| Step | Hyp | Ref | Expression | 
|---|
| 1 |  | mdetuni.a | . . . . 5
⊢ 𝐴 = (𝑁 Mat 𝑅) | 
| 2 |  | mdetuni.b | . . . . 5
⊢ 𝐵 = (Base‘𝐴) | 
| 3 |  | mdetuni.k | . . . . 5
⊢ 𝐾 = (Base‘𝑅) | 
| 4 |  | mdetuni.0g | . . . . 5
⊢  0 =
(0g‘𝑅) | 
| 5 |  | mdetuni.1r | . . . . 5
⊢  1 =
(1r‘𝑅) | 
| 6 |  | mdetuni.pg | . . . . 5
⊢  + =
(+g‘𝑅) | 
| 7 |  | mdetuni.tg | . . . . 5
⊢  · =
(.r‘𝑅) | 
| 8 |  | mdetuni.n | . . . . 5
⊢ (𝜑 → 𝑁 ∈ Fin) | 
| 9 |  | mdetuni.r | . . . . 5
⊢ (𝜑 → 𝑅 ∈ Ring) | 
| 10 |  | ringgrp 20235 | . . . . . . . . 9
⊢ (𝑅 ∈ Ring → 𝑅 ∈ Grp) | 
| 11 | 9, 10 | syl 17 | . . . . . . . 8
⊢ (𝜑 → 𝑅 ∈ Grp) | 
| 12 | 11 | adantr 480 | . . . . . . 7
⊢ ((𝜑 ∧ 𝑎 ∈ 𝐵) → 𝑅 ∈ Grp) | 
| 13 |  | mdetuni.ff | . . . . . . . 8
⊢ (𝜑 → 𝐷:𝐵⟶𝐾) | 
| 14 | 13 | ffvelcdmda 7104 | . . . . . . 7
⊢ ((𝜑 ∧ 𝑎 ∈ 𝐵) → (𝐷‘𝑎) ∈ 𝐾) | 
| 15 | 9 | adantr 480 | . . . . . . . 8
⊢ ((𝜑 ∧ 𝑎 ∈ 𝐵) → 𝑅 ∈ Ring) | 
| 16 | 8, 9 | jca 511 | . . . . . . . . . . 11
⊢ (𝜑 → (𝑁 ∈ Fin ∧ 𝑅 ∈ Ring)) | 
| 17 | 1 | matring 22449 | . . . . . . . . . . 11
⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝐴 ∈ Ring) | 
| 18 |  | eqid 2737 | . . . . . . . . . . . 12
⊢
(1r‘𝐴) = (1r‘𝐴) | 
| 19 | 2, 18 | ringidcl 20262 | . . . . . . . . . . 11
⊢ (𝐴 ∈ Ring →
(1r‘𝐴)
∈ 𝐵) | 
| 20 | 16, 17, 19 | 3syl 18 | . . . . . . . . . 10
⊢ (𝜑 → (1r‘𝐴) ∈ 𝐵) | 
| 21 | 13, 20 | ffvelcdmd 7105 | . . . . . . . . 9
⊢ (𝜑 → (𝐷‘(1r‘𝐴)) ∈ 𝐾) | 
| 22 | 21 | adantr 480 | . . . . . . . 8
⊢ ((𝜑 ∧ 𝑎 ∈ 𝐵) → (𝐷‘(1r‘𝐴)) ∈ 𝐾) | 
| 23 |  | mdetuni.cr | . . . . . . . . . 10
⊢ (𝜑 → 𝑅 ∈ CRing) | 
| 24 |  | mdetuni.e | . . . . . . . . . . 11
⊢ 𝐸 = (𝑁 maDet 𝑅) | 
| 25 | 24, 1, 2, 3 | mdetf 22601 | . . . . . . . . . 10
⊢ (𝑅 ∈ CRing → 𝐸:𝐵⟶𝐾) | 
| 26 | 23, 25 | syl 17 | . . . . . . . . 9
⊢ (𝜑 → 𝐸:𝐵⟶𝐾) | 
| 27 | 26 | ffvelcdmda 7104 | . . . . . . . 8
⊢ ((𝜑 ∧ 𝑎 ∈ 𝐵) → (𝐸‘𝑎) ∈ 𝐾) | 
| 28 | 3, 7 | ringcl 20247 | . . . . . . . 8
⊢ ((𝑅 ∈ Ring ∧ (𝐷‘(1r‘𝐴)) ∈ 𝐾 ∧ (𝐸‘𝑎) ∈ 𝐾) → ((𝐷‘(1r‘𝐴)) · (𝐸‘𝑎)) ∈ 𝐾) | 
| 29 | 15, 22, 27, 28 | syl3anc 1373 | . . . . . . 7
⊢ ((𝜑 ∧ 𝑎 ∈ 𝐵) → ((𝐷‘(1r‘𝐴)) · (𝐸‘𝑎)) ∈ 𝐾) | 
| 30 |  | eqid 2737 | . . . . . . . 8
⊢
(-g‘𝑅) = (-g‘𝑅) | 
| 31 | 3, 30 | grpsubcl 19038 | . . . . . . 7
⊢ ((𝑅 ∈ Grp ∧ (𝐷‘𝑎) ∈ 𝐾 ∧ ((𝐷‘(1r‘𝐴)) · (𝐸‘𝑎)) ∈ 𝐾) → ((𝐷‘𝑎)(-g‘𝑅)((𝐷‘(1r‘𝐴)) · (𝐸‘𝑎))) ∈ 𝐾) | 
| 32 | 12, 14, 29, 31 | syl3anc 1373 | . . . . . 6
⊢ ((𝜑 ∧ 𝑎 ∈ 𝐵) → ((𝐷‘𝑎)(-g‘𝑅)((𝐷‘(1r‘𝐴)) · (𝐸‘𝑎))) ∈ 𝐾) | 
| 33 | 32 | fmpttd 7135 | . . . . 5
⊢ (𝜑 → (𝑎 ∈ 𝐵 ↦ ((𝐷‘𝑎)(-g‘𝑅)((𝐷‘(1r‘𝐴)) · (𝐸‘𝑎)))):𝐵⟶𝐾) | 
| 34 |  | simpr1 1195 | . . . . . . . . . 10
⊢ ((𝜑 ∧ (𝑏 ∈ 𝐵 ∧ 𝑐 ∈ 𝑁 ∧ 𝑑 ∈ 𝑁)) → 𝑏 ∈ 𝐵) | 
| 35 |  | fveq2 6906 | . . . . . . . . . . . 12
⊢ (𝑎 = 𝑏 → (𝐷‘𝑎) = (𝐷‘𝑏)) | 
| 36 |  | fveq2 6906 | . . . . . . . . . . . . 13
⊢ (𝑎 = 𝑏 → (𝐸‘𝑎) = (𝐸‘𝑏)) | 
| 37 | 36 | oveq2d 7447 | . . . . . . . . . . . 12
⊢ (𝑎 = 𝑏 → ((𝐷‘(1r‘𝐴)) · (𝐸‘𝑎)) = ((𝐷‘(1r‘𝐴)) · (𝐸‘𝑏))) | 
| 38 | 35, 37 | oveq12d 7449 | . . . . . . . . . . 11
⊢ (𝑎 = 𝑏 → ((𝐷‘𝑎)(-g‘𝑅)((𝐷‘(1r‘𝐴)) · (𝐸‘𝑎))) = ((𝐷‘𝑏)(-g‘𝑅)((𝐷‘(1r‘𝐴)) · (𝐸‘𝑏)))) | 
| 39 |  | eqid 2737 | . . . . . . . . . . 11
⊢ (𝑎 ∈ 𝐵 ↦ ((𝐷‘𝑎)(-g‘𝑅)((𝐷‘(1r‘𝐴)) · (𝐸‘𝑎)))) = (𝑎 ∈ 𝐵 ↦ ((𝐷‘𝑎)(-g‘𝑅)((𝐷‘(1r‘𝐴)) · (𝐸‘𝑎)))) | 
| 40 |  | ovex 7464 | . . . . . . . . . . 11
⊢ ((𝐷‘𝑏)(-g‘𝑅)((𝐷‘(1r‘𝐴)) · (𝐸‘𝑏))) ∈ V | 
| 41 | 38, 39, 40 | fvmpt 7016 | . . . . . . . . . 10
⊢ (𝑏 ∈ 𝐵 → ((𝑎 ∈ 𝐵 ↦ ((𝐷‘𝑎)(-g‘𝑅)((𝐷‘(1r‘𝐴)) · (𝐸‘𝑎))))‘𝑏) = ((𝐷‘𝑏)(-g‘𝑅)((𝐷‘(1r‘𝐴)) · (𝐸‘𝑏)))) | 
| 42 | 34, 41 | syl 17 | . . . . . . . . 9
⊢ ((𝜑 ∧ (𝑏 ∈ 𝐵 ∧ 𝑐 ∈ 𝑁 ∧ 𝑑 ∈ 𝑁)) → ((𝑎 ∈ 𝐵 ↦ ((𝐷‘𝑎)(-g‘𝑅)((𝐷‘(1r‘𝐴)) · (𝐸‘𝑎))))‘𝑏) = ((𝐷‘𝑏)(-g‘𝑅)((𝐷‘(1r‘𝐴)) · (𝐸‘𝑏)))) | 
| 43 | 42 | 3adant3 1133 | . . . . . . . 8
⊢ ((𝜑 ∧ (𝑏 ∈ 𝐵 ∧ 𝑐 ∈ 𝑁 ∧ 𝑑 ∈ 𝑁) ∧ (𝑐 ≠ 𝑑 ∧ ∀𝑒 ∈ 𝑁 (𝑐𝑏𝑒) = (𝑑𝑏𝑒))) → ((𝑎 ∈ 𝐵 ↦ ((𝐷‘𝑎)(-g‘𝑅)((𝐷‘(1r‘𝐴)) · (𝐸‘𝑎))))‘𝑏) = ((𝐷‘𝑏)(-g‘𝑅)((𝐷‘(1r‘𝐴)) · (𝐸‘𝑏)))) | 
| 44 |  | simp1 1137 | . . . . . . . . . 10
⊢ ((𝜑 ∧ (𝑏 ∈ 𝐵 ∧ 𝑐 ∈ 𝑁 ∧ 𝑑 ∈ 𝑁) ∧ (𝑐 ≠ 𝑑 ∧ ∀𝑒 ∈ 𝑁 (𝑐𝑏𝑒) = (𝑑𝑏𝑒))) → 𝜑) | 
| 45 |  | simp21 1207 | . . . . . . . . . 10
⊢ ((𝜑 ∧ (𝑏 ∈ 𝐵 ∧ 𝑐 ∈ 𝑁 ∧ 𝑑 ∈ 𝑁) ∧ (𝑐 ≠ 𝑑 ∧ ∀𝑒 ∈ 𝑁 (𝑐𝑏𝑒) = (𝑑𝑏𝑒))) → 𝑏 ∈ 𝐵) | 
| 46 |  | simp3r 1203 | . . . . . . . . . . 11
⊢ ((𝜑 ∧ (𝑏 ∈ 𝐵 ∧ 𝑐 ∈ 𝑁 ∧ 𝑑 ∈ 𝑁) ∧ (𝑐 ≠ 𝑑 ∧ ∀𝑒 ∈ 𝑁 (𝑐𝑏𝑒) = (𝑑𝑏𝑒))) → ∀𝑒 ∈ 𝑁 (𝑐𝑏𝑒) = (𝑑𝑏𝑒)) | 
| 47 |  | oveq2 7439 | . . . . . . . . . . . . 13
⊢ (𝑒 = 𝑤 → (𝑐𝑏𝑒) = (𝑐𝑏𝑤)) | 
| 48 |  | oveq2 7439 | . . . . . . . . . . . . 13
⊢ (𝑒 = 𝑤 → (𝑑𝑏𝑒) = (𝑑𝑏𝑤)) | 
| 49 | 47, 48 | eqeq12d 2753 | . . . . . . . . . . . 12
⊢ (𝑒 = 𝑤 → ((𝑐𝑏𝑒) = (𝑑𝑏𝑒) ↔ (𝑐𝑏𝑤) = (𝑑𝑏𝑤))) | 
| 50 | 49 | cbvralvw 3237 | . . . . . . . . . . 11
⊢
(∀𝑒 ∈
𝑁 (𝑐𝑏𝑒) = (𝑑𝑏𝑒) ↔ ∀𝑤 ∈ 𝑁 (𝑐𝑏𝑤) = (𝑑𝑏𝑤)) | 
| 51 | 46, 50 | sylib 218 | . . . . . . . . . 10
⊢ ((𝜑 ∧ (𝑏 ∈ 𝐵 ∧ 𝑐 ∈ 𝑁 ∧ 𝑑 ∈ 𝑁) ∧ (𝑐 ≠ 𝑑 ∧ ∀𝑒 ∈ 𝑁 (𝑐𝑏𝑒) = (𝑑𝑏𝑒))) → ∀𝑤 ∈ 𝑁 (𝑐𝑏𝑤) = (𝑑𝑏𝑤)) | 
| 52 |  | simp22 1208 | . . . . . . . . . 10
⊢ ((𝜑 ∧ (𝑏 ∈ 𝐵 ∧ 𝑐 ∈ 𝑁 ∧ 𝑑 ∈ 𝑁) ∧ (𝑐 ≠ 𝑑 ∧ ∀𝑒 ∈ 𝑁 (𝑐𝑏𝑒) = (𝑑𝑏𝑒))) → 𝑐 ∈ 𝑁) | 
| 53 |  | simp23 1209 | . . . . . . . . . 10
⊢ ((𝜑 ∧ (𝑏 ∈ 𝐵 ∧ 𝑐 ∈ 𝑁 ∧ 𝑑 ∈ 𝑁) ∧ (𝑐 ≠ 𝑑 ∧ ∀𝑒 ∈ 𝑁 (𝑐𝑏𝑒) = (𝑑𝑏𝑒))) → 𝑑 ∈ 𝑁) | 
| 54 |  | simp3l 1202 | . . . . . . . . . 10
⊢ ((𝜑 ∧ (𝑏 ∈ 𝐵 ∧ 𝑐 ∈ 𝑁 ∧ 𝑑 ∈ 𝑁) ∧ (𝑐 ≠ 𝑑 ∧ ∀𝑒 ∈ 𝑁 (𝑐𝑏𝑒) = (𝑑𝑏𝑒))) → 𝑐 ≠ 𝑑) | 
| 55 |  | mdetuni.al | . . . . . . . . . . 11
⊢ (𝜑 → ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝑁 ∀𝑧 ∈ 𝑁 ((𝑦 ≠ 𝑧 ∧ ∀𝑤 ∈ 𝑁 (𝑦𝑥𝑤) = (𝑧𝑥𝑤)) → (𝐷‘𝑥) = 0 )) | 
| 56 |  | mdetuni.li | . . . . . . . . . . 11
⊢ (𝜑 → ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐵 ∀𝑤 ∈ 𝑁 (((𝑥 ↾ ({𝑤} × 𝑁)) = ((𝑦 ↾ ({𝑤} × 𝑁)) ∘f + (𝑧 ↾ ({𝑤} × 𝑁))) ∧ (𝑥 ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) = (𝑦 ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) ∧ (𝑥 ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) = (𝑧 ↾ ((𝑁 ∖ {𝑤}) × 𝑁))) → (𝐷‘𝑥) = ((𝐷‘𝑦) + (𝐷‘𝑧)))) | 
| 57 |  | mdetuni.sc | . . . . . . . . . . 11
⊢ (𝜑 → ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐾 ∀𝑧 ∈ 𝐵 ∀𝑤 ∈ 𝑁 (((𝑥 ↾ ({𝑤} × 𝑁)) = ((({𝑤} × 𝑁) × {𝑦}) ∘f · (𝑧 ↾ ({𝑤} × 𝑁))) ∧ (𝑥 ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) = (𝑧 ↾ ((𝑁 ∖ {𝑤}) × 𝑁))) → (𝐷‘𝑥) = (𝑦 · (𝐷‘𝑧)))) | 
| 58 | 1, 2, 3, 4, 5, 6, 7, 8, 9, 13,
55, 56, 57 | mdetunilem1 22618 | . . . . . . . . . 10
⊢ (((𝜑 ∧ 𝑏 ∈ 𝐵 ∧ ∀𝑤 ∈ 𝑁 (𝑐𝑏𝑤) = (𝑑𝑏𝑤)) ∧ (𝑐 ∈ 𝑁 ∧ 𝑑 ∈ 𝑁 ∧ 𝑐 ≠ 𝑑)) → (𝐷‘𝑏) = 0 ) | 
| 59 | 44, 45, 51, 52, 53, 54, 58 | syl33anc 1387 | . . . . . . . . 9
⊢ ((𝜑 ∧ (𝑏 ∈ 𝐵 ∧ 𝑐 ∈ 𝑁 ∧ 𝑑 ∈ 𝑁) ∧ (𝑐 ≠ 𝑑 ∧ ∀𝑒 ∈ 𝑁 (𝑐𝑏𝑒) = (𝑑𝑏𝑒))) → (𝐷‘𝑏) = 0 ) | 
| 60 | 23 | 3ad2ant1 1134 | . . . . . . . . . . 11
⊢ ((𝜑 ∧ (𝑏 ∈ 𝐵 ∧ 𝑐 ∈ 𝑁 ∧ 𝑑 ∈ 𝑁) ∧ (𝑐 ≠ 𝑑 ∧ ∀𝑒 ∈ 𝑁 (𝑐𝑏𝑒) = (𝑑𝑏𝑒))) → 𝑅 ∈ CRing) | 
| 61 | 24, 1, 2, 4, 60, 45, 52, 53, 54, 46 | mdetralt 22614 | . . . . . . . . . 10
⊢ ((𝜑 ∧ (𝑏 ∈ 𝐵 ∧ 𝑐 ∈ 𝑁 ∧ 𝑑 ∈ 𝑁) ∧ (𝑐 ≠ 𝑑 ∧ ∀𝑒 ∈ 𝑁 (𝑐𝑏𝑒) = (𝑑𝑏𝑒))) → (𝐸‘𝑏) = 0 ) | 
| 62 | 61 | oveq2d 7447 | . . . . . . . . 9
⊢ ((𝜑 ∧ (𝑏 ∈ 𝐵 ∧ 𝑐 ∈ 𝑁 ∧ 𝑑 ∈ 𝑁) ∧ (𝑐 ≠ 𝑑 ∧ ∀𝑒 ∈ 𝑁 (𝑐𝑏𝑒) = (𝑑𝑏𝑒))) → ((𝐷‘(1r‘𝐴)) · (𝐸‘𝑏)) = ((𝐷‘(1r‘𝐴)) · 0 )) | 
| 63 | 59, 62 | oveq12d 7449 | . . . . . . . 8
⊢ ((𝜑 ∧ (𝑏 ∈ 𝐵 ∧ 𝑐 ∈ 𝑁 ∧ 𝑑 ∈ 𝑁) ∧ (𝑐 ≠ 𝑑 ∧ ∀𝑒 ∈ 𝑁 (𝑐𝑏𝑒) = (𝑑𝑏𝑒))) → ((𝐷‘𝑏)(-g‘𝑅)((𝐷‘(1r‘𝐴)) · (𝐸‘𝑏))) = ( 0 (-g‘𝑅)((𝐷‘(1r‘𝐴)) · 0 ))) | 
| 64 | 3, 7, 4 | ringrz 20291 | . . . . . . . . . . . 12
⊢ ((𝑅 ∈ Ring ∧ (𝐷‘(1r‘𝐴)) ∈ 𝐾) → ((𝐷‘(1r‘𝐴)) · 0 ) = 0 ) | 
| 65 | 9, 21, 64 | syl2anc 584 | . . . . . . . . . . 11
⊢ (𝜑 → ((𝐷‘(1r‘𝐴)) · 0 ) = 0 ) | 
| 66 | 65 | oveq2d 7447 | . . . . . . . . . 10
⊢ (𝜑 → ( 0 (-g‘𝑅)((𝐷‘(1r‘𝐴)) · 0 )) = ( 0 (-g‘𝑅) 0 )) | 
| 67 | 3, 4 | grpidcl 18983 | . . . . . . . . . . 11
⊢ (𝑅 ∈ Grp → 0 ∈ 𝐾) | 
| 68 | 3, 4, 30 | grpsubid 19042 | . . . . . . . . . . 11
⊢ ((𝑅 ∈ Grp ∧ 0 ∈ 𝐾) → ( 0 (-g‘𝑅) 0 ) = 0 ) | 
| 69 | 11, 67, 68 | syl2anc2 585 | . . . . . . . . . 10
⊢ (𝜑 → ( 0 (-g‘𝑅) 0 ) = 0 ) | 
| 70 | 66, 69 | eqtrd 2777 | . . . . . . . . 9
⊢ (𝜑 → ( 0 (-g‘𝑅)((𝐷‘(1r‘𝐴)) · 0 )) = 0 ) | 
| 71 | 70 | 3ad2ant1 1134 | . . . . . . . 8
⊢ ((𝜑 ∧ (𝑏 ∈ 𝐵 ∧ 𝑐 ∈ 𝑁 ∧ 𝑑 ∈ 𝑁) ∧ (𝑐 ≠ 𝑑 ∧ ∀𝑒 ∈ 𝑁 (𝑐𝑏𝑒) = (𝑑𝑏𝑒))) → ( 0 (-g‘𝑅)((𝐷‘(1r‘𝐴)) · 0 )) = 0 ) | 
| 72 | 43, 63, 71 | 3eqtrd 2781 | . . . . . . 7
⊢ ((𝜑 ∧ (𝑏 ∈ 𝐵 ∧ 𝑐 ∈ 𝑁 ∧ 𝑑 ∈ 𝑁) ∧ (𝑐 ≠ 𝑑 ∧ ∀𝑒 ∈ 𝑁 (𝑐𝑏𝑒) = (𝑑𝑏𝑒))) → ((𝑎 ∈ 𝐵 ↦ ((𝐷‘𝑎)(-g‘𝑅)((𝐷‘(1r‘𝐴)) · (𝐸‘𝑎))))‘𝑏) = 0 ) | 
| 73 | 72 | 3expia 1122 | . . . . . 6
⊢ ((𝜑 ∧ (𝑏 ∈ 𝐵 ∧ 𝑐 ∈ 𝑁 ∧ 𝑑 ∈ 𝑁)) → ((𝑐 ≠ 𝑑 ∧ ∀𝑒 ∈ 𝑁 (𝑐𝑏𝑒) = (𝑑𝑏𝑒)) → ((𝑎 ∈ 𝐵 ↦ ((𝐷‘𝑎)(-g‘𝑅)((𝐷‘(1r‘𝐴)) · (𝐸‘𝑎))))‘𝑏) = 0 )) | 
| 74 | 73 | ralrimivvva 3205 | . . . . 5
⊢ (𝜑 → ∀𝑏 ∈ 𝐵 ∀𝑐 ∈ 𝑁 ∀𝑑 ∈ 𝑁 ((𝑐 ≠ 𝑑 ∧ ∀𝑒 ∈ 𝑁 (𝑐𝑏𝑒) = (𝑑𝑏𝑒)) → ((𝑎 ∈ 𝐵 ↦ ((𝐷‘𝑎)(-g‘𝑅)((𝐷‘(1r‘𝐴)) · (𝐸‘𝑎))))‘𝑏) = 0 )) | 
| 75 |  | simp1 1137 | . . . . . . . . . . . 12
⊢ ((𝜑 ∧ ((𝑏 ∈ 𝐵 ∧ 𝑐 ∈ 𝐵) ∧ (𝑑 ∈ 𝐵 ∧ 𝑒 ∈ 𝑁)) ∧ ((𝑏 ↾ ({𝑒} × 𝑁)) = ((𝑐 ↾ ({𝑒} × 𝑁)) ∘f + (𝑑 ↾ ({𝑒} × 𝑁))) ∧ (𝑏 ↾ ((𝑁 ∖ {𝑒}) × 𝑁)) = (𝑐 ↾ ((𝑁 ∖ {𝑒}) × 𝑁)) ∧ (𝑏 ↾ ((𝑁 ∖ {𝑒}) × 𝑁)) = (𝑑 ↾ ((𝑁 ∖ {𝑒}) × 𝑁)))) → 𝜑) | 
| 76 |  | simp2ll 1241 | . . . . . . . . . . . 12
⊢ ((𝜑 ∧ ((𝑏 ∈ 𝐵 ∧ 𝑐 ∈ 𝐵) ∧ (𝑑 ∈ 𝐵 ∧ 𝑒 ∈ 𝑁)) ∧ ((𝑏 ↾ ({𝑒} × 𝑁)) = ((𝑐 ↾ ({𝑒} × 𝑁)) ∘f + (𝑑 ↾ ({𝑒} × 𝑁))) ∧ (𝑏 ↾ ((𝑁 ∖ {𝑒}) × 𝑁)) = (𝑐 ↾ ((𝑁 ∖ {𝑒}) × 𝑁)) ∧ (𝑏 ↾ ((𝑁 ∖ {𝑒}) × 𝑁)) = (𝑑 ↾ ((𝑁 ∖ {𝑒}) × 𝑁)))) → 𝑏 ∈ 𝐵) | 
| 77 |  | simp2lr 1242 | . . . . . . . . . . . 12
⊢ ((𝜑 ∧ ((𝑏 ∈ 𝐵 ∧ 𝑐 ∈ 𝐵) ∧ (𝑑 ∈ 𝐵 ∧ 𝑒 ∈ 𝑁)) ∧ ((𝑏 ↾ ({𝑒} × 𝑁)) = ((𝑐 ↾ ({𝑒} × 𝑁)) ∘f + (𝑑 ↾ ({𝑒} × 𝑁))) ∧ (𝑏 ↾ ((𝑁 ∖ {𝑒}) × 𝑁)) = (𝑐 ↾ ((𝑁 ∖ {𝑒}) × 𝑁)) ∧ (𝑏 ↾ ((𝑁 ∖ {𝑒}) × 𝑁)) = (𝑑 ↾ ((𝑁 ∖ {𝑒}) × 𝑁)))) → 𝑐 ∈ 𝐵) | 
| 78 |  | simp2rl 1243 | . . . . . . . . . . . 12
⊢ ((𝜑 ∧ ((𝑏 ∈ 𝐵 ∧ 𝑐 ∈ 𝐵) ∧ (𝑑 ∈ 𝐵 ∧ 𝑒 ∈ 𝑁)) ∧ ((𝑏 ↾ ({𝑒} × 𝑁)) = ((𝑐 ↾ ({𝑒} × 𝑁)) ∘f + (𝑑 ↾ ({𝑒} × 𝑁))) ∧ (𝑏 ↾ ((𝑁 ∖ {𝑒}) × 𝑁)) = (𝑐 ↾ ((𝑁 ∖ {𝑒}) × 𝑁)) ∧ (𝑏 ↾ ((𝑁 ∖ {𝑒}) × 𝑁)) = (𝑑 ↾ ((𝑁 ∖ {𝑒}) × 𝑁)))) → 𝑑 ∈ 𝐵) | 
| 79 |  | simp2rr 1244 | . . . . . . . . . . . 12
⊢ ((𝜑 ∧ ((𝑏 ∈ 𝐵 ∧ 𝑐 ∈ 𝐵) ∧ (𝑑 ∈ 𝐵 ∧ 𝑒 ∈ 𝑁)) ∧ ((𝑏 ↾ ({𝑒} × 𝑁)) = ((𝑐 ↾ ({𝑒} × 𝑁)) ∘f + (𝑑 ↾ ({𝑒} × 𝑁))) ∧ (𝑏 ↾ ((𝑁 ∖ {𝑒}) × 𝑁)) = (𝑐 ↾ ((𝑁 ∖ {𝑒}) × 𝑁)) ∧ (𝑏 ↾ ((𝑁 ∖ {𝑒}) × 𝑁)) = (𝑑 ↾ ((𝑁 ∖ {𝑒}) × 𝑁)))) → 𝑒 ∈ 𝑁) | 
| 80 |  | simp31 1210 | . . . . . . . . . . . 12
⊢ ((𝜑 ∧ ((𝑏 ∈ 𝐵 ∧ 𝑐 ∈ 𝐵) ∧ (𝑑 ∈ 𝐵 ∧ 𝑒 ∈ 𝑁)) ∧ ((𝑏 ↾ ({𝑒} × 𝑁)) = ((𝑐 ↾ ({𝑒} × 𝑁)) ∘f + (𝑑 ↾ ({𝑒} × 𝑁))) ∧ (𝑏 ↾ ((𝑁 ∖ {𝑒}) × 𝑁)) = (𝑐 ↾ ((𝑁 ∖ {𝑒}) × 𝑁)) ∧ (𝑏 ↾ ((𝑁 ∖ {𝑒}) × 𝑁)) = (𝑑 ↾ ((𝑁 ∖ {𝑒}) × 𝑁)))) → (𝑏 ↾ ({𝑒} × 𝑁)) = ((𝑐 ↾ ({𝑒} × 𝑁)) ∘f + (𝑑 ↾ ({𝑒} × 𝑁)))) | 
| 81 |  | simp32 1211 | . . . . . . . . . . . 12
⊢ ((𝜑 ∧ ((𝑏 ∈ 𝐵 ∧ 𝑐 ∈ 𝐵) ∧ (𝑑 ∈ 𝐵 ∧ 𝑒 ∈ 𝑁)) ∧ ((𝑏 ↾ ({𝑒} × 𝑁)) = ((𝑐 ↾ ({𝑒} × 𝑁)) ∘f + (𝑑 ↾ ({𝑒} × 𝑁))) ∧ (𝑏 ↾ ((𝑁 ∖ {𝑒}) × 𝑁)) = (𝑐 ↾ ((𝑁 ∖ {𝑒}) × 𝑁)) ∧ (𝑏 ↾ ((𝑁 ∖ {𝑒}) × 𝑁)) = (𝑑 ↾ ((𝑁 ∖ {𝑒}) × 𝑁)))) → (𝑏 ↾ ((𝑁 ∖ {𝑒}) × 𝑁)) = (𝑐 ↾ ((𝑁 ∖ {𝑒}) × 𝑁))) | 
| 82 |  | simp33 1212 | . . . . . . . . . . . 12
⊢ ((𝜑 ∧ ((𝑏 ∈ 𝐵 ∧ 𝑐 ∈ 𝐵) ∧ (𝑑 ∈ 𝐵 ∧ 𝑒 ∈ 𝑁)) ∧ ((𝑏 ↾ ({𝑒} × 𝑁)) = ((𝑐 ↾ ({𝑒} × 𝑁)) ∘f + (𝑑 ↾ ({𝑒} × 𝑁))) ∧ (𝑏 ↾ ((𝑁 ∖ {𝑒}) × 𝑁)) = (𝑐 ↾ ((𝑁 ∖ {𝑒}) × 𝑁)) ∧ (𝑏 ↾ ((𝑁 ∖ {𝑒}) × 𝑁)) = (𝑑 ↾ ((𝑁 ∖ {𝑒}) × 𝑁)))) → (𝑏 ↾ ((𝑁 ∖ {𝑒}) × 𝑁)) = (𝑑 ↾ ((𝑁 ∖ {𝑒}) × 𝑁))) | 
| 83 | 1, 2, 3, 4, 5, 6, 7, 8, 9, 13,
55, 56, 57 | mdetunilem3 22620 | . . . . . . . . . . . 12
⊢ (((𝜑 ∧ 𝑏 ∈ 𝐵 ∧ 𝑐 ∈ 𝐵) ∧ (𝑑 ∈ 𝐵 ∧ 𝑒 ∈ 𝑁 ∧ (𝑏 ↾ ({𝑒} × 𝑁)) = ((𝑐 ↾ ({𝑒} × 𝑁)) ∘f + (𝑑 ↾ ({𝑒} × 𝑁)))) ∧ ((𝑏 ↾ ((𝑁 ∖ {𝑒}) × 𝑁)) = (𝑐 ↾ ((𝑁 ∖ {𝑒}) × 𝑁)) ∧ (𝑏 ↾ ((𝑁 ∖ {𝑒}) × 𝑁)) = (𝑑 ↾ ((𝑁 ∖ {𝑒}) × 𝑁)))) → (𝐷‘𝑏) = ((𝐷‘𝑐) + (𝐷‘𝑑))) | 
| 84 | 75, 76, 77, 78, 79, 80, 81, 82, 83 | syl332anc 1403 | . . . . . . . . . . 11
⊢ ((𝜑 ∧ ((𝑏 ∈ 𝐵 ∧ 𝑐 ∈ 𝐵) ∧ (𝑑 ∈ 𝐵 ∧ 𝑒 ∈ 𝑁)) ∧ ((𝑏 ↾ ({𝑒} × 𝑁)) = ((𝑐 ↾ ({𝑒} × 𝑁)) ∘f + (𝑑 ↾ ({𝑒} × 𝑁))) ∧ (𝑏 ↾ ((𝑁 ∖ {𝑒}) × 𝑁)) = (𝑐 ↾ ((𝑁 ∖ {𝑒}) × 𝑁)) ∧ (𝑏 ↾ ((𝑁 ∖ {𝑒}) × 𝑁)) = (𝑑 ↾ ((𝑁 ∖ {𝑒}) × 𝑁)))) → (𝐷‘𝑏) = ((𝐷‘𝑐) + (𝐷‘𝑑))) | 
| 85 | 23 | 3ad2ant1 1134 | . . . . . . . . . . . . 13
⊢ ((𝜑 ∧ ((𝑏 ∈ 𝐵 ∧ 𝑐 ∈ 𝐵) ∧ (𝑑 ∈ 𝐵 ∧ 𝑒 ∈ 𝑁)) ∧ ((𝑏 ↾ ({𝑒} × 𝑁)) = ((𝑐 ↾ ({𝑒} × 𝑁)) ∘f + (𝑑 ↾ ({𝑒} × 𝑁))) ∧ (𝑏 ↾ ((𝑁 ∖ {𝑒}) × 𝑁)) = (𝑐 ↾ ((𝑁 ∖ {𝑒}) × 𝑁)) ∧ (𝑏 ↾ ((𝑁 ∖ {𝑒}) × 𝑁)) = (𝑑 ↾ ((𝑁 ∖ {𝑒}) × 𝑁)))) → 𝑅 ∈ CRing) | 
| 86 | 24, 1, 2, 6, 85, 76, 77, 78, 79, 80, 81, 82 | mdetrlin 22608 | . . . . . . . . . . . 12
⊢ ((𝜑 ∧ ((𝑏 ∈ 𝐵 ∧ 𝑐 ∈ 𝐵) ∧ (𝑑 ∈ 𝐵 ∧ 𝑒 ∈ 𝑁)) ∧ ((𝑏 ↾ ({𝑒} × 𝑁)) = ((𝑐 ↾ ({𝑒} × 𝑁)) ∘f + (𝑑 ↾ ({𝑒} × 𝑁))) ∧ (𝑏 ↾ ((𝑁 ∖ {𝑒}) × 𝑁)) = (𝑐 ↾ ((𝑁 ∖ {𝑒}) × 𝑁)) ∧ (𝑏 ↾ ((𝑁 ∖ {𝑒}) × 𝑁)) = (𝑑 ↾ ((𝑁 ∖ {𝑒}) × 𝑁)))) → (𝐸‘𝑏) = ((𝐸‘𝑐) + (𝐸‘𝑑))) | 
| 87 | 86 | oveq2d 7447 | . . . . . . . . . . 11
⊢ ((𝜑 ∧ ((𝑏 ∈ 𝐵 ∧ 𝑐 ∈ 𝐵) ∧ (𝑑 ∈ 𝐵 ∧ 𝑒 ∈ 𝑁)) ∧ ((𝑏 ↾ ({𝑒} × 𝑁)) = ((𝑐 ↾ ({𝑒} × 𝑁)) ∘f + (𝑑 ↾ ({𝑒} × 𝑁))) ∧ (𝑏 ↾ ((𝑁 ∖ {𝑒}) × 𝑁)) = (𝑐 ↾ ((𝑁 ∖ {𝑒}) × 𝑁)) ∧ (𝑏 ↾ ((𝑁 ∖ {𝑒}) × 𝑁)) = (𝑑 ↾ ((𝑁 ∖ {𝑒}) × 𝑁)))) → ((𝐷‘(1r‘𝐴)) · (𝐸‘𝑏)) = ((𝐷‘(1r‘𝐴)) · ((𝐸‘𝑐) + (𝐸‘𝑑)))) | 
| 88 | 84, 87 | oveq12d 7449 | . . . . . . . . . 10
⊢ ((𝜑 ∧ ((𝑏 ∈ 𝐵 ∧ 𝑐 ∈ 𝐵) ∧ (𝑑 ∈ 𝐵 ∧ 𝑒 ∈ 𝑁)) ∧ ((𝑏 ↾ ({𝑒} × 𝑁)) = ((𝑐 ↾ ({𝑒} × 𝑁)) ∘f + (𝑑 ↾ ({𝑒} × 𝑁))) ∧ (𝑏 ↾ ((𝑁 ∖ {𝑒}) × 𝑁)) = (𝑐 ↾ ((𝑁 ∖ {𝑒}) × 𝑁)) ∧ (𝑏 ↾ ((𝑁 ∖ {𝑒}) × 𝑁)) = (𝑑 ↾ ((𝑁 ∖ {𝑒}) × 𝑁)))) → ((𝐷‘𝑏)(-g‘𝑅)((𝐷‘(1r‘𝐴)) · (𝐸‘𝑏))) = (((𝐷‘𝑐) + (𝐷‘𝑑))(-g‘𝑅)((𝐷‘(1r‘𝐴)) · ((𝐸‘𝑐) + (𝐸‘𝑑))))) | 
| 89 |  | simprll 779 | . . . . . . . . . . . 12
⊢ ((𝜑 ∧ ((𝑏 ∈ 𝐵 ∧ 𝑐 ∈ 𝐵) ∧ (𝑑 ∈ 𝐵 ∧ 𝑒 ∈ 𝑁))) → 𝑏 ∈ 𝐵) | 
| 90 | 89, 41 | syl 17 | . . . . . . . . . . 11
⊢ ((𝜑 ∧ ((𝑏 ∈ 𝐵 ∧ 𝑐 ∈ 𝐵) ∧ (𝑑 ∈ 𝐵 ∧ 𝑒 ∈ 𝑁))) → ((𝑎 ∈ 𝐵 ↦ ((𝐷‘𝑎)(-g‘𝑅)((𝐷‘(1r‘𝐴)) · (𝐸‘𝑎))))‘𝑏) = ((𝐷‘𝑏)(-g‘𝑅)((𝐷‘(1r‘𝐴)) · (𝐸‘𝑏)))) | 
| 91 | 90 | 3adant3 1133 | . . . . . . . . . 10
⊢ ((𝜑 ∧ ((𝑏 ∈ 𝐵 ∧ 𝑐 ∈ 𝐵) ∧ (𝑑 ∈ 𝐵 ∧ 𝑒 ∈ 𝑁)) ∧ ((𝑏 ↾ ({𝑒} × 𝑁)) = ((𝑐 ↾ ({𝑒} × 𝑁)) ∘f + (𝑑 ↾ ({𝑒} × 𝑁))) ∧ (𝑏 ↾ ((𝑁 ∖ {𝑒}) × 𝑁)) = (𝑐 ↾ ((𝑁 ∖ {𝑒}) × 𝑁)) ∧ (𝑏 ↾ ((𝑁 ∖ {𝑒}) × 𝑁)) = (𝑑 ↾ ((𝑁 ∖ {𝑒}) × 𝑁)))) → ((𝑎 ∈ 𝐵 ↦ ((𝐷‘𝑎)(-g‘𝑅)((𝐷‘(1r‘𝐴)) · (𝐸‘𝑎))))‘𝑏) = ((𝐷‘𝑏)(-g‘𝑅)((𝐷‘(1r‘𝐴)) · (𝐸‘𝑏)))) | 
| 92 |  | simprlr 780 | . . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ ((𝑏 ∈ 𝐵 ∧ 𝑐 ∈ 𝐵) ∧ (𝑑 ∈ 𝐵 ∧ 𝑒 ∈ 𝑁))) → 𝑐 ∈ 𝐵) | 
| 93 |  | fveq2 6906 | . . . . . . . . . . . . . . . 16
⊢ (𝑎 = 𝑐 → (𝐷‘𝑎) = (𝐷‘𝑐)) | 
| 94 |  | fveq2 6906 | . . . . . . . . . . . . . . . . 17
⊢ (𝑎 = 𝑐 → (𝐸‘𝑎) = (𝐸‘𝑐)) | 
| 95 | 94 | oveq2d 7447 | . . . . . . . . . . . . . . . 16
⊢ (𝑎 = 𝑐 → ((𝐷‘(1r‘𝐴)) · (𝐸‘𝑎)) = ((𝐷‘(1r‘𝐴)) · (𝐸‘𝑐))) | 
| 96 | 93, 95 | oveq12d 7449 | . . . . . . . . . . . . . . 15
⊢ (𝑎 = 𝑐 → ((𝐷‘𝑎)(-g‘𝑅)((𝐷‘(1r‘𝐴)) · (𝐸‘𝑎))) = ((𝐷‘𝑐)(-g‘𝑅)((𝐷‘(1r‘𝐴)) · (𝐸‘𝑐)))) | 
| 97 |  | ovex 7464 | . . . . . . . . . . . . . . 15
⊢ ((𝐷‘𝑐)(-g‘𝑅)((𝐷‘(1r‘𝐴)) · (𝐸‘𝑐))) ∈ V | 
| 98 | 96, 39, 97 | fvmpt 7016 | . . . . . . . . . . . . . 14
⊢ (𝑐 ∈ 𝐵 → ((𝑎 ∈ 𝐵 ↦ ((𝐷‘𝑎)(-g‘𝑅)((𝐷‘(1r‘𝐴)) · (𝐸‘𝑎))))‘𝑐) = ((𝐷‘𝑐)(-g‘𝑅)((𝐷‘(1r‘𝐴)) · (𝐸‘𝑐)))) | 
| 99 | 92, 98 | syl 17 | . . . . . . . . . . . . 13
⊢ ((𝜑 ∧ ((𝑏 ∈ 𝐵 ∧ 𝑐 ∈ 𝐵) ∧ (𝑑 ∈ 𝐵 ∧ 𝑒 ∈ 𝑁))) → ((𝑎 ∈ 𝐵 ↦ ((𝐷‘𝑎)(-g‘𝑅)((𝐷‘(1r‘𝐴)) · (𝐸‘𝑎))))‘𝑐) = ((𝐷‘𝑐)(-g‘𝑅)((𝐷‘(1r‘𝐴)) · (𝐸‘𝑐)))) | 
| 100 |  | simprrl 781 | . . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ ((𝑏 ∈ 𝐵 ∧ 𝑐 ∈ 𝐵) ∧ (𝑑 ∈ 𝐵 ∧ 𝑒 ∈ 𝑁))) → 𝑑 ∈ 𝐵) | 
| 101 |  | fveq2 6906 | . . . . . . . . . . . . . . . 16
⊢ (𝑎 = 𝑑 → (𝐷‘𝑎) = (𝐷‘𝑑)) | 
| 102 |  | fveq2 6906 | . . . . . . . . . . . . . . . . 17
⊢ (𝑎 = 𝑑 → (𝐸‘𝑎) = (𝐸‘𝑑)) | 
| 103 | 102 | oveq2d 7447 | . . . . . . . . . . . . . . . 16
⊢ (𝑎 = 𝑑 → ((𝐷‘(1r‘𝐴)) · (𝐸‘𝑎)) = ((𝐷‘(1r‘𝐴)) · (𝐸‘𝑑))) | 
| 104 | 101, 103 | oveq12d 7449 | . . . . . . . . . . . . . . 15
⊢ (𝑎 = 𝑑 → ((𝐷‘𝑎)(-g‘𝑅)((𝐷‘(1r‘𝐴)) · (𝐸‘𝑎))) = ((𝐷‘𝑑)(-g‘𝑅)((𝐷‘(1r‘𝐴)) · (𝐸‘𝑑)))) | 
| 105 |  | ovex 7464 | . . . . . . . . . . . . . . 15
⊢ ((𝐷‘𝑑)(-g‘𝑅)((𝐷‘(1r‘𝐴)) · (𝐸‘𝑑))) ∈ V | 
| 106 | 104, 39, 105 | fvmpt 7016 | . . . . . . . . . . . . . 14
⊢ (𝑑 ∈ 𝐵 → ((𝑎 ∈ 𝐵 ↦ ((𝐷‘𝑎)(-g‘𝑅)((𝐷‘(1r‘𝐴)) · (𝐸‘𝑎))))‘𝑑) = ((𝐷‘𝑑)(-g‘𝑅)((𝐷‘(1r‘𝐴)) · (𝐸‘𝑑)))) | 
| 107 | 100, 106 | syl 17 | . . . . . . . . . . . . 13
⊢ ((𝜑 ∧ ((𝑏 ∈ 𝐵 ∧ 𝑐 ∈ 𝐵) ∧ (𝑑 ∈ 𝐵 ∧ 𝑒 ∈ 𝑁))) → ((𝑎 ∈ 𝐵 ↦ ((𝐷‘𝑎)(-g‘𝑅)((𝐷‘(1r‘𝐴)) · (𝐸‘𝑎))))‘𝑑) = ((𝐷‘𝑑)(-g‘𝑅)((𝐷‘(1r‘𝐴)) · (𝐸‘𝑑)))) | 
| 108 | 99, 107 | oveq12d 7449 | . . . . . . . . . . . 12
⊢ ((𝜑 ∧ ((𝑏 ∈ 𝐵 ∧ 𝑐 ∈ 𝐵) ∧ (𝑑 ∈ 𝐵 ∧ 𝑒 ∈ 𝑁))) → (((𝑎 ∈ 𝐵 ↦ ((𝐷‘𝑎)(-g‘𝑅)((𝐷‘(1r‘𝐴)) · (𝐸‘𝑎))))‘𝑐) + ((𝑎 ∈ 𝐵 ↦ ((𝐷‘𝑎)(-g‘𝑅)((𝐷‘(1r‘𝐴)) · (𝐸‘𝑎))))‘𝑑)) = (((𝐷‘𝑐)(-g‘𝑅)((𝐷‘(1r‘𝐴)) · (𝐸‘𝑐))) + ((𝐷‘𝑑)(-g‘𝑅)((𝐷‘(1r‘𝐴)) · (𝐸‘𝑑))))) | 
| 109 |  | ringabl 20278 | . . . . . . . . . . . . . . 15
⊢ (𝑅 ∈ Ring → 𝑅 ∈ Abel) | 
| 110 | 9, 109 | syl 17 | . . . . . . . . . . . . . 14
⊢ (𝜑 → 𝑅 ∈ Abel) | 
| 111 | 110 | adantr 480 | . . . . . . . . . . . . 13
⊢ ((𝜑 ∧ ((𝑏 ∈ 𝐵 ∧ 𝑐 ∈ 𝐵) ∧ (𝑑 ∈ 𝐵 ∧ 𝑒 ∈ 𝑁))) → 𝑅 ∈ Abel) | 
| 112 | 13 | adantr 480 | . . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ ((𝑏 ∈ 𝐵 ∧ 𝑐 ∈ 𝐵) ∧ (𝑑 ∈ 𝐵 ∧ 𝑒 ∈ 𝑁))) → 𝐷:𝐵⟶𝐾) | 
| 113 | 112, 92 | ffvelcdmd 7105 | . . . . . . . . . . . . 13
⊢ ((𝜑 ∧ ((𝑏 ∈ 𝐵 ∧ 𝑐 ∈ 𝐵) ∧ (𝑑 ∈ 𝐵 ∧ 𝑒 ∈ 𝑁))) → (𝐷‘𝑐) ∈ 𝐾) | 
| 114 | 112, 100 | ffvelcdmd 7105 | . . . . . . . . . . . . 13
⊢ ((𝜑 ∧ ((𝑏 ∈ 𝐵 ∧ 𝑐 ∈ 𝐵) ∧ (𝑑 ∈ 𝐵 ∧ 𝑒 ∈ 𝑁))) → (𝐷‘𝑑) ∈ 𝐾) | 
| 115 | 9 | adantr 480 | . . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ ((𝑏 ∈ 𝐵 ∧ 𝑐 ∈ 𝐵) ∧ (𝑑 ∈ 𝐵 ∧ 𝑒 ∈ 𝑁))) → 𝑅 ∈ Ring) | 
| 116 | 21 | adantr 480 | . . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ ((𝑏 ∈ 𝐵 ∧ 𝑐 ∈ 𝐵) ∧ (𝑑 ∈ 𝐵 ∧ 𝑒 ∈ 𝑁))) → (𝐷‘(1r‘𝐴)) ∈ 𝐾) | 
| 117 | 26 | adantr 480 | . . . . . . . . . . . . . . 15
⊢ ((𝜑 ∧ ((𝑏 ∈ 𝐵 ∧ 𝑐 ∈ 𝐵) ∧ (𝑑 ∈ 𝐵 ∧ 𝑒 ∈ 𝑁))) → 𝐸:𝐵⟶𝐾) | 
| 118 | 117, 92 | ffvelcdmd 7105 | . . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ ((𝑏 ∈ 𝐵 ∧ 𝑐 ∈ 𝐵) ∧ (𝑑 ∈ 𝐵 ∧ 𝑒 ∈ 𝑁))) → (𝐸‘𝑐) ∈ 𝐾) | 
| 119 | 3, 7 | ringcl 20247 | . . . . . . . . . . . . . 14
⊢ ((𝑅 ∈ Ring ∧ (𝐷‘(1r‘𝐴)) ∈ 𝐾 ∧ (𝐸‘𝑐) ∈ 𝐾) → ((𝐷‘(1r‘𝐴)) · (𝐸‘𝑐)) ∈ 𝐾) | 
| 120 | 115, 116,
118, 119 | syl3anc 1373 | . . . . . . . . . . . . 13
⊢ ((𝜑 ∧ ((𝑏 ∈ 𝐵 ∧ 𝑐 ∈ 𝐵) ∧ (𝑑 ∈ 𝐵 ∧ 𝑒 ∈ 𝑁))) → ((𝐷‘(1r‘𝐴)) · (𝐸‘𝑐)) ∈ 𝐾) | 
| 121 | 117, 100 | ffvelcdmd 7105 | . . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ ((𝑏 ∈ 𝐵 ∧ 𝑐 ∈ 𝐵) ∧ (𝑑 ∈ 𝐵 ∧ 𝑒 ∈ 𝑁))) → (𝐸‘𝑑) ∈ 𝐾) | 
| 122 | 3, 7 | ringcl 20247 | . . . . . . . . . . . . . 14
⊢ ((𝑅 ∈ Ring ∧ (𝐷‘(1r‘𝐴)) ∈ 𝐾 ∧ (𝐸‘𝑑) ∈ 𝐾) → ((𝐷‘(1r‘𝐴)) · (𝐸‘𝑑)) ∈ 𝐾) | 
| 123 | 115, 116,
121, 122 | syl3anc 1373 | . . . . . . . . . . . . 13
⊢ ((𝜑 ∧ ((𝑏 ∈ 𝐵 ∧ 𝑐 ∈ 𝐵) ∧ (𝑑 ∈ 𝐵 ∧ 𝑒 ∈ 𝑁))) → ((𝐷‘(1r‘𝐴)) · (𝐸‘𝑑)) ∈ 𝐾) | 
| 124 | 3, 6, 30 | ablsub4 19828 | . . . . . . . . . . . . 13
⊢ ((𝑅 ∈ Abel ∧ ((𝐷‘𝑐) ∈ 𝐾 ∧ (𝐷‘𝑑) ∈ 𝐾) ∧ (((𝐷‘(1r‘𝐴)) · (𝐸‘𝑐)) ∈ 𝐾 ∧ ((𝐷‘(1r‘𝐴)) · (𝐸‘𝑑)) ∈ 𝐾)) → (((𝐷‘𝑐) + (𝐷‘𝑑))(-g‘𝑅)(((𝐷‘(1r‘𝐴)) · (𝐸‘𝑐)) + ((𝐷‘(1r‘𝐴)) · (𝐸‘𝑑)))) = (((𝐷‘𝑐)(-g‘𝑅)((𝐷‘(1r‘𝐴)) · (𝐸‘𝑐))) + ((𝐷‘𝑑)(-g‘𝑅)((𝐷‘(1r‘𝐴)) · (𝐸‘𝑑))))) | 
| 125 | 111, 113,
114, 120, 123, 124 | syl122anc 1381 | . . . . . . . . . . . 12
⊢ ((𝜑 ∧ ((𝑏 ∈ 𝐵 ∧ 𝑐 ∈ 𝐵) ∧ (𝑑 ∈ 𝐵 ∧ 𝑒 ∈ 𝑁))) → (((𝐷‘𝑐) + (𝐷‘𝑑))(-g‘𝑅)(((𝐷‘(1r‘𝐴)) · (𝐸‘𝑐)) + ((𝐷‘(1r‘𝐴)) · (𝐸‘𝑑)))) = (((𝐷‘𝑐)(-g‘𝑅)((𝐷‘(1r‘𝐴)) · (𝐸‘𝑐))) + ((𝐷‘𝑑)(-g‘𝑅)((𝐷‘(1r‘𝐴)) · (𝐸‘𝑑))))) | 
| 126 | 3, 6, 7 | ringdi 20258 | . . . . . . . . . . . . . . 15
⊢ ((𝑅 ∈ Ring ∧ ((𝐷‘(1r‘𝐴)) ∈ 𝐾 ∧ (𝐸‘𝑐) ∈ 𝐾 ∧ (𝐸‘𝑑) ∈ 𝐾)) → ((𝐷‘(1r‘𝐴)) · ((𝐸‘𝑐) + (𝐸‘𝑑))) = (((𝐷‘(1r‘𝐴)) · (𝐸‘𝑐)) + ((𝐷‘(1r‘𝐴)) · (𝐸‘𝑑)))) | 
| 127 | 115, 116,
118, 121, 126 | syl13anc 1374 | . . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ ((𝑏 ∈ 𝐵 ∧ 𝑐 ∈ 𝐵) ∧ (𝑑 ∈ 𝐵 ∧ 𝑒 ∈ 𝑁))) → ((𝐷‘(1r‘𝐴)) · ((𝐸‘𝑐) + (𝐸‘𝑑))) = (((𝐷‘(1r‘𝐴)) · (𝐸‘𝑐)) + ((𝐷‘(1r‘𝐴)) · (𝐸‘𝑑)))) | 
| 128 | 127 | eqcomd 2743 | . . . . . . . . . . . . 13
⊢ ((𝜑 ∧ ((𝑏 ∈ 𝐵 ∧ 𝑐 ∈ 𝐵) ∧ (𝑑 ∈ 𝐵 ∧ 𝑒 ∈ 𝑁))) → (((𝐷‘(1r‘𝐴)) · (𝐸‘𝑐)) + ((𝐷‘(1r‘𝐴)) · (𝐸‘𝑑))) = ((𝐷‘(1r‘𝐴)) · ((𝐸‘𝑐) + (𝐸‘𝑑)))) | 
| 129 | 128 | oveq2d 7447 | . . . . . . . . . . . 12
⊢ ((𝜑 ∧ ((𝑏 ∈ 𝐵 ∧ 𝑐 ∈ 𝐵) ∧ (𝑑 ∈ 𝐵 ∧ 𝑒 ∈ 𝑁))) → (((𝐷‘𝑐) + (𝐷‘𝑑))(-g‘𝑅)(((𝐷‘(1r‘𝐴)) · (𝐸‘𝑐)) + ((𝐷‘(1r‘𝐴)) · (𝐸‘𝑑)))) = (((𝐷‘𝑐) + (𝐷‘𝑑))(-g‘𝑅)((𝐷‘(1r‘𝐴)) · ((𝐸‘𝑐) + (𝐸‘𝑑))))) | 
| 130 | 108, 125,
129 | 3eqtr2d 2783 | . . . . . . . . . . 11
⊢ ((𝜑 ∧ ((𝑏 ∈ 𝐵 ∧ 𝑐 ∈ 𝐵) ∧ (𝑑 ∈ 𝐵 ∧ 𝑒 ∈ 𝑁))) → (((𝑎 ∈ 𝐵 ↦ ((𝐷‘𝑎)(-g‘𝑅)((𝐷‘(1r‘𝐴)) · (𝐸‘𝑎))))‘𝑐) + ((𝑎 ∈ 𝐵 ↦ ((𝐷‘𝑎)(-g‘𝑅)((𝐷‘(1r‘𝐴)) · (𝐸‘𝑎))))‘𝑑)) = (((𝐷‘𝑐) + (𝐷‘𝑑))(-g‘𝑅)((𝐷‘(1r‘𝐴)) · ((𝐸‘𝑐) + (𝐸‘𝑑))))) | 
| 131 | 130 | 3adant3 1133 | . . . . . . . . . 10
⊢ ((𝜑 ∧ ((𝑏 ∈ 𝐵 ∧ 𝑐 ∈ 𝐵) ∧ (𝑑 ∈ 𝐵 ∧ 𝑒 ∈ 𝑁)) ∧ ((𝑏 ↾ ({𝑒} × 𝑁)) = ((𝑐 ↾ ({𝑒} × 𝑁)) ∘f + (𝑑 ↾ ({𝑒} × 𝑁))) ∧ (𝑏 ↾ ((𝑁 ∖ {𝑒}) × 𝑁)) = (𝑐 ↾ ((𝑁 ∖ {𝑒}) × 𝑁)) ∧ (𝑏 ↾ ((𝑁 ∖ {𝑒}) × 𝑁)) = (𝑑 ↾ ((𝑁 ∖ {𝑒}) × 𝑁)))) → (((𝑎 ∈ 𝐵 ↦ ((𝐷‘𝑎)(-g‘𝑅)((𝐷‘(1r‘𝐴)) · (𝐸‘𝑎))))‘𝑐) + ((𝑎 ∈ 𝐵 ↦ ((𝐷‘𝑎)(-g‘𝑅)((𝐷‘(1r‘𝐴)) · (𝐸‘𝑎))))‘𝑑)) = (((𝐷‘𝑐) + (𝐷‘𝑑))(-g‘𝑅)((𝐷‘(1r‘𝐴)) · ((𝐸‘𝑐) + (𝐸‘𝑑))))) | 
| 132 | 88, 91, 131 | 3eqtr4d 2787 | . . . . . . . . 9
⊢ ((𝜑 ∧ ((𝑏 ∈ 𝐵 ∧ 𝑐 ∈ 𝐵) ∧ (𝑑 ∈ 𝐵 ∧ 𝑒 ∈ 𝑁)) ∧ ((𝑏 ↾ ({𝑒} × 𝑁)) = ((𝑐 ↾ ({𝑒} × 𝑁)) ∘f + (𝑑 ↾ ({𝑒} × 𝑁))) ∧ (𝑏 ↾ ((𝑁 ∖ {𝑒}) × 𝑁)) = (𝑐 ↾ ((𝑁 ∖ {𝑒}) × 𝑁)) ∧ (𝑏 ↾ ((𝑁 ∖ {𝑒}) × 𝑁)) = (𝑑 ↾ ((𝑁 ∖ {𝑒}) × 𝑁)))) → ((𝑎 ∈ 𝐵 ↦ ((𝐷‘𝑎)(-g‘𝑅)((𝐷‘(1r‘𝐴)) · (𝐸‘𝑎))))‘𝑏) = (((𝑎 ∈ 𝐵 ↦ ((𝐷‘𝑎)(-g‘𝑅)((𝐷‘(1r‘𝐴)) · (𝐸‘𝑎))))‘𝑐) + ((𝑎 ∈ 𝐵 ↦ ((𝐷‘𝑎)(-g‘𝑅)((𝐷‘(1r‘𝐴)) · (𝐸‘𝑎))))‘𝑑))) | 
| 133 | 132 | 3expia 1122 | . . . . . . . 8
⊢ ((𝜑 ∧ ((𝑏 ∈ 𝐵 ∧ 𝑐 ∈ 𝐵) ∧ (𝑑 ∈ 𝐵 ∧ 𝑒 ∈ 𝑁))) → (((𝑏 ↾ ({𝑒} × 𝑁)) = ((𝑐 ↾ ({𝑒} × 𝑁)) ∘f + (𝑑 ↾ ({𝑒} × 𝑁))) ∧ (𝑏 ↾ ((𝑁 ∖ {𝑒}) × 𝑁)) = (𝑐 ↾ ((𝑁 ∖ {𝑒}) × 𝑁)) ∧ (𝑏 ↾ ((𝑁 ∖ {𝑒}) × 𝑁)) = (𝑑 ↾ ((𝑁 ∖ {𝑒}) × 𝑁))) → ((𝑎 ∈ 𝐵 ↦ ((𝐷‘𝑎)(-g‘𝑅)((𝐷‘(1r‘𝐴)) · (𝐸‘𝑎))))‘𝑏) = (((𝑎 ∈ 𝐵 ↦ ((𝐷‘𝑎)(-g‘𝑅)((𝐷‘(1r‘𝐴)) · (𝐸‘𝑎))))‘𝑐) + ((𝑎 ∈ 𝐵 ↦ ((𝐷‘𝑎)(-g‘𝑅)((𝐷‘(1r‘𝐴)) · (𝐸‘𝑎))))‘𝑑)))) | 
| 134 | 133 | anassrs 467 | . . . . . . 7
⊢ (((𝜑 ∧ (𝑏 ∈ 𝐵 ∧ 𝑐 ∈ 𝐵)) ∧ (𝑑 ∈ 𝐵 ∧ 𝑒 ∈ 𝑁)) → (((𝑏 ↾ ({𝑒} × 𝑁)) = ((𝑐 ↾ ({𝑒} × 𝑁)) ∘f + (𝑑 ↾ ({𝑒} × 𝑁))) ∧ (𝑏 ↾ ((𝑁 ∖ {𝑒}) × 𝑁)) = (𝑐 ↾ ((𝑁 ∖ {𝑒}) × 𝑁)) ∧ (𝑏 ↾ ((𝑁 ∖ {𝑒}) × 𝑁)) = (𝑑 ↾ ((𝑁 ∖ {𝑒}) × 𝑁))) → ((𝑎 ∈ 𝐵 ↦ ((𝐷‘𝑎)(-g‘𝑅)((𝐷‘(1r‘𝐴)) · (𝐸‘𝑎))))‘𝑏) = (((𝑎 ∈ 𝐵 ↦ ((𝐷‘𝑎)(-g‘𝑅)((𝐷‘(1r‘𝐴)) · (𝐸‘𝑎))))‘𝑐) + ((𝑎 ∈ 𝐵 ↦ ((𝐷‘𝑎)(-g‘𝑅)((𝐷‘(1r‘𝐴)) · (𝐸‘𝑎))))‘𝑑)))) | 
| 135 | 134 | ralrimivva 3202 | . . . . . 6
⊢ ((𝜑 ∧ (𝑏 ∈ 𝐵 ∧ 𝑐 ∈ 𝐵)) → ∀𝑑 ∈ 𝐵 ∀𝑒 ∈ 𝑁 (((𝑏 ↾ ({𝑒} × 𝑁)) = ((𝑐 ↾ ({𝑒} × 𝑁)) ∘f + (𝑑 ↾ ({𝑒} × 𝑁))) ∧ (𝑏 ↾ ((𝑁 ∖ {𝑒}) × 𝑁)) = (𝑐 ↾ ((𝑁 ∖ {𝑒}) × 𝑁)) ∧ (𝑏 ↾ ((𝑁 ∖ {𝑒}) × 𝑁)) = (𝑑 ↾ ((𝑁 ∖ {𝑒}) × 𝑁))) → ((𝑎 ∈ 𝐵 ↦ ((𝐷‘𝑎)(-g‘𝑅)((𝐷‘(1r‘𝐴)) · (𝐸‘𝑎))))‘𝑏) = (((𝑎 ∈ 𝐵 ↦ ((𝐷‘𝑎)(-g‘𝑅)((𝐷‘(1r‘𝐴)) · (𝐸‘𝑎))))‘𝑐) + ((𝑎 ∈ 𝐵 ↦ ((𝐷‘𝑎)(-g‘𝑅)((𝐷‘(1r‘𝐴)) · (𝐸‘𝑎))))‘𝑑)))) | 
| 136 | 135 | ralrimivva 3202 | . . . . 5
⊢ (𝜑 → ∀𝑏 ∈ 𝐵 ∀𝑐 ∈ 𝐵 ∀𝑑 ∈ 𝐵 ∀𝑒 ∈ 𝑁 (((𝑏 ↾ ({𝑒} × 𝑁)) = ((𝑐 ↾ ({𝑒} × 𝑁)) ∘f + (𝑑 ↾ ({𝑒} × 𝑁))) ∧ (𝑏 ↾ ((𝑁 ∖ {𝑒}) × 𝑁)) = (𝑐 ↾ ((𝑁 ∖ {𝑒}) × 𝑁)) ∧ (𝑏 ↾ ((𝑁 ∖ {𝑒}) × 𝑁)) = (𝑑 ↾ ((𝑁 ∖ {𝑒}) × 𝑁))) → ((𝑎 ∈ 𝐵 ↦ ((𝐷‘𝑎)(-g‘𝑅)((𝐷‘(1r‘𝐴)) · (𝐸‘𝑎))))‘𝑏) = (((𝑎 ∈ 𝐵 ↦ ((𝐷‘𝑎)(-g‘𝑅)((𝐷‘(1r‘𝐴)) · (𝐸‘𝑎))))‘𝑐) + ((𝑎 ∈ 𝐵 ↦ ((𝐷‘𝑎)(-g‘𝑅)((𝐷‘(1r‘𝐴)) · (𝐸‘𝑎))))‘𝑑)))) | 
| 137 |  | simp1 1137 | . . . . . . . . . . . 12
⊢ ((𝜑 ∧ ((𝑏 ∈ 𝐵 ∧ 𝑐 ∈ 𝐾) ∧ (𝑑 ∈ 𝐵 ∧ 𝑒 ∈ 𝑁)) ∧ ((𝑏 ↾ ({𝑒} × 𝑁)) = ((({𝑒} × 𝑁) × {𝑐}) ∘f · (𝑑 ↾ ({𝑒} × 𝑁))) ∧ (𝑏 ↾ ((𝑁 ∖ {𝑒}) × 𝑁)) = (𝑑 ↾ ((𝑁 ∖ {𝑒}) × 𝑁)))) → 𝜑) | 
| 138 |  | simp2ll 1241 | . . . . . . . . . . . 12
⊢ ((𝜑 ∧ ((𝑏 ∈ 𝐵 ∧ 𝑐 ∈ 𝐾) ∧ (𝑑 ∈ 𝐵 ∧ 𝑒 ∈ 𝑁)) ∧ ((𝑏 ↾ ({𝑒} × 𝑁)) = ((({𝑒} × 𝑁) × {𝑐}) ∘f · (𝑑 ↾ ({𝑒} × 𝑁))) ∧ (𝑏 ↾ ((𝑁 ∖ {𝑒}) × 𝑁)) = (𝑑 ↾ ((𝑁 ∖ {𝑒}) × 𝑁)))) → 𝑏 ∈ 𝐵) | 
| 139 |  | simp2lr 1242 | . . . . . . . . . . . 12
⊢ ((𝜑 ∧ ((𝑏 ∈ 𝐵 ∧ 𝑐 ∈ 𝐾) ∧ (𝑑 ∈ 𝐵 ∧ 𝑒 ∈ 𝑁)) ∧ ((𝑏 ↾ ({𝑒} × 𝑁)) = ((({𝑒} × 𝑁) × {𝑐}) ∘f · (𝑑 ↾ ({𝑒} × 𝑁))) ∧ (𝑏 ↾ ((𝑁 ∖ {𝑒}) × 𝑁)) = (𝑑 ↾ ((𝑁 ∖ {𝑒}) × 𝑁)))) → 𝑐 ∈ 𝐾) | 
| 140 |  | simp2rl 1243 | . . . . . . . . . . . 12
⊢ ((𝜑 ∧ ((𝑏 ∈ 𝐵 ∧ 𝑐 ∈ 𝐾) ∧ (𝑑 ∈ 𝐵 ∧ 𝑒 ∈ 𝑁)) ∧ ((𝑏 ↾ ({𝑒} × 𝑁)) = ((({𝑒} × 𝑁) × {𝑐}) ∘f · (𝑑 ↾ ({𝑒} × 𝑁))) ∧ (𝑏 ↾ ((𝑁 ∖ {𝑒}) × 𝑁)) = (𝑑 ↾ ((𝑁 ∖ {𝑒}) × 𝑁)))) → 𝑑 ∈ 𝐵) | 
| 141 |  | simp2rr 1244 | . . . . . . . . . . . 12
⊢ ((𝜑 ∧ ((𝑏 ∈ 𝐵 ∧ 𝑐 ∈ 𝐾) ∧ (𝑑 ∈ 𝐵 ∧ 𝑒 ∈ 𝑁)) ∧ ((𝑏 ↾ ({𝑒} × 𝑁)) = ((({𝑒} × 𝑁) × {𝑐}) ∘f · (𝑑 ↾ ({𝑒} × 𝑁))) ∧ (𝑏 ↾ ((𝑁 ∖ {𝑒}) × 𝑁)) = (𝑑 ↾ ((𝑁 ∖ {𝑒}) × 𝑁)))) → 𝑒 ∈ 𝑁) | 
| 142 |  | simp3l 1202 | . . . . . . . . . . . 12
⊢ ((𝜑 ∧ ((𝑏 ∈ 𝐵 ∧ 𝑐 ∈ 𝐾) ∧ (𝑑 ∈ 𝐵 ∧ 𝑒 ∈ 𝑁)) ∧ ((𝑏 ↾ ({𝑒} × 𝑁)) = ((({𝑒} × 𝑁) × {𝑐}) ∘f · (𝑑 ↾ ({𝑒} × 𝑁))) ∧ (𝑏 ↾ ((𝑁 ∖ {𝑒}) × 𝑁)) = (𝑑 ↾ ((𝑁 ∖ {𝑒}) × 𝑁)))) → (𝑏 ↾ ({𝑒} × 𝑁)) = ((({𝑒} × 𝑁) × {𝑐}) ∘f · (𝑑 ↾ ({𝑒} × 𝑁)))) | 
| 143 |  | simp3r 1203 | . . . . . . . . . . . 12
⊢ ((𝜑 ∧ ((𝑏 ∈ 𝐵 ∧ 𝑐 ∈ 𝐾) ∧ (𝑑 ∈ 𝐵 ∧ 𝑒 ∈ 𝑁)) ∧ ((𝑏 ↾ ({𝑒} × 𝑁)) = ((({𝑒} × 𝑁) × {𝑐}) ∘f · (𝑑 ↾ ({𝑒} × 𝑁))) ∧ (𝑏 ↾ ((𝑁 ∖ {𝑒}) × 𝑁)) = (𝑑 ↾ ((𝑁 ∖ {𝑒}) × 𝑁)))) → (𝑏 ↾ ((𝑁 ∖ {𝑒}) × 𝑁)) = (𝑑 ↾ ((𝑁 ∖ {𝑒}) × 𝑁))) | 
| 144 | 1, 2, 3, 4, 5, 6, 7, 8, 9, 13,
55, 56, 57 | mdetunilem4 22621 | . . . . . . . . . . . 12
⊢ ((𝜑 ∧ (𝑏 ∈ 𝐵 ∧ 𝑐 ∈ 𝐾 ∧ 𝑑 ∈ 𝐵) ∧ (𝑒 ∈ 𝑁 ∧ (𝑏 ↾ ({𝑒} × 𝑁)) = ((({𝑒} × 𝑁) × {𝑐}) ∘f · (𝑑 ↾ ({𝑒} × 𝑁))) ∧ (𝑏 ↾ ((𝑁 ∖ {𝑒}) × 𝑁)) = (𝑑 ↾ ((𝑁 ∖ {𝑒}) × 𝑁)))) → (𝐷‘𝑏) = (𝑐 · (𝐷‘𝑑))) | 
| 145 | 137, 138,
139, 140, 141, 142, 143, 144 | syl133anc 1395 | . . . . . . . . . . 11
⊢ ((𝜑 ∧ ((𝑏 ∈ 𝐵 ∧ 𝑐 ∈ 𝐾) ∧ (𝑑 ∈ 𝐵 ∧ 𝑒 ∈ 𝑁)) ∧ ((𝑏 ↾ ({𝑒} × 𝑁)) = ((({𝑒} × 𝑁) × {𝑐}) ∘f · (𝑑 ↾ ({𝑒} × 𝑁))) ∧ (𝑏 ↾ ((𝑁 ∖ {𝑒}) × 𝑁)) = (𝑑 ↾ ((𝑁 ∖ {𝑒}) × 𝑁)))) → (𝐷‘𝑏) = (𝑐 · (𝐷‘𝑑))) | 
| 146 | 23 | 3ad2ant1 1134 | . . . . . . . . . . . . 13
⊢ ((𝜑 ∧ ((𝑏 ∈ 𝐵 ∧ 𝑐 ∈ 𝐾) ∧ (𝑑 ∈ 𝐵 ∧ 𝑒 ∈ 𝑁)) ∧ ((𝑏 ↾ ({𝑒} × 𝑁)) = ((({𝑒} × 𝑁) × {𝑐}) ∘f · (𝑑 ↾ ({𝑒} × 𝑁))) ∧ (𝑏 ↾ ((𝑁 ∖ {𝑒}) × 𝑁)) = (𝑑 ↾ ((𝑁 ∖ {𝑒}) × 𝑁)))) → 𝑅 ∈ CRing) | 
| 147 | 24, 1, 2, 3, 7, 146, 138, 139, 140, 141, 142, 143 | mdetrsca 22609 | . . . . . . . . . . . 12
⊢ ((𝜑 ∧ ((𝑏 ∈ 𝐵 ∧ 𝑐 ∈ 𝐾) ∧ (𝑑 ∈ 𝐵 ∧ 𝑒 ∈ 𝑁)) ∧ ((𝑏 ↾ ({𝑒} × 𝑁)) = ((({𝑒} × 𝑁) × {𝑐}) ∘f · (𝑑 ↾ ({𝑒} × 𝑁))) ∧ (𝑏 ↾ ((𝑁 ∖ {𝑒}) × 𝑁)) = (𝑑 ↾ ((𝑁 ∖ {𝑒}) × 𝑁)))) → (𝐸‘𝑏) = (𝑐 · (𝐸‘𝑑))) | 
| 148 | 147 | oveq2d 7447 | . . . . . . . . . . 11
⊢ ((𝜑 ∧ ((𝑏 ∈ 𝐵 ∧ 𝑐 ∈ 𝐾) ∧ (𝑑 ∈ 𝐵 ∧ 𝑒 ∈ 𝑁)) ∧ ((𝑏 ↾ ({𝑒} × 𝑁)) = ((({𝑒} × 𝑁) × {𝑐}) ∘f · (𝑑 ↾ ({𝑒} × 𝑁))) ∧ (𝑏 ↾ ((𝑁 ∖ {𝑒}) × 𝑁)) = (𝑑 ↾ ((𝑁 ∖ {𝑒}) × 𝑁)))) → ((𝐷‘(1r‘𝐴)) · (𝐸‘𝑏)) = ((𝐷‘(1r‘𝐴)) · (𝑐 · (𝐸‘𝑑)))) | 
| 149 | 145, 148 | oveq12d 7449 | . . . . . . . . . 10
⊢ ((𝜑 ∧ ((𝑏 ∈ 𝐵 ∧ 𝑐 ∈ 𝐾) ∧ (𝑑 ∈ 𝐵 ∧ 𝑒 ∈ 𝑁)) ∧ ((𝑏 ↾ ({𝑒} × 𝑁)) = ((({𝑒} × 𝑁) × {𝑐}) ∘f · (𝑑 ↾ ({𝑒} × 𝑁))) ∧ (𝑏 ↾ ((𝑁 ∖ {𝑒}) × 𝑁)) = (𝑑 ↾ ((𝑁 ∖ {𝑒}) × 𝑁)))) → ((𝐷‘𝑏)(-g‘𝑅)((𝐷‘(1r‘𝐴)) · (𝐸‘𝑏))) = ((𝑐 · (𝐷‘𝑑))(-g‘𝑅)((𝐷‘(1r‘𝐴)) · (𝑐 · (𝐸‘𝑑))))) | 
| 150 |  | simprll 779 | . . . . . . . . . . . 12
⊢ ((𝜑 ∧ ((𝑏 ∈ 𝐵 ∧ 𝑐 ∈ 𝐾) ∧ (𝑑 ∈ 𝐵 ∧ 𝑒 ∈ 𝑁))) → 𝑏 ∈ 𝐵) | 
| 151 | 150, 41 | syl 17 | . . . . . . . . . . 11
⊢ ((𝜑 ∧ ((𝑏 ∈ 𝐵 ∧ 𝑐 ∈ 𝐾) ∧ (𝑑 ∈ 𝐵 ∧ 𝑒 ∈ 𝑁))) → ((𝑎 ∈ 𝐵 ↦ ((𝐷‘𝑎)(-g‘𝑅)((𝐷‘(1r‘𝐴)) · (𝐸‘𝑎))))‘𝑏) = ((𝐷‘𝑏)(-g‘𝑅)((𝐷‘(1r‘𝐴)) · (𝐸‘𝑏)))) | 
| 152 | 151 | 3adant3 1133 | . . . . . . . . . 10
⊢ ((𝜑 ∧ ((𝑏 ∈ 𝐵 ∧ 𝑐 ∈ 𝐾) ∧ (𝑑 ∈ 𝐵 ∧ 𝑒 ∈ 𝑁)) ∧ ((𝑏 ↾ ({𝑒} × 𝑁)) = ((({𝑒} × 𝑁) × {𝑐}) ∘f · (𝑑 ↾ ({𝑒} × 𝑁))) ∧ (𝑏 ↾ ((𝑁 ∖ {𝑒}) × 𝑁)) = (𝑑 ↾ ((𝑁 ∖ {𝑒}) × 𝑁)))) → ((𝑎 ∈ 𝐵 ↦ ((𝐷‘𝑎)(-g‘𝑅)((𝐷‘(1r‘𝐴)) · (𝐸‘𝑎))))‘𝑏) = ((𝐷‘𝑏)(-g‘𝑅)((𝐷‘(1r‘𝐴)) · (𝐸‘𝑏)))) | 
| 153 |  | simprrl 781 | . . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ ((𝑏 ∈ 𝐵 ∧ 𝑐 ∈ 𝐾) ∧ (𝑑 ∈ 𝐵 ∧ 𝑒 ∈ 𝑁))) → 𝑑 ∈ 𝐵) | 
| 154 | 153, 106 | syl 17 | . . . . . . . . . . . . 13
⊢ ((𝜑 ∧ ((𝑏 ∈ 𝐵 ∧ 𝑐 ∈ 𝐾) ∧ (𝑑 ∈ 𝐵 ∧ 𝑒 ∈ 𝑁))) → ((𝑎 ∈ 𝐵 ↦ ((𝐷‘𝑎)(-g‘𝑅)((𝐷‘(1r‘𝐴)) · (𝐸‘𝑎))))‘𝑑) = ((𝐷‘𝑑)(-g‘𝑅)((𝐷‘(1r‘𝐴)) · (𝐸‘𝑑)))) | 
| 155 | 154 | oveq2d 7447 | . . . . . . . . . . . 12
⊢ ((𝜑 ∧ ((𝑏 ∈ 𝐵 ∧ 𝑐 ∈ 𝐾) ∧ (𝑑 ∈ 𝐵 ∧ 𝑒 ∈ 𝑁))) → (𝑐 · ((𝑎 ∈ 𝐵 ↦ ((𝐷‘𝑎)(-g‘𝑅)((𝐷‘(1r‘𝐴)) · (𝐸‘𝑎))))‘𝑑)) = (𝑐 · ((𝐷‘𝑑)(-g‘𝑅)((𝐷‘(1r‘𝐴)) · (𝐸‘𝑑))))) | 
| 156 | 9 | adantr 480 | . . . . . . . . . . . . 13
⊢ ((𝜑 ∧ ((𝑏 ∈ 𝐵 ∧ 𝑐 ∈ 𝐾) ∧ (𝑑 ∈ 𝐵 ∧ 𝑒 ∈ 𝑁))) → 𝑅 ∈ Ring) | 
| 157 |  | simprlr 780 | . . . . . . . . . . . . 13
⊢ ((𝜑 ∧ ((𝑏 ∈ 𝐵 ∧ 𝑐 ∈ 𝐾) ∧ (𝑑 ∈ 𝐵 ∧ 𝑒 ∈ 𝑁))) → 𝑐 ∈ 𝐾) | 
| 158 | 13 | adantr 480 | . . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ ((𝑏 ∈ 𝐵 ∧ 𝑐 ∈ 𝐾) ∧ (𝑑 ∈ 𝐵 ∧ 𝑒 ∈ 𝑁))) → 𝐷:𝐵⟶𝐾) | 
| 159 | 158, 153 | ffvelcdmd 7105 | . . . . . . . . . . . . 13
⊢ ((𝜑 ∧ ((𝑏 ∈ 𝐵 ∧ 𝑐 ∈ 𝐾) ∧ (𝑑 ∈ 𝐵 ∧ 𝑒 ∈ 𝑁))) → (𝐷‘𝑑) ∈ 𝐾) | 
| 160 | 21 | adantr 480 | . . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ ((𝑏 ∈ 𝐵 ∧ 𝑐 ∈ 𝐾) ∧ (𝑑 ∈ 𝐵 ∧ 𝑒 ∈ 𝑁))) → (𝐷‘(1r‘𝐴)) ∈ 𝐾) | 
| 161 | 26 | adantr 480 | . . . . . . . . . . . . . . 15
⊢ ((𝜑 ∧ ((𝑏 ∈ 𝐵 ∧ 𝑐 ∈ 𝐾) ∧ (𝑑 ∈ 𝐵 ∧ 𝑒 ∈ 𝑁))) → 𝐸:𝐵⟶𝐾) | 
| 162 | 161, 153 | ffvelcdmd 7105 | . . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ ((𝑏 ∈ 𝐵 ∧ 𝑐 ∈ 𝐾) ∧ (𝑑 ∈ 𝐵 ∧ 𝑒 ∈ 𝑁))) → (𝐸‘𝑑) ∈ 𝐾) | 
| 163 | 156, 160,
162, 122 | syl3anc 1373 | . . . . . . . . . . . . 13
⊢ ((𝜑 ∧ ((𝑏 ∈ 𝐵 ∧ 𝑐 ∈ 𝐾) ∧ (𝑑 ∈ 𝐵 ∧ 𝑒 ∈ 𝑁))) → ((𝐷‘(1r‘𝐴)) · (𝐸‘𝑑)) ∈ 𝐾) | 
| 164 | 3, 7, 30, 156, 157, 159, 163 | ringsubdi 20304 | . . . . . . . . . . . 12
⊢ ((𝜑 ∧ ((𝑏 ∈ 𝐵 ∧ 𝑐 ∈ 𝐾) ∧ (𝑑 ∈ 𝐵 ∧ 𝑒 ∈ 𝑁))) → (𝑐 · ((𝐷‘𝑑)(-g‘𝑅)((𝐷‘(1r‘𝐴)) · (𝐸‘𝑑)))) = ((𝑐 · (𝐷‘𝑑))(-g‘𝑅)(𝑐 · ((𝐷‘(1r‘𝐴)) · (𝐸‘𝑑))))) | 
| 165 |  | eqid 2737 | . . . . . . . . . . . . . . . . 17
⊢
(mulGrp‘𝑅) =
(mulGrp‘𝑅) | 
| 166 | 165 | crngmgp 20238 | . . . . . . . . . . . . . . . 16
⊢ (𝑅 ∈ CRing →
(mulGrp‘𝑅) ∈
CMnd) | 
| 167 | 23, 166 | syl 17 | . . . . . . . . . . . . . . 15
⊢ (𝜑 → (mulGrp‘𝑅) ∈ CMnd) | 
| 168 | 167 | adantr 480 | . . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ ((𝑏 ∈ 𝐵 ∧ 𝑐 ∈ 𝐾) ∧ (𝑑 ∈ 𝐵 ∧ 𝑒 ∈ 𝑁))) → (mulGrp‘𝑅) ∈ CMnd) | 
| 169 | 165, 3 | mgpbas 20142 | . . . . . . . . . . . . . . 15
⊢ 𝐾 =
(Base‘(mulGrp‘𝑅)) | 
| 170 | 165, 7 | mgpplusg 20141 | . . . . . . . . . . . . . . 15
⊢  · =
(+g‘(mulGrp‘𝑅)) | 
| 171 | 169, 170 | cmn12 19820 | . . . . . . . . . . . . . 14
⊢
(((mulGrp‘𝑅)
∈ CMnd ∧ (𝑐 ∈
𝐾 ∧ (𝐷‘(1r‘𝐴)) ∈ 𝐾 ∧ (𝐸‘𝑑) ∈ 𝐾)) → (𝑐 · ((𝐷‘(1r‘𝐴)) · (𝐸‘𝑑))) = ((𝐷‘(1r‘𝐴)) · (𝑐 · (𝐸‘𝑑)))) | 
| 172 | 168, 157,
160, 162, 171 | syl13anc 1374 | . . . . . . . . . . . . 13
⊢ ((𝜑 ∧ ((𝑏 ∈ 𝐵 ∧ 𝑐 ∈ 𝐾) ∧ (𝑑 ∈ 𝐵 ∧ 𝑒 ∈ 𝑁))) → (𝑐 · ((𝐷‘(1r‘𝐴)) · (𝐸‘𝑑))) = ((𝐷‘(1r‘𝐴)) · (𝑐 · (𝐸‘𝑑)))) | 
| 173 | 172 | oveq2d 7447 | . . . . . . . . . . . 12
⊢ ((𝜑 ∧ ((𝑏 ∈ 𝐵 ∧ 𝑐 ∈ 𝐾) ∧ (𝑑 ∈ 𝐵 ∧ 𝑒 ∈ 𝑁))) → ((𝑐 · (𝐷‘𝑑))(-g‘𝑅)(𝑐 · ((𝐷‘(1r‘𝐴)) · (𝐸‘𝑑)))) = ((𝑐 · (𝐷‘𝑑))(-g‘𝑅)((𝐷‘(1r‘𝐴)) · (𝑐 · (𝐸‘𝑑))))) | 
| 174 | 155, 164,
173 | 3eqtrd 2781 | . . . . . . . . . . 11
⊢ ((𝜑 ∧ ((𝑏 ∈ 𝐵 ∧ 𝑐 ∈ 𝐾) ∧ (𝑑 ∈ 𝐵 ∧ 𝑒 ∈ 𝑁))) → (𝑐 · ((𝑎 ∈ 𝐵 ↦ ((𝐷‘𝑎)(-g‘𝑅)((𝐷‘(1r‘𝐴)) · (𝐸‘𝑎))))‘𝑑)) = ((𝑐 · (𝐷‘𝑑))(-g‘𝑅)((𝐷‘(1r‘𝐴)) · (𝑐 · (𝐸‘𝑑))))) | 
| 175 | 174 | 3adant3 1133 | . . . . . . . . . 10
⊢ ((𝜑 ∧ ((𝑏 ∈ 𝐵 ∧ 𝑐 ∈ 𝐾) ∧ (𝑑 ∈ 𝐵 ∧ 𝑒 ∈ 𝑁)) ∧ ((𝑏 ↾ ({𝑒} × 𝑁)) = ((({𝑒} × 𝑁) × {𝑐}) ∘f · (𝑑 ↾ ({𝑒} × 𝑁))) ∧ (𝑏 ↾ ((𝑁 ∖ {𝑒}) × 𝑁)) = (𝑑 ↾ ((𝑁 ∖ {𝑒}) × 𝑁)))) → (𝑐 · ((𝑎 ∈ 𝐵 ↦ ((𝐷‘𝑎)(-g‘𝑅)((𝐷‘(1r‘𝐴)) · (𝐸‘𝑎))))‘𝑑)) = ((𝑐 · (𝐷‘𝑑))(-g‘𝑅)((𝐷‘(1r‘𝐴)) · (𝑐 · (𝐸‘𝑑))))) | 
| 176 | 149, 152,
175 | 3eqtr4d 2787 | . . . . . . . . 9
⊢ ((𝜑 ∧ ((𝑏 ∈ 𝐵 ∧ 𝑐 ∈ 𝐾) ∧ (𝑑 ∈ 𝐵 ∧ 𝑒 ∈ 𝑁)) ∧ ((𝑏 ↾ ({𝑒} × 𝑁)) = ((({𝑒} × 𝑁) × {𝑐}) ∘f · (𝑑 ↾ ({𝑒} × 𝑁))) ∧ (𝑏 ↾ ((𝑁 ∖ {𝑒}) × 𝑁)) = (𝑑 ↾ ((𝑁 ∖ {𝑒}) × 𝑁)))) → ((𝑎 ∈ 𝐵 ↦ ((𝐷‘𝑎)(-g‘𝑅)((𝐷‘(1r‘𝐴)) · (𝐸‘𝑎))))‘𝑏) = (𝑐 · ((𝑎 ∈ 𝐵 ↦ ((𝐷‘𝑎)(-g‘𝑅)((𝐷‘(1r‘𝐴)) · (𝐸‘𝑎))))‘𝑑))) | 
| 177 | 176 | 3expia 1122 | . . . . . . . 8
⊢ ((𝜑 ∧ ((𝑏 ∈ 𝐵 ∧ 𝑐 ∈ 𝐾) ∧ (𝑑 ∈ 𝐵 ∧ 𝑒 ∈ 𝑁))) → (((𝑏 ↾ ({𝑒} × 𝑁)) = ((({𝑒} × 𝑁) × {𝑐}) ∘f · (𝑑 ↾ ({𝑒} × 𝑁))) ∧ (𝑏 ↾ ((𝑁 ∖ {𝑒}) × 𝑁)) = (𝑑 ↾ ((𝑁 ∖ {𝑒}) × 𝑁))) → ((𝑎 ∈ 𝐵 ↦ ((𝐷‘𝑎)(-g‘𝑅)((𝐷‘(1r‘𝐴)) · (𝐸‘𝑎))))‘𝑏) = (𝑐 · ((𝑎 ∈ 𝐵 ↦ ((𝐷‘𝑎)(-g‘𝑅)((𝐷‘(1r‘𝐴)) · (𝐸‘𝑎))))‘𝑑)))) | 
| 178 | 177 | anassrs 467 | . . . . . . 7
⊢ (((𝜑 ∧ (𝑏 ∈ 𝐵 ∧ 𝑐 ∈ 𝐾)) ∧ (𝑑 ∈ 𝐵 ∧ 𝑒 ∈ 𝑁)) → (((𝑏 ↾ ({𝑒} × 𝑁)) = ((({𝑒} × 𝑁) × {𝑐}) ∘f · (𝑑 ↾ ({𝑒} × 𝑁))) ∧ (𝑏 ↾ ((𝑁 ∖ {𝑒}) × 𝑁)) = (𝑑 ↾ ((𝑁 ∖ {𝑒}) × 𝑁))) → ((𝑎 ∈ 𝐵 ↦ ((𝐷‘𝑎)(-g‘𝑅)((𝐷‘(1r‘𝐴)) · (𝐸‘𝑎))))‘𝑏) = (𝑐 · ((𝑎 ∈ 𝐵 ↦ ((𝐷‘𝑎)(-g‘𝑅)((𝐷‘(1r‘𝐴)) · (𝐸‘𝑎))))‘𝑑)))) | 
| 179 | 178 | ralrimivva 3202 | . . . . . 6
⊢ ((𝜑 ∧ (𝑏 ∈ 𝐵 ∧ 𝑐 ∈ 𝐾)) → ∀𝑑 ∈ 𝐵 ∀𝑒 ∈ 𝑁 (((𝑏 ↾ ({𝑒} × 𝑁)) = ((({𝑒} × 𝑁) × {𝑐}) ∘f · (𝑑 ↾ ({𝑒} × 𝑁))) ∧ (𝑏 ↾ ((𝑁 ∖ {𝑒}) × 𝑁)) = (𝑑 ↾ ((𝑁 ∖ {𝑒}) × 𝑁))) → ((𝑎 ∈ 𝐵 ↦ ((𝐷‘𝑎)(-g‘𝑅)((𝐷‘(1r‘𝐴)) · (𝐸‘𝑎))))‘𝑏) = (𝑐 · ((𝑎 ∈ 𝐵 ↦ ((𝐷‘𝑎)(-g‘𝑅)((𝐷‘(1r‘𝐴)) · (𝐸‘𝑎))))‘𝑑)))) | 
| 180 | 179 | ralrimivva 3202 | . . . . 5
⊢ (𝜑 → ∀𝑏 ∈ 𝐵 ∀𝑐 ∈ 𝐾 ∀𝑑 ∈ 𝐵 ∀𝑒 ∈ 𝑁 (((𝑏 ↾ ({𝑒} × 𝑁)) = ((({𝑒} × 𝑁) × {𝑐}) ∘f · (𝑑 ↾ ({𝑒} × 𝑁))) ∧ (𝑏 ↾ ((𝑁 ∖ {𝑒}) × 𝑁)) = (𝑑 ↾ ((𝑁 ∖ {𝑒}) × 𝑁))) → ((𝑎 ∈ 𝐵 ↦ ((𝐷‘𝑎)(-g‘𝑅)((𝐷‘(1r‘𝐴)) · (𝐸‘𝑎))))‘𝑏) = (𝑐 · ((𝑎 ∈ 𝐵 ↦ ((𝐷‘𝑎)(-g‘𝑅)((𝐷‘(1r‘𝐴)) · (𝐸‘𝑎))))‘𝑑)))) | 
| 181 |  | eqidd 2738 | . . . . . 6
⊢ (𝜑 → (𝑎 ∈ 𝐵 ↦ ((𝐷‘𝑎)(-g‘𝑅)((𝐷‘(1r‘𝐴)) · (𝐸‘𝑎)))) = (𝑎 ∈ 𝐵 ↦ ((𝐷‘𝑎)(-g‘𝑅)((𝐷‘(1r‘𝐴)) · (𝐸‘𝑎))))) | 
| 182 |  | fveq2 6906 | . . . . . . . 8
⊢ (𝑎 = (1r‘𝐴) → (𝐷‘𝑎) = (𝐷‘(1r‘𝐴))) | 
| 183 |  | fveq2 6906 | . . . . . . . . 9
⊢ (𝑎 = (1r‘𝐴) → (𝐸‘𝑎) = (𝐸‘(1r‘𝐴))) | 
| 184 | 183 | oveq2d 7447 | . . . . . . . 8
⊢ (𝑎 = (1r‘𝐴) → ((𝐷‘(1r‘𝐴)) · (𝐸‘𝑎)) = ((𝐷‘(1r‘𝐴)) · (𝐸‘(1r‘𝐴)))) | 
| 185 | 182, 184 | oveq12d 7449 | . . . . . . 7
⊢ (𝑎 = (1r‘𝐴) → ((𝐷‘𝑎)(-g‘𝑅)((𝐷‘(1r‘𝐴)) · (𝐸‘𝑎))) = ((𝐷‘(1r‘𝐴))(-g‘𝑅)((𝐷‘(1r‘𝐴)) · (𝐸‘(1r‘𝐴))))) | 
| 186 | 24, 1, 18, 5 | mdet1 22607 | . . . . . . . . . . . 12
⊢ ((𝑅 ∈ CRing ∧ 𝑁 ∈ Fin) → (𝐸‘(1r‘𝐴)) = 1 ) | 
| 187 | 23, 8, 186 | syl2anc 584 | . . . . . . . . . . 11
⊢ (𝜑 → (𝐸‘(1r‘𝐴)) = 1 ) | 
| 188 | 187 | oveq2d 7447 | . . . . . . . . . 10
⊢ (𝜑 → ((𝐷‘(1r‘𝐴)) · (𝐸‘(1r‘𝐴))) = ((𝐷‘(1r‘𝐴)) · 1 )) | 
| 189 | 3, 7, 5 | ringridm 20267 | . . . . . . . . . . 11
⊢ ((𝑅 ∈ Ring ∧ (𝐷‘(1r‘𝐴)) ∈ 𝐾) → ((𝐷‘(1r‘𝐴)) · 1 ) = (𝐷‘(1r‘𝐴))) | 
| 190 | 9, 21, 189 | syl2anc 584 | . . . . . . . . . 10
⊢ (𝜑 → ((𝐷‘(1r‘𝐴)) · 1 ) = (𝐷‘(1r‘𝐴))) | 
| 191 | 188, 190 | eqtrd 2777 | . . . . . . . . 9
⊢ (𝜑 → ((𝐷‘(1r‘𝐴)) · (𝐸‘(1r‘𝐴))) = (𝐷‘(1r‘𝐴))) | 
| 192 | 191 | oveq2d 7447 | . . . . . . . 8
⊢ (𝜑 → ((𝐷‘(1r‘𝐴))(-g‘𝑅)((𝐷‘(1r‘𝐴)) · (𝐸‘(1r‘𝐴)))) = ((𝐷‘(1r‘𝐴))(-g‘𝑅)(𝐷‘(1r‘𝐴)))) | 
| 193 | 3, 4, 30 | grpsubid 19042 | . . . . . . . . 9
⊢ ((𝑅 ∈ Grp ∧ (𝐷‘(1r‘𝐴)) ∈ 𝐾) → ((𝐷‘(1r‘𝐴))(-g‘𝑅)(𝐷‘(1r‘𝐴))) = 0 ) | 
| 194 | 11, 21, 193 | syl2anc 584 | . . . . . . . 8
⊢ (𝜑 → ((𝐷‘(1r‘𝐴))(-g‘𝑅)(𝐷‘(1r‘𝐴))) = 0 ) | 
| 195 | 192, 194 | eqtrd 2777 | . . . . . . 7
⊢ (𝜑 → ((𝐷‘(1r‘𝐴))(-g‘𝑅)((𝐷‘(1r‘𝐴)) · (𝐸‘(1r‘𝐴)))) = 0 ) | 
| 196 | 185, 195 | sylan9eqr 2799 | . . . . . 6
⊢ ((𝜑 ∧ 𝑎 = (1r‘𝐴)) → ((𝐷‘𝑎)(-g‘𝑅)((𝐷‘(1r‘𝐴)) · (𝐸‘𝑎))) = 0 ) | 
| 197 | 4 | fvexi 6920 | . . . . . . 7
⊢  0 ∈
V | 
| 198 | 197 | a1i 11 | . . . . . 6
⊢ (𝜑 → 0 ∈ V) | 
| 199 | 181, 196,
20, 198 | fvmptd 7023 | . . . . 5
⊢ (𝜑 → ((𝑎 ∈ 𝐵 ↦ ((𝐷‘𝑎)(-g‘𝑅)((𝐷‘(1r‘𝐴)) · (𝐸‘𝑎))))‘(1r‘𝐴)) = 0 ) | 
| 200 |  | eqid 2737 | . . . . 5
⊢ {𝑏 ∣ ∀𝑐 ∈ 𝐵 ∀𝑑 ∈ (𝑁 ↑m 𝑁)(∀𝑒 ∈ 𝑏 (𝑐‘𝑒) = if(𝑒 ∈ 𝑑, 1 , 0 ) → ((𝑎 ∈ 𝐵 ↦ ((𝐷‘𝑎)(-g‘𝑅)((𝐷‘(1r‘𝐴)) · (𝐸‘𝑎))))‘𝑐) = 0 )} = {𝑏 ∣ ∀𝑐 ∈ 𝐵 ∀𝑑 ∈ (𝑁 ↑m 𝑁)(∀𝑒 ∈ 𝑏 (𝑐‘𝑒) = if(𝑒 ∈ 𝑑, 1 , 0 ) → ((𝑎 ∈ 𝐵 ↦ ((𝐷‘𝑎)(-g‘𝑅)((𝐷‘(1r‘𝐴)) · (𝐸‘𝑎))))‘𝑐) = 0 )} | 
| 201 | 1, 2, 3, 4, 5, 6, 7, 8, 9, 33,
74, 136, 180, 199, 200 | mdetunilem9 22626 | . . . 4
⊢ (𝜑 → (𝑎 ∈ 𝐵 ↦ ((𝐷‘𝑎)(-g‘𝑅)((𝐷‘(1r‘𝐴)) · (𝐸‘𝑎)))) = (𝐵 × { 0 })) | 
| 202 | 201 | fveq1d 6908 | . . 3
⊢ (𝜑 → ((𝑎 ∈ 𝐵 ↦ ((𝐷‘𝑎)(-g‘𝑅)((𝐷‘(1r‘𝐴)) · (𝐸‘𝑎))))‘𝐹) = ((𝐵 × { 0 })‘𝐹)) | 
| 203 |  | fveq2 6906 | . . . . . 6
⊢ (𝑎 = 𝐹 → (𝐷‘𝑎) = (𝐷‘𝐹)) | 
| 204 |  | fveq2 6906 | . . . . . . 7
⊢ (𝑎 = 𝐹 → (𝐸‘𝑎) = (𝐸‘𝐹)) | 
| 205 | 204 | oveq2d 7447 | . . . . . 6
⊢ (𝑎 = 𝐹 → ((𝐷‘(1r‘𝐴)) · (𝐸‘𝑎)) = ((𝐷‘(1r‘𝐴)) · (𝐸‘𝐹))) | 
| 206 | 203, 205 | oveq12d 7449 | . . . . 5
⊢ (𝑎 = 𝐹 → ((𝐷‘𝑎)(-g‘𝑅)((𝐷‘(1r‘𝐴)) · (𝐸‘𝑎))) = ((𝐷‘𝐹)(-g‘𝑅)((𝐷‘(1r‘𝐴)) · (𝐸‘𝐹)))) | 
| 207 | 206 | adantl 481 | . . . 4
⊢ ((𝜑 ∧ 𝑎 = 𝐹) → ((𝐷‘𝑎)(-g‘𝑅)((𝐷‘(1r‘𝐴)) · (𝐸‘𝑎))) = ((𝐷‘𝐹)(-g‘𝑅)((𝐷‘(1r‘𝐴)) · (𝐸‘𝐹)))) | 
| 208 |  | mdetuni.f | . . . 4
⊢ (𝜑 → 𝐹 ∈ 𝐵) | 
| 209 |  | ovexd 7466 | . . . 4
⊢ (𝜑 → ((𝐷‘𝐹)(-g‘𝑅)((𝐷‘(1r‘𝐴)) · (𝐸‘𝐹))) ∈ V) | 
| 210 | 181, 207,
208, 209 | fvmptd 7023 | . . 3
⊢ (𝜑 → ((𝑎 ∈ 𝐵 ↦ ((𝐷‘𝑎)(-g‘𝑅)((𝐷‘(1r‘𝐴)) · (𝐸‘𝑎))))‘𝐹) = ((𝐷‘𝐹)(-g‘𝑅)((𝐷‘(1r‘𝐴)) · (𝐸‘𝐹)))) | 
| 211 | 197 | fvconst2 7224 | . . . 4
⊢ (𝐹 ∈ 𝐵 → ((𝐵 × { 0 })‘𝐹) = 0 ) | 
| 212 | 208, 211 | syl 17 | . . 3
⊢ (𝜑 → ((𝐵 × { 0 })‘𝐹) = 0 ) | 
| 213 | 202, 210,
212 | 3eqtr3d 2785 | . 2
⊢ (𝜑 → ((𝐷‘𝐹)(-g‘𝑅)((𝐷‘(1r‘𝐴)) · (𝐸‘𝐹))) = 0 ) | 
| 214 | 13, 208 | ffvelcdmd 7105 | . . 3
⊢ (𝜑 → (𝐷‘𝐹) ∈ 𝐾) | 
| 215 | 26, 208 | ffvelcdmd 7105 | . . . 4
⊢ (𝜑 → (𝐸‘𝐹) ∈ 𝐾) | 
| 216 | 3, 7 | ringcl 20247 | . . . 4
⊢ ((𝑅 ∈ Ring ∧ (𝐷‘(1r‘𝐴)) ∈ 𝐾 ∧ (𝐸‘𝐹) ∈ 𝐾) → ((𝐷‘(1r‘𝐴)) · (𝐸‘𝐹)) ∈ 𝐾) | 
| 217 | 9, 21, 215, 216 | syl3anc 1373 | . . 3
⊢ (𝜑 → ((𝐷‘(1r‘𝐴)) · (𝐸‘𝐹)) ∈ 𝐾) | 
| 218 | 3, 4, 30 | grpsubeq0 19044 | . . 3
⊢ ((𝑅 ∈ Grp ∧ (𝐷‘𝐹) ∈ 𝐾 ∧ ((𝐷‘(1r‘𝐴)) · (𝐸‘𝐹)) ∈ 𝐾) → (((𝐷‘𝐹)(-g‘𝑅)((𝐷‘(1r‘𝐴)) · (𝐸‘𝐹))) = 0 ↔ (𝐷‘𝐹) = ((𝐷‘(1r‘𝐴)) · (𝐸‘𝐹)))) | 
| 219 | 11, 214, 217, 218 | syl3anc 1373 | . 2
⊢ (𝜑 → (((𝐷‘𝐹)(-g‘𝑅)((𝐷‘(1r‘𝐴)) · (𝐸‘𝐹))) = 0 ↔ (𝐷‘𝐹) = ((𝐷‘(1r‘𝐴)) · (𝐸‘𝐹)))) | 
| 220 | 213, 219 | mpbid 232 | 1
⊢ (𝜑 → (𝐷‘𝐹) = ((𝐷‘(1r‘𝐴)) · (𝐸‘𝐹))) |