| Step | Hyp | Ref | Expression | 
|---|
| 1 |  | eqidd 2737 | . . . . 5
⊢ (((𝑅 ∈ Ring ∧ (𝑈 ∈ 𝐵 ∧ 𝑊 ∈ 𝐵) ∧ 𝐾 ∈ ℕ0) ∧ (𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁)) → (𝑥 ∈ 𝑁, 𝑦 ∈ 𝑁 ↦ (𝑅 Σg (𝑘 ∈ (0...𝐾) ↦ (𝑅 Σg (𝑡 ∈ 𝑁 ↦ ((𝑥(𝑈 decompPMat 𝑘)𝑡)(.r‘𝑅)(𝑡(𝑊 decompPMat (𝐾 − 𝑘))𝑦))))))) = (𝑥 ∈ 𝑁, 𝑦 ∈ 𝑁 ↦ (𝑅 Σg (𝑘 ∈ (0...𝐾) ↦ (𝑅 Σg (𝑡 ∈ 𝑁 ↦ ((𝑥(𝑈 decompPMat 𝑘)𝑡)(.r‘𝑅)(𝑡(𝑊 decompPMat (𝐾 − 𝑘))𝑦)))))))) | 
| 2 |  | oveq1 7439 | . . . . . . . . . . 11
⊢ (𝑥 = 𝑖 → (𝑥(𝑈 decompPMat 𝑘)𝑡) = (𝑖(𝑈 decompPMat 𝑘)𝑡)) | 
| 3 |  | oveq2 7440 | . . . . . . . . . . 11
⊢ (𝑦 = 𝑗 → (𝑡(𝑊 decompPMat (𝐾 − 𝑘))𝑦) = (𝑡(𝑊 decompPMat (𝐾 − 𝑘))𝑗)) | 
| 4 | 2, 3 | oveqan12d 7451 | . . . . . . . . . 10
⊢ ((𝑥 = 𝑖 ∧ 𝑦 = 𝑗) → ((𝑥(𝑈 decompPMat 𝑘)𝑡)(.r‘𝑅)(𝑡(𝑊 decompPMat (𝐾 − 𝑘))𝑦)) = ((𝑖(𝑈 decompPMat 𝑘)𝑡)(.r‘𝑅)(𝑡(𝑊 decompPMat (𝐾 − 𝑘))𝑗))) | 
| 5 | 4 | mpteq2dv 5243 | . . . . . . . . 9
⊢ ((𝑥 = 𝑖 ∧ 𝑦 = 𝑗) → (𝑡 ∈ 𝑁 ↦ ((𝑥(𝑈 decompPMat 𝑘)𝑡)(.r‘𝑅)(𝑡(𝑊 decompPMat (𝐾 − 𝑘))𝑦))) = (𝑡 ∈ 𝑁 ↦ ((𝑖(𝑈 decompPMat 𝑘)𝑡)(.r‘𝑅)(𝑡(𝑊 decompPMat (𝐾 − 𝑘))𝑗)))) | 
| 6 | 5 | oveq2d 7448 | . . . . . . . 8
⊢ ((𝑥 = 𝑖 ∧ 𝑦 = 𝑗) → (𝑅 Σg (𝑡 ∈ 𝑁 ↦ ((𝑥(𝑈 decompPMat 𝑘)𝑡)(.r‘𝑅)(𝑡(𝑊 decompPMat (𝐾 − 𝑘))𝑦)))) = (𝑅 Σg (𝑡 ∈ 𝑁 ↦ ((𝑖(𝑈 decompPMat 𝑘)𝑡)(.r‘𝑅)(𝑡(𝑊 decompPMat (𝐾 − 𝑘))𝑗))))) | 
| 7 | 6 | mpteq2dv 5243 | . . . . . . 7
⊢ ((𝑥 = 𝑖 ∧ 𝑦 = 𝑗) → (𝑘 ∈ (0...𝐾) ↦ (𝑅 Σg (𝑡 ∈ 𝑁 ↦ ((𝑥(𝑈 decompPMat 𝑘)𝑡)(.r‘𝑅)(𝑡(𝑊 decompPMat (𝐾 − 𝑘))𝑦))))) = (𝑘 ∈ (0...𝐾) ↦ (𝑅 Σg (𝑡 ∈ 𝑁 ↦ ((𝑖(𝑈 decompPMat 𝑘)𝑡)(.r‘𝑅)(𝑡(𝑊 decompPMat (𝐾 − 𝑘))𝑗)))))) | 
| 8 | 7 | oveq2d 7448 | . . . . . 6
⊢ ((𝑥 = 𝑖 ∧ 𝑦 = 𝑗) → (𝑅 Σg (𝑘 ∈ (0...𝐾) ↦ (𝑅 Σg (𝑡 ∈ 𝑁 ↦ ((𝑥(𝑈 decompPMat 𝑘)𝑡)(.r‘𝑅)(𝑡(𝑊 decompPMat (𝐾 − 𝑘))𝑦)))))) = (𝑅 Σg (𝑘 ∈ (0...𝐾) ↦ (𝑅 Σg (𝑡 ∈ 𝑁 ↦ ((𝑖(𝑈 decompPMat 𝑘)𝑡)(.r‘𝑅)(𝑡(𝑊 decompPMat (𝐾 − 𝑘))𝑗))))))) | 
| 9 | 8 | adantl 481 | . . . . 5
⊢ ((((𝑅 ∈ Ring ∧ (𝑈 ∈ 𝐵 ∧ 𝑊 ∈ 𝐵) ∧ 𝐾 ∈ ℕ0) ∧ (𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁)) ∧ (𝑥 = 𝑖 ∧ 𝑦 = 𝑗)) → (𝑅 Σg (𝑘 ∈ (0...𝐾) ↦ (𝑅 Σg (𝑡 ∈ 𝑁 ↦ ((𝑥(𝑈 decompPMat 𝑘)𝑡)(.r‘𝑅)(𝑡(𝑊 decompPMat (𝐾 − 𝑘))𝑦)))))) = (𝑅 Σg (𝑘 ∈ (0...𝐾) ↦ (𝑅 Σg (𝑡 ∈ 𝑁 ↦ ((𝑖(𝑈 decompPMat 𝑘)𝑡)(.r‘𝑅)(𝑡(𝑊 decompPMat (𝐾 − 𝑘))𝑗))))))) | 
| 10 |  | simprl 770 | . . . . 5
⊢ (((𝑅 ∈ Ring ∧ (𝑈 ∈ 𝐵 ∧ 𝑊 ∈ 𝐵) ∧ 𝐾 ∈ ℕ0) ∧ (𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁)) → 𝑖 ∈ 𝑁) | 
| 11 |  | simprr 772 | . . . . 5
⊢ (((𝑅 ∈ Ring ∧ (𝑈 ∈ 𝐵 ∧ 𝑊 ∈ 𝐵) ∧ 𝐾 ∈ ℕ0) ∧ (𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁)) → 𝑗 ∈ 𝑁) | 
| 12 |  | ovexd 7467 | . . . . 5
⊢ (((𝑅 ∈ Ring ∧ (𝑈 ∈ 𝐵 ∧ 𝑊 ∈ 𝐵) ∧ 𝐾 ∈ ℕ0) ∧ (𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁)) → (𝑅 Σg (𝑘 ∈ (0...𝐾) ↦ (𝑅 Σg (𝑡 ∈ 𝑁 ↦ ((𝑖(𝑈 decompPMat 𝑘)𝑡)(.r‘𝑅)(𝑡(𝑊 decompPMat (𝐾 − 𝑘))𝑗)))))) ∈ V) | 
| 13 | 1, 9, 10, 11, 12 | ovmpod 7586 | . . . 4
⊢ (((𝑅 ∈ Ring ∧ (𝑈 ∈ 𝐵 ∧ 𝑊 ∈ 𝐵) ∧ 𝐾 ∈ ℕ0) ∧ (𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁)) → (𝑖(𝑥 ∈ 𝑁, 𝑦 ∈ 𝑁 ↦ (𝑅 Σg (𝑘 ∈ (0...𝐾) ↦ (𝑅 Σg (𝑡 ∈ 𝑁 ↦ ((𝑥(𝑈 decompPMat 𝑘)𝑡)(.r‘𝑅)(𝑡(𝑊 decompPMat (𝐾 − 𝑘))𝑦)))))))𝑗) = (𝑅 Σg (𝑘 ∈ (0...𝐾) ↦ (𝑅 Σg (𝑡 ∈ 𝑁 ↦ ((𝑖(𝑈 decompPMat 𝑘)𝑡)(.r‘𝑅)(𝑡(𝑊 decompPMat (𝐾 − 𝑘))𝑗))))))) | 
| 14 |  | decpmatmul.c | . . . . . . . . . . . . . . . . . . . 20
⊢ 𝐶 = (𝑁 Mat 𝑃) | 
| 15 |  | decpmatmul.b | . . . . . . . . . . . . . . . . . . . 20
⊢ 𝐵 = (Base‘𝐶) | 
| 16 | 14, 15 | matrcl 22417 | . . . . . . . . . . . . . . . . . . 19
⊢ (𝑈 ∈ 𝐵 → (𝑁 ∈ Fin ∧ 𝑃 ∈ V)) | 
| 17 | 16 | simpld 494 | . . . . . . . . . . . . . . . . . 18
⊢ (𝑈 ∈ 𝐵 → 𝑁 ∈ Fin) | 
| 18 | 17 | adantr 480 | . . . . . . . . . . . . . . . . 17
⊢ ((𝑈 ∈ 𝐵 ∧ 𝑊 ∈ 𝐵) → 𝑁 ∈ Fin) | 
| 19 | 18 | anim2i 617 | . . . . . . . . . . . . . . . 16
⊢ ((𝑅 ∈ Ring ∧ (𝑈 ∈ 𝐵 ∧ 𝑊 ∈ 𝐵)) → (𝑅 ∈ Ring ∧ 𝑁 ∈ Fin)) | 
| 20 | 19 | ancomd 461 | . . . . . . . . . . . . . . 15
⊢ ((𝑅 ∈ Ring ∧ (𝑈 ∈ 𝐵 ∧ 𝑊 ∈ 𝐵)) → (𝑁 ∈ Fin ∧ 𝑅 ∈ Ring)) | 
| 21 | 20 | 3adant3 1132 | . . . . . . . . . . . . . 14
⊢ ((𝑅 ∈ Ring ∧ (𝑈 ∈ 𝐵 ∧ 𝑊 ∈ 𝐵) ∧ 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〉) | 
| 24 | 22, 23 | matmulr 22445 | . . . . . . . . . . . . . 14
⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → (𝑅 maMul 〈𝑁, 𝑁, 𝑁〉) = (.r‘𝐴)) | 
| 25 | 21, 24 | syl 17 | . . . . . . . . . . . . 13
⊢ ((𝑅 ∈ Ring ∧ (𝑈 ∈ 𝐵 ∧ 𝑊 ∈ 𝐵) ∧ 𝐾 ∈ ℕ0) → (𝑅 maMul 〈𝑁, 𝑁, 𝑁〉) = (.r‘𝐴)) | 
| 26 | 25 | adantr 480 | . . . . . . . . . . . 12
⊢ (((𝑅 ∈ Ring ∧ (𝑈 ∈ 𝐵 ∧ 𝑊 ∈ 𝐵) ∧ 𝐾 ∈ ℕ0) ∧ (𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁)) → (𝑅 maMul 〈𝑁, 𝑁, 𝑁〉) = (.r‘𝐴)) | 
| 27 | 26 | adantr 480 | . . . . . . . . . . 11
⊢ ((((𝑅 ∈ Ring ∧ (𝑈 ∈ 𝐵 ∧ 𝑊 ∈ 𝐵) ∧ 𝐾 ∈ ℕ0) ∧ (𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁)) ∧ 𝑘 ∈ (0...𝐾)) → (𝑅 maMul 〈𝑁, 𝑁, 𝑁〉) = (.r‘𝐴)) | 
| 28 | 27 | eqcomd 2742 | . . . . . . . . . 10
⊢ ((((𝑅 ∈ Ring ∧ (𝑈 ∈ 𝐵 ∧ 𝑊 ∈ 𝐵) ∧ 𝐾 ∈ ℕ0) ∧ (𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁)) ∧ 𝑘 ∈ (0...𝐾)) → (.r‘𝐴) = (𝑅 maMul 〈𝑁, 𝑁, 𝑁〉)) | 
| 29 | 28 | oveqd 7449 | . . . . . . . . 9
⊢ ((((𝑅 ∈ Ring ∧ (𝑈 ∈ 𝐵 ∧ 𝑊 ∈ 𝐵) ∧ 𝐾 ∈ ℕ0) ∧ (𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁)) ∧ 𝑘 ∈ (0...𝐾)) → ((𝑈 decompPMat 𝑘)(.r‘𝐴)(𝑊 decompPMat (𝐾 − 𝑘))) = ((𝑈 decompPMat 𝑘)(𝑅 maMul 〈𝑁, 𝑁, 𝑁〉)(𝑊 decompPMat (𝐾 − 𝑘)))) | 
| 30 |  | eqid 2736 | . . . . . . . . . 10
⊢
(Base‘𝑅) =
(Base‘𝑅) | 
| 31 |  | eqid 2736 | . . . . . . . . . 10
⊢
(.r‘𝑅) = (.r‘𝑅) | 
| 32 |  | simp1 1136 | . . . . . . . . . . . 12
⊢ ((𝑅 ∈ Ring ∧ (𝑈 ∈ 𝐵 ∧ 𝑊 ∈ 𝐵) ∧ 𝐾 ∈ ℕ0) → 𝑅 ∈ Ring) | 
| 33 | 32 | adantr 480 | . . . . . . . . . . 11
⊢ (((𝑅 ∈ Ring ∧ (𝑈 ∈ 𝐵 ∧ 𝑊 ∈ 𝐵) ∧ 𝐾 ∈ ℕ0) ∧ (𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁)) → 𝑅 ∈ Ring) | 
| 34 | 33 | adantr 480 | . . . . . . . . . 10
⊢ ((((𝑅 ∈ Ring ∧ (𝑈 ∈ 𝐵 ∧ 𝑊 ∈ 𝐵) ∧ 𝐾 ∈ ℕ0) ∧ (𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁)) ∧ 𝑘 ∈ (0...𝐾)) → 𝑅 ∈ Ring) | 
| 35 | 21 | simpld 494 | . . . . . . . . . . . 12
⊢ ((𝑅 ∈ Ring ∧ (𝑈 ∈ 𝐵 ∧ 𝑊 ∈ 𝐵) ∧ 𝐾 ∈ ℕ0) → 𝑁 ∈ Fin) | 
| 36 | 35 | adantr 480 | . . . . . . . . . . 11
⊢ (((𝑅 ∈ Ring ∧ (𝑈 ∈ 𝐵 ∧ 𝑊 ∈ 𝐵) ∧ 𝐾 ∈ ℕ0) ∧ (𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁)) → 𝑁 ∈ Fin) | 
| 37 | 36 | adantr 480 | . . . . . . . . . 10
⊢ ((((𝑅 ∈ Ring ∧ (𝑈 ∈ 𝐵 ∧ 𝑊 ∈ 𝐵) ∧ 𝐾 ∈ ℕ0) ∧ (𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁)) ∧ 𝑘 ∈ (0...𝐾)) → 𝑁 ∈ Fin) | 
| 38 |  | simpl2l 1226 | . . . . . . . . . . . . . 14
⊢ (((𝑅 ∈ Ring ∧ (𝑈 ∈ 𝐵 ∧ 𝑊 ∈ 𝐵) ∧ 𝐾 ∈ ℕ0) ∧ (𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁)) → 𝑈 ∈ 𝐵) | 
| 39 | 38 | adantr 480 | . . . . . . . . . . . . 13
⊢ ((((𝑅 ∈ Ring ∧ (𝑈 ∈ 𝐵 ∧ 𝑊 ∈ 𝐵) ∧ 𝐾 ∈ ℕ0) ∧ (𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁)) ∧ 𝑘 ∈ (0...𝐾)) → 𝑈 ∈ 𝐵) | 
| 40 |  | elfznn0 13661 | . . . . . . . . . . . . . 14
⊢ (𝑘 ∈ (0...𝐾) → 𝑘 ∈ ℕ0) | 
| 41 | 40 | adantl 481 | . . . . . . . . . . . . 13
⊢ ((((𝑅 ∈ Ring ∧ (𝑈 ∈ 𝐵 ∧ 𝑊 ∈ 𝐵) ∧ 𝐾 ∈ ℕ0) ∧ (𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁)) ∧ 𝑘 ∈ (0...𝐾)) → 𝑘 ∈ ℕ0) | 
| 42 | 34, 39, 41 | 3jca 1128 | . . . . . . . . . . . 12
⊢ ((((𝑅 ∈ Ring ∧ (𝑈 ∈ 𝐵 ∧ 𝑊 ∈ 𝐵) ∧ 𝐾 ∈ ℕ0) ∧ (𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁)) ∧ 𝑘 ∈ (0...𝐾)) → (𝑅 ∈ Ring ∧ 𝑈 ∈ 𝐵 ∧ 𝑘 ∈
ℕ0)) | 
| 43 |  | decpmatmul.p | . . . . . . . . . . . . 13
⊢ 𝑃 = (Poly1‘𝑅) | 
| 44 |  | eqid 2736 | . . . . . . . . . . . . 13
⊢
(Base‘𝐴) =
(Base‘𝐴) | 
| 45 | 43, 14, 15, 22, 44 | decpmatcl 22774 | . . . . . . . . . . . 12
⊢ ((𝑅 ∈ Ring ∧ 𝑈 ∈ 𝐵 ∧ 𝑘 ∈ ℕ0) → (𝑈 decompPMat 𝑘) ∈ (Base‘𝐴)) | 
| 46 | 42, 45 | syl 17 | . . . . . . . . . . 11
⊢ ((((𝑅 ∈ Ring ∧ (𝑈 ∈ 𝐵 ∧ 𝑊 ∈ 𝐵) ∧ 𝐾 ∈ ℕ0) ∧ (𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁)) ∧ 𝑘 ∈ (0...𝐾)) → (𝑈 decompPMat 𝑘) ∈ (Base‘𝐴)) | 
| 47 | 22, 30, 44 | matbas2i 22429 | . . . . . . . . . . 11
⊢ ((𝑈 decompPMat 𝑘) ∈ (Base‘𝐴) → (𝑈 decompPMat 𝑘) ∈ ((Base‘𝑅) ↑m (𝑁 × 𝑁))) | 
| 48 | 46, 47 | syl 17 | . . . . . . . . . 10
⊢ ((((𝑅 ∈ Ring ∧ (𝑈 ∈ 𝐵 ∧ 𝑊 ∈ 𝐵) ∧ 𝐾 ∈ ℕ0) ∧ (𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁)) ∧ 𝑘 ∈ (0...𝐾)) → (𝑈 decompPMat 𝑘) ∈ ((Base‘𝑅) ↑m (𝑁 × 𝑁))) | 
| 49 |  | simpl2r 1227 | . . . . . . . . . . . . . 14
⊢ (((𝑅 ∈ Ring ∧ (𝑈 ∈ 𝐵 ∧ 𝑊 ∈ 𝐵) ∧ 𝐾 ∈ ℕ0) ∧ (𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁)) → 𝑊 ∈ 𝐵) | 
| 50 | 49 | adantr 480 | . . . . . . . . . . . . 13
⊢ ((((𝑅 ∈ Ring ∧ (𝑈 ∈ 𝐵 ∧ 𝑊 ∈ 𝐵) ∧ 𝐾 ∈ ℕ0) ∧ (𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁)) ∧ 𝑘 ∈ (0...𝐾)) → 𝑊 ∈ 𝐵) | 
| 51 |  | fznn0sub 13597 | . . . . . . . . . . . . . 14
⊢ (𝑘 ∈ (0...𝐾) → (𝐾 − 𝑘) ∈
ℕ0) | 
| 52 | 51 | adantl 481 | . . . . . . . . . . . . 13
⊢ ((((𝑅 ∈ Ring ∧ (𝑈 ∈ 𝐵 ∧ 𝑊 ∈ 𝐵) ∧ 𝐾 ∈ ℕ0) ∧ (𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁)) ∧ 𝑘 ∈ (0...𝐾)) → (𝐾 − 𝑘) ∈
ℕ0) | 
| 53 | 34, 50, 52 | 3jca 1128 | . . . . . . . . . . . 12
⊢ ((((𝑅 ∈ Ring ∧ (𝑈 ∈ 𝐵 ∧ 𝑊 ∈ 𝐵) ∧ 𝐾 ∈ ℕ0) ∧ (𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁)) ∧ 𝑘 ∈ (0...𝐾)) → (𝑅 ∈ Ring ∧ 𝑊 ∈ 𝐵 ∧ (𝐾 − 𝑘) ∈
ℕ0)) | 
| 54 | 43, 14, 15, 22, 44 | decpmatcl 22774 | . . . . . . . . . . . 12
⊢ ((𝑅 ∈ Ring ∧ 𝑊 ∈ 𝐵 ∧ (𝐾 − 𝑘) ∈ ℕ0) → (𝑊 decompPMat (𝐾 − 𝑘)) ∈ (Base‘𝐴)) | 
| 55 | 53, 54 | syl 17 | . . . . . . . . . . 11
⊢ ((((𝑅 ∈ Ring ∧ (𝑈 ∈ 𝐵 ∧ 𝑊 ∈ 𝐵) ∧ 𝐾 ∈ ℕ0) ∧ (𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁)) ∧ 𝑘 ∈ (0...𝐾)) → (𝑊 decompPMat (𝐾 − 𝑘)) ∈ (Base‘𝐴)) | 
| 56 | 22, 30, 44 | matbas2i 22429 | . . . . . . . . . . 11
⊢ ((𝑊 decompPMat (𝐾 − 𝑘)) ∈ (Base‘𝐴) → (𝑊 decompPMat (𝐾 − 𝑘)) ∈ ((Base‘𝑅) ↑m (𝑁 × 𝑁))) | 
| 57 | 55, 56 | syl 17 | . . . . . . . . . 10
⊢ ((((𝑅 ∈ Ring ∧ (𝑈 ∈ 𝐵 ∧ 𝑊 ∈ 𝐵) ∧ 𝐾 ∈ ℕ0) ∧ (𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁)) ∧ 𝑘 ∈ (0...𝐾)) → (𝑊 decompPMat (𝐾 − 𝑘)) ∈ ((Base‘𝑅) ↑m (𝑁 × 𝑁))) | 
| 58 | 23, 30, 31, 34, 37, 37, 37, 48, 57 | mamuval 22398 | . . . . . . . . 9
⊢ ((((𝑅 ∈ Ring ∧ (𝑈 ∈ 𝐵 ∧ 𝑊 ∈ 𝐵) ∧ 𝐾 ∈ ℕ0) ∧ (𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁)) ∧ 𝑘 ∈ (0...𝐾)) → ((𝑈 decompPMat 𝑘)(𝑅 maMul 〈𝑁, 𝑁, 𝑁〉)(𝑊 decompPMat (𝐾 − 𝑘))) = (𝑥 ∈ 𝑁, 𝑦 ∈ 𝑁 ↦ (𝑅 Σg (𝑡 ∈ 𝑁 ↦ ((𝑥(𝑈 decompPMat 𝑘)𝑡)(.r‘𝑅)(𝑡(𝑊 decompPMat (𝐾 − 𝑘))𝑦)))))) | 
| 59 | 29, 58 | eqtrd 2776 | . . . . . . . 8
⊢ ((((𝑅 ∈ Ring ∧ (𝑈 ∈ 𝐵 ∧ 𝑊 ∈ 𝐵) ∧ 𝐾 ∈ ℕ0) ∧ (𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁)) ∧ 𝑘 ∈ (0...𝐾)) → ((𝑈 decompPMat 𝑘)(.r‘𝐴)(𝑊 decompPMat (𝐾 − 𝑘))) = (𝑥 ∈ 𝑁, 𝑦 ∈ 𝑁 ↦ (𝑅 Σg (𝑡 ∈ 𝑁 ↦ ((𝑥(𝑈 decompPMat 𝑘)𝑡)(.r‘𝑅)(𝑡(𝑊 decompPMat (𝐾 − 𝑘))𝑦)))))) | 
| 60 | 59 | mpteq2dva 5241 | . . . . . . 7
⊢ (((𝑅 ∈ Ring ∧ (𝑈 ∈ 𝐵 ∧ 𝑊 ∈ 𝐵) ∧ 𝐾 ∈ ℕ0) ∧ (𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁)) → (𝑘 ∈ (0...𝐾) ↦ ((𝑈 decompPMat 𝑘)(.r‘𝐴)(𝑊 decompPMat (𝐾 − 𝑘)))) = (𝑘 ∈ (0...𝐾) ↦ (𝑥 ∈ 𝑁, 𝑦 ∈ 𝑁 ↦ (𝑅 Σg (𝑡 ∈ 𝑁 ↦ ((𝑥(𝑈 decompPMat 𝑘)𝑡)(.r‘𝑅)(𝑡(𝑊 decompPMat (𝐾 − 𝑘))𝑦))))))) | 
| 61 | 60 | oveq2d 7448 | . . . . . 6
⊢ (((𝑅 ∈ Ring ∧ (𝑈 ∈ 𝐵 ∧ 𝑊 ∈ 𝐵) ∧ 𝐾 ∈ ℕ0) ∧ (𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁)) → (𝐴 Σg (𝑘 ∈ (0...𝐾) ↦ ((𝑈 decompPMat 𝑘)(.r‘𝐴)(𝑊 decompPMat (𝐾 − 𝑘))))) = (𝐴 Σg (𝑘 ∈ (0...𝐾) ↦ (𝑥 ∈ 𝑁, 𝑦 ∈ 𝑁 ↦ (𝑅 Σg (𝑡 ∈ 𝑁 ↦ ((𝑥(𝑈 decompPMat 𝑘)𝑡)(.r‘𝑅)(𝑡(𝑊 decompPMat (𝐾 − 𝑘))𝑦)))))))) | 
| 62 |  | eqid 2736 | . . . . . . 7
⊢
(0g‘𝐴) = (0g‘𝐴) | 
| 63 |  | ovexd 7467 | . . . . . . 7
⊢ (((𝑅 ∈ Ring ∧ (𝑈 ∈ 𝐵 ∧ 𝑊 ∈ 𝐵) ∧ 𝐾 ∈ ℕ0) ∧ (𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁)) → (0...𝐾) ∈ V) | 
| 64 |  | ringcmn 20280 | . . . . . . . . . . . . 13
⊢ (𝑅 ∈ Ring → 𝑅 ∈ CMnd) | 
| 65 | 32, 64 | syl 17 | . . . . . . . . . . . 12
⊢ ((𝑅 ∈ Ring ∧ (𝑈 ∈ 𝐵 ∧ 𝑊 ∈ 𝐵) ∧ 𝐾 ∈ ℕ0) → 𝑅 ∈ CMnd) | 
| 66 | 65 | adantr 480 | . . . . . . . . . . 11
⊢ (((𝑅 ∈ Ring ∧ (𝑈 ∈ 𝐵 ∧ 𝑊 ∈ 𝐵) ∧ 𝐾 ∈ ℕ0) ∧ (𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁)) → 𝑅 ∈ CMnd) | 
| 67 | 66 | adantr 480 | . . . . . . . . . 10
⊢ ((((𝑅 ∈ Ring ∧ (𝑈 ∈ 𝐵 ∧ 𝑊 ∈ 𝐵) ∧ 𝐾 ∈ ℕ0) ∧ (𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁)) ∧ 𝑘 ∈ (0...𝐾)) → 𝑅 ∈ CMnd) | 
| 68 | 67 | 3ad2ant1 1133 | . . . . . . . . 9
⊢
(((((𝑅 ∈ Ring
∧ (𝑈 ∈ 𝐵 ∧ 𝑊 ∈ 𝐵) ∧ 𝐾 ∈ ℕ0) ∧ (𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁)) ∧ 𝑘 ∈ (0...𝐾)) ∧ 𝑥 ∈ 𝑁 ∧ 𝑦 ∈ 𝑁) → 𝑅 ∈ CMnd) | 
| 69 | 37 | 3ad2ant1 1133 | . . . . . . . . 9
⊢
(((((𝑅 ∈ Ring
∧ (𝑈 ∈ 𝐵 ∧ 𝑊 ∈ 𝐵) ∧ 𝐾 ∈ ℕ0) ∧ (𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁)) ∧ 𝑘 ∈ (0...𝐾)) ∧ 𝑥 ∈ 𝑁 ∧ 𝑦 ∈ 𝑁) → 𝑁 ∈ Fin) | 
| 70 | 34 | 3ad2ant1 1133 | . . . . . . . . . . . 12
⊢
(((((𝑅 ∈ Ring
∧ (𝑈 ∈ 𝐵 ∧ 𝑊 ∈ 𝐵) ∧ 𝐾 ∈ ℕ0) ∧ (𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁)) ∧ 𝑘 ∈ (0...𝐾)) ∧ 𝑥 ∈ 𝑁 ∧ 𝑦 ∈ 𝑁) → 𝑅 ∈ Ring) | 
| 71 | 70 | adantr 480 | . . . . . . . . . . 11
⊢
((((((𝑅 ∈ Ring
∧ (𝑈 ∈ 𝐵 ∧ 𝑊 ∈ 𝐵) ∧ 𝐾 ∈ ℕ0) ∧ (𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁)) ∧ 𝑘 ∈ (0...𝐾)) ∧ 𝑥 ∈ 𝑁 ∧ 𝑦 ∈ 𝑁) ∧ 𝑡 ∈ 𝑁) → 𝑅 ∈ Ring) | 
| 72 |  | simpl2 1192 | . . . . . . . . . . . 12
⊢
((((((𝑅 ∈ Ring
∧ (𝑈 ∈ 𝐵 ∧ 𝑊 ∈ 𝐵) ∧ 𝐾 ∈ ℕ0) ∧ (𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁)) ∧ 𝑘 ∈ (0...𝐾)) ∧ 𝑥 ∈ 𝑁 ∧ 𝑦 ∈ 𝑁) ∧ 𝑡 ∈ 𝑁) → 𝑥 ∈ 𝑁) | 
| 73 |  | simpr 484 | . . . . . . . . . . . 12
⊢
((((((𝑅 ∈ Ring
∧ (𝑈 ∈ 𝐵 ∧ 𝑊 ∈ 𝐵) ∧ 𝐾 ∈ ℕ0) ∧ (𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁)) ∧ 𝑘 ∈ (0...𝐾)) ∧ 𝑥 ∈ 𝑁 ∧ 𝑦 ∈ 𝑁) ∧ 𝑡 ∈ 𝑁) → 𝑡 ∈ 𝑁) | 
| 74 | 42 | 3ad2ant1 1133 | . . . . . . . . . . . . . 14
⊢
(((((𝑅 ∈ Ring
∧ (𝑈 ∈ 𝐵 ∧ 𝑊 ∈ 𝐵) ∧ 𝐾 ∈ ℕ0) ∧ (𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁)) ∧ 𝑘 ∈ (0...𝐾)) ∧ 𝑥 ∈ 𝑁 ∧ 𝑦 ∈ 𝑁) → (𝑅 ∈ Ring ∧ 𝑈 ∈ 𝐵 ∧ 𝑘 ∈
ℕ0)) | 
| 75 | 74 | adantr 480 | . . . . . . . . . . . . 13
⊢
((((((𝑅 ∈ Ring
∧ (𝑈 ∈ 𝐵 ∧ 𝑊 ∈ 𝐵) ∧ 𝐾 ∈ ℕ0) ∧ (𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁)) ∧ 𝑘 ∈ (0...𝐾)) ∧ 𝑥 ∈ 𝑁 ∧ 𝑦 ∈ 𝑁) ∧ 𝑡 ∈ 𝑁) → (𝑅 ∈ Ring ∧ 𝑈 ∈ 𝐵 ∧ 𝑘 ∈
ℕ0)) | 
| 76 | 75, 45 | syl 17 | . . . . . . . . . . . 12
⊢
((((((𝑅 ∈ Ring
∧ (𝑈 ∈ 𝐵 ∧ 𝑊 ∈ 𝐵) ∧ 𝐾 ∈ ℕ0) ∧ (𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁)) ∧ 𝑘 ∈ (0...𝐾)) ∧ 𝑥 ∈ 𝑁 ∧ 𝑦 ∈ 𝑁) ∧ 𝑡 ∈ 𝑁) → (𝑈 decompPMat 𝑘) ∈ (Base‘𝐴)) | 
| 77 | 22, 30, 44, 72, 73, 76 | matecld 22433 | . . . . . . . . . . 11
⊢
((((((𝑅 ∈ Ring
∧ (𝑈 ∈ 𝐵 ∧ 𝑊 ∈ 𝐵) ∧ 𝐾 ∈ ℕ0) ∧ (𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁)) ∧ 𝑘 ∈ (0...𝐾)) ∧ 𝑥 ∈ 𝑁 ∧ 𝑦 ∈ 𝑁) ∧ 𝑡 ∈ 𝑁) → (𝑥(𝑈 decompPMat 𝑘)𝑡) ∈ (Base‘𝑅)) | 
| 78 |  | simpl3 1193 | . . . . . . . . . . . 12
⊢
((((((𝑅 ∈ Ring
∧ (𝑈 ∈ 𝐵 ∧ 𝑊 ∈ 𝐵) ∧ 𝐾 ∈ ℕ0) ∧ (𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁)) ∧ 𝑘 ∈ (0...𝐾)) ∧ 𝑥 ∈ 𝑁 ∧ 𝑦 ∈ 𝑁) ∧ 𝑡 ∈ 𝑁) → 𝑦 ∈ 𝑁) | 
| 79 | 55 | 3ad2ant1 1133 | . . . . . . . . . . . . 13
⊢
(((((𝑅 ∈ Ring
∧ (𝑈 ∈ 𝐵 ∧ 𝑊 ∈ 𝐵) ∧ 𝐾 ∈ ℕ0) ∧ (𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁)) ∧ 𝑘 ∈ (0...𝐾)) ∧ 𝑥 ∈ 𝑁 ∧ 𝑦 ∈ 𝑁) → (𝑊 decompPMat (𝐾 − 𝑘)) ∈ (Base‘𝐴)) | 
| 80 | 79 | adantr 480 | . . . . . . . . . . . 12
⊢
((((((𝑅 ∈ Ring
∧ (𝑈 ∈ 𝐵 ∧ 𝑊 ∈ 𝐵) ∧ 𝐾 ∈ ℕ0) ∧ (𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁)) ∧ 𝑘 ∈ (0...𝐾)) ∧ 𝑥 ∈ 𝑁 ∧ 𝑦 ∈ 𝑁) ∧ 𝑡 ∈ 𝑁) → (𝑊 decompPMat (𝐾 − 𝑘)) ∈ (Base‘𝐴)) | 
| 81 | 22, 30, 44, 73, 78, 80 | matecld 22433 | . . . . . . . . . . 11
⊢
((((((𝑅 ∈ Ring
∧ (𝑈 ∈ 𝐵 ∧ 𝑊 ∈ 𝐵) ∧ 𝐾 ∈ ℕ0) ∧ (𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁)) ∧ 𝑘 ∈ (0...𝐾)) ∧ 𝑥 ∈ 𝑁 ∧ 𝑦 ∈ 𝑁) ∧ 𝑡 ∈ 𝑁) → (𝑡(𝑊 decompPMat (𝐾 − 𝑘))𝑦) ∈ (Base‘𝑅)) | 
| 82 | 30, 31 | ringcl 20248 | . . . . . . . . . . 11
⊢ ((𝑅 ∈ Ring ∧ (𝑥(𝑈 decompPMat 𝑘)𝑡) ∈ (Base‘𝑅) ∧ (𝑡(𝑊 decompPMat (𝐾 − 𝑘))𝑦) ∈ (Base‘𝑅)) → ((𝑥(𝑈 decompPMat 𝑘)𝑡)(.r‘𝑅)(𝑡(𝑊 decompPMat (𝐾 − 𝑘))𝑦)) ∈ (Base‘𝑅)) | 
| 83 | 71, 77, 81, 82 | syl3anc 1372 | . . . . . . . . . 10
⊢
((((((𝑅 ∈ Ring
∧ (𝑈 ∈ 𝐵 ∧ 𝑊 ∈ 𝐵) ∧ 𝐾 ∈ ℕ0) ∧ (𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁)) ∧ 𝑘 ∈ (0...𝐾)) ∧ 𝑥 ∈ 𝑁 ∧ 𝑦 ∈ 𝑁) ∧ 𝑡 ∈ 𝑁) → ((𝑥(𝑈 decompPMat 𝑘)𝑡)(.r‘𝑅)(𝑡(𝑊 decompPMat (𝐾 − 𝑘))𝑦)) ∈ (Base‘𝑅)) | 
| 84 | 83 | ralrimiva 3145 | . . . . . . . . 9
⊢
(((((𝑅 ∈ Ring
∧ (𝑈 ∈ 𝐵 ∧ 𝑊 ∈ 𝐵) ∧ 𝐾 ∈ ℕ0) ∧ (𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁)) ∧ 𝑘 ∈ (0...𝐾)) ∧ 𝑥 ∈ 𝑁 ∧ 𝑦 ∈ 𝑁) → ∀𝑡 ∈ 𝑁 ((𝑥(𝑈 decompPMat 𝑘)𝑡)(.r‘𝑅)(𝑡(𝑊 decompPMat (𝐾 − 𝑘))𝑦)) ∈ (Base‘𝑅)) | 
| 85 | 30, 68, 69, 84 | gsummptcl 19986 | . . . . . . . 8
⊢
(((((𝑅 ∈ Ring
∧ (𝑈 ∈ 𝐵 ∧ 𝑊 ∈ 𝐵) ∧ 𝐾 ∈ ℕ0) ∧ (𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁)) ∧ 𝑘 ∈ (0...𝐾)) ∧ 𝑥 ∈ 𝑁 ∧ 𝑦 ∈ 𝑁) → (𝑅 Σg (𝑡 ∈ 𝑁 ↦ ((𝑥(𝑈 decompPMat 𝑘)𝑡)(.r‘𝑅)(𝑡(𝑊 decompPMat (𝐾 − 𝑘))𝑦)))) ∈ (Base‘𝑅)) | 
| 86 | 22, 30, 44, 37, 34, 85 | matbas2d 22430 | . . . . . . 7
⊢ ((((𝑅 ∈ Ring ∧ (𝑈 ∈ 𝐵 ∧ 𝑊 ∈ 𝐵) ∧ 𝐾 ∈ ℕ0) ∧ (𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁)) ∧ 𝑘 ∈ (0...𝐾)) → (𝑥 ∈ 𝑁, 𝑦 ∈ 𝑁 ↦ (𝑅 Σg (𝑡 ∈ 𝑁 ↦ ((𝑥(𝑈 decompPMat 𝑘)𝑡)(.r‘𝑅)(𝑡(𝑊 decompPMat (𝐾 − 𝑘))𝑦))))) ∈ (Base‘𝐴)) | 
| 87 |  | eqid 2736 | . . . . . . . 8
⊢ (𝑘 ∈ (0...𝐾) ↦ (𝑥 ∈ 𝑁, 𝑦 ∈ 𝑁 ↦ (𝑅 Σg (𝑡 ∈ 𝑁 ↦ ((𝑥(𝑈 decompPMat 𝑘)𝑡)(.r‘𝑅)(𝑡(𝑊 decompPMat (𝐾 − 𝑘))𝑦)))))) = (𝑘 ∈ (0...𝐾) ↦ (𝑥 ∈ 𝑁, 𝑦 ∈ 𝑁 ↦ (𝑅 Σg (𝑡 ∈ 𝑁 ↦ ((𝑥(𝑈 decompPMat 𝑘)𝑡)(.r‘𝑅)(𝑡(𝑊 decompPMat (𝐾 − 𝑘))𝑦)))))) | 
| 88 |  | fzfid 14015 | . . . . . . . 8
⊢ (((𝑅 ∈ Ring ∧ (𝑈 ∈ 𝐵 ∧ 𝑊 ∈ 𝐵) ∧ 𝐾 ∈ ℕ0) ∧ (𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁)) → (0...𝐾) ∈ Fin) | 
| 89 |  | simpl 482 | . . . . . . . . . . . . . . 15
⊢ ((𝑁 ∈ Fin ∧ 𝑃 ∈ V) → 𝑁 ∈ Fin) | 
| 90 | 89, 89 | jca 511 | . . . . . . . . . . . . . 14
⊢ ((𝑁 ∈ Fin ∧ 𝑃 ∈ V) → (𝑁 ∈ Fin ∧ 𝑁 ∈ Fin)) | 
| 91 | 16, 90 | syl 17 | . . . . . . . . . . . . 13
⊢ (𝑈 ∈ 𝐵 → (𝑁 ∈ Fin ∧ 𝑁 ∈ Fin)) | 
| 92 | 91 | adantr 480 | . . . . . . . . . . . 12
⊢ ((𝑈 ∈ 𝐵 ∧ 𝑊 ∈ 𝐵) → (𝑁 ∈ Fin ∧ 𝑁 ∈ Fin)) | 
| 93 | 92 | 3ad2ant2 1134 | . . . . . . . . . . 11
⊢ ((𝑅 ∈ Ring ∧ (𝑈 ∈ 𝐵 ∧ 𝑊 ∈ 𝐵) ∧ 𝐾 ∈ ℕ0) → (𝑁 ∈ Fin ∧ 𝑁 ∈ Fin)) | 
| 94 | 93 | adantr 480 | . . . . . . . . . 10
⊢ (((𝑅 ∈ Ring ∧ (𝑈 ∈ 𝐵 ∧ 𝑊 ∈ 𝐵) ∧ 𝐾 ∈ ℕ0) ∧ (𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁)) → (𝑁 ∈ Fin ∧ 𝑁 ∈ Fin)) | 
| 95 | 94 | adantr 480 | . . . . . . . . 9
⊢ ((((𝑅 ∈ Ring ∧ (𝑈 ∈ 𝐵 ∧ 𝑊 ∈ 𝐵) ∧ 𝐾 ∈ ℕ0) ∧ (𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁)) ∧ 𝑘 ∈ (0...𝐾)) → (𝑁 ∈ Fin ∧ 𝑁 ∈ Fin)) | 
| 96 |  | mpoexga 8103 | . . . . . . . . 9
⊢ ((𝑁 ∈ Fin ∧ 𝑁 ∈ Fin) → (𝑥 ∈ 𝑁, 𝑦 ∈ 𝑁 ↦ (𝑅 Σg (𝑡 ∈ 𝑁 ↦ ((𝑥(𝑈 decompPMat 𝑘)𝑡)(.r‘𝑅)(𝑡(𝑊 decompPMat (𝐾 − 𝑘))𝑦))))) ∈ V) | 
| 97 | 95, 96 | syl 17 | . . . . . . . 8
⊢ ((((𝑅 ∈ Ring ∧ (𝑈 ∈ 𝐵 ∧ 𝑊 ∈ 𝐵) ∧ 𝐾 ∈ ℕ0) ∧ (𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁)) ∧ 𝑘 ∈ (0...𝐾)) → (𝑥 ∈ 𝑁, 𝑦 ∈ 𝑁 ↦ (𝑅 Σg (𝑡 ∈ 𝑁 ↦ ((𝑥(𝑈 decompPMat 𝑘)𝑡)(.r‘𝑅)(𝑡(𝑊 decompPMat (𝐾 − 𝑘))𝑦))))) ∈ V) | 
| 98 |  | fvexd 6920 | . . . . . . . 8
⊢ (((𝑅 ∈ Ring ∧ (𝑈 ∈ 𝐵 ∧ 𝑊 ∈ 𝐵) ∧ 𝐾 ∈ ℕ0) ∧ (𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁)) → (0g‘𝐴) ∈ V) | 
| 99 | 87, 88, 97, 98 | fsuppmptdm 9417 | . . . . . . 7
⊢ (((𝑅 ∈ Ring ∧ (𝑈 ∈ 𝐵 ∧ 𝑊 ∈ 𝐵) ∧ 𝐾 ∈ ℕ0) ∧ (𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁)) → (𝑘 ∈ (0...𝐾) ↦ (𝑥 ∈ 𝑁, 𝑦 ∈ 𝑁 ↦ (𝑅 Σg (𝑡 ∈ 𝑁 ↦ ((𝑥(𝑈 decompPMat 𝑘)𝑡)(.r‘𝑅)(𝑡(𝑊 decompPMat (𝐾 − 𝑘))𝑦)))))) finSupp (0g‘𝐴)) | 
| 100 | 22, 44, 62, 36, 63, 33, 86, 99 | matgsum 22444 | . . . . . 6
⊢ (((𝑅 ∈ Ring ∧ (𝑈 ∈ 𝐵 ∧ 𝑊 ∈ 𝐵) ∧ 𝐾 ∈ ℕ0) ∧ (𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁)) → (𝐴 Σg (𝑘 ∈ (0...𝐾) ↦ (𝑥 ∈ 𝑁, 𝑦 ∈ 𝑁 ↦ (𝑅 Σg (𝑡 ∈ 𝑁 ↦ ((𝑥(𝑈 decompPMat 𝑘)𝑡)(.r‘𝑅)(𝑡(𝑊 decompPMat (𝐾 − 𝑘))𝑦))))))) = (𝑥 ∈ 𝑁, 𝑦 ∈ 𝑁 ↦ (𝑅 Σg (𝑘 ∈ (0...𝐾) ↦ (𝑅 Σg (𝑡 ∈ 𝑁 ↦ ((𝑥(𝑈 decompPMat 𝑘)𝑡)(.r‘𝑅)(𝑡(𝑊 decompPMat (𝐾 − 𝑘))𝑦)))))))) | 
| 101 | 61, 100 | eqtrd 2776 | . . . . 5
⊢ (((𝑅 ∈ Ring ∧ (𝑈 ∈ 𝐵 ∧ 𝑊 ∈ 𝐵) ∧ 𝐾 ∈ ℕ0) ∧ (𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁)) → (𝐴 Σg (𝑘 ∈ (0...𝐾) ↦ ((𝑈 decompPMat 𝑘)(.r‘𝐴)(𝑊 decompPMat (𝐾 − 𝑘))))) = (𝑥 ∈ 𝑁, 𝑦 ∈ 𝑁 ↦ (𝑅 Σg (𝑘 ∈ (0...𝐾) ↦ (𝑅 Σg (𝑡 ∈ 𝑁 ↦ ((𝑥(𝑈 decompPMat 𝑘)𝑡)(.r‘𝑅)(𝑡(𝑊 decompPMat (𝐾 − 𝑘))𝑦)))))))) | 
| 102 | 101 | oveqd 7449 | . . . 4
⊢ (((𝑅 ∈ Ring ∧ (𝑈 ∈ 𝐵 ∧ 𝑊 ∈ 𝐵) ∧ 𝐾 ∈ ℕ0) ∧ (𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁)) → (𝑖(𝐴 Σg (𝑘 ∈ (0...𝐾) ↦ ((𝑈 decompPMat 𝑘)(.r‘𝐴)(𝑊 decompPMat (𝐾 − 𝑘)))))𝑗) = (𝑖(𝑥 ∈ 𝑁, 𝑦 ∈ 𝑁 ↦ (𝑅 Σg (𝑘 ∈ (0...𝐾) ↦ (𝑅 Σg (𝑡 ∈ 𝑁 ↦ ((𝑥(𝑈 decompPMat 𝑘)𝑡)(.r‘𝑅)(𝑡(𝑊 decompPMat (𝐾 − 𝑘))𝑦)))))))𝑗)) | 
| 103 |  | simpl2 1192 | . . . . . 6
⊢ (((𝑅 ∈ Ring ∧ (𝑈 ∈ 𝐵 ∧ 𝑊 ∈ 𝐵) ∧ 𝐾 ∈ ℕ0) ∧ (𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁)) → (𝑈 ∈ 𝐵 ∧ 𝑊 ∈ 𝐵)) | 
| 104 |  | simpl3 1193 | . . . . . 6
⊢ (((𝑅 ∈ Ring ∧ (𝑈 ∈ 𝐵 ∧ 𝑊 ∈ 𝐵) ∧ 𝐾 ∈ ℕ0) ∧ (𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁)) → 𝐾 ∈
ℕ0) | 
| 105 | 43, 14, 15 | decpmatmullem 22778 | . . . . . 6
⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑈 ∈ 𝐵 ∧ 𝑊 ∈ 𝐵) ∧ (𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁 ∧ 𝐾 ∈ ℕ0)) → (𝑖((𝑈(.r‘𝐶)𝑊) decompPMat 𝐾)𝑗) = (𝑅 Σg (𝑡 ∈ 𝑁 ↦ (𝑅 Σg (𝑘 ∈ (0...𝐾) ↦ (((coe1‘(𝑖𝑈𝑡))‘𝑘)(.r‘𝑅)((coe1‘(𝑡𝑊𝑗))‘(𝐾 − 𝑘)))))))) | 
| 106 | 36, 33, 103, 10, 11, 104, 105 | syl213anc 1390 | . . . . 5
⊢ (((𝑅 ∈ Ring ∧ (𝑈 ∈ 𝐵 ∧ 𝑊 ∈ 𝐵) ∧ 𝐾 ∈ ℕ0) ∧ (𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁)) → (𝑖((𝑈(.r‘𝐶)𝑊) decompPMat 𝐾)𝑗) = (𝑅 Σg (𝑡 ∈ 𝑁 ↦ (𝑅 Σg (𝑘 ∈ (0...𝐾) ↦ (((coe1‘(𝑖𝑈𝑡))‘𝑘)(.r‘𝑅)((coe1‘(𝑡𝑊𝑗))‘(𝐾 − 𝑘)))))))) | 
| 107 |  | simpll1 1212 | . . . . . . 7
⊢ ((((𝑅 ∈ Ring ∧ (𝑈 ∈ 𝐵 ∧ 𝑊 ∈ 𝐵) ∧ 𝐾 ∈ ℕ0) ∧ (𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁)) ∧ (𝑡 ∈ 𝑁 ∧ 𝑘 ∈ (0...𝐾))) → 𝑅 ∈ Ring) | 
| 108 |  | simplrl 776 | . . . . . . . . 9
⊢ ((((𝑅 ∈ Ring ∧ (𝑈 ∈ 𝐵 ∧ 𝑊 ∈ 𝐵) ∧ 𝐾 ∈ ℕ0) ∧ (𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁)) ∧ (𝑡 ∈ 𝑁 ∧ 𝑘 ∈ (0...𝐾))) → 𝑖 ∈ 𝑁) | 
| 109 |  | simprl 770 | . . . . . . . . 9
⊢ ((((𝑅 ∈ Ring ∧ (𝑈 ∈ 𝐵 ∧ 𝑊 ∈ 𝐵) ∧ 𝐾 ∈ ℕ0) ∧ (𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁)) ∧ (𝑡 ∈ 𝑁 ∧ 𝑘 ∈ (0...𝐾))) → 𝑡 ∈ 𝑁) | 
| 110 | 15 | eleq2i 2832 | . . . . . . . . . . . . . 14
⊢ (𝑈 ∈ 𝐵 ↔ 𝑈 ∈ (Base‘𝐶)) | 
| 111 | 110 | biimpi 216 | . . . . . . . . . . . . 13
⊢ (𝑈 ∈ 𝐵 → 𝑈 ∈ (Base‘𝐶)) | 
| 112 | 111 | adantr 480 | . . . . . . . . . . . 12
⊢ ((𝑈 ∈ 𝐵 ∧ 𝑊 ∈ 𝐵) → 𝑈 ∈ (Base‘𝐶)) | 
| 113 | 112 | 3ad2ant2 1134 | . . . . . . . . . . 11
⊢ ((𝑅 ∈ Ring ∧ (𝑈 ∈ 𝐵 ∧ 𝑊 ∈ 𝐵) ∧ 𝐾 ∈ ℕ0) → 𝑈 ∈ (Base‘𝐶)) | 
| 114 | 113 | adantr 480 | . . . . . . . . . 10
⊢ (((𝑅 ∈ Ring ∧ (𝑈 ∈ 𝐵 ∧ 𝑊 ∈ 𝐵) ∧ 𝐾 ∈ ℕ0) ∧ (𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁)) → 𝑈 ∈ (Base‘𝐶)) | 
| 115 | 114 | adantr 480 | . . . . . . . . 9
⊢ ((((𝑅 ∈ Ring ∧ (𝑈 ∈ 𝐵 ∧ 𝑊 ∈ 𝐵) ∧ 𝐾 ∈ ℕ0) ∧ (𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁)) ∧ (𝑡 ∈ 𝑁 ∧ 𝑘 ∈ (0...𝐾))) → 𝑈 ∈ (Base‘𝐶)) | 
| 116 |  | eqid 2736 | . . . . . . . . . 10
⊢
(Base‘𝑃) =
(Base‘𝑃) | 
| 117 | 14, 116 | matecl 22432 | . . . . . . . . 9
⊢ ((𝑖 ∈ 𝑁 ∧ 𝑡 ∈ 𝑁 ∧ 𝑈 ∈ (Base‘𝐶)) → (𝑖𝑈𝑡) ∈ (Base‘𝑃)) | 
| 118 | 108, 109,
115, 117 | syl3anc 1372 | . . . . . . . 8
⊢ ((((𝑅 ∈ Ring ∧ (𝑈 ∈ 𝐵 ∧ 𝑊 ∈ 𝐵) ∧ 𝐾 ∈ ℕ0) ∧ (𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁)) ∧ (𝑡 ∈ 𝑁 ∧ 𝑘 ∈ (0...𝐾))) → (𝑖𝑈𝑡) ∈ (Base‘𝑃)) | 
| 119 | 40 | ad2antll 729 | . . . . . . . 8
⊢ ((((𝑅 ∈ Ring ∧ (𝑈 ∈ 𝐵 ∧ 𝑊 ∈ 𝐵) ∧ 𝐾 ∈ ℕ0) ∧ (𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁)) ∧ (𝑡 ∈ 𝑁 ∧ 𝑘 ∈ (0...𝐾))) → 𝑘 ∈ ℕ0) | 
| 120 |  | eqid 2736 | . . . . . . . . 9
⊢
(coe1‘(𝑖𝑈𝑡)) = (coe1‘(𝑖𝑈𝑡)) | 
| 121 | 120, 116,
43, 30 | coe1fvalcl 22215 | . . . . . . . 8
⊢ (((𝑖𝑈𝑡) ∈ (Base‘𝑃) ∧ 𝑘 ∈ ℕ0) →
((coe1‘(𝑖𝑈𝑡))‘𝑘) ∈ (Base‘𝑅)) | 
| 122 | 118, 119,
121 | syl2anc 584 | . . . . . . 7
⊢ ((((𝑅 ∈ Ring ∧ (𝑈 ∈ 𝐵 ∧ 𝑊 ∈ 𝐵) ∧ 𝐾 ∈ ℕ0) ∧ (𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁)) ∧ (𝑡 ∈ 𝑁 ∧ 𝑘 ∈ (0...𝐾))) → ((coe1‘(𝑖𝑈𝑡))‘𝑘) ∈ (Base‘𝑅)) | 
| 123 |  | simplrr 777 | . . . . . . . . 9
⊢ ((((𝑅 ∈ Ring ∧ (𝑈 ∈ 𝐵 ∧ 𝑊 ∈ 𝐵) ∧ 𝐾 ∈ ℕ0) ∧ (𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁)) ∧ (𝑡 ∈ 𝑁 ∧ 𝑘 ∈ (0...𝐾))) → 𝑗 ∈ 𝑁) | 
| 124 | 49 | adantr 480 | . . . . . . . . 9
⊢ ((((𝑅 ∈ Ring ∧ (𝑈 ∈ 𝐵 ∧ 𝑊 ∈ 𝐵) ∧ 𝐾 ∈ ℕ0) ∧ (𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁)) ∧ (𝑡 ∈ 𝑁 ∧ 𝑘 ∈ (0...𝐾))) → 𝑊 ∈ 𝐵) | 
| 125 | 14, 116, 15, 109, 123, 124 | matecld 22433 | . . . . . . . 8
⊢ ((((𝑅 ∈ Ring ∧ (𝑈 ∈ 𝐵 ∧ 𝑊 ∈ 𝐵) ∧ 𝐾 ∈ ℕ0) ∧ (𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁)) ∧ (𝑡 ∈ 𝑁 ∧ 𝑘 ∈ (0...𝐾))) → (𝑡𝑊𝑗) ∈ (Base‘𝑃)) | 
| 126 | 51 | ad2antll 729 | . . . . . . . 8
⊢ ((((𝑅 ∈ Ring ∧ (𝑈 ∈ 𝐵 ∧ 𝑊 ∈ 𝐵) ∧ 𝐾 ∈ ℕ0) ∧ (𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁)) ∧ (𝑡 ∈ 𝑁 ∧ 𝑘 ∈ (0...𝐾))) → (𝐾 − 𝑘) ∈
ℕ0) | 
| 127 |  | eqid 2736 | . . . . . . . . 9
⊢
(coe1‘(𝑡𝑊𝑗)) = (coe1‘(𝑡𝑊𝑗)) | 
| 128 | 127, 116,
43, 30 | coe1fvalcl 22215 | . . . . . . . 8
⊢ (((𝑡𝑊𝑗) ∈ (Base‘𝑃) ∧ (𝐾 − 𝑘) ∈ ℕ0) →
((coe1‘(𝑡𝑊𝑗))‘(𝐾 − 𝑘)) ∈ (Base‘𝑅)) | 
| 129 | 125, 126,
128 | syl2anc 584 | . . . . . . 7
⊢ ((((𝑅 ∈ Ring ∧ (𝑈 ∈ 𝐵 ∧ 𝑊 ∈ 𝐵) ∧ 𝐾 ∈ ℕ0) ∧ (𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁)) ∧ (𝑡 ∈ 𝑁 ∧ 𝑘 ∈ (0...𝐾))) → ((coe1‘(𝑡𝑊𝑗))‘(𝐾 − 𝑘)) ∈ (Base‘𝑅)) | 
| 130 | 30, 31 | ringcl 20248 | . . . . . . 7
⊢ ((𝑅 ∈ Ring ∧
((coe1‘(𝑖𝑈𝑡))‘𝑘) ∈ (Base‘𝑅) ∧ ((coe1‘(𝑡𝑊𝑗))‘(𝐾 − 𝑘)) ∈ (Base‘𝑅)) → (((coe1‘(𝑖𝑈𝑡))‘𝑘)(.r‘𝑅)((coe1‘(𝑡𝑊𝑗))‘(𝐾 − 𝑘))) ∈ (Base‘𝑅)) | 
| 131 | 107, 122,
129, 130 | syl3anc 1372 | . . . . . 6
⊢ ((((𝑅 ∈ Ring ∧ (𝑈 ∈ 𝐵 ∧ 𝑊 ∈ 𝐵) ∧ 𝐾 ∈ ℕ0) ∧ (𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁)) ∧ (𝑡 ∈ 𝑁 ∧ 𝑘 ∈ (0...𝐾))) → (((coe1‘(𝑖𝑈𝑡))‘𝑘)(.r‘𝑅)((coe1‘(𝑡𝑊𝑗))‘(𝐾 − 𝑘))) ∈ (Base‘𝑅)) | 
| 132 | 30, 66, 36, 88, 131 | gsumcom3fi 19998 | . . . . 5
⊢ (((𝑅 ∈ Ring ∧ (𝑈 ∈ 𝐵 ∧ 𝑊 ∈ 𝐵) ∧ 𝐾 ∈ ℕ0) ∧ (𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁)) → (𝑅 Σg (𝑡 ∈ 𝑁 ↦ (𝑅 Σg (𝑘 ∈ (0...𝐾) ↦ (((coe1‘(𝑖𝑈𝑡))‘𝑘)(.r‘𝑅)((coe1‘(𝑡𝑊𝑗))‘(𝐾 − 𝑘))))))) = (𝑅 Σg (𝑘 ∈ (0...𝐾) ↦ (𝑅 Σg (𝑡 ∈ 𝑁 ↦ (((coe1‘(𝑖𝑈𝑡))‘𝑘)(.r‘𝑅)((coe1‘(𝑡𝑊𝑗))‘(𝐾 − 𝑘)))))))) | 
| 133 | 10 | adantr 480 | . . . . . . . . . . . . 13
⊢ ((((𝑅 ∈ Ring ∧ (𝑈 ∈ 𝐵 ∧ 𝑊 ∈ 𝐵) ∧ 𝐾 ∈ ℕ0) ∧ (𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁)) ∧ 𝑘 ∈ (0...𝐾)) → 𝑖 ∈ 𝑁) | 
| 134 | 133 | anim1i 615 | . . . . . . . . . . . 12
⊢
(((((𝑅 ∈ Ring
∧ (𝑈 ∈ 𝐵 ∧ 𝑊 ∈ 𝐵) ∧ 𝐾 ∈ ℕ0) ∧ (𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁)) ∧ 𝑘 ∈ (0...𝐾)) ∧ 𝑡 ∈ 𝑁) → (𝑖 ∈ 𝑁 ∧ 𝑡 ∈ 𝑁)) | 
| 135 | 43, 14, 15 | decpmate 22773 | . . . . . . . . . . . 12
⊢ (((𝑅 ∈ Ring ∧ 𝑈 ∈ 𝐵 ∧ 𝑘 ∈ ℕ0) ∧ (𝑖 ∈ 𝑁 ∧ 𝑡 ∈ 𝑁)) → (𝑖(𝑈 decompPMat 𝑘)𝑡) = ((coe1‘(𝑖𝑈𝑡))‘𝑘)) | 
| 136 | 42, 134, 135 | syl2an2r 685 | . . . . . . . . . . 11
⊢
(((((𝑅 ∈ Ring
∧ (𝑈 ∈ 𝐵 ∧ 𝑊 ∈ 𝐵) ∧ 𝐾 ∈ ℕ0) ∧ (𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁)) ∧ 𝑘 ∈ (0...𝐾)) ∧ 𝑡 ∈ 𝑁) → (𝑖(𝑈 decompPMat 𝑘)𝑡) = ((coe1‘(𝑖𝑈𝑡))‘𝑘)) | 
| 137 |  | simplrr 777 | . . . . . . . . . . . . 13
⊢ ((((𝑅 ∈ Ring ∧ (𝑈 ∈ 𝐵 ∧ 𝑊 ∈ 𝐵) ∧ 𝐾 ∈ ℕ0) ∧ (𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁)) ∧ 𝑘 ∈ (0...𝐾)) → 𝑗 ∈ 𝑁) | 
| 138 | 137 | anim1ci 616 | . . . . . . . . . . . 12
⊢
(((((𝑅 ∈ Ring
∧ (𝑈 ∈ 𝐵 ∧ 𝑊 ∈ 𝐵) ∧ 𝐾 ∈ ℕ0) ∧ (𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁)) ∧ 𝑘 ∈ (0...𝐾)) ∧ 𝑡 ∈ 𝑁) → (𝑡 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁)) | 
| 139 | 43, 14, 15 | decpmate 22773 | . . . . . . . . . . . 12
⊢ (((𝑅 ∈ Ring ∧ 𝑊 ∈ 𝐵 ∧ (𝐾 − 𝑘) ∈ ℕ0) ∧ (𝑡 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁)) → (𝑡(𝑊 decompPMat (𝐾 − 𝑘))𝑗) = ((coe1‘(𝑡𝑊𝑗))‘(𝐾 − 𝑘))) | 
| 140 | 53, 138, 139 | syl2an2r 685 | . . . . . . . . . . 11
⊢
(((((𝑅 ∈ Ring
∧ (𝑈 ∈ 𝐵 ∧ 𝑊 ∈ 𝐵) ∧ 𝐾 ∈ ℕ0) ∧ (𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁)) ∧ 𝑘 ∈ (0...𝐾)) ∧ 𝑡 ∈ 𝑁) → (𝑡(𝑊 decompPMat (𝐾 − 𝑘))𝑗) = ((coe1‘(𝑡𝑊𝑗))‘(𝐾 − 𝑘))) | 
| 141 | 136, 140 | oveq12d 7450 | . . . . . . . . . 10
⊢
(((((𝑅 ∈ Ring
∧ (𝑈 ∈ 𝐵 ∧ 𝑊 ∈ 𝐵) ∧ 𝐾 ∈ ℕ0) ∧ (𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁)) ∧ 𝑘 ∈ (0...𝐾)) ∧ 𝑡 ∈ 𝑁) → ((𝑖(𝑈 decompPMat 𝑘)𝑡)(.r‘𝑅)(𝑡(𝑊 decompPMat (𝐾 − 𝑘))𝑗)) = (((coe1‘(𝑖𝑈𝑡))‘𝑘)(.r‘𝑅)((coe1‘(𝑡𝑊𝑗))‘(𝐾 − 𝑘)))) | 
| 142 | 141 | eqcomd 2742 | . . . . . . . . 9
⊢
(((((𝑅 ∈ Ring
∧ (𝑈 ∈ 𝐵 ∧ 𝑊 ∈ 𝐵) ∧ 𝐾 ∈ ℕ0) ∧ (𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁)) ∧ 𝑘 ∈ (0...𝐾)) ∧ 𝑡 ∈ 𝑁) → (((coe1‘(𝑖𝑈𝑡))‘𝑘)(.r‘𝑅)((coe1‘(𝑡𝑊𝑗))‘(𝐾 − 𝑘))) = ((𝑖(𝑈 decompPMat 𝑘)𝑡)(.r‘𝑅)(𝑡(𝑊 decompPMat (𝐾 − 𝑘))𝑗))) | 
| 143 | 142 | mpteq2dva 5241 | . . . . . . . 8
⊢ ((((𝑅 ∈ Ring ∧ (𝑈 ∈ 𝐵 ∧ 𝑊 ∈ 𝐵) ∧ 𝐾 ∈ ℕ0) ∧ (𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁)) ∧ 𝑘 ∈ (0...𝐾)) → (𝑡 ∈ 𝑁 ↦ (((coe1‘(𝑖𝑈𝑡))‘𝑘)(.r‘𝑅)((coe1‘(𝑡𝑊𝑗))‘(𝐾 − 𝑘)))) = (𝑡 ∈ 𝑁 ↦ ((𝑖(𝑈 decompPMat 𝑘)𝑡)(.r‘𝑅)(𝑡(𝑊 decompPMat (𝐾 − 𝑘))𝑗)))) | 
| 144 | 143 | oveq2d 7448 | . . . . . . 7
⊢ ((((𝑅 ∈ Ring ∧ (𝑈 ∈ 𝐵 ∧ 𝑊 ∈ 𝐵) ∧ 𝐾 ∈ ℕ0) ∧ (𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁)) ∧ 𝑘 ∈ (0...𝐾)) → (𝑅 Σg (𝑡 ∈ 𝑁 ↦ (((coe1‘(𝑖𝑈𝑡))‘𝑘)(.r‘𝑅)((coe1‘(𝑡𝑊𝑗))‘(𝐾 − 𝑘))))) = (𝑅 Σg (𝑡 ∈ 𝑁 ↦ ((𝑖(𝑈 decompPMat 𝑘)𝑡)(.r‘𝑅)(𝑡(𝑊 decompPMat (𝐾 − 𝑘))𝑗))))) | 
| 145 | 144 | mpteq2dva 5241 | . . . . . 6
⊢ (((𝑅 ∈ Ring ∧ (𝑈 ∈ 𝐵 ∧ 𝑊 ∈ 𝐵) ∧ 𝐾 ∈ ℕ0) ∧ (𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁)) → (𝑘 ∈ (0...𝐾) ↦ (𝑅 Σg (𝑡 ∈ 𝑁 ↦ (((coe1‘(𝑖𝑈𝑡))‘𝑘)(.r‘𝑅)((coe1‘(𝑡𝑊𝑗))‘(𝐾 − 𝑘)))))) = (𝑘 ∈ (0...𝐾) ↦ (𝑅 Σg (𝑡 ∈ 𝑁 ↦ ((𝑖(𝑈 decompPMat 𝑘)𝑡)(.r‘𝑅)(𝑡(𝑊 decompPMat (𝐾 − 𝑘))𝑗)))))) | 
| 146 | 145 | oveq2d 7448 | . . . . 5
⊢ (((𝑅 ∈ Ring ∧ (𝑈 ∈ 𝐵 ∧ 𝑊 ∈ 𝐵) ∧ 𝐾 ∈ ℕ0) ∧ (𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁)) → (𝑅 Σg (𝑘 ∈ (0...𝐾) ↦ (𝑅 Σg (𝑡 ∈ 𝑁 ↦ (((coe1‘(𝑖𝑈𝑡))‘𝑘)(.r‘𝑅)((coe1‘(𝑡𝑊𝑗))‘(𝐾 − 𝑘))))))) = (𝑅 Σg (𝑘 ∈ (0...𝐾) ↦ (𝑅 Σg (𝑡 ∈ 𝑁 ↦ ((𝑖(𝑈 decompPMat 𝑘)𝑡)(.r‘𝑅)(𝑡(𝑊 decompPMat (𝐾 − 𝑘))𝑗))))))) | 
| 147 | 106, 132,
146 | 3eqtrd 2780 | . . . 4
⊢ (((𝑅 ∈ Ring ∧ (𝑈 ∈ 𝐵 ∧ 𝑊 ∈ 𝐵) ∧ 𝐾 ∈ ℕ0) ∧ (𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁)) → (𝑖((𝑈(.r‘𝐶)𝑊) decompPMat 𝐾)𝑗) = (𝑅 Σg (𝑘 ∈ (0...𝐾) ↦ (𝑅 Σg (𝑡 ∈ 𝑁 ↦ ((𝑖(𝑈 decompPMat 𝑘)𝑡)(.r‘𝑅)(𝑡(𝑊 decompPMat (𝐾 − 𝑘))𝑗))))))) | 
| 148 | 13, 102, 147 | 3eqtr4rd 2787 | . . 3
⊢ (((𝑅 ∈ Ring ∧ (𝑈 ∈ 𝐵 ∧ 𝑊 ∈ 𝐵) ∧ 𝐾 ∈ ℕ0) ∧ (𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁)) → (𝑖((𝑈(.r‘𝐶)𝑊) decompPMat 𝐾)𝑗) = (𝑖(𝐴 Σg (𝑘 ∈ (0...𝐾) ↦ ((𝑈 decompPMat 𝑘)(.r‘𝐴)(𝑊 decompPMat (𝐾 − 𝑘)))))𝑗)) | 
| 149 | 148 | ralrimivva 3201 | . 2
⊢ ((𝑅 ∈ Ring ∧ (𝑈 ∈ 𝐵 ∧ 𝑊 ∈ 𝐵) ∧ 𝐾 ∈ ℕ0) →
∀𝑖 ∈ 𝑁 ∀𝑗 ∈ 𝑁 (𝑖((𝑈(.r‘𝐶)𝑊) decompPMat 𝐾)𝑗) = (𝑖(𝐴 Σg (𝑘 ∈ (0...𝐾) ↦ ((𝑈 decompPMat 𝑘)(.r‘𝐴)(𝑊 decompPMat (𝐾 − 𝑘)))))𝑗)) | 
| 150 | 43, 14 | pmatring 22699 | . . . . . . 7
⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝐶 ∈ Ring) | 
| 151 | 20, 150 | syl 17 | . . . . . 6
⊢ ((𝑅 ∈ Ring ∧ (𝑈 ∈ 𝐵 ∧ 𝑊 ∈ 𝐵)) → 𝐶 ∈ Ring) | 
| 152 |  | simprl 770 | . . . . . 6
⊢ ((𝑅 ∈ Ring ∧ (𝑈 ∈ 𝐵 ∧ 𝑊 ∈ 𝐵)) → 𝑈 ∈ 𝐵) | 
| 153 |  | simprr 772 | . . . . . 6
⊢ ((𝑅 ∈ Ring ∧ (𝑈 ∈ 𝐵 ∧ 𝑊 ∈ 𝐵)) → 𝑊 ∈ 𝐵) | 
| 154 |  | eqid 2736 | . . . . . . 7
⊢
(.r‘𝐶) = (.r‘𝐶) | 
| 155 | 15, 154 | ringcl 20248 | . . . . . 6
⊢ ((𝐶 ∈ Ring ∧ 𝑈 ∈ 𝐵 ∧ 𝑊 ∈ 𝐵) → (𝑈(.r‘𝐶)𝑊) ∈ 𝐵) | 
| 156 | 151, 152,
153, 155 | syl3anc 1372 | . . . . 5
⊢ ((𝑅 ∈ Ring ∧ (𝑈 ∈ 𝐵 ∧ 𝑊 ∈ 𝐵)) → (𝑈(.r‘𝐶)𝑊) ∈ 𝐵) | 
| 157 | 156 | 3adant3 1132 | . . . 4
⊢ ((𝑅 ∈ Ring ∧ (𝑈 ∈ 𝐵 ∧ 𝑊 ∈ 𝐵) ∧ 𝐾 ∈ ℕ0) → (𝑈(.r‘𝐶)𝑊) ∈ 𝐵) | 
| 158 | 43, 14, 15, 22, 44 | decpmatcl 22774 | . . . 4
⊢ ((𝑅 ∈ Ring ∧ (𝑈(.r‘𝐶)𝑊) ∈ 𝐵 ∧ 𝐾 ∈ ℕ0) → ((𝑈(.r‘𝐶)𝑊) decompPMat 𝐾) ∈ (Base‘𝐴)) | 
| 159 | 157, 158 | syld3an2 1412 | . . 3
⊢ ((𝑅 ∈ Ring ∧ (𝑈 ∈ 𝐵 ∧ 𝑊 ∈ 𝐵) ∧ 𝐾 ∈ ℕ0) → ((𝑈(.r‘𝐶)𝑊) decompPMat 𝐾) ∈ (Base‘𝐴)) | 
| 160 | 22 | matring 22450 | . . . . . 6
⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝐴 ∈ Ring) | 
| 161 | 21, 160 | syl 17 | . . . . 5
⊢ ((𝑅 ∈ Ring ∧ (𝑈 ∈ 𝐵 ∧ 𝑊 ∈ 𝐵) ∧ 𝐾 ∈ ℕ0) → 𝐴 ∈ Ring) | 
| 162 |  | ringcmn 20280 | . . . . 5
⊢ (𝐴 ∈ Ring → 𝐴 ∈ CMnd) | 
| 163 | 161, 162 | syl 17 | . . . 4
⊢ ((𝑅 ∈ Ring ∧ (𝑈 ∈ 𝐵 ∧ 𝑊 ∈ 𝐵) ∧ 𝐾 ∈ ℕ0) → 𝐴 ∈ CMnd) | 
| 164 |  | fzfid 14015 | . . . 4
⊢ ((𝑅 ∈ Ring ∧ (𝑈 ∈ 𝐵 ∧ 𝑊 ∈ 𝐵) ∧ 𝐾 ∈ ℕ0) →
(0...𝐾) ∈
Fin) | 
| 165 | 161 | adantr 480 | . . . . . 6
⊢ (((𝑅 ∈ Ring ∧ (𝑈 ∈ 𝐵 ∧ 𝑊 ∈ 𝐵) ∧ 𝐾 ∈ ℕ0) ∧ 𝑘 ∈ (0...𝐾)) → 𝐴 ∈ Ring) | 
| 166 | 32 | adantr 480 | . . . . . . . 8
⊢ (((𝑅 ∈ Ring ∧ (𝑈 ∈ 𝐵 ∧ 𝑊 ∈ 𝐵) ∧ 𝐾 ∈ ℕ0) ∧ 𝑘 ∈ (0...𝐾)) → 𝑅 ∈ Ring) | 
| 167 |  | simpl2l 1226 | . . . . . . . 8
⊢ (((𝑅 ∈ Ring ∧ (𝑈 ∈ 𝐵 ∧ 𝑊 ∈ 𝐵) ∧ 𝐾 ∈ ℕ0) ∧ 𝑘 ∈ (0...𝐾)) → 𝑈 ∈ 𝐵) | 
| 168 | 40 | adantl 481 | . . . . . . . 8
⊢ (((𝑅 ∈ Ring ∧ (𝑈 ∈ 𝐵 ∧ 𝑊 ∈ 𝐵) ∧ 𝐾 ∈ ℕ0) ∧ 𝑘 ∈ (0...𝐾)) → 𝑘 ∈ ℕ0) | 
| 169 | 166, 167,
168 | 3jca 1128 | . . . . . . 7
⊢ (((𝑅 ∈ Ring ∧ (𝑈 ∈ 𝐵 ∧ 𝑊 ∈ 𝐵) ∧ 𝐾 ∈ ℕ0) ∧ 𝑘 ∈ (0...𝐾)) → (𝑅 ∈ Ring ∧ 𝑈 ∈ 𝐵 ∧ 𝑘 ∈
ℕ0)) | 
| 170 | 169, 45 | syl 17 | . . . . . 6
⊢ (((𝑅 ∈ Ring ∧ (𝑈 ∈ 𝐵 ∧ 𝑊 ∈ 𝐵) ∧ 𝐾 ∈ ℕ0) ∧ 𝑘 ∈ (0...𝐾)) → (𝑈 decompPMat 𝑘) ∈ (Base‘𝐴)) | 
| 171 |  | simpl2r 1227 | . . . . . . . 8
⊢ (((𝑅 ∈ Ring ∧ (𝑈 ∈ 𝐵 ∧ 𝑊 ∈ 𝐵) ∧ 𝐾 ∈ ℕ0) ∧ 𝑘 ∈ (0...𝐾)) → 𝑊 ∈ 𝐵) | 
| 172 | 51 | adantl 481 | . . . . . . . 8
⊢ (((𝑅 ∈ Ring ∧ (𝑈 ∈ 𝐵 ∧ 𝑊 ∈ 𝐵) ∧ 𝐾 ∈ ℕ0) ∧ 𝑘 ∈ (0...𝐾)) → (𝐾 − 𝑘) ∈
ℕ0) | 
| 173 | 166, 171,
172 | 3jca 1128 | . . . . . . 7
⊢ (((𝑅 ∈ Ring ∧ (𝑈 ∈ 𝐵 ∧ 𝑊 ∈ 𝐵) ∧ 𝐾 ∈ ℕ0) ∧ 𝑘 ∈ (0...𝐾)) → (𝑅 ∈ Ring ∧ 𝑊 ∈ 𝐵 ∧ (𝐾 − 𝑘) ∈
ℕ0)) | 
| 174 | 173, 54 | syl 17 | . . . . . 6
⊢ (((𝑅 ∈ Ring ∧ (𝑈 ∈ 𝐵 ∧ 𝑊 ∈ 𝐵) ∧ 𝐾 ∈ ℕ0) ∧ 𝑘 ∈ (0...𝐾)) → (𝑊 decompPMat (𝐾 − 𝑘)) ∈ (Base‘𝐴)) | 
| 175 |  | eqid 2736 | . . . . . . 7
⊢
(.r‘𝐴) = (.r‘𝐴) | 
| 176 | 44, 175 | ringcl 20248 | . . . . . 6
⊢ ((𝐴 ∈ Ring ∧ (𝑈 decompPMat 𝑘) ∈ (Base‘𝐴) ∧ (𝑊 decompPMat (𝐾 − 𝑘)) ∈ (Base‘𝐴)) → ((𝑈 decompPMat 𝑘)(.r‘𝐴)(𝑊 decompPMat (𝐾 − 𝑘))) ∈ (Base‘𝐴)) | 
| 177 | 165, 170,
174, 176 | syl3anc 1372 | . . . . 5
⊢ (((𝑅 ∈ Ring ∧ (𝑈 ∈ 𝐵 ∧ 𝑊 ∈ 𝐵) ∧ 𝐾 ∈ ℕ0) ∧ 𝑘 ∈ (0...𝐾)) → ((𝑈 decompPMat 𝑘)(.r‘𝐴)(𝑊 decompPMat (𝐾 − 𝑘))) ∈ (Base‘𝐴)) | 
| 178 | 177 | ralrimiva 3145 | . . . 4
⊢ ((𝑅 ∈ Ring ∧ (𝑈 ∈ 𝐵 ∧ 𝑊 ∈ 𝐵) ∧ 𝐾 ∈ ℕ0) →
∀𝑘 ∈ (0...𝐾)((𝑈 decompPMat 𝑘)(.r‘𝐴)(𝑊 decompPMat (𝐾 − 𝑘))) ∈ (Base‘𝐴)) | 
| 179 | 44, 163, 164, 178 | gsummptcl 19986 | . . 3
⊢ ((𝑅 ∈ Ring ∧ (𝑈 ∈ 𝐵 ∧ 𝑊 ∈ 𝐵) ∧ 𝐾 ∈ ℕ0) → (𝐴 Σg
(𝑘 ∈ (0...𝐾) ↦ ((𝑈 decompPMat 𝑘)(.r‘𝐴)(𝑊 decompPMat (𝐾 − 𝑘))))) ∈ (Base‘𝐴)) | 
| 180 | 22, 44 | eqmat 22431 | . . 3
⊢ ((((𝑈(.r‘𝐶)𝑊) decompPMat 𝐾) ∈ (Base‘𝐴) ∧ (𝐴 Σg (𝑘 ∈ (0...𝐾) ↦ ((𝑈 decompPMat 𝑘)(.r‘𝐴)(𝑊 decompPMat (𝐾 − 𝑘))))) ∈ (Base‘𝐴)) → (((𝑈(.r‘𝐶)𝑊) decompPMat 𝐾) = (𝐴 Σg (𝑘 ∈ (0...𝐾) ↦ ((𝑈 decompPMat 𝑘)(.r‘𝐴)(𝑊 decompPMat (𝐾 − 𝑘))))) ↔ ∀𝑖 ∈ 𝑁 ∀𝑗 ∈ 𝑁 (𝑖((𝑈(.r‘𝐶)𝑊) decompPMat 𝐾)𝑗) = (𝑖(𝐴 Σg (𝑘 ∈ (0...𝐾) ↦ ((𝑈 decompPMat 𝑘)(.r‘𝐴)(𝑊 decompPMat (𝐾 − 𝑘)))))𝑗))) | 
| 181 | 159, 179,
180 | syl2anc 584 | . 2
⊢ ((𝑅 ∈ Ring ∧ (𝑈 ∈ 𝐵 ∧ 𝑊 ∈ 𝐵) ∧ 𝐾 ∈ ℕ0) → (((𝑈(.r‘𝐶)𝑊) decompPMat 𝐾) = (𝐴 Σg (𝑘 ∈ (0...𝐾) ↦ ((𝑈 decompPMat 𝑘)(.r‘𝐴)(𝑊 decompPMat (𝐾 − 𝑘))))) ↔ ∀𝑖 ∈ 𝑁 ∀𝑗 ∈ 𝑁 (𝑖((𝑈(.r‘𝐶)𝑊) decompPMat 𝐾)𝑗) = (𝑖(𝐴 Σg (𝑘 ∈ (0...𝐾) ↦ ((𝑈 decompPMat 𝑘)(.r‘𝐴)(𝑊 decompPMat (𝐾 − 𝑘)))))𝑗))) | 
| 182 | 149, 181 | mpbird 257 | 1
⊢ ((𝑅 ∈ Ring ∧ (𝑈 ∈ 𝐵 ∧ 𝑊 ∈ 𝐵) ∧ 𝐾 ∈ ℕ0) → ((𝑈(.r‘𝐶)𝑊) decompPMat 𝐾) = (𝐴 Σg (𝑘 ∈ (0...𝐾) ↦ ((𝑈 decompPMat 𝑘)(.r‘𝐴)(𝑊 decompPMat (𝐾 − 𝑘)))))) |