32, 125, 129, 131gsummoncoe1 20464 . . . . . . . . . . . . 13 (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥𝐵𝑦𝐵)) ∧ 𝑛 ∈ ℕ0) ∧ 𝑙 ∈ (0...𝑛)) → ((coe1‘(𝑄 Σg (𝑘 ∈ ℕ0 ↦ ((𝑥 decompPMat 𝑘)( ·𝑠𝑄)(𝑘(.g‘(mulGrp‘𝑄))(var1𝐴))))))‘𝑙) = 𝑙 / 𝑘(𝑥 decompPMat 𝑘))
133 csbov2g 7194 . . . . . . . . . . . . . . 15 (𝑙 ∈ (0...𝑛) → 𝑙 / 𝑘(𝑥 decompPMat 𝑘) = (𝑥 decompPMat 𝑙 / 𝑘𝑘))
134 csbvarg 4381 . . . . . . . . . . . . . . . 16 (𝑙 ∈ (0...𝑛) → 𝑙 / 𝑘𝑘 = 𝑙)
135134oveq2d 7164 . . . . . . . . . . . . . . 15 (𝑙 ∈ (0...𝑛) → (𝑥 decompPMat 𝑙 / 𝑘𝑘) = (𝑥 decompPMat 𝑙))
136133, 135eqtrd 2854 . . . . . . . . . . . . . 14 (𝑙 ∈ (0...𝑛) → 𝑙 / 𝑘(𝑥 decompPMat 𝑘) = (𝑥 decompPMat 𝑙))
137136adantl 484 . . . . . . . . . . . . 13 (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥𝐵𝑦𝐵)) ∧ 𝑛 ∈ ℕ0) ∧ 𝑙 ∈ (0...𝑛)) → 𝑙 / 𝑘(𝑥 decompPMat 𝑘) = (𝑥 decompPMat 𝑙))
138132, 137eqtr2d 2855 . . . . . . . . . . . 12 (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥𝐵𝑦𝐵)) ∧ 𝑛 ∈ ℕ0) ∧ 𝑙 ∈ (0...𝑛)) → (𝑥 decompPMat 𝑙) = ((coe1‘(𝑄 Σg (𝑘 ∈ ℕ0 ↦ ((𝑥 decompPMat 𝑘)( ·𝑠𝑄)(𝑘(.g‘(mulGrp‘𝑄))(var1𝐴))))))‘𝑙))
13910ad3antrrr 728 . . . . . . . . . . . . . . . 16 ((((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥𝐵𝑦𝐵)) ∧ 𝑛 ∈ ℕ0) ∧ 𝑙 ∈ (0...𝑛)) ∧ 𝑘 ∈ ℕ0) → 𝑦𝐵)
1403, 4, 11, 18, 31decpmatcl 21367 . . . . . . . . . . . . . . . 16 ((𝑅 ∈ Ring ∧ 𝑦𝐵𝑘 ∈ ℕ0) → (𝑦 decompPMat 𝑘) ∈ (Base‘𝐴))
141120, 139, 122, 140syl3anc 1365 . . . . . . . . . . . . . . 15 ((((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥𝐵𝑦𝐵)) ∧ 𝑛 ∈ ℕ0) ∧ 𝑙 ∈ (0...𝑛)) ∧ 𝑘 ∈ ℕ0) → (𝑦 decompPMat 𝑘) ∈ (Base‘𝐴))
142141ralrimiva 3180 . . . . . . . . . . . . . 14 (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥𝐵𝑦𝐵)) ∧ 𝑛 ∈ ℕ0) ∧ 𝑙 ∈ (0...𝑛)) → ∀𝑘 ∈ ℕ0 (𝑦 decompPMat 𝑘) ∈ (Base‘𝐴))
1432, 10jca 514 . . . . . . . . . . . . . . . 16 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥𝐵𝑦𝐵)) → (𝑅 ∈ Ring ∧ 𝑦𝐵))
144143ad2antrr 724 . . . . . . . . . . . . . . 15 (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥𝐵𝑦𝐵)) ∧ 𝑛 ∈ ℕ0) ∧ 𝑙 ∈ (0...𝑛)) → (𝑅 ∈ Ring ∧ 𝑦𝐵))
1453, 4, 11, 18, 32decpmatfsupp 21369 . . . . . . . . . . . . . . 15 ((𝑅 ∈ Ring ∧ 𝑦𝐵) → (𝑘 ∈ ℕ0 ↦ (𝑦 decompPMat 𝑘)) finSupp (0g𝐴))
146144, 145syl 17 . . . . . . . . . . . . . 14 (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥𝐵𝑦𝐵)) ∧ 𝑛 ∈ ℕ0) ∧ 𝑙 ∈ (0...𝑛)) → (𝑘 ∈ ℕ0 ↦ (𝑦 decompPMat 𝑘)) finSupp (0g𝐴))
147 fznn0sub 12931 . . . . . . . . . . . . . . 15 (𝑙 ∈ (0...𝑛) → (𝑛𝑙) ∈ ℕ0)
148147adantl 484 . . . . . . . . . . . . . 14 (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥𝐵𝑦𝐵)) ∧ 𝑛 ∈ ℕ0) ∧ 𝑙 ∈ (0...𝑛)) → (𝑛𝑙) ∈ ℕ0)
14919, 28, 17, 16, 119, 31, 15, 32, 142, 146, 148gsummoncoe1 20464 . . . . . . . . . . . . 13 (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥𝐵𝑦𝐵)) ∧ 𝑛 ∈ ℕ0) ∧ 𝑙 ∈ (0...𝑛)) → ((coe1‘(𝑄 Σg (𝑘 ∈ ℕ0 ↦ ((𝑦 decompPMat 𝑘)( ·𝑠𝑄)(𝑘(.g‘(mulGrp‘𝑄))(var1𝐴))))))‘(𝑛𝑙)) = (𝑛𝑙) / 𝑘(𝑦 decompPMat 𝑘))
150 ovex 7181 . . . . . . . . . . . . . 14 (𝑛𝑙) ∈ V
151 csbov2g 7194 . . . . . . . . . . . . . 14 ((𝑛𝑙) ∈ V → (𝑛𝑙) / 𝑘(𝑦 decompPMat 𝑘) = (𝑦 decompPMat (𝑛𝑙) / 𝑘𝑘))
152150, 151mp1i 13 . . . . . . . . . . . . 13 (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥𝐵𝑦𝐵)) ∧ 𝑛 ∈ ℕ0) ∧ 𝑙 ∈ (0...𝑛)) → (𝑛𝑙) / 𝑘(𝑦 decompPMat 𝑘) = (𝑦 decompPMat (𝑛𝑙) / 𝑘𝑘))
153 csbvarg 4381 . . . . . . . . . . . . . . 15 ((𝑛𝑙) ∈ V → (𝑛𝑙) / 𝑘𝑘 = (𝑛𝑙))
154150, 153mp1i 13 . . . . . . . . . . . . . 14 (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥𝐵𝑦𝐵)) ∧ 𝑛 ∈ ℕ0) ∧ 𝑙 ∈ (0...𝑛)) → (𝑛𝑙) / 𝑘𝑘 = (𝑛𝑙))
155154oveq2d 7164 . . . . . . . . . . . . 13 (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥𝐵𝑦𝐵)) ∧ 𝑛 ∈ ℕ0) ∧ 𝑙 ∈ (0...𝑛)) → (𝑦 decompPMat (𝑛𝑙) / 𝑘𝑘) = (𝑦 decompPMat (𝑛𝑙)))
156149, 152, 1553eqtrrd 2859 . . . . . . . . . . . 12 (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥𝐵𝑦𝐵)) ∧ 𝑛 ∈ ℕ0) ∧ 𝑙 ∈ (0...𝑛)) → (𝑦 decompPMat (𝑛𝑙)) = ((coe1‘(𝑄 Σg (𝑘 ∈ ℕ0 ↦ ((𝑦 decompPMat 𝑘)( ·𝑠𝑄)(𝑘(.g‘(mulGrp‘𝑄))(var1𝐴))))))‘(𝑛𝑙)))
157138, 156oveq12d 7166 . . . . . . . . . . 11 (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥𝐵𝑦𝐵)) ∧ 𝑛 ∈ ℕ0) ∧ 𝑙 ∈ (0...𝑛)) → ((𝑥 decompPMat 𝑙)(.r𝐴)(𝑦 decompPMat (𝑛𝑙))) = (((coe1‘(𝑄 Σg (𝑘 ∈ ℕ0 ↦ ((𝑥 decompPMat 𝑘)( ·𝑠𝑄)(𝑘(.g‘(mulGrp‘𝑄))(var1𝐴))))))‘𝑙)(.r𝐴)((coe1‘(𝑄 Σg (𝑘 ∈ ℕ0 ↦ ((𝑦 decompPMat 𝑘)( ·𝑠𝑄)(𝑘(.g‘(mulGrp‘𝑄))(var1𝐴))))))‘(𝑛𝑙))))
158157mpteq2dva 5152 . . . . . . . . . 10 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥𝐵𝑦𝐵)) ∧ 𝑛 ∈ ℕ0) → (𝑙 ∈ (0...𝑛) ↦ ((𝑥 decompPMat 𝑙)(.r𝐴)(𝑦 decompPMat (𝑛𝑙)))) = (𝑙 ∈ (0...𝑛) ↦ (((coe1‘(𝑄 Σg (𝑘 ∈ ℕ0 ↦ ((𝑥 decompPMat 𝑘)( ·𝑠𝑄)(𝑘(.g‘(mulGrp‘𝑄))(var1𝐴))))))‘𝑙)(.r𝐴)((coe1‘(𝑄 Σg (𝑘 ∈ ℕ0 ↦ ((𝑦 decompPMat 𝑘)( ·𝑠𝑄)(𝑘(.g‘(mulGrp‘𝑄))(var1𝐴))))))‘(𝑛𝑙)))))
159118, 158syl5eq 2866 . . . . . . . . 9 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥𝐵𝑦𝐵)) ∧ 𝑛 ∈ ℕ0) → (𝑧 ∈ (0...𝑛) ↦ ((𝑥 decompPMat 𝑧)(.r𝐴)(𝑦 decompPMat (𝑛𝑧)))) = (𝑙 ∈ (0...𝑛) ↦ (((coe1‘(𝑄 Σg (𝑘 ∈ ℕ0 ↦ ((𝑥 decompPMat 𝑘)( ·𝑠𝑄)(𝑘(.g‘(mulGrp‘𝑄))(var1𝐴))))))‘𝑙)(.r𝐴)((coe1‘(𝑄 Σg (𝑘 ∈ ℕ0 ↦ ((𝑦 decompPMat 𝑘)( ·𝑠𝑄)(𝑘(.g‘(mulGrp‘𝑄))(var1𝐴))))))‘(𝑛𝑙)))))
160159oveq2d 7164 . . . . . . . 8 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥𝐵𝑦𝐵)) ∧ 𝑛 ∈ ℕ0) → (𝐴 Σg (𝑧 ∈ (0...𝑛) ↦ ((𝑥 decompPMat 𝑧)(.r𝐴)(𝑦 decompPMat (𝑛𝑧))))) = (𝐴 Σg (𝑙 ∈ (0...𝑛) ↦ (((coe1‘(𝑄 Σg (𝑘 ∈ ℕ0 ↦ ((𝑥 decompPMat 𝑘)( ·𝑠𝑄)(𝑘(.g‘(mulGrp‘𝑄))(var1𝐴))))))‘𝑙)(.r𝐴)((coe1‘(𝑄 Σg (𝑘 ∈ ℕ0 ↦ ((𝑦 decompPMat 𝑘)( ·𝑠𝑄)(𝑘(.g‘(mulGrp‘𝑄))(var1𝐴))))))‘(𝑛𝑙))))))
16179, 113, 1603eqtr4rd 2865 . . . . . . 7 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥𝐵𝑦𝐵)) ∧ 𝑛 ∈ ℕ0) → (𝐴 Σg (𝑧 ∈ (0...𝑛) ↦ ((𝑥 decompPMat 𝑧)(.r𝐴)(𝑦 decompPMat (𝑛𝑧))))) = ((coe1‘((𝑄 Σg (𝑘 ∈ ℕ0 ↦ ((𝑥 decompPMat 𝑘)( ·𝑠𝑄)(𝑘(.g‘(mulGrp‘𝑄))(var1𝐴)))))(.r𝑄)(𝑄 Σg (𝑘 ∈ ℕ0 ↦ ((𝑦 decompPMat 𝑘)( ·𝑠𝑄)(𝑘(.g‘(mulGrp‘𝑄))(var1𝐴)))))))‘𝑛))
16258, 70, 1613eqtrd 2858 . . . . . 6 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥𝐵𝑦𝐵)) ∧ 𝑛 ∈ ℕ0) → ((coe1‘(𝑄 Σg (𝑘 ∈ ℕ0 ↦ ((𝐴 Σg (𝑧 ∈ (0...𝑘) ↦ ((𝑥 decompPMat 𝑧)(.r𝐴)(𝑦 decompPMat (𝑘𝑧)))))( ·𝑠𝑄)(𝑘(.g‘(mulGrp‘𝑄))(var1𝐴))))))‘𝑛) = ((coe1‘((𝑄 Σg (𝑘 ∈ ℕ0 ↦ ((𝑥 decompPMat 𝑘)( ·𝑠𝑄)(𝑘(.g‘(mulGrp‘𝑄))(var1𝐴)))))(.r𝑄)(𝑄 Σg (𝑘 ∈ ℕ0 ↦ ((𝑦 decompPMat 𝑘)( ·𝑠𝑄)(𝑘(.g‘(mulGrp‘𝑄))(var1𝐴)))))))‘𝑛))
163162ralrimiva 3180 . . . . 5 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥𝐵𝑦𝐵)) → ∀𝑛 ∈ ℕ0 ((coe1‘(𝑄 Σg (𝑘 ∈ ℕ0 ↦ ((𝐴 Σg (𝑧 ∈ (0...𝑘) ↦ ((𝑥 decompPMat 𝑧)(.r𝐴)(𝑦 decompPMat (𝑘𝑧)))))( ·𝑠𝑄)(𝑘(.g‘(mulGrp‘𝑄))(var1𝐴))))))‘𝑛) = ((coe1‘((𝑄 Σg (𝑘 ∈ ℕ0 ↦ ((𝑥 decompPMat 𝑘)( ·𝑠𝑄)(𝑘(.g‘(mulGrp‘𝑄))(var1𝐴)))))(.r𝑄)(𝑄 Σg (𝑘 ∈ ℕ0 ↦ ((𝑦 decompPMat 𝑘)( ·𝑠𝑄)(𝑘(.g‘(mulGrp‘𝑄))(var1𝐴)))))))‘𝑛))
16429adantr 483 . . . . . 6 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥𝐵𝑦𝐵)) → 𝐴 ∈ Ring)
16584adantr 483 . . . . . . 7 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥𝐵𝑦𝐵)) → 𝑄 ∈ CMnd)
16686a1i 11 . . . . . . 7 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥𝐵𝑦𝐵)) → ℕ0 ∈ V)
16719ply1lmod 20412 . . . . . . . . . . 11 (𝐴 ∈ Ring → 𝑄 ∈ LMod)
16829, 167syl 17 . . . . . . . . . 10 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝑄 ∈ LMod)
169168ad2antrr 724 . . . . . . . . 9 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥𝐵𝑦𝐵)) ∧ 𝑘 ∈ ℕ0) → 𝑄 ∈ LMod)
17034ad2antrr 724 . . . . . . . . . . 11 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥𝐵𝑦𝐵)) ∧ 𝑘 ∈ ℕ0) → 𝐴 ∈ CMnd)
171 fzfid 13333 . . . . . . . . . . 11 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥𝐵𝑦𝐵)) ∧ 𝑘 ∈ ℕ0) → (0...𝑘) ∈ Fin)
17229ad3antrrr 728 . . . . . . . . . . . . 13 (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥𝐵𝑦𝐵)) ∧ 𝑘 ∈ ℕ0) ∧ 𝑧 ∈ (0...𝑘)) → 𝐴 ∈ Ring)
173 simp-4r 782 . . . . . . . . . . . . . 14 (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥𝐵𝑦𝐵)) ∧ 𝑘 ∈ ℕ0) ∧ 𝑧 ∈ (0...𝑘)) → 𝑅 ∈ Ring)
174 simplrl 775 . . . . . . . . . . . . . . 15 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥𝐵𝑦𝐵)) ∧ 𝑘 ∈ ℕ0) → 𝑥𝐵)
175174adantr 483 . . . . . . . . . . . . . 14 (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥𝐵𝑦𝐵)) ∧ 𝑘 ∈ ℕ0) ∧ 𝑧 ∈ (0...𝑘)) → 𝑥𝐵)
17640adantl 484 . . . . . . . . . . . . . 14 (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥𝐵𝑦𝐵)) ∧ 𝑘 ∈ ℕ0) ∧ 𝑧 ∈ (0...𝑘)) → 𝑧 ∈ ℕ0)
177173, 175, 176, 42syl3anc 1365 . . . . . . . . . . . . 13 (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥𝐵𝑦𝐵)) ∧ 𝑘 ∈ ℕ0) ∧ 𝑧 ∈ (0...𝑘)) → (𝑥 decompPMat 𝑧) ∈ (Base‘𝐴))
178 simplrr 776 . . . . . . . . . . . . . . 15 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥𝐵𝑦𝐵)) ∧ 𝑘 ∈ ℕ0) → 𝑦𝐵)
179178adantr 483 . . . . . . . . . . . . . 14 (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥𝐵𝑦𝐵)) ∧ 𝑘 ∈ ℕ0) ∧ 𝑧 ∈ (0...𝑘)) → 𝑦𝐵)
18045adantl 484 . . . . . . . . . . . . . 14 (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥𝐵𝑦𝐵)) ∧ 𝑘 ∈ ℕ0) ∧ 𝑧 ∈ (0...𝑘)) → (𝑘𝑧) ∈ ℕ0)
181173, 179, 180, 47syl3anc 1365 . . . . . . . . . . . . 13 (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥𝐵𝑦𝐵)) ∧ 𝑘 ∈ ℕ0) ∧ 𝑧 ∈ (0...𝑘)) → (𝑦 decompPMat (𝑘𝑧)) ∈ (Base‘𝐴))
182172, 177, 181, 50syl3anc 1365 . . . . . . . . . . . 12 (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥𝐵𝑦𝐵)) ∧ 𝑘 ∈ ℕ0) ∧ 𝑧 ∈ (0...𝑘)) → ((𝑥 decompPMat 𝑧)(.r𝐴)(𝑦 decompPMat (𝑘𝑧))) ∈ (Base‘𝐴))
183182ralrimiva 3180 . . . . . . . . . . 11 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥𝐵𝑦𝐵)) ∧ 𝑘 ∈ ℕ0) → ∀𝑧 ∈ (0...𝑘)((𝑥 decompPMat 𝑧)(.r𝐴)(𝑦 decompPMat (𝑘𝑧))) ∈ (Base‘𝐴))
18431, 170, 171, 183gsummptcl 19079 . . . . . . . . . 10 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥𝐵𝑦𝐵)) ∧ 𝑘 ∈ ℕ0) → (𝐴 Σg (𝑧 ∈ (0...𝑘) ↦ ((𝑥 decompPMat 𝑧)(.r𝐴)(𝑦 decompPMat (𝑘𝑧))))) ∈ (Base‘𝐴))
18529ad2antrr 724 . . . . . . . . . . . . 13 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥𝐵𝑦𝐵)) ∧ 𝑘 ∈ ℕ0) → 𝐴 ∈ Ring)
18619ply1sca 20413 . . . . . . . . . . . . 13 (𝐴 ∈ Ring → 𝐴 = (Scalar‘𝑄))
187185, 186syl 17 . . . . . . . . . . . 12 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥𝐵𝑦𝐵)) ∧ 𝑘 ∈ ℕ0) → 𝐴 = (Scalar‘𝑄))
188187eqcomd 2825 . . . . . . . . . . 11 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥𝐵𝑦𝐵)) ∧ 𝑘 ∈ ℕ0) → (Scalar‘𝑄) = 𝐴)
189188fveq2d 6667 . . . . . . . . . 10 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥𝐵𝑦𝐵)) ∧ 𝑘 ∈ ℕ0) → (Base‘(Scalar‘𝑄)) = (Base‘𝐴))
190184, 189eleqtrrd 2914 . . . . . . . . 9 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥𝐵𝑦𝐵)) ∧ 𝑘 ∈ ℕ0) → (𝐴 Σg (𝑧 ∈ (0...𝑘) ↦ ((𝑥 decompPMat 𝑧)(.r𝐴)(𝑦 decompPMat (𝑘𝑧))))) ∈ (Base‘(Scalar‘𝑄)))
191 eqid 2819 . . . . . . . . . . 11 (mulGrp‘𝑄) = (mulGrp‘𝑄)
19219, 17, 191, 16, 28ply1moncl 20431 . . . . . . . . . 10 ((𝐴 ∈ Ring ∧ 𝑘 ∈ ℕ0) → (𝑘(.g‘(mulGrp‘𝑄))(var1𝐴)) ∈ (Base‘𝑄))
193185, 192sylancom 590 . . . . . . . . 9 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥𝐵𝑦𝐵)) ∧ 𝑘 ∈ ℕ0) → (𝑘(.g‘(mulGrp‘𝑄))(var1𝐴)) ∈ (Base‘𝑄))
194 eqid 2819 . . . . . . . . . 10 (Scalar‘𝑄) = (Scalar‘𝑄)
195 eqid 2819 . . . . . . . . . 10 (Base‘(Scalar‘𝑄)) = (Base‘(Scalar‘𝑄))
19628, 194, 15, 195lmodvscl 19643 . . . . . . . . 9 ((𝑄 ∈ LMod ∧ (𝐴 Σg (𝑧 ∈ (0...𝑘) ↦ ((𝑥 decompPMat 𝑧)(.r𝐴)(𝑦 decompPMat (𝑘𝑧))))) ∈ (Base‘(Scalar‘𝑄)) ∧ (𝑘(.g‘(mulGrp‘𝑄))(var1𝐴)) ∈ (Base‘𝑄)) → ((𝐴 Σg (𝑧 ∈ (0...𝑘) ↦ ((𝑥 decompPMat 𝑧)(.r𝐴)(𝑦 decompPMat (𝑘𝑧)))))( ·𝑠𝑄)(𝑘(.g‘(mulGrp‘𝑄))(var1𝐴))) ∈ (Base‘𝑄))
197169, 190, 193, 196syl3anc 1365 . . . . . . . 8 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥𝐵𝑦𝐵)) ∧ 𝑘 ∈ ℕ0) → ((𝐴 Σg (𝑧 ∈ (0...𝑘) ↦ ((𝑥 decompPMat 𝑧)(.r𝐴)(𝑦 decompPMat (𝑘𝑧)))))( ·𝑠𝑄)(𝑘(.g‘(mulGrp‘𝑄))(var1𝐴))) ∈ (Base‘𝑄))
198197fmpttd 6872 . . . . . . 7 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥𝐵𝑦𝐵)) → (𝑘 ∈ ℕ0 ↦ ((𝐴 Σg (𝑧 ∈ (0...𝑘) ↦ ((𝑥 decompPMat 𝑧)(.r𝐴)(𝑦 decompPMat (𝑘𝑧)))))( ·𝑠𝑄)(𝑘(.g‘(mulGrp‘𝑄))(var1𝐴)))):ℕ0⟶(Base‘𝑄))
1993, 4, 11, 15, 16, 17, 18, 19, 28, 20pm2mpmhmlem1 21418 . . . . . . 7 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥𝐵𝑦𝐵)) → (𝑘 ∈ ℕ0 ↦ ((𝐴 Σg (𝑧 ∈ (0...𝑘) ↦ ((𝑥 decompPMat 𝑧)(.r𝐴)(𝑦 decompPMat (𝑘𝑧)))))( ·𝑠𝑄)(𝑘(.g‘(mulGrp‘𝑄))(var1𝐴)))) finSupp (0g𝑄))
20028, 80, 165, 166, 198, 199gsumcl 19027 . . . . . 6 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥𝐵𝑦𝐵)) → (𝑄 Σg (𝑘 ∈ ℕ0 ↦ ((𝐴 Σg (𝑧 ∈ (0...𝑘) ↦ ((𝑥 decompPMat 𝑧)(.r𝐴)(𝑦 decompPMat (𝑘𝑧)))))( ·𝑠𝑄)(𝑘(.g‘(mulGrp‘𝑄))(var1𝐴))))) ∈ (Base‘𝑄))
20182adantr 483 . . . . . . 7 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥𝐵𝑦𝐵)) → 𝑄 ∈ Ring)
20290, 92sylan 582 . . . . . . . . 9 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥𝐵𝑦𝐵)) ∧ 𝑘 ∈ ℕ0) → ((𝑥 decompPMat 𝑘)( ·𝑠𝑄)(𝑘(.g‘(mulGrp‘𝑄))(var1𝐴))) ∈ (Base‘𝑄))
203202fmpttd 6872 . . . . . . . 8 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥𝐵𝑦𝐵)) → (𝑘 ∈ ℕ0 ↦ ((𝑥 decompPMat 𝑘)( ·𝑠𝑄)(𝑘(.g‘(mulGrp‘𝑄))(var1𝐴)))):ℕ0⟶(Base‘𝑄))
20490, 95syl 17 . . . . . . . 8 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥𝐵𝑦𝐵)) → (𝑘 ∈ ℕ0 ↦ ((𝑥 decompPMat 𝑘)( ·𝑠𝑄)(𝑘(.g‘(mulGrp‘𝑄))(var1𝐴)))) finSupp (0g𝑄))
20528, 80, 165, 166, 203, 204gsumcl 19027 . . . . . . 7 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥𝐵𝑦𝐵)) → (𝑄 Σg (𝑘 ∈ ℕ0 ↦ ((𝑥 decompPMat 𝑘)( ·𝑠𝑄)(𝑘(.g‘(mulGrp‘𝑄))(var1𝐴))))) ∈ (Base‘𝑄))
206100, 102sylan 582 . . . . . . . . 9 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥𝐵𝑦𝐵)) ∧ 𝑘 ∈ ℕ0) → ((𝑦 decompPMat 𝑘)( ·𝑠𝑄)(𝑘(.g‘(mulGrp‘𝑄))(var1𝐴))) ∈ (Base‘𝑄))
207206fmpttd 6872 . . . . . . . 8 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥𝐵𝑦𝐵)) → (𝑘 ∈ ℕ0 ↦ ((𝑦 decompPMat 𝑘)( ·𝑠𝑄)(𝑘(.g‘(mulGrp‘𝑄))(var1𝐴)))):ℕ0⟶(Base‘𝑄))
2081, 2, 10, 107syl3anc 1365 . . . . . . . 8 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥𝐵𝑦𝐵)) → (𝑘 ∈ ℕ0 ↦ ((𝑦 decompPMat 𝑘)( ·𝑠𝑄)(𝑘(.g‘(mulGrp‘𝑄))(var1𝐴)))) finSupp (0g𝑄))
20928, 80, 165, 166, 207, 208gsumcl 19027 . . . . . . 7 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥𝐵𝑦𝐵)) → (𝑄 Σg (𝑘 ∈ ℕ0 ↦ ((𝑦 decompPMat 𝑘)( ·𝑠𝑄)(𝑘(.g‘(mulGrp‘𝑄))(var1𝐴))))) ∈ (Base‘𝑄))
21028, 110ringcl 19303 . . . . . . 7 ((𝑄 ∈ Ring ∧ (𝑄 Σg (𝑘 ∈ ℕ0 ↦ ((𝑥 decompPMat 𝑘)( ·𝑠𝑄)(𝑘(.g‘(mulGrp‘𝑄))(var1𝐴))))) ∈ (Base‘𝑄) ∧ (𝑄 Σg (𝑘 ∈ ℕ0 ↦ ((𝑦 decompPMat 𝑘)( ·𝑠𝑄)(𝑘(.g‘(mulGrp‘𝑄))(var1𝐴))))) ∈ (Base‘𝑄)) → ((𝑄 Σg (𝑘 ∈ ℕ0 ↦ ((𝑥 decompPMat 𝑘)( ·𝑠𝑄)(𝑘(.g‘(mulGrp‘𝑄))(var1𝐴)))))(.r𝑄)(𝑄 Σg (𝑘 ∈ ℕ0 ↦ ((𝑦 decompPMat 𝑘)( ·𝑠𝑄)(𝑘(.g‘(mulGrp‘𝑄))(var1𝐴)))))) ∈ (Base‘𝑄))
211201, 205, 209, 210syl3anc 1365 . . . . . 6 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥𝐵𝑦𝐵)) → ((𝑄 Σg (𝑘 ∈ ℕ0 ↦ ((𝑥 decompPMat 𝑘)( ·𝑠𝑄)(𝑘(.g‘(mulGrp‘𝑄))(var1𝐴)))))(.r𝑄)(𝑄 Σg (𝑘 ∈ ℕ0 ↦ ((𝑦 decompPMat 𝑘)( ·𝑠𝑄)(𝑘(.g‘(mulGrp‘𝑄))(var1𝐴)))))) ∈ (Base‘𝑄))
212 eqid 2819 . . . . . . 7 (coe1‘(𝑄 Σg (𝑘 ∈ ℕ0 ↦ ((𝐴 Σg (𝑧 ∈ (0...𝑘) ↦ ((𝑥 decompPMat 𝑧)(.r𝐴)(𝑦 decompPMat (𝑘𝑧)))))( ·𝑠𝑄)(𝑘(.g‘(mulGrp‘𝑄))(var1𝐴)))))) = (coe1‘(𝑄 Σg (𝑘 ∈ ℕ0 ↦ ((𝐴 Σg (𝑧 ∈ (0...𝑘) ↦ ((𝑥 decompPMat 𝑧)(.r𝐴)(𝑦 decompPMat (𝑘𝑧)))))( ·𝑠𝑄)(𝑘(.g‘(mulGrp‘𝑄))(var1𝐴))))))
213 eqid 2819 . . . . . . 7 (coe1‘((𝑄 Σg (𝑘 ∈ ℕ0 ↦ ((𝑥 decompPMat 𝑘)( ·𝑠𝑄)(𝑘(.g‘(mulGrp‘𝑄))(var1𝐴)))))(.r𝑄)(𝑄 Σg (𝑘 ∈ ℕ0 ↦ ((𝑦 decompPMat 𝑘)( ·𝑠𝑄)(𝑘(.g‘(mulGrp‘𝑄))(var1𝐴))))))) = (coe1‘((𝑄 Σg (𝑘 ∈ ℕ0 ↦ ((𝑥 decompPMat 𝑘)( ·𝑠𝑄)(𝑘(.g‘(mulGrp‘𝑄))(var1𝐴)))))(.r𝑄)(𝑄 Σg (𝑘 ∈ ℕ0 ↦ ((𝑦 decompPMat 𝑘)( ·𝑠𝑄)(𝑘(.g‘(mulGrp‘𝑄))(var1𝐴)))))))
21419, 28, 212, 213ply1coe1eq 20458 . . . . . 6 ((𝐴 ∈ Ring ∧ (𝑄 Σg (𝑘 ∈ ℕ0 ↦ ((𝐴 Σg (𝑧 ∈ (0...𝑘) ↦ ((𝑥 decompPMat 𝑧)(.r𝐴)(𝑦 decompPMat (𝑘𝑧)))))( ·𝑠𝑄)(𝑘(.g‘(mulGrp‘𝑄))(var1𝐴))))) ∈ (Base‘𝑄) ∧ ((𝑄 Σg (𝑘 ∈ ℕ0 ↦ ((𝑥 decompPMat 𝑘)( ·𝑠𝑄)(𝑘(.g‘(mulGrp‘𝑄))(var1𝐴)))))(.r𝑄)(𝑄 Σg (𝑘 ∈ ℕ0 ↦ ((𝑦 decompPMat 𝑘)( ·𝑠𝑄)(𝑘(.g‘(mulGrp‘𝑄))(var1𝐴)))))) ∈ (Base‘𝑄)) → (∀𝑛 ∈ ℕ0 ((coe1‘(𝑄 Σg (𝑘 ∈ ℕ0 ↦ ((𝐴 Σg (𝑧 ∈ (0...𝑘) ↦ ((𝑥 decompPMat 𝑧)(.r𝐴)(𝑦 decompPMat (𝑘𝑧)))))( ·𝑠𝑄)(𝑘(.g‘(mulGrp‘𝑄))(var1𝐴))))))‘𝑛) = ((coe1‘((𝑄 Σg (𝑘 ∈ ℕ0 ↦ ((𝑥 decompPMat 𝑘)( ·𝑠𝑄)(𝑘(.g‘(mulGrp‘𝑄))(var1𝐴)))))(.r𝑄)(𝑄 Σg (𝑘 ∈ ℕ0 ↦ ((𝑦 decompPMat 𝑘)( ·𝑠𝑄)(𝑘(.g‘(mulGrp‘𝑄))(var1𝐴)))))))‘𝑛) ↔ (𝑄 Σg (𝑘 ∈ ℕ0 ↦ ((𝐴 Σg (𝑧 ∈ (0...𝑘) ↦ ((𝑥 decompPMat 𝑧)(.r𝐴)(𝑦 decompPMat (𝑘𝑧)))))( ·𝑠𝑄)(𝑘(.g‘(mulGrp‘𝑄))(var1𝐴))))) = ((𝑄 Σg (𝑘 ∈ ℕ0 ↦ ((𝑥 decompPMat 𝑘)( ·𝑠𝑄)(𝑘(.g‘(mulGrp‘𝑄))(var1𝐴)))))(.r𝑄)(𝑄 Σg (𝑘 ∈ ℕ0 ↦ ((𝑦 decompPMat 𝑘)( ·𝑠𝑄)(𝑘(.g‘(mulGrp‘𝑄))(var1𝐴))))))))
215164, 200, 211, 214syl3anc 1365 . . . . 5 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥𝐵𝑦𝐵)) → (∀𝑛 ∈ ℕ0 ((coe1‘(𝑄 Σg (𝑘 ∈ ℕ0 ↦ ((𝐴 Σg (𝑧 ∈ (0...𝑘) ↦ ((𝑥 decompPMat 𝑧)(.r𝐴)(𝑦 decompPMat (𝑘𝑧)))))( ·𝑠𝑄)(𝑘(.g‘(mulGrp‘𝑄))(var1𝐴))))))‘𝑛) = ((coe1‘((𝑄 Σg (𝑘 ∈ ℕ0 ↦ ((𝑥 decompPMat 𝑘)( ·𝑠𝑄)(𝑘(.g‘(mulGrp‘𝑄))(var1𝐴)))))(.r𝑄)(𝑄 Σg (𝑘 ∈ ℕ0 ↦ ((𝑦 decompPMat 𝑘)( ·𝑠𝑄)(𝑘(.g‘(mulGrp‘𝑄))(var1𝐴)))))))‘𝑛) ↔ (𝑄 Σg (𝑘 ∈ ℕ0 ↦ ((𝐴 Σg (𝑧 ∈ (0...𝑘) ↦ ((𝑥 decompPMat 𝑧)(.r𝐴)(𝑦 decompPMat (𝑘𝑧)))))( ·𝑠𝑄)(𝑘(.g‘(mulGrp‘𝑄))(var1𝐴))))) = ((𝑄 Σg (𝑘 ∈ ℕ0 ↦ ((𝑥 decompPMat 𝑘)( ·𝑠𝑄)(𝑘(.g‘(mulGrp‘𝑄))(var1𝐴)))))(.r𝑄)(𝑄 Σg (𝑘 ∈ ℕ0 ↦ ((𝑦 decompPMat 𝑘)( ·𝑠𝑄)(𝑘(.g‘(mulGrp‘𝑄))(var1𝐴))))))))
216163, 215mpbid 234 . . . 4 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥𝐵𝑦𝐵)) → (𝑄 Σg (𝑘 ∈ ℕ0 ↦ ((𝐴 Σg (𝑧 ∈ (0...𝑘) ↦ ((𝑥 decompPMat 𝑧)(.r𝐴)(𝑦 decompPMat (𝑘𝑧)))))( ·𝑠𝑄)(𝑘(.g‘(mulGrp‘𝑄))(var1𝐴))))) = ((𝑄 Σg (𝑘 ∈ ℕ0 ↦ ((𝑥 decompPMat 𝑘)( ·𝑠𝑄)(𝑘(.g‘(mulGrp‘𝑄))(var1𝐴)))))(.r𝑄)(𝑄 Σg (𝑘 ∈ ℕ0 ↦ ((𝑦 decompPMat 𝑘)( ·𝑠𝑄)(𝑘(.g‘(mulGrp‘𝑄))(var1𝐴)))))))
21722, 27, 2163eqtrd 2858 . . 3 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥𝐵𝑦𝐵)) → (𝑇‘(𝑥(.r𝐶)𝑦)) = ((𝑄 Σg (𝑘 ∈ ℕ0 ↦ ((𝑥 decompPMat 𝑘)( ·𝑠𝑄)(𝑘(.g‘(mulGrp‘𝑄))(var1𝐴)))))(.r𝑄)(𝑄 Σg (𝑘 ∈ ℕ0 ↦ ((𝑦 decompPMat 𝑘)( ·𝑠𝑄)(𝑘(.g‘(mulGrp‘𝑄))(var1𝐴)))))))
2183, 4, 11, 15, 16, 17, 18, 19, 20pm2mpfval 21396 . . . . 5 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑥𝐵) → (𝑇𝑥) = (𝑄 Σg (𝑘 ∈ ℕ0 ↦ ((𝑥 decompPMat 𝑘)( ·𝑠𝑄)(𝑘(.g‘(mulGrp‘𝑄))(var1𝐴))))))
2191, 2, 8, 218syl3anc 1365 . . . 4 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥𝐵𝑦𝐵)) → (𝑇𝑥) = (𝑄 Σg (𝑘 ∈ ℕ0 ↦ ((𝑥 decompPMat 𝑘)( ·𝑠𝑄)(𝑘(.g‘(mulGrp‘𝑄))(var1𝐴))))))
2203, 4, 11, 15, 16, 17, 18, 19, 20pm2mpfval 21396 . . . . 5 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑦𝐵) → (𝑇𝑦) = (𝑄 Σg (𝑘 ∈ ℕ0 ↦ ((𝑦 decompPMat 𝑘)( ·𝑠𝑄)(𝑘(.g‘(mulGrp‘𝑄))(var1𝐴))))))
2211, 2, 10, 220syl3anc 1365 . . . 4 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥𝐵𝑦𝐵)) → (𝑇𝑦) = (𝑄 Σg (𝑘 ∈ ℕ0 ↦ ((𝑦 decompPMat 𝑘)( ·𝑠𝑄)(𝑘(.g‘(mulGrp‘𝑄))(var1𝐴))))))
222219, 221oveq12d 7166 . . 3 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥𝐵𝑦𝐵)) → ((𝑇𝑥)(.r𝑄)(𝑇𝑦)) = ((𝑄 Σg (𝑘 ∈ ℕ0 ↦ ((𝑥 decompPMat 𝑘)( ·𝑠𝑄)(𝑘(.g‘(mulGrp‘𝑄))(var1𝐴)))))(.r𝑄)(𝑄 Σg (𝑘 ∈ ℕ0 ↦ ((𝑦 decompPMat 𝑘)( ·𝑠𝑄)(𝑘(.g‘(mulGrp‘𝑄))(var1𝐴)))))))
223217, 222eqtr4d 2857 . 2 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥𝐵𝑦𝐵)) → (𝑇‘(𝑥(.r𝐶)𝑦)) = ((𝑇𝑥)(.r𝑄)(𝑇𝑦)))
224223ralrimivva 3189 1 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → ∀𝑥𝐵𝑦𝐵 (𝑇‘(𝑥(.r𝐶)𝑦)) = ((𝑇𝑥)(.r𝑄)(𝑇𝑦)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 208  wa 398  w3a 1081   = wceq 1530  wcel 2107  wral 3136  Vcvv 3493  csb 3881   class class class wbr 5057  cmpt 5137  cfv 6348  (class class class)co 7148  Fincfn 8501   finSupp cfsupp 8825  0cc0 10529  cmin 10862  0cn0 11889  ...cfz 12884  Basecbs 16475  .rcmulr 16558  Scalarcsca 16560   ·𝑠 cvsca 16561  0gc0g 16705   Σg cgsu 16706  .gcmg 18216  CMndccmn 18898  mulGrpcmgp 19231  Ringcrg 19289  LModclmod 19626  var1cv1 20336  Poly1cpl1 20337  coe1cco1 20338   Mat cmat 21008   decompPMat cdecpmat 21362   pMatToMatPoly cpm2mp 21392
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2791  ax-rep 5181  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7453  ax-cnex 10585  ax-resscn 10586  ax-1cn 10587  ax-icn 10588  ax-addcl 10589  ax-addrcl 10590  ax-mulcl 10591  ax-mulrcl 10592  ax-mulcom 10593  ax-addass 10594  ax-mulass 10595  ax-distr 10596  ax-i2m1 10597  ax-1ne0 10598  ax-1rid 10599  ax-rnegex 10600  ax-rrecex 10601  ax-cnre 10602  ax-pre-lttri 10603  ax-pre-lttrn 10604  ax-pre-ltadd 10605  ax-pre-mulgt0 10606
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1082  df-3an 1083  df-tru 1533  df-fal 1543  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2616  df-eu 2648  df-clab 2798  df-cleq 2812  df-clel 2891  df-nfc 2961  df-ne 3015  df-nel 3122  df-ral 3141  df-rex 3142  df-reu 3143  df-rmo 3144  df-rab 3145  df-v 3495  df-sbc 3771  df-csb 3882  df-dif 3937  df-un 3939  df-in 3941  df-ss 3950  df-pss 3952  df-nul 4290  df-if 4466  df-pw 4539  df-sn 4560  df-pr 4562  df-tp 4564  df-op 4566  df-ot 4568  df-uni 4831  df-int 4868  df-iun 4912  df-iin 4913  df-br 5058  df-opab 5120  df-mpt 5138  df-tr 5164  df-id 5453  df-eprel 5458  df-po 5467  df-so 5468  df-fr 5507  df-se 5508  df-we 5509  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-pred 6141  df-ord 6187  df-on 6188  df-lim 6189  df-suc 6190  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-f1 6353  df-fo 6354  df-f1o 6355  df-fv 6356  df-isom 6357  df-riota 7106  df-ov 7151  df-oprab 7152  df-mpo 7153  df-of 7401  df-ofr 7402  df-om 7573  df-1st 7681  df-2nd 7682  df-supp 7823  df-wrecs 7939  df-recs 8000  df-rdg 8038  df-1o 8094  df-2o 8095  df-oadd 8098  df-er 8281  df-map 8400  df-pm 8401  df-ixp 8454  df-en 8502  df-dom 8503  df-sdom 8504  df-fin 8505  df-fsupp 8826  df-sup 8898  df-oi 8966  df-card 9360  df-pnf 10669  df-mnf 10670  df-xr 10671  df-ltxr 10672  df-le 10673  df-sub 10864  df-neg 10865  df-nn 11631  df-2 11692  df-3 11693  df-4 11694  df-5 11695  df-6 11696  df-7 11697  df-8 11698  df-9 11699  df-n0 11890  df-z 11974  df-dec 12091  df-uz 12236  df-fz 12885  df-fzo 13026  df-seq 13362  df-hash 13683  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-hom 16581  df-cco 16582  df-0g 16707  df-gsum 16708  df-prds 16713  df-pws 16715  df-mre 16849  df-mrc 16850  df-acs 16852  df-mgm 17844  df-sgrp 17893  df-mnd 17904  df-mhm 17948  df-submnd 17949  df-grp 18098  df-minusg 18099  df-sbg 18100  df-mulg 18217  df-subg 18268  df-ghm 18348  df-cntz 18439  df-cmn 18900  df-abl 18901  df-mgp 19232  df-ur 19244  df-srg 19248  df-ring 19291  df-subrg 19525  df-lmod 19628  df-lss 19696  df-sra 19936  df-rgmod 19937  df-psr 20128  df-mvr 20129  df-mpl 20130  df-opsr 20132  df-psr1 20340  df-vr1 20341  df-ply1 20342  df-coe1 20343  df-dsmm 20868  df-frlm 20883  df-mamu 20987  df-mat 21009  df-decpmat 21363  df-pm2mp 21393
This theorem is referenced by:  pm2mpmhm  21420
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