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Theorem decpmatmul 21372
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 2820 . . . . 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 7155 . . . . . . . . . . 11 (𝑥 = 𝑖 → (𝑥(𝑈 decompPMat 𝑘)𝑡) = (𝑖(𝑈 decompPMat 𝑘)𝑡))
3 oveq2 7156 . . . . . . . . . . 11 (𝑦 = 𝑗 → (𝑡(𝑊 decompPMat (𝐾𝑘))𝑦) = (𝑡(𝑊 decompPMat (𝐾𝑘))𝑗))
42, 3oveqan12d 7167 . . . . . . . . . 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 5153 . . . . . . . . 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 7164 . . . . . . . 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 5153 . . . . . . 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 7164 . . . . . 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 484 . . . . 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 7183 . . . . 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 7294 . . . 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 21013 . . . . . . . . . . . . . . . . . . 19 (𝑈𝐵 → (𝑁 ∈ Fin ∧ 𝑃 ∈ V))
1716simpld 497 . . . . . . . . . . . . . . . . . 18 (𝑈𝐵𝑁 ∈ Fin)
1817adantr 483 . . . . . . . . . . . . . . . . 17 ((𝑈𝐵𝑊𝐵) → 𝑁 ∈ Fin)
1918anim2i 618 . . . . . . . . . . . . . . . 16 ((𝑅 ∈ Ring ∧ (𝑈𝐵𝑊𝐵)) → (𝑅 ∈ Ring ∧ 𝑁 ∈ Fin))
2019ancomd 464 . . . . . . . . . . . . . . 15 ((𝑅 ∈ Ring ∧ (𝑈𝐵𝑊𝐵)) → (𝑁 ∈ Fin ∧ 𝑅 ∈ Ring))
21203adant3 1127 . . . . . . . . . . . . . 14 ((𝑅 ∈ Ring ∧ (𝑈𝐵𝑊𝐵) ∧ 𝐾 ∈ ℕ0) → (𝑁 ∈ Fin ∧ 𝑅 ∈ Ring))
22 decpmatmul.a . . . . . . . . . . . . . . 15 𝐴 = (𝑁 Mat 𝑅)
23 eqid 2819 . . . . . . . . . . . . . . 15 (𝑅 maMul ⟨𝑁, 𝑁, 𝑁⟩) = (𝑅 maMul ⟨𝑁, 𝑁, 𝑁⟩)
2422, 23matmulr 21039 . . . . . . . . . . . . . 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 483 . . . . . . . . . . . 12 (((𝑅 ∈ Ring ∧ (𝑈𝐵𝑊𝐵) ∧ 𝐾 ∈ ℕ0) ∧ (𝑖𝑁𝑗𝑁)) → (𝑅 maMul ⟨𝑁, 𝑁, 𝑁⟩) = (.r𝐴))
2726adantr 483 . . . . . . . . . . 11 ((((𝑅 ∈ Ring ∧ (𝑈𝐵𝑊𝐵) ∧ 𝐾 ∈ ℕ0) ∧ (𝑖𝑁𝑗𝑁)) ∧ 𝑘 ∈ (0...𝐾)) → (𝑅 maMul ⟨𝑁, 𝑁, 𝑁⟩) = (.r𝐴))
2827eqcomd 2825 . . . . . . . . . 10 ((((𝑅 ∈ Ring ∧ (𝑈𝐵𝑊𝐵) ∧ 𝐾 ∈ ℕ0) ∧ (𝑖𝑁𝑗𝑁)) ∧ 𝑘 ∈ (0...𝐾)) → (.r𝐴) = (𝑅 maMul ⟨𝑁, 𝑁, 𝑁⟩))
2928oveqd 7165 . . . . . . . . 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 2819 . . . . . . . . . 10 (Base‘𝑅) = (Base‘𝑅)
31 eqid 2819 . . . . . . . . . 10 (.r𝑅) = (.r𝑅)
32 simp1 1131 . . . . . . . . . . . 12 ((𝑅 ∈ Ring ∧ (𝑈𝐵𝑊𝐵) ∧ 𝐾 ∈ ℕ0) → 𝑅 ∈ Ring)
3332adantr 483 . . . . . . . . . . 11 (((𝑅 ∈ Ring ∧ (𝑈𝐵𝑊𝐵) ∧ 𝐾 ∈ ℕ0) ∧ (𝑖𝑁𝑗𝑁)) → 𝑅 ∈ Ring)
3433adantr 483 . . . . . . . . . 10 ((((𝑅 ∈ Ring ∧ (𝑈𝐵𝑊𝐵) ∧ 𝐾 ∈ ℕ0) ∧ (𝑖𝑁𝑗𝑁)) ∧ 𝑘 ∈ (0...𝐾)) → 𝑅 ∈ Ring)
3521simpld 497 . . . . . . . . . . . 12 ((𝑅 ∈ Ring ∧ (𝑈𝐵𝑊𝐵) ∧ 𝐾 ∈ ℕ0) → 𝑁 ∈ Fin)
3635adantr 483 . . . . . . . . . . 11 (((𝑅 ∈ Ring ∧ (𝑈𝐵𝑊𝐵) ∧ 𝐾 ∈ ℕ0) ∧ (𝑖𝑁𝑗𝑁)) → 𝑁 ∈ Fin)
3736adantr 483 . . . . . . . . . 10 ((((𝑅 ∈ Ring ∧ (𝑈𝐵𝑊𝐵) ∧ 𝐾 ∈ ℕ0) ∧ (𝑖𝑁𝑗𝑁)) ∧ 𝑘 ∈ (0...𝐾)) → 𝑁 ∈ Fin)
38 simpl2l 1221 . . . . . . . . . . . . . 14 (((𝑅 ∈ Ring ∧ (𝑈𝐵𝑊𝐵) ∧ 𝐾 ∈ ℕ0) ∧ (𝑖𝑁𝑗𝑁)) → 𝑈𝐵)
3938adantr 483 . . . . . . . . . . . . 13 ((((𝑅 ∈ Ring ∧ (𝑈𝐵𝑊𝐵) ∧ 𝐾 ∈ ℕ0) ∧ (𝑖𝑁𝑗𝑁)) ∧ 𝑘 ∈ (0...𝐾)) → 𝑈𝐵)
40 elfznn0 12992 . . . . . . . . . . . . . 14 (𝑘 ∈ (0...𝐾) → 𝑘 ∈ ℕ0)
4140adantl 484 . . . . . . . . . . . . 13 ((((𝑅 ∈ Ring ∧ (𝑈𝐵𝑊𝐵) ∧ 𝐾 ∈ ℕ0) ∧ (𝑖𝑁𝑗𝑁)) ∧ 𝑘 ∈ (0...𝐾)) → 𝑘 ∈ ℕ0)
4234, 39, 413jca 1123 . . . . . . . . . . . 12 ((((𝑅 ∈ Ring ∧ (𝑈𝐵𝑊𝐵) ∧ 𝐾 ∈ ℕ0) ∧ (𝑖𝑁𝑗𝑁)) ∧ 𝑘 ∈ (0...𝐾)) → (𝑅 ∈ Ring ∧ 𝑈𝐵𝑘 ∈ ℕ0))
43 decpmatmul.p . . . . . . . . . . . . 13 𝑃 = (Poly1𝑅)
44 eqid 2819 . . . . . . . . . . . . 13 (Base‘𝐴) = (Base‘𝐴)
4543, 14, 15, 22, 44decpmatcl 21367 . . . . . . . . . . . 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 21023 . . . . . . . . . . 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 1222 . . . . . . . . . . . . . 14 (((𝑅 ∈ Ring ∧ (𝑈𝐵𝑊𝐵) ∧ 𝐾 ∈ ℕ0) ∧ (𝑖𝑁𝑗𝑁)) → 𝑊𝐵)
5049adantr 483 . . . . . . . . . . . . 13 ((((𝑅 ∈ Ring ∧ (𝑈𝐵𝑊𝐵) ∧ 𝐾 ∈ ℕ0) ∧ (𝑖𝑁𝑗𝑁)) ∧ 𝑘 ∈ (0...𝐾)) → 𝑊𝐵)
51 fznn0sub 12931 . . . . . . . . . . . . . 14 (𝑘 ∈ (0...𝐾) → (𝐾𝑘) ∈ ℕ0)
5251adantl 484 . . . . . . . . . . . . 13 ((((𝑅 ∈ Ring ∧ (𝑈𝐵𝑊𝐵) ∧ 𝐾 ∈ ℕ0) ∧ (𝑖𝑁𝑗𝑁)) ∧ 𝑘 ∈ (0...𝐾)) → (𝐾𝑘) ∈ ℕ0)
5334, 50, 523jca 1123 . . . . . . . . . . . 12 ((((𝑅 ∈ Ring ∧ (𝑈𝐵𝑊𝐵) ∧ 𝐾 ∈ ℕ0) ∧ (𝑖𝑁𝑗𝑁)) ∧ 𝑘 ∈ (0...𝐾)) → (𝑅 ∈ Ring ∧ 𝑊𝐵 ∧ (𝐾𝑘) ∈ ℕ0))
5443, 14, 15, 22, 44decpmatcl 21367 . . . . . . . . . . . 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 21023 . . . . . . . . . . 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 20989 . . . . . . . . 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 2854 . . . . . . . 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 5152 . . . . . . 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 7164 . . . . . 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 2819 . . . . . . 7 (0g𝐴) = (0g𝐴)
63 ovexd 7183 . . . . . . 7 (((𝑅 ∈ Ring ∧ (𝑈𝐵𝑊𝐵) ∧ 𝐾 ∈ ℕ0) ∧ (𝑖𝑁𝑗𝑁)) → (0...𝐾) ∈ V)
64 ringcmn 19323 . . . . . . . . . . . . 13 (𝑅 ∈ Ring → 𝑅 ∈ CMnd)
6532, 64syl 17 . . . . . . . . . . . 12 ((𝑅 ∈ Ring ∧ (𝑈𝐵𝑊𝐵) ∧ 𝐾 ∈ ℕ0) → 𝑅 ∈ CMnd)
6665adantr 483 . . . . . . . . . . 11 (((𝑅 ∈ Ring ∧ (𝑈𝐵𝑊𝐵) ∧ 𝐾 ∈ ℕ0) ∧ (𝑖𝑁𝑗𝑁)) → 𝑅 ∈ CMnd)
6766adantr 483 . . . . . . . . . 10 ((((𝑅 ∈ Ring ∧ (𝑈𝐵𝑊𝐵) ∧ 𝐾 ∈ ℕ0) ∧ (𝑖𝑁𝑗𝑁)) ∧ 𝑘 ∈ (0...𝐾)) → 𝑅 ∈ CMnd)
68673ad2ant1 1128 . . . . . . . . 9 (((((𝑅 ∈ Ring ∧ (𝑈𝐵𝑊𝐵) ∧ 𝐾 ∈ ℕ0) ∧ (𝑖𝑁𝑗𝑁)) ∧ 𝑘 ∈ (0...𝐾)) ∧ 𝑥𝑁𝑦𝑁) → 𝑅 ∈ CMnd)
69373ad2ant1 1128 . . . . . . . . 9 (((((𝑅 ∈ Ring ∧ (𝑈𝐵𝑊𝐵) ∧ 𝐾 ∈ ℕ0) ∧ (𝑖𝑁𝑗𝑁)) ∧ 𝑘 ∈ (0...𝐾)) ∧ 𝑥𝑁𝑦𝑁) → 𝑁 ∈ Fin)
70343ad2ant1 1128 . . . . . . . . . . . 12 (((((𝑅 ∈ Ring ∧ (𝑈𝐵𝑊𝐵) ∧ 𝐾 ∈ ℕ0) ∧ (𝑖𝑁𝑗𝑁)) ∧ 𝑘 ∈ (0...𝐾)) ∧ 𝑥𝑁𝑦𝑁) → 𝑅 ∈ Ring)
7170adantr 483 . . . . . . . . . . 11 ((((((𝑅 ∈ Ring ∧ (𝑈𝐵𝑊𝐵) ∧ 𝐾 ∈ ℕ0) ∧ (𝑖𝑁𝑗𝑁)) ∧ 𝑘 ∈ (0...𝐾)) ∧ 𝑥𝑁𝑦𝑁) ∧ 𝑡𝑁) → 𝑅 ∈ Ring)
72 simpl2 1187 . . . . . . . . . . . 12 ((((((𝑅 ∈ Ring ∧ (𝑈𝐵𝑊𝐵) ∧ 𝐾 ∈ ℕ0) ∧ (𝑖𝑁𝑗𝑁)) ∧ 𝑘 ∈ (0...𝐾)) ∧ 𝑥𝑁𝑦𝑁) ∧ 𝑡𝑁) → 𝑥𝑁)
73 simpr 487 . . . . . . . . . . . 12 ((((((𝑅 ∈ Ring ∧ (𝑈𝐵𝑊𝐵) ∧ 𝐾 ∈ ℕ0) ∧ (𝑖𝑁𝑗𝑁)) ∧ 𝑘 ∈ (0...𝐾)) ∧ 𝑥𝑁𝑦𝑁) ∧ 𝑡𝑁) → 𝑡𝑁)
74423ad2ant1 1128 . . . . . . . . . . . . . 14 (((((𝑅 ∈ Ring ∧ (𝑈𝐵𝑊𝐵) ∧ 𝐾 ∈ ℕ0) ∧ (𝑖𝑁𝑗𝑁)) ∧ 𝑘 ∈ (0...𝐾)) ∧ 𝑥𝑁𝑦𝑁) → (𝑅 ∈ Ring ∧ 𝑈𝐵𝑘 ∈ ℕ0))
7574adantr 483 . . . . . . . . . . . . 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 21027 . . . . . . . . . . 11 ((((((𝑅 ∈ Ring ∧ (𝑈𝐵𝑊𝐵) ∧ 𝐾 ∈ ℕ0) ∧ (𝑖𝑁𝑗𝑁)) ∧ 𝑘 ∈ (0...𝐾)) ∧ 𝑥𝑁𝑦𝑁) ∧ 𝑡𝑁) → (𝑥(𝑈 decompPMat 𝑘)𝑡) ∈ (Base‘𝑅))
78 simpl3 1188 . . . . . . . . . . . 12 ((((((𝑅 ∈ Ring ∧ (𝑈𝐵𝑊𝐵) ∧ 𝐾 ∈ ℕ0) ∧ (𝑖𝑁𝑗𝑁)) ∧ 𝑘 ∈ (0...𝐾)) ∧ 𝑥𝑁𝑦𝑁) ∧ 𝑡𝑁) → 𝑦𝑁)
79553ad2ant1 1128 . . . . . . . . . . . . 13 (((((𝑅 ∈ Ring ∧ (𝑈𝐵𝑊𝐵) ∧ 𝐾 ∈ ℕ0) ∧ (𝑖𝑁𝑗𝑁)) ∧ 𝑘 ∈ (0...𝐾)) ∧ 𝑥𝑁𝑦𝑁) → (𝑊 decompPMat (𝐾𝑘)) ∈ (Base‘𝐴))
8079adantr 483 . . . . . . . . . . . 12 ((((((𝑅 ∈ Ring ∧ (𝑈𝐵𝑊𝐵) ∧ 𝐾 ∈ ℕ0) ∧ (𝑖𝑁𝑗𝑁)) ∧ 𝑘 ∈ (0...𝐾)) ∧ 𝑥𝑁𝑦𝑁) ∧ 𝑡𝑁) → (𝑊 decompPMat (𝐾𝑘)) ∈ (Base‘𝐴))
8122, 30, 44, 73, 78, 80matecld 21027 . . . . . . . . . . 11 ((((((𝑅 ∈ Ring ∧ (𝑈𝐵𝑊𝐵) ∧ 𝐾 ∈ ℕ0) ∧ (𝑖𝑁𝑗𝑁)) ∧ 𝑘 ∈ (0...𝐾)) ∧ 𝑥𝑁𝑦𝑁) ∧ 𝑡𝑁) → (𝑡(𝑊 decompPMat (𝐾𝑘))𝑦) ∈ (Base‘𝑅))
8230, 31ringcl 19303 . . . . . . . . . . 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 1366 . . . . . . . . . 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 3180 . . . . . . . . 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 19079 . . . . . . . 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 21024 . . . . . . 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 2819 . . . . . . . 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 13333 . . . . . . . 8 (((𝑅 ∈ Ring ∧ (𝑈𝐵𝑊𝐵) ∧ 𝐾 ∈ ℕ0) ∧ (𝑖𝑁𝑗𝑁)) → (0...𝐾) ∈ Fin)
89 simpl 485 . . . . . . . . . . . . . . 15 ((𝑁 ∈ Fin ∧ 𝑃 ∈ V) → 𝑁 ∈ Fin)
9089, 89jca 514 . . . . . . . . . . . . . 14 ((𝑁 ∈ Fin ∧ 𝑃 ∈ V) → (𝑁 ∈ Fin ∧ 𝑁 ∈ Fin))
9116, 90syl 17 . . . . . . . . . . . . 13 (𝑈𝐵 → (𝑁 ∈ Fin ∧ 𝑁 ∈ Fin))
9291adantr 483 . . . . . . . . . . . 12 ((𝑈𝐵𝑊𝐵) → (𝑁 ∈ Fin ∧ 𝑁 ∈ Fin))
93923ad2ant2 1129 . . . . . . . . . . 11 ((𝑅 ∈ Ring ∧ (𝑈𝐵𝑊𝐵) ∧ 𝐾 ∈ ℕ0) → (𝑁 ∈ Fin ∧ 𝑁 ∈ Fin))
9493adantr 483 . . . . . . . . . 10 (((𝑅 ∈ Ring ∧ (𝑈𝐵𝑊𝐵) ∧ 𝐾 ∈ ℕ0) ∧ (𝑖𝑁𝑗𝑁)) → (𝑁 ∈ Fin ∧ 𝑁 ∈ Fin))
9594adantr 483 . . . . . . . . 9 ((((𝑅 ∈ Ring ∧ (𝑈𝐵𝑊𝐵) ∧ 𝐾 ∈ ℕ0) ∧ (𝑖𝑁𝑗𝑁)) ∧ 𝑘 ∈ (0...𝐾)) → (𝑁 ∈ Fin ∧ 𝑁 ∈ Fin))
96 mpoexga 7767 . . . . . . . . 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 6678 . . . . . . . 8 (((𝑅 ∈ Ring ∧ (𝑈𝐵𝑊𝐵) ∧ 𝐾 ∈ ℕ0) ∧ (𝑖𝑁𝑗𝑁)) → (0g𝐴) ∈ V)
9987, 88, 97, 98fsuppmptdm 8836 . . . . . . 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 21038 . . . . . 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 2854 . . . . 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 7165 . . . 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 1187 . . . . . 6 (((𝑅 ∈ Ring ∧ (𝑈𝐵𝑊𝐵) ∧ 𝐾 ∈ ℕ0) ∧ (𝑖𝑁𝑗𝑁)) → (𝑈𝐵𝑊𝐵))
104 simpl3 1188 . . . . . 6 (((𝑅 ∈ Ring ∧ (𝑈𝐵𝑊𝐵) ∧ 𝐾 ∈ ℕ0) ∧ (𝑖𝑁𝑗𝑁)) → 𝐾 ∈ ℕ0)
10543, 14, 15decpmatmullem 21371 . . . . . 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 1384 . . . . 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 1207 . . . . . . 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 2902 . . . . . . . . . . . . . 14 (𝑈𝐵𝑈 ∈ (Base‘𝐶))
111110biimpi 218 . . . . . . . . . . . . 13 (𝑈𝐵𝑈 ∈ (Base‘𝐶))
112111adantr 483 . . . . . . . . . . . 12 ((𝑈𝐵𝑊𝐵) → 𝑈 ∈ (Base‘𝐶))
1131123ad2ant2 1129 . . . . . . . . . . 11 ((𝑅 ∈ Ring ∧ (𝑈𝐵𝑊𝐵) ∧ 𝐾 ∈ ℕ0) → 𝑈 ∈ (Base‘𝐶))
114113adantr 483 . . . . . . . . . 10 (((𝑅 ∈ Ring ∧ (𝑈𝐵𝑊𝐵) ∧ 𝐾 ∈ ℕ0) ∧ (𝑖𝑁𝑗𝑁)) → 𝑈 ∈ (Base‘𝐶))
115114adantr 483 . . . . . . . . 9 ((((𝑅 ∈ Ring ∧ (𝑈𝐵𝑊𝐵) ∧ 𝐾 ∈ ℕ0) ∧ (𝑖𝑁𝑗𝑁)) ∧ (𝑡𝑁𝑘 ∈ (0...𝐾))) → 𝑈 ∈ (Base‘𝐶))
116 eqid 2819 . . . . . . . . . 10 (Base‘𝑃) = (Base‘𝑃)
11714, 116matecl 21026 . . . . . . . . 9 ((𝑖𝑁𝑡𝑁𝑈 ∈ (Base‘𝐶)) → (𝑖𝑈𝑡) ∈ (Base‘𝑃))
118108, 109, 115, 117syl3anc 1366 . . . . . . . 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 2819 . . . . . . . . 9 (coe1‘(𝑖𝑈𝑡)) = (coe1‘(𝑖𝑈𝑡))
121120, 116, 43, 30coe1fvalcl 20372 . . . . . . . 8 (((𝑖𝑈𝑡) ∈ (Base‘𝑃) ∧ 𝑘 ∈ ℕ0) → ((coe1‘(𝑖𝑈𝑡))‘𝑘) ∈ (Base‘𝑅))
122118, 119, 121syl2anc 586 . . . . . . 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 483 . . . . . . . . 9 ((((𝑅 ∈ Ring ∧ (𝑈𝐵𝑊𝐵) ∧ 𝐾 ∈ ℕ0) ∧ (𝑖𝑁𝑗𝑁)) ∧ (𝑡𝑁𝑘 ∈ (0...𝐾))) → 𝑊𝐵)
12514, 116, 15, 109, 123, 124matecld 21027 . . . . . . . 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 2819 . . . . . . . . 9 (coe1‘(𝑡𝑊𝑗)) = (coe1‘(𝑡𝑊𝑗))
128127, 116, 43, 30coe1fvalcl 20372 . . . . . . . 8 (((𝑡𝑊𝑗) ∈ (Base‘𝑃) ∧ (𝐾𝑘) ∈ ℕ0) → ((coe1‘(𝑡𝑊𝑗))‘(𝐾𝑘)) ∈ (Base‘𝑅))
129125, 126, 128syl2anc 586 . . . . . . 7 ((((𝑅 ∈ Ring ∧ (𝑈𝐵𝑊𝐵) ∧ 𝐾 ∈ ℕ0) ∧ (𝑖𝑁𝑗𝑁)) ∧ (𝑡𝑁𝑘 ∈ (0...𝐾))) → ((coe1‘(𝑡𝑊𝑗))‘(𝐾𝑘)) ∈ (Base‘𝑅))
13030, 31ringcl 19303 . . . . . . 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 1366 . . . . . 6 ((((𝑅 ∈ Ring ∧ (𝑈𝐵𝑊𝐵) ∧ 𝐾 ∈ ℕ0) ∧ (𝑖𝑁𝑗𝑁)) ∧ (𝑡𝑁𝑘 ∈ (0...𝐾))) → (((coe1‘(𝑖𝑈𝑡))‘𝑘)(.r𝑅)((coe1‘(𝑡𝑊𝑗))‘(𝐾𝑘))) ∈ (Base‘𝑅))
13230, 66, 36, 88, 131gsumcom3fi 19091 . . . . 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 483 . . . . . . . . . . . . 13 ((((𝑅 ∈ Ring ∧ (𝑈𝐵𝑊𝐵) ∧ 𝐾 ∈ ℕ0) ∧ (𝑖𝑁𝑗𝑁)) ∧ 𝑘 ∈ (0...𝐾)) → 𝑖𝑁)
134133anim1i 616 . . . . . . . . . . . 12 (((((𝑅 ∈ Ring ∧ (𝑈𝐵𝑊𝐵) ∧ 𝐾 ∈ ℕ0) ∧ (𝑖𝑁𝑗𝑁)) ∧ 𝑘 ∈ (0...𝐾)) ∧ 𝑡𝑁) → (𝑖𝑁𝑡𝑁))
13543, 14, 15decpmate 21366 . . . . . . . . . . . 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 617 . . . . . . . . . . . 12 (((((𝑅 ∈ Ring ∧ (𝑈𝐵𝑊𝐵) ∧ 𝐾 ∈ ℕ0) ∧ (𝑖𝑁𝑗𝑁)) ∧ 𝑘 ∈ (0...𝐾)) ∧ 𝑡𝑁) → (𝑡𝑁𝑗𝑁))
13943, 14, 15decpmate 21366 . . . . . . . . . . . 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 7166 . . . . . . . . . 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 2825 . . . . . . . . 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 5152 . . . . . . . 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 7164 . . . . . . 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 5152 . . . . . 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 7164 . . . . 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 2858 . . . 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 2865 . . 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 3189 . 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 21293 . . . . . . 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 2819 . . . . . . 7 (.r𝐶) = (.r𝐶)
15515, 154ringcl 19303 . . . . . 6 ((𝐶 ∈ Ring ∧ 𝑈𝐵𝑊𝐵) → (𝑈(.r𝐶)𝑊) ∈ 𝐵)
156151, 152, 153, 155syl3anc 1366 . . . . 5 ((𝑅 ∈ Ring ∧ (𝑈𝐵𝑊𝐵)) → (𝑈(.r𝐶)𝑊) ∈ 𝐵)
1571563adant3 1127 . . . 4 ((𝑅 ∈ Ring ∧ (𝑈𝐵𝑊𝐵) ∧ 𝐾 ∈ ℕ0) → (𝑈(.r𝐶)𝑊) ∈ 𝐵)
15843, 14, 15, 22, 44decpmatcl 21367 . . . 4 ((𝑅 ∈ Ring ∧ (𝑈(.r𝐶)𝑊) ∈ 𝐵𝐾 ∈ ℕ0) → ((𝑈(.r𝐶)𝑊) decompPMat 𝐾) ∈ (Base‘𝐴))
159157, 158syld3an2 1406 . . 3 ((𝑅 ∈ Ring ∧ (𝑈𝐵𝑊𝐵) ∧ 𝐾 ∈ ℕ0) → ((𝑈(.r𝐶)𝑊) decompPMat 𝐾) ∈ (Base‘𝐴))
16022matring 21044 . . . . . 6 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝐴 ∈ Ring)
16121, 160syl 17 . . . . 5 ((𝑅 ∈ Ring ∧ (𝑈𝐵𝑊𝐵) ∧ 𝐾 ∈ ℕ0) → 𝐴 ∈ Ring)
162 ringcmn 19323 . . . . 5 (𝐴 ∈ Ring → 𝐴 ∈ CMnd)
163161, 162syl 17 . . . 4 ((𝑅 ∈ Ring ∧ (𝑈𝐵𝑊𝐵) ∧ 𝐾 ∈ ℕ0) → 𝐴 ∈ CMnd)
164 fzfid 13333 . . . 4 ((𝑅 ∈ Ring ∧ (𝑈𝐵𝑊𝐵) ∧ 𝐾 ∈ ℕ0) → (0...𝐾) ∈ Fin)
165161adantr 483 . . . . . 6 (((𝑅 ∈ Ring ∧ (𝑈𝐵𝑊𝐵) ∧ 𝐾 ∈ ℕ0) ∧ 𝑘 ∈ (0...𝐾)) → 𝐴 ∈ Ring)
16632adantr 483 . . . . . . . 8 (((𝑅 ∈ Ring ∧ (𝑈𝐵𝑊𝐵) ∧ 𝐾 ∈ ℕ0) ∧ 𝑘 ∈ (0...𝐾)) → 𝑅 ∈ Ring)
167 simpl2l 1221 . . . . . . . 8 (((𝑅 ∈ Ring ∧ (𝑈𝐵𝑊𝐵) ∧ 𝐾 ∈ ℕ0) ∧ 𝑘 ∈ (0...𝐾)) → 𝑈𝐵)
16840adantl 484 . . . . . . . 8 (((𝑅 ∈ Ring ∧ (𝑈𝐵𝑊𝐵) ∧ 𝐾 ∈ ℕ0) ∧ 𝑘 ∈ (0...𝐾)) → 𝑘 ∈ ℕ0)
169166, 167, 1683jca 1123 . . . . . . 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 1222 . . . . . . . 8 (((𝑅 ∈ Ring ∧ (𝑈𝐵𝑊𝐵) ∧ 𝐾 ∈ ℕ0) ∧ 𝑘 ∈ (0...𝐾)) → 𝑊𝐵)
17251adantl 484 . . . . . . . 8 (((𝑅 ∈ Ring ∧ (𝑈𝐵𝑊𝐵) ∧ 𝐾 ∈ ℕ0) ∧ 𝑘 ∈ (0...𝐾)) → (𝐾𝑘) ∈ ℕ0)
173166, 171, 1723jca 1123 . . . . . . 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 2819 . . . . . . 7 (.r𝐴) = (.r𝐴)
17644, 175ringcl 19303 . . . . . 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 1366 . . . . 5 (((𝑅 ∈ Ring ∧ (𝑈𝐵𝑊𝐵) ∧ 𝐾 ∈ ℕ0) ∧ 𝑘 ∈ (0...𝐾)) → ((𝑈 decompPMat 𝑘)(.r𝐴)(𝑊 decompPMat (𝐾𝑘))) ∈ (Base‘𝐴))
178177ralrimiva 3180 . . . 4 ((𝑅 ∈ Ring ∧ (𝑈𝐵𝑊𝐵) ∧ 𝐾 ∈ ℕ0) → ∀𝑘 ∈ (0...𝐾)((𝑈 decompPMat 𝑘)(.r𝐴)(𝑊 decompPMat (𝐾𝑘))) ∈ (Base‘𝐴))
17944, 163, 164, 178gsummptcl 19079 . . 3 ((𝑅 ∈ Ring ∧ (𝑈𝐵𝑊𝐵) ∧ 𝐾 ∈ ℕ0) → (𝐴 Σg (𝑘 ∈ (0...𝐾) ↦ ((𝑈 decompPMat 𝑘)(.r𝐴)(𝑊 decompPMat (𝐾𝑘))))) ∈ (Base‘𝐴))
18022, 44eqmat 21025 . . 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 586 . 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 259 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 208  wa 398  w3a 1082   = wceq 1531  wcel 2108  wral 3136  Vcvv 3493  cotp 4567  cmpt 5137   × cxp 5546  cfv 6348  (class class class)co 7148  cmpo 7150  m cmap 8398  Fincfn 8501  0cc0 10529  cmin 10862  0cn0 11889  ...cfz 12884  Basecbs 16475  .rcmulr 16558  0gc0g 16705   Σg cgsu 16706  CMndccmn 18898  Ringcrg 19289  Poly1cpl1 20337  coe1cco1 20338   maMul cmmul 20986   Mat cmat 21008   decompPMat cdecpmat 21362
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 1905  ax-6 1964  ax-7 2009  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2154  ax-12 2170  ax-ext 2791  ax-rep 5181  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7453  ax-cnex 10585  ax-resscn 10586  ax-1cn 10587  ax-icn 10588  ax-addcl 10589  ax-addrcl 10590  ax-mulcl 10591  ax-mulrcl 10592  ax-mulcom 10593  ax-addass 10594  ax-mulass 10595  ax-distr 10596  ax-i2m1 10597  ax-1ne0 10598  ax-1rid 10599  ax-rnegex 10600  ax-rrecex 10601  ax-cnre 10602  ax-pre-lttri 10603  ax-pre-lttrn 10604  ax-pre-ltadd 10605  ax-pre-mulgt0 10606
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1083  df-3an 1084  df-tru 1534  df-fal 1544  df-ex 1775  df-nf 1779  df-sb 2064  df-mo 2616  df-eu 2648  df-clab 2798  df-cleq 2812  df-clel 2891  df-nfc 2961  df-ne 3015  df-nel 3122  df-ral 3141  df-rex 3142  df-reu 3143  df-rmo 3144  df-rab 3145  df-v 3495  df-sbc 3771  df-csb 3882  df-dif 3937  df-un 3939  df-in 3941  df-ss 3950  df-pss 3952  df-nul 4290  df-if 4466  df-pw 4539  df-sn 4560  df-pr 4562  df-tp 4564  df-op 4566  df-ot 4568  df-uni 4831  df-int 4868  df-iun 4912  df-iin 4913  df-br 5058  df-opab 5120  df-mpt 5138  df-tr 5164  df-id 5453  df-eprel 5458  df-po 5467  df-so 5468  df-fr 5507  df-se 5508  df-we 5509  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-pred 6141  df-ord 6187  df-on 6188  df-lim 6189  df-suc 6190  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-f1 6353  df-fo 6354  df-f1o 6355  df-fv 6356  df-isom 6357  df-riota 7106  df-ov 7151  df-oprab 7152  df-mpo 7153  df-of 7401  df-ofr 7402  df-om 7573  df-1st 7681  df-2nd 7682  df-supp 7823  df-wrecs 7939  df-recs 8000  df-rdg 8038  df-1o 8094  df-2o 8095  df-oadd 8098  df-er 8281  df-map 8400  df-pm 8401  df-ixp 8454  df-en 8502  df-dom 8503  df-sdom 8504  df-fin 8505  df-fsupp 8826  df-sup 8898  df-oi 8966  df-card 9360  df-pnf 10669  df-mnf 10670  df-xr 10671  df-ltxr 10672  df-le 10673  df-sub 10864  df-neg 10865  df-nn 11631  df-2 11692  df-3 11693  df-4 11694  df-5 11695  df-6 11696  df-7 11697  df-8 11698  df-9 11699  df-n0 11890  df-z 11974  df-dec 12091  df-uz 12236  df-fz 12885  df-fzo 13026  df-seq 13362  df-hash 13683  df-struct 16477  df-ndx 16478  df-slot 16479  df-base 16481  df-sets 16482  df-ress 16483  df-plusg 16570  df-mulr 16571  df-sca 16573  df-vsca 16574  df-ip 16575  df-tset 16576  df-ple 16577  df-ds 16579  df-hom 16581  df-cco 16582  df-0g 16707  df-gsum 16708  df-prds 16713  df-pws 16715  df-mre 16849  df-mrc 16850  df-acs 16852  df-mgm 17844  df-sgrp 17893  df-mnd 17904  df-mhm 17948  df-submnd 17949  df-grp 18098  df-minusg 18099  df-sbg 18100  df-mulg 18217  df-subg 18268  df-ghm 18348  df-cntz 18439  df-cmn 18900  df-abl 18901  df-mgp 19232  df-ur 19244  df-ring 19291  df-subrg 19525  df-lmod 19628  df-lss 19696  df-sra 19936  df-rgmod 19937  df-psr 20128  df-mpl 20130  df-opsr 20132  df-psr1 20340  df-ply1 20342  df-coe1 20343  df-dsmm 20868  df-frlm 20883  df-mamu 20987  df-mat 21009  df-decpmat 21363
This theorem is referenced by:  decpmatmulsumfsupp  21373  pm2mpmhmlem1  21418  pm2mpmhmlem2  21419
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