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Theorem pm2mpmhmlem2 22825
Description: Lemma 2 for pm2mpmhm 22826. (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 767 . . . . 5 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥𝐵𝑦𝐵)) → 𝑁 ∈ Fin)
2 simplr 769 . . . . 5 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥𝐵𝑦𝐵)) → 𝑅 ∈ Ring)
3 pm2mpmhm.p . . . . . . . 8 𝑃 = (Poly1𝑅)
4 pm2mpmhm.c . . . . . . . 8 𝐶 = (𝑁 Mat 𝑃)
53, 4pmatring 22698 . . . . . . 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 2737 . . . . . . 7 (.r𝐶) = (.r𝐶)
1311, 12ringcl 20247 . . . . . 6 ((𝐶 ∈ Ring ∧ 𝑥𝐵𝑦𝐵) → (𝑥(.r𝐶)𝑦) ∈ 𝐵)
146, 8, 10, 13syl3anc 1373 . . . . 5 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥𝐵𝑦𝐵)) → (𝑥(.r𝐶)𝑦) ∈ 𝐵)
15 eqid 2737 . . . . . 6 ( ·𝑠𝑄) = ( ·𝑠𝑄)
16 eqid 2737 . . . . . 6 (.g‘(mulGrp‘𝑄)) = (.g‘(mulGrp‘𝑄))
17 eqid 2737 . . . . . 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 22802 . . . . 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 1373 . . . 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 22778 . . . . . . . 8 ((𝑅 ∈ Ring ∧ (𝑥𝐵𝑦𝐵) ∧ 𝑘 ∈ ℕ0) → ((𝑥(.r𝐶)𝑦) decompPMat 𝑘) = (𝐴 Σg (𝑧 ∈ (0...𝑘) ↦ ((𝑥 decompPMat 𝑧)(.r𝐴)(𝑦 decompPMat (𝑘𝑧))))))
2423ad4ant234 1176 . . . . . . 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 7446 . . . . . 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 5242 . . . . 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 7447 . . . 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 2737 . . . . . . . 8 (Base‘𝑄) = (Base‘𝑄)
2918matring 22449 . . . . . . . . 9 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝐴 ∈ Ring)
3029ad2antrr 726 . . . . . . . 8 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥𝐵𝑦𝐵)) ∧ 𝑛 ∈ ℕ0) → 𝐴 ∈ Ring)
31 eqid 2737 . . . . . . . 8 (Base‘𝐴) = (Base‘𝐴)
32 eqid 2737 . . . . . . . 8 (0g𝐴) = (0g𝐴)
33 ringcmn 20279 . . . . . . . . . . . 12 (𝐴 ∈ Ring → 𝐴 ∈ CMnd)
3429, 33syl 17 . . . . . . . . . . 11 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝐴 ∈ CMnd)
3534ad3antrrr 730 . . . . . . . . . 10 (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥𝐵𝑦𝐵)) ∧ 𝑛 ∈ ℕ0) ∧ 𝑘 ∈ ℕ0) → 𝐴 ∈ CMnd)
36 fzfid 14014 . . . . . . . . . 10 (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥𝐵𝑦𝐵)) ∧ 𝑛 ∈ ℕ0) ∧ 𝑘 ∈ ℕ0) → (0...𝑘) ∈ Fin)
3730ad2antrr 726 . . . . . . . . . . . 12 ((((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥𝐵𝑦𝐵)) ∧ 𝑛 ∈ ℕ0) ∧ 𝑘 ∈ ℕ0) ∧ 𝑧 ∈ (0...𝑘)) → 𝐴 ∈ Ring)
38 simp-5r 786 . . . . . . . . . . . . 13 ((((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥𝐵𝑦𝐵)) ∧ 𝑛 ∈ ℕ0) ∧ 𝑘 ∈ ℕ0) ∧ 𝑧 ∈ (0...𝑘)) → 𝑅 ∈ Ring)
398ad3antrrr 730 . . . . . . . . . . . . 13 ((((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥𝐵𝑦𝐵)) ∧ 𝑛 ∈ ℕ0) ∧ 𝑘 ∈ ℕ0) ∧ 𝑧 ∈ (0...𝑘)) → 𝑥𝐵)
40 elfznn0 13660 . . . . . . . . . . . . . 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 22773 . . . . . . . . . . . . 13 ((𝑅 ∈ Ring ∧ 𝑥𝐵𝑧 ∈ ℕ0) → (𝑥 decompPMat 𝑧) ∈ (Base‘𝐴))
4338, 39, 41, 42syl3anc 1373 . . . . . . . . . . . 12 ((((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥𝐵𝑦𝐵)) ∧ 𝑛 ∈ ℕ0) ∧ 𝑘 ∈ ℕ0) ∧ 𝑧 ∈ (0...𝑘)) → (𝑥 decompPMat 𝑧) ∈ (Base‘𝐴))
4410ad3antrrr 730 . . . . . . . . . . . . 13 ((((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥𝐵𝑦𝐵)) ∧ 𝑛 ∈ ℕ0) ∧ 𝑘 ∈ ℕ0) ∧ 𝑧 ∈ (0...𝑘)) → 𝑦𝐵)
45 fznn0sub 13596 . . . . . . . . . . . . . 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 22773 . . . . . . . . . . . . 13 ((𝑅 ∈ Ring ∧ 𝑦𝐵 ∧ (𝑘𝑧) ∈ ℕ0) → (𝑦 decompPMat (𝑘𝑧)) ∈ (Base‘𝐴))
4838, 44, 46, 47syl3anc 1373 . . . . . . . . . . . 12 ((((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥𝐵𝑦𝐵)) ∧ 𝑛 ∈ ℕ0) ∧ 𝑘 ∈ ℕ0) ∧ 𝑧 ∈ (0...𝑘)) → (𝑦 decompPMat (𝑘𝑧)) ∈ (Base‘𝐴))
49 eqid 2737 . . . . . . . . . . . . 13 (.r𝐴) = (.r𝐴)
5031, 49ringcl 20247 . . . . . . . . . . . 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 1373 . . . . . . . . . . 11 ((((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥𝐵𝑦𝐵)) ∧ 𝑛 ∈ ℕ0) ∧ 𝑘 ∈ ℕ0) ∧ 𝑧 ∈ (0...𝑘)) → ((𝑥 decompPMat 𝑧)(.r𝐴)(𝑦 decompPMat (𝑘𝑧))) ∈ (Base‘𝐴))
5251ralrimiva 3146 . . . . . . . . . 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 19985 . . . . . . . . 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 3146 . . . . . . . 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 22779 . . . . . . . . 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 22312 . . . . . . 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 7479 . . . . . . . . 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 7439 . . . . . . . . . . . . 13 (𝑘 = 𝑛 → (0...𝑘) = (0...𝑛))
62 oveq1 7438 . . . . . . . . . . . . . . 15 (𝑘 = 𝑛 → (𝑘𝑧) = (𝑛𝑧))
6362oveq2d 7447 . . . . . . . . . . . . . 14 (𝑘 = 𝑛 → (𝑦 decompPMat (𝑘𝑧)) = (𝑦 decompPMat (𝑛𝑧)))
6463oveq2d 7447 . . . . . . . . . . . . 13 (𝑘 = 𝑛 → ((𝑥 decompPMat 𝑧)(.r𝐴)(𝑦 decompPMat (𝑘𝑧))) = ((𝑥 decompPMat 𝑧)(.r𝐴)(𝑦 decompPMat (𝑛𝑧))))
6561, 64mpteq12dv 5233 . . . . . . . . . . . 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 3935 . . . . . . . . . 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 7447 . . . . . . . . 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 2777 . . . . . . . 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 2738 . . . . . . . . 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 7439 . . . . . . . . . . . 12 (𝑟 = 𝑛 → (0...𝑟) = (0...𝑛))
73 fvoveq1 7454 . . . . . . . . . . . . 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 7447 . . . . . . . . . . . 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 5233 . . . . . . . . . . 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 7447 . . . . . . . . . 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 7466 . . . . . . . . 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 7023 . . . . . . . 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 2737 . . . . . . . . . 10 (0g𝑄) = (0g𝑄)
8119ply1ring 22249 . . . . . . . . . . . . 13 (𝐴 ∈ Ring → 𝑄 ∈ Ring)
8229, 81syl 17 . . . . . . . . . . . 12 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝑄 ∈ Ring)
83 ringcmn 20279 . . . . . . . . . . . 12 (𝑄 ∈ Ring → 𝑄 ∈ CMnd)
8482, 83syl 17 . . . . . . . . . . 11 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝑄 ∈ CMnd)
8584ad2antrr 726 . . . . . . . . . 10 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥𝐵𝑦𝐵)) ∧ 𝑛 ∈ ℕ0) → 𝑄 ∈ CMnd)
86 nn0ex 12532 . . . . . . . . . . 11 0 ∈ V
8786a1i 11 . . . . . . . . . 10 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥𝐵𝑦𝐵)) ∧ 𝑛 ∈ ℕ0) → ℕ0 ∈ V)
887anim2i 617 . . . . . . . . . . . . . 14 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥𝐵𝑦𝐵)) → ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝑥𝐵))
89 df-3an 1089 . . . . . . . . . . . . . 14 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑥𝐵) ↔ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝑥𝐵))
9088, 89sylibr 234 . . . . . . . . . . . . 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 22819 . . . . . . . . . . . 12 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑥𝐵) ∧ 𝑘 ∈ ℕ0) → ((𝑥 decompPMat 𝑘)( ·𝑠𝑄)(𝑘(.g‘(mulGrp‘𝑄))(var1𝐴))) ∈ (Base‘𝑄))
9391, 92sylan 580 . . . . . . . . . . 11 (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥𝐵𝑦𝐵)) ∧ 𝑛 ∈ ℕ0) ∧ 𝑘 ∈ ℕ0) → ((𝑥 decompPMat 𝑘)( ·𝑠𝑄)(𝑘(.g‘(mulGrp‘𝑄))(var1𝐴))) ∈ (Base‘𝑄))
9493fmpttd 7135 . . . . . . . . . 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 22818 . . . . . . . . . . 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 19933 . . . . . . . . 9 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥𝐵𝑦𝐵)) ∧ 𝑛 ∈ ℕ0) → (𝑄 Σg (𝑘 ∈ ℕ0 ↦ ((𝑥 decompPMat 𝑘)( ·𝑠𝑄)(𝑘(.g‘(mulGrp‘𝑄))(var1𝐴))))) ∈ (Base‘𝑄))
989anim2i 617 . . . . . . . . . . . . . 14 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥𝐵𝑦𝐵)) → ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝑦𝐵))
99 df-3an 1089 . . . . . . . . . . . . . 14 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑦𝐵) ↔ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝑦𝐵))
10098, 99sylibr 234 . . . . . . . . . . . . 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 22819 . . . . . . . . . . . 12 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑦𝐵) ∧ 𝑘 ∈ ℕ0) → ((𝑦 decompPMat 𝑘)( ·𝑠𝑄)(𝑘(.g‘(mulGrp‘𝑄))(var1𝐴))) ∈ (Base‘𝑄))
103101, 102sylan 580 . . . . . . . . . . 11 (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥𝐵𝑦𝐵)) ∧ 𝑛 ∈ ℕ0) ∧ 𝑘 ∈ ℕ0) → ((𝑦 decompPMat 𝑘)( ·𝑠𝑄)(𝑘(.g‘(mulGrp‘𝑄))(var1𝐴))) ∈ (Base‘𝑄))
104103fmpttd 7135 . . . . . . . . . 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 1129 . . . . . . . . . . . 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 22818 . . . . . . . . . . 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 19933 . . . . . . . . 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 2737 . . . . . . . . . . 11 (.r𝑄) = (.r𝑄)
11119, 110, 49, 28coe1mul 22273 . . . . . . . . . 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 6908 . . . . . . . . 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 1373 . . . . . . . 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 7439 . . . . . . . . . . . 12 (𝑧 = 𝑙 → (𝑥 decompPMat 𝑧) = (𝑥 decompPMat 𝑙))
115 oveq2 7439 . . . . . . . . . . . . 13 (𝑧 = 𝑙 → (𝑛𝑧) = (𝑛𝑙))
116115oveq2d 7447 . . . . . . . . . . . 12 (𝑧 = 𝑙 → (𝑦 decompPMat (𝑛𝑧)) = (𝑦 decompPMat (𝑛𝑙)))
117114, 116oveq12d 7449 . . . . . . . . . . 11 (𝑧 = 𝑙 → ((𝑥 decompPMat 𝑧)(.r𝐴)(𝑦 decompPMat (𝑛𝑧))) = ((𝑥 decompPMat 𝑙)(.r𝐴)(𝑦 decompPMat (𝑛𝑙))))
118117cbvmptv 5255 . . . . . . . . . 10 (𝑧 ∈ (0...𝑛) ↦ ((𝑥 decompPMat 𝑧)(.r𝐴)(𝑦 decompPMat (𝑛𝑧)))) = (𝑙 ∈ (0...𝑛) ↦ ((𝑥 decompPMat 𝑙)(.r𝐴)(𝑦 decompPMat (𝑛𝑙))))
11929ad3antrrr 730 . . . . . . . . . . . . . 14 (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥𝐵𝑦𝐵)) ∧ 𝑛 ∈ ℕ0) ∧ 𝑙 ∈ (0...𝑛)) → 𝐴 ∈ Ring)
120 simp-5r 786 . . . . . . . . . . . . . . . 16 ((((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥𝐵𝑦𝐵)) ∧ 𝑛 ∈ ℕ0) ∧ 𝑙 ∈ (0...𝑛)) ∧ 𝑘 ∈ ℕ0) → 𝑅 ∈ Ring)
1218ad3antrrr 730 . . . . . . . . . . . . . . . 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 22773 . . . . . . . . . . . . . . . 16 ((𝑅 ∈ Ring ∧ 𝑥𝐵𝑘 ∈ ℕ0) → (𝑥 decompPMat 𝑘) ∈ (Base‘𝐴))
124120, 121, 122, 123syl3anc 1373 . . . . . . . . . . . . . . 15 ((((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥𝐵𝑦𝐵)) ∧ 𝑛 ∈ ℕ0) ∧ 𝑙 ∈ (0...𝑛)) ∧ 𝑘 ∈ ℕ0) → (𝑥 decompPMat 𝑘) ∈ (Base‘𝐴))
125124ralrimiva 3146 . . . . . . . . . . . . . 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 726 . . . . . . . . . . . . . . 15 (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥𝐵𝑦𝐵)) ∧ 𝑛 ∈ ℕ0) ∧ 𝑙 ∈ (0...𝑛)) → (𝑅 ∈ Ring ∧ 𝑥𝐵))
1283, 4, 11, 18, 32decpmatfsupp 22775 . . . . . . . . . . . . . . 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 13660 . . . . . . . . . . . . . . 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 22312 . . . . . . . . . . . . 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 7479 . . . . . . . . . . . . . . 15 (𝑙 ∈ (0...𝑛) → 𝑙 / 𝑘(𝑥 decompPMat 𝑘) = (𝑥 decompPMat 𝑙 / 𝑘𝑘))
134 csbvarg 4434 . . . . . . . . . . . . . . . 16 (𝑙 ∈ (0...𝑛) → 𝑙 / 𝑘𝑘 = 𝑙)
135134oveq2d 7447 . . . . . . . . . . . . . . 15 (𝑙 ∈ (0...𝑛) → (𝑥 decompPMat 𝑙 / 𝑘𝑘) = (𝑥 decompPMat 𝑙))
136133, 135eqtrd 2777 . . . . . . . . . . . . . 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 2778 . . . . . . . . . . . 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 730 . . . . . . . . . . . . . . . 16 ((((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥𝐵𝑦𝐵)) ∧ 𝑛 ∈ ℕ0) ∧ 𝑙 ∈ (0...𝑛)) ∧ 𝑘 ∈ ℕ0) → 𝑦𝐵)
1403, 4, 11, 18, 31decpmatcl 22773 . . . . . . . . . . . . . . . 16 ((𝑅 ∈ Ring ∧ 𝑦𝐵𝑘 ∈ ℕ0) → (𝑦 decompPMat 𝑘) ∈ (Base‘𝐴))
141120, 139, 122, 140syl3anc 1373 . . . . . . . . . . . . . . 15 ((((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥𝐵𝑦𝐵)) ∧ 𝑛 ∈ ℕ0) ∧ 𝑙 ∈ (0...𝑛)) ∧ 𝑘 ∈ ℕ0) → (𝑦 decompPMat 𝑘) ∈ (Base‘𝐴))
142141ralrimiva 3146 . . . . . . . . . . . . . 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 726 . . . . . . . . . . . . . . 15 (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥𝐵𝑦𝐵)) ∧ 𝑛 ∈ ℕ0) ∧ 𝑙 ∈ (0...𝑛)) → (𝑅 ∈ Ring ∧ 𝑦𝐵))
1453, 4, 11, 18, 32decpmatfsupp 22775 . . . . . . . . . . . . . . 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 13596 . . . . . . . . . . . . . . 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 22312 . . . . . . . . . . . . 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 7464 . . . . . . . . . . . . . 14 (𝑛𝑙) ∈ V
151 csbov2g 7479 . . . . . . . . . . . . . 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 4434 . . . . . . . . . . . . . . 15 ((𝑛𝑙) ∈ V → (𝑛𝑙) / 𝑘𝑘 = (𝑛𝑙))
154150, 153mp1i 13 . . . . . . . . . . . . . 14 (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥𝐵𝑦𝐵)) ∧ 𝑛 ∈ ℕ0) ∧ 𝑙 ∈ (0...𝑛)) → (𝑛𝑙) / 𝑘𝑘 = (𝑛𝑙))
155154oveq2d 7447 . . . . . . . . . . . . 13 (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥𝐵𝑦𝐵)) ∧ 𝑛 ∈ ℕ0) ∧ 𝑙 ∈ (0...𝑛)) → (𝑦 decompPMat (𝑛𝑙) / 𝑘𝑘) = (𝑦 decompPMat (𝑛𝑙)))
156149, 152, 1553eqtrrd 2782 . . . . . . . . . . . 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 7449 . . . . . . . . . . 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 5242 . . . . . . . . . 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 2789 . . . . . . . . 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 7447 . . . . . . . 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 2788 . . . . . . 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 2781 . . . . . 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 3146 . . . . 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 22253 . . . . . . . . . . 11 (𝐴 ∈ Ring → 𝑄 ∈ LMod)
16829, 167syl 17 . . . . . . . . . 10 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝑄 ∈ LMod)
169168ad2antrr 726 . . . . . . . . 9 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥𝐵𝑦𝐵)) ∧ 𝑘 ∈ ℕ0) → 𝑄 ∈ LMod)
17034ad2antrr 726 . . . . . . . . . . 11 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥𝐵𝑦𝐵)) ∧ 𝑘 ∈ ℕ0) → 𝐴 ∈ CMnd)
171 fzfid 14014 . . . . . . . . . . 11 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥𝐵𝑦𝐵)) ∧ 𝑘 ∈ ℕ0) → (0...𝑘) ∈ Fin)
17229ad3antrrr 730 . . . . . . . . . . . . 13 (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥𝐵𝑦𝐵)) ∧ 𝑘 ∈ ℕ0) ∧ 𝑧 ∈ (0...𝑘)) → 𝐴 ∈ Ring)
173 simp-4r 784 . . . . . . . . . . . . . 14 (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥𝐵𝑦𝐵)) ∧ 𝑘 ∈ ℕ0) ∧ 𝑧 ∈ (0...𝑘)) → 𝑅 ∈ Ring)
174 simplrl 777 . . . . . . . . . . . . . . 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 1373 . . . . . . . . . . . . 13 (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥𝐵𝑦𝐵)) ∧ 𝑘 ∈ ℕ0) ∧ 𝑧 ∈ (0...𝑘)) → (𝑥 decompPMat 𝑧) ∈ (Base‘𝐴))
178 simplrr 778 . . . . . . . . . . . . . . 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 1373 . . . . . . . . . . . . 13 (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥𝐵𝑦𝐵)) ∧ 𝑘 ∈ ℕ0) ∧ 𝑧 ∈ (0...𝑘)) → (𝑦 decompPMat (𝑘𝑧)) ∈ (Base‘𝐴))
182172, 177, 181, 50syl3anc 1373 . . . . . . . . . . . 12 (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥𝐵𝑦𝐵)) ∧ 𝑘 ∈ ℕ0) ∧ 𝑧 ∈ (0...𝑘)) → ((𝑥 decompPMat 𝑧)(.r𝐴)(𝑦 decompPMat (𝑘𝑧))) ∈ (Base‘𝐴))
183182ralrimiva 3146 . . . . . . . . . . 11 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥𝐵𝑦𝐵)) ∧ 𝑘 ∈ ℕ0) → ∀𝑧 ∈ (0...𝑘)((𝑥 decompPMat 𝑧)(.r𝐴)(𝑦 decompPMat (𝑘𝑧))) ∈ (Base‘𝐴))
18431, 170, 171, 183gsummptcl 19985 . . . . . . . . . 10 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥𝐵𝑦𝐵)) ∧ 𝑘 ∈ ℕ0) → (𝐴 Σg (𝑧 ∈ (0...𝑘) ↦ ((𝑥 decompPMat 𝑧)(.r𝐴)(𝑦 decompPMat (𝑘𝑧))))) ∈ (Base‘𝐴))
18529ad2antrr 726 . . . . . . . . . . . . 13 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥𝐵𝑦𝐵)) ∧ 𝑘 ∈ ℕ0) → 𝐴 ∈ Ring)
18619ply1sca 22254 . . . . . . . . . . . . 13 (𝐴 ∈ Ring → 𝐴 = (Scalar‘𝑄))
187185, 186syl 17 . . . . . . . . . . . 12 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥𝐵𝑦𝐵)) ∧ 𝑘 ∈ ℕ0) → 𝐴 = (Scalar‘𝑄))
188187eqcomd 2743 . . . . . . . . . . 11 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥𝐵𝑦𝐵)) ∧ 𝑘 ∈ ℕ0) → (Scalar‘𝑄) = 𝐴)
189188fveq2d 6910 . . . . . . . . . 10 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥𝐵𝑦𝐵)) ∧ 𝑘 ∈ ℕ0) → (Base‘(Scalar‘𝑄)) = (Base‘𝐴))
190184, 189eleqtrrd 2844 . . . . . . . . 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 2737 . . . . . . . . . . 11 (mulGrp‘𝑄) = (mulGrp‘𝑄)
19219, 17, 191, 16, 28ply1moncl 22274 . . . . . . . . . 10 ((𝐴 ∈ Ring ∧ 𝑘 ∈ ℕ0) → (𝑘(.g‘(mulGrp‘𝑄))(var1𝐴)) ∈ (Base‘𝑄))
193185, 192sylancom 588 . . . . . . . . 9 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥𝐵𝑦𝐵)) ∧ 𝑘 ∈ ℕ0) → (𝑘(.g‘(mulGrp‘𝑄))(var1𝐴)) ∈ (Base‘𝑄))
194 eqid 2737 . . . . . . . . . 10 (Scalar‘𝑄) = (Scalar‘𝑄)
195 eqid 2737 . . . . . . . . . 10 (Base‘(Scalar‘𝑄)) = (Base‘(Scalar‘𝑄))
19628, 194, 15, 195lmodvscl 20876 . . . . . . . . 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 1373 . . . . . . . 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 7135 . . . . . . 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 22824 . . . . . . 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 19933 . . . . . 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 580 . . . . . . . . 9 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥𝐵𝑦𝐵)) ∧ 𝑘 ∈ ℕ0) → ((𝑥 decompPMat 𝑘)( ·𝑠𝑄)(𝑘(.g‘(mulGrp‘𝑄))(var1𝐴))) ∈ (Base‘𝑄))
203202fmpttd 7135 . . . . . . . 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 19933 . . . . . . 7 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥𝐵𝑦𝐵)) → (𝑄 Σg (𝑘 ∈ ℕ0 ↦ ((𝑥 decompPMat 𝑘)( ·𝑠𝑄)(𝑘(.g‘(mulGrp‘𝑄))(var1𝐴))))) ∈ (Base‘𝑄))
206100, 102sylan 580 . . . . . . . . 9 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥𝐵𝑦𝐵)) ∧ 𝑘 ∈ ℕ0) → ((𝑦 decompPMat 𝑘)( ·𝑠𝑄)(𝑘(.g‘(mulGrp‘𝑄))(var1𝐴))) ∈ (Base‘𝑄))
207206fmpttd 7135 . . . . . . . 8 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥𝐵𝑦𝐵)) → (𝑘 ∈ ℕ0 ↦ ((𝑦 decompPMat 𝑘)( ·𝑠𝑄)(𝑘(.g‘(mulGrp‘𝑄))(var1𝐴)))):ℕ0⟶(Base‘𝑄))
2081, 2, 10, 107syl3anc 1373 . . . . . . . 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 19933 . . . . . . 7 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥𝐵𝑦𝐵)) → (𝑄 Σg (𝑘 ∈ ℕ0 ↦ ((𝑦 decompPMat 𝑘)( ·𝑠𝑄)(𝑘(.g‘(mulGrp‘𝑄))(var1𝐴))))) ∈ (Base‘𝑄))
21028, 110ringcl 20247 . . . . . . 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 1373 . . . . . 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 2737 . . . . . . 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 2737 . . . . . . 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 22304 . . . . . 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 1373 . . . . 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 232 . . . 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 2781 . . 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 22802 . . . . 5 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑥𝐵) → (𝑇𝑥) = (𝑄 Σg (𝑘 ∈ ℕ0 ↦ ((𝑥 decompPMat 𝑘)( ·𝑠𝑄)(𝑘(.g‘(mulGrp‘𝑄))(var1𝐴))))))
2191, 2, 8, 218syl3anc 1373 . . . 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 22802 . . . . 5 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑦𝐵) → (𝑇𝑦) = (𝑄 Σg (𝑘 ∈ ℕ0 ↦ ((𝑦 decompPMat 𝑘)( ·𝑠𝑄)(𝑘(.g‘(mulGrp‘𝑄))(var1𝐴))))))
2211, 2, 10, 220syl3anc 1373 . . . 4 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥𝐵𝑦𝐵)) → (𝑇𝑦) = (𝑄 Σg (𝑘 ∈ ℕ0 ↦ ((𝑦 decompPMat 𝑘)( ·𝑠𝑄)(𝑘(.g‘(mulGrp‘𝑄))(var1𝐴))))))
222219, 221oveq12d 7449 . . 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 2780 . 2 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥𝐵𝑦𝐵)) → (𝑇‘(𝑥(.r𝐶)𝑦)) = ((𝑇𝑥)(.r𝑄)(𝑇𝑦)))
224223ralrimivva 3202 1 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → ∀𝑥𝐵𝑦𝐵 (𝑇‘(𝑥(.r𝐶)𝑦)) = ((𝑇𝑥)(.r𝑄)(𝑇𝑦)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395  w3a 1087   = wceq 1540  wcel 2108  wral 3061  Vcvv 3480  csb 3899   class class class wbr 5143  cmpt 5225  cfv 6561  (class class class)co 7431  Fincfn 8985   finSupp cfsupp 9401  0cc0 11155  cmin 11492  0cn0 12526  ...cfz 13547  Basecbs 17247  .rcmulr 17298  Scalarcsca 17300   ·𝑠 cvsca 17301  0gc0g 17484   Σg cgsu 17485  .gcmg 19085  CMndccmn 19798  mulGrpcmgp 20137  Ringcrg 20230  LModclmod 20858  var1cv1 22177  Poly1cpl1 22178  coe1cco1 22179   Mat cmat 22411   decompPMat cdecpmat 22768   pMatToMatPoly cpm2mp 22798
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755  ax-cnex 11211  ax-resscn 11212  ax-1cn 11213  ax-icn 11214  ax-addcl 11215  ax-addrcl 11216  ax-mulcl 11217  ax-mulrcl 11218  ax-mulcom 11219  ax-addass 11220  ax-mulass 11221  ax-distr 11222  ax-i2m1 11223  ax-1ne0 11224  ax-1rid 11225  ax-rnegex 11226  ax-rrecex 11227  ax-cnre 11228  ax-pre-lttri 11229  ax-pre-lttrn 11230  ax-pre-ltadd 11231  ax-pre-mulgt0 11232
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-rmo 3380  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-tp 4631  df-op 4633  df-ot 4635  df-uni 4908  df-int 4947  df-iun 4993  df-iin 4994  df-br 5144  df-opab 5206  df-mpt 5226  df-tr 5260  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-se 5638  df-we 5639  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-pred 6321  df-ord 6387  df-on 6388  df-lim 6389  df-suc 6390  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-isom 6570  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-of 7697  df-ofr 7698  df-om 7888  df-1st 8014  df-2nd 8015  df-supp 8186  df-frecs 8306  df-wrecs 8337  df-recs 8411  df-rdg 8450  df-1o 8506  df-2o 8507  df-er 8745  df-map 8868  df-pm 8869  df-ixp 8938  df-en 8986  df-dom 8987  df-sdom 8988  df-fin 8989  df-fsupp 9402  df-sup 9482  df-oi 9550  df-card 9979  df-pnf 11297  df-mnf 11298  df-xr 11299  df-ltxr 11300  df-le 11301  df-sub 11494  df-neg 11495  df-nn 12267  df-2 12329  df-3 12330  df-4 12331  df-5 12332  df-6 12333  df-7 12334  df-8 12335  df-9 12336  df-n0 12527  df-z 12614  df-dec 12734  df-uz 12879  df-fz 13548  df-fzo 13695  df-seq 14043  df-hash 14370  df-struct 17184  df-sets 17201  df-slot 17219  df-ndx 17231  df-base 17248  df-ress 17275  df-plusg 17310  df-mulr 17311  df-sca 17313  df-vsca 17314  df-ip 17315  df-tset 17316  df-ple 17317  df-ds 17319  df-hom 17321  df-cco 17322  df-0g 17486  df-gsum 17487  df-prds 17492  df-pws 17494  df-mre 17629  df-mrc 17630  df-acs 17632  df-mgm 18653  df-sgrp 18732  df-mnd 18748  df-mhm 18796  df-submnd 18797  df-grp 18954  df-minusg 18955  df-sbg 18956  df-mulg 19086  df-subg 19141  df-ghm 19231  df-cntz 19335  df-cmn 19800  df-abl 19801  df-mgp 20138  df-rng 20150  df-ur 20179  df-srg 20184  df-ring 20232  df-subrng 20546  df-subrg 20570  df-lmod 20860  df-lss 20930  df-sra 21172  df-rgmod 21173  df-dsmm 21752  df-frlm 21767  df-psr 21929  df-mvr 21930  df-mpl 21931  df-opsr 21933  df-psr1 22181  df-vr1 22182  df-ply1 22183  df-coe1 22184  df-mamu 22395  df-mat 22412  df-decpmat 22769  df-pm2mp 22799
This theorem is referenced by:  pm2mpmhm  22826
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