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Theorem pm2mpmhmlem2 20818
Description: Lemma 2 for pm2mpmhm 20819. (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 807 . . . . 5 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥𝐵𝑦𝐵)) → 𝑁 ∈ Fin)
2 simplr 809 . . . . 5 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥𝐵𝑦𝐵)) → 𝑅 ∈ Ring)
3 pm2mpmhm.p . . . . . . . 8 𝑃 = (Poly1𝑅)
4 pm2mpmhm.c . . . . . . . 8 𝐶 = (𝑁 Mat 𝑃)
53, 4pmatring 20692 . . . . . . 7 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝐶 ∈ Ring)
65adantr 472 . . . . . 6 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥𝐵𝑦𝐵)) → 𝐶 ∈ Ring)
7 simpl 474 . . . . . . 7 ((𝑥𝐵𝑦𝐵) → 𝑥𝐵)
87adantl 473 . . . . . 6 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥𝐵𝑦𝐵)) → 𝑥𝐵)
9 simpr 479 . . . . . . 7 ((𝑥𝐵𝑦𝐵) → 𝑦𝐵)
109adantl 473 . . . . . 6 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥𝐵𝑦𝐵)) → 𝑦𝐵)
11 pm2mpmhm.b . . . . . . 7 𝐵 = (Base‘𝐶)
12 eqid 2752 . . . . . . 7 (.r𝐶) = (.r𝐶)
1311, 12ringcl 18753 . . . . . 6 ((𝐶 ∈ Ring ∧ 𝑥𝐵𝑦𝐵) → (𝑥(.r𝐶)𝑦) ∈ 𝐵)
146, 8, 10, 13syl3anc 1473 . . . . 5 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥𝐵𝑦𝐵)) → (𝑥(.r𝐶)𝑦) ∈ 𝐵)
15 eqid 2752 . . . . . 6 ( ·𝑠𝑄) = ( ·𝑠𝑄)
16 eqid 2752 . . . . . 6 (.g‘(mulGrp‘𝑄)) = (.g‘(mulGrp‘𝑄))
17 eqid 2752 . . . . . 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 20795 . . . . 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 1473 . . . 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))))))
23 simpllr 817 . . . . . . . 8 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥𝐵𝑦𝐵)) ∧ 𝑘 ∈ ℕ0) → 𝑅 ∈ Ring)
24 simplr 809 . . . . . . . 8 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥𝐵𝑦𝐵)) ∧ 𝑘 ∈ ℕ0) → (𝑥𝐵𝑦𝐵))
25 simpr 479 . . . . . . . 8 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥𝐵𝑦𝐵)) ∧ 𝑘 ∈ ℕ0) → 𝑘 ∈ ℕ0)
263, 4, 11, 18decpmatmul 20771 . . . . . . . 8 ((𝑅 ∈ Ring ∧ (𝑥𝐵𝑦𝐵) ∧ 𝑘 ∈ ℕ0) → ((𝑥(.r𝐶)𝑦) decompPMat 𝑘) = (𝐴 Σg (𝑧 ∈ (0...𝑘) ↦ ((𝑥 decompPMat 𝑧)(.r𝐴)(𝑦 decompPMat (𝑘𝑧))))))
2723, 24, 25, 26syl3anc 1473 . . . . . . 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))))))
2827oveq1d 6820 . . . . . 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))))
2928mpteq2dva 4888 . . . . 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)))))
3029oveq2d 6821 . . . 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))))))
31 eqid 2752 . . . . . . . 8 (Base‘𝑄) = (Base‘𝑄)
3218matring 20443 . . . . . . . . 9 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝐴 ∈ Ring)
3332ad2antrr 764 . . . . . . . 8 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥𝐵𝑦𝐵)) ∧ 𝑛 ∈ ℕ0) → 𝐴 ∈ Ring)
34 eqid 2752 . . . . . . . 8 (Base‘𝐴) = (Base‘𝐴)
35 eqid 2752 . . . . . . . 8 (0g𝐴) = (0g𝐴)
36 ringcmn 18773 . . . . . . . . . . . 12 (𝐴 ∈ Ring → 𝐴 ∈ CMnd)
3732, 36syl 17 . . . . . . . . . . 11 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝐴 ∈ CMnd)
3837ad3antrrr 768 . . . . . . . . . 10 (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥𝐵𝑦𝐵)) ∧ 𝑛 ∈ ℕ0) ∧ 𝑘 ∈ ℕ0) → 𝐴 ∈ CMnd)
39 fzfid 12958 . . . . . . . . . 10 (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥𝐵𝑦𝐵)) ∧ 𝑛 ∈ ℕ0) ∧ 𝑘 ∈ ℕ0) → (0...𝑘) ∈ Fin)
4033ad2antrr 764 . . . . . . . . . . . 12 ((((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥𝐵𝑦𝐵)) ∧ 𝑛 ∈ ℕ0) ∧ 𝑘 ∈ ℕ0) ∧ 𝑧 ∈ (0...𝑘)) → 𝐴 ∈ Ring)
41 simp-5r 831 . . . . . . . . . . . . 13 ((((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥𝐵𝑦𝐵)) ∧ 𝑛 ∈ ℕ0) ∧ 𝑘 ∈ ℕ0) ∧ 𝑧 ∈ (0...𝑘)) → 𝑅 ∈ Ring)
428ad3antrrr 768 . . . . . . . . . . . . 13 ((((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥𝐵𝑦𝐵)) ∧ 𝑛 ∈ ℕ0) ∧ 𝑘 ∈ ℕ0) ∧ 𝑧 ∈ (0...𝑘)) → 𝑥𝐵)
43 elfznn0 12618 . . . . . . . . . . . . . 14 (𝑧 ∈ (0...𝑘) → 𝑧 ∈ ℕ0)
4443adantl 473 . . . . . . . . . . . . 13 ((((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥𝐵𝑦𝐵)) ∧ 𝑛 ∈ ℕ0) ∧ 𝑘 ∈ ℕ0) ∧ 𝑧 ∈ (0...𝑘)) → 𝑧 ∈ ℕ0)
453, 4, 11, 18, 34decpmatcl 20766 . . . . . . . . . . . . 13 ((𝑅 ∈ Ring ∧ 𝑥𝐵𝑧 ∈ ℕ0) → (𝑥 decompPMat 𝑧) ∈ (Base‘𝐴))
4641, 42, 44, 45syl3anc 1473 . . . . . . . . . . . 12 ((((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥𝐵𝑦𝐵)) ∧ 𝑛 ∈ ℕ0) ∧ 𝑘 ∈ ℕ0) ∧ 𝑧 ∈ (0...𝑘)) → (𝑥 decompPMat 𝑧) ∈ (Base‘𝐴))
4710ad3antrrr 768 . . . . . . . . . . . . 13 ((((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥𝐵𝑦𝐵)) ∧ 𝑛 ∈ ℕ0) ∧ 𝑘 ∈ ℕ0) ∧ 𝑧 ∈ (0...𝑘)) → 𝑦𝐵)
48 fznn0sub 12558 . . . . . . . . . . . . . 14 (𝑧 ∈ (0...𝑘) → (𝑘𝑧) ∈ ℕ0)
4948adantl 473 . . . . . . . . . . . . 13 ((((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥𝐵𝑦𝐵)) ∧ 𝑛 ∈ ℕ0) ∧ 𝑘 ∈ ℕ0) ∧ 𝑧 ∈ (0...𝑘)) → (𝑘𝑧) ∈ ℕ0)
503, 4, 11, 18, 34decpmatcl 20766 . . . . . . . . . . . . 13 ((𝑅 ∈ Ring ∧ 𝑦𝐵 ∧ (𝑘𝑧) ∈ ℕ0) → (𝑦 decompPMat (𝑘𝑧)) ∈ (Base‘𝐴))
5141, 47, 49, 50syl3anc 1473 . . . . . . . . . . . 12 ((((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥𝐵𝑦𝐵)) ∧ 𝑛 ∈ ℕ0) ∧ 𝑘 ∈ ℕ0) ∧ 𝑧 ∈ (0...𝑘)) → (𝑦 decompPMat (𝑘𝑧)) ∈ (Base‘𝐴))
52 eqid 2752 . . . . . . . . . . . . 13 (.r𝐴) = (.r𝐴)
5334, 52ringcl 18753 . . . . . . . . . . . 12 ((𝐴 ∈ Ring ∧ (𝑥 decompPMat 𝑧) ∈ (Base‘𝐴) ∧ (𝑦 decompPMat (𝑘𝑧)) ∈ (Base‘𝐴)) → ((𝑥 decompPMat 𝑧)(.r𝐴)(𝑦 decompPMat (𝑘𝑧))) ∈ (Base‘𝐴))
5440, 46, 51, 53syl3anc 1473 . . . . . . . . . . 11 ((((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥𝐵𝑦𝐵)) ∧ 𝑛 ∈ ℕ0) ∧ 𝑘 ∈ ℕ0) ∧ 𝑧 ∈ (0...𝑘)) → ((𝑥 decompPMat 𝑧)(.r𝐴)(𝑦 decompPMat (𝑘𝑧))) ∈ (Base‘𝐴))
5554ralrimiva 3096 . . . . . . . . . 10 (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥𝐵𝑦𝐵)) ∧ 𝑛 ∈ ℕ0) ∧ 𝑘 ∈ ℕ0) → ∀𝑧 ∈ (0...𝑘)((𝑥 decompPMat 𝑧)(.r𝐴)(𝑦 decompPMat (𝑘𝑧))) ∈ (Base‘𝐴))
5634, 38, 39, 55gsummptcl 18558 . . . . . . . . 9 (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥𝐵𝑦𝐵)) ∧ 𝑛 ∈ ℕ0) ∧ 𝑘 ∈ ℕ0) → (𝐴 Σg (𝑧 ∈ (0...𝑘) ↦ ((𝑥 decompPMat 𝑧)(.r𝐴)(𝑦 decompPMat (𝑘𝑧))))) ∈ (Base‘𝐴))
5756ralrimiva 3096 . . . . . . . 8 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥𝐵𝑦𝐵)) ∧ 𝑛 ∈ ℕ0) → ∀𝑘 ∈ ℕ0 (𝐴 Σg (𝑧 ∈ (0...𝑘) ↦ ((𝑥 decompPMat 𝑧)(.r𝐴)(𝑦 decompPMat (𝑘𝑧))))) ∈ (Base‘𝐴))
583, 4, 11, 18, 52, 35decpmatmulsumfsupp 20772 . . . . . . . . 9 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥𝐵𝑦𝐵)) → (𝑘 ∈ ℕ0 ↦ (𝐴 Σg (𝑧 ∈ (0...𝑘) ↦ ((𝑥 decompPMat 𝑧)(.r𝐴)(𝑦 decompPMat (𝑘𝑧)))))) finSupp (0g𝐴))
5958adantr 472 . . . . . . . 8 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥𝐵𝑦𝐵)) ∧ 𝑛 ∈ ℕ0) → (𝑘 ∈ ℕ0 ↦ (𝐴 Σg (𝑧 ∈ (0...𝑘) ↦ ((𝑥 decompPMat 𝑧)(.r𝐴)(𝑦 decompPMat (𝑘𝑧)))))) finSupp (0g𝐴))
60 simpr 479 . . . . . . . 8 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥𝐵𝑦𝐵)) ∧ 𝑛 ∈ ℕ0) → 𝑛 ∈ ℕ0)
6119, 31, 17, 16, 33, 34, 15, 35, 57, 59, 60gsummoncoe1 19868 . . . . . . 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))))))
62 csbov2g 6846 . . . . . . . . 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))))))
63 id 22 . . . . . . . . . . 11 (𝑛 ∈ ℕ0𝑛 ∈ ℕ0)
64 oveq2 6813 . . . . . . . . . . . . 13 (𝑘 = 𝑛 → (0...𝑘) = (0...𝑛))
65 oveq1 6812 . . . . . . . . . . . . . . 15 (𝑘 = 𝑛 → (𝑘𝑧) = (𝑛𝑧))
6665oveq2d 6821 . . . . . . . . . . . . . 14 (𝑘 = 𝑛 → (𝑦 decompPMat (𝑘𝑧)) = (𝑦 decompPMat (𝑛𝑧)))
6766oveq2d 6821 . . . . . . . . . . . . 13 (𝑘 = 𝑛 → ((𝑥 decompPMat 𝑧)(.r𝐴)(𝑦 decompPMat (𝑘𝑧))) = ((𝑥 decompPMat 𝑧)(.r𝐴)(𝑦 decompPMat (𝑛𝑧))))
6864, 67mpteq12dv 4877 . . . . . . . . . . . 12 (𝑘 = 𝑛 → (𝑧 ∈ (0...𝑘) ↦ ((𝑥 decompPMat 𝑧)(.r𝐴)(𝑦 decompPMat (𝑘𝑧)))) = (𝑧 ∈ (0...𝑛) ↦ ((𝑥 decompPMat 𝑧)(.r𝐴)(𝑦 decompPMat (𝑛𝑧)))))
6968adantl 473 . . . . . . . . . . 11 ((𝑛 ∈ ℕ0𝑘 = 𝑛) → (𝑧 ∈ (0...𝑘) ↦ ((𝑥 decompPMat 𝑧)(.r𝐴)(𝑦 decompPMat (𝑘𝑧)))) = (𝑧 ∈ (0...𝑛) ↦ ((𝑥 decompPMat 𝑧)(.r𝐴)(𝑦 decompPMat (𝑛𝑧)))))
7063, 69csbied 3693 . . . . . . . . . 10 (𝑛 ∈ ℕ0𝑛 / 𝑘(𝑧 ∈ (0...𝑘) ↦ ((𝑥 decompPMat 𝑧)(.r𝐴)(𝑦 decompPMat (𝑘𝑧)))) = (𝑧 ∈ (0...𝑛) ↦ ((𝑥 decompPMat 𝑧)(.r𝐴)(𝑦 decompPMat (𝑛𝑧)))))
7170oveq2d 6821 . . . . . . . . 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))))))
7262, 71eqtrd 2786 . . . . . . . 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))))))
7372adantl 473 . . . . . . 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))))))
74 eqidd 2753 . . . . . . . . 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)))))))
75 oveq2 6813 . . . . . . . . . . . 12 (𝑟 = 𝑛 → (0...𝑟) = (0...𝑛))
76 oveq1 6812 . . . . . . . . . . . . . 14 (𝑟 = 𝑛 → (𝑟𝑙) = (𝑛𝑙))
7776fveq2d 6348 . . . . . . . . . . . . 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)))
7877oveq2d 6821 . . . . . . . . . . . 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))))
7975, 78mpteq12dv 4877 . . . . . . . . . . 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)))))
8079oveq2d 6821 . . . . . . . . . 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))))))
8180adantl 473 . . . . . . . . 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))))))
82 ovexd 6835 . . . . . . . . 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)
8374, 81, 60, 82fvmptd 6442 . . . . . . . 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))))))
84 eqid 2752 . . . . . . . . . 10 (0g𝑄) = (0g𝑄)
8519ply1ring 19812 . . . . . . . . . . . . 13 (𝐴 ∈ Ring → 𝑄 ∈ Ring)
8632, 85syl 17 . . . . . . . . . . . 12 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝑄 ∈ Ring)
87 ringcmn 18773 . . . . . . . . . . . 12 (𝑄 ∈ Ring → 𝑄 ∈ CMnd)
8886, 87syl 17 . . . . . . . . . . 11 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝑄 ∈ CMnd)
8988ad2antrr 764 . . . . . . . . . 10 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥𝐵𝑦𝐵)) ∧ 𝑛 ∈ ℕ0) → 𝑄 ∈ CMnd)
90 nn0ex 11482 . . . . . . . . . . 11 0 ∈ V
9190a1i 11 . . . . . . . . . 10 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥𝐵𝑦𝐵)) ∧ 𝑛 ∈ ℕ0) → ℕ0 ∈ V)
927anim2i 594 . . . . . . . . . . . . . 14 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥𝐵𝑦𝐵)) → ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝑥𝐵))
93 df-3an 1074 . . . . . . . . . . . . . 14 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑥𝐵) ↔ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝑥𝐵))
9492, 93sylibr 224 . . . . . . . . . . . . 13 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥𝐵𝑦𝐵)) → (𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑥𝐵))
9594adantr 472 . . . . . . . . . . . 12 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥𝐵𝑦𝐵)) ∧ 𝑛 ∈ ℕ0) → (𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑥𝐵))
963, 4, 11, 15, 16, 17, 18, 19, 31pm2mpghmlem1 20812 . . . . . . . . . . . 12 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑥𝐵) ∧ 𝑘 ∈ ℕ0) → ((𝑥 decompPMat 𝑘)( ·𝑠𝑄)(𝑘(.g‘(mulGrp‘𝑄))(var1𝐴))) ∈ (Base‘𝑄))
9795, 96sylan 489 . . . . . . . . . . 11 (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥𝐵𝑦𝐵)) ∧ 𝑛 ∈ ℕ0) ∧ 𝑘 ∈ ℕ0) → ((𝑥 decompPMat 𝑘)( ·𝑠𝑄)(𝑘(.g‘(mulGrp‘𝑄))(var1𝐴))) ∈ (Base‘𝑄))
98 eqid 2752 . . . . . . . . . . 11 (𝑘 ∈ ℕ0 ↦ ((𝑥 decompPMat 𝑘)( ·𝑠𝑄)(𝑘(.g‘(mulGrp‘𝑄))(var1𝐴)))) = (𝑘 ∈ ℕ0 ↦ ((𝑥 decompPMat 𝑘)( ·𝑠𝑄)(𝑘(.g‘(mulGrp‘𝑄))(var1𝐴))))
9997, 98fmptd 6540 . . . . . . . . . 10 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥𝐵𝑦𝐵)) ∧ 𝑛 ∈ ℕ0) → (𝑘 ∈ ℕ0 ↦ ((𝑥 decompPMat 𝑘)( ·𝑠𝑄)(𝑘(.g‘(mulGrp‘𝑄))(var1𝐴)))):ℕ0⟶(Base‘𝑄))
1003, 4, 11, 15, 16, 17, 18, 19pm2mpghmlem2 20811 . . . . . . . . . . 11 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑥𝐵) → (𝑘 ∈ ℕ0 ↦ ((𝑥 decompPMat 𝑘)( ·𝑠𝑄)(𝑘(.g‘(mulGrp‘𝑄))(var1𝐴)))) finSupp (0g𝑄))
10195, 100syl 17 . . . . . . . . . 10 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥𝐵𝑦𝐵)) ∧ 𝑛 ∈ ℕ0) → (𝑘 ∈ ℕ0 ↦ ((𝑥 decompPMat 𝑘)( ·𝑠𝑄)(𝑘(.g‘(mulGrp‘𝑄))(var1𝐴)))) finSupp (0g𝑄))
10231, 84, 89, 91, 99, 101gsumcl 18508 . . . . . . . . 9 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥𝐵𝑦𝐵)) ∧ 𝑛 ∈ ℕ0) → (𝑄 Σg (𝑘 ∈ ℕ0 ↦ ((𝑥 decompPMat 𝑘)( ·𝑠𝑄)(𝑘(.g‘(mulGrp‘𝑄))(var1𝐴))))) ∈ (Base‘𝑄))
1039anim2i 594 . . . . . . . . . . . . . 14 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥𝐵𝑦𝐵)) → ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝑦𝐵))
104 df-3an 1074 . . . . . . . . . . . . . 14 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑦𝐵) ↔ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝑦𝐵))
105103, 104sylibr 224 . . . . . . . . . . . . 13 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥𝐵𝑦𝐵)) → (𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑦𝐵))
106105adantr 472 . . . . . . . . . . . 12 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥𝐵𝑦𝐵)) ∧ 𝑛 ∈ ℕ0) → (𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑦𝐵))
1073, 4, 11, 15, 16, 17, 18, 19, 31pm2mpghmlem1 20812 . . . . . . . . . . . 12 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑦𝐵) ∧ 𝑘 ∈ ℕ0) → ((𝑦 decompPMat 𝑘)( ·𝑠𝑄)(𝑘(.g‘(mulGrp‘𝑄))(var1𝐴))) ∈ (Base‘𝑄))
108106, 107sylan 489 . . . . . . . . . . 11 (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥𝐵𝑦𝐵)) ∧ 𝑛 ∈ ℕ0) ∧ 𝑘 ∈ ℕ0) → ((𝑦 decompPMat 𝑘)( ·𝑠𝑄)(𝑘(.g‘(mulGrp‘𝑄))(var1𝐴))) ∈ (Base‘𝑄))
109 eqid 2752 . . . . . . . . . . 11 (𝑘 ∈ ℕ0 ↦ ((𝑦 decompPMat 𝑘)( ·𝑠𝑄)(𝑘(.g‘(mulGrp‘𝑄))(var1𝐴)))) = (𝑘 ∈ ℕ0 ↦ ((𝑦 decompPMat 𝑘)( ·𝑠𝑄)(𝑘(.g‘(mulGrp‘𝑄))(var1𝐴))))
110108, 109fmptd 6540 . . . . . . . . . 10 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥𝐵𝑦𝐵)) ∧ 𝑛 ∈ ℕ0) → (𝑘 ∈ ℕ0 ↦ ((𝑦 decompPMat 𝑘)( ·𝑠𝑄)(𝑘(.g‘(mulGrp‘𝑄))(var1𝐴)))):ℕ0⟶(Base‘𝑄))
1111, 2, 103jca 1122 . . . . . . . . . . . 12 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥𝐵𝑦𝐵)) → (𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑦𝐵))
112111adantr 472 . . . . . . . . . . 11 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥𝐵𝑦𝐵)) ∧ 𝑛 ∈ ℕ0) → (𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑦𝐵))
1133, 4, 11, 15, 16, 17, 18, 19pm2mpghmlem2 20811 . . . . . . . . . . 11 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑦𝐵) → (𝑘 ∈ ℕ0 ↦ ((𝑦 decompPMat 𝑘)( ·𝑠𝑄)(𝑘(.g‘(mulGrp‘𝑄))(var1𝐴)))) finSupp (0g𝑄))
114112, 113syl 17 . . . . . . . . . 10 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥𝐵𝑦𝐵)) ∧ 𝑛 ∈ ℕ0) → (𝑘 ∈ ℕ0 ↦ ((𝑦 decompPMat 𝑘)( ·𝑠𝑄)(𝑘(.g‘(mulGrp‘𝑄))(var1𝐴)))) finSupp (0g𝑄))
11531, 84, 89, 91, 110, 114gsumcl 18508 . . . . . . . . 9 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥𝐵𝑦𝐵)) ∧ 𝑛 ∈ ℕ0) → (𝑄 Σg (𝑘 ∈ ℕ0 ↦ ((𝑦 decompPMat 𝑘)( ·𝑠𝑄)(𝑘(.g‘(mulGrp‘𝑄))(var1𝐴))))) ∈ (Base‘𝑄))
116 eqid 2752 . . . . . . . . . . 11 (.r𝑄) = (.r𝑄)
11719, 116, 52, 31coe1mul 19834 . . . . . . . . . 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)))))))
118117fveq1d 6346 . . . . . . . . 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))
11933, 102, 115, 118syl3anc 1473 . . . . . . . 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))
120 oveq2 6813 . . . . . . . . . . . 12 (𝑧 = 𝑙 → (𝑥 decompPMat 𝑧) = (𝑥 decompPMat 𝑙))
121 oveq2 6813 . . . . . . . . . . . . 13 (𝑧 = 𝑙 → (𝑛𝑧) = (𝑛𝑙))
122121oveq2d 6821 . . . . . . . . . . . 12 (𝑧 = 𝑙 → (𝑦 decompPMat (𝑛𝑧)) = (𝑦 decompPMat (𝑛𝑙)))
123120, 122oveq12d 6823 . . . . . . . . . . 11 (𝑧 = 𝑙 → ((𝑥 decompPMat 𝑧)(.r𝐴)(𝑦 decompPMat (𝑛𝑧))) = ((𝑥 decompPMat 𝑙)(.r𝐴)(𝑦 decompPMat (𝑛𝑙))))
124123cbvmptv 4894 . . . . . . . . . 10 (𝑧 ∈ (0...𝑛) ↦ ((𝑥 decompPMat 𝑧)(.r𝐴)(𝑦 decompPMat (𝑛𝑧)))) = (𝑙 ∈ (0...𝑛) ↦ ((𝑥 decompPMat 𝑙)(.r𝐴)(𝑦 decompPMat (𝑛𝑙))))
12532ad3antrrr 768 . . . . . . . . . . . . . 14 (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥𝐵𝑦𝐵)) ∧ 𝑛 ∈ ℕ0) ∧ 𝑙 ∈ (0...𝑛)) → 𝐴 ∈ Ring)
126 simp-5r 831 . . . . . . . . . . . . . . . 16 ((((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥𝐵𝑦𝐵)) ∧ 𝑛 ∈ ℕ0) ∧ 𝑙 ∈ (0...𝑛)) ∧ 𝑘 ∈ ℕ0) → 𝑅 ∈ Ring)
1278ad3antrrr 768 . . . . . . . . . . . . . . . 16 ((((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥𝐵𝑦𝐵)) ∧ 𝑛 ∈ ℕ0) ∧ 𝑙 ∈ (0...𝑛)) ∧ 𝑘 ∈ ℕ0) → 𝑥𝐵)
128 simpr 479 . . . . . . . . . . . . . . . 16 ((((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥𝐵𝑦𝐵)) ∧ 𝑛 ∈ ℕ0) ∧ 𝑙 ∈ (0...𝑛)) ∧ 𝑘 ∈ ℕ0) → 𝑘 ∈ ℕ0)
1293, 4, 11, 18, 34decpmatcl 20766 . . . . . . . . . . . . . . . 16 ((𝑅 ∈ Ring ∧ 𝑥𝐵𝑘 ∈ ℕ0) → (𝑥 decompPMat 𝑘) ∈ (Base‘𝐴))
130126, 127, 128, 129syl3anc 1473 . . . . . . . . . . . . . . 15 ((((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥𝐵𝑦𝐵)) ∧ 𝑛 ∈ ℕ0) ∧ 𝑙 ∈ (0...𝑛)) ∧ 𝑘 ∈ ℕ0) → (𝑥 decompPMat 𝑘) ∈ (Base‘𝐴))
131130ralrimiva 3096 . . . . . . . . . . . . . 14 (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥𝐵𝑦𝐵)) ∧ 𝑛 ∈ ℕ0) ∧ 𝑙 ∈ (0...𝑛)) → ∀𝑘 ∈ ℕ0 (𝑥 decompPMat 𝑘) ∈ (Base‘𝐴))
1322, 8jca 555 . . . . . . . . . . . . . . . 16 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥𝐵𝑦𝐵)) → (𝑅 ∈ Ring ∧ 𝑥𝐵))
133132ad2antrr 764 . . . . . . . . . . . . . . 15 (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥𝐵𝑦𝐵)) ∧ 𝑛 ∈ ℕ0) ∧ 𝑙 ∈ (0...𝑛)) → (𝑅 ∈ Ring ∧ 𝑥𝐵))
1343, 4, 11, 18, 35decpmatfsupp 20768 . . . . . . . . . . . . . . 15 ((𝑅 ∈ Ring ∧ 𝑥𝐵) → (𝑘 ∈ ℕ0 ↦ (𝑥 decompPMat 𝑘)) finSupp (0g𝐴))
135133, 134syl 17 . . . . . . . . . . . . . 14 (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥𝐵𝑦𝐵)) ∧ 𝑛 ∈ ℕ0) ∧ 𝑙 ∈ (0...𝑛)) → (𝑘 ∈ ℕ0 ↦ (𝑥 decompPMat 𝑘)) finSupp (0g𝐴))
136 elfznn0 12618 . . . . . . . . . . . . . . 15 (𝑙 ∈ (0...𝑛) → 𝑙 ∈ ℕ0)
137136adantl 473 . . . . . . . . . . . . . 14 (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥𝐵𝑦𝐵)) ∧ 𝑛 ∈ ℕ0) ∧ 𝑙 ∈ (0...𝑛)) → 𝑙 ∈ ℕ0)
13819, 31, 17, 16, 125, 34, 15, 35, 131, 135, 137gsummoncoe1 19868 . . . . . . . . . . . . 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))
139 csbov2g 6846 . . . . . . . . . . . . . . 15 (𝑙 ∈ (0...𝑛) → 𝑙 / 𝑘(𝑥 decompPMat 𝑘) = (𝑥 decompPMat 𝑙 / 𝑘𝑘))
140 csbvarg 4138 . . . . . . . . . . . . . . . 16 (𝑙 ∈ (0...𝑛) → 𝑙 / 𝑘𝑘 = 𝑙)
141140oveq2d 6821 . . . . . . . . . . . . . . 15 (𝑙 ∈ (0...𝑛) → (𝑥 decompPMat 𝑙 / 𝑘𝑘) = (𝑥 decompPMat 𝑙))
142139, 141eqtrd 2786 . . . . . . . . . . . . . 14 (𝑙 ∈ (0...𝑛) → 𝑙 / 𝑘(𝑥 decompPMat 𝑘) = (𝑥 decompPMat 𝑙))
143142adantl 473 . . . . . . . . . . . . 13 (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥𝐵𝑦𝐵)) ∧ 𝑛 ∈ ℕ0) ∧ 𝑙 ∈ (0...𝑛)) → 𝑙 / 𝑘(𝑥 decompPMat 𝑘) = (𝑥 decompPMat 𝑙))
144138, 143eqtr2d 2787 . . . . . . . . . . . 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))
14510ad3antrrr 768 . . . . . . . . . . . . . . . 16 ((((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥𝐵𝑦𝐵)) ∧ 𝑛 ∈ ℕ0) ∧ 𝑙 ∈ (0...𝑛)) ∧ 𝑘 ∈ ℕ0) → 𝑦𝐵)
1463, 4, 11, 18, 34decpmatcl 20766 . . . . . . . . . . . . . . . 16 ((𝑅 ∈ Ring ∧ 𝑦𝐵𝑘 ∈ ℕ0) → (𝑦 decompPMat 𝑘) ∈ (Base‘𝐴))
147126, 145, 128, 146syl3anc 1473 . . . . . . . . . . . . . . 15 ((((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥𝐵𝑦𝐵)) ∧ 𝑛 ∈ ℕ0) ∧ 𝑙 ∈ (0...𝑛)) ∧ 𝑘 ∈ ℕ0) → (𝑦 decompPMat 𝑘) ∈ (Base‘𝐴))
148147ralrimiva 3096 . . . . . . . . . . . . . 14 (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥𝐵𝑦𝐵)) ∧ 𝑛 ∈ ℕ0) ∧ 𝑙 ∈ (0...𝑛)) → ∀𝑘 ∈ ℕ0 (𝑦 decompPMat 𝑘) ∈ (Base‘𝐴))
1492, 10jca 555 . . . . . . . . . . . . . . . 16 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥𝐵𝑦𝐵)) → (𝑅 ∈ Ring ∧ 𝑦𝐵))
150149ad2antrr 764 . . . . . . . . . . . . . . 15 (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥𝐵𝑦𝐵)) ∧ 𝑛 ∈ ℕ0) ∧ 𝑙 ∈ (0...𝑛)) → (𝑅 ∈ Ring ∧ 𝑦𝐵))
1513, 4, 11, 18, 35decpmatfsupp 20768 . . . . . . . . . . . . . . 15 ((𝑅 ∈ Ring ∧ 𝑦𝐵) → (𝑘 ∈ ℕ0 ↦ (𝑦 decompPMat 𝑘)) finSupp (0g𝐴))
152150, 151syl 17 . . . . . . . . . . . . . 14 (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥𝐵𝑦𝐵)) ∧ 𝑛 ∈ ℕ0) ∧ 𝑙 ∈ (0...𝑛)) → (𝑘 ∈ ℕ0 ↦ (𝑦 decompPMat 𝑘)) finSupp (0g𝐴))
153 fznn0sub 12558 . . . . . . . . . . . . . . 15 (𝑙 ∈ (0...𝑛) → (𝑛𝑙) ∈ ℕ0)
154153adantl 473 . . . . . . . . . . . . . 14 (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥𝐵𝑦𝐵)) ∧ 𝑛 ∈ ℕ0) ∧ 𝑙 ∈ (0...𝑛)) → (𝑛𝑙) ∈ ℕ0)
15519, 31, 17, 16, 125, 34, 15, 35, 148, 152, 154gsummoncoe1 19868 . . . . . . . . . . . . 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))
156 ovex 6833 . . . . . . . . . . . . . 14 (𝑛𝑙) ∈ V
157 csbov2g 6846 . . . . . . . . . . . . . 14 ((𝑛𝑙) ∈ V → (𝑛𝑙) / 𝑘(𝑦 decompPMat 𝑘) = (𝑦 decompPMat (𝑛𝑙) / 𝑘𝑘))
158156, 157mp1i 13 . . . . . . . . . . . . 13 (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥𝐵𝑦𝐵)) ∧ 𝑛 ∈ ℕ0) ∧ 𝑙 ∈ (0...𝑛)) → (𝑛𝑙) / 𝑘(𝑦 decompPMat 𝑘) = (𝑦 decompPMat (𝑛𝑙) / 𝑘𝑘))
159 csbvarg 4138 . . . . . . . . . . . . . . 15 ((𝑛𝑙) ∈ V → (𝑛𝑙) / 𝑘𝑘 = (𝑛𝑙))
160156, 159mp1i 13 . . . . . . . . . . . . . 14 (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥𝐵𝑦𝐵)) ∧ 𝑛 ∈ ℕ0) ∧ 𝑙 ∈ (0...𝑛)) → (𝑛𝑙) / 𝑘𝑘 = (𝑛𝑙))
161160oveq2d 6821 . . . . . . . . . . . . 13 (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥𝐵𝑦𝐵)) ∧ 𝑛 ∈ ℕ0) ∧ 𝑙 ∈ (0...𝑛)) → (𝑦 decompPMat (𝑛𝑙) / 𝑘𝑘) = (𝑦 decompPMat (𝑛𝑙)))
162155, 158, 1613eqtrrd 2791 . . . . . . . . . . . 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)))
163144, 162oveq12d 6823 . . . . . . . . . . 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))))
164163mpteq2dva 4888 . . . . . . . . . 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)))))
165124, 164syl5eq 2798 . . . . . . . . 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)))))
166165oveq2d 6821 . . . . . . . 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))))))
16783, 119, 1663eqtr4rd 2797 . . . . . . 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))
16861, 73, 1673eqtrd 2790 . . . . . 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))
169168ralrimiva 3096 . . . . 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))
17032adantr 472 . . . . . 6 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥𝐵𝑦𝐵)) → 𝐴 ∈ Ring)
17188adantr 472 . . . . . . 7 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥𝐵𝑦𝐵)) → 𝑄 ∈ CMnd)
17290a1i 11 . . . . . . 7 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥𝐵𝑦𝐵)) → ℕ0 ∈ V)
17319ply1lmod 19816 . . . . . . . . . . 11 (𝐴 ∈ Ring → 𝑄 ∈ LMod)
17432, 173syl 17 . . . . . . . . . 10 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝑄 ∈ LMod)
175174ad2antrr 764 . . . . . . . . 9 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥𝐵𝑦𝐵)) ∧ 𝑘 ∈ ℕ0) → 𝑄 ∈ LMod)
17637ad2antrr 764 . . . . . . . . . . 11 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥𝐵𝑦𝐵)) ∧ 𝑘 ∈ ℕ0) → 𝐴 ∈ CMnd)
177 fzfid 12958 . . . . . . . . . . 11 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥𝐵𝑦𝐵)) ∧ 𝑘 ∈ ℕ0) → (0...𝑘) ∈ Fin)
17832ad3antrrr 768 . . . . . . . . . . . . 13 (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥𝐵𝑦𝐵)) ∧ 𝑘 ∈ ℕ0) ∧ 𝑧 ∈ (0...𝑘)) → 𝐴 ∈ Ring)
179 simp-4r 827 . . . . . . . . . . . . . 14 (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥𝐵𝑦𝐵)) ∧ 𝑘 ∈ ℕ0) ∧ 𝑧 ∈ (0...𝑘)) → 𝑅 ∈ Ring)
180 simplrl 819 . . . . . . . . . . . . . . 15 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥𝐵𝑦𝐵)) ∧ 𝑘 ∈ ℕ0) → 𝑥𝐵)
181180adantr 472 . . . . . . . . . . . . . 14 (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥𝐵𝑦𝐵)) ∧ 𝑘 ∈ ℕ0) ∧ 𝑧 ∈ (0...𝑘)) → 𝑥𝐵)
18243adantl 473 . . . . . . . . . . . . . 14 (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥𝐵𝑦𝐵)) ∧ 𝑘 ∈ ℕ0) ∧ 𝑧 ∈ (0...𝑘)) → 𝑧 ∈ ℕ0)
183179, 181, 182, 45syl3anc 1473 . . . . . . . . . . . . 13 (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥𝐵𝑦𝐵)) ∧ 𝑘 ∈ ℕ0) ∧ 𝑧 ∈ (0...𝑘)) → (𝑥 decompPMat 𝑧) ∈ (Base‘𝐴))
184 simplrr 820 . . . . . . . . . . . . . . 15 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥𝐵𝑦𝐵)) ∧ 𝑘 ∈ ℕ0) → 𝑦𝐵)
185184adantr 472 . . . . . . . . . . . . . 14 (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥𝐵𝑦𝐵)) ∧ 𝑘 ∈ ℕ0) ∧ 𝑧 ∈ (0...𝑘)) → 𝑦𝐵)
18648adantl 473 . . . . . . . . . . . . . 14 (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥𝐵𝑦𝐵)) ∧ 𝑘 ∈ ℕ0) ∧ 𝑧 ∈ (0...𝑘)) → (𝑘𝑧) ∈ ℕ0)
187179, 185, 186, 50syl3anc 1473 . . . . . . . . . . . . 13 (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥𝐵𝑦𝐵)) ∧ 𝑘 ∈ ℕ0) ∧ 𝑧 ∈ (0...𝑘)) → (𝑦 decompPMat (𝑘𝑧)) ∈ (Base‘𝐴))
188178, 183, 187, 53syl3anc 1473 . . . . . . . . . . . 12 (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥𝐵𝑦𝐵)) ∧ 𝑘 ∈ ℕ0) ∧ 𝑧 ∈ (0...𝑘)) → ((𝑥 decompPMat 𝑧)(.r𝐴)(𝑦 decompPMat (𝑘𝑧))) ∈ (Base‘𝐴))
189188ralrimiva 3096 . . . . . . . . . . 11 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥𝐵𝑦𝐵)) ∧ 𝑘 ∈ ℕ0) → ∀𝑧 ∈ (0...𝑘)((𝑥 decompPMat 𝑧)(.r𝐴)(𝑦 decompPMat (𝑘𝑧))) ∈ (Base‘𝐴))
19034, 176, 177, 189gsummptcl 18558 . . . . . . . . . 10 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥𝐵𝑦𝐵)) ∧ 𝑘 ∈ ℕ0) → (𝐴 Σg (𝑧 ∈ (0...𝑘) ↦ ((𝑥 decompPMat 𝑧)(.r𝐴)(𝑦 decompPMat (𝑘𝑧))))) ∈ (Base‘𝐴))
19132ad2antrr 764 . . . . . . . . . . . . 13 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥𝐵𝑦𝐵)) ∧ 𝑘 ∈ ℕ0) → 𝐴 ∈ Ring)
19219ply1sca 19817 . . . . . . . . . . . . 13 (𝐴 ∈ Ring → 𝐴 = (Scalar‘𝑄))
193191, 192syl 17 . . . . . . . . . . . 12 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥𝐵𝑦𝐵)) ∧ 𝑘 ∈ ℕ0) → 𝐴 = (Scalar‘𝑄))
194193eqcomd 2758 . . . . . . . . . . 11 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥𝐵𝑦𝐵)) ∧ 𝑘 ∈ ℕ0) → (Scalar‘𝑄) = 𝐴)
195194fveq2d 6348 . . . . . . . . . 10 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥𝐵𝑦𝐵)) ∧ 𝑘 ∈ ℕ0) → (Base‘(Scalar‘𝑄)) = (Base‘𝐴))
196190, 195eleqtrrd 2834 . . . . . . . . 9 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥𝐵𝑦𝐵)) ∧ 𝑘 ∈ ℕ0) → (𝐴 Σg (𝑧 ∈ (0...𝑘) ↦ ((𝑥 decompPMat 𝑧)(.r𝐴)(𝑦 decompPMat (𝑘𝑧))))) ∈ (Base‘(Scalar‘𝑄)))
197 eqid 2752 . . . . . . . . . . 11 (mulGrp‘𝑄) = (mulGrp‘𝑄)
19819, 17, 197, 16, 31ply1moncl 19835 . . . . . . . . . 10 ((𝐴 ∈ Ring ∧ 𝑘 ∈ ℕ0) → (𝑘(.g‘(mulGrp‘𝑄))(var1𝐴)) ∈ (Base‘𝑄))
199191, 198sylancom 704 . . . . . . . . 9 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥𝐵𝑦𝐵)) ∧ 𝑘 ∈ ℕ0) → (𝑘(.g‘(mulGrp‘𝑄))(var1𝐴)) ∈ (Base‘𝑄))
200 eqid 2752 . . . . . . . . . 10 (Scalar‘𝑄) = (Scalar‘𝑄)
201 eqid 2752 . . . . . . . . . 10 (Base‘(Scalar‘𝑄)) = (Base‘(Scalar‘𝑄))
20231, 200, 15, 201lmodvscl 19074 . . . . . . . . 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))
203175, 196, 199, 202syl3anc 1473 . . . . . . . 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))
204 eqid 2752 . . . . . . . 8 (𝑘 ∈ ℕ0 ↦ ((𝐴 Σg (𝑧 ∈ (0...𝑘) ↦ ((𝑥 decompPMat 𝑧)(.r𝐴)(𝑦 decompPMat (𝑘𝑧)))))( ·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))))
205203, 204fmptd 6540 . . . . . . 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))
2063, 4, 11, 15, 16, 17, 18, 19, 31, 20pm2mpmhmlem1 20817 . . . . . . 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))
20731, 84, 171, 172, 205, 206gsumcl 18508 . . . . . 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))
20886adantr 472 . . . . . . 7 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥𝐵𝑦𝐵)) → 𝑄 ∈ Ring)
20994, 96sylan 489 . . . . . . . . 9 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥𝐵𝑦𝐵)) ∧ 𝑘 ∈ ℕ0) → ((𝑥 decompPMat 𝑘)( ·𝑠𝑄)(𝑘(.g‘(mulGrp‘𝑄))(var1𝐴))) ∈ (Base‘𝑄))
210209, 98fmptd 6540 . . . . . . . 8 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥𝐵𝑦𝐵)) → (𝑘 ∈ ℕ0 ↦ ((𝑥 decompPMat 𝑘)( ·𝑠𝑄)(𝑘(.g‘(mulGrp‘𝑄))(var1𝐴)))):ℕ0⟶(Base‘𝑄))
21194, 100syl 17 . . . . . . . 8 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥𝐵𝑦𝐵)) → (𝑘 ∈ ℕ0 ↦ ((𝑥 decompPMat 𝑘)( ·𝑠𝑄)(𝑘(.g‘(mulGrp‘𝑄))(var1𝐴)))) finSupp (0g𝑄))
21231, 84, 171, 172, 210, 211gsumcl 18508 . . . . . . 7 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥𝐵𝑦𝐵)) → (𝑄 Σg (𝑘 ∈ ℕ0 ↦ ((𝑥 decompPMat 𝑘)( ·𝑠𝑄)(𝑘(.g‘(mulGrp‘𝑄))(var1𝐴))))) ∈ (Base‘𝑄))
213105, 107sylan 489 . . . . . . . . 9 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥𝐵𝑦𝐵)) ∧ 𝑘 ∈ ℕ0) → ((𝑦 decompPMat 𝑘)( ·𝑠𝑄)(𝑘(.g‘(mulGrp‘𝑄))(var1𝐴))) ∈ (Base‘𝑄))
214213, 109fmptd 6540 . . . . . . . 8 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥𝐵𝑦𝐵)) → (𝑘 ∈ ℕ0 ↦ ((𝑦 decompPMat 𝑘)( ·𝑠𝑄)(𝑘(.g‘(mulGrp‘𝑄))(var1𝐴)))):ℕ0⟶(Base‘𝑄))
2151, 2, 10, 113syl3anc 1473 . . . . . . . 8 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥𝐵𝑦𝐵)) → (𝑘 ∈ ℕ0 ↦ ((𝑦 decompPMat 𝑘)( ·𝑠𝑄)(𝑘(.g‘(mulGrp‘𝑄))(var1𝐴)))) finSupp (0g𝑄))
21631, 84, 171, 172, 214, 215gsumcl 18508 . . . . . . 7 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥𝐵𝑦𝐵)) → (𝑄 Σg (𝑘 ∈ ℕ0 ↦ ((𝑦 decompPMat 𝑘)( ·𝑠𝑄)(𝑘(.g‘(mulGrp‘𝑄))(var1𝐴))))) ∈ (Base‘𝑄))
21731, 116ringcl 18753 . . . . . . 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))
218208, 212, 216, 217syl3anc 1473 . . . . . 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))
219 eqid 2752 . . . . . . 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))))))
220 eqid 2752 . . . . . . 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)))))))
22119, 31, 219, 220ply1coe1eq 19862 . . . . . 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))))))))
222170, 207, 218, 221syl3anc 1473 . . . . 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))))))))
223169, 222mpbid 222 . . . 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)))))))
22422, 30, 2233eqtrd 2790 . . 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)))))))
2253, 4, 11, 15, 16, 17, 18, 19, 20pm2mpfval 20795 . . . . 5 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑥𝐵) → (𝑇𝑥) = (𝑄 Σg (𝑘 ∈ ℕ0 ↦ ((𝑥 decompPMat 𝑘)( ·𝑠𝑄)(𝑘(.g‘(mulGrp‘𝑄))(var1𝐴))))))
2261, 2, 8, 225syl3anc 1473 . . . 4 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥𝐵𝑦𝐵)) → (𝑇𝑥) = (𝑄 Σg (𝑘 ∈ ℕ0 ↦ ((𝑥 decompPMat 𝑘)( ·𝑠𝑄)(𝑘(.g‘(mulGrp‘𝑄))(var1𝐴))))))
2273, 4, 11, 15, 16, 17, 18, 19, 20pm2mpfval 20795 . . . . 5 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑦𝐵) → (𝑇𝑦) = (𝑄 Σg (𝑘 ∈ ℕ0 ↦ ((𝑦 decompPMat 𝑘)( ·𝑠𝑄)(𝑘(.g‘(mulGrp‘𝑄))(var1𝐴))))))
2281, 2, 10, 227syl3anc 1473 . . . 4 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥𝐵𝑦𝐵)) → (𝑇𝑦) = (𝑄 Σg (𝑘 ∈ ℕ0 ↦ ((𝑦 decompPMat 𝑘)( ·𝑠𝑄)(𝑘(.g‘(mulGrp‘𝑄))(var1𝐴))))))
229226, 228oveq12d 6823 . . 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)))))))
230224, 229eqtr4d 2789 . 2 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥𝐵𝑦𝐵)) → (𝑇‘(𝑥(.r𝐶)𝑦)) = ((𝑇𝑥)(.r𝑄)(𝑇𝑦)))
231230ralrimivva 3101 1 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → ∀𝑥𝐵𝑦𝐵 (𝑇‘(𝑥(.r𝐶)𝑦)) = ((𝑇𝑥)(.r𝑄)(𝑇𝑦)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 383  w3a 1072   = wceq 1624  wcel 2131  wral 3042  Vcvv 3332  csb 3666   class class class wbr 4796  cmpt 4873  cfv 6041  (class class class)co 6805  Fincfn 8113   finSupp cfsupp 8432  0cc0 10120  cmin 10450  0cn0 11476  ...cfz 12511  Basecbs 16051  .rcmulr 16136  Scalarcsca 16138   ·𝑠 cvsca 16139  0gc0g 16294   Σg cgsu 16295  .gcmg 17733  CMndccmn 18385  mulGrpcmgp 18681  Ringcrg 18739  LModclmod 19057  var1cv1 19740  Poly1cpl1 19741  coe1cco1 19742   Mat cmat 20407   decompPMat cdecpmat 20761   pMatToMatPoly cpm2mp 20791
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1863  ax-4 1878  ax-5 1980  ax-6 2046  ax-7 2082  ax-8 2133  ax-9 2140  ax-10 2160  ax-11 2175  ax-12 2188  ax-13 2383  ax-ext 2732  ax-rep 4915  ax-sep 4925  ax-nul 4933  ax-pow 4984  ax-pr 5047  ax-un 7106  ax-inf2 8703  ax-cnex 10176  ax-resscn 10177  ax-1cn 10178  ax-icn 10179  ax-addcl 10180  ax-addrcl 10181  ax-mulcl 10182  ax-mulrcl 10183  ax-mulcom 10184  ax-addass 10185  ax-mulass 10186  ax-distr 10187  ax-i2m1 10188  ax-1ne0 10189  ax-1rid 10190  ax-rnegex 10191  ax-rrecex 10192  ax-cnre 10193  ax-pre-lttri 10194  ax-pre-lttrn 10195  ax-pre-ltadd 10196  ax-pre-mulgt0 10197
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1073  df-3an 1074  df-tru 1627  df-fal 1630  df-ex 1846  df-nf 1851  df-sb 2039  df-eu 2603  df-mo 2604  df-clab 2739  df-cleq 2745  df-clel 2748  df-nfc 2883  df-ne 2925  df-nel 3028  df-ral 3047  df-rex 3048  df-reu 3049  df-rmo 3050  df-rab 3051  df-v 3334  df-sbc 3569  df-csb 3667  df-dif 3710  df-un 3712  df-in 3714  df-ss 3721  df-pss 3723  df-nul 4051  df-if 4223  df-pw 4296  df-sn 4314  df-pr 4316  df-tp 4318  df-op 4320  df-ot 4322  df-uni 4581  df-int 4620  df-iun 4666  df-iin 4667  df-br 4797  df-opab 4857  df-mpt 4874  df-tr 4897  df-id 5166  df-eprel 5171  df-po 5179  df-so 5180  df-fr 5217  df-se 5218  df-we 5219  df-xp 5264  df-rel 5265  df-cnv 5266  df-co 5267  df-dm 5268  df-rn 5269  df-res 5270  df-ima 5271  df-pred 5833  df-ord 5879  df-on 5880  df-lim 5881  df-suc 5882  df-iota 6004  df-fun 6043  df-fn 6044  df-f 6045  df-f1 6046  df-fo 6047  df-f1o 6048  df-fv 6049  df-isom 6050  df-riota 6766  df-ov 6808  df-oprab 6809  df-mpt2 6810  df-of 7054  df-ofr 7055  df-om 7223  df-1st 7325  df-2nd 7326  df-supp 7456  df-wrecs 7568  df-recs 7629  df-rdg 7667  df-1o 7721  df-2o 7722  df-oadd 7725  df-er 7903  df-map 8017  df-pm 8018  df-ixp 8067  df-en 8114  df-dom 8115  df-sdom 8116  df-fin 8117  df-fsupp 8433  df-sup 8505  df-oi 8572  df-card 8947  df-pnf 10260  df-mnf 10261  df-xr 10262  df-ltxr 10263  df-le 10264  df-sub 10452  df-neg 10453  df-nn 11205  df-2 11263  df-3 11264  df-4 11265  df-5 11266  df-6 11267  df-7 11268  df-8 11269  df-9 11270  df-n0 11477  df-z 11562  df-dec 11678  df-uz 11872  df-fz 12512  df-fzo 12652  df-seq 12988  df-hash 13304  df-struct 16053  df-ndx 16054  df-slot 16055  df-base 16057  df-sets 16058  df-ress 16059  df-plusg 16148  df-mulr 16149  df-sca 16151  df-vsca 16152  df-ip 16153  df-tset 16154  df-ple 16155  df-ds 16158  df-hom 16160  df-cco 16161  df-0g 16296  df-gsum 16297  df-prds 16302  df-pws 16304  df-mre 16440  df-mrc 16441  df-acs 16443  df-mgm 17435  df-sgrp 17477  df-mnd 17488  df-mhm 17528  df-submnd 17529  df-grp 17618  df-minusg 17619  df-sbg 17620  df-mulg 17734  df-subg 17784  df-ghm 17851  df-cntz 17942  df-cmn 18387  df-abl 18388  df-mgp 18682  df-ur 18694  df-srg 18698  df-ring 18741  df-subrg 18972  df-lmod 19059  df-lss 19127  df-sra 19366  df-rgmod 19367  df-psr 19550  df-mvr 19551  df-mpl 19552  df-opsr 19554  df-psr1 19744  df-vr1 19745  df-ply1 19746  df-coe1 19747  df-dsmm 20270  df-frlm 20285  df-mamu 20384  df-mat 20408  df-decpmat 20762  df-pm2mp 20792
This theorem is referenced by:  pm2mpmhm  20819
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