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Theorem pm2mpmhmlem2 22640
Description: Lemma 2 for pm2mpmhm 22641. (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 764 . . . . 5 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥𝐵𝑦𝐵)) → 𝑁 ∈ Fin)
2 simplr 766 . . . . 5 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥𝐵𝑦𝐵)) → 𝑅 ∈ Ring)
3 pm2mpmhm.p . . . . . . . 8 𝑃 = (Poly1𝑅)
4 pm2mpmhm.c . . . . . . . 8 𝐶 = (𝑁 Mat 𝑃)
53, 4pmatring 22513 . . . . . . 7 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝐶 ∈ Ring)
65adantr 480 . . . . . 6 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥𝐵𝑦𝐵)) → 𝐶 ∈ Ring)
7 simpl 482 . . . . . . 7 ((𝑥𝐵𝑦𝐵) → 𝑥𝐵)
87adantl 481 . . . . . 6 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥𝐵𝑦𝐵)) → 𝑥𝐵)
9 simpr 484 . . . . . . 7 ((𝑥𝐵𝑦𝐵) → 𝑦𝐵)
109adantl 481 . . . . . 6 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥𝐵𝑦𝐵)) → 𝑦𝐵)
11 pm2mpmhm.b . . . . . . 7 𝐵 = (Base‘𝐶)
12 eqid 2731 . . . . . . 7 (.r𝐶) = (.r𝐶)
1311, 12ringcl 20151 . . . . . 6 ((𝐶 ∈ Ring ∧ 𝑥𝐵𝑦𝐵) → (𝑥(.r𝐶)𝑦) ∈ 𝐵)
146, 8, 10, 13syl3anc 1370 . . . . 5 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥𝐵𝑦𝐵)) → (𝑥(.r𝐶)𝑦) ∈ 𝐵)
15 eqid 2731 . . . . . 6 ( ·𝑠𝑄) = ( ·𝑠𝑄)
16 eqid 2731 . . . . . 6 (.g‘(mulGrp‘𝑄)) = (.g‘(mulGrp‘𝑄))
17 eqid 2731 . . . . . 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 22617 . . . . 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 1370 . . . 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 22593 . . . . . . . 8 ((𝑅 ∈ Ring ∧ (𝑥𝐵𝑦𝐵) ∧ 𝑘 ∈ ℕ0) → ((𝑥(.r𝐶)𝑦) decompPMat 𝑘) = (𝐴 Σg (𝑧 ∈ (0...𝑘) ↦ ((𝑥 decompPMat 𝑧)(.r𝐴)(𝑦 decompPMat (𝑘𝑧))))))
2423ad4ant234 1174 . . . . . . 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 7427 . . . . . 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 5248 . . . . 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 7428 . . . 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 2731 . . . . . . . 8 (Base‘𝑄) = (Base‘𝑄)
2918matring 22264 . . . . . . . . 9 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝐴 ∈ Ring)
3029ad2antrr 723 . . . . . . . 8 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥𝐵𝑦𝐵)) ∧ 𝑛 ∈ ℕ0) → 𝐴 ∈ Ring)
31 eqid 2731 . . . . . . . 8 (Base‘𝐴) = (Base‘𝐴)
32 eqid 2731 . . . . . . . 8 (0g𝐴) = (0g𝐴)
33 ringcmn 20177 . . . . . . . . . . . 12 (𝐴 ∈ Ring → 𝐴 ∈ CMnd)
3429, 33syl 17 . . . . . . . . . . 11 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝐴 ∈ CMnd)
3534ad3antrrr 727 . . . . . . . . . 10 (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥𝐵𝑦𝐵)) ∧ 𝑛 ∈ ℕ0) ∧ 𝑘 ∈ ℕ0) → 𝐴 ∈ CMnd)
36 fzfid 13945 . . . . . . . . . 10 (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥𝐵𝑦𝐵)) ∧ 𝑛 ∈ ℕ0) ∧ 𝑘 ∈ ℕ0) → (0...𝑘) ∈ Fin)
3730ad2antrr 723 . . . . . . . . . . . 12 ((((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥𝐵𝑦𝐵)) ∧ 𝑛 ∈ ℕ0) ∧ 𝑘 ∈ ℕ0) ∧ 𝑧 ∈ (0...𝑘)) → 𝐴 ∈ Ring)
38 simp-5r 783 . . . . . . . . . . . . 13 ((((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥𝐵𝑦𝐵)) ∧ 𝑛 ∈ ℕ0) ∧ 𝑘 ∈ ℕ0) ∧ 𝑧 ∈ (0...𝑘)) → 𝑅 ∈ Ring)
398ad3antrrr 727 . . . . . . . . . . . . 13 ((((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥𝐵𝑦𝐵)) ∧ 𝑛 ∈ ℕ0) ∧ 𝑘 ∈ ℕ0) ∧ 𝑧 ∈ (0...𝑘)) → 𝑥𝐵)
40 elfznn0 13601 . . . . . . . . . . . . . 14 (𝑧 ∈ (0...𝑘) → 𝑧 ∈ ℕ0)
4140adantl 481 . . . . . . . . . . . . 13 ((((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥𝐵𝑦𝐵)) ∧ 𝑛 ∈ ℕ0) ∧ 𝑘 ∈ ℕ0) ∧ 𝑧 ∈ (0...𝑘)) → 𝑧 ∈ ℕ0)
423, 4, 11, 18, 31decpmatcl 22588 . . . . . . . . . . . . 13 ((𝑅 ∈ Ring ∧ 𝑥𝐵𝑧 ∈ ℕ0) → (𝑥 decompPMat 𝑧) ∈ (Base‘𝐴))
4338, 39, 41, 42syl3anc 1370 . . . . . . . . . . . 12 ((((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥𝐵𝑦𝐵)) ∧ 𝑛 ∈ ℕ0) ∧ 𝑘 ∈ ℕ0) ∧ 𝑧 ∈ (0...𝑘)) → (𝑥 decompPMat 𝑧) ∈ (Base‘𝐴))
4410ad3antrrr 727 . . . . . . . . . . . . 13 ((((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥𝐵𝑦𝐵)) ∧ 𝑛 ∈ ℕ0) ∧ 𝑘 ∈ ℕ0) ∧ 𝑧 ∈ (0...𝑘)) → 𝑦𝐵)
45 fznn0sub 13540 . . . . . . . . . . . . . 14 (𝑧 ∈ (0...𝑘) → (𝑘𝑧) ∈ ℕ0)
4645adantl 481 . . . . . . . . . . . . 13 ((((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥𝐵𝑦𝐵)) ∧ 𝑛 ∈ ℕ0) ∧ 𝑘 ∈ ℕ0) ∧ 𝑧 ∈ (0...𝑘)) → (𝑘𝑧) ∈ ℕ0)
473, 4, 11, 18, 31decpmatcl 22588 . . . . . . . . . . . . 13 ((𝑅 ∈ Ring ∧ 𝑦𝐵 ∧ (𝑘𝑧) ∈ ℕ0) → (𝑦 decompPMat (𝑘𝑧)) ∈ (Base‘𝐴))
4838, 44, 46, 47syl3anc 1370 . . . . . . . . . . . 12 ((((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥𝐵𝑦𝐵)) ∧ 𝑛 ∈ ℕ0) ∧ 𝑘 ∈ ℕ0) ∧ 𝑧 ∈ (0...𝑘)) → (𝑦 decompPMat (𝑘𝑧)) ∈ (Base‘𝐴))
49 eqid 2731 . . . . . . . . . . . . 13 (.r𝐴) = (.r𝐴)
5031, 49ringcl 20151 . . . . . . . . . . . 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 1370 . . . . . . . . . . 11 ((((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥𝐵𝑦𝐵)) ∧ 𝑛 ∈ ℕ0) ∧ 𝑘 ∈ ℕ0) ∧ 𝑧 ∈ (0...𝑘)) → ((𝑥 decompPMat 𝑧)(.r𝐴)(𝑦 decompPMat (𝑘𝑧))) ∈ (Base‘𝐴))
5251ralrimiva 3145 . . . . . . . . . 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 19883 . . . . . . . . 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 3145 . . . . . . . 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 22594 . . . . . . . . 9 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥𝐵𝑦𝐵)) → (𝑘 ∈ ℕ0 ↦ (𝐴 Σg (𝑧 ∈ (0...𝑘) ↦ ((𝑥 decompPMat 𝑧)(.r𝐴)(𝑦 decompPMat (𝑘𝑧)))))) finSupp (0g𝐴))
5655adantr 480 . . . . . . . 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 484 . . . . . . . 8 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥𝐵𝑦𝐵)) ∧ 𝑛 ∈ ℕ0) → 𝑛 ∈ ℕ0)
5819, 28, 17, 16, 30, 31, 15, 32, 54, 56, 57gsummoncoe1 22147 . . . . . . 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 7458 . . . . . . . . 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 7420 . . . . . . . . . . . . 13 (𝑘 = 𝑛 → (0...𝑘) = (0...𝑛))
62 oveq1 7419 . . . . . . . . . . . . . . 15 (𝑘 = 𝑛 → (𝑘𝑧) = (𝑛𝑧))
6362oveq2d 7428 . . . . . . . . . . . . . 14 (𝑘 = 𝑛 → (𝑦 decompPMat (𝑘𝑧)) = (𝑦 decompPMat (𝑛𝑧)))
6463oveq2d 7428 . . . . . . . . . . . . 13 (𝑘 = 𝑛 → ((𝑥 decompPMat 𝑧)(.r𝐴)(𝑦 decompPMat (𝑘𝑧))) = ((𝑥 decompPMat 𝑧)(.r𝐴)(𝑦 decompPMat (𝑛𝑧))))
6561, 64mpteq12dv 5239 . . . . . . . . . . . 12 (𝑘 = 𝑛 → (𝑧 ∈ (0...𝑘) ↦ ((𝑥 decompPMat 𝑧)(.r𝐴)(𝑦 decompPMat (𝑘𝑧)))) = (𝑧 ∈ (0...𝑛) ↦ ((𝑥 decompPMat 𝑧)(.r𝐴)(𝑦 decompPMat (𝑛𝑧)))))
6665adantl 481 . . . . . . . . . . 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 3931 . . . . . . . . . 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 7428 . . . . . . . . 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 2771 . . . . . . . 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 481 . . . . . . 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 2732 . . . . . . . . 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 7420 . . . . . . . . . . . 12 (𝑟 = 𝑛 → (0...𝑟) = (0...𝑛))
73 fvoveq1 7435 . . . . . . . . . . . . 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 7428 . . . . . . . . . . . 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 5239 . . . . . . . . . . 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 7428 . . . . . . . . . 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 481 . . . . . . . . 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 7447 . . . . . . . . 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 7005 . . . . . . . 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 2731 . . . . . . . . . 10 (0g𝑄) = (0g𝑄)
8119ply1ring 22089 . . . . . . . . . . . . 13 (𝐴 ∈ Ring → 𝑄 ∈ Ring)
8229, 81syl 17 . . . . . . . . . . . 12 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝑄 ∈ Ring)
83 ringcmn 20177 . . . . . . . . . . . 12 (𝑄 ∈ Ring → 𝑄 ∈ CMnd)
8482, 83syl 17 . . . . . . . . . . 11 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝑄 ∈ CMnd)
8584ad2antrr 723 . . . . . . . . . 10 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥𝐵𝑦𝐵)) ∧ 𝑛 ∈ ℕ0) → 𝑄 ∈ CMnd)
86 nn0ex 12485 . . . . . . . . . . 11 0 ∈ V
8786a1i 11 . . . . . . . . . 10 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥𝐵𝑦𝐵)) ∧ 𝑛 ∈ ℕ0) → ℕ0 ∈ V)
887anim2i 616 . . . . . . . . . . . . . 14 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥𝐵𝑦𝐵)) → ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝑥𝐵))
89 df-3an 1088 . . . . . . . . . . . . . 14 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑥𝐵) ↔ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝑥𝐵))
9088, 89sylibr 233 . . . . . . . . . . . . 13 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥𝐵𝑦𝐵)) → (𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑥𝐵))
9190adantr 480 . . . . . . . . . . . 12 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥𝐵𝑦𝐵)) ∧ 𝑛 ∈ ℕ0) → (𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑥𝐵))
923, 4, 11, 15, 16, 17, 18, 19, 28pm2mpghmlem1 22634 . . . . . . . . . . . 12 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑥𝐵) ∧ 𝑘 ∈ ℕ0) → ((𝑥 decompPMat 𝑘)( ·𝑠𝑄)(𝑘(.g‘(mulGrp‘𝑄))(var1𝐴))) ∈ (Base‘𝑄))
9391, 92sylan 579 . . . . . . . . . . 11 (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥𝐵𝑦𝐵)) ∧ 𝑛 ∈ ℕ0) ∧ 𝑘 ∈ ℕ0) → ((𝑥 decompPMat 𝑘)( ·𝑠𝑄)(𝑘(.g‘(mulGrp‘𝑄))(var1𝐴))) ∈ (Base‘𝑄))
9493fmpttd 7116 . . . . . . . . . 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 22633 . . . . . . . . . . 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 19831 . . . . . . . . 9 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥𝐵𝑦𝐵)) ∧ 𝑛 ∈ ℕ0) → (𝑄 Σg (𝑘 ∈ ℕ0 ↦ ((𝑥 decompPMat 𝑘)( ·𝑠𝑄)(𝑘(.g‘(mulGrp‘𝑄))(var1𝐴))))) ∈ (Base‘𝑄))
989anim2i 616 . . . . . . . . . . . . . 14 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥𝐵𝑦𝐵)) → ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝑦𝐵))
99 df-3an 1088 . . . . . . . . . . . . . 14 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑦𝐵) ↔ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝑦𝐵))
10098, 99sylibr 233 . . . . . . . . . . . . 13 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥𝐵𝑦𝐵)) → (𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑦𝐵))
101100adantr 480 . . . . . . . . . . . 12 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥𝐵𝑦𝐵)) ∧ 𝑛 ∈ ℕ0) → (𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑦𝐵))
1023, 4, 11, 15, 16, 17, 18, 19, 28pm2mpghmlem1 22634 . . . . . . . . . . . 12 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑦𝐵) ∧ 𝑘 ∈ ℕ0) → ((𝑦 decompPMat 𝑘)( ·𝑠𝑄)(𝑘(.g‘(mulGrp‘𝑄))(var1𝐴))) ∈ (Base‘𝑄))
103101, 102sylan 579 . . . . . . . . . . 11 (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥𝐵𝑦𝐵)) ∧ 𝑛 ∈ ℕ0) ∧ 𝑘 ∈ ℕ0) → ((𝑦 decompPMat 𝑘)( ·𝑠𝑄)(𝑘(.g‘(mulGrp‘𝑄))(var1𝐴))) ∈ (Base‘𝑄))
104103fmpttd 7116 . . . . . . . . . 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 1127 . . . . . . . . . . . 12 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥𝐵𝑦𝐵)) → (𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑦𝐵))
106105adantr 480 . . . . . . . . . . 11 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥𝐵𝑦𝐵)) ∧ 𝑛 ∈ ℕ0) → (𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑦𝐵))
1073, 4, 11, 15, 16, 17, 18, 19pm2mpghmlem2 22633 . . . . . . . . . . 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 19831 . . . . . . . . 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 2731 . . . . . . . . . . 11 (.r𝑄) = (.r𝑄)
11119, 110, 49, 28coe1mul 22111 . . . . . . . . . 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 6893 . . . . . . . . 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 1370 . . . . . . . 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 7420 . . . . . . . . . . . 12 (𝑧 = 𝑙 → (𝑥 decompPMat 𝑧) = (𝑥 decompPMat 𝑙))
115 oveq2 7420 . . . . . . . . . . . . 13 (𝑧 = 𝑙 → (𝑛𝑧) = (𝑛𝑙))
116115oveq2d 7428 . . . . . . . . . . . 12 (𝑧 = 𝑙 → (𝑦 decompPMat (𝑛𝑧)) = (𝑦 decompPMat (𝑛𝑙)))
117114, 116oveq12d 7430 . . . . . . . . . . 11 (𝑧 = 𝑙 → ((𝑥 decompPMat 𝑧)(.r𝐴)(𝑦 decompPMat (𝑛𝑧))) = ((𝑥 decompPMat 𝑙)(.r𝐴)(𝑦 decompPMat (𝑛𝑙))))
118117cbvmptv 5261 . . . . . . . . . 10 (𝑧 ∈ (0...𝑛) ↦ ((𝑥 decompPMat 𝑧)(.r𝐴)(𝑦 decompPMat (𝑛𝑧)))) = (𝑙 ∈ (0...𝑛) ↦ ((𝑥 decompPMat 𝑙)(.r𝐴)(𝑦 decompPMat (𝑛𝑙))))
11929ad3antrrr 727 . . . . . . . . . . . . . 14 (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥𝐵𝑦𝐵)) ∧ 𝑛 ∈ ℕ0) ∧ 𝑙 ∈ (0...𝑛)) → 𝐴 ∈ Ring)
120 simp-5r 783 . . . . . . . . . . . . . . . 16 ((((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥𝐵𝑦𝐵)) ∧ 𝑛 ∈ ℕ0) ∧ 𝑙 ∈ (0...𝑛)) ∧ 𝑘 ∈ ℕ0) → 𝑅 ∈ Ring)
1218ad3antrrr 727 . . . . . . . . . . . . . . . 16 ((((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥𝐵𝑦𝐵)) ∧ 𝑛 ∈ ℕ0) ∧ 𝑙 ∈ (0...𝑛)) ∧ 𝑘 ∈ ℕ0) → 𝑥𝐵)
122 simpr 484 . . . . . . . . . . . . . . . 16 ((((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥𝐵𝑦𝐵)) ∧ 𝑛 ∈ ℕ0) ∧ 𝑙 ∈ (0...𝑛)) ∧ 𝑘 ∈ ℕ0) → 𝑘 ∈ ℕ0)
1233, 4, 11, 18, 31decpmatcl 22588 . . . . . . . . . . . . . . . 16 ((𝑅 ∈ Ring ∧ 𝑥𝐵𝑘 ∈ ℕ0) → (𝑥 decompPMat 𝑘) ∈ (Base‘𝐴))
124120, 121, 122, 123syl3anc 1370 . . . . . . . . . . . . . . 15 ((((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥𝐵𝑦𝐵)) ∧ 𝑛 ∈ ℕ0) ∧ 𝑙 ∈ (0...𝑛)) ∧ 𝑘 ∈ ℕ0) → (𝑥 decompPMat 𝑘) ∈ (Base‘𝐴))
125124ralrimiva 3145 . . . . . . . . . . . . . 14 (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥𝐵𝑦𝐵)) ∧ 𝑛 ∈ ℕ0) ∧ 𝑙 ∈ (0...𝑛)) → ∀𝑘 ∈ ℕ0 (𝑥 decompPMat 𝑘) ∈ (Base‘𝐴))
1262, 8jca 511 . . . . . . . . . . . . . . . 16 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥𝐵𝑦𝐵)) → (𝑅 ∈ Ring ∧ 𝑥𝐵))
127126ad2antrr 723 . . . . . . . . . . . . . . 15 (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥𝐵𝑦𝐵)) ∧ 𝑛 ∈ ℕ0) ∧ 𝑙 ∈ (0...𝑛)) → (𝑅 ∈ Ring ∧ 𝑥𝐵))
1283, 4, 11, 18, 32decpmatfsupp 22590 . . . . . . . . . . . . . . 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 13601 . . . . . . . . . . . . . . 15 (𝑙 ∈ (0...𝑛) → 𝑙 ∈ ℕ0)
131130adantl 481 . . . . . . . . . . . . . 14 (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥𝐵𝑦𝐵)) ∧ 𝑛 ∈ ℕ0) ∧ 𝑙 ∈ (0...𝑛)) → 𝑙 ∈ ℕ0)
13219, 28, 17, 16, 119, 31, 15, 32, 125, 129, 131gsummoncoe1 22147 . . . . . . . . . . . . 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 7458 . . . . . . . . . . . . . . 15 (𝑙 ∈ (0...𝑛) → 𝑙 / 𝑘(𝑥 decompPMat 𝑘) = (𝑥 decompPMat 𝑙 / 𝑘𝑘))
134 csbvarg 4431 . . . . . . . . . . . . . . . 16 (𝑙 ∈ (0...𝑛) → 𝑙 / 𝑘𝑘 = 𝑙)
135134oveq2d 7428 . . . . . . . . . . . . . . 15 (𝑙 ∈ (0...𝑛) → (𝑥 decompPMat 𝑙 / 𝑘𝑘) = (𝑥 decompPMat 𝑙))
136133, 135eqtrd 2771 . . . . . . . . . . . . . 14 (𝑙 ∈ (0...𝑛) → 𝑙 / 𝑘(𝑥 decompPMat 𝑘) = (𝑥 decompPMat 𝑙))
137136adantl 481 . . . . . . . . . . . . 13 (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥𝐵𝑦𝐵)) ∧ 𝑛 ∈ ℕ0) ∧ 𝑙 ∈ (0...𝑛)) → 𝑙 / 𝑘(𝑥 decompPMat 𝑘) = (𝑥 decompPMat 𝑙))
138132, 137eqtr2d 2772 . . . . . . . . . . . 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 727 . . . . . . . . . . . . . . . 16 ((((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥𝐵𝑦𝐵)) ∧ 𝑛 ∈ ℕ0) ∧ 𝑙 ∈ (0...𝑛)) ∧ 𝑘 ∈ ℕ0) → 𝑦𝐵)
1403, 4, 11, 18, 31decpmatcl 22588 . . . . . . . . . . . . . . . 16 ((𝑅 ∈ Ring ∧ 𝑦𝐵𝑘 ∈ ℕ0) → (𝑦 decompPMat 𝑘) ∈ (Base‘𝐴))
141120, 139, 122, 140syl3anc 1370 . . . . . . . . . . . . . . 15 ((((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥𝐵𝑦𝐵)) ∧ 𝑛 ∈ ℕ0) ∧ 𝑙 ∈ (0...𝑛)) ∧ 𝑘 ∈ ℕ0) → (𝑦 decompPMat 𝑘) ∈ (Base‘𝐴))
142141ralrimiva 3145 . . . . . . . . . . . . . 14 (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥𝐵𝑦𝐵)) ∧ 𝑛 ∈ ℕ0) ∧ 𝑙 ∈ (0...𝑛)) → ∀𝑘 ∈ ℕ0 (𝑦 decompPMat 𝑘) ∈ (Base‘𝐴))
1432, 10jca 511 . . . . . . . . . . . . . . . 16 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥𝐵𝑦𝐵)) → (𝑅 ∈ Ring ∧ 𝑦𝐵))
144143ad2antrr 723 . . . . . . . . . . . . . . 15 (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥𝐵𝑦𝐵)) ∧ 𝑛 ∈ ℕ0) ∧ 𝑙 ∈ (0...𝑛)) → (𝑅 ∈ Ring ∧ 𝑦𝐵))
1453, 4, 11, 18, 32decpmatfsupp 22590 . . . . . . . . . . . . . . 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 13540 . . . . . . . . . . . . . . 15 (𝑙 ∈ (0...𝑛) → (𝑛𝑙) ∈ ℕ0)
148147adantl 481 . . . . . . . . . . . . . 14 (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥𝐵𝑦𝐵)) ∧ 𝑛 ∈ ℕ0) ∧ 𝑙 ∈ (0...𝑛)) → (𝑛𝑙) ∈ ℕ0)
14919, 28, 17, 16, 119, 31, 15, 32, 142, 146, 148gsummoncoe1 22147 . . . . . . . . . . . . 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 7445 . . . . . . . . . . . . . 14 (𝑛𝑙) ∈ V
151 csbov2g 7458 . . . . . . . . . . . . . 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 4431 . . . . . . . . . . . . . . 15 ((𝑛𝑙) ∈ V → (𝑛𝑙) / 𝑘𝑘 = (𝑛𝑙))
154150, 153mp1i 13 . . . . . . . . . . . . . 14 (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥𝐵𝑦𝐵)) ∧ 𝑛 ∈ ℕ0) ∧ 𝑙 ∈ (0...𝑛)) → (𝑛𝑙) / 𝑘𝑘 = (𝑛𝑙))
155154oveq2d 7428 . . . . . . . . . . . . 13 (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥𝐵𝑦𝐵)) ∧ 𝑛 ∈ ℕ0) ∧ 𝑙 ∈ (0...𝑛)) → (𝑦 decompPMat (𝑛𝑙) / 𝑘𝑘) = (𝑦 decompPMat (𝑛𝑙)))
156149, 152, 1553eqtrrd 2776 . . . . . . . . . . . 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 7430 . . . . . . . . . . 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 5248 . . . . . . . . . 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, 158eqtrid 2783 . . . . . . . . 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 7428 . . . . . . . 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 2782 . . . . . . 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 2775 . . . . . 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 3145 . . . . 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 480 . . . . . 6 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥𝐵𝑦𝐵)) → 𝐴 ∈ Ring)
16584adantr 480 . . . . . . 7 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥𝐵𝑦𝐵)) → 𝑄 ∈ CMnd)
16686a1i 11 . . . . . . 7 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥𝐵𝑦𝐵)) → ℕ0 ∈ V)
16719ply1lmod 22093 . . . . . . . . . . 11 (𝐴 ∈ Ring → 𝑄 ∈ LMod)
16829, 167syl 17 . . . . . . . . . 10 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝑄 ∈ LMod)
169168ad2antrr 723 . . . . . . . . 9 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥𝐵𝑦𝐵)) ∧ 𝑘 ∈ ℕ0) → 𝑄 ∈ LMod)
17034ad2antrr 723 . . . . . . . . . . 11 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥𝐵𝑦𝐵)) ∧ 𝑘 ∈ ℕ0) → 𝐴 ∈ CMnd)
171 fzfid 13945 . . . . . . . . . . 11 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥𝐵𝑦𝐵)) ∧ 𝑘 ∈ ℕ0) → (0...𝑘) ∈ Fin)
17229ad3antrrr 727 . . . . . . . . . . . . 13 (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥𝐵𝑦𝐵)) ∧ 𝑘 ∈ ℕ0) ∧ 𝑧 ∈ (0...𝑘)) → 𝐴 ∈ Ring)
173 simp-4r 781 . . . . . . . . . . . . . 14 (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥𝐵𝑦𝐵)) ∧ 𝑘 ∈ ℕ0) ∧ 𝑧 ∈ (0...𝑘)) → 𝑅 ∈ Ring)
174 simplrl 774 . . . . . . . . . . . . . . 15 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥𝐵𝑦𝐵)) ∧ 𝑘 ∈ ℕ0) → 𝑥𝐵)
175174adantr 480 . . . . . . . . . . . . . 14 (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥𝐵𝑦𝐵)) ∧ 𝑘 ∈ ℕ0) ∧ 𝑧 ∈ (0...𝑘)) → 𝑥𝐵)
17640adantl 481 . . . . . . . . . . . . . 14 (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥𝐵𝑦𝐵)) ∧ 𝑘 ∈ ℕ0) ∧ 𝑧 ∈ (0...𝑘)) → 𝑧 ∈ ℕ0)
177173, 175, 176, 42syl3anc 1370 . . . . . . . . . . . . 13 (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥𝐵𝑦𝐵)) ∧ 𝑘 ∈ ℕ0) ∧ 𝑧 ∈ (0...𝑘)) → (𝑥 decompPMat 𝑧) ∈ (Base‘𝐴))
178 simplrr 775 . . . . . . . . . . . . . . 15 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥𝐵𝑦𝐵)) ∧ 𝑘 ∈ ℕ0) → 𝑦𝐵)
179178adantr 480 . . . . . . . . . . . . . 14 (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥𝐵𝑦𝐵)) ∧ 𝑘 ∈ ℕ0) ∧ 𝑧 ∈ (0...𝑘)) → 𝑦𝐵)
18045adantl 481 . . . . . . . . . . . . . 14 (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥𝐵𝑦𝐵)) ∧ 𝑘 ∈ ℕ0) ∧ 𝑧 ∈ (0...𝑘)) → (𝑘𝑧) ∈ ℕ0)
181173, 179, 180, 47syl3anc 1370 . . . . . . . . . . . . 13 (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥𝐵𝑦𝐵)) ∧ 𝑘 ∈ ℕ0) ∧ 𝑧 ∈ (0...𝑘)) → (𝑦 decompPMat (𝑘𝑧)) ∈ (Base‘𝐴))
182172, 177, 181, 50syl3anc 1370 . . . . . . . . . . . 12 (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥𝐵𝑦𝐵)) ∧ 𝑘 ∈ ℕ0) ∧ 𝑧 ∈ (0...𝑘)) → ((𝑥 decompPMat 𝑧)(.r𝐴)(𝑦 decompPMat (𝑘𝑧))) ∈ (Base‘𝐴))
183182ralrimiva 3145 . . . . . . . . . . 11 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥𝐵𝑦𝐵)) ∧ 𝑘 ∈ ℕ0) → ∀𝑧 ∈ (0...𝑘)((𝑥 decompPMat 𝑧)(.r𝐴)(𝑦 decompPMat (𝑘𝑧))) ∈ (Base‘𝐴))
18431, 170, 171, 183gsummptcl 19883 . . . . . . . . . 10 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥𝐵𝑦𝐵)) ∧ 𝑘 ∈ ℕ0) → (𝐴 Σg (𝑧 ∈ (0...𝑘) ↦ ((𝑥 decompPMat 𝑧)(.r𝐴)(𝑦 decompPMat (𝑘𝑧))))) ∈ (Base‘𝐴))
18529ad2antrr 723 . . . . . . . . . . . . 13 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥𝐵𝑦𝐵)) ∧ 𝑘 ∈ ℕ0) → 𝐴 ∈ Ring)
18619ply1sca 22094 . . . . . . . . . . . . 13 (𝐴 ∈ Ring → 𝐴 = (Scalar‘𝑄))
187185, 186syl 17 . . . . . . . . . . . 12 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥𝐵𝑦𝐵)) ∧ 𝑘 ∈ ℕ0) → 𝐴 = (Scalar‘𝑄))
188187eqcomd 2737 . . . . . . . . . . 11 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥𝐵𝑦𝐵)) ∧ 𝑘 ∈ ℕ0) → (Scalar‘𝑄) = 𝐴)
189188fveq2d 6895 . . . . . . . . . 10 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥𝐵𝑦𝐵)) ∧ 𝑘 ∈ ℕ0) → (Base‘(Scalar‘𝑄)) = (Base‘𝐴))
190184, 189eleqtrrd 2835 . . . . . . . . 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 2731 . . . . . . . . . . 11 (mulGrp‘𝑄) = (mulGrp‘𝑄)
19219, 17, 191, 16, 28ply1moncl 22112 . . . . . . . . . 10 ((𝐴 ∈ Ring ∧ 𝑘 ∈ ℕ0) → (𝑘(.g‘(mulGrp‘𝑄))(var1𝐴)) ∈ (Base‘𝑄))
193185, 192sylancom 587 . . . . . . . . 9 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥𝐵𝑦𝐵)) ∧ 𝑘 ∈ ℕ0) → (𝑘(.g‘(mulGrp‘𝑄))(var1𝐴)) ∈ (Base‘𝑄))
194 eqid 2731 . . . . . . . . . 10 (Scalar‘𝑄) = (Scalar‘𝑄)
195 eqid 2731 . . . . . . . . . 10 (Base‘(Scalar‘𝑄)) = (Base‘(Scalar‘𝑄))
19628, 194, 15, 195lmodvscl 20720 . . . . . . . . 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 1370 . . . . . . . 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 7116 . . . . . . 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 22639 . . . . . . 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 19831 . . . . . 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 480 . . . . . . 7 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥𝐵𝑦𝐵)) → 𝑄 ∈ Ring)
20290, 92sylan 579 . . . . . . . . 9 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥𝐵𝑦𝐵)) ∧ 𝑘 ∈ ℕ0) → ((𝑥 decompPMat 𝑘)( ·𝑠𝑄)(𝑘(.g‘(mulGrp‘𝑄))(var1𝐴))) ∈ (Base‘𝑄))
203202fmpttd 7116 . . . . . . . 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 19831 . . . . . . 7 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥𝐵𝑦𝐵)) → (𝑄 Σg (𝑘 ∈ ℕ0 ↦ ((𝑥 decompPMat 𝑘)( ·𝑠𝑄)(𝑘(.g‘(mulGrp‘𝑄))(var1𝐴))))) ∈ (Base‘𝑄))
206100, 102sylan 579 . . . . . . . . 9 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥𝐵𝑦𝐵)) ∧ 𝑘 ∈ ℕ0) → ((𝑦 decompPMat 𝑘)( ·𝑠𝑄)(𝑘(.g‘(mulGrp‘𝑄))(var1𝐴))) ∈ (Base‘𝑄))
207206fmpttd 7116 . . . . . . . 8 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥𝐵𝑦𝐵)) → (𝑘 ∈ ℕ0 ↦ ((𝑦 decompPMat 𝑘)( ·𝑠𝑄)(𝑘(.g‘(mulGrp‘𝑄))(var1𝐴)))):ℕ0⟶(Base‘𝑄))
2081, 2, 10, 107syl3anc 1370 . . . . . . . 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 19831 . . . . . . 7 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥𝐵𝑦𝐵)) → (𝑄 Σg (𝑘 ∈ ℕ0 ↦ ((𝑦 decompPMat 𝑘)( ·𝑠𝑄)(𝑘(.g‘(mulGrp‘𝑄))(var1𝐴))))) ∈ (Base‘𝑄))
21028, 110ringcl 20151 . . . . . . 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 1370 . . . . . 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 2731 . . . . . . 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 2731 . . . . . . 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 22141 . . . . . 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 1370 . . . . 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 231 . . . 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 2775 . . 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 22617 . . . . 5 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑥𝐵) → (𝑇𝑥) = (𝑄 Σg (𝑘 ∈ ℕ0 ↦ ((𝑥 decompPMat 𝑘)( ·𝑠𝑄)(𝑘(.g‘(mulGrp‘𝑄))(var1𝐴))))))
2191, 2, 8, 218syl3anc 1370 . . . 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 22617 . . . . 5 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑦𝐵) → (𝑇𝑦) = (𝑄 Σg (𝑘 ∈ ℕ0 ↦ ((𝑦 decompPMat 𝑘)( ·𝑠𝑄)(𝑘(.g‘(mulGrp‘𝑄))(var1𝐴))))))
2211, 2, 10, 220syl3anc 1370 . . . 4 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥𝐵𝑦𝐵)) → (𝑇𝑦) = (𝑄 Σg (𝑘 ∈ ℕ0 ↦ ((𝑦 decompPMat 𝑘)( ·𝑠𝑄)(𝑘(.g‘(mulGrp‘𝑄))(var1𝐴))))))
222219, 221oveq12d 7430 . . 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 2774 . 2 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥𝐵𝑦𝐵)) → (𝑇‘(𝑥(.r𝐶)𝑦)) = ((𝑇𝑥)(.r𝑄)(𝑇𝑦)))
224223ralrimivva 3199 1 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → ∀𝑥𝐵𝑦𝐵 (𝑇‘(𝑥(.r𝐶)𝑦)) = ((𝑇𝑥)(.r𝑄)(𝑇𝑦)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 395  w3a 1086   = wceq 1540  wcel 2105  wral 3060  Vcvv 3473  csb 3893   class class class wbr 5148  cmpt 5231  cfv 6543  (class class class)co 7412  Fincfn 8945   finSupp cfsupp 9367  0cc0 11116  cmin 11451  0cn0 12479  ...cfz 13491  Basecbs 17151  .rcmulr 17205  Scalarcsca 17207   ·𝑠 cvsca 17208  0gc0g 17392   Σg cgsu 17393  .gcmg 18993  CMndccmn 19696  mulGrpcmgp 20035  Ringcrg 20134  LModclmod 20702  var1cv1 22018  Poly1cpl1 22019  coe1cco1 22020   Mat cmat 22226   decompPMat cdecpmat 22583   pMatToMatPoly cpm2mp 22613
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2702  ax-rep 5285  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7729  ax-cnex 11172  ax-resscn 11173  ax-1cn 11174  ax-icn 11175  ax-addcl 11176  ax-addrcl 11177  ax-mulcl 11178  ax-mulrcl 11179  ax-mulcom 11180  ax-addass 11181  ax-mulass 11182  ax-distr 11183  ax-i2m1 11184  ax-1ne0 11185  ax-1rid 11186  ax-rnegex 11187  ax-rrecex 11188  ax-cnre 11189  ax-pre-lttri 11190  ax-pre-lttrn 11191  ax-pre-ltadd 11192  ax-pre-mulgt0 11193
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ne 2940  df-nel 3046  df-ral 3061  df-rex 3070  df-rmo 3375  df-reu 3376  df-rab 3432  df-v 3475  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-pss 3967  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-tp 4633  df-op 4635  df-ot 4637  df-uni 4909  df-int 4951  df-iun 4999  df-iin 5000  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5574  df-eprel 5580  df-po 5588  df-so 5589  df-fr 5631  df-se 5632  df-we 5633  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-pred 6300  df-ord 6367  df-on 6368  df-lim 6369  df-suc 6370  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-isom 6552  df-riota 7368  df-ov 7415  df-oprab 7416  df-mpo 7417  df-of 7674  df-ofr 7675  df-om 7860  df-1st 7979  df-2nd 7980  df-supp 8152  df-frecs 8272  df-wrecs 8303  df-recs 8377  df-rdg 8416  df-1o 8472  df-er 8709  df-map 8828  df-pm 8829  df-ixp 8898  df-en 8946  df-dom 8947  df-sdom 8948  df-fin 8949  df-fsupp 9368  df-sup 9443  df-oi 9511  df-card 9940  df-pnf 11257  df-mnf 11258  df-xr 11259  df-ltxr 11260  df-le 11261  df-sub 11453  df-neg 11454  df-nn 12220  df-2 12282  df-3 12283  df-4 12284  df-5 12285  df-6 12286  df-7 12287  df-8 12288  df-9 12289  df-n0 12480  df-z 12566  df-dec 12685  df-uz 12830  df-fz 13492  df-fzo 13635  df-seq 13974  df-hash 14298  df-struct 17087  df-sets 17104  df-slot 17122  df-ndx 17134  df-base 17152  df-ress 17181  df-plusg 17217  df-mulr 17218  df-sca 17220  df-vsca 17221  df-ip 17222  df-tset 17223  df-ple 17224  df-ds 17226  df-hom 17228  df-cco 17229  df-0g 17394  df-gsum 17395  df-prds 17400  df-pws 17402  df-mre 17537  df-mrc 17538  df-acs 17540  df-mgm 18571  df-sgrp 18650  df-mnd 18666  df-mhm 18711  df-submnd 18712  df-grp 18864  df-minusg 18865  df-sbg 18866  df-mulg 18994  df-subg 19046  df-ghm 19135  df-cntz 19229  df-cmn 19698  df-abl 19699  df-mgp 20036  df-rng 20054  df-ur 20083  df-srg 20088  df-ring 20136  df-subrng 20442  df-subrg 20467  df-lmod 20704  df-lss 20775  df-sra 21018  df-rgmod 21019  df-dsmm 21596  df-frlm 21611  df-psr 21771  df-mvr 21772  df-mpl 21773  df-opsr 21775  df-psr1 22022  df-vr1 22023  df-ply1 22024  df-coe1 22025  df-mamu 22205  df-mat 22227  df-decpmat 22584  df-pm2mp 22614
This theorem is referenced by:  pm2mpmhm  22641
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