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Theorem pm2mpmhmlem2 21001
Description: Lemma 2 for pm2mpmhm 21002. (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 783 . . . . 5 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥𝐵𝑦𝐵)) → 𝑁 ∈ Fin)
2 simplr 785 . . . . 5 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥𝐵𝑦𝐵)) → 𝑅 ∈ Ring)
3 pm2mpmhm.p . . . . . . . 8 𝑃 = (Poly1𝑅)
4 pm2mpmhm.c . . . . . . . 8 𝐶 = (𝑁 Mat 𝑃)
53, 4pmatring 20875 . . . . . . 7 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝐶 ∈ Ring)
65adantr 474 . . . . . 6 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥𝐵𝑦𝐵)) → 𝐶 ∈ Ring)
7 simpl 476 . . . . . . 7 ((𝑥𝐵𝑦𝐵) → 𝑥𝐵)
87adantl 475 . . . . . 6 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥𝐵𝑦𝐵)) → 𝑥𝐵)
9 simpr 479 . . . . . . 7 ((𝑥𝐵𝑦𝐵) → 𝑦𝐵)
109adantl 475 . . . . . 6 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥𝐵𝑦𝐵)) → 𝑦𝐵)
11 pm2mpmhm.b . . . . . . 7 𝐵 = (Base‘𝐶)
12 eqid 2825 . . . . . . 7 (.r𝐶) = (.r𝐶)
1311, 12ringcl 18922 . . . . . 6 ((𝐶 ∈ Ring ∧ 𝑥𝐵𝑦𝐵) → (𝑥(.r𝐶)𝑦) ∈ 𝐵)
146, 8, 10, 13syl3anc 1494 . . . . 5 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥𝐵𝑦𝐵)) → (𝑥(.r𝐶)𝑦) ∈ 𝐵)
15 eqid 2825 . . . . . 6 ( ·𝑠𝑄) = ( ·𝑠𝑄)
16 eqid 2825 . . . . . 6 (.g‘(mulGrp‘𝑄)) = (.g‘(mulGrp‘𝑄))
17 eqid 2825 . . . . . 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 20978 . . . . 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 1494 . . . 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 793 . . . . . . . 8 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥𝐵𝑦𝐵)) ∧ 𝑘 ∈ ℕ0) → 𝑅 ∈ Ring)
24 simplr 785 . . . . . . . 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 20954 . . . . . . . 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 1494 . . . . . . 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 6925 . . . . . 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 4969 . . . . 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 6926 . . . 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 2825 . . . . . . . 8 (Base‘𝑄) = (Base‘𝑄)
3218matring 20623 . . . . . . . . 9 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝐴 ∈ Ring)
3332ad2antrr 717 . . . . . . . 8 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥𝐵𝑦𝐵)) ∧ 𝑛 ∈ ℕ0) → 𝐴 ∈ Ring)
34 eqid 2825 . . . . . . . 8 (Base‘𝐴) = (Base‘𝐴)
35 eqid 2825 . . . . . . . 8 (0g𝐴) = (0g𝐴)
36 ringcmn 18942 . . . . . . . . . . . 12 (𝐴 ∈ Ring → 𝐴 ∈ CMnd)
3732, 36syl 17 . . . . . . . . . . 11 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝐴 ∈ CMnd)
3837ad3antrrr 721 . . . . . . . . . 10 (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥𝐵𝑦𝐵)) ∧ 𝑛 ∈ ℕ0) ∧ 𝑘 ∈ ℕ0) → 𝐴 ∈ CMnd)
39 fzfid 13074 . . . . . . . . . 10 (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥𝐵𝑦𝐵)) ∧ 𝑛 ∈ ℕ0) ∧ 𝑘 ∈ ℕ0) → (0...𝑘) ∈ Fin)
4033ad2antrr 717 . . . . . . . . . . . 12 ((((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥𝐵𝑦𝐵)) ∧ 𝑛 ∈ ℕ0) ∧ 𝑘 ∈ ℕ0) ∧ 𝑧 ∈ (0...𝑘)) → 𝐴 ∈ Ring)
41 simp-5r 807 . . . . . . . . . . . . 13 ((((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥𝐵𝑦𝐵)) ∧ 𝑛 ∈ ℕ0) ∧ 𝑘 ∈ ℕ0) ∧ 𝑧 ∈ (0...𝑘)) → 𝑅 ∈ Ring)
428ad3antrrr 721 . . . . . . . . . . . . 13 ((((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥𝐵𝑦𝐵)) ∧ 𝑛 ∈ ℕ0) ∧ 𝑘 ∈ ℕ0) ∧ 𝑧 ∈ (0...𝑘)) → 𝑥𝐵)
43 elfznn0 12734 . . . . . . . . . . . . . 14 (𝑧 ∈ (0...𝑘) → 𝑧 ∈ ℕ0)
4443adantl 475 . . . . . . . . . . . . 13 ((((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥𝐵𝑦𝐵)) ∧ 𝑛 ∈ ℕ0) ∧ 𝑘 ∈ ℕ0) ∧ 𝑧 ∈ (0...𝑘)) → 𝑧 ∈ ℕ0)
453, 4, 11, 18, 34decpmatcl 20949 . . . . . . . . . . . . 13 ((𝑅 ∈ Ring ∧ 𝑥𝐵𝑧 ∈ ℕ0) → (𝑥 decompPMat 𝑧) ∈ (Base‘𝐴))
4641, 42, 44, 45syl3anc 1494 . . . . . . . . . . . 12 ((((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥𝐵𝑦𝐵)) ∧ 𝑛 ∈ ℕ0) ∧ 𝑘 ∈ ℕ0) ∧ 𝑧 ∈ (0...𝑘)) → (𝑥 decompPMat 𝑧) ∈ (Base‘𝐴))
4710ad3antrrr 721 . . . . . . . . . . . . 13 ((((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥𝐵𝑦𝐵)) ∧ 𝑛 ∈ ℕ0) ∧ 𝑘 ∈ ℕ0) ∧ 𝑧 ∈ (0...𝑘)) → 𝑦𝐵)
48 fznn0sub 12673 . . . . . . . . . . . . . 14 (𝑧 ∈ (0...𝑘) → (𝑘𝑧) ∈ ℕ0)
4948adantl 475 . . . . . . . . . . . . 13 ((((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥𝐵𝑦𝐵)) ∧ 𝑛 ∈ ℕ0) ∧ 𝑘 ∈ ℕ0) ∧ 𝑧 ∈ (0...𝑘)) → (𝑘𝑧) ∈ ℕ0)
503, 4, 11, 18, 34decpmatcl 20949 . . . . . . . . . . . . 13 ((𝑅 ∈ Ring ∧ 𝑦𝐵 ∧ (𝑘𝑧) ∈ ℕ0) → (𝑦 decompPMat (𝑘𝑧)) ∈ (Base‘𝐴))
5141, 47, 49, 50syl3anc 1494 . . . . . . . . . . . 12 ((((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥𝐵𝑦𝐵)) ∧ 𝑛 ∈ ℕ0) ∧ 𝑘 ∈ ℕ0) ∧ 𝑧 ∈ (0...𝑘)) → (𝑦 decompPMat (𝑘𝑧)) ∈ (Base‘𝐴))
52 eqid 2825 . . . . . . . . . . . . 13 (.r𝐴) = (.r𝐴)
5334, 52ringcl 18922 . . . . . . . . . . . 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 1494 . . . . . . . . . . 11 ((((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥𝐵𝑦𝐵)) ∧ 𝑛 ∈ ℕ0) ∧ 𝑘 ∈ ℕ0) ∧ 𝑧 ∈ (0...𝑘)) → ((𝑥 decompPMat 𝑧)(.r𝐴)(𝑦 decompPMat (𝑘𝑧))) ∈ (Base‘𝐴))
5554ralrimiva 3175 . . . . . . . . . 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 18726 . . . . . . . . 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 3175 . . . . . . . 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 20955 . . . . . . . . 9 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥𝐵𝑦𝐵)) → (𝑘 ∈ ℕ0 ↦ (𝐴 Σg (𝑧 ∈ (0...𝑘) ↦ ((𝑥 decompPMat 𝑧)(.r𝐴)(𝑦 decompPMat (𝑘𝑧)))))) finSupp (0g𝐴))
5958adantr 474 . . . . . . . 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 20041 . . . . . . 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 6955 . . . . . . . . 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 6918 . . . . . . . . . . . . 13 (𝑘 = 𝑛 → (0...𝑘) = (0...𝑛))
65 oveq1 6917 . . . . . . . . . . . . . . 15 (𝑘 = 𝑛 → (𝑘𝑧) = (𝑛𝑧))
6665oveq2d 6926 . . . . . . . . . . . . . 14 (𝑘 = 𝑛 → (𝑦 decompPMat (𝑘𝑧)) = (𝑦 decompPMat (𝑛𝑧)))
6766oveq2d 6926 . . . . . . . . . . . . 13 (𝑘 = 𝑛 → ((𝑥 decompPMat 𝑧)(.r𝐴)(𝑦 decompPMat (𝑘𝑧))) = ((𝑥 decompPMat 𝑧)(.r𝐴)(𝑦 decompPMat (𝑛𝑧))))
6864, 67mpteq12dv 4958 . . . . . . . . . . . 12 (𝑘 = 𝑛 → (𝑧 ∈ (0...𝑘) ↦ ((𝑥 decompPMat 𝑧)(.r𝐴)(𝑦 decompPMat (𝑘𝑧)))) = (𝑧 ∈ (0...𝑛) ↦ ((𝑥 decompPMat 𝑧)(.r𝐴)(𝑦 decompPMat (𝑛𝑧)))))
6968adantl 475 . . . . . . . . . . 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 3784 . . . . . . . . . 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 6926 . . . . . . . . 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 2861 . . . . . . . 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 475 . . . . . . 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 2826 . . . . . . . . 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 6918 . . . . . . . . . . . 12 (𝑟 = 𝑛 → (0...𝑟) = (0...𝑛))
76 fvoveq1 6933 . . . . . . . . . . . . 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)))
7776oveq2d 6926 . . . . . . . . . . . 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))))
7875, 77mpteq12dv 4958 . . . . . . . . . . 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)))))
7978oveq2d 6926 . . . . . . . . . 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))))))
8079adantl 475 . . . . . . . . 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))))))
81 ovexd 6944 . . . . . . . . 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)
8274, 80, 60, 81fvmptd 6539 . . . . . . . 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))))))
83 eqid 2825 . . . . . . . . . 10 (0g𝑄) = (0g𝑄)
8419ply1ring 19985 . . . . . . . . . . . . 13 (𝐴 ∈ Ring → 𝑄 ∈ Ring)
8532, 84syl 17 . . . . . . . . . . . 12 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝑄 ∈ Ring)
86 ringcmn 18942 . . . . . . . . . . . 12 (𝑄 ∈ Ring → 𝑄 ∈ CMnd)
8785, 86syl 17 . . . . . . . . . . 11 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝑄 ∈ CMnd)
8887ad2antrr 717 . . . . . . . . . 10 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥𝐵𝑦𝐵)) ∧ 𝑛 ∈ ℕ0) → 𝑄 ∈ CMnd)
89 nn0ex 11632 . . . . . . . . . . 11 0 ∈ V
9089a1i 11 . . . . . . . . . 10 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥𝐵𝑦𝐵)) ∧ 𝑛 ∈ ℕ0) → ℕ0 ∈ V)
917anim2i 610 . . . . . . . . . . . . . 14 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥𝐵𝑦𝐵)) → ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝑥𝐵))
92 df-3an 1113 . . . . . . . . . . . . . 14 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑥𝐵) ↔ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝑥𝐵))
9391, 92sylibr 226 . . . . . . . . . . . . 13 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥𝐵𝑦𝐵)) → (𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑥𝐵))
9493adantr 474 . . . . . . . . . . . 12 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥𝐵𝑦𝐵)) ∧ 𝑛 ∈ ℕ0) → (𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑥𝐵))
953, 4, 11, 15, 16, 17, 18, 19, 31pm2mpghmlem1 20995 . . . . . . . . . . . 12 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑥𝐵) ∧ 𝑘 ∈ ℕ0) → ((𝑥 decompPMat 𝑘)( ·𝑠𝑄)(𝑘(.g‘(mulGrp‘𝑄))(var1𝐴))) ∈ (Base‘𝑄))
9694, 95sylan 575 . . . . . . . . . . 11 (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥𝐵𝑦𝐵)) ∧ 𝑛 ∈ ℕ0) ∧ 𝑘 ∈ ℕ0) → ((𝑥 decompPMat 𝑘)( ·𝑠𝑄)(𝑘(.g‘(mulGrp‘𝑄))(var1𝐴))) ∈ (Base‘𝑄))
9796fmpttd 6639 . . . . . . . . . 10 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥𝐵𝑦𝐵)) ∧ 𝑛 ∈ ℕ0) → (𝑘 ∈ ℕ0 ↦ ((𝑥 decompPMat 𝑘)( ·𝑠𝑄)(𝑘(.g‘(mulGrp‘𝑄))(var1𝐴)))):ℕ0⟶(Base‘𝑄))
983, 4, 11, 15, 16, 17, 18, 19pm2mpghmlem2 20994 . . . . . . . . . . 11 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑥𝐵) → (𝑘 ∈ ℕ0 ↦ ((𝑥 decompPMat 𝑘)( ·𝑠𝑄)(𝑘(.g‘(mulGrp‘𝑄))(var1𝐴)))) finSupp (0g𝑄))
9994, 98syl 17 . . . . . . . . . 10 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥𝐵𝑦𝐵)) ∧ 𝑛 ∈ ℕ0) → (𝑘 ∈ ℕ0 ↦ ((𝑥 decompPMat 𝑘)( ·𝑠𝑄)(𝑘(.g‘(mulGrp‘𝑄))(var1𝐴)))) finSupp (0g𝑄))
10031, 83, 88, 90, 97, 99gsumcl 18676 . . . . . . . . 9 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥𝐵𝑦𝐵)) ∧ 𝑛 ∈ ℕ0) → (𝑄 Σg (𝑘 ∈ ℕ0 ↦ ((𝑥 decompPMat 𝑘)( ·𝑠𝑄)(𝑘(.g‘(mulGrp‘𝑄))(var1𝐴))))) ∈ (Base‘𝑄))
1019anim2i 610 . . . . . . . . . . . . . 14 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥𝐵𝑦𝐵)) → ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝑦𝐵))
102 df-3an 1113 . . . . . . . . . . . . . 14 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑦𝐵) ↔ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝑦𝐵))
103101, 102sylibr 226 . . . . . . . . . . . . 13 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥𝐵𝑦𝐵)) → (𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑦𝐵))
104103adantr 474 . . . . . . . . . . . 12 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥𝐵𝑦𝐵)) ∧ 𝑛 ∈ ℕ0) → (𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑦𝐵))
1053, 4, 11, 15, 16, 17, 18, 19, 31pm2mpghmlem1 20995 . . . . . . . . . . . 12 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑦𝐵) ∧ 𝑘 ∈ ℕ0) → ((𝑦 decompPMat 𝑘)( ·𝑠𝑄)(𝑘(.g‘(mulGrp‘𝑄))(var1𝐴))) ∈ (Base‘𝑄))
106104, 105sylan 575 . . . . . . . . . . 11 (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥𝐵𝑦𝐵)) ∧ 𝑛 ∈ ℕ0) ∧ 𝑘 ∈ ℕ0) → ((𝑦 decompPMat 𝑘)( ·𝑠𝑄)(𝑘(.g‘(mulGrp‘𝑄))(var1𝐴))) ∈ (Base‘𝑄))
107106fmpttd 6639 . . . . . . . . . 10 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥𝐵𝑦𝐵)) ∧ 𝑛 ∈ ℕ0) → (𝑘 ∈ ℕ0 ↦ ((𝑦 decompPMat 𝑘)( ·𝑠𝑄)(𝑘(.g‘(mulGrp‘𝑄))(var1𝐴)))):ℕ0⟶(Base‘𝑄))
1081, 2, 103jca 1162 . . . . . . . . . . . 12 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥𝐵𝑦𝐵)) → (𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑦𝐵))
109108adantr 474 . . . . . . . . . . 11 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥𝐵𝑦𝐵)) ∧ 𝑛 ∈ ℕ0) → (𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑦𝐵))
1103, 4, 11, 15, 16, 17, 18, 19pm2mpghmlem2 20994 . . . . . . . . . . 11 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑦𝐵) → (𝑘 ∈ ℕ0 ↦ ((𝑦 decompPMat 𝑘)( ·𝑠𝑄)(𝑘(.g‘(mulGrp‘𝑄))(var1𝐴)))) finSupp (0g𝑄))
111109, 110syl 17 . . . . . . . . . 10 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥𝐵𝑦𝐵)) ∧ 𝑛 ∈ ℕ0) → (𝑘 ∈ ℕ0 ↦ ((𝑦 decompPMat 𝑘)( ·𝑠𝑄)(𝑘(.g‘(mulGrp‘𝑄))(var1𝐴)))) finSupp (0g𝑄))
11231, 83, 88, 90, 107, 111gsumcl 18676 . . . . . . . . 9 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥𝐵𝑦𝐵)) ∧ 𝑛 ∈ ℕ0) → (𝑄 Σg (𝑘 ∈ ℕ0 ↦ ((𝑦 decompPMat 𝑘)( ·𝑠𝑄)(𝑘(.g‘(mulGrp‘𝑄))(var1𝐴))))) ∈ (Base‘𝑄))
113 eqid 2825 . . . . . . . . . . 11 (.r𝑄) = (.r𝑄)
11419, 113, 52, 31coe1mul 20007 . . . . . . . . . 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)))))))
115114fveq1d 6439 . . . . . . . . 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))
11633, 100, 112, 115syl3anc 1494 . . . . . . . 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))
117 oveq2 6918 . . . . . . . . . . . 12 (𝑧 = 𝑙 → (𝑥 decompPMat 𝑧) = (𝑥 decompPMat 𝑙))
118 oveq2 6918 . . . . . . . . . . . . 13 (𝑧 = 𝑙 → (𝑛𝑧) = (𝑛𝑙))
119118oveq2d 6926 . . . . . . . . . . . 12 (𝑧 = 𝑙 → (𝑦 decompPMat (𝑛𝑧)) = (𝑦 decompPMat (𝑛𝑙)))
120117, 119oveq12d 6928 . . . . . . . . . . 11 (𝑧 = 𝑙 → ((𝑥 decompPMat 𝑧)(.r𝐴)(𝑦 decompPMat (𝑛𝑧))) = ((𝑥 decompPMat 𝑙)(.r𝐴)(𝑦 decompPMat (𝑛𝑙))))
121120cbvmptv 4975 . . . . . . . . . 10 (𝑧 ∈ (0...𝑛) ↦ ((𝑥 decompPMat 𝑧)(.r𝐴)(𝑦 decompPMat (𝑛𝑧)))) = (𝑙 ∈ (0...𝑛) ↦ ((𝑥 decompPMat 𝑙)(.r𝐴)(𝑦 decompPMat (𝑛𝑙))))
12232ad3antrrr 721 . . . . . . . . . . . . . 14 (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥𝐵𝑦𝐵)) ∧ 𝑛 ∈ ℕ0) ∧ 𝑙 ∈ (0...𝑛)) → 𝐴 ∈ Ring)
123 simp-5r 807 . . . . . . . . . . . . . . . 16 ((((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥𝐵𝑦𝐵)) ∧ 𝑛 ∈ ℕ0) ∧ 𝑙 ∈ (0...𝑛)) ∧ 𝑘 ∈ ℕ0) → 𝑅 ∈ Ring)
1248ad3antrrr 721 . . . . . . . . . . . . . . . 16 ((((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥𝐵𝑦𝐵)) ∧ 𝑛 ∈ ℕ0) ∧ 𝑙 ∈ (0...𝑛)) ∧ 𝑘 ∈ ℕ0) → 𝑥𝐵)
125 simpr 479 . . . . . . . . . . . . . . . 16 ((((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥𝐵𝑦𝐵)) ∧ 𝑛 ∈ ℕ0) ∧ 𝑙 ∈ (0...𝑛)) ∧ 𝑘 ∈ ℕ0) → 𝑘 ∈ ℕ0)
1263, 4, 11, 18, 34decpmatcl 20949 . . . . . . . . . . . . . . . 16 ((𝑅 ∈ Ring ∧ 𝑥𝐵𝑘 ∈ ℕ0) → (𝑥 decompPMat 𝑘) ∈ (Base‘𝐴))
127123, 124, 125, 126syl3anc 1494 . . . . . . . . . . . . . . 15 ((((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥𝐵𝑦𝐵)) ∧ 𝑛 ∈ ℕ0) ∧ 𝑙 ∈ (0...𝑛)) ∧ 𝑘 ∈ ℕ0) → (𝑥 decompPMat 𝑘) ∈ (Base‘𝐴))
128127ralrimiva 3175 . . . . . . . . . . . . . 14 (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥𝐵𝑦𝐵)) ∧ 𝑛 ∈ ℕ0) ∧ 𝑙 ∈ (0...𝑛)) → ∀𝑘 ∈ ℕ0 (𝑥 decompPMat 𝑘) ∈ (Base‘𝐴))
1292, 8jca 507 . . . . . . . . . . . . . . . 16 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥𝐵𝑦𝐵)) → (𝑅 ∈ Ring ∧ 𝑥𝐵))
130129ad2antrr 717 . . . . . . . . . . . . . . 15 (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥𝐵𝑦𝐵)) ∧ 𝑛 ∈ ℕ0) ∧ 𝑙 ∈ (0...𝑛)) → (𝑅 ∈ Ring ∧ 𝑥𝐵))
1313, 4, 11, 18, 35decpmatfsupp 20951 . . . . . . . . . . . . . . 15 ((𝑅 ∈ Ring ∧ 𝑥𝐵) → (𝑘 ∈ ℕ0 ↦ (𝑥 decompPMat 𝑘)) finSupp (0g𝐴))
132130, 131syl 17 . . . . . . . . . . . . . 14 (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥𝐵𝑦𝐵)) ∧ 𝑛 ∈ ℕ0) ∧ 𝑙 ∈ (0...𝑛)) → (𝑘 ∈ ℕ0 ↦ (𝑥 decompPMat 𝑘)) finSupp (0g𝐴))
133 elfznn0 12734 . . . . . . . . . . . . . . 15 (𝑙 ∈ (0...𝑛) → 𝑙 ∈ ℕ0)
134133adantl 475 . . . . . . . . . . . . . 14 (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥𝐵𝑦𝐵)) ∧ 𝑛 ∈ ℕ0) ∧ 𝑙 ∈ (0...𝑛)) → 𝑙 ∈ ℕ0)
13519, 31, 17, 16, 122, 34, 15, 35, 128, 132, 134gsummoncoe1 20041 . . . . . . . . . . . . 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))
136 csbov2g 6955 . . . . . . . . . . . . . . 15 (𝑙 ∈ (0...𝑛) → 𝑙 / 𝑘(𝑥 decompPMat 𝑘) = (𝑥 decompPMat 𝑙 / 𝑘𝑘))
137 csbvarg 4229 . . . . . . . . . . . . . . . 16 (𝑙 ∈ (0...𝑛) → 𝑙 / 𝑘𝑘 = 𝑙)
138137oveq2d 6926 . . . . . . . . . . . . . . 15 (𝑙 ∈ (0...𝑛) → (𝑥 decompPMat 𝑙 / 𝑘𝑘) = (𝑥 decompPMat 𝑙))
139136, 138eqtrd 2861 . . . . . . . . . . . . . 14 (𝑙 ∈ (0...𝑛) → 𝑙 / 𝑘(𝑥 decompPMat 𝑘) = (𝑥 decompPMat 𝑙))
140139adantl 475 . . . . . . . . . . . . 13 (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥𝐵𝑦𝐵)) ∧ 𝑛 ∈ ℕ0) ∧ 𝑙 ∈ (0...𝑛)) → 𝑙 / 𝑘(𝑥 decompPMat 𝑘) = (𝑥 decompPMat 𝑙))
141135, 140eqtr2d 2862 . . . . . . . . . . . 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))
14210ad3antrrr 721 . . . . . . . . . . . . . . . 16 ((((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥𝐵𝑦𝐵)) ∧ 𝑛 ∈ ℕ0) ∧ 𝑙 ∈ (0...𝑛)) ∧ 𝑘 ∈ ℕ0) → 𝑦𝐵)
1433, 4, 11, 18, 34decpmatcl 20949 . . . . . . . . . . . . . . . 16 ((𝑅 ∈ Ring ∧ 𝑦𝐵𝑘 ∈ ℕ0) → (𝑦 decompPMat 𝑘) ∈ (Base‘𝐴))
144123, 142, 125, 143syl3anc 1494 . . . . . . . . . . . . . . 15 ((((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥𝐵𝑦𝐵)) ∧ 𝑛 ∈ ℕ0) ∧ 𝑙 ∈ (0...𝑛)) ∧ 𝑘 ∈ ℕ0) → (𝑦 decompPMat 𝑘) ∈ (Base‘𝐴))
145144ralrimiva 3175 . . . . . . . . . . . . . 14 (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥𝐵𝑦𝐵)) ∧ 𝑛 ∈ ℕ0) ∧ 𝑙 ∈ (0...𝑛)) → ∀𝑘 ∈ ℕ0 (𝑦 decompPMat 𝑘) ∈ (Base‘𝐴))
1462, 10jca 507 . . . . . . . . . . . . . . . 16 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥𝐵𝑦𝐵)) → (𝑅 ∈ Ring ∧ 𝑦𝐵))
147146ad2antrr 717 . . . . . . . . . . . . . . 15 (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥𝐵𝑦𝐵)) ∧ 𝑛 ∈ ℕ0) ∧ 𝑙 ∈ (0...𝑛)) → (𝑅 ∈ Ring ∧ 𝑦𝐵))
1483, 4, 11, 18, 35decpmatfsupp 20951 . . . . . . . . . . . . . . 15 ((𝑅 ∈ Ring ∧ 𝑦𝐵) → (𝑘 ∈ ℕ0 ↦ (𝑦 decompPMat 𝑘)) finSupp (0g𝐴))
149147, 148syl 17 . . . . . . . . . . . . . 14 (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥𝐵𝑦𝐵)) ∧ 𝑛 ∈ ℕ0) ∧ 𝑙 ∈ (0...𝑛)) → (𝑘 ∈ ℕ0 ↦ (𝑦 decompPMat 𝑘)) finSupp (0g𝐴))
150 fznn0sub 12673 . . . . . . . . . . . . . . 15 (𝑙 ∈ (0...𝑛) → (𝑛𝑙) ∈ ℕ0)
151150adantl 475 . . . . . . . . . . . . . 14 (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥𝐵𝑦𝐵)) ∧ 𝑛 ∈ ℕ0) ∧ 𝑙 ∈ (0...𝑛)) → (𝑛𝑙) ∈ ℕ0)
15219, 31, 17, 16, 122, 34, 15, 35, 145, 149, 151gsummoncoe1 20041 . . . . . . . . . . . . 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))
153 ovex 6942 . . . . . . . . . . . . . 14 (𝑛𝑙) ∈ V
154 csbov2g 6955 . . . . . . . . . . . . . 14 ((𝑛𝑙) ∈ V → (𝑛𝑙) / 𝑘(𝑦 decompPMat 𝑘) = (𝑦 decompPMat (𝑛𝑙) / 𝑘𝑘))
155153, 154mp1i 13 . . . . . . . . . . . . 13 (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥𝐵𝑦𝐵)) ∧ 𝑛 ∈ ℕ0) ∧ 𝑙 ∈ (0...𝑛)) → (𝑛𝑙) / 𝑘(𝑦 decompPMat 𝑘) = (𝑦 decompPMat (𝑛𝑙) / 𝑘𝑘))
156 csbvarg 4229 . . . . . . . . . . . . . . 15 ((𝑛𝑙) ∈ V → (𝑛𝑙) / 𝑘𝑘 = (𝑛𝑙))
157153, 156mp1i 13 . . . . . . . . . . . . . 14 (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥𝐵𝑦𝐵)) ∧ 𝑛 ∈ ℕ0) ∧ 𝑙 ∈ (0...𝑛)) → (𝑛𝑙) / 𝑘𝑘 = (𝑛𝑙))
158157oveq2d 6926 . . . . . . . . . . . . 13 (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥𝐵𝑦𝐵)) ∧ 𝑛 ∈ ℕ0) ∧ 𝑙 ∈ (0...𝑛)) → (𝑦 decompPMat (𝑛𝑙) / 𝑘𝑘) = (𝑦 decompPMat (𝑛𝑙)))
159152, 155, 1583eqtrrd 2866 . . . . . . . . . . . 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)))
160141, 159oveq12d 6928 . . . . . . . . . . 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))))
161160mpteq2dva 4969 . . . . . . . . . 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)))))
162121, 161syl5eq 2873 . . . . . . . . 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)))))
163162oveq2d 6926 . . . . . . . 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))))))
16482, 116, 1633eqtr4rd 2872 . . . . . . 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))
16561, 73, 1643eqtrd 2865 . . . . . 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))
166165ralrimiva 3175 . . . . 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))
16732adantr 474 . . . . . 6 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥𝐵𝑦𝐵)) → 𝐴 ∈ Ring)
16887adantr 474 . . . . . . 7 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥𝐵𝑦𝐵)) → 𝑄 ∈ CMnd)
16989a1i 11 . . . . . . 7 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥𝐵𝑦𝐵)) → ℕ0 ∈ V)
17019ply1lmod 19989 . . . . . . . . . . 11 (𝐴 ∈ Ring → 𝑄 ∈ LMod)
17132, 170syl 17 . . . . . . . . . 10 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝑄 ∈ LMod)
172171ad2antrr 717 . . . . . . . . 9 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥𝐵𝑦𝐵)) ∧ 𝑘 ∈ ℕ0) → 𝑄 ∈ LMod)
17337ad2antrr 717 . . . . . . . . . . 11 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥𝐵𝑦𝐵)) ∧ 𝑘 ∈ ℕ0) → 𝐴 ∈ CMnd)
174 fzfid 13074 . . . . . . . . . . 11 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥𝐵𝑦𝐵)) ∧ 𝑘 ∈ ℕ0) → (0...𝑘) ∈ Fin)
17532ad3antrrr 721 . . . . . . . . . . . . 13 (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥𝐵𝑦𝐵)) ∧ 𝑘 ∈ ℕ0) ∧ 𝑧 ∈ (0...𝑘)) → 𝐴 ∈ Ring)
176 simp-4r 803 . . . . . . . . . . . . . 14 (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥𝐵𝑦𝐵)) ∧ 𝑘 ∈ ℕ0) ∧ 𝑧 ∈ (0...𝑘)) → 𝑅 ∈ Ring)
177 simplrl 795 . . . . . . . . . . . . . . 15 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥𝐵𝑦𝐵)) ∧ 𝑘 ∈ ℕ0) → 𝑥𝐵)
178177adantr 474 . . . . . . . . . . . . . 14 (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥𝐵𝑦𝐵)) ∧ 𝑘 ∈ ℕ0) ∧ 𝑧 ∈ (0...𝑘)) → 𝑥𝐵)
17943adantl 475 . . . . . . . . . . . . . 14 (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥𝐵𝑦𝐵)) ∧ 𝑘 ∈ ℕ0) ∧ 𝑧 ∈ (0...𝑘)) → 𝑧 ∈ ℕ0)
180176, 178, 179, 45syl3anc 1494 . . . . . . . . . . . . 13 (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥𝐵𝑦𝐵)) ∧ 𝑘 ∈ ℕ0) ∧ 𝑧 ∈ (0...𝑘)) → (𝑥 decompPMat 𝑧) ∈ (Base‘𝐴))
181 simplrr 796 . . . . . . . . . . . . . . 15 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥𝐵𝑦𝐵)) ∧ 𝑘 ∈ ℕ0) → 𝑦𝐵)
182181adantr 474 . . . . . . . . . . . . . 14 (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥𝐵𝑦𝐵)) ∧ 𝑘 ∈ ℕ0) ∧ 𝑧 ∈ (0...𝑘)) → 𝑦𝐵)
18348adantl 475 . . . . . . . . . . . . . 14 (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥𝐵𝑦𝐵)) ∧ 𝑘 ∈ ℕ0) ∧ 𝑧 ∈ (0...𝑘)) → (𝑘𝑧) ∈ ℕ0)
184176, 182, 183, 50syl3anc 1494 . . . . . . . . . . . . 13 (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥𝐵𝑦𝐵)) ∧ 𝑘 ∈ ℕ0) ∧ 𝑧 ∈ (0...𝑘)) → (𝑦 decompPMat (𝑘𝑧)) ∈ (Base‘𝐴))
185175, 180, 184, 53syl3anc 1494 . . . . . . . . . . . 12 (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥𝐵𝑦𝐵)) ∧ 𝑘 ∈ ℕ0) ∧ 𝑧 ∈ (0...𝑘)) → ((𝑥 decompPMat 𝑧)(.r𝐴)(𝑦 decompPMat (𝑘𝑧))) ∈ (Base‘𝐴))
186185ralrimiva 3175 . . . . . . . . . . 11 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥𝐵𝑦𝐵)) ∧ 𝑘 ∈ ℕ0) → ∀𝑧 ∈ (0...𝑘)((𝑥 decompPMat 𝑧)(.r𝐴)(𝑦 decompPMat (𝑘𝑧))) ∈ (Base‘𝐴))
18734, 173, 174, 186gsummptcl 18726 . . . . . . . . . 10 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥𝐵𝑦𝐵)) ∧ 𝑘 ∈ ℕ0) → (𝐴 Σg (𝑧 ∈ (0...𝑘) ↦ ((𝑥 decompPMat 𝑧)(.r𝐴)(𝑦 decompPMat (𝑘𝑧))))) ∈ (Base‘𝐴))
18832ad2antrr 717 . . . . . . . . . . . . 13 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥𝐵𝑦𝐵)) ∧ 𝑘 ∈ ℕ0) → 𝐴 ∈ Ring)
18919ply1sca 19990 . . . . . . . . . . . . 13 (𝐴 ∈ Ring → 𝐴 = (Scalar‘𝑄))
190188, 189syl 17 . . . . . . . . . . . 12 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥𝐵𝑦𝐵)) ∧ 𝑘 ∈ ℕ0) → 𝐴 = (Scalar‘𝑄))
191190eqcomd 2831 . . . . . . . . . . 11 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥𝐵𝑦𝐵)) ∧ 𝑘 ∈ ℕ0) → (Scalar‘𝑄) = 𝐴)
192191fveq2d 6441 . . . . . . . . . 10 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥𝐵𝑦𝐵)) ∧ 𝑘 ∈ ℕ0) → (Base‘(Scalar‘𝑄)) = (Base‘𝐴))
193187, 192eleqtrrd 2909 . . . . . . . . 9 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥𝐵𝑦𝐵)) ∧ 𝑘 ∈ ℕ0) → (𝐴 Σg (𝑧 ∈ (0...𝑘) ↦ ((𝑥 decompPMat 𝑧)(.r𝐴)(𝑦 decompPMat (𝑘𝑧))))) ∈ (Base‘(Scalar‘𝑄)))
194 eqid 2825 . . . . . . . . . . 11 (mulGrp‘𝑄) = (mulGrp‘𝑄)
19519, 17, 194, 16, 31ply1moncl 20008 . . . . . . . . . 10 ((𝐴 ∈ Ring ∧ 𝑘 ∈ ℕ0) → (𝑘(.g‘(mulGrp‘𝑄))(var1𝐴)) ∈ (Base‘𝑄))
196188, 195sylancom 582 . . . . . . . . 9 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥𝐵𝑦𝐵)) ∧ 𝑘 ∈ ℕ0) → (𝑘(.g‘(mulGrp‘𝑄))(var1𝐴)) ∈ (Base‘𝑄))
197 eqid 2825 . . . . . . . . . 10 (Scalar‘𝑄) = (Scalar‘𝑄)
198 eqid 2825 . . . . . . . . . 10 (Base‘(Scalar‘𝑄)) = (Base‘(Scalar‘𝑄))
19931, 197, 15, 198lmodvscl 19243 . . . . . . . . 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))
200172, 193, 196, 199syl3anc 1494 . . . . . . . 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))
201200fmpttd 6639 . . . . . . 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))
2023, 4, 11, 15, 16, 17, 18, 19, 31, 20pm2mpmhmlem1 21000 . . . . . . 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))
20331, 83, 168, 169, 201, 202gsumcl 18676 . . . . . 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))
20485adantr 474 . . . . . . 7 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥𝐵𝑦𝐵)) → 𝑄 ∈ Ring)
20593, 95sylan 575 . . . . . . . . 9 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥𝐵𝑦𝐵)) ∧ 𝑘 ∈ ℕ0) → ((𝑥 decompPMat 𝑘)( ·𝑠𝑄)(𝑘(.g‘(mulGrp‘𝑄))(var1𝐴))) ∈ (Base‘𝑄))
206205fmpttd 6639 . . . . . . . 8 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥𝐵𝑦𝐵)) → (𝑘 ∈ ℕ0 ↦ ((𝑥 decompPMat 𝑘)( ·𝑠𝑄)(𝑘(.g‘(mulGrp‘𝑄))(var1𝐴)))):ℕ0⟶(Base‘𝑄))
20793, 98syl 17 . . . . . . . 8 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥𝐵𝑦𝐵)) → (𝑘 ∈ ℕ0 ↦ ((𝑥 decompPMat 𝑘)( ·𝑠𝑄)(𝑘(.g‘(mulGrp‘𝑄))(var1𝐴)))) finSupp (0g𝑄))
20831, 83, 168, 169, 206, 207gsumcl 18676 . . . . . . 7 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥𝐵𝑦𝐵)) → (𝑄 Σg (𝑘 ∈ ℕ0 ↦ ((𝑥 decompPMat 𝑘)( ·𝑠𝑄)(𝑘(.g‘(mulGrp‘𝑄))(var1𝐴))))) ∈ (Base‘𝑄))
209103, 105sylan 575 . . . . . . . . 9 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥𝐵𝑦𝐵)) ∧ 𝑘 ∈ ℕ0) → ((𝑦 decompPMat 𝑘)( ·𝑠𝑄)(𝑘(.g‘(mulGrp‘𝑄))(var1𝐴))) ∈ (Base‘𝑄))
210209fmpttd 6639 . . . . . . . 8 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥𝐵𝑦𝐵)) → (𝑘 ∈ ℕ0 ↦ ((𝑦 decompPMat 𝑘)( ·𝑠𝑄)(𝑘(.g‘(mulGrp‘𝑄))(var1𝐴)))):ℕ0⟶(Base‘𝑄))
2111, 2, 10, 110syl3anc 1494 . . . . . . . 8 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥𝐵𝑦𝐵)) → (𝑘 ∈ ℕ0 ↦ ((𝑦 decompPMat 𝑘)( ·𝑠𝑄)(𝑘(.g‘(mulGrp‘𝑄))(var1𝐴)))) finSupp (0g𝑄))
21231, 83, 168, 169, 210, 211gsumcl 18676 . . . . . . 7 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥𝐵𝑦𝐵)) → (𝑄 Σg (𝑘 ∈ ℕ0 ↦ ((𝑦 decompPMat 𝑘)( ·𝑠𝑄)(𝑘(.g‘(mulGrp‘𝑄))(var1𝐴))))) ∈ (Base‘𝑄))
21331, 113ringcl 18922 . . . . . . 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))
214204, 208, 212, 213syl3anc 1494 . . . . . 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))
215 eqid 2825 . . . . . . 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))))))
216 eqid 2825 . . . . . . 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)))))))
21719, 31, 215, 216ply1coe1eq 20035 . . . . . 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))))))))
218167, 203, 214, 217syl3anc 1494 . . . . 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))))))))
219166, 218mpbid 224 . . . 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)))))))
22022, 30, 2193eqtrd 2865 . . 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)))))))
2213, 4, 11, 15, 16, 17, 18, 19, 20pm2mpfval 20978 . . . . 5 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑥𝐵) → (𝑇𝑥) = (𝑄 Σg (𝑘 ∈ ℕ0 ↦ ((𝑥 decompPMat 𝑘)( ·𝑠𝑄)(𝑘(.g‘(mulGrp‘𝑄))(var1𝐴))))))
2221, 2, 8, 221syl3anc 1494 . . . 4 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥𝐵𝑦𝐵)) → (𝑇𝑥) = (𝑄 Σg (𝑘 ∈ ℕ0 ↦ ((𝑥 decompPMat 𝑘)( ·𝑠𝑄)(𝑘(.g‘(mulGrp‘𝑄))(var1𝐴))))))
2233, 4, 11, 15, 16, 17, 18, 19, 20pm2mpfval 20978 . . . . 5 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑦𝐵) → (𝑇𝑦) = (𝑄 Σg (𝑘 ∈ ℕ0 ↦ ((𝑦 decompPMat 𝑘)( ·𝑠𝑄)(𝑘(.g‘(mulGrp‘𝑄))(var1𝐴))))))
2241, 2, 10, 223syl3anc 1494 . . . 4 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥𝐵𝑦𝐵)) → (𝑇𝑦) = (𝑄 Σg (𝑘 ∈ ℕ0 ↦ ((𝑦 decompPMat 𝑘)( ·𝑠𝑄)(𝑘(.g‘(mulGrp‘𝑄))(var1𝐴))))))
225222, 224oveq12d 6928 . . 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)))))))
226220, 225eqtr4d 2864 . 2 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥𝐵𝑦𝐵)) → (𝑇‘(𝑥(.r𝐶)𝑦)) = ((𝑇𝑥)(.r𝑄)(𝑇𝑦)))
227226ralrimivva 3180 1 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → ∀𝑥𝐵𝑦𝐵 (𝑇‘(𝑥(.r𝐶)𝑦)) = ((𝑇𝑥)(.r𝑄)(𝑇𝑦)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 198  wa 386  w3a 1111   = wceq 1656  wcel 2164  wral 3117  Vcvv 3414  csb 3757   class class class wbr 4875  cmpt 4954  cfv 6127  (class class class)co 6910  Fincfn 8228   finSupp cfsupp 8550  0cc0 10259  cmin 10592  0cn0 11625  ...cfz 12626  Basecbs 16229  .rcmulr 16313  Scalarcsca 16315   ·𝑠 cvsca 16316  0gc0g 16460   Σg cgsu 16461  .gcmg 17901  CMndccmn 18553  mulGrpcmgp 18850  Ringcrg 18908  LModclmod 19226  var1cv1 19913  Poly1cpl1 19914  coe1cco1 19915   Mat cmat 20587   decompPMat cdecpmat 20944   pMatToMatPoly cpm2mp 20974
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1894  ax-4 1908  ax-5 2009  ax-6 2075  ax-7 2112  ax-8 2166  ax-9 2173  ax-10 2192  ax-11 2207  ax-12 2220  ax-13 2389  ax-ext 2803  ax-rep 4996  ax-sep 5007  ax-nul 5015  ax-pow 5067  ax-pr 5129  ax-un 7214  ax-inf2 8822  ax-cnex 10315  ax-resscn 10316  ax-1cn 10317  ax-icn 10318  ax-addcl 10319  ax-addrcl 10320  ax-mulcl 10321  ax-mulrcl 10322  ax-mulcom 10323  ax-addass 10324  ax-mulass 10325  ax-distr 10326  ax-i2m1 10327  ax-1ne0 10328  ax-1rid 10329  ax-rnegex 10330  ax-rrecex 10331  ax-cnre 10332  ax-pre-lttri 10333  ax-pre-lttrn 10334  ax-pre-ltadd 10335  ax-pre-mulgt0 10336
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 879  df-3or 1112  df-3an 1113  df-tru 1660  df-fal 1670  df-ex 1879  df-nf 1883  df-sb 2068  df-mo 2605  df-eu 2640  df-clab 2812  df-cleq 2818  df-clel 2821  df-nfc 2958  df-ne 3000  df-nel 3103  df-ral 3122  df-rex 3123  df-reu 3124  df-rmo 3125  df-rab 3126  df-v 3416  df-sbc 3663  df-csb 3758  df-dif 3801  df-un 3803  df-in 3805  df-ss 3812  df-pss 3814  df-nul 4147  df-if 4309  df-pw 4382  df-sn 4400  df-pr 4402  df-tp 4404  df-op 4406  df-ot 4408  df-uni 4661  df-int 4700  df-iun 4744  df-iin 4745  df-br 4876  df-opab 4938  df-mpt 4955  df-tr 4978  df-id 5252  df-eprel 5257  df-po 5265  df-so 5266  df-fr 5305  df-se 5306  df-we 5307  df-xp 5352  df-rel 5353  df-cnv 5354  df-co 5355  df-dm 5356  df-rn 5357  df-res 5358  df-ima 5359  df-pred 5924  df-ord 5970  df-on 5971  df-lim 5972  df-suc 5973  df-iota 6090  df-fun 6129  df-fn 6130  df-f 6131  df-f1 6132  df-fo 6133  df-f1o 6134  df-fv 6135  df-isom 6136  df-riota 6871  df-ov 6913  df-oprab 6914  df-mpt2 6915  df-of 7162  df-ofr 7163  df-om 7332  df-1st 7433  df-2nd 7434  df-supp 7565  df-wrecs 7677  df-recs 7739  df-rdg 7777  df-1o 7831  df-2o 7832  df-oadd 7835  df-er 8014  df-map 8129  df-pm 8130  df-ixp 8182  df-en 8229  df-dom 8230  df-sdom 8231  df-fin 8232  df-fsupp 8551  df-sup 8623  df-oi 8691  df-card 9085  df-pnf 10400  df-mnf 10401  df-xr 10402  df-ltxr 10403  df-le 10404  df-sub 10594  df-neg 10595  df-nn 11358  df-2 11421  df-3 11422  df-4 11423  df-5 11424  df-6 11425  df-7 11426  df-8 11427  df-9 11428  df-n0 11626  df-z 11712  df-dec 11829  df-uz 11976  df-fz 12627  df-fzo 12768  df-seq 13103  df-hash 13418  df-struct 16231  df-ndx 16232  df-slot 16233  df-base 16235  df-sets 16236  df-ress 16237  df-plusg 16325  df-mulr 16326  df-sca 16328  df-vsca 16329  df-ip 16330  df-tset 16331  df-ple 16332  df-ds 16334  df-hom 16336  df-cco 16337  df-0g 16462  df-gsum 16463  df-prds 16468  df-pws 16470  df-mre 16606  df-mrc 16607  df-acs 16609  df-mgm 17602  df-sgrp 17644  df-mnd 17655  df-mhm 17695  df-submnd 17696  df-grp 17786  df-minusg 17787  df-sbg 17788  df-mulg 17902  df-subg 17949  df-ghm 18016  df-cntz 18107  df-cmn 18555  df-abl 18556  df-mgp 18851  df-ur 18863  df-srg 18867  df-ring 18910  df-subrg 19141  df-lmod 19228  df-lss 19296  df-sra 19540  df-rgmod 19541  df-psr 19724  df-mvr 19725  df-mpl 19726  df-opsr 19728  df-psr1 19917  df-vr1 19918  df-ply1 19919  df-coe1 19920  df-dsmm 20446  df-frlm 20461  df-mamu 20564  df-mat 20588  df-decpmat 20945  df-pm2mp 20975
This theorem is referenced by:  pm2mpmhm  21002
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