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Theorem decpmatmul 22687
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 2729 . . . . 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 7427 . . . . . . . . . . 11 (𝑥 = 𝑖 → (𝑥(𝑈 decompPMat 𝑘)𝑡) = (𝑖(𝑈 decompPMat 𝑘)𝑡))
3 oveq2 7428 . . . . . . . . . . 11 (𝑦 = 𝑗 → (𝑡(𝑊 decompPMat (𝐾𝑘))𝑦) = (𝑡(𝑊 decompPMat (𝐾𝑘))𝑗))
42, 3oveqan12d 7439 . . . . . . . . . 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 5250 . . . . . . . . 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 7436 . . . . . . . 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 5250 . . . . . . 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 7436 . . . . . 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 7455 . . . . 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 7573 . . . 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 22325 . . . . . . . . . . . . . . . . . . 19 (𝑈𝐵 → (𝑁 ∈ Fin ∧ 𝑃 ∈ V))
1716simpld 494 . . . . . . . . . . . . . . . . . 18 (𝑈𝐵𝑁 ∈ Fin)
1817adantr 480 . . . . . . . . . . . . . . . . 17 ((𝑈𝐵𝑊𝐵) → 𝑁 ∈ Fin)
1918anim2i 616 . . . . . . . . . . . . . . . 16 ((𝑅 ∈ Ring ∧ (𝑈𝐵𝑊𝐵)) → (𝑅 ∈ Ring ∧ 𝑁 ∈ Fin))
2019ancomd 461 . . . . . . . . . . . . . . 15 ((𝑅 ∈ Ring ∧ (𝑈𝐵𝑊𝐵)) → (𝑁 ∈ Fin ∧ 𝑅 ∈ Ring))
21203adant3 1130 . . . . . . . . . . . . . 14 ((𝑅 ∈ Ring ∧ (𝑈𝐵𝑊𝐵) ∧ 𝐾 ∈ ℕ0) → (𝑁 ∈ Fin ∧ 𝑅 ∈ Ring))
22 decpmatmul.a . . . . . . . . . . . . . . 15 𝐴 = (𝑁 Mat 𝑅)
23 eqid 2728 . . . . . . . . . . . . . . 15 (𝑅 maMul ⟨𝑁, 𝑁, 𝑁⟩) = (𝑅 maMul ⟨𝑁, 𝑁, 𝑁⟩)
2422, 23matmulr 22353 . . . . . . . . . . . . . 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 2734 . . . . . . . . . 10 ((((𝑅 ∈ Ring ∧ (𝑈𝐵𝑊𝐵) ∧ 𝐾 ∈ ℕ0) ∧ (𝑖𝑁𝑗𝑁)) ∧ 𝑘 ∈ (0...𝐾)) → (.r𝐴) = (𝑅 maMul ⟨𝑁, 𝑁, 𝑁⟩))
2928oveqd 7437 . . . . . . . . 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 2728 . . . . . . . . . 10 (Base‘𝑅) = (Base‘𝑅)
31 eqid 2728 . . . . . . . . . 10 (.r𝑅) = (.r𝑅)
32 simp1 1134 . . . . . . . . . . . 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 1224 . . . . . . . . . . . . . 14 (((𝑅 ∈ Ring ∧ (𝑈𝐵𝑊𝐵) ∧ 𝐾 ∈ ℕ0) ∧ (𝑖𝑁𝑗𝑁)) → 𝑈𝐵)
3938adantr 480 . . . . . . . . . . . . 13 ((((𝑅 ∈ Ring ∧ (𝑈𝐵𝑊𝐵) ∧ 𝐾 ∈ ℕ0) ∧ (𝑖𝑁𝑗𝑁)) ∧ 𝑘 ∈ (0...𝐾)) → 𝑈𝐵)
40 elfznn0 13627 . . . . . . . . . . . . . 14 (𝑘 ∈ (0...𝐾) → 𝑘 ∈ ℕ0)
4140adantl 481 . . . . . . . . . . . . 13 ((((𝑅 ∈ Ring ∧ (𝑈𝐵𝑊𝐵) ∧ 𝐾 ∈ ℕ0) ∧ (𝑖𝑁𝑗𝑁)) ∧ 𝑘 ∈ (0...𝐾)) → 𝑘 ∈ ℕ0)
4234, 39, 413jca 1126 . . . . . . . . . . . 12 ((((𝑅 ∈ Ring ∧ (𝑈𝐵𝑊𝐵) ∧ 𝐾 ∈ ℕ0) ∧ (𝑖𝑁𝑗𝑁)) ∧ 𝑘 ∈ (0...𝐾)) → (𝑅 ∈ Ring ∧ 𝑈𝐵𝑘 ∈ ℕ0))
43 decpmatmul.p . . . . . . . . . . . . 13 𝑃 = (Poly1𝑅)
44 eqid 2728 . . . . . . . . . . . . 13 (Base‘𝐴) = (Base‘𝐴)
4543, 14, 15, 22, 44decpmatcl 22682 . . . . . . . . . . . 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 22337 . . . . . . . . . . 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 1225 . . . . . . . . . . . . . 14 (((𝑅 ∈ Ring ∧ (𝑈𝐵𝑊𝐵) ∧ 𝐾 ∈ ℕ0) ∧ (𝑖𝑁𝑗𝑁)) → 𝑊𝐵)
5049adantr 480 . . . . . . . . . . . . 13 ((((𝑅 ∈ Ring ∧ (𝑈𝐵𝑊𝐵) ∧ 𝐾 ∈ ℕ0) ∧ (𝑖𝑁𝑗𝑁)) ∧ 𝑘 ∈ (0...𝐾)) → 𝑊𝐵)
51 fznn0sub 13566 . . . . . . . . . . . . . 14 (𝑘 ∈ (0...𝐾) → (𝐾𝑘) ∈ ℕ0)
5251adantl 481 . . . . . . . . . . . . 13 ((((𝑅 ∈ Ring ∧ (𝑈𝐵𝑊𝐵) ∧ 𝐾 ∈ ℕ0) ∧ (𝑖𝑁𝑗𝑁)) ∧ 𝑘 ∈ (0...𝐾)) → (𝐾𝑘) ∈ ℕ0)
5334, 50, 523jca 1126 . . . . . . . . . . . 12 ((((𝑅 ∈ Ring ∧ (𝑈𝐵𝑊𝐵) ∧ 𝐾 ∈ ℕ0) ∧ (𝑖𝑁𝑗𝑁)) ∧ 𝑘 ∈ (0...𝐾)) → (𝑅 ∈ Ring ∧ 𝑊𝐵 ∧ (𝐾𝑘) ∈ ℕ0))
5443, 14, 15, 22, 44decpmatcl 22682 . . . . . . . . . . . 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 22337 . . . . . . . . . . 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 22301 . . . . . . . . 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 2768 . . . . . . . 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 5248 . . . . . . 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 7436 . . . . . 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 2728 . . . . . . 7 (0g𝐴) = (0g𝐴)
63 ovexd 7455 . . . . . . 7 (((𝑅 ∈ Ring ∧ (𝑈𝐵𝑊𝐵) ∧ 𝐾 ∈ ℕ0) ∧ (𝑖𝑁𝑗𝑁)) → (0...𝐾) ∈ V)
64 ringcmn 20218 . . . . . . . . . . . . 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 1131 . . . . . . . . 9 (((((𝑅 ∈ Ring ∧ (𝑈𝐵𝑊𝐵) ∧ 𝐾 ∈ ℕ0) ∧ (𝑖𝑁𝑗𝑁)) ∧ 𝑘 ∈ (0...𝐾)) ∧ 𝑥𝑁𝑦𝑁) → 𝑅 ∈ CMnd)
69373ad2ant1 1131 . . . . . . . . 9 (((((𝑅 ∈ Ring ∧ (𝑈𝐵𝑊𝐵) ∧ 𝐾 ∈ ℕ0) ∧ (𝑖𝑁𝑗𝑁)) ∧ 𝑘 ∈ (0...𝐾)) ∧ 𝑥𝑁𝑦𝑁) → 𝑁 ∈ Fin)
70343ad2ant1 1131 . . . . . . . . . . . 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 1190 . . . . . . . . . . . 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 1131 . . . . . . . . . . . . . 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 22341 . . . . . . . . . . 11 ((((((𝑅 ∈ Ring ∧ (𝑈𝐵𝑊𝐵) ∧ 𝐾 ∈ ℕ0) ∧ (𝑖𝑁𝑗𝑁)) ∧ 𝑘 ∈ (0...𝐾)) ∧ 𝑥𝑁𝑦𝑁) ∧ 𝑡𝑁) → (𝑥(𝑈 decompPMat 𝑘)𝑡) ∈ (Base‘𝑅))
78 simpl3 1191 . . . . . . . . . . . 12 ((((((𝑅 ∈ Ring ∧ (𝑈𝐵𝑊𝐵) ∧ 𝐾 ∈ ℕ0) ∧ (𝑖𝑁𝑗𝑁)) ∧ 𝑘 ∈ (0...𝐾)) ∧ 𝑥𝑁𝑦𝑁) ∧ 𝑡𝑁) → 𝑦𝑁)
79553ad2ant1 1131 . . . . . . . . . . . . 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 22341 . . . . . . . . . . 11 ((((((𝑅 ∈ Ring ∧ (𝑈𝐵𝑊𝐵) ∧ 𝐾 ∈ ℕ0) ∧ (𝑖𝑁𝑗𝑁)) ∧ 𝑘 ∈ (0...𝐾)) ∧ 𝑥𝑁𝑦𝑁) ∧ 𝑡𝑁) → (𝑡(𝑊 decompPMat (𝐾𝑘))𝑦) ∈ (Base‘𝑅))
8230, 31ringcl 20190 . . . . . . . . . . 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 1369 . . . . . . . . . 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 19922 . . . . . . . 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 22338 . . . . . . 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 2728 . . . . . . . 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 13971 . . . . . . . 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 1132 . . . . . . . . . . 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 8082 . . . . . . . . 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 6912 . . . . . . . 8 (((𝑅 ∈ Ring ∧ (𝑈𝐵𝑊𝐵) ∧ 𝐾 ∈ ℕ0) ∧ (𝑖𝑁𝑗𝑁)) → (0g𝐴) ∈ V)
9987, 88, 97, 98fsuppmptdm 9400 . . . . . . 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 22352 . . . . . 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 2768 . . . . 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 7437 . . . 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 1190 . . . . . 6 (((𝑅 ∈ Ring ∧ (𝑈𝐵𝑊𝐵) ∧ 𝐾 ∈ ℕ0) ∧ (𝑖𝑁𝑗𝑁)) → (𝑈𝐵𝑊𝐵))
104 simpl3 1191 . . . . . 6 (((𝑅 ∈ Ring ∧ (𝑈𝐵𝑊𝐵) ∧ 𝐾 ∈ ℕ0) ∧ (𝑖𝑁𝑗𝑁)) → 𝐾 ∈ ℕ0)
10543, 14, 15decpmatmullem 22686 . . . . . 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 1387 . . . . 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 1210 . . . . . . 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 2821 . . . . . . . . . . . . . 14 (𝑈𝐵𝑈 ∈ (Base‘𝐶))
111110biimpi 215 . . . . . . . . . . . . 13 (𝑈𝐵𝑈 ∈ (Base‘𝐶))
112111adantr 480 . . . . . . . . . . . 12 ((𝑈𝐵𝑊𝐵) → 𝑈 ∈ (Base‘𝐶))
1131123ad2ant2 1132 . . . . . . . . . . 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 2728 . . . . . . . . . 10 (Base‘𝑃) = (Base‘𝑃)
11714, 116matecl 22340 . . . . . . . . 9 ((𝑖𝑁𝑡𝑁𝑈 ∈ (Base‘𝐶)) → (𝑖𝑈𝑡) ∈ (Base‘𝑃))
118108, 109, 115, 117syl3anc 1369 . . . . . . . 8 ((((𝑅 ∈ Ring ∧ (𝑈𝐵𝑊𝐵) ∧ 𝐾 ∈ ℕ0) ∧ (𝑖𝑁𝑗𝑁)) ∧ (𝑡𝑁𝑘 ∈ (0...𝐾))) → (𝑖𝑈𝑡) ∈ (Base‘𝑃))
11940ad2antll 728 . . . . . . . 8 ((((𝑅 ∈ Ring ∧ (𝑈𝐵𝑊𝐵) ∧ 𝐾 ∈ ℕ0) ∧ (𝑖𝑁𝑗𝑁)) ∧ (𝑡𝑁𝑘 ∈ (0...𝐾))) → 𝑘 ∈ ℕ0)
120 eqid 2728 . . . . . . . . 9 (coe1‘(𝑖𝑈𝑡)) = (coe1‘(𝑖𝑈𝑡))
121120, 116, 43, 30coe1fvalcl 22131 . . . . . . . 8 (((𝑖𝑈𝑡) ∈ (Base‘𝑃) ∧ 𝑘 ∈ ℕ0) → ((coe1‘(𝑖𝑈𝑡))‘𝑘) ∈ (Base‘𝑅))
122118, 119, 121syl2anc 583 . . . . . . 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 22341 . . . . . . . 8 ((((𝑅 ∈ Ring ∧ (𝑈𝐵𝑊𝐵) ∧ 𝐾 ∈ ℕ0) ∧ (𝑖𝑁𝑗𝑁)) ∧ (𝑡𝑁𝑘 ∈ (0...𝐾))) → (𝑡𝑊𝑗) ∈ (Base‘𝑃))
12651ad2antll 728 . . . . . . . 8 ((((𝑅 ∈ Ring ∧ (𝑈𝐵𝑊𝐵) ∧ 𝐾 ∈ ℕ0) ∧ (𝑖𝑁𝑗𝑁)) ∧ (𝑡𝑁𝑘 ∈ (0...𝐾))) → (𝐾𝑘) ∈ ℕ0)
127 eqid 2728 . . . . . . . . 9 (coe1‘(𝑡𝑊𝑗)) = (coe1‘(𝑡𝑊𝑗))
128127, 116, 43, 30coe1fvalcl 22131 . . . . . . . 8 (((𝑡𝑊𝑗) ∈ (Base‘𝑃) ∧ (𝐾𝑘) ∈ ℕ0) → ((coe1‘(𝑡𝑊𝑗))‘(𝐾𝑘)) ∈ (Base‘𝑅))
129125, 126, 128syl2anc 583 . . . . . . 7 ((((𝑅 ∈ Ring ∧ (𝑈𝐵𝑊𝐵) ∧ 𝐾 ∈ ℕ0) ∧ (𝑖𝑁𝑗𝑁)) ∧ (𝑡𝑁𝑘 ∈ (0...𝐾))) → ((coe1‘(𝑡𝑊𝑗))‘(𝐾𝑘)) ∈ (Base‘𝑅))
13030, 31ringcl 20190 . . . . . . 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 1369 . . . . . 6 ((((𝑅 ∈ Ring ∧ (𝑈𝐵𝑊𝐵) ∧ 𝐾 ∈ ℕ0) ∧ (𝑖𝑁𝑗𝑁)) ∧ (𝑡𝑁𝑘 ∈ (0...𝐾))) → (((coe1‘(𝑖𝑈𝑡))‘𝑘)(.r𝑅)((coe1‘(𝑡𝑊𝑗))‘(𝐾𝑘))) ∈ (Base‘𝑅))
13230, 66, 36, 88, 131gsumcom3fi 19934 . . . . 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 614 . . . . . . . . . . . 12 (((((𝑅 ∈ Ring ∧ (𝑈𝐵𝑊𝐵) ∧ 𝐾 ∈ ℕ0) ∧ (𝑖𝑁𝑗𝑁)) ∧ 𝑘 ∈ (0...𝐾)) ∧ 𝑡𝑁) → (𝑖𝑁𝑡𝑁))
13543, 14, 15decpmate 22681 . . . . . . . . . . . 12 (((𝑅 ∈ Ring ∧ 𝑈𝐵𝑘 ∈ ℕ0) ∧ (𝑖𝑁𝑡𝑁)) → (𝑖(𝑈 decompPMat 𝑘)𝑡) = ((coe1‘(𝑖𝑈𝑡))‘𝑘))
13642, 134, 135syl2an2r 684 . . . . . . . . . . 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 615 . . . . . . . . . . . 12 (((((𝑅 ∈ Ring ∧ (𝑈𝐵𝑊𝐵) ∧ 𝐾 ∈ ℕ0) ∧ (𝑖𝑁𝑗𝑁)) ∧ 𝑘 ∈ (0...𝐾)) ∧ 𝑡𝑁) → (𝑡𝑁𝑗𝑁))
13943, 14, 15decpmate 22681 . . . . . . . . . . . 12 (((𝑅 ∈ Ring ∧ 𝑊𝐵 ∧ (𝐾𝑘) ∈ ℕ0) ∧ (𝑡𝑁𝑗𝑁)) → (𝑡(𝑊 decompPMat (𝐾𝑘))𝑗) = ((coe1‘(𝑡𝑊𝑗))‘(𝐾𝑘)))
14053, 138, 139syl2an2r 684 . . . . . . . . . . 11 (((((𝑅 ∈ Ring ∧ (𝑈𝐵𝑊𝐵) ∧ 𝐾 ∈ ℕ0) ∧ (𝑖𝑁𝑗𝑁)) ∧ 𝑘 ∈ (0...𝐾)) ∧ 𝑡𝑁) → (𝑡(𝑊 decompPMat (𝐾𝑘))𝑗) = ((coe1‘(𝑡𝑊𝑗))‘(𝐾𝑘)))
141136, 140oveq12d 7438 . . . . . . . . . 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 2734 . . . . . . . . 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 5248 . . . . . . . 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 7436 . . . . . . 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 5248 . . . . . 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 7436 . . . . 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 2772 . . . 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 2779 . . 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 22607 . . . . . . 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 2728 . . . . . . 7 (.r𝐶) = (.r𝐶)
15515, 154ringcl 20190 . . . . . 6 ((𝐶 ∈ Ring ∧ 𝑈𝐵𝑊𝐵) → (𝑈(.r𝐶)𝑊) ∈ 𝐵)
156151, 152, 153, 155syl3anc 1369 . . . . 5 ((𝑅 ∈ Ring ∧ (𝑈𝐵𝑊𝐵)) → (𝑈(.r𝐶)𝑊) ∈ 𝐵)
1571563adant3 1130 . . . 4 ((𝑅 ∈ Ring ∧ (𝑈𝐵𝑊𝐵) ∧ 𝐾 ∈ ℕ0) → (𝑈(.r𝐶)𝑊) ∈ 𝐵)
15843, 14, 15, 22, 44decpmatcl 22682 . . . 4 ((𝑅 ∈ Ring ∧ (𝑈(.r𝐶)𝑊) ∈ 𝐵𝐾 ∈ ℕ0) → ((𝑈(.r𝐶)𝑊) decompPMat 𝐾) ∈ (Base‘𝐴))
159157, 158syld3an2 1409 . . 3 ((𝑅 ∈ Ring ∧ (𝑈𝐵𝑊𝐵) ∧ 𝐾 ∈ ℕ0) → ((𝑈(.r𝐶)𝑊) decompPMat 𝐾) ∈ (Base‘𝐴))
16022matring 22358 . . . . . 6 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝐴 ∈ Ring)
16121, 160syl 17 . . . . 5 ((𝑅 ∈ Ring ∧ (𝑈𝐵𝑊𝐵) ∧ 𝐾 ∈ ℕ0) → 𝐴 ∈ Ring)
162 ringcmn 20218 . . . . 5 (𝐴 ∈ Ring → 𝐴 ∈ CMnd)
163161, 162syl 17 . . . 4 ((𝑅 ∈ Ring ∧ (𝑈𝐵𝑊𝐵) ∧ 𝐾 ∈ ℕ0) → 𝐴 ∈ CMnd)
164 fzfid 13971 . . . 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 1224 . . . . . . . 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 1126 . . . . . . 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 1225 . . . . . . . 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 1126 . . . . . . 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 2728 . . . . . . 7 (.r𝐴) = (.r𝐴)
17644, 175ringcl 20190 . . . . . 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 1369 . . . . 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 19922 . . 3 ((𝑅 ∈ Ring ∧ (𝑈𝐵𝑊𝐵) ∧ 𝐾 ∈ ℕ0) → (𝐴 Σg (𝑘 ∈ (0...𝐾) ↦ ((𝑈 decompPMat 𝑘)(.r𝐴)(𝑊 decompPMat (𝐾𝑘))))) ∈ (Base‘𝐴))
18022, 44eqmat 22339 . . 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 583 . 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 205  wa 395  w3a 1085   = wceq 1534  wcel 2099  wral 3058  Vcvv 3471  cotp 4637  cmpt 5231   × cxp 5676  cfv 6548  (class class class)co 7420  cmpo 7422  m cmap 8845  Fincfn 8964  0cc0 11139  cmin 11475  0cn0 12503  ...cfz 13517  Basecbs 17180  .rcmulr 17234  0gc0g 17421   Σg cgsu 17422  CMndccmn 19735  Ringcrg 20173  Poly1cpl1 22096  coe1cco1 22097   maMul cmmul 22298   Mat cmat 22320   decompPMat cdecpmat 22677
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2699  ax-rep 5285  ax-sep 5299  ax-nul 5306  ax-pow 5365  ax-pr 5429  ax-un 7740  ax-cnex 11195  ax-resscn 11196  ax-1cn 11197  ax-icn 11198  ax-addcl 11199  ax-addrcl 11200  ax-mulcl 11201  ax-mulrcl 11202  ax-mulcom 11203  ax-addass 11204  ax-mulass 11205  ax-distr 11206  ax-i2m1 11207  ax-1ne0 11208  ax-1rid 11209  ax-rnegex 11210  ax-rrecex 11211  ax-cnre 11212  ax-pre-lttri 11213  ax-pre-lttrn 11214  ax-pre-ltadd 11215  ax-pre-mulgt0 11216
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3or 1086  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2530  df-eu 2559  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2938  df-nel 3044  df-ral 3059  df-rex 3068  df-rmo 3373  df-reu 3374  df-rab 3430  df-v 3473  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-pss 3966  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-tp 4634  df-op 4636  df-ot 4638  df-uni 4909  df-int 4950  df-iun 4998  df-iin 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5576  df-eprel 5582  df-po 5590  df-so 5591  df-fr 5633  df-se 5634  df-we 5635  df-xp 5684  df-rel 5685  df-cnv 5686  df-co 5687  df-dm 5688  df-rn 5689  df-res 5690  df-ima 5691  df-pred 6305  df-ord 6372  df-on 6373  df-lim 6374  df-suc 6375  df-iota 6500  df-fun 6550  df-fn 6551  df-f 6552  df-f1 6553  df-fo 6554  df-f1o 6555  df-fv 6556  df-isom 6557  df-riota 7376  df-ov 7423  df-oprab 7424  df-mpo 7425  df-of 7685  df-ofr 7686  df-om 7871  df-1st 7993  df-2nd 7994  df-supp 8166  df-frecs 8287  df-wrecs 8318  df-recs 8392  df-rdg 8431  df-1o 8487  df-er 8725  df-map 8847  df-pm 8848  df-ixp 8917  df-en 8965  df-dom 8966  df-sdom 8967  df-fin 8968  df-fsupp 9387  df-sup 9466  df-oi 9534  df-card 9963  df-pnf 11281  df-mnf 11282  df-xr 11283  df-ltxr 11284  df-le 11285  df-sub 11477  df-neg 11478  df-nn 12244  df-2 12306  df-3 12307  df-4 12308  df-5 12309  df-6 12310  df-7 12311  df-8 12312  df-9 12313  df-n0 12504  df-z 12590  df-dec 12709  df-uz 12854  df-fz 13518  df-fzo 13661  df-seq 14000  df-hash 14323  df-struct 17116  df-sets 17133  df-slot 17151  df-ndx 17163  df-base 17181  df-ress 17210  df-plusg 17246  df-mulr 17247  df-sca 17249  df-vsca 17250  df-ip 17251  df-tset 17252  df-ple 17253  df-ds 17255  df-hom 17257  df-cco 17258  df-0g 17423  df-gsum 17424  df-prds 17429  df-pws 17431  df-mre 17566  df-mrc 17567  df-acs 17569  df-mgm 18600  df-sgrp 18679  df-mnd 18695  df-mhm 18740  df-submnd 18741  df-grp 18893  df-minusg 18894  df-sbg 18895  df-mulg 19024  df-subg 19078  df-ghm 19168  df-cntz 19268  df-cmn 19737  df-abl 19738  df-mgp 20075  df-rng 20093  df-ur 20122  df-ring 20175  df-subrng 20483  df-subrg 20508  df-lmod 20745  df-lss 20816  df-sra 21058  df-rgmod 21059  df-dsmm 21666  df-frlm 21681  df-psr 21842  df-mpl 21844  df-opsr 21846  df-psr1 22099  df-ply1 22101  df-coe1 22102  df-mamu 22299  df-mat 22321  df-decpmat 22678
This theorem is referenced by:  decpmatmulsumfsupp  22688  pm2mpmhmlem1  22733  pm2mpmhmlem2  22734
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