MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  decpmatmul Structured version   Visualization version   GIF version

Theorem decpmatmul 22121
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 7364 . . . . . . . . . . 11 (𝑥 = 𝑖 → (𝑥(𝑈 decompPMat 𝑘)𝑡) = (𝑖(𝑈 decompPMat 𝑘)𝑡))
3 oveq2 7365 . . . . . . . . . . 11 (𝑦 = 𝑗 → (𝑡(𝑊 decompPMat (𝐾𝑘))𝑦) = (𝑡(𝑊 decompPMat (𝐾𝑘))𝑗))
42, 3oveqan12d 7376 . . . . . . . . . 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 5207 . . . . . . . . 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 7373 . . . . . . . 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 5207 . . . . . . 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 7373 . . . . . 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 482 . . . . 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 769 . . . . 5 (((𝑅 ∈ Ring ∧ (𝑈𝐵𝑊𝐵) ∧ 𝐾 ∈ ℕ0) ∧ (𝑖𝑁𝑗𝑁)) → 𝑖𝑁)
11 simprr 771 . . . . 5 (((𝑅 ∈ Ring ∧ (𝑈𝐵𝑊𝐵) ∧ 𝐾 ∈ ℕ0) ∧ (𝑖𝑁𝑗𝑁)) → 𝑗𝑁)
12 ovexd 7392 . . . . 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 7507 . . . 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 21759 . . . . . . . . . . . . . . . . . . 19 (𝑈𝐵 → (𝑁 ∈ Fin ∧ 𝑃 ∈ V))
1716simpld 495 . . . . . . . . . . . . . . . . . 18 (𝑈𝐵𝑁 ∈ Fin)
1817adantr 481 . . . . . . . . . . . . . . . . 17 ((𝑈𝐵𝑊𝐵) → 𝑁 ∈ Fin)
1918anim2i 617 . . . . . . . . . . . . . . . 16 ((𝑅 ∈ Ring ∧ (𝑈𝐵𝑊𝐵)) → (𝑅 ∈ Ring ∧ 𝑁 ∈ Fin))
2019ancomd 462 . . . . . . . . . . . . . . 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 21787 . . . . . . . . . . . . . 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 481 . . . . . . . . . . . 12 (((𝑅 ∈ Ring ∧ (𝑈𝐵𝑊𝐵) ∧ 𝐾 ∈ ℕ0) ∧ (𝑖𝑁𝑗𝑁)) → (𝑅 maMul ⟨𝑁, 𝑁, 𝑁⟩) = (.r𝐴))
2726adantr 481 . . . . . . . . . . 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 7374 . . . . . . . . 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 481 . . . . . . . . . . 11 (((𝑅 ∈ Ring ∧ (𝑈𝐵𝑊𝐵) ∧ 𝐾 ∈ ℕ0) ∧ (𝑖𝑁𝑗𝑁)) → 𝑅 ∈ Ring)
3433adantr 481 . . . . . . . . . 10 ((((𝑅 ∈ Ring ∧ (𝑈𝐵𝑊𝐵) ∧ 𝐾 ∈ ℕ0) ∧ (𝑖𝑁𝑗𝑁)) ∧ 𝑘 ∈ (0...𝐾)) → 𝑅 ∈ Ring)
3521simpld 495 . . . . . . . . . . . 12 ((𝑅 ∈ Ring ∧ (𝑈𝐵𝑊𝐵) ∧ 𝐾 ∈ ℕ0) → 𝑁 ∈ Fin)
3635adantr 481 . . . . . . . . . . 11 (((𝑅 ∈ Ring ∧ (𝑈𝐵𝑊𝐵) ∧ 𝐾 ∈ ℕ0) ∧ (𝑖𝑁𝑗𝑁)) → 𝑁 ∈ Fin)
3736adantr 481 . . . . . . . . . 10 ((((𝑅 ∈ Ring ∧ (𝑈𝐵𝑊𝐵) ∧ 𝐾 ∈ ℕ0) ∧ (𝑖𝑁𝑗𝑁)) ∧ 𝑘 ∈ (0...𝐾)) → 𝑁 ∈ Fin)
38 simpl2l 1226 . . . . . . . . . . . . . 14 (((𝑅 ∈ Ring ∧ (𝑈𝐵𝑊𝐵) ∧ 𝐾 ∈ ℕ0) ∧ (𝑖𝑁𝑗𝑁)) → 𝑈𝐵)
3938adantr 481 . . . . . . . . . . . . 13 ((((𝑅 ∈ Ring ∧ (𝑈𝐵𝑊𝐵) ∧ 𝐾 ∈ ℕ0) ∧ (𝑖𝑁𝑗𝑁)) ∧ 𝑘 ∈ (0...𝐾)) → 𝑈𝐵)
40 elfznn0 13534 . . . . . . . . . . . . . 14 (𝑘 ∈ (0...𝐾) → 𝑘 ∈ ℕ0)
4140adantl 482 . . . . . . . . . . . . 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 22116 . . . . . . . . . . . 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 21771 . . . . . . . . . . 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 481 . . . . . . . . . . . . 13 ((((𝑅 ∈ Ring ∧ (𝑈𝐵𝑊𝐵) ∧ 𝐾 ∈ ℕ0) ∧ (𝑖𝑁𝑗𝑁)) ∧ 𝑘 ∈ (0...𝐾)) → 𝑊𝐵)
51 fznn0sub 13473 . . . . . . . . . . . . . 14 (𝑘 ∈ (0...𝐾) → (𝐾𝑘) ∈ ℕ0)
5251adantl 482 . . . . . . . . . . . . 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 22116 . . . . . . . . . . . 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 21771 . . . . . . . . . . 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 21735 . . . . . . . . 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 5205 . . . . . . 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 7373 . . . . . 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 7392 . . . . . . 7 (((𝑅 ∈ Ring ∧ (𝑈𝐵𝑊𝐵) ∧ 𝐾 ∈ ℕ0) ∧ (𝑖𝑁𝑗𝑁)) → (0...𝐾) ∈ V)
64 ringcmn 20003 . . . . . . . . . . . . 13 (𝑅 ∈ Ring → 𝑅 ∈ CMnd)
6532, 64syl 17 . . . . . . . . . . . 12 ((𝑅 ∈ Ring ∧ (𝑈𝐵𝑊𝐵) ∧ 𝐾 ∈ ℕ0) → 𝑅 ∈ CMnd)
6665adantr 481 . . . . . . . . . . 11 (((𝑅 ∈ Ring ∧ (𝑈𝐵𝑊𝐵) ∧ 𝐾 ∈ ℕ0) ∧ (𝑖𝑁𝑗𝑁)) → 𝑅 ∈ CMnd)
6766adantr 481 . . . . . . . . . 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 481 . . . . . . . . . . 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 485 . . . . . . . . . . . 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 481 . . . . . . . . . . . . 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 21775 . . . . . . . . . . 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 481 . . . . . . . . . . . 12 ((((((𝑅 ∈ Ring ∧ (𝑈𝐵𝑊𝐵) ∧ 𝐾 ∈ ℕ0) ∧ (𝑖𝑁𝑗𝑁)) ∧ 𝑘 ∈ (0...𝐾)) ∧ 𝑥𝑁𝑦𝑁) ∧ 𝑡𝑁) → (𝑊 decompPMat (𝐾𝑘)) ∈ (Base‘𝐴))
8122, 30, 44, 73, 78, 80matecld 21775 . . . . . . . . . . 11 ((((((𝑅 ∈ Ring ∧ (𝑈𝐵𝑊𝐵) ∧ 𝐾 ∈ ℕ0) ∧ (𝑖𝑁𝑗𝑁)) ∧ 𝑘 ∈ (0...𝐾)) ∧ 𝑥𝑁𝑦𝑁) ∧ 𝑡𝑁) → (𝑡(𝑊 decompPMat (𝐾𝑘))𝑦) ∈ (Base‘𝑅))
8230, 31ringcl 19981 . . . . . . . . . . 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 1371 . . . . . . . . . 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 3143 . . . . . . . . 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 19744 . . . . . . . 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 21772 . . . . . . 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 13878 . . . . . . . 8 (((𝑅 ∈ Ring ∧ (𝑈𝐵𝑊𝐵) ∧ 𝐾 ∈ ℕ0) ∧ (𝑖𝑁𝑗𝑁)) → (0...𝐾) ∈ Fin)
89 simpl 483 . . . . . . . . . . . . . . 15 ((𝑁 ∈ Fin ∧ 𝑃 ∈ V) → 𝑁 ∈ Fin)
9089, 89jca 512 . . . . . . . . . . . . . 14 ((𝑁 ∈ Fin ∧ 𝑃 ∈ V) → (𝑁 ∈ Fin ∧ 𝑁 ∈ Fin))
9116, 90syl 17 . . . . . . . . . . . . 13 (𝑈𝐵 → (𝑁 ∈ Fin ∧ 𝑁 ∈ Fin))
9291adantr 481 . . . . . . . . . . . 12 ((𝑈𝐵𝑊𝐵) → (𝑁 ∈ Fin ∧ 𝑁 ∈ Fin))
93923ad2ant2 1134 . . . . . . . . . . 11 ((𝑅 ∈ Ring ∧ (𝑈𝐵𝑊𝐵) ∧ 𝐾 ∈ ℕ0) → (𝑁 ∈ Fin ∧ 𝑁 ∈ Fin))
9493adantr 481 . . . . . . . . . 10 (((𝑅 ∈ Ring ∧ (𝑈𝐵𝑊𝐵) ∧ 𝐾 ∈ ℕ0) ∧ (𝑖𝑁𝑗𝑁)) → (𝑁 ∈ Fin ∧ 𝑁 ∈ Fin))
9594adantr 481 . . . . . . . . 9 ((((𝑅 ∈ Ring ∧ (𝑈𝐵𝑊𝐵) ∧ 𝐾 ∈ ℕ0) ∧ (𝑖𝑁𝑗𝑁)) ∧ 𝑘 ∈ (0...𝐾)) → (𝑁 ∈ Fin ∧ 𝑁 ∈ Fin))
96 mpoexga 8010 . . . . . . . . 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 6857 . . . . . . . 8 (((𝑅 ∈ Ring ∧ (𝑈𝐵𝑊𝐵) ∧ 𝐾 ∈ ℕ0) ∧ (𝑖𝑁𝑗𝑁)) → (0g𝐴) ∈ V)
9987, 88, 97, 98fsuppmptdm 9316 . . . . . . 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 21786 . . . . . 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 7374 . . . 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 22120 . . . . . 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 1389 . . . . 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 775 . . . . . . . . 9 ((((𝑅 ∈ Ring ∧ (𝑈𝐵𝑊𝐵) ∧ 𝐾 ∈ ℕ0) ∧ (𝑖𝑁𝑗𝑁)) ∧ (𝑡𝑁𝑘 ∈ (0...𝐾))) → 𝑖𝑁)
109 simprl 769 . . . . . . . . 9 ((((𝑅 ∈ Ring ∧ (𝑈𝐵𝑊𝐵) ∧ 𝐾 ∈ ℕ0) ∧ (𝑖𝑁𝑗𝑁)) ∧ (𝑡𝑁𝑘 ∈ (0...𝐾))) → 𝑡𝑁)
11015eleq2i 2829 . . . . . . . . . . . . . 14 (𝑈𝐵𝑈 ∈ (Base‘𝐶))
111110biimpi 215 . . . . . . . . . . . . 13 (𝑈𝐵𝑈 ∈ (Base‘𝐶))
112111adantr 481 . . . . . . . . . . . 12 ((𝑈𝐵𝑊𝐵) → 𝑈 ∈ (Base‘𝐶))
1131123ad2ant2 1134 . . . . . . . . . . 11 ((𝑅 ∈ Ring ∧ (𝑈𝐵𝑊𝐵) ∧ 𝐾 ∈ ℕ0) → 𝑈 ∈ (Base‘𝐶))
114113adantr 481 . . . . . . . . . 10 (((𝑅 ∈ Ring ∧ (𝑈𝐵𝑊𝐵) ∧ 𝐾 ∈ ℕ0) ∧ (𝑖𝑁𝑗𝑁)) → 𝑈 ∈ (Base‘𝐶))
115114adantr 481 . . . . . . . . 9 ((((𝑅 ∈ Ring ∧ (𝑈𝐵𝑊𝐵) ∧ 𝐾 ∈ ℕ0) ∧ (𝑖𝑁𝑗𝑁)) ∧ (𝑡𝑁𝑘 ∈ (0...𝐾))) → 𝑈 ∈ (Base‘𝐶))
116 eqid 2736 . . . . . . . . . 10 (Base‘𝑃) = (Base‘𝑃)
11714, 116matecl 21774 . . . . . . . . 9 ((𝑖𝑁𝑡𝑁𝑈 ∈ (Base‘𝐶)) → (𝑖𝑈𝑡) ∈ (Base‘𝑃))
118108, 109, 115, 117syl3anc 1371 . . . . . . . 8 ((((𝑅 ∈ Ring ∧ (𝑈𝐵𝑊𝐵) ∧ 𝐾 ∈ ℕ0) ∧ (𝑖𝑁𝑗𝑁)) ∧ (𝑡𝑁𝑘 ∈ (0...𝐾))) → (𝑖𝑈𝑡) ∈ (Base‘𝑃))
11940ad2antll 727 . . . . . . . 8 ((((𝑅 ∈ Ring ∧ (𝑈𝐵𝑊𝐵) ∧ 𝐾 ∈ ℕ0) ∧ (𝑖𝑁𝑗𝑁)) ∧ (𝑡𝑁𝑘 ∈ (0...𝐾))) → 𝑘 ∈ ℕ0)
120 eqid 2736 . . . . . . . . 9 (coe1‘(𝑖𝑈𝑡)) = (coe1‘(𝑖𝑈𝑡))
121120, 116, 43, 30coe1fvalcl 21583 . . . . . . . 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 776 . . . . . . . . 9 ((((𝑅 ∈ Ring ∧ (𝑈𝐵𝑊𝐵) ∧ 𝐾 ∈ ℕ0) ∧ (𝑖𝑁𝑗𝑁)) ∧ (𝑡𝑁𝑘 ∈ (0...𝐾))) → 𝑗𝑁)
12449adantr 481 . . . . . . . . 9 ((((𝑅 ∈ Ring ∧ (𝑈𝐵𝑊𝐵) ∧ 𝐾 ∈ ℕ0) ∧ (𝑖𝑁𝑗𝑁)) ∧ (𝑡𝑁𝑘 ∈ (0...𝐾))) → 𝑊𝐵)
12514, 116, 15, 109, 123, 124matecld 21775 . . . . . . . 8 ((((𝑅 ∈ Ring ∧ (𝑈𝐵𝑊𝐵) ∧ 𝐾 ∈ ℕ0) ∧ (𝑖𝑁𝑗𝑁)) ∧ (𝑡𝑁𝑘 ∈ (0...𝐾))) → (𝑡𝑊𝑗) ∈ (Base‘𝑃))
12651ad2antll 727 . . . . . . . 8 ((((𝑅 ∈ Ring ∧ (𝑈𝐵𝑊𝐵) ∧ 𝐾 ∈ ℕ0) ∧ (𝑖𝑁𝑗𝑁)) ∧ (𝑡𝑁𝑘 ∈ (0...𝐾))) → (𝐾𝑘) ∈ ℕ0)
127 eqid 2736 . . . . . . . . 9 (coe1‘(𝑡𝑊𝑗)) = (coe1‘(𝑡𝑊𝑗))
128127, 116, 43, 30coe1fvalcl 21583 . . . . . . . 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 19981 . . . . . . 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 1371 . . . . . 6 ((((𝑅 ∈ Ring ∧ (𝑈𝐵𝑊𝐵) ∧ 𝐾 ∈ ℕ0) ∧ (𝑖𝑁𝑗𝑁)) ∧ (𝑡𝑁𝑘 ∈ (0...𝐾))) → (((coe1‘(𝑖𝑈𝑡))‘𝑘)(.r𝑅)((coe1‘(𝑡𝑊𝑗))‘(𝐾𝑘))) ∈ (Base‘𝑅))
13230, 66, 36, 88, 131gsumcom3fi 19756 . . . . 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 481 . . . . . . . . . . . . 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 22115 . . . . . . . . . . . 12 (((𝑅 ∈ Ring ∧ 𝑈𝐵𝑘 ∈ ℕ0) ∧ (𝑖𝑁𝑡𝑁)) → (𝑖(𝑈 decompPMat 𝑘)𝑡) = ((coe1‘(𝑖𝑈𝑡))‘𝑘))
13642, 134, 135syl2an2r 683 . . . . . . . . . . 11 (((((𝑅 ∈ Ring ∧ (𝑈𝐵𝑊𝐵) ∧ 𝐾 ∈ ℕ0) ∧ (𝑖𝑁𝑗𝑁)) ∧ 𝑘 ∈ (0...𝐾)) ∧ 𝑡𝑁) → (𝑖(𝑈 decompPMat 𝑘)𝑡) = ((coe1‘(𝑖𝑈𝑡))‘𝑘))
137 simplrr 776 . . . . . . . . . . . . 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 22115 . . . . . . . . . . . 12 (((𝑅 ∈ Ring ∧ 𝑊𝐵 ∧ (𝐾𝑘) ∈ ℕ0) ∧ (𝑡𝑁𝑗𝑁)) → (𝑡(𝑊 decompPMat (𝐾𝑘))𝑗) = ((coe1‘(𝑡𝑊𝑗))‘(𝐾𝑘)))
14053, 138, 139syl2an2r 683 . . . . . . . . . . 11 (((((𝑅 ∈ Ring ∧ (𝑈𝐵𝑊𝐵) ∧ 𝐾 ∈ ℕ0) ∧ (𝑖𝑁𝑗𝑁)) ∧ 𝑘 ∈ (0...𝐾)) ∧ 𝑡𝑁) → (𝑡(𝑊 decompPMat (𝐾𝑘))𝑗) = ((coe1‘(𝑡𝑊𝑗))‘(𝐾𝑘)))
141136, 140oveq12d 7375 . . . . . . . . . 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 5205 . . . . . . . 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 7373 . . . . . . 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 5205 . . . . . 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 7373 . . . . 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 3197 . 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 22041 . . . . . . 7 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝐶 ∈ Ring)
15120, 150syl 17 . . . . . 6 ((𝑅 ∈ Ring ∧ (𝑈𝐵𝑊𝐵)) → 𝐶 ∈ Ring)
152 simprl 769 . . . . . 6 ((𝑅 ∈ Ring ∧ (𝑈𝐵𝑊𝐵)) → 𝑈𝐵)
153 simprr 771 . . . . . 6 ((𝑅 ∈ Ring ∧ (𝑈𝐵𝑊𝐵)) → 𝑊𝐵)
154 eqid 2736 . . . . . . 7 (.r𝐶) = (.r𝐶)
15515, 154ringcl 19981 . . . . . 6 ((𝐶 ∈ Ring ∧ 𝑈𝐵𝑊𝐵) → (𝑈(.r𝐶)𝑊) ∈ 𝐵)
156151, 152, 153, 155syl3anc 1371 . . . . 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 22116 . . . 4 ((𝑅 ∈ Ring ∧ (𝑈(.r𝐶)𝑊) ∈ 𝐵𝐾 ∈ ℕ0) → ((𝑈(.r𝐶)𝑊) decompPMat 𝐾) ∈ (Base‘𝐴))
159157, 158syld3an2 1411 . . 3 ((𝑅 ∈ Ring ∧ (𝑈𝐵𝑊𝐵) ∧ 𝐾 ∈ ℕ0) → ((𝑈(.r𝐶)𝑊) decompPMat 𝐾) ∈ (Base‘𝐴))
16022matring 21792 . . . . . 6 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝐴 ∈ Ring)
16121, 160syl 17 . . . . 5 ((𝑅 ∈ Ring ∧ (𝑈𝐵𝑊𝐵) ∧ 𝐾 ∈ ℕ0) → 𝐴 ∈ Ring)
162 ringcmn 20003 . . . . 5 (𝐴 ∈ Ring → 𝐴 ∈ CMnd)
163161, 162syl 17 . . . 4 ((𝑅 ∈ Ring ∧ (𝑈𝐵𝑊𝐵) ∧ 𝐾 ∈ ℕ0) → 𝐴 ∈ CMnd)
164 fzfid 13878 . . . 4 ((𝑅 ∈ Ring ∧ (𝑈𝐵𝑊𝐵) ∧ 𝐾 ∈ ℕ0) → (0...𝐾) ∈ Fin)
165161adantr 481 . . . . . 6 (((𝑅 ∈ Ring ∧ (𝑈𝐵𝑊𝐵) ∧ 𝐾 ∈ ℕ0) ∧ 𝑘 ∈ (0...𝐾)) → 𝐴 ∈ Ring)
16632adantr 481 . . . . . . . 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 482 . . . . . . . 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 482 . . . . . . . 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 19981 . . . . . 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 1371 . . . . 5 (((𝑅 ∈ Ring ∧ (𝑈𝐵𝑊𝐵) ∧ 𝐾 ∈ ℕ0) ∧ 𝑘 ∈ (0...𝐾)) → ((𝑈 decompPMat 𝑘)(.r𝐴)(𝑊 decompPMat (𝐾𝑘))) ∈ (Base‘𝐴))
178177ralrimiva 3143 . . . 4 ((𝑅 ∈ Ring ∧ (𝑈𝐵𝑊𝐵) ∧ 𝐾 ∈ ℕ0) → ∀𝑘 ∈ (0...𝐾)((𝑈 decompPMat 𝑘)(.r𝐴)(𝑊 decompPMat (𝐾𝑘))) ∈ (Base‘𝐴))
17944, 163, 164, 178gsummptcl 19744 . . 3 ((𝑅 ∈ Ring ∧ (𝑈𝐵𝑊𝐵) ∧ 𝐾 ∈ ℕ0) → (𝐴 Σg (𝑘 ∈ (0...𝐾) ↦ ((𝑈 decompPMat 𝑘)(.r𝐴)(𝑊 decompPMat (𝐾𝑘))))) ∈ (Base‘𝐴))
18022, 44eqmat 21773 . . 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 256 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 205  wa 396  w3a 1087   = wceq 1541  wcel 2106  wral 3064  Vcvv 3445  cotp 4594  cmpt 5188   × cxp 5631  cfv 6496  (class class class)co 7357  cmpo 7359  m cmap 8765  Fincfn 8883  0cc0 11051  cmin 11385  0cn0 12413  ...cfz 13424  Basecbs 17083  .rcmulr 17134  0gc0g 17321   Σg cgsu 17322  CMndccmn 19562  Ringcrg 19964  Poly1cpl1 21548  coe1cco1 21549   maMul cmmul 21732   Mat cmat 21754   decompPMat cdecpmat 22111
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-rep 5242  ax-sep 5256  ax-nul 5263  ax-pow 5320  ax-pr 5384  ax-un 7672  ax-cnex 11107  ax-resscn 11108  ax-1cn 11109  ax-icn 11110  ax-addcl 11111  ax-addrcl 11112  ax-mulcl 11113  ax-mulrcl 11114  ax-mulcom 11115  ax-addass 11116  ax-mulass 11117  ax-distr 11118  ax-i2m1 11119  ax-1ne0 11120  ax-1rid 11121  ax-rnegex 11122  ax-rrecex 11123  ax-cnre 11124  ax-pre-lttri 11125  ax-pre-lttrn 11126  ax-pre-ltadd 11127  ax-pre-mulgt0 11128
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-nel 3050  df-ral 3065  df-rex 3074  df-rmo 3353  df-reu 3354  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-pss 3929  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-tp 4591  df-op 4593  df-ot 4595  df-uni 4866  df-int 4908  df-iun 4956  df-iin 4957  df-br 5106  df-opab 5168  df-mpt 5189  df-tr 5223  df-id 5531  df-eprel 5537  df-po 5545  df-so 5546  df-fr 5588  df-se 5589  df-we 5590  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-pred 6253  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-f1 6501  df-fo 6502  df-f1o 6503  df-fv 6504  df-isom 6505  df-riota 7313  df-ov 7360  df-oprab 7361  df-mpo 7362  df-of 7617  df-ofr 7618  df-om 7803  df-1st 7921  df-2nd 7922  df-supp 8093  df-frecs 8212  df-wrecs 8243  df-recs 8317  df-rdg 8356  df-1o 8412  df-er 8648  df-map 8767  df-pm 8768  df-ixp 8836  df-en 8884  df-dom 8885  df-sdom 8886  df-fin 8887  df-fsupp 9306  df-sup 9378  df-oi 9446  df-card 9875  df-pnf 11191  df-mnf 11192  df-xr 11193  df-ltxr 11194  df-le 11195  df-sub 11387  df-neg 11388  df-nn 12154  df-2 12216  df-3 12217  df-4 12218  df-5 12219  df-6 12220  df-7 12221  df-8 12222  df-9 12223  df-n0 12414  df-z 12500  df-dec 12619  df-uz 12764  df-fz 13425  df-fzo 13568  df-seq 13907  df-hash 14231  df-struct 17019  df-sets 17036  df-slot 17054  df-ndx 17066  df-base 17084  df-ress 17113  df-plusg 17146  df-mulr 17147  df-sca 17149  df-vsca 17150  df-ip 17151  df-tset 17152  df-ple 17153  df-ds 17155  df-hom 17157  df-cco 17158  df-0g 17323  df-gsum 17324  df-prds 17329  df-pws 17331  df-mre 17466  df-mrc 17467  df-acs 17469  df-mgm 18497  df-sgrp 18546  df-mnd 18557  df-mhm 18601  df-submnd 18602  df-grp 18751  df-minusg 18752  df-sbg 18753  df-mulg 18873  df-subg 18925  df-ghm 19006  df-cntz 19097  df-cmn 19564  df-abl 19565  df-mgp 19897  df-ur 19914  df-ring 19966  df-subrg 20220  df-lmod 20324  df-lss 20393  df-sra 20633  df-rgmod 20634  df-dsmm 21138  df-frlm 21153  df-psr 21311  df-mpl 21313  df-opsr 21315  df-psr1 21551  df-ply1 21553  df-coe1 21554  df-mamu 21733  df-mat 21755  df-decpmat 22112
This theorem is referenced by:  decpmatmulsumfsupp  22122  pm2mpmhmlem1  22167  pm2mpmhmlem2  22168
  Copyright terms: Public domain W3C validator