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Theorem decpmatmul 20700
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 2725 . . . . 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 6772 . . . . . . . . . . 11 (𝑥 = 𝑖 → (𝑥(𝑈 decompPMat 𝑘)𝑡) = (𝑖(𝑈 decompPMat 𝑘)𝑡))
3 oveq2 6773 . . . . . . . . . . 11 (𝑦 = 𝑗 → (𝑡(𝑊 decompPMat (𝐾𝑘))𝑦) = (𝑡(𝑊 decompPMat (𝐾𝑘))𝑗))
42, 3oveqan12d 6784 . . . . . . . . . 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 4853 . . . . . . . . 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 6781 . . . . . . . 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 4853 . . . . . . 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 6781 . . . . . 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 473 . . . . 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 811 . . . . 5 (((𝑅 ∈ Ring ∧ (𝑈𝐵𝑊𝐵) ∧ 𝐾 ∈ ℕ0) ∧ (𝑖𝑁𝑗𝑁)) → 𝑖𝑁)
11 simprr 813 . . . . 5 (((𝑅 ∈ Ring ∧ (𝑈𝐵𝑊𝐵) ∧ 𝐾 ∈ ℕ0) ∧ (𝑖𝑁𝑗𝑁)) → 𝑗𝑁)
12 ovexd 6795 . . . . 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, 12ovmpt2d 6905 . . . 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 20341 . . . . . . . . . . . . . . . . . . 19 (𝑈𝐵 → (𝑁 ∈ Fin ∧ 𝑃 ∈ V))
1716simpld 477 . . . . . . . . . . . . . . . . . 18 (𝑈𝐵𝑁 ∈ Fin)
1817adantr 472 . . . . . . . . . . . . . . . . 17 ((𝑈𝐵𝑊𝐵) → 𝑁 ∈ Fin)
1918anim2i 594 . . . . . . . . . . . . . . . 16 ((𝑅 ∈ Ring ∧ (𝑈𝐵𝑊𝐵)) → (𝑅 ∈ Ring ∧ 𝑁 ∈ Fin))
2019ancomd 466 . . . . . . . . . . . . . . 15 ((𝑅 ∈ Ring ∧ (𝑈𝐵𝑊𝐵)) → (𝑁 ∈ Fin ∧ 𝑅 ∈ Ring))
21203adant3 1124 . . . . . . . . . . . . . 14 ((𝑅 ∈ Ring ∧ (𝑈𝐵𝑊𝐵) ∧ 𝐾 ∈ ℕ0) → (𝑁 ∈ Fin ∧ 𝑅 ∈ Ring))
22 decpmatmul.a . . . . . . . . . . . . . . 15 𝐴 = (𝑁 Mat 𝑅)
23 eqid 2724 . . . . . . . . . . . . . . 15 (𝑅 maMul ⟨𝑁, 𝑁, 𝑁⟩) = (𝑅 maMul ⟨𝑁, 𝑁, 𝑁⟩)
2422, 23matmulr 20367 . . . . . . . . . . . . . 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 472 . . . . . . . . . . . 12 (((𝑅 ∈ Ring ∧ (𝑈𝐵𝑊𝐵) ∧ 𝐾 ∈ ℕ0) ∧ (𝑖𝑁𝑗𝑁)) → (𝑅 maMul ⟨𝑁, 𝑁, 𝑁⟩) = (.r𝐴))
2726adantr 472 . . . . . . . . . . 11 ((((𝑅 ∈ Ring ∧ (𝑈𝐵𝑊𝐵) ∧ 𝐾 ∈ ℕ0) ∧ (𝑖𝑁𝑗𝑁)) ∧ 𝑘 ∈ (0...𝐾)) → (𝑅 maMul ⟨𝑁, 𝑁, 𝑁⟩) = (.r𝐴))
2827eqcomd 2730 . . . . . . . . . 10 ((((𝑅 ∈ Ring ∧ (𝑈𝐵𝑊𝐵) ∧ 𝐾 ∈ ℕ0) ∧ (𝑖𝑁𝑗𝑁)) ∧ 𝑘 ∈ (0...𝐾)) → (.r𝐴) = (𝑅 maMul ⟨𝑁, 𝑁, 𝑁⟩))
2928oveqd 6782 . . . . . . . . 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 2724 . . . . . . . . . 10 (Base‘𝑅) = (Base‘𝑅)
31 eqid 2724 . . . . . . . . . 10 (.r𝑅) = (.r𝑅)
32 simp1 1128 . . . . . . . . . . . 12 ((𝑅 ∈ Ring ∧ (𝑈𝐵𝑊𝐵) ∧ 𝐾 ∈ ℕ0) → 𝑅 ∈ Ring)
3332adantr 472 . . . . . . . . . . 11 (((𝑅 ∈ Ring ∧ (𝑈𝐵𝑊𝐵) ∧ 𝐾 ∈ ℕ0) ∧ (𝑖𝑁𝑗𝑁)) → 𝑅 ∈ Ring)
3433adantr 472 . . . . . . . . . 10 ((((𝑅 ∈ Ring ∧ (𝑈𝐵𝑊𝐵) ∧ 𝐾 ∈ ℕ0) ∧ (𝑖𝑁𝑗𝑁)) ∧ 𝑘 ∈ (0...𝐾)) → 𝑅 ∈ Ring)
3521simpld 477 . . . . . . . . . . . 12 ((𝑅 ∈ Ring ∧ (𝑈𝐵𝑊𝐵) ∧ 𝐾 ∈ ℕ0) → 𝑁 ∈ Fin)
3635adantr 472 . . . . . . . . . . 11 (((𝑅 ∈ Ring ∧ (𝑈𝐵𝑊𝐵) ∧ 𝐾 ∈ ℕ0) ∧ (𝑖𝑁𝑗𝑁)) → 𝑁 ∈ Fin)
3736adantr 472 . . . . . . . . . 10 ((((𝑅 ∈ Ring ∧ (𝑈𝐵𝑊𝐵) ∧ 𝐾 ∈ ℕ0) ∧ (𝑖𝑁𝑗𝑁)) ∧ 𝑘 ∈ (0...𝐾)) → 𝑁 ∈ Fin)
38 simpl2l 1259 . . . . . . . . . . . . . 14 (((𝑅 ∈ Ring ∧ (𝑈𝐵𝑊𝐵) ∧ 𝐾 ∈ ℕ0) ∧ (𝑖𝑁𝑗𝑁)) → 𝑈𝐵)
3938adantr 472 . . . . . . . . . . . . 13 ((((𝑅 ∈ Ring ∧ (𝑈𝐵𝑊𝐵) ∧ 𝐾 ∈ ℕ0) ∧ (𝑖𝑁𝑗𝑁)) ∧ 𝑘 ∈ (0...𝐾)) → 𝑈𝐵)
40 elfznn0 12547 . . . . . . . . . . . . . 14 (𝑘 ∈ (0...𝐾) → 𝑘 ∈ ℕ0)
4140adantl 473 . . . . . . . . . . . . 13 ((((𝑅 ∈ Ring ∧ (𝑈𝐵𝑊𝐵) ∧ 𝐾 ∈ ℕ0) ∧ (𝑖𝑁𝑗𝑁)) ∧ 𝑘 ∈ (0...𝐾)) → 𝑘 ∈ ℕ0)
4234, 39, 413jca 1379 . . . . . . . . . . . 12 ((((𝑅 ∈ Ring ∧ (𝑈𝐵𝑊𝐵) ∧ 𝐾 ∈ ℕ0) ∧ (𝑖𝑁𝑗𝑁)) ∧ 𝑘 ∈ (0...𝐾)) → (𝑅 ∈ Ring ∧ 𝑈𝐵𝑘 ∈ ℕ0))
43 decpmatmul.p . . . . . . . . . . . . 13 𝑃 = (Poly1𝑅)
44 eqid 2724 . . . . . . . . . . . . 13 (Base‘𝐴) = (Base‘𝐴)
4543, 14, 15, 22, 44decpmatcl 20695 . . . . . . . . . . . 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 20351 . . . . . . . . . . 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 1261 . . . . . . . . . . . . . 14 (((𝑅 ∈ Ring ∧ (𝑈𝐵𝑊𝐵) ∧ 𝐾 ∈ ℕ0) ∧ (𝑖𝑁𝑗𝑁)) → 𝑊𝐵)
5049adantr 472 . . . . . . . . . . . . 13 ((((𝑅 ∈ Ring ∧ (𝑈𝐵𝑊𝐵) ∧ 𝐾 ∈ ℕ0) ∧ (𝑖𝑁𝑗𝑁)) ∧ 𝑘 ∈ (0...𝐾)) → 𝑊𝐵)
51 fznn0sub 12487 . . . . . . . . . . . . . 14 (𝑘 ∈ (0...𝐾) → (𝐾𝑘) ∈ ℕ0)
5251adantl 473 . . . . . . . . . . . . 13 ((((𝑅 ∈ Ring ∧ (𝑈𝐵𝑊𝐵) ∧ 𝐾 ∈ ℕ0) ∧ (𝑖𝑁𝑗𝑁)) ∧ 𝑘 ∈ (0...𝐾)) → (𝐾𝑘) ∈ ℕ0)
5334, 50, 523jca 1379 . . . . . . . . . . . 12 ((((𝑅 ∈ Ring ∧ (𝑈𝐵𝑊𝐵) ∧ 𝐾 ∈ ℕ0) ∧ (𝑖𝑁𝑗𝑁)) ∧ 𝑘 ∈ (0...𝐾)) → (𝑅 ∈ Ring ∧ 𝑊𝐵 ∧ (𝐾𝑘) ∈ ℕ0))
5443, 14, 15, 22, 44decpmatcl 20695 . . . . . . . . . . . 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 20351 . . . . . . . . . . 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 20315 . . . . . . . . 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 2758 . . . . . . . 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 4852 . . . . . . 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 6781 . . . . . 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 2724 . . . . . . 7 (0g𝐴) = (0g𝐴)
63 ovexd 6795 . . . . . . 7 (((𝑅 ∈ Ring ∧ (𝑈𝐵𝑊𝐵) ∧ 𝐾 ∈ ℕ0) ∧ (𝑖𝑁𝑗𝑁)) → (0...𝐾) ∈ V)
64 ringcmn 18702 . . . . . . . . . . . . 13 (𝑅 ∈ Ring → 𝑅 ∈ CMnd)
6532, 64syl 17 . . . . . . . . . . . 12 ((𝑅 ∈ Ring ∧ (𝑈𝐵𝑊𝐵) ∧ 𝐾 ∈ ℕ0) → 𝑅 ∈ CMnd)
6665adantr 472 . . . . . . . . . . 11 (((𝑅 ∈ Ring ∧ (𝑈𝐵𝑊𝐵) ∧ 𝐾 ∈ ℕ0) ∧ (𝑖𝑁𝑗𝑁)) → 𝑅 ∈ CMnd)
6766adantr 472 . . . . . . . . . 10 ((((𝑅 ∈ Ring ∧ (𝑈𝐵𝑊𝐵) ∧ 𝐾 ∈ ℕ0) ∧ (𝑖𝑁𝑗𝑁)) ∧ 𝑘 ∈ (0...𝐾)) → 𝑅 ∈ CMnd)
68673ad2ant1 1125 . . . . . . . . 9 (((((𝑅 ∈ Ring ∧ (𝑈𝐵𝑊𝐵) ∧ 𝐾 ∈ ℕ0) ∧ (𝑖𝑁𝑗𝑁)) ∧ 𝑘 ∈ (0...𝐾)) ∧ 𝑥𝑁𝑦𝑁) → 𝑅 ∈ CMnd)
69373ad2ant1 1125 . . . . . . . . 9 (((((𝑅 ∈ Ring ∧ (𝑈𝐵𝑊𝐵) ∧ 𝐾 ∈ ℕ0) ∧ (𝑖𝑁𝑗𝑁)) ∧ 𝑘 ∈ (0...𝐾)) ∧ 𝑥𝑁𝑦𝑁) → 𝑁 ∈ Fin)
70343ad2ant1 1125 . . . . . . . . . . . 12 (((((𝑅 ∈ Ring ∧ (𝑈𝐵𝑊𝐵) ∧ 𝐾 ∈ ℕ0) ∧ (𝑖𝑁𝑗𝑁)) ∧ 𝑘 ∈ (0...𝐾)) ∧ 𝑥𝑁𝑦𝑁) → 𝑅 ∈ Ring)
7170adantr 472 . . . . . . . . . . 11 ((((((𝑅 ∈ Ring ∧ (𝑈𝐵𝑊𝐵) ∧ 𝐾 ∈ ℕ0) ∧ (𝑖𝑁𝑗𝑁)) ∧ 𝑘 ∈ (0...𝐾)) ∧ 𝑥𝑁𝑦𝑁) ∧ 𝑡𝑁) → 𝑅 ∈ Ring)
72 simpl2 1206 . . . . . . . . . . . 12 ((((((𝑅 ∈ Ring ∧ (𝑈𝐵𝑊𝐵) ∧ 𝐾 ∈ ℕ0) ∧ (𝑖𝑁𝑗𝑁)) ∧ 𝑘 ∈ (0...𝐾)) ∧ 𝑥𝑁𝑦𝑁) ∧ 𝑡𝑁) → 𝑥𝑁)
73 simpr 479 . . . . . . . . . . . 12 ((((((𝑅 ∈ Ring ∧ (𝑈𝐵𝑊𝐵) ∧ 𝐾 ∈ ℕ0) ∧ (𝑖𝑁𝑗𝑁)) ∧ 𝑘 ∈ (0...𝐾)) ∧ 𝑥𝑁𝑦𝑁) ∧ 𝑡𝑁) → 𝑡𝑁)
74423ad2ant1 1125 . . . . . . . . . . . . . 14 (((((𝑅 ∈ Ring ∧ (𝑈𝐵𝑊𝐵) ∧ 𝐾 ∈ ℕ0) ∧ (𝑖𝑁𝑗𝑁)) ∧ 𝑘 ∈ (0...𝐾)) ∧ 𝑥𝑁𝑦𝑁) → (𝑅 ∈ Ring ∧ 𝑈𝐵𝑘 ∈ ℕ0))
7574adantr 472 . . . . . . . . . . . . 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 20355 . . . . . . . . . . 11 ((((((𝑅 ∈ Ring ∧ (𝑈𝐵𝑊𝐵) ∧ 𝐾 ∈ ℕ0) ∧ (𝑖𝑁𝑗𝑁)) ∧ 𝑘 ∈ (0...𝐾)) ∧ 𝑥𝑁𝑦𝑁) ∧ 𝑡𝑁) → (𝑥(𝑈 decompPMat 𝑘)𝑡) ∈ (Base‘𝑅))
78 simpl3 1208 . . . . . . . . . . . 12 ((((((𝑅 ∈ Ring ∧ (𝑈𝐵𝑊𝐵) ∧ 𝐾 ∈ ℕ0) ∧ (𝑖𝑁𝑗𝑁)) ∧ 𝑘 ∈ (0...𝐾)) ∧ 𝑥𝑁𝑦𝑁) ∧ 𝑡𝑁) → 𝑦𝑁)
79553ad2ant1 1125 . . . . . . . . . . . . 13 (((((𝑅 ∈ Ring ∧ (𝑈𝐵𝑊𝐵) ∧ 𝐾 ∈ ℕ0) ∧ (𝑖𝑁𝑗𝑁)) ∧ 𝑘 ∈ (0...𝐾)) ∧ 𝑥𝑁𝑦𝑁) → (𝑊 decompPMat (𝐾𝑘)) ∈ (Base‘𝐴))
8079adantr 472 . . . . . . . . . . . 12 ((((((𝑅 ∈ Ring ∧ (𝑈𝐵𝑊𝐵) ∧ 𝐾 ∈ ℕ0) ∧ (𝑖𝑁𝑗𝑁)) ∧ 𝑘 ∈ (0...𝐾)) ∧ 𝑥𝑁𝑦𝑁) ∧ 𝑡𝑁) → (𝑊 decompPMat (𝐾𝑘)) ∈ (Base‘𝐴))
8122, 30, 44, 73, 78, 80matecld 20355 . . . . . . . . . . 11 ((((((𝑅 ∈ Ring ∧ (𝑈𝐵𝑊𝐵) ∧ 𝐾 ∈ ℕ0) ∧ (𝑖𝑁𝑗𝑁)) ∧ 𝑘 ∈ (0...𝐾)) ∧ 𝑥𝑁𝑦𝑁) ∧ 𝑡𝑁) → (𝑡(𝑊 decompPMat (𝐾𝑘))𝑦) ∈ (Base‘𝑅))
8230, 31ringcl 18682 . . . . . . . . . . 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 1439 . . . . . . . . . 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 3068 . . . . . . . . 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 18487 . . . . . . . 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 20352 . . . . . . 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 2724 . . . . . . . 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 12887 . . . . . . . 8 (((𝑅 ∈ Ring ∧ (𝑈𝐵𝑊𝐵) ∧ 𝐾 ∈ ℕ0) ∧ (𝑖𝑁𝑗𝑁)) → (0...𝐾) ∈ Fin)
89 simpl 474 . . . . . . . . . . . . . . 15 ((𝑁 ∈ Fin ∧ 𝑃 ∈ V) → 𝑁 ∈ Fin)
9089, 89jca 555 . . . . . . . . . . . . . 14 ((𝑁 ∈ Fin ∧ 𝑃 ∈ V) → (𝑁 ∈ Fin ∧ 𝑁 ∈ Fin))
9116, 90syl 17 . . . . . . . . . . . . 13 (𝑈𝐵 → (𝑁 ∈ Fin ∧ 𝑁 ∈ Fin))
9291adantr 472 . . . . . . . . . . . 12 ((𝑈𝐵𝑊𝐵) → (𝑁 ∈ Fin ∧ 𝑁 ∈ Fin))
93923ad2ant2 1126 . . . . . . . . . . 11 ((𝑅 ∈ Ring ∧ (𝑈𝐵𝑊𝐵) ∧ 𝐾 ∈ ℕ0) → (𝑁 ∈ Fin ∧ 𝑁 ∈ Fin))
9493adantr 472 . . . . . . . . . 10 (((𝑅 ∈ Ring ∧ (𝑈𝐵𝑊𝐵) ∧ 𝐾 ∈ ℕ0) ∧ (𝑖𝑁𝑗𝑁)) → (𝑁 ∈ Fin ∧ 𝑁 ∈ Fin))
9594adantr 472 . . . . . . . . 9 ((((𝑅 ∈ Ring ∧ (𝑈𝐵𝑊𝐵) ∧ 𝐾 ∈ ℕ0) ∧ (𝑖𝑁𝑗𝑁)) ∧ 𝑘 ∈ (0...𝐾)) → (𝑁 ∈ Fin ∧ 𝑁 ∈ Fin))
96 mpt2exga 7366 . . . . . . . . 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 6316 . . . . . . . 8 (((𝑅 ∈ Ring ∧ (𝑈𝐵𝑊𝐵) ∧ 𝐾 ∈ ℕ0) ∧ (𝑖𝑁𝑗𝑁)) → (0g𝐴) ∈ V)
9987, 88, 97, 98fsuppmptdm 8402 . . . . . . 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 20366 . . . . . 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 2758 . . . . 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 6782 . . . 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 1206 . . . . . 6 (((𝑅 ∈ Ring ∧ (𝑈𝐵𝑊𝐵) ∧ 𝐾 ∈ ℕ0) ∧ (𝑖𝑁𝑗𝑁)) → (𝑈𝐵𝑊𝐵))
104 simpl3 1208 . . . . . 6 (((𝑅 ∈ Ring ∧ (𝑈𝐵𝑊𝐵) ∧ 𝐾 ∈ ℕ0) ∧ (𝑖𝑁𝑗𝑁)) → 𝐾 ∈ ℕ0)
10543, 14, 15decpmatmullem 20699 . . . . . 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 1458 . . . . 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 1231 . . . . . . 7 ((((𝑅 ∈ Ring ∧ (𝑈𝐵𝑊𝐵) ∧ 𝐾 ∈ ℕ0) ∧ (𝑖𝑁𝑗𝑁)) ∧ (𝑡𝑁𝑘 ∈ (0...𝐾))) → 𝑅 ∈ Ring)
108 simplrl 819 . . . . . . . . 9 ((((𝑅 ∈ Ring ∧ (𝑈𝐵𝑊𝐵) ∧ 𝐾 ∈ ℕ0) ∧ (𝑖𝑁𝑗𝑁)) ∧ (𝑡𝑁𝑘 ∈ (0...𝐾))) → 𝑖𝑁)
109 simprl 811 . . . . . . . . 9 ((((𝑅 ∈ Ring ∧ (𝑈𝐵𝑊𝐵) ∧ 𝐾 ∈ ℕ0) ∧ (𝑖𝑁𝑗𝑁)) ∧ (𝑡𝑁𝑘 ∈ (0...𝐾))) → 𝑡𝑁)
11015eleq2i 2795 . . . . . . . . . . . . . 14 (𝑈𝐵𝑈 ∈ (Base‘𝐶))
111110biimpi 206 . . . . . . . . . . . . 13 (𝑈𝐵𝑈 ∈ (Base‘𝐶))
112111adantr 472 . . . . . . . . . . . 12 ((𝑈𝐵𝑊𝐵) → 𝑈 ∈ (Base‘𝐶))
1131123ad2ant2 1126 . . . . . . . . . . 11 ((𝑅 ∈ Ring ∧ (𝑈𝐵𝑊𝐵) ∧ 𝐾 ∈ ℕ0) → 𝑈 ∈ (Base‘𝐶))
114113adantr 472 . . . . . . . . . 10 (((𝑅 ∈ Ring ∧ (𝑈𝐵𝑊𝐵) ∧ 𝐾 ∈ ℕ0) ∧ (𝑖𝑁𝑗𝑁)) → 𝑈 ∈ (Base‘𝐶))
115114adantr 472 . . . . . . . . 9 ((((𝑅 ∈ Ring ∧ (𝑈𝐵𝑊𝐵) ∧ 𝐾 ∈ ℕ0) ∧ (𝑖𝑁𝑗𝑁)) ∧ (𝑡𝑁𝑘 ∈ (0...𝐾))) → 𝑈 ∈ (Base‘𝐶))
116 eqid 2724 . . . . . . . . . 10 (Base‘𝑃) = (Base‘𝑃)
11714, 116matecl 20354 . . . . . . . . 9 ((𝑖𝑁𝑡𝑁𝑈 ∈ (Base‘𝐶)) → (𝑖𝑈𝑡) ∈ (Base‘𝑃))
118108, 109, 115, 117syl3anc 1439 . . . . . . . 8 ((((𝑅 ∈ Ring ∧ (𝑈𝐵𝑊𝐵) ∧ 𝐾 ∈ ℕ0) ∧ (𝑖𝑁𝑗𝑁)) ∧ (𝑡𝑁𝑘 ∈ (0...𝐾))) → (𝑖𝑈𝑡) ∈ (Base‘𝑃))
11940ad2antll 767 . . . . . . . 8 ((((𝑅 ∈ Ring ∧ (𝑈𝐵𝑊𝐵) ∧ 𝐾 ∈ ℕ0) ∧ (𝑖𝑁𝑗𝑁)) ∧ (𝑡𝑁𝑘 ∈ (0...𝐾))) → 𝑘 ∈ ℕ0)
120 eqid 2724 . . . . . . . . 9 (coe1‘(𝑖𝑈𝑡)) = (coe1‘(𝑖𝑈𝑡))
121120, 116, 43, 30coe1fvalcl 19705 . . . . . . . 8 (((𝑖𝑈𝑡) ∈ (Base‘𝑃) ∧ 𝑘 ∈ ℕ0) → ((coe1‘(𝑖𝑈𝑡))‘𝑘) ∈ (Base‘𝑅))
122118, 119, 121syl2anc 696 . . . . . . 7 ((((𝑅 ∈ Ring ∧ (𝑈𝐵𝑊𝐵) ∧ 𝐾 ∈ ℕ0) ∧ (𝑖𝑁𝑗𝑁)) ∧ (𝑡𝑁𝑘 ∈ (0...𝐾))) → ((coe1‘(𝑖𝑈𝑡))‘𝑘) ∈ (Base‘𝑅))
123 simplrr 820 . . . . . . . . 9 ((((𝑅 ∈ Ring ∧ (𝑈𝐵𝑊𝐵) ∧ 𝐾 ∈ ℕ0) ∧ (𝑖𝑁𝑗𝑁)) ∧ (𝑡𝑁𝑘 ∈ (0...𝐾))) → 𝑗𝑁)
12449adantr 472 . . . . . . . . 9 ((((𝑅 ∈ Ring ∧ (𝑈𝐵𝑊𝐵) ∧ 𝐾 ∈ ℕ0) ∧ (𝑖𝑁𝑗𝑁)) ∧ (𝑡𝑁𝑘 ∈ (0...𝐾))) → 𝑊𝐵)
12514, 116, 15, 109, 123, 124matecld 20355 . . . . . . . 8 ((((𝑅 ∈ Ring ∧ (𝑈𝐵𝑊𝐵) ∧ 𝐾 ∈ ℕ0) ∧ (𝑖𝑁𝑗𝑁)) ∧ (𝑡𝑁𝑘 ∈ (0...𝐾))) → (𝑡𝑊𝑗) ∈ (Base‘𝑃))
12651ad2antll 767 . . . . . . . 8 ((((𝑅 ∈ Ring ∧ (𝑈𝐵𝑊𝐵) ∧ 𝐾 ∈ ℕ0) ∧ (𝑖𝑁𝑗𝑁)) ∧ (𝑡𝑁𝑘 ∈ (0...𝐾))) → (𝐾𝑘) ∈ ℕ0)
127 eqid 2724 . . . . . . . . 9 (coe1‘(𝑡𝑊𝑗)) = (coe1‘(𝑡𝑊𝑗))
128127, 116, 43, 30coe1fvalcl 19705 . . . . . . . 8 (((𝑡𝑊𝑗) ∈ (Base‘𝑃) ∧ (𝐾𝑘) ∈ ℕ0) → ((coe1‘(𝑡𝑊𝑗))‘(𝐾𝑘)) ∈ (Base‘𝑅))
129125, 126, 128syl2anc 696 . . . . . . 7 ((((𝑅 ∈ Ring ∧ (𝑈𝐵𝑊𝐵) ∧ 𝐾 ∈ ℕ0) ∧ (𝑖𝑁𝑗𝑁)) ∧ (𝑡𝑁𝑘 ∈ (0...𝐾))) → ((coe1‘(𝑡𝑊𝑗))‘(𝐾𝑘)) ∈ (Base‘𝑅))
13030, 31ringcl 18682 . . . . . . 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 1439 . . . . . 6 ((((𝑅 ∈ Ring ∧ (𝑈𝐵𝑊𝐵) ∧ 𝐾 ∈ ℕ0) ∧ (𝑖𝑁𝑗𝑁)) ∧ (𝑡𝑁𝑘 ∈ (0...𝐾))) → (((coe1‘(𝑖𝑈𝑡))‘𝑘)(.r𝑅)((coe1‘(𝑡𝑊𝑗))‘(𝐾𝑘))) ∈ (Base‘𝑅))
13230, 66, 36, 88, 131gsumcom3fi 20329 . . . . 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))))))))
13342adantr 472 . . . . . . . . . . . 12 (((((𝑅 ∈ Ring ∧ (𝑈𝐵𝑊𝐵) ∧ 𝐾 ∈ ℕ0) ∧ (𝑖𝑁𝑗𝑁)) ∧ 𝑘 ∈ (0...𝐾)) ∧ 𝑡𝑁) → (𝑅 ∈ Ring ∧ 𝑈𝐵𝑘 ∈ ℕ0))
13410adantr 472 . . . . . . . . . . . . 13 ((((𝑅 ∈ Ring ∧ (𝑈𝐵𝑊𝐵) ∧ 𝐾 ∈ ℕ0) ∧ (𝑖𝑁𝑗𝑁)) ∧ 𝑘 ∈ (0...𝐾)) → 𝑖𝑁)
135134anim1i 593 . . . . . . . . . . . 12 (((((𝑅 ∈ Ring ∧ (𝑈𝐵𝑊𝐵) ∧ 𝐾 ∈ ℕ0) ∧ (𝑖𝑁𝑗𝑁)) ∧ 𝑘 ∈ (0...𝐾)) ∧ 𝑡𝑁) → (𝑖𝑁𝑡𝑁))
13643, 14, 15decpmate 20694 . . . . . . . . . . . 12 (((𝑅 ∈ Ring ∧ 𝑈𝐵𝑘 ∈ ℕ0) ∧ (𝑖𝑁𝑡𝑁)) → (𝑖(𝑈 decompPMat 𝑘)𝑡) = ((coe1‘(𝑖𝑈𝑡))‘𝑘))
137133, 135, 136syl2anc 696 . . . . . . . . . . 11 (((((𝑅 ∈ Ring ∧ (𝑈𝐵𝑊𝐵) ∧ 𝐾 ∈ ℕ0) ∧ (𝑖𝑁𝑗𝑁)) ∧ 𝑘 ∈ (0...𝐾)) ∧ 𝑡𝑁) → (𝑖(𝑈 decompPMat 𝑘)𝑡) = ((coe1‘(𝑖𝑈𝑡))‘𝑘))
13853adantr 472 . . . . . . . . . . . 12 (((((𝑅 ∈ Ring ∧ (𝑈𝐵𝑊𝐵) ∧ 𝐾 ∈ ℕ0) ∧ (𝑖𝑁𝑗𝑁)) ∧ 𝑘 ∈ (0...𝐾)) ∧ 𝑡𝑁) → (𝑅 ∈ Ring ∧ 𝑊𝐵 ∧ (𝐾𝑘) ∈ ℕ0))
139 simplrr 820 . . . . . . . . . . . . . 14 ((((𝑅 ∈ Ring ∧ (𝑈𝐵𝑊𝐵) ∧ 𝐾 ∈ ℕ0) ∧ (𝑖𝑁𝑗𝑁)) ∧ 𝑘 ∈ (0...𝐾)) → 𝑗𝑁)
140139anim1i 593 . . . . . . . . . . . . 13 (((((𝑅 ∈ Ring ∧ (𝑈𝐵𝑊𝐵) ∧ 𝐾 ∈ ℕ0) ∧ (𝑖𝑁𝑗𝑁)) ∧ 𝑘 ∈ (0...𝐾)) ∧ 𝑡𝑁) → (𝑗𝑁𝑡𝑁))
141140ancomd 466 . . . . . . . . . . . 12 (((((𝑅 ∈ Ring ∧ (𝑈𝐵𝑊𝐵) ∧ 𝐾 ∈ ℕ0) ∧ (𝑖𝑁𝑗𝑁)) ∧ 𝑘 ∈ (0...𝐾)) ∧ 𝑡𝑁) → (𝑡𝑁𝑗𝑁))
14243, 14, 15decpmate 20694 . . . . . . . . . . . 12 (((𝑅 ∈ Ring ∧ 𝑊𝐵 ∧ (𝐾𝑘) ∈ ℕ0) ∧ (𝑡𝑁𝑗𝑁)) → (𝑡(𝑊 decompPMat (𝐾𝑘))𝑗) = ((coe1‘(𝑡𝑊𝑗))‘(𝐾𝑘)))
143138, 141, 142syl2anc 696 . . . . . . . . . . 11 (((((𝑅 ∈ Ring ∧ (𝑈𝐵𝑊𝐵) ∧ 𝐾 ∈ ℕ0) ∧ (𝑖𝑁𝑗𝑁)) ∧ 𝑘 ∈ (0...𝐾)) ∧ 𝑡𝑁) → (𝑡(𝑊 decompPMat (𝐾𝑘))𝑗) = ((coe1‘(𝑡𝑊𝑗))‘(𝐾𝑘)))
144137, 143oveq12d 6783 . . . . . . . . . 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))))
145144eqcomd 2730 . . . . . . . . 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)))
146145mpteq2dva 4852 . . . . . . . 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))))
147146oveq2d 6781 . . . . . . 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)))))
148147mpteq2dva 4852 . . . . . 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))))))
149148oveq2d 6781 . . . . 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)))))))
150106, 132, 1493eqtrd 2762 . . . 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)))))))
15113, 102, 1503eqtr4rd 2769 . . 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))
152151ralrimivva 3073 . 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))
15343, 14pmatring 20621 . . . . . . 7 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝐶 ∈ Ring)
15420, 153syl 17 . . . . . 6 ((𝑅 ∈ Ring ∧ (𝑈𝐵𝑊𝐵)) → 𝐶 ∈ Ring)
155 simprl 811 . . . . . 6 ((𝑅 ∈ Ring ∧ (𝑈𝐵𝑊𝐵)) → 𝑈𝐵)
156 simprr 813 . . . . . 6 ((𝑅 ∈ Ring ∧ (𝑈𝐵𝑊𝐵)) → 𝑊𝐵)
157 eqid 2724 . . . . . . 7 (.r𝐶) = (.r𝐶)
15815, 157ringcl 18682 . . . . . 6 ((𝐶 ∈ Ring ∧ 𝑈𝐵𝑊𝐵) → (𝑈(.r𝐶)𝑊) ∈ 𝐵)
159154, 155, 156, 158syl3anc 1439 . . . . 5 ((𝑅 ∈ Ring ∧ (𝑈𝐵𝑊𝐵)) → (𝑈(.r𝐶)𝑊) ∈ 𝐵)
1601593adant3 1124 . . . 4 ((𝑅 ∈ Ring ∧ (𝑈𝐵𝑊𝐵) ∧ 𝐾 ∈ ℕ0) → (𝑈(.r𝐶)𝑊) ∈ 𝐵)
16143, 14, 15, 22, 44decpmatcl 20695 . . . 4 ((𝑅 ∈ Ring ∧ (𝑈(.r𝐶)𝑊) ∈ 𝐵𝐾 ∈ ℕ0) → ((𝑈(.r𝐶)𝑊) decompPMat 𝐾) ∈ (Base‘𝐴))
162160, 161syld3an2 1486 . . 3 ((𝑅 ∈ Ring ∧ (𝑈𝐵𝑊𝐵) ∧ 𝐾 ∈ ℕ0) → ((𝑈(.r𝐶)𝑊) decompPMat 𝐾) ∈ (Base‘𝐴))
16322matring 20372 . . . . . 6 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝐴 ∈ Ring)
16421, 163syl 17 . . . . 5 ((𝑅 ∈ Ring ∧ (𝑈𝐵𝑊𝐵) ∧ 𝐾 ∈ ℕ0) → 𝐴 ∈ Ring)
165 ringcmn 18702 . . . . 5 (𝐴 ∈ Ring → 𝐴 ∈ CMnd)
166164, 165syl 17 . . . 4 ((𝑅 ∈ Ring ∧ (𝑈𝐵𝑊𝐵) ∧ 𝐾 ∈ ℕ0) → 𝐴 ∈ CMnd)
167 fzfid 12887 . . . 4 ((𝑅 ∈ Ring ∧ (𝑈𝐵𝑊𝐵) ∧ 𝐾 ∈ ℕ0) → (0...𝐾) ∈ Fin)
168164adantr 472 . . . . . 6 (((𝑅 ∈ Ring ∧ (𝑈𝐵𝑊𝐵) ∧ 𝐾 ∈ ℕ0) ∧ 𝑘 ∈ (0...𝐾)) → 𝐴 ∈ Ring)
16932adantr 472 . . . . . . . 8 (((𝑅 ∈ Ring ∧ (𝑈𝐵𝑊𝐵) ∧ 𝐾 ∈ ℕ0) ∧ 𝑘 ∈ (0...𝐾)) → 𝑅 ∈ Ring)
170 simpl2l 1259 . . . . . . . 8 (((𝑅 ∈ Ring ∧ (𝑈𝐵𝑊𝐵) ∧ 𝐾 ∈ ℕ0) ∧ 𝑘 ∈ (0...𝐾)) → 𝑈𝐵)
17140adantl 473 . . . . . . . 8 (((𝑅 ∈ Ring ∧ (𝑈𝐵𝑊𝐵) ∧ 𝐾 ∈ ℕ0) ∧ 𝑘 ∈ (0...𝐾)) → 𝑘 ∈ ℕ0)
172169, 170, 1713jca 1379 . . . . . . 7 (((𝑅 ∈ Ring ∧ (𝑈𝐵𝑊𝐵) ∧ 𝐾 ∈ ℕ0) ∧ 𝑘 ∈ (0...𝐾)) → (𝑅 ∈ Ring ∧ 𝑈𝐵𝑘 ∈ ℕ0))
173172, 45syl 17 . . . . . 6 (((𝑅 ∈ Ring ∧ (𝑈𝐵𝑊𝐵) ∧ 𝐾 ∈ ℕ0) ∧ 𝑘 ∈ (0...𝐾)) → (𝑈 decompPMat 𝑘) ∈ (Base‘𝐴))
174 simpl2r 1261 . . . . . . . 8 (((𝑅 ∈ Ring ∧ (𝑈𝐵𝑊𝐵) ∧ 𝐾 ∈ ℕ0) ∧ 𝑘 ∈ (0...𝐾)) → 𝑊𝐵)
17551adantl 473 . . . . . . . 8 (((𝑅 ∈ Ring ∧ (𝑈𝐵𝑊𝐵) ∧ 𝐾 ∈ ℕ0) ∧ 𝑘 ∈ (0...𝐾)) → (𝐾𝑘) ∈ ℕ0)
176169, 174, 1753jca 1379 . . . . . . 7 (((𝑅 ∈ Ring ∧ (𝑈𝐵𝑊𝐵) ∧ 𝐾 ∈ ℕ0) ∧ 𝑘 ∈ (0...𝐾)) → (𝑅 ∈ Ring ∧ 𝑊𝐵 ∧ (𝐾𝑘) ∈ ℕ0))
177176, 54syl 17 . . . . . 6 (((𝑅 ∈ Ring ∧ (𝑈𝐵𝑊𝐵) ∧ 𝐾 ∈ ℕ0) ∧ 𝑘 ∈ (0...𝐾)) → (𝑊 decompPMat (𝐾𝑘)) ∈ (Base‘𝐴))
178 eqid 2724 . . . . . . 7 (.r𝐴) = (.r𝐴)
17944, 178ringcl 18682 . . . . . 6 ((𝐴 ∈ Ring ∧ (𝑈 decompPMat 𝑘) ∈ (Base‘𝐴) ∧ (𝑊 decompPMat (𝐾𝑘)) ∈ (Base‘𝐴)) → ((𝑈 decompPMat 𝑘)(.r𝐴)(𝑊 decompPMat (𝐾𝑘))) ∈ (Base‘𝐴))
180168, 173, 177, 179syl3anc 1439 . . . . 5 (((𝑅 ∈ Ring ∧ (𝑈𝐵𝑊𝐵) ∧ 𝐾 ∈ ℕ0) ∧ 𝑘 ∈ (0...𝐾)) → ((𝑈 decompPMat 𝑘)(.r𝐴)(𝑊 decompPMat (𝐾𝑘))) ∈ (Base‘𝐴))
181180ralrimiva 3068 . . . 4 ((𝑅 ∈ Ring ∧ (𝑈𝐵𝑊𝐵) ∧ 𝐾 ∈ ℕ0) → ∀𝑘 ∈ (0...𝐾)((𝑈 decompPMat 𝑘)(.r𝐴)(𝑊 decompPMat (𝐾𝑘))) ∈ (Base‘𝐴))
18244, 166, 167, 181gsummptcl 18487 . . 3 ((𝑅 ∈ Ring ∧ (𝑈𝐵𝑊𝐵) ∧ 𝐾 ∈ ℕ0) → (𝐴 Σg (𝑘 ∈ (0...𝐾) ↦ ((𝑈 decompPMat 𝑘)(.r𝐴)(𝑊 decompPMat (𝐾𝑘))))) ∈ (Base‘𝐴))
18322, 44eqmat 20353 . . 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)))
184162, 182, 183syl2anc 696 . 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)))
185152, 184mpbird 247 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 196  wa 383  w3a 1072   = wceq 1596  wcel 2103  wral 3014  Vcvv 3304  cotp 4293  cmpt 4837   × cxp 5216  cfv 6001  (class class class)co 6765  cmpt2 6767  𝑚 cmap 7974  Fincfn 8072  0cc0 10049  cmin 10379  0cn0 11405  ...cfz 12440  Basecbs 15980  .rcmulr 16065  0gc0g 16223   Σg cgsu 16224  CMndccmn 18314  Ringcrg 18668  Poly1cpl1 19670  coe1cco1 19671   maMul cmmul 20312   Mat cmat 20336   decompPMat cdecpmat 20690
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1835  ax-4 1850  ax-5 1952  ax-6 2018  ax-7 2054  ax-8 2105  ax-9 2112  ax-10 2132  ax-11 2147  ax-12 2160  ax-13 2355  ax-ext 2704  ax-rep 4879  ax-sep 4889  ax-nul 4897  ax-pow 4948  ax-pr 5011  ax-un 7066  ax-inf2 8651  ax-cnex 10105  ax-resscn 10106  ax-1cn 10107  ax-icn 10108  ax-addcl 10109  ax-addrcl 10110  ax-mulcl 10111  ax-mulrcl 10112  ax-mulcom 10113  ax-addass 10114  ax-mulass 10115  ax-distr 10116  ax-i2m1 10117  ax-1ne0 10118  ax-1rid 10119  ax-rnegex 10120  ax-rrecex 10121  ax-cnre 10122  ax-pre-lttri 10123  ax-pre-lttrn 10124  ax-pre-ltadd 10125  ax-pre-mulgt0 10126
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1073  df-3an 1074  df-tru 1599  df-fal 1602  df-ex 1818  df-nf 1823  df-sb 2011  df-eu 2575  df-mo 2576  df-clab 2711  df-cleq 2717  df-clel 2720  df-nfc 2855  df-ne 2897  df-nel 3000  df-ral 3019  df-rex 3020  df-reu 3021  df-rmo 3022  df-rab 3023  df-v 3306  df-sbc 3542  df-csb 3640  df-dif 3683  df-un 3685  df-in 3687  df-ss 3694  df-pss 3696  df-nul 4024  df-if 4195  df-pw 4268  df-sn 4286  df-pr 4288  df-tp 4290  df-op 4292  df-ot 4294  df-uni 4545  df-int 4584  df-iun 4630  df-iin 4631  df-br 4761  df-opab 4821  df-mpt 4838  df-tr 4861  df-id 5128  df-eprel 5133  df-po 5139  df-so 5140  df-fr 5177  df-se 5178  df-we 5179  df-xp 5224  df-rel 5225  df-cnv 5226  df-co 5227  df-dm 5228  df-rn 5229  df-res 5230  df-ima 5231  df-pred 5793  df-ord 5839  df-on 5840  df-lim 5841  df-suc 5842  df-iota 5964  df-fun 6003  df-fn 6004  df-f 6005  df-f1 6006  df-fo 6007  df-f1o 6008  df-fv 6009  df-isom 6010  df-riota 6726  df-ov 6768  df-oprab 6769  df-mpt2 6770  df-of 7014  df-ofr 7015  df-om 7183  df-1st 7285  df-2nd 7286  df-supp 7416  df-wrecs 7527  df-recs 7588  df-rdg 7626  df-1o 7680  df-2o 7681  df-oadd 7684  df-er 7862  df-map 7976  df-pm 7977  df-ixp 8026  df-en 8073  df-dom 8074  df-sdom 8075  df-fin 8076  df-fsupp 8392  df-sup 8464  df-oi 8531  df-card 8878  df-pnf 10189  df-mnf 10190  df-xr 10191  df-ltxr 10192  df-le 10193  df-sub 10381  df-neg 10382  df-nn 11134  df-2 11192  df-3 11193  df-4 11194  df-5 11195  df-6 11196  df-7 11197  df-8 11198  df-9 11199  df-n0 11406  df-z 11491  df-dec 11607  df-uz 11801  df-fz 12441  df-fzo 12581  df-seq 12917  df-hash 13233  df-struct 15982  df-ndx 15983  df-slot 15984  df-base 15986  df-sets 15987  df-ress 15988  df-plusg 16077  df-mulr 16078  df-sca 16080  df-vsca 16081  df-ip 16082  df-tset 16083  df-ple 16084  df-ds 16087  df-hom 16089  df-cco 16090  df-0g 16225  df-gsum 16226  df-prds 16231  df-pws 16233  df-mre 16369  df-mrc 16370  df-acs 16372  df-mgm 17364  df-sgrp 17406  df-mnd 17417  df-mhm 17457  df-submnd 17458  df-grp 17547  df-minusg 17548  df-sbg 17549  df-mulg 17663  df-subg 17713  df-ghm 17780  df-cntz 17871  df-cmn 18316  df-abl 18317  df-mgp 18611  df-ur 18623  df-ring 18670  df-subrg 18901  df-lmod 18988  df-lss 19056  df-sra 19295  df-rgmod 19296  df-psr 19479  df-mpl 19481  df-opsr 19483  df-psr1 19673  df-ply1 19675  df-coe1 19676  df-dsmm 20199  df-frlm 20214  df-mamu 20313  df-mat 20337  df-decpmat 20691
This theorem is referenced by:  decpmatmulsumfsupp  20701  pm2mpmhmlem1  20746  pm2mpmhmlem2  20747
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