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Theorem decpmatmul 22779
Description: The matrix consisting of the coefficients in the polynomial entries of the product of two polynomial matrices is a sum of products of the matrices consisting of the coefficients in the polynomial entries of the polynomial matrices for the same power. (Contributed by AV, 21-Oct-2019.) (Revised by AV, 3-Dec-2019.)
Hypotheses
Ref Expression
decpmatmul.p 𝑃 = (Poly1𝑅)
decpmatmul.c 𝐶 = (𝑁 Mat 𝑃)
decpmatmul.b 𝐵 = (Base‘𝐶)
decpmatmul.a 𝐴 = (𝑁 Mat 𝑅)
Assertion
Ref Expression
decpmatmul ((𝑅 ∈ Ring ∧ (𝑈𝐵𝑊𝐵) ∧ 𝐾 ∈ ℕ0) → ((𝑈(.r𝐶)𝑊) decompPMat 𝐾) = (𝐴 Σg (𝑘 ∈ (0...𝐾) ↦ ((𝑈 decompPMat 𝑘)(.r𝐴)(𝑊 decompPMat (𝐾𝑘))))))
Distinct variable groups:   𝐵,𝑘   𝑘,𝐾   𝑘,𝑁   𝑃,𝑘   𝑅,𝑘   𝑈,𝑘   𝑘,𝑊   𝐴,𝑘
Allowed substitution hint:   𝐶(𝑘)

Proof of Theorem decpmatmul
Dummy variables 𝑡 𝑖 𝑗 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqidd 2737 . . . . 5 (((𝑅 ∈ Ring ∧ (𝑈𝐵𝑊𝐵) ∧ 𝐾 ∈ ℕ0) ∧ (𝑖𝑁𝑗𝑁)) → (𝑥𝑁, 𝑦𝑁 ↦ (𝑅 Σg (𝑘 ∈ (0...𝐾) ↦ (𝑅 Σg (𝑡𝑁 ↦ ((𝑥(𝑈 decompPMat 𝑘)𝑡)(.r𝑅)(𝑡(𝑊 decompPMat (𝐾𝑘))𝑦))))))) = (𝑥𝑁, 𝑦𝑁 ↦ (𝑅 Σg (𝑘 ∈ (0...𝐾) ↦ (𝑅 Σg (𝑡𝑁 ↦ ((𝑥(𝑈 decompPMat 𝑘)𝑡)(.r𝑅)(𝑡(𝑊 decompPMat (𝐾𝑘))𝑦))))))))
2 oveq1 7439 . . . . . . . . . . 11 (𝑥 = 𝑖 → (𝑥(𝑈 decompPMat 𝑘)𝑡) = (𝑖(𝑈 decompPMat 𝑘)𝑡))
3 oveq2 7440 . . . . . . . . . . 11 (𝑦 = 𝑗 → (𝑡(𝑊 decompPMat (𝐾𝑘))𝑦) = (𝑡(𝑊 decompPMat (𝐾𝑘))𝑗))
42, 3oveqan12d 7451 . . . . . . . . . 10 ((𝑥 = 𝑖𝑦 = 𝑗) → ((𝑥(𝑈 decompPMat 𝑘)𝑡)(.r𝑅)(𝑡(𝑊 decompPMat (𝐾𝑘))𝑦)) = ((𝑖(𝑈 decompPMat 𝑘)𝑡)(.r𝑅)(𝑡(𝑊 decompPMat (𝐾𝑘))𝑗)))
54mpteq2dv 5243 . . . . . . . . 9 ((𝑥 = 𝑖𝑦 = 𝑗) → (𝑡𝑁 ↦ ((𝑥(𝑈 decompPMat 𝑘)𝑡)(.r𝑅)(𝑡(𝑊 decompPMat (𝐾𝑘))𝑦))) = (𝑡𝑁 ↦ ((𝑖(𝑈 decompPMat 𝑘)𝑡)(.r𝑅)(𝑡(𝑊 decompPMat (𝐾𝑘))𝑗))))
65oveq2d 7448 . . . . . . . 8 ((𝑥 = 𝑖𝑦 = 𝑗) → (𝑅 Σg (𝑡𝑁 ↦ ((𝑥(𝑈 decompPMat 𝑘)𝑡)(.r𝑅)(𝑡(𝑊 decompPMat (𝐾𝑘))𝑦)))) = (𝑅 Σg (𝑡𝑁 ↦ ((𝑖(𝑈 decompPMat 𝑘)𝑡)(.r𝑅)(𝑡(𝑊 decompPMat (𝐾𝑘))𝑗)))))
76mpteq2dv 5243 . . . . . . 7 ((𝑥 = 𝑖𝑦 = 𝑗) → (𝑘 ∈ (0...𝐾) ↦ (𝑅 Σg (𝑡𝑁 ↦ ((𝑥(𝑈 decompPMat 𝑘)𝑡)(.r𝑅)(𝑡(𝑊 decompPMat (𝐾𝑘))𝑦))))) = (𝑘 ∈ (0...𝐾) ↦ (𝑅 Σg (𝑡𝑁 ↦ ((𝑖(𝑈 decompPMat 𝑘)𝑡)(.r𝑅)(𝑡(𝑊 decompPMat (𝐾𝑘))𝑗))))))
87oveq2d 7448 . . . . . 6 ((𝑥 = 𝑖𝑦 = 𝑗) → (𝑅 Σg (𝑘 ∈ (0...𝐾) ↦ (𝑅 Σg (𝑡𝑁 ↦ ((𝑥(𝑈 decompPMat 𝑘)𝑡)(.r𝑅)(𝑡(𝑊 decompPMat (𝐾𝑘))𝑦)))))) = (𝑅 Σg (𝑘 ∈ (0...𝐾) ↦ (𝑅 Σg (𝑡𝑁 ↦ ((𝑖(𝑈 decompPMat 𝑘)𝑡)(.r𝑅)(𝑡(𝑊 decompPMat (𝐾𝑘))𝑗)))))))
98adantl 481 . . . . 5 ((((𝑅 ∈ Ring ∧ (𝑈𝐵𝑊𝐵) ∧ 𝐾 ∈ ℕ0) ∧ (𝑖𝑁𝑗𝑁)) ∧ (𝑥 = 𝑖𝑦 = 𝑗)) → (𝑅 Σg (𝑘 ∈ (0...𝐾) ↦ (𝑅 Σg (𝑡𝑁 ↦ ((𝑥(𝑈 decompPMat 𝑘)𝑡)(.r𝑅)(𝑡(𝑊 decompPMat (𝐾𝑘))𝑦)))))) = (𝑅 Σg (𝑘 ∈ (0...𝐾) ↦ (𝑅 Σg (𝑡𝑁 ↦ ((𝑖(𝑈 decompPMat 𝑘)𝑡)(.r𝑅)(𝑡(𝑊 decompPMat (𝐾𝑘))𝑗)))))))
10 simprl 770 . . . . 5 (((𝑅 ∈ Ring ∧ (𝑈𝐵𝑊𝐵) ∧ 𝐾 ∈ ℕ0) ∧ (𝑖𝑁𝑗𝑁)) → 𝑖𝑁)
11 simprr 772 . . . . 5 (((𝑅 ∈ Ring ∧ (𝑈𝐵𝑊𝐵) ∧ 𝐾 ∈ ℕ0) ∧ (𝑖𝑁𝑗𝑁)) → 𝑗𝑁)
12 ovexd 7467 . . . . 5 (((𝑅 ∈ Ring ∧ (𝑈𝐵𝑊𝐵) ∧ 𝐾 ∈ ℕ0) ∧ (𝑖𝑁𝑗𝑁)) → (𝑅 Σg (𝑘 ∈ (0...𝐾) ↦ (𝑅 Σg (𝑡𝑁 ↦ ((𝑖(𝑈 decompPMat 𝑘)𝑡)(.r𝑅)(𝑡(𝑊 decompPMat (𝐾𝑘))𝑗)))))) ∈ V)
131, 9, 10, 11, 12ovmpod 7586 . . . 4 (((𝑅 ∈ Ring ∧ (𝑈𝐵𝑊𝐵) ∧ 𝐾 ∈ ℕ0) ∧ (𝑖𝑁𝑗𝑁)) → (𝑖(𝑥𝑁, 𝑦𝑁 ↦ (𝑅 Σg (𝑘 ∈ (0...𝐾) ↦ (𝑅 Σg (𝑡𝑁 ↦ ((𝑥(𝑈 decompPMat 𝑘)𝑡)(.r𝑅)(𝑡(𝑊 decompPMat (𝐾𝑘))𝑦)))))))𝑗) = (𝑅 Σg (𝑘 ∈ (0...𝐾) ↦ (𝑅 Σg (𝑡𝑁 ↦ ((𝑖(𝑈 decompPMat 𝑘)𝑡)(.r𝑅)(𝑡(𝑊 decompPMat (𝐾𝑘))𝑗)))))))
14 decpmatmul.c . . . . . . . . . . . . . . . . . . . 20 𝐶 = (𝑁 Mat 𝑃)
15 decpmatmul.b . . . . . . . . . . . . . . . . . . . 20 𝐵 = (Base‘𝐶)
1614, 15matrcl 22417 . . . . . . . . . . . . . . . . . . 19 (𝑈𝐵 → (𝑁 ∈ Fin ∧ 𝑃 ∈ V))
1716simpld 494 . . . . . . . . . . . . . . . . . 18 (𝑈𝐵𝑁 ∈ Fin)
1817adantr 480 . . . . . . . . . . . . . . . . 17 ((𝑈𝐵𝑊𝐵) → 𝑁 ∈ Fin)
1918anim2i 617 . . . . . . . . . . . . . . . 16 ((𝑅 ∈ Ring ∧ (𝑈𝐵𝑊𝐵)) → (𝑅 ∈ Ring ∧ 𝑁 ∈ Fin))
2019ancomd 461 . . . . . . . . . . . . . . 15 ((𝑅 ∈ Ring ∧ (𝑈𝐵𝑊𝐵)) → (𝑁 ∈ Fin ∧ 𝑅 ∈ Ring))
21203adant3 1132 . . . . . . . . . . . . . 14 ((𝑅 ∈ Ring ∧ (𝑈𝐵𝑊𝐵) ∧ 𝐾 ∈ ℕ0) → (𝑁 ∈ Fin ∧ 𝑅 ∈ Ring))
22 decpmatmul.a . . . . . . . . . . . . . . 15 𝐴 = (𝑁 Mat 𝑅)
23 eqid 2736 . . . . . . . . . . . . . . 15 (𝑅 maMul ⟨𝑁, 𝑁, 𝑁⟩) = (𝑅 maMul ⟨𝑁, 𝑁, 𝑁⟩)
2422, 23matmulr 22445 . . . . . . . . . . . . . 14 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → (𝑅 maMul ⟨𝑁, 𝑁, 𝑁⟩) = (.r𝐴))
2521, 24syl 17 . . . . . . . . . . . . 13 ((𝑅 ∈ Ring ∧ (𝑈𝐵𝑊𝐵) ∧ 𝐾 ∈ ℕ0) → (𝑅 maMul ⟨𝑁, 𝑁, 𝑁⟩) = (.r𝐴))
2625adantr 480 . . . . . . . . . . . 12 (((𝑅 ∈ Ring ∧ (𝑈𝐵𝑊𝐵) ∧ 𝐾 ∈ ℕ0) ∧ (𝑖𝑁𝑗𝑁)) → (𝑅 maMul ⟨𝑁, 𝑁, 𝑁⟩) = (.r𝐴))
2726adantr 480 . . . . . . . . . . 11 ((((𝑅 ∈ Ring ∧ (𝑈𝐵𝑊𝐵) ∧ 𝐾 ∈ ℕ0) ∧ (𝑖𝑁𝑗𝑁)) ∧ 𝑘 ∈ (0...𝐾)) → (𝑅 maMul ⟨𝑁, 𝑁, 𝑁⟩) = (.r𝐴))
2827eqcomd 2742 . . . . . . . . . 10 ((((𝑅 ∈ Ring ∧ (𝑈𝐵𝑊𝐵) ∧ 𝐾 ∈ ℕ0) ∧ (𝑖𝑁𝑗𝑁)) ∧ 𝑘 ∈ (0...𝐾)) → (.r𝐴) = (𝑅 maMul ⟨𝑁, 𝑁, 𝑁⟩))
2928oveqd 7449 . . . . . . . . 9 ((((𝑅 ∈ Ring ∧ (𝑈𝐵𝑊𝐵) ∧ 𝐾 ∈ ℕ0) ∧ (𝑖𝑁𝑗𝑁)) ∧ 𝑘 ∈ (0...𝐾)) → ((𝑈 decompPMat 𝑘)(.r𝐴)(𝑊 decompPMat (𝐾𝑘))) = ((𝑈 decompPMat 𝑘)(𝑅 maMul ⟨𝑁, 𝑁, 𝑁⟩)(𝑊 decompPMat (𝐾𝑘))))
30 eqid 2736 . . . . . . . . . 10 (Base‘𝑅) = (Base‘𝑅)
31 eqid 2736 . . . . . . . . . 10 (.r𝑅) = (.r𝑅)
32 simp1 1136 . . . . . . . . . . . 12 ((𝑅 ∈ Ring ∧ (𝑈𝐵𝑊𝐵) ∧ 𝐾 ∈ ℕ0) → 𝑅 ∈ Ring)
3332adantr 480 . . . . . . . . . . 11 (((𝑅 ∈ Ring ∧ (𝑈𝐵𝑊𝐵) ∧ 𝐾 ∈ ℕ0) ∧ (𝑖𝑁𝑗𝑁)) → 𝑅 ∈ Ring)
3433adantr 480 . . . . . . . . . 10 ((((𝑅 ∈ Ring ∧ (𝑈𝐵𝑊𝐵) ∧ 𝐾 ∈ ℕ0) ∧ (𝑖𝑁𝑗𝑁)) ∧ 𝑘 ∈ (0...𝐾)) → 𝑅 ∈ Ring)
3521simpld 494 . . . . . . . . . . . 12 ((𝑅 ∈ Ring ∧ (𝑈𝐵𝑊𝐵) ∧ 𝐾 ∈ ℕ0) → 𝑁 ∈ Fin)
3635adantr 480 . . . . . . . . . . 11 (((𝑅 ∈ Ring ∧ (𝑈𝐵𝑊𝐵) ∧ 𝐾 ∈ ℕ0) ∧ (𝑖𝑁𝑗𝑁)) → 𝑁 ∈ Fin)
3736adantr 480 . . . . . . . . . 10 ((((𝑅 ∈ Ring ∧ (𝑈𝐵𝑊𝐵) ∧ 𝐾 ∈ ℕ0) ∧ (𝑖𝑁𝑗𝑁)) ∧ 𝑘 ∈ (0...𝐾)) → 𝑁 ∈ Fin)
38 simpl2l 1226 . . . . . . . . . . . . . 14 (((𝑅 ∈ Ring ∧ (𝑈𝐵𝑊𝐵) ∧ 𝐾 ∈ ℕ0) ∧ (𝑖𝑁𝑗𝑁)) → 𝑈𝐵)
3938adantr 480 . . . . . . . . . . . . 13 ((((𝑅 ∈ Ring ∧ (𝑈𝐵𝑊𝐵) ∧ 𝐾 ∈ ℕ0) ∧ (𝑖𝑁𝑗𝑁)) ∧ 𝑘 ∈ (0...𝐾)) → 𝑈𝐵)
40 elfznn0 13661 . . . . . . . . . . . . . 14 (𝑘 ∈ (0...𝐾) → 𝑘 ∈ ℕ0)
4140adantl 481 . . . . . . . . . . . . 13 ((((𝑅 ∈ Ring ∧ (𝑈𝐵𝑊𝐵) ∧ 𝐾 ∈ ℕ0) ∧ (𝑖𝑁𝑗𝑁)) ∧ 𝑘 ∈ (0...𝐾)) → 𝑘 ∈ ℕ0)
4234, 39, 413jca 1128 . . . . . . . . . . . 12 ((((𝑅 ∈ Ring ∧ (𝑈𝐵𝑊𝐵) ∧ 𝐾 ∈ ℕ0) ∧ (𝑖𝑁𝑗𝑁)) ∧ 𝑘 ∈ (0...𝐾)) → (𝑅 ∈ Ring ∧ 𝑈𝐵𝑘 ∈ ℕ0))
43 decpmatmul.p . . . . . . . . . . . . 13 𝑃 = (Poly1𝑅)
44 eqid 2736 . . . . . . . . . . . . 13 (Base‘𝐴) = (Base‘𝐴)
4543, 14, 15, 22, 44decpmatcl 22774 . . . . . . . . . . . 12 ((𝑅 ∈ Ring ∧ 𝑈𝐵𝑘 ∈ ℕ0) → (𝑈 decompPMat 𝑘) ∈ (Base‘𝐴))
4642, 45syl 17 . . . . . . . . . . 11 ((((𝑅 ∈ Ring ∧ (𝑈𝐵𝑊𝐵) ∧ 𝐾 ∈ ℕ0) ∧ (𝑖𝑁𝑗𝑁)) ∧ 𝑘 ∈ (0...𝐾)) → (𝑈 decompPMat 𝑘) ∈ (Base‘𝐴))
4722, 30, 44matbas2i 22429 . . . . . . . . . . 11 ((𝑈 decompPMat 𝑘) ∈ (Base‘𝐴) → (𝑈 decompPMat 𝑘) ∈ ((Base‘𝑅) ↑m (𝑁 × 𝑁)))
4846, 47syl 17 . . . . . . . . . 10 ((((𝑅 ∈ Ring ∧ (𝑈𝐵𝑊𝐵) ∧ 𝐾 ∈ ℕ0) ∧ (𝑖𝑁𝑗𝑁)) ∧ 𝑘 ∈ (0...𝐾)) → (𝑈 decompPMat 𝑘) ∈ ((Base‘𝑅) ↑m (𝑁 × 𝑁)))
49 simpl2r 1227 . . . . . . . . . . . . . 14 (((𝑅 ∈ Ring ∧ (𝑈𝐵𝑊𝐵) ∧ 𝐾 ∈ ℕ0) ∧ (𝑖𝑁𝑗𝑁)) → 𝑊𝐵)
5049adantr 480 . . . . . . . . . . . . 13 ((((𝑅 ∈ Ring ∧ (𝑈𝐵𝑊𝐵) ∧ 𝐾 ∈ ℕ0) ∧ (𝑖𝑁𝑗𝑁)) ∧ 𝑘 ∈ (0...𝐾)) → 𝑊𝐵)
51 fznn0sub 13597 . . . . . . . . . . . . . 14 (𝑘 ∈ (0...𝐾) → (𝐾𝑘) ∈ ℕ0)
5251adantl 481 . . . . . . . . . . . . 13 ((((𝑅 ∈ Ring ∧ (𝑈𝐵𝑊𝐵) ∧ 𝐾 ∈ ℕ0) ∧ (𝑖𝑁𝑗𝑁)) ∧ 𝑘 ∈ (0...𝐾)) → (𝐾𝑘) ∈ ℕ0)
5334, 50, 523jca 1128 . . . . . . . . . . . 12 ((((𝑅 ∈ Ring ∧ (𝑈𝐵𝑊𝐵) ∧ 𝐾 ∈ ℕ0) ∧ (𝑖𝑁𝑗𝑁)) ∧ 𝑘 ∈ (0...𝐾)) → (𝑅 ∈ Ring ∧ 𝑊𝐵 ∧ (𝐾𝑘) ∈ ℕ0))
5443, 14, 15, 22, 44decpmatcl 22774 . . . . . . . . . . . 12 ((𝑅 ∈ Ring ∧ 𝑊𝐵 ∧ (𝐾𝑘) ∈ ℕ0) → (𝑊 decompPMat (𝐾𝑘)) ∈ (Base‘𝐴))
5553, 54syl 17 . . . . . . . . . . 11 ((((𝑅 ∈ Ring ∧ (𝑈𝐵𝑊𝐵) ∧ 𝐾 ∈ ℕ0) ∧ (𝑖𝑁𝑗𝑁)) ∧ 𝑘 ∈ (0...𝐾)) → (𝑊 decompPMat (𝐾𝑘)) ∈ (Base‘𝐴))
5622, 30, 44matbas2i 22429 . . . . . . . . . . 11 ((𝑊 decompPMat (𝐾𝑘)) ∈ (Base‘𝐴) → (𝑊 decompPMat (𝐾𝑘)) ∈ ((Base‘𝑅) ↑m (𝑁 × 𝑁)))
5755, 56syl 17 . . . . . . . . . 10 ((((𝑅 ∈ Ring ∧ (𝑈𝐵𝑊𝐵) ∧ 𝐾 ∈ ℕ0) ∧ (𝑖𝑁𝑗𝑁)) ∧ 𝑘 ∈ (0...𝐾)) → (𝑊 decompPMat (𝐾𝑘)) ∈ ((Base‘𝑅) ↑m (𝑁 × 𝑁)))
5823, 30, 31, 34, 37, 37, 37, 48, 57mamuval 22398 . . . . . . . . 9 ((((𝑅 ∈ Ring ∧ (𝑈𝐵𝑊𝐵) ∧ 𝐾 ∈ ℕ0) ∧ (𝑖𝑁𝑗𝑁)) ∧ 𝑘 ∈ (0...𝐾)) → ((𝑈 decompPMat 𝑘)(𝑅 maMul ⟨𝑁, 𝑁, 𝑁⟩)(𝑊 decompPMat (𝐾𝑘))) = (𝑥𝑁, 𝑦𝑁 ↦ (𝑅 Σg (𝑡𝑁 ↦ ((𝑥(𝑈 decompPMat 𝑘)𝑡)(.r𝑅)(𝑡(𝑊 decompPMat (𝐾𝑘))𝑦))))))
5929, 58eqtrd 2776 . . . . . . . 8 ((((𝑅 ∈ Ring ∧ (𝑈𝐵𝑊𝐵) ∧ 𝐾 ∈ ℕ0) ∧ (𝑖𝑁𝑗𝑁)) ∧ 𝑘 ∈ (0...𝐾)) → ((𝑈 decompPMat 𝑘)(.r𝐴)(𝑊 decompPMat (𝐾𝑘))) = (𝑥𝑁, 𝑦𝑁 ↦ (𝑅 Σg (𝑡𝑁 ↦ ((𝑥(𝑈 decompPMat 𝑘)𝑡)(.r𝑅)(𝑡(𝑊 decompPMat (𝐾𝑘))𝑦))))))
6059mpteq2dva 5241 . . . . . . 7 (((𝑅 ∈ Ring ∧ (𝑈𝐵𝑊𝐵) ∧ 𝐾 ∈ ℕ0) ∧ (𝑖𝑁𝑗𝑁)) → (𝑘 ∈ (0...𝐾) ↦ ((𝑈 decompPMat 𝑘)(.r𝐴)(𝑊 decompPMat (𝐾𝑘)))) = (𝑘 ∈ (0...𝐾) ↦ (𝑥𝑁, 𝑦𝑁 ↦ (𝑅 Σg (𝑡𝑁 ↦ ((𝑥(𝑈 decompPMat 𝑘)𝑡)(.r𝑅)(𝑡(𝑊 decompPMat (𝐾𝑘))𝑦)))))))
6160oveq2d 7448 . . . . . 6 (((𝑅 ∈ Ring ∧ (𝑈𝐵𝑊𝐵) ∧ 𝐾 ∈ ℕ0) ∧ (𝑖𝑁𝑗𝑁)) → (𝐴 Σg (𝑘 ∈ (0...𝐾) ↦ ((𝑈 decompPMat 𝑘)(.r𝐴)(𝑊 decompPMat (𝐾𝑘))))) = (𝐴 Σg (𝑘 ∈ (0...𝐾) ↦ (𝑥𝑁, 𝑦𝑁 ↦ (𝑅 Σg (𝑡𝑁 ↦ ((𝑥(𝑈 decompPMat 𝑘)𝑡)(.r𝑅)(𝑡(𝑊 decompPMat (𝐾𝑘))𝑦))))))))
62 eqid 2736 . . . . . . 7 (0g𝐴) = (0g𝐴)
63 ovexd 7467 . . . . . . 7 (((𝑅 ∈ Ring ∧ (𝑈𝐵𝑊𝐵) ∧ 𝐾 ∈ ℕ0) ∧ (𝑖𝑁𝑗𝑁)) → (0...𝐾) ∈ V)
64 ringcmn 20280 . . . . . . . . . . . . 13 (𝑅 ∈ Ring → 𝑅 ∈ CMnd)
6532, 64syl 17 . . . . . . . . . . . 12 ((𝑅 ∈ Ring ∧ (𝑈𝐵𝑊𝐵) ∧ 𝐾 ∈ ℕ0) → 𝑅 ∈ CMnd)
6665adantr 480 . . . . . . . . . . 11 (((𝑅 ∈ Ring ∧ (𝑈𝐵𝑊𝐵) ∧ 𝐾 ∈ ℕ0) ∧ (𝑖𝑁𝑗𝑁)) → 𝑅 ∈ CMnd)
6766adantr 480 . . . . . . . . . 10 ((((𝑅 ∈ Ring ∧ (𝑈𝐵𝑊𝐵) ∧ 𝐾 ∈ ℕ0) ∧ (𝑖𝑁𝑗𝑁)) ∧ 𝑘 ∈ (0...𝐾)) → 𝑅 ∈ CMnd)
68673ad2ant1 1133 . . . . . . . . 9 (((((𝑅 ∈ Ring ∧ (𝑈𝐵𝑊𝐵) ∧ 𝐾 ∈ ℕ0) ∧ (𝑖𝑁𝑗𝑁)) ∧ 𝑘 ∈ (0...𝐾)) ∧ 𝑥𝑁𝑦𝑁) → 𝑅 ∈ CMnd)
69373ad2ant1 1133 . . . . . . . . 9 (((((𝑅 ∈ Ring ∧ (𝑈𝐵𝑊𝐵) ∧ 𝐾 ∈ ℕ0) ∧ (𝑖𝑁𝑗𝑁)) ∧ 𝑘 ∈ (0...𝐾)) ∧ 𝑥𝑁𝑦𝑁) → 𝑁 ∈ Fin)
70343ad2ant1 1133 . . . . . . . . . . . 12 (((((𝑅 ∈ Ring ∧ (𝑈𝐵𝑊𝐵) ∧ 𝐾 ∈ ℕ0) ∧ (𝑖𝑁𝑗𝑁)) ∧ 𝑘 ∈ (0...𝐾)) ∧ 𝑥𝑁𝑦𝑁) → 𝑅 ∈ Ring)
7170adantr 480 . . . . . . . . . . 11 ((((((𝑅 ∈ Ring ∧ (𝑈𝐵𝑊𝐵) ∧ 𝐾 ∈ ℕ0) ∧ (𝑖𝑁𝑗𝑁)) ∧ 𝑘 ∈ (0...𝐾)) ∧ 𝑥𝑁𝑦𝑁) ∧ 𝑡𝑁) → 𝑅 ∈ Ring)
72 simpl2 1192 . . . . . . . . . . . 12 ((((((𝑅 ∈ Ring ∧ (𝑈𝐵𝑊𝐵) ∧ 𝐾 ∈ ℕ0) ∧ (𝑖𝑁𝑗𝑁)) ∧ 𝑘 ∈ (0...𝐾)) ∧ 𝑥𝑁𝑦𝑁) ∧ 𝑡𝑁) → 𝑥𝑁)
73 simpr 484 . . . . . . . . . . . 12 ((((((𝑅 ∈ Ring ∧ (𝑈𝐵𝑊𝐵) ∧ 𝐾 ∈ ℕ0) ∧ (𝑖𝑁𝑗𝑁)) ∧ 𝑘 ∈ (0...𝐾)) ∧ 𝑥𝑁𝑦𝑁) ∧ 𝑡𝑁) → 𝑡𝑁)
74423ad2ant1 1133 . . . . . . . . . . . . . 14 (((((𝑅 ∈ Ring ∧ (𝑈𝐵𝑊𝐵) ∧ 𝐾 ∈ ℕ0) ∧ (𝑖𝑁𝑗𝑁)) ∧ 𝑘 ∈ (0...𝐾)) ∧ 𝑥𝑁𝑦𝑁) → (𝑅 ∈ Ring ∧ 𝑈𝐵𝑘 ∈ ℕ0))
7574adantr 480 . . . . . . . . . . . . 13 ((((((𝑅 ∈ Ring ∧ (𝑈𝐵𝑊𝐵) ∧ 𝐾 ∈ ℕ0) ∧ (𝑖𝑁𝑗𝑁)) ∧ 𝑘 ∈ (0...𝐾)) ∧ 𝑥𝑁𝑦𝑁) ∧ 𝑡𝑁) → (𝑅 ∈ Ring ∧ 𝑈𝐵𝑘 ∈ ℕ0))
7675, 45syl 17 . . . . . . . . . . . 12 ((((((𝑅 ∈ Ring ∧ (𝑈𝐵𝑊𝐵) ∧ 𝐾 ∈ ℕ0) ∧ (𝑖𝑁𝑗𝑁)) ∧ 𝑘 ∈ (0...𝐾)) ∧ 𝑥𝑁𝑦𝑁) ∧ 𝑡𝑁) → (𝑈 decompPMat 𝑘) ∈ (Base‘𝐴))
7722, 30, 44, 72, 73, 76matecld 22433 . . . . . . . . . . 11 ((((((𝑅 ∈ Ring ∧ (𝑈𝐵𝑊𝐵) ∧ 𝐾 ∈ ℕ0) ∧ (𝑖𝑁𝑗𝑁)) ∧ 𝑘 ∈ (0...𝐾)) ∧ 𝑥𝑁𝑦𝑁) ∧ 𝑡𝑁) → (𝑥(𝑈 decompPMat 𝑘)𝑡) ∈ (Base‘𝑅))
78 simpl3 1193 . . . . . . . . . . . 12 ((((((𝑅 ∈ Ring ∧ (𝑈𝐵𝑊𝐵) ∧ 𝐾 ∈ ℕ0) ∧ (𝑖𝑁𝑗𝑁)) ∧ 𝑘 ∈ (0...𝐾)) ∧ 𝑥𝑁𝑦𝑁) ∧ 𝑡𝑁) → 𝑦𝑁)
79553ad2ant1 1133 . . . . . . . . . . . . 13 (((((𝑅 ∈ Ring ∧ (𝑈𝐵𝑊𝐵) ∧ 𝐾 ∈ ℕ0) ∧ (𝑖𝑁𝑗𝑁)) ∧ 𝑘 ∈ (0...𝐾)) ∧ 𝑥𝑁𝑦𝑁) → (𝑊 decompPMat (𝐾𝑘)) ∈ (Base‘𝐴))
8079adantr 480 . . . . . . . . . . . 12 ((((((𝑅 ∈ Ring ∧ (𝑈𝐵𝑊𝐵) ∧ 𝐾 ∈ ℕ0) ∧ (𝑖𝑁𝑗𝑁)) ∧ 𝑘 ∈ (0...𝐾)) ∧ 𝑥𝑁𝑦𝑁) ∧ 𝑡𝑁) → (𝑊 decompPMat (𝐾𝑘)) ∈ (Base‘𝐴))
8122, 30, 44, 73, 78, 80matecld 22433 . . . . . . . . . . 11 ((((((𝑅 ∈ Ring ∧ (𝑈𝐵𝑊𝐵) ∧ 𝐾 ∈ ℕ0) ∧ (𝑖𝑁𝑗𝑁)) ∧ 𝑘 ∈ (0...𝐾)) ∧ 𝑥𝑁𝑦𝑁) ∧ 𝑡𝑁) → (𝑡(𝑊 decompPMat (𝐾𝑘))𝑦) ∈ (Base‘𝑅))
8230, 31ringcl 20248 . . . . . . . . . . 11 ((𝑅 ∈ Ring ∧ (𝑥(𝑈 decompPMat 𝑘)𝑡) ∈ (Base‘𝑅) ∧ (𝑡(𝑊 decompPMat (𝐾𝑘))𝑦) ∈ (Base‘𝑅)) → ((𝑥(𝑈 decompPMat 𝑘)𝑡)(.r𝑅)(𝑡(𝑊 decompPMat (𝐾𝑘))𝑦)) ∈ (Base‘𝑅))
8371, 77, 81, 82syl3anc 1372 . . . . . . . . . 10 ((((((𝑅 ∈ Ring ∧ (𝑈𝐵𝑊𝐵) ∧ 𝐾 ∈ ℕ0) ∧ (𝑖𝑁𝑗𝑁)) ∧ 𝑘 ∈ (0...𝐾)) ∧ 𝑥𝑁𝑦𝑁) ∧ 𝑡𝑁) → ((𝑥(𝑈 decompPMat 𝑘)𝑡)(.r𝑅)(𝑡(𝑊 decompPMat (𝐾𝑘))𝑦)) ∈ (Base‘𝑅))
8483ralrimiva 3145 . . . . . . . . 9 (((((𝑅 ∈ Ring ∧ (𝑈𝐵𝑊𝐵) ∧ 𝐾 ∈ ℕ0) ∧ (𝑖𝑁𝑗𝑁)) ∧ 𝑘 ∈ (0...𝐾)) ∧ 𝑥𝑁𝑦𝑁) → ∀𝑡𝑁 ((𝑥(𝑈 decompPMat 𝑘)𝑡)(.r𝑅)(𝑡(𝑊 decompPMat (𝐾𝑘))𝑦)) ∈ (Base‘𝑅))
8530, 68, 69, 84gsummptcl 19986 . . . . . . . 8 (((((𝑅 ∈ Ring ∧ (𝑈𝐵𝑊𝐵) ∧ 𝐾 ∈ ℕ0) ∧ (𝑖𝑁𝑗𝑁)) ∧ 𝑘 ∈ (0...𝐾)) ∧ 𝑥𝑁𝑦𝑁) → (𝑅 Σg (𝑡𝑁 ↦ ((𝑥(𝑈 decompPMat 𝑘)𝑡)(.r𝑅)(𝑡(𝑊 decompPMat (𝐾𝑘))𝑦)))) ∈ (Base‘𝑅))
8622, 30, 44, 37, 34, 85matbas2d 22430 . . . . . . 7 ((((𝑅 ∈ Ring ∧ (𝑈𝐵𝑊𝐵) ∧ 𝐾 ∈ ℕ0) ∧ (𝑖𝑁𝑗𝑁)) ∧ 𝑘 ∈ (0...𝐾)) → (𝑥𝑁, 𝑦𝑁 ↦ (𝑅 Σg (𝑡𝑁 ↦ ((𝑥(𝑈 decompPMat 𝑘)𝑡)(.r𝑅)(𝑡(𝑊 decompPMat (𝐾𝑘))𝑦))))) ∈ (Base‘𝐴))
87 eqid 2736 . . . . . . . 8 (𝑘 ∈ (0...𝐾) ↦ (𝑥𝑁, 𝑦𝑁 ↦ (𝑅 Σg (𝑡𝑁 ↦ ((𝑥(𝑈 decompPMat 𝑘)𝑡)(.r𝑅)(𝑡(𝑊 decompPMat (𝐾𝑘))𝑦)))))) = (𝑘 ∈ (0...𝐾) ↦ (𝑥𝑁, 𝑦𝑁 ↦ (𝑅 Σg (𝑡𝑁 ↦ ((𝑥(𝑈 decompPMat 𝑘)𝑡)(.r𝑅)(𝑡(𝑊 decompPMat (𝐾𝑘))𝑦))))))
88 fzfid 14015 . . . . . . . 8 (((𝑅 ∈ Ring ∧ (𝑈𝐵𝑊𝐵) ∧ 𝐾 ∈ ℕ0) ∧ (𝑖𝑁𝑗𝑁)) → (0...𝐾) ∈ Fin)
89 simpl 482 . . . . . . . . . . . . . . 15 ((𝑁 ∈ Fin ∧ 𝑃 ∈ V) → 𝑁 ∈ Fin)
9089, 89jca 511 . . . . . . . . . . . . . 14 ((𝑁 ∈ Fin ∧ 𝑃 ∈ V) → (𝑁 ∈ Fin ∧ 𝑁 ∈ Fin))
9116, 90syl 17 . . . . . . . . . . . . 13 (𝑈𝐵 → (𝑁 ∈ Fin ∧ 𝑁 ∈ Fin))
9291adantr 480 . . . . . . . . . . . 12 ((𝑈𝐵𝑊𝐵) → (𝑁 ∈ Fin ∧ 𝑁 ∈ Fin))
93923ad2ant2 1134 . . . . . . . . . . 11 ((𝑅 ∈ Ring ∧ (𝑈𝐵𝑊𝐵) ∧ 𝐾 ∈ ℕ0) → (𝑁 ∈ Fin ∧ 𝑁 ∈ Fin))
9493adantr 480 . . . . . . . . . 10 (((𝑅 ∈ Ring ∧ (𝑈𝐵𝑊𝐵) ∧ 𝐾 ∈ ℕ0) ∧ (𝑖𝑁𝑗𝑁)) → (𝑁 ∈ Fin ∧ 𝑁 ∈ Fin))
9594adantr 480 . . . . . . . . 9 ((((𝑅 ∈ Ring ∧ (𝑈𝐵𝑊𝐵) ∧ 𝐾 ∈ ℕ0) ∧ (𝑖𝑁𝑗𝑁)) ∧ 𝑘 ∈ (0...𝐾)) → (𝑁 ∈ Fin ∧ 𝑁 ∈ Fin))
96 mpoexga 8103 . . . . . . . . 9 ((𝑁 ∈ Fin ∧ 𝑁 ∈ Fin) → (𝑥𝑁, 𝑦𝑁 ↦ (𝑅 Σg (𝑡𝑁 ↦ ((𝑥(𝑈 decompPMat 𝑘)𝑡)(.r𝑅)(𝑡(𝑊 decompPMat (𝐾𝑘))𝑦))))) ∈ V)
9795, 96syl 17 . . . . . . . 8 ((((𝑅 ∈ Ring ∧ (𝑈𝐵𝑊𝐵) ∧ 𝐾 ∈ ℕ0) ∧ (𝑖𝑁𝑗𝑁)) ∧ 𝑘 ∈ (0...𝐾)) → (𝑥𝑁, 𝑦𝑁 ↦ (𝑅 Σg (𝑡𝑁 ↦ ((𝑥(𝑈 decompPMat 𝑘)𝑡)(.r𝑅)(𝑡(𝑊 decompPMat (𝐾𝑘))𝑦))))) ∈ V)
98 fvexd 6920 . . . . . . . 8 (((𝑅 ∈ Ring ∧ (𝑈𝐵𝑊𝐵) ∧ 𝐾 ∈ ℕ0) ∧ (𝑖𝑁𝑗𝑁)) → (0g𝐴) ∈ V)
9987, 88, 97, 98fsuppmptdm 9417 . . . . . . 7 (((𝑅 ∈ Ring ∧ (𝑈𝐵𝑊𝐵) ∧ 𝐾 ∈ ℕ0) ∧ (𝑖𝑁𝑗𝑁)) → (𝑘 ∈ (0...𝐾) ↦ (𝑥𝑁, 𝑦𝑁 ↦ (𝑅 Σg (𝑡𝑁 ↦ ((𝑥(𝑈 decompPMat 𝑘)𝑡)(.r𝑅)(𝑡(𝑊 decompPMat (𝐾𝑘))𝑦)))))) finSupp (0g𝐴))
10022, 44, 62, 36, 63, 33, 86, 99matgsum 22444 . . . . . 6 (((𝑅 ∈ Ring ∧ (𝑈𝐵𝑊𝐵) ∧ 𝐾 ∈ ℕ0) ∧ (𝑖𝑁𝑗𝑁)) → (𝐴 Σg (𝑘 ∈ (0...𝐾) ↦ (𝑥𝑁, 𝑦𝑁 ↦ (𝑅 Σg (𝑡𝑁 ↦ ((𝑥(𝑈 decompPMat 𝑘)𝑡)(.r𝑅)(𝑡(𝑊 decompPMat (𝐾𝑘))𝑦))))))) = (𝑥𝑁, 𝑦𝑁 ↦ (𝑅 Σg (𝑘 ∈ (0...𝐾) ↦ (𝑅 Σg (𝑡𝑁 ↦ ((𝑥(𝑈 decompPMat 𝑘)𝑡)(.r𝑅)(𝑡(𝑊 decompPMat (𝐾𝑘))𝑦))))))))
10161, 100eqtrd 2776 . . . . 5 (((𝑅 ∈ Ring ∧ (𝑈𝐵𝑊𝐵) ∧ 𝐾 ∈ ℕ0) ∧ (𝑖𝑁𝑗𝑁)) → (𝐴 Σg (𝑘 ∈ (0...𝐾) ↦ ((𝑈 decompPMat 𝑘)(.r𝐴)(𝑊 decompPMat (𝐾𝑘))))) = (𝑥𝑁, 𝑦𝑁 ↦ (𝑅 Σg (𝑘 ∈ (0...𝐾) ↦ (𝑅 Σg (𝑡𝑁 ↦ ((𝑥(𝑈 decompPMat 𝑘)𝑡)(.r𝑅)(𝑡(𝑊 decompPMat (𝐾𝑘))𝑦))))))))
102101oveqd 7449 . . . 4 (((𝑅 ∈ Ring ∧ (𝑈𝐵𝑊𝐵) ∧ 𝐾 ∈ ℕ0) ∧ (𝑖𝑁𝑗𝑁)) → (𝑖(𝐴 Σg (𝑘 ∈ (0...𝐾) ↦ ((𝑈 decompPMat 𝑘)(.r𝐴)(𝑊 decompPMat (𝐾𝑘)))))𝑗) = (𝑖(𝑥𝑁, 𝑦𝑁 ↦ (𝑅 Σg (𝑘 ∈ (0...𝐾) ↦ (𝑅 Σg (𝑡𝑁 ↦ ((𝑥(𝑈 decompPMat 𝑘)𝑡)(.r𝑅)(𝑡(𝑊 decompPMat (𝐾𝑘))𝑦)))))))𝑗))
103 simpl2 1192 . . . . . 6 (((𝑅 ∈ Ring ∧ (𝑈𝐵𝑊𝐵) ∧ 𝐾 ∈ ℕ0) ∧ (𝑖𝑁𝑗𝑁)) → (𝑈𝐵𝑊𝐵))
104 simpl3 1193 . . . . . 6 (((𝑅 ∈ Ring ∧ (𝑈𝐵𝑊𝐵) ∧ 𝐾 ∈ ℕ0) ∧ (𝑖𝑁𝑗𝑁)) → 𝐾 ∈ ℕ0)
10543, 14, 15decpmatmullem 22778 . . . . . 6 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑈𝐵𝑊𝐵) ∧ (𝑖𝑁𝑗𝑁𝐾 ∈ ℕ0)) → (𝑖((𝑈(.r𝐶)𝑊) decompPMat 𝐾)𝑗) = (𝑅 Σg (𝑡𝑁 ↦ (𝑅 Σg (𝑘 ∈ (0...𝐾) ↦ (((coe1‘(𝑖𝑈𝑡))‘𝑘)(.r𝑅)((coe1‘(𝑡𝑊𝑗))‘(𝐾𝑘))))))))
10636, 33, 103, 10, 11, 104, 105syl213anc 1390 . . . . 5 (((𝑅 ∈ Ring ∧ (𝑈𝐵𝑊𝐵) ∧ 𝐾 ∈ ℕ0) ∧ (𝑖𝑁𝑗𝑁)) → (𝑖((𝑈(.r𝐶)𝑊) decompPMat 𝐾)𝑗) = (𝑅 Σg (𝑡𝑁 ↦ (𝑅 Σg (𝑘 ∈ (0...𝐾) ↦ (((coe1‘(𝑖𝑈𝑡))‘𝑘)(.r𝑅)((coe1‘(𝑡𝑊𝑗))‘(𝐾𝑘))))))))
107 simpll1 1212 . . . . . . 7 ((((𝑅 ∈ Ring ∧ (𝑈𝐵𝑊𝐵) ∧ 𝐾 ∈ ℕ0) ∧ (𝑖𝑁𝑗𝑁)) ∧ (𝑡𝑁𝑘 ∈ (0...𝐾))) → 𝑅 ∈ Ring)
108 simplrl 776 . . . . . . . . 9 ((((𝑅 ∈ Ring ∧ (𝑈𝐵𝑊𝐵) ∧ 𝐾 ∈ ℕ0) ∧ (𝑖𝑁𝑗𝑁)) ∧ (𝑡𝑁𝑘 ∈ (0...𝐾))) → 𝑖𝑁)
109 simprl 770 . . . . . . . . 9 ((((𝑅 ∈ Ring ∧ (𝑈𝐵𝑊𝐵) ∧ 𝐾 ∈ ℕ0) ∧ (𝑖𝑁𝑗𝑁)) ∧ (𝑡𝑁𝑘 ∈ (0...𝐾))) → 𝑡𝑁)
11015eleq2i 2832 . . . . . . . . . . . . . 14 (𝑈𝐵𝑈 ∈ (Base‘𝐶))
111110biimpi 216 . . . . . . . . . . . . 13 (𝑈𝐵𝑈 ∈ (Base‘𝐶))
112111adantr 480 . . . . . . . . . . . 12 ((𝑈𝐵𝑊𝐵) → 𝑈 ∈ (Base‘𝐶))
1131123ad2ant2 1134 . . . . . . . . . . 11 ((𝑅 ∈ Ring ∧ (𝑈𝐵𝑊𝐵) ∧ 𝐾 ∈ ℕ0) → 𝑈 ∈ (Base‘𝐶))
114113adantr 480 . . . . . . . . . 10 (((𝑅 ∈ Ring ∧ (𝑈𝐵𝑊𝐵) ∧ 𝐾 ∈ ℕ0) ∧ (𝑖𝑁𝑗𝑁)) → 𝑈 ∈ (Base‘𝐶))
115114adantr 480 . . . . . . . . 9 ((((𝑅 ∈ Ring ∧ (𝑈𝐵𝑊𝐵) ∧ 𝐾 ∈ ℕ0) ∧ (𝑖𝑁𝑗𝑁)) ∧ (𝑡𝑁𝑘 ∈ (0...𝐾))) → 𝑈 ∈ (Base‘𝐶))
116 eqid 2736 . . . . . . . . . 10 (Base‘𝑃) = (Base‘𝑃)
11714, 116matecl 22432 . . . . . . . . 9 ((𝑖𝑁𝑡𝑁𝑈 ∈ (Base‘𝐶)) → (𝑖𝑈𝑡) ∈ (Base‘𝑃))
118108, 109, 115, 117syl3anc 1372 . . . . . . . 8 ((((𝑅 ∈ Ring ∧ (𝑈𝐵𝑊𝐵) ∧ 𝐾 ∈ ℕ0) ∧ (𝑖𝑁𝑗𝑁)) ∧ (𝑡𝑁𝑘 ∈ (0...𝐾))) → (𝑖𝑈𝑡) ∈ (Base‘𝑃))
11940ad2antll 729 . . . . . . . 8 ((((𝑅 ∈ Ring ∧ (𝑈𝐵𝑊𝐵) ∧ 𝐾 ∈ ℕ0) ∧ (𝑖𝑁𝑗𝑁)) ∧ (𝑡𝑁𝑘 ∈ (0...𝐾))) → 𝑘 ∈ ℕ0)
120 eqid 2736 . . . . . . . . 9 (coe1‘(𝑖𝑈𝑡)) = (coe1‘(𝑖𝑈𝑡))
121120, 116, 43, 30coe1fvalcl 22215 . . . . . . . 8 (((𝑖𝑈𝑡) ∈ (Base‘𝑃) ∧ 𝑘 ∈ ℕ0) → ((coe1‘(𝑖𝑈𝑡))‘𝑘) ∈ (Base‘𝑅))
122118, 119, 121syl2anc 584 . . . . . . 7 ((((𝑅 ∈ Ring ∧ (𝑈𝐵𝑊𝐵) ∧ 𝐾 ∈ ℕ0) ∧ (𝑖𝑁𝑗𝑁)) ∧ (𝑡𝑁𝑘 ∈ (0...𝐾))) → ((coe1‘(𝑖𝑈𝑡))‘𝑘) ∈ (Base‘𝑅))
123 simplrr 777 . . . . . . . . 9 ((((𝑅 ∈ Ring ∧ (𝑈𝐵𝑊𝐵) ∧ 𝐾 ∈ ℕ0) ∧ (𝑖𝑁𝑗𝑁)) ∧ (𝑡𝑁𝑘 ∈ (0...𝐾))) → 𝑗𝑁)
12449adantr 480 . . . . . . . . 9 ((((𝑅 ∈ Ring ∧ (𝑈𝐵𝑊𝐵) ∧ 𝐾 ∈ ℕ0) ∧ (𝑖𝑁𝑗𝑁)) ∧ (𝑡𝑁𝑘 ∈ (0...𝐾))) → 𝑊𝐵)
12514, 116, 15, 109, 123, 124matecld 22433 . . . . . . . 8 ((((𝑅 ∈ Ring ∧ (𝑈𝐵𝑊𝐵) ∧ 𝐾 ∈ ℕ0) ∧ (𝑖𝑁𝑗𝑁)) ∧ (𝑡𝑁𝑘 ∈ (0...𝐾))) → (𝑡𝑊𝑗) ∈ (Base‘𝑃))
12651ad2antll 729 . . . . . . . 8 ((((𝑅 ∈ Ring ∧ (𝑈𝐵𝑊𝐵) ∧ 𝐾 ∈ ℕ0) ∧ (𝑖𝑁𝑗𝑁)) ∧ (𝑡𝑁𝑘 ∈ (0...𝐾))) → (𝐾𝑘) ∈ ℕ0)
127 eqid 2736 . . . . . . . . 9 (coe1‘(𝑡𝑊𝑗)) = (coe1‘(𝑡𝑊𝑗))
128127, 116, 43, 30coe1fvalcl 22215 . . . . . . . 8 (((𝑡𝑊𝑗) ∈ (Base‘𝑃) ∧ (𝐾𝑘) ∈ ℕ0) → ((coe1‘(𝑡𝑊𝑗))‘(𝐾𝑘)) ∈ (Base‘𝑅))
129125, 126, 128syl2anc 584 . . . . . . 7 ((((𝑅 ∈ Ring ∧ (𝑈𝐵𝑊𝐵) ∧ 𝐾 ∈ ℕ0) ∧ (𝑖𝑁𝑗𝑁)) ∧ (𝑡𝑁𝑘 ∈ (0...𝐾))) → ((coe1‘(𝑡𝑊𝑗))‘(𝐾𝑘)) ∈ (Base‘𝑅))
13030, 31ringcl 20248 . . . . . . 7 ((𝑅 ∈ Ring ∧ ((coe1‘(𝑖𝑈𝑡))‘𝑘) ∈ (Base‘𝑅) ∧ ((coe1‘(𝑡𝑊𝑗))‘(𝐾𝑘)) ∈ (Base‘𝑅)) → (((coe1‘(𝑖𝑈𝑡))‘𝑘)(.r𝑅)((coe1‘(𝑡𝑊𝑗))‘(𝐾𝑘))) ∈ (Base‘𝑅))
131107, 122, 129, 130syl3anc 1372 . . . . . 6 ((((𝑅 ∈ Ring ∧ (𝑈𝐵𝑊𝐵) ∧ 𝐾 ∈ ℕ0) ∧ (𝑖𝑁𝑗𝑁)) ∧ (𝑡𝑁𝑘 ∈ (0...𝐾))) → (((coe1‘(𝑖𝑈𝑡))‘𝑘)(.r𝑅)((coe1‘(𝑡𝑊𝑗))‘(𝐾𝑘))) ∈ (Base‘𝑅))
13230, 66, 36, 88, 131gsumcom3fi 19998 . . . . 5 (((𝑅 ∈ Ring ∧ (𝑈𝐵𝑊𝐵) ∧ 𝐾 ∈ ℕ0) ∧ (𝑖𝑁𝑗𝑁)) → (𝑅 Σg (𝑡𝑁 ↦ (𝑅 Σg (𝑘 ∈ (0...𝐾) ↦ (((coe1‘(𝑖𝑈𝑡))‘𝑘)(.r𝑅)((coe1‘(𝑡𝑊𝑗))‘(𝐾𝑘))))))) = (𝑅 Σg (𝑘 ∈ (0...𝐾) ↦ (𝑅 Σg (𝑡𝑁 ↦ (((coe1‘(𝑖𝑈𝑡))‘𝑘)(.r𝑅)((coe1‘(𝑡𝑊𝑗))‘(𝐾𝑘))))))))
13310adantr 480 . . . . . . . . . . . . 13 ((((𝑅 ∈ Ring ∧ (𝑈𝐵𝑊𝐵) ∧ 𝐾 ∈ ℕ0) ∧ (𝑖𝑁𝑗𝑁)) ∧ 𝑘 ∈ (0...𝐾)) → 𝑖𝑁)
134133anim1i 615 . . . . . . . . . . . 12 (((((𝑅 ∈ Ring ∧ (𝑈𝐵𝑊𝐵) ∧ 𝐾 ∈ ℕ0) ∧ (𝑖𝑁𝑗𝑁)) ∧ 𝑘 ∈ (0...𝐾)) ∧ 𝑡𝑁) → (𝑖𝑁𝑡𝑁))
13543, 14, 15decpmate 22773 . . . . . . . . . . . 12 (((𝑅 ∈ Ring ∧ 𝑈𝐵𝑘 ∈ ℕ0) ∧ (𝑖𝑁𝑡𝑁)) → (𝑖(𝑈 decompPMat 𝑘)𝑡) = ((coe1‘(𝑖𝑈𝑡))‘𝑘))
13642, 134, 135syl2an2r 685 . . . . . . . . . . 11 (((((𝑅 ∈ Ring ∧ (𝑈𝐵𝑊𝐵) ∧ 𝐾 ∈ ℕ0) ∧ (𝑖𝑁𝑗𝑁)) ∧ 𝑘 ∈ (0...𝐾)) ∧ 𝑡𝑁) → (𝑖(𝑈 decompPMat 𝑘)𝑡) = ((coe1‘(𝑖𝑈𝑡))‘𝑘))
137 simplrr 777 . . . . . . . . . . . . 13 ((((𝑅 ∈ Ring ∧ (𝑈𝐵𝑊𝐵) ∧ 𝐾 ∈ ℕ0) ∧ (𝑖𝑁𝑗𝑁)) ∧ 𝑘 ∈ (0...𝐾)) → 𝑗𝑁)
138137anim1ci 616 . . . . . . . . . . . 12 (((((𝑅 ∈ Ring ∧ (𝑈𝐵𝑊𝐵) ∧ 𝐾 ∈ ℕ0) ∧ (𝑖𝑁𝑗𝑁)) ∧ 𝑘 ∈ (0...𝐾)) ∧ 𝑡𝑁) → (𝑡𝑁𝑗𝑁))
13943, 14, 15decpmate 22773 . . . . . . . . . . . 12 (((𝑅 ∈ Ring ∧ 𝑊𝐵 ∧ (𝐾𝑘) ∈ ℕ0) ∧ (𝑡𝑁𝑗𝑁)) → (𝑡(𝑊 decompPMat (𝐾𝑘))𝑗) = ((coe1‘(𝑡𝑊𝑗))‘(𝐾𝑘)))
14053, 138, 139syl2an2r 685 . . . . . . . . . . 11 (((((𝑅 ∈ Ring ∧ (𝑈𝐵𝑊𝐵) ∧ 𝐾 ∈ ℕ0) ∧ (𝑖𝑁𝑗𝑁)) ∧ 𝑘 ∈ (0...𝐾)) ∧ 𝑡𝑁) → (𝑡(𝑊 decompPMat (𝐾𝑘))𝑗) = ((coe1‘(𝑡𝑊𝑗))‘(𝐾𝑘)))
141136, 140oveq12d 7450 . . . . . . . . . 10 (((((𝑅 ∈ Ring ∧ (𝑈𝐵𝑊𝐵) ∧ 𝐾 ∈ ℕ0) ∧ (𝑖𝑁𝑗𝑁)) ∧ 𝑘 ∈ (0...𝐾)) ∧ 𝑡𝑁) → ((𝑖(𝑈 decompPMat 𝑘)𝑡)(.r𝑅)(𝑡(𝑊 decompPMat (𝐾𝑘))𝑗)) = (((coe1‘(𝑖𝑈𝑡))‘𝑘)(.r𝑅)((coe1‘(𝑡𝑊𝑗))‘(𝐾𝑘))))
142141eqcomd 2742 . . . . . . . . 9 (((((𝑅 ∈ Ring ∧ (𝑈𝐵𝑊𝐵) ∧ 𝐾 ∈ ℕ0) ∧ (𝑖𝑁𝑗𝑁)) ∧ 𝑘 ∈ (0...𝐾)) ∧ 𝑡𝑁) → (((coe1‘(𝑖𝑈𝑡))‘𝑘)(.r𝑅)((coe1‘(𝑡𝑊𝑗))‘(𝐾𝑘))) = ((𝑖(𝑈 decompPMat 𝑘)𝑡)(.r𝑅)(𝑡(𝑊 decompPMat (𝐾𝑘))𝑗)))
143142mpteq2dva 5241 . . . . . . . 8 ((((𝑅 ∈ Ring ∧ (𝑈𝐵𝑊𝐵) ∧ 𝐾 ∈ ℕ0) ∧ (𝑖𝑁𝑗𝑁)) ∧ 𝑘 ∈ (0...𝐾)) → (𝑡𝑁 ↦ (((coe1‘(𝑖𝑈𝑡))‘𝑘)(.r𝑅)((coe1‘(𝑡𝑊𝑗))‘(𝐾𝑘)))) = (𝑡𝑁 ↦ ((𝑖(𝑈 decompPMat 𝑘)𝑡)(.r𝑅)(𝑡(𝑊 decompPMat (𝐾𝑘))𝑗))))
144143oveq2d 7448 . . . . . . 7 ((((𝑅 ∈ Ring ∧ (𝑈𝐵𝑊𝐵) ∧ 𝐾 ∈ ℕ0) ∧ (𝑖𝑁𝑗𝑁)) ∧ 𝑘 ∈ (0...𝐾)) → (𝑅 Σg (𝑡𝑁 ↦ (((coe1‘(𝑖𝑈𝑡))‘𝑘)(.r𝑅)((coe1‘(𝑡𝑊𝑗))‘(𝐾𝑘))))) = (𝑅 Σg (𝑡𝑁 ↦ ((𝑖(𝑈 decompPMat 𝑘)𝑡)(.r𝑅)(𝑡(𝑊 decompPMat (𝐾𝑘))𝑗)))))
145144mpteq2dva 5241 . . . . . 6 (((𝑅 ∈ Ring ∧ (𝑈𝐵𝑊𝐵) ∧ 𝐾 ∈ ℕ0) ∧ (𝑖𝑁𝑗𝑁)) → (𝑘 ∈ (0...𝐾) ↦ (𝑅 Σg (𝑡𝑁 ↦ (((coe1‘(𝑖𝑈𝑡))‘𝑘)(.r𝑅)((coe1‘(𝑡𝑊𝑗))‘(𝐾𝑘)))))) = (𝑘 ∈ (0...𝐾) ↦ (𝑅 Σg (𝑡𝑁 ↦ ((𝑖(𝑈 decompPMat 𝑘)𝑡)(.r𝑅)(𝑡(𝑊 decompPMat (𝐾𝑘))𝑗))))))
146145oveq2d 7448 . . . . 5 (((𝑅 ∈ Ring ∧ (𝑈𝐵𝑊𝐵) ∧ 𝐾 ∈ ℕ0) ∧ (𝑖𝑁𝑗𝑁)) → (𝑅 Σg (𝑘 ∈ (0...𝐾) ↦ (𝑅 Σg (𝑡𝑁 ↦ (((coe1‘(𝑖𝑈𝑡))‘𝑘)(.r𝑅)((coe1‘(𝑡𝑊𝑗))‘(𝐾𝑘))))))) = (𝑅 Σg (𝑘 ∈ (0...𝐾) ↦ (𝑅 Σg (𝑡𝑁 ↦ ((𝑖(𝑈 decompPMat 𝑘)𝑡)(.r𝑅)(𝑡(𝑊 decompPMat (𝐾𝑘))𝑗)))))))
147106, 132, 1463eqtrd 2780 . . . 4 (((𝑅 ∈ Ring ∧ (𝑈𝐵𝑊𝐵) ∧ 𝐾 ∈ ℕ0) ∧ (𝑖𝑁𝑗𝑁)) → (𝑖((𝑈(.r𝐶)𝑊) decompPMat 𝐾)𝑗) = (𝑅 Σg (𝑘 ∈ (0...𝐾) ↦ (𝑅 Σg (𝑡𝑁 ↦ ((𝑖(𝑈 decompPMat 𝑘)𝑡)(.r𝑅)(𝑡(𝑊 decompPMat (𝐾𝑘))𝑗)))))))
14813, 102, 1473eqtr4rd 2787 . . 3 (((𝑅 ∈ Ring ∧ (𝑈𝐵𝑊𝐵) ∧ 𝐾 ∈ ℕ0) ∧ (𝑖𝑁𝑗𝑁)) → (𝑖((𝑈(.r𝐶)𝑊) decompPMat 𝐾)𝑗) = (𝑖(𝐴 Σg (𝑘 ∈ (0...𝐾) ↦ ((𝑈 decompPMat 𝑘)(.r𝐴)(𝑊 decompPMat (𝐾𝑘)))))𝑗))
149148ralrimivva 3201 . 2 ((𝑅 ∈ Ring ∧ (𝑈𝐵𝑊𝐵) ∧ 𝐾 ∈ ℕ0) → ∀𝑖𝑁𝑗𝑁 (𝑖((𝑈(.r𝐶)𝑊) decompPMat 𝐾)𝑗) = (𝑖(𝐴 Σg (𝑘 ∈ (0...𝐾) ↦ ((𝑈 decompPMat 𝑘)(.r𝐴)(𝑊 decompPMat (𝐾𝑘)))))𝑗))
15043, 14pmatring 22699 . . . . . . 7 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝐶 ∈ Ring)
15120, 150syl 17 . . . . . 6 ((𝑅 ∈ Ring ∧ (𝑈𝐵𝑊𝐵)) → 𝐶 ∈ Ring)
152 simprl 770 . . . . . 6 ((𝑅 ∈ Ring ∧ (𝑈𝐵𝑊𝐵)) → 𝑈𝐵)
153 simprr 772 . . . . . 6 ((𝑅 ∈ Ring ∧ (𝑈𝐵𝑊𝐵)) → 𝑊𝐵)
154 eqid 2736 . . . . . . 7 (.r𝐶) = (.r𝐶)
15515, 154ringcl 20248 . . . . . 6 ((𝐶 ∈ Ring ∧ 𝑈𝐵𝑊𝐵) → (𝑈(.r𝐶)𝑊) ∈ 𝐵)
156151, 152, 153, 155syl3anc 1372 . . . . 5 ((𝑅 ∈ Ring ∧ (𝑈𝐵𝑊𝐵)) → (𝑈(.r𝐶)𝑊) ∈ 𝐵)
1571563adant3 1132 . . . 4 ((𝑅 ∈ Ring ∧ (𝑈𝐵𝑊𝐵) ∧ 𝐾 ∈ ℕ0) → (𝑈(.r𝐶)𝑊) ∈ 𝐵)
15843, 14, 15, 22, 44decpmatcl 22774 . . . 4 ((𝑅 ∈ Ring ∧ (𝑈(.r𝐶)𝑊) ∈ 𝐵𝐾 ∈ ℕ0) → ((𝑈(.r𝐶)𝑊) decompPMat 𝐾) ∈ (Base‘𝐴))
159157, 158syld3an2 1412 . . 3 ((𝑅 ∈ Ring ∧ (𝑈𝐵𝑊𝐵) ∧ 𝐾 ∈ ℕ0) → ((𝑈(.r𝐶)𝑊) decompPMat 𝐾) ∈ (Base‘𝐴))
16022matring 22450 . . . . . 6 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝐴 ∈ Ring)
16121, 160syl 17 . . . . 5 ((𝑅 ∈ Ring ∧ (𝑈𝐵𝑊𝐵) ∧ 𝐾 ∈ ℕ0) → 𝐴 ∈ Ring)
162 ringcmn 20280 . . . . 5 (𝐴 ∈ Ring → 𝐴 ∈ CMnd)
163161, 162syl 17 . . . 4 ((𝑅 ∈ Ring ∧ (𝑈𝐵𝑊𝐵) ∧ 𝐾 ∈ ℕ0) → 𝐴 ∈ CMnd)
164 fzfid 14015 . . . 4 ((𝑅 ∈ Ring ∧ (𝑈𝐵𝑊𝐵) ∧ 𝐾 ∈ ℕ0) → (0...𝐾) ∈ Fin)
165161adantr 480 . . . . . 6 (((𝑅 ∈ Ring ∧ (𝑈𝐵𝑊𝐵) ∧ 𝐾 ∈ ℕ0) ∧ 𝑘 ∈ (0...𝐾)) → 𝐴 ∈ Ring)
16632adantr 480 . . . . . . . 8 (((𝑅 ∈ Ring ∧ (𝑈𝐵𝑊𝐵) ∧ 𝐾 ∈ ℕ0) ∧ 𝑘 ∈ (0...𝐾)) → 𝑅 ∈ Ring)
167 simpl2l 1226 . . . . . . . 8 (((𝑅 ∈ Ring ∧ (𝑈𝐵𝑊𝐵) ∧ 𝐾 ∈ ℕ0) ∧ 𝑘 ∈ (0...𝐾)) → 𝑈𝐵)
16840adantl 481 . . . . . . . 8 (((𝑅 ∈ Ring ∧ (𝑈𝐵𝑊𝐵) ∧ 𝐾 ∈ ℕ0) ∧ 𝑘 ∈ (0...𝐾)) → 𝑘 ∈ ℕ0)
169166, 167, 1683jca 1128 . . . . . . 7 (((𝑅 ∈ Ring ∧ (𝑈𝐵𝑊𝐵) ∧ 𝐾 ∈ ℕ0) ∧ 𝑘 ∈ (0...𝐾)) → (𝑅 ∈ Ring ∧ 𝑈𝐵𝑘 ∈ ℕ0))
170169, 45syl 17 . . . . . 6 (((𝑅 ∈ Ring ∧ (𝑈𝐵𝑊𝐵) ∧ 𝐾 ∈ ℕ0) ∧ 𝑘 ∈ (0...𝐾)) → (𝑈 decompPMat 𝑘) ∈ (Base‘𝐴))
171 simpl2r 1227 . . . . . . . 8 (((𝑅 ∈ Ring ∧ (𝑈𝐵𝑊𝐵) ∧ 𝐾 ∈ ℕ0) ∧ 𝑘 ∈ (0...𝐾)) → 𝑊𝐵)
17251adantl 481 . . . . . . . 8 (((𝑅 ∈ Ring ∧ (𝑈𝐵𝑊𝐵) ∧ 𝐾 ∈ ℕ0) ∧ 𝑘 ∈ (0...𝐾)) → (𝐾𝑘) ∈ ℕ0)
173166, 171, 1723jca 1128 . . . . . . 7 (((𝑅 ∈ Ring ∧ (𝑈𝐵𝑊𝐵) ∧ 𝐾 ∈ ℕ0) ∧ 𝑘 ∈ (0...𝐾)) → (𝑅 ∈ Ring ∧ 𝑊𝐵 ∧ (𝐾𝑘) ∈ ℕ0))
174173, 54syl 17 . . . . . 6 (((𝑅 ∈ Ring ∧ (𝑈𝐵𝑊𝐵) ∧ 𝐾 ∈ ℕ0) ∧ 𝑘 ∈ (0...𝐾)) → (𝑊 decompPMat (𝐾𝑘)) ∈ (Base‘𝐴))
175 eqid 2736 . . . . . . 7 (.r𝐴) = (.r𝐴)
17644, 175ringcl 20248 . . . . . 6 ((𝐴 ∈ Ring ∧ (𝑈 decompPMat 𝑘) ∈ (Base‘𝐴) ∧ (𝑊 decompPMat (𝐾𝑘)) ∈ (Base‘𝐴)) → ((𝑈 decompPMat 𝑘)(.r𝐴)(𝑊 decompPMat (𝐾𝑘))) ∈ (Base‘𝐴))
177165, 170, 174, 176syl3anc 1372 . . . . 5 (((𝑅 ∈ Ring ∧ (𝑈𝐵𝑊𝐵) ∧ 𝐾 ∈ ℕ0) ∧ 𝑘 ∈ (0...𝐾)) → ((𝑈 decompPMat 𝑘)(.r𝐴)(𝑊 decompPMat (𝐾𝑘))) ∈ (Base‘𝐴))
178177ralrimiva 3145 . . . 4 ((𝑅 ∈ Ring ∧ (𝑈𝐵𝑊𝐵) ∧ 𝐾 ∈ ℕ0) → ∀𝑘 ∈ (0...𝐾)((𝑈 decompPMat 𝑘)(.r𝐴)(𝑊 decompPMat (𝐾𝑘))) ∈ (Base‘𝐴))
17944, 163, 164, 178gsummptcl 19986 . . 3 ((𝑅 ∈ Ring ∧ (𝑈𝐵𝑊𝐵) ∧ 𝐾 ∈ ℕ0) → (𝐴 Σg (𝑘 ∈ (0...𝐾) ↦ ((𝑈 decompPMat 𝑘)(.r𝐴)(𝑊 decompPMat (𝐾𝑘))))) ∈ (Base‘𝐴))
18022, 44eqmat 22431 . . 3 ((((𝑈(.r𝐶)𝑊) decompPMat 𝐾) ∈ (Base‘𝐴) ∧ (𝐴 Σg (𝑘 ∈ (0...𝐾) ↦ ((𝑈 decompPMat 𝑘)(.r𝐴)(𝑊 decompPMat (𝐾𝑘))))) ∈ (Base‘𝐴)) → (((𝑈(.r𝐶)𝑊) decompPMat 𝐾) = (𝐴 Σg (𝑘 ∈ (0...𝐾) ↦ ((𝑈 decompPMat 𝑘)(.r𝐴)(𝑊 decompPMat (𝐾𝑘))))) ↔ ∀𝑖𝑁𝑗𝑁 (𝑖((𝑈(.r𝐶)𝑊) decompPMat 𝐾)𝑗) = (𝑖(𝐴 Σg (𝑘 ∈ (0...𝐾) ↦ ((𝑈 decompPMat 𝑘)(.r𝐴)(𝑊 decompPMat (𝐾𝑘)))))𝑗)))
181159, 179, 180syl2anc 584 . 2 ((𝑅 ∈ Ring ∧ (𝑈𝐵𝑊𝐵) ∧ 𝐾 ∈ ℕ0) → (((𝑈(.r𝐶)𝑊) decompPMat 𝐾) = (𝐴 Σg (𝑘 ∈ (0...𝐾) ↦ ((𝑈 decompPMat 𝑘)(.r𝐴)(𝑊 decompPMat (𝐾𝑘))))) ↔ ∀𝑖𝑁𝑗𝑁 (𝑖((𝑈(.r𝐶)𝑊) decompPMat 𝐾)𝑗) = (𝑖(𝐴 Σg (𝑘 ∈ (0...𝐾) ↦ ((𝑈 decompPMat 𝑘)(.r𝐴)(𝑊 decompPMat (𝐾𝑘)))))𝑗)))
182149, 181mpbird 257 1 ((𝑅 ∈ Ring ∧ (𝑈𝐵𝑊𝐵) ∧ 𝐾 ∈ ℕ0) → ((𝑈(.r𝐶)𝑊) decompPMat 𝐾) = (𝐴 Σg (𝑘 ∈ (0...𝐾) ↦ ((𝑈 decompPMat 𝑘)(.r𝐴)(𝑊 decompPMat (𝐾𝑘))))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395  w3a 1086   = wceq 1539  wcel 2107  wral 3060  Vcvv 3479  cotp 4633  cmpt 5224   × cxp 5682  cfv 6560  (class class class)co 7432  cmpo 7434  m cmap 8867  Fincfn 8986  0cc0 11156  cmin 11493  0cn0 12528  ...cfz 13548  Basecbs 17248  .rcmulr 17299  0gc0g 17485   Σg cgsu 17486  CMndccmn 19799  Ringcrg 20231  Poly1cpl1 22179  coe1cco1 22180   maMul cmmul 22395   Mat cmat 22412   decompPMat cdecpmat 22769
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2707  ax-rep 5278  ax-sep 5295  ax-nul 5305  ax-pow 5364  ax-pr 5431  ax-un 7756  ax-cnex 11212  ax-resscn 11213  ax-1cn 11214  ax-icn 11215  ax-addcl 11216  ax-addrcl 11217  ax-mulcl 11218  ax-mulrcl 11219  ax-mulcom 11220  ax-addass 11221  ax-mulass 11222  ax-distr 11223  ax-i2m1 11224  ax-1ne0 11225  ax-1rid 11226  ax-rnegex 11227  ax-rrecex 11228  ax-cnre 11229  ax-pre-lttri 11230  ax-pre-lttrn 11231  ax-pre-ltadd 11232  ax-pre-mulgt0 11233
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2728  df-clel 2815  df-nfc 2891  df-ne 2940  df-nel 3046  df-ral 3061  df-rex 3070  df-rmo 3379  df-reu 3380  df-rab 3436  df-v 3481  df-sbc 3788  df-csb 3899  df-dif 3953  df-un 3955  df-in 3957  df-ss 3967  df-pss 3970  df-nul 4333  df-if 4525  df-pw 4601  df-sn 4626  df-pr 4628  df-tp 4630  df-op 4632  df-ot 4634  df-uni 4907  df-int 4946  df-iun 4992  df-iin 4993  df-br 5143  df-opab 5205  df-mpt 5225  df-tr 5259  df-id 5577  df-eprel 5583  df-po 5591  df-so 5592  df-fr 5636  df-se 5637  df-we 5638  df-xp 5690  df-rel 5691  df-cnv 5692  df-co 5693  df-dm 5694  df-rn 5695  df-res 5696  df-ima 5697  df-pred 6320  df-ord 6386  df-on 6387  df-lim 6388  df-suc 6389  df-iota 6513  df-fun 6562  df-fn 6563  df-f 6564  df-f1 6565  df-fo 6566  df-f1o 6567  df-fv 6568  df-isom 6569  df-riota 7389  df-ov 7435  df-oprab 7436  df-mpo 7437  df-of 7698  df-ofr 7699  df-om 7889  df-1st 8015  df-2nd 8016  df-supp 8187  df-frecs 8307  df-wrecs 8338  df-recs 8412  df-rdg 8451  df-1o 8507  df-2o 8508  df-er 8746  df-map 8869  df-pm 8870  df-ixp 8939  df-en 8987  df-dom 8988  df-sdom 8989  df-fin 8990  df-fsupp 9403  df-sup 9483  df-oi 9551  df-card 9980  df-pnf 11298  df-mnf 11299  df-xr 11300  df-ltxr 11301  df-le 11302  df-sub 11495  df-neg 11496  df-nn 12268  df-2 12330  df-3 12331  df-4 12332  df-5 12333  df-6 12334  df-7 12335  df-8 12336  df-9 12337  df-n0 12529  df-z 12616  df-dec 12736  df-uz 12880  df-fz 13549  df-fzo 13696  df-seq 14044  df-hash 14371  df-struct 17185  df-sets 17202  df-slot 17220  df-ndx 17232  df-base 17249  df-ress 17276  df-plusg 17311  df-mulr 17312  df-sca 17314  df-vsca 17315  df-ip 17316  df-tset 17317  df-ple 17318  df-ds 17320  df-hom 17322  df-cco 17323  df-0g 17487  df-gsum 17488  df-prds 17493  df-pws 17495  df-mre 17630  df-mrc 17631  df-acs 17633  df-mgm 18654  df-sgrp 18733  df-mnd 18749  df-mhm 18797  df-submnd 18798  df-grp 18955  df-minusg 18956  df-sbg 18957  df-mulg 19087  df-subg 19142  df-ghm 19232  df-cntz 19336  df-cmn 19801  df-abl 19802  df-mgp 20139  df-rng 20151  df-ur 20180  df-ring 20233  df-subrng 20547  df-subrg 20571  df-lmod 20861  df-lss 20931  df-sra 21173  df-rgmod 21174  df-dsmm 21753  df-frlm 21768  df-psr 21930  df-mpl 21932  df-opsr 21934  df-psr1 22182  df-ply1 22184  df-coe1 22185  df-mamu 22396  df-mat 22413  df-decpmat 22770
This theorem is referenced by:  decpmatmulsumfsupp  22780  pm2mpmhmlem1  22825  pm2mpmhmlem2  22826
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