Step | Hyp | Ref
| Expression |
1 | | eqid 2740 |
. . 3
⊢ (𝑅 maMul 〈𝑁, 𝑁, 𝑁〉) = (𝑅 maMul 〈𝑁, 𝑁, 𝑁〉) |
2 | | eqid 2740 |
. . 3
⊢
(Base‘𝑅) =
(Base‘𝑅) |
3 | | eqid 2740 |
. . 3
⊢
(.r‘𝑅) = (.r‘𝑅) |
4 | | simpr 485 |
. . 3
⊢ ((𝑀 ∈ 𝐵 ∧ 𝑅 ∈ CRing) → 𝑅 ∈ CRing) |
5 | | madurid.a |
. . . . . 6
⊢ 𝐴 = (𝑁 Mat 𝑅) |
6 | | madurid.b |
. . . . . 6
⊢ 𝐵 = (Base‘𝐴) |
7 | 5, 6 | matrcl 21555 |
. . . . 5
⊢ (𝑀 ∈ 𝐵 → (𝑁 ∈ Fin ∧ 𝑅 ∈ V)) |
8 | 7 | simpld 495 |
. . . 4
⊢ (𝑀 ∈ 𝐵 → 𝑁 ∈ Fin) |
9 | 8 | adantr 481 |
. . 3
⊢ ((𝑀 ∈ 𝐵 ∧ 𝑅 ∈ CRing) → 𝑁 ∈ Fin) |
10 | 5, 2, 6 | matbas2i 21567 |
. . . 4
⊢ (𝑀 ∈ 𝐵 → 𝑀 ∈ ((Base‘𝑅) ↑m (𝑁 × 𝑁))) |
11 | 10 | adantr 481 |
. . 3
⊢ ((𝑀 ∈ 𝐵 ∧ 𝑅 ∈ CRing) → 𝑀 ∈ ((Base‘𝑅) ↑m (𝑁 × 𝑁))) |
12 | | madurid.j |
. . . . . . 7
⊢ 𝐽 = (𝑁 maAdju 𝑅) |
13 | 5, 12, 6 | maduf 21786 |
. . . . . 6
⊢ (𝑅 ∈ CRing → 𝐽:𝐵⟶𝐵) |
14 | 13 | adantl 482 |
. . . . 5
⊢ ((𝑀 ∈ 𝐵 ∧ 𝑅 ∈ CRing) → 𝐽:𝐵⟶𝐵) |
15 | | simpl 483 |
. . . . 5
⊢ ((𝑀 ∈ 𝐵 ∧ 𝑅 ∈ CRing) → 𝑀 ∈ 𝐵) |
16 | 14, 15 | ffvelrnd 6957 |
. . . 4
⊢ ((𝑀 ∈ 𝐵 ∧ 𝑅 ∈ CRing) → (𝐽‘𝑀) ∈ 𝐵) |
17 | 5, 2, 6 | matbas2i 21567 |
. . . 4
⊢ ((𝐽‘𝑀) ∈ 𝐵 → (𝐽‘𝑀) ∈ ((Base‘𝑅) ↑m (𝑁 × 𝑁))) |
18 | 16, 17 | syl 17 |
. . 3
⊢ ((𝑀 ∈ 𝐵 ∧ 𝑅 ∈ CRing) → (𝐽‘𝑀) ∈ ((Base‘𝑅) ↑m (𝑁 × 𝑁))) |
19 | 1, 2, 3, 4, 9, 9, 9, 11, 18 | mamuval 21531 |
. 2
⊢ ((𝑀 ∈ 𝐵 ∧ 𝑅 ∈ CRing) → (𝑀(𝑅 maMul 〈𝑁, 𝑁, 𝑁〉)(𝐽‘𝑀)) = (𝑎 ∈ 𝑁, 𝑏 ∈ 𝑁 ↦ (𝑅 Σg (𝑐 ∈ 𝑁 ↦ ((𝑎𝑀𝑐)(.r‘𝑅)(𝑐(𝐽‘𝑀)𝑏)))))) |
20 | 5, 1 | matmulr 21583 |
. . . . 5
⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) → (𝑅 maMul 〈𝑁, 𝑁, 𝑁〉) = (.r‘𝐴)) |
21 | 8, 20 | sylan 580 |
. . . 4
⊢ ((𝑀 ∈ 𝐵 ∧ 𝑅 ∈ CRing) → (𝑅 maMul 〈𝑁, 𝑁, 𝑁〉) = (.r‘𝐴)) |
22 | | madurid.t |
. . . 4
⊢ · =
(.r‘𝐴) |
23 | 21, 22 | eqtr4di 2798 |
. . 3
⊢ ((𝑀 ∈ 𝐵 ∧ 𝑅 ∈ CRing) → (𝑅 maMul 〈𝑁, 𝑁, 𝑁〉) = · ) |
24 | 23 | oveqd 7286 |
. 2
⊢ ((𝑀 ∈ 𝐵 ∧ 𝑅 ∈ CRing) → (𝑀(𝑅 maMul 〈𝑁, 𝑁, 𝑁〉)(𝐽‘𝑀)) = (𝑀 · (𝐽‘𝑀))) |
25 | | madurid.d |
. . . . . 6
⊢ 𝐷 = (𝑁 maDet 𝑅) |
26 | | simp1l 1196 |
. . . . . 6
⊢ (((𝑀 ∈ 𝐵 ∧ 𝑅 ∈ CRing) ∧ 𝑎 ∈ 𝑁 ∧ 𝑏 ∈ 𝑁) → 𝑀 ∈ 𝐵) |
27 | | simp1r 1197 |
. . . . . 6
⊢ (((𝑀 ∈ 𝐵 ∧ 𝑅 ∈ CRing) ∧ 𝑎 ∈ 𝑁 ∧ 𝑏 ∈ 𝑁) → 𝑅 ∈ CRing) |
28 | | elmapi 8618 |
. . . . . . . . . 10
⊢ (𝑀 ∈ ((Base‘𝑅) ↑m (𝑁 × 𝑁)) → 𝑀:(𝑁 × 𝑁)⟶(Base‘𝑅)) |
29 | 11, 28 | syl 17 |
. . . . . . . . 9
⊢ ((𝑀 ∈ 𝐵 ∧ 𝑅 ∈ CRing) → 𝑀:(𝑁 × 𝑁)⟶(Base‘𝑅)) |
30 | 29 | 3ad2ant1 1132 |
. . . . . . . 8
⊢ (((𝑀 ∈ 𝐵 ∧ 𝑅 ∈ CRing) ∧ 𝑎 ∈ 𝑁 ∧ 𝑏 ∈ 𝑁) → 𝑀:(𝑁 × 𝑁)⟶(Base‘𝑅)) |
31 | 30 | adantr 481 |
. . . . . . 7
⊢ ((((𝑀 ∈ 𝐵 ∧ 𝑅 ∈ CRing) ∧ 𝑎 ∈ 𝑁 ∧ 𝑏 ∈ 𝑁) ∧ 𝑐 ∈ 𝑁) → 𝑀:(𝑁 × 𝑁)⟶(Base‘𝑅)) |
32 | | simpl2 1191 |
. . . . . . 7
⊢ ((((𝑀 ∈ 𝐵 ∧ 𝑅 ∈ CRing) ∧ 𝑎 ∈ 𝑁 ∧ 𝑏 ∈ 𝑁) ∧ 𝑐 ∈ 𝑁) → 𝑎 ∈ 𝑁) |
33 | | simpr 485 |
. . . . . . 7
⊢ ((((𝑀 ∈ 𝐵 ∧ 𝑅 ∈ CRing) ∧ 𝑎 ∈ 𝑁 ∧ 𝑏 ∈ 𝑁) ∧ 𝑐 ∈ 𝑁) → 𝑐 ∈ 𝑁) |
34 | 31, 32, 33 | fovrnd 7436 |
. . . . . 6
⊢ ((((𝑀 ∈ 𝐵 ∧ 𝑅 ∈ CRing) ∧ 𝑎 ∈ 𝑁 ∧ 𝑏 ∈ 𝑁) ∧ 𝑐 ∈ 𝑁) → (𝑎𝑀𝑐) ∈ (Base‘𝑅)) |
35 | | simp3 1137 |
. . . . . 6
⊢ (((𝑀 ∈ 𝐵 ∧ 𝑅 ∈ CRing) ∧ 𝑎 ∈ 𝑁 ∧ 𝑏 ∈ 𝑁) → 𝑏 ∈ 𝑁) |
36 | 5, 12, 6, 25, 3, 2,
26, 27, 34, 35 | madugsum 21788 |
. . . . 5
⊢ (((𝑀 ∈ 𝐵 ∧ 𝑅 ∈ CRing) ∧ 𝑎 ∈ 𝑁 ∧ 𝑏 ∈ 𝑁) → (𝑅 Σg (𝑐 ∈ 𝑁 ↦ ((𝑎𝑀𝑐)(.r‘𝑅)(𝑐(𝐽‘𝑀)𝑏)))) = (𝐷‘(𝑑 ∈ 𝑁, 𝑐 ∈ 𝑁 ↦ if(𝑑 = 𝑏, (𝑎𝑀𝑐), (𝑑𝑀𝑐))))) |
37 | | iftrue 4471 |
. . . . . . . . 9
⊢ (𝑎 = 𝑏 → if(𝑎 = 𝑏, (𝐷‘𝑀), (0g‘𝑅)) = (𝐷‘𝑀)) |
38 | 37 | adantl 482 |
. . . . . . . 8
⊢ (((𝑀 ∈ 𝐵 ∧ 𝑅 ∈ CRing) ∧ 𝑎 = 𝑏) → if(𝑎 = 𝑏, (𝐷‘𝑀), (0g‘𝑅)) = (𝐷‘𝑀)) |
39 | 29 | ffnd 6598 |
. . . . . . . . . . . 12
⊢ ((𝑀 ∈ 𝐵 ∧ 𝑅 ∈ CRing) → 𝑀 Fn (𝑁 × 𝑁)) |
40 | | fnov 7397 |
. . . . . . . . . . . 12
⊢ (𝑀 Fn (𝑁 × 𝑁) ↔ 𝑀 = (𝑑 ∈ 𝑁, 𝑐 ∈ 𝑁 ↦ (𝑑𝑀𝑐))) |
41 | 39, 40 | sylib 217 |
. . . . . . . . . . 11
⊢ ((𝑀 ∈ 𝐵 ∧ 𝑅 ∈ CRing) → 𝑀 = (𝑑 ∈ 𝑁, 𝑐 ∈ 𝑁 ↦ (𝑑𝑀𝑐))) |
42 | 41 | adantr 481 |
. . . . . . . . . 10
⊢ (((𝑀 ∈ 𝐵 ∧ 𝑅 ∈ CRing) ∧ 𝑎 = 𝑏) → 𝑀 = (𝑑 ∈ 𝑁, 𝑐 ∈ 𝑁 ↦ (𝑑𝑀𝑐))) |
43 | | equtr2 2034 |
. . . . . . . . . . . . . . 15
⊢ ((𝑎 = 𝑏 ∧ 𝑑 = 𝑏) → 𝑎 = 𝑑) |
44 | 43 | oveq1d 7284 |
. . . . . . . . . . . . . 14
⊢ ((𝑎 = 𝑏 ∧ 𝑑 = 𝑏) → (𝑎𝑀𝑐) = (𝑑𝑀𝑐)) |
45 | 44 | ifeq1da 4496 |
. . . . . . . . . . . . 13
⊢ (𝑎 = 𝑏 → if(𝑑 = 𝑏, (𝑎𝑀𝑐), (𝑑𝑀𝑐)) = if(𝑑 = 𝑏, (𝑑𝑀𝑐), (𝑑𝑀𝑐))) |
46 | | ifid 4505 |
. . . . . . . . . . . . 13
⊢ if(𝑑 = 𝑏, (𝑑𝑀𝑐), (𝑑𝑀𝑐)) = (𝑑𝑀𝑐) |
47 | 45, 46 | eqtrdi 2796 |
. . . . . . . . . . . 12
⊢ (𝑎 = 𝑏 → if(𝑑 = 𝑏, (𝑎𝑀𝑐), (𝑑𝑀𝑐)) = (𝑑𝑀𝑐)) |
48 | 47 | adantl 482 |
. . . . . . . . . . 11
⊢ (((𝑀 ∈ 𝐵 ∧ 𝑅 ∈ CRing) ∧ 𝑎 = 𝑏) → if(𝑑 = 𝑏, (𝑎𝑀𝑐), (𝑑𝑀𝑐)) = (𝑑𝑀𝑐)) |
49 | 48 | mpoeq3dv 7346 |
. . . . . . . . . 10
⊢ (((𝑀 ∈ 𝐵 ∧ 𝑅 ∈ CRing) ∧ 𝑎 = 𝑏) → (𝑑 ∈ 𝑁, 𝑐 ∈ 𝑁 ↦ if(𝑑 = 𝑏, (𝑎𝑀𝑐), (𝑑𝑀𝑐))) = (𝑑 ∈ 𝑁, 𝑐 ∈ 𝑁 ↦ (𝑑𝑀𝑐))) |
50 | 42, 49 | eqtr4d 2783 |
. . . . . . . . 9
⊢ (((𝑀 ∈ 𝐵 ∧ 𝑅 ∈ CRing) ∧ 𝑎 = 𝑏) → 𝑀 = (𝑑 ∈ 𝑁, 𝑐 ∈ 𝑁 ↦ if(𝑑 = 𝑏, (𝑎𝑀𝑐), (𝑑𝑀𝑐)))) |
51 | 50 | fveq2d 6773 |
. . . . . . . 8
⊢ (((𝑀 ∈ 𝐵 ∧ 𝑅 ∈ CRing) ∧ 𝑎 = 𝑏) → (𝐷‘𝑀) = (𝐷‘(𝑑 ∈ 𝑁, 𝑐 ∈ 𝑁 ↦ if(𝑑 = 𝑏, (𝑎𝑀𝑐), (𝑑𝑀𝑐))))) |
52 | 38, 51 | eqtr2d 2781 |
. . . . . . 7
⊢ (((𝑀 ∈ 𝐵 ∧ 𝑅 ∈ CRing) ∧ 𝑎 = 𝑏) → (𝐷‘(𝑑 ∈ 𝑁, 𝑐 ∈ 𝑁 ↦ if(𝑑 = 𝑏, (𝑎𝑀𝑐), (𝑑𝑀𝑐)))) = if(𝑎 = 𝑏, (𝐷‘𝑀), (0g‘𝑅))) |
53 | 52 | 3ad2antl1 1184 |
. . . . . 6
⊢ ((((𝑀 ∈ 𝐵 ∧ 𝑅 ∈ CRing) ∧ 𝑎 ∈ 𝑁 ∧ 𝑏 ∈ 𝑁) ∧ 𝑎 = 𝑏) → (𝐷‘(𝑑 ∈ 𝑁, 𝑐 ∈ 𝑁 ↦ if(𝑑 = 𝑏, (𝑎𝑀𝑐), (𝑑𝑀𝑐)))) = if(𝑎 = 𝑏, (𝐷‘𝑀), (0g‘𝑅))) |
54 | | eqid 2740 |
. . . . . . . 8
⊢
(0g‘𝑅) = (0g‘𝑅) |
55 | | simpl1r 1224 |
. . . . . . . 8
⊢ ((((𝑀 ∈ 𝐵 ∧ 𝑅 ∈ CRing) ∧ 𝑎 ∈ 𝑁 ∧ 𝑏 ∈ 𝑁) ∧ ¬ 𝑎 = 𝑏) → 𝑅 ∈ CRing) |
56 | 9 | 3ad2ant1 1132 |
. . . . . . . . 9
⊢ (((𝑀 ∈ 𝐵 ∧ 𝑅 ∈ CRing) ∧ 𝑎 ∈ 𝑁 ∧ 𝑏 ∈ 𝑁) → 𝑁 ∈ Fin) |
57 | 56 | adantr 481 |
. . . . . . . 8
⊢ ((((𝑀 ∈ 𝐵 ∧ 𝑅 ∈ CRing) ∧ 𝑎 ∈ 𝑁 ∧ 𝑏 ∈ 𝑁) ∧ ¬ 𝑎 = 𝑏) → 𝑁 ∈ Fin) |
58 | 30 | ad2antrr 723 |
. . . . . . . . 9
⊢
(((((𝑀 ∈ 𝐵 ∧ 𝑅 ∈ CRing) ∧ 𝑎 ∈ 𝑁 ∧ 𝑏 ∈ 𝑁) ∧ ¬ 𝑎 = 𝑏) ∧ 𝑐 ∈ 𝑁) → 𝑀:(𝑁 × 𝑁)⟶(Base‘𝑅)) |
59 | | simpll2 1212 |
. . . . . . . . 9
⊢
(((((𝑀 ∈ 𝐵 ∧ 𝑅 ∈ CRing) ∧ 𝑎 ∈ 𝑁 ∧ 𝑏 ∈ 𝑁) ∧ ¬ 𝑎 = 𝑏) ∧ 𝑐 ∈ 𝑁) → 𝑎 ∈ 𝑁) |
60 | | simpr 485 |
. . . . . . . . 9
⊢
(((((𝑀 ∈ 𝐵 ∧ 𝑅 ∈ CRing) ∧ 𝑎 ∈ 𝑁 ∧ 𝑏 ∈ 𝑁) ∧ ¬ 𝑎 = 𝑏) ∧ 𝑐 ∈ 𝑁) → 𝑐 ∈ 𝑁) |
61 | 58, 59, 60 | fovrnd 7436 |
. . . . . . . 8
⊢
(((((𝑀 ∈ 𝐵 ∧ 𝑅 ∈ CRing) ∧ 𝑎 ∈ 𝑁 ∧ 𝑏 ∈ 𝑁) ∧ ¬ 𝑎 = 𝑏) ∧ 𝑐 ∈ 𝑁) → (𝑎𝑀𝑐) ∈ (Base‘𝑅)) |
62 | 30 | adantr 481 |
. . . . . . . . . 10
⊢ ((((𝑀 ∈ 𝐵 ∧ 𝑅 ∈ CRing) ∧ 𝑎 ∈ 𝑁 ∧ 𝑏 ∈ 𝑁) ∧ ¬ 𝑎 = 𝑏) → 𝑀:(𝑁 × 𝑁)⟶(Base‘𝑅)) |
63 | 62 | fovrnda 7435 |
. . . . . . . . 9
⊢
(((((𝑀 ∈ 𝐵 ∧ 𝑅 ∈ CRing) ∧ 𝑎 ∈ 𝑁 ∧ 𝑏 ∈ 𝑁) ∧ ¬ 𝑎 = 𝑏) ∧ (𝑑 ∈ 𝑁 ∧ 𝑐 ∈ 𝑁)) → (𝑑𝑀𝑐) ∈ (Base‘𝑅)) |
64 | 63 | 3impb 1114 |
. . . . . . . 8
⊢
(((((𝑀 ∈ 𝐵 ∧ 𝑅 ∈ CRing) ∧ 𝑎 ∈ 𝑁 ∧ 𝑏 ∈ 𝑁) ∧ ¬ 𝑎 = 𝑏) ∧ 𝑑 ∈ 𝑁 ∧ 𝑐 ∈ 𝑁) → (𝑑𝑀𝑐) ∈ (Base‘𝑅)) |
65 | | simpl3 1192 |
. . . . . . . 8
⊢ ((((𝑀 ∈ 𝐵 ∧ 𝑅 ∈ CRing) ∧ 𝑎 ∈ 𝑁 ∧ 𝑏 ∈ 𝑁) ∧ ¬ 𝑎 = 𝑏) → 𝑏 ∈ 𝑁) |
66 | | simpl2 1191 |
. . . . . . . 8
⊢ ((((𝑀 ∈ 𝐵 ∧ 𝑅 ∈ CRing) ∧ 𝑎 ∈ 𝑁 ∧ 𝑏 ∈ 𝑁) ∧ ¬ 𝑎 = 𝑏) → 𝑎 ∈ 𝑁) |
67 | | neqne 2953 |
. . . . . . . . . 10
⊢ (¬
𝑎 = 𝑏 → 𝑎 ≠ 𝑏) |
68 | 67 | necomd 3001 |
. . . . . . . . 9
⊢ (¬
𝑎 = 𝑏 → 𝑏 ≠ 𝑎) |
69 | 68 | adantl 482 |
. . . . . . . 8
⊢ ((((𝑀 ∈ 𝐵 ∧ 𝑅 ∈ CRing) ∧ 𝑎 ∈ 𝑁 ∧ 𝑏 ∈ 𝑁) ∧ ¬ 𝑎 = 𝑏) → 𝑏 ≠ 𝑎) |
70 | 25, 2, 54, 55, 57, 61, 64, 65, 66, 69 | mdetralt2 21754 |
. . . . . . 7
⊢ ((((𝑀 ∈ 𝐵 ∧ 𝑅 ∈ CRing) ∧ 𝑎 ∈ 𝑁 ∧ 𝑏 ∈ 𝑁) ∧ ¬ 𝑎 = 𝑏) → (𝐷‘(𝑑 ∈ 𝑁, 𝑐 ∈ 𝑁 ↦ if(𝑑 = 𝑏, (𝑎𝑀𝑐), if(𝑑 = 𝑎, (𝑎𝑀𝑐), (𝑑𝑀𝑐))))) = (0g‘𝑅)) |
71 | | ifid 4505 |
. . . . . . . . . . 11
⊢ if(𝑑 = 𝑎, (𝑑𝑀𝑐), (𝑑𝑀𝑐)) = (𝑑𝑀𝑐) |
72 | | oveq1 7276 |
. . . . . . . . . . . . 13
⊢ (𝑑 = 𝑎 → (𝑑𝑀𝑐) = (𝑎𝑀𝑐)) |
73 | 72 | adantl 482 |
. . . . . . . . . . . 12
⊢
(((((𝑀 ∈ 𝐵 ∧ 𝑅 ∈ CRing) ∧ 𝑎 ∈ 𝑁 ∧ 𝑏 ∈ 𝑁) ∧ ¬ 𝑎 = 𝑏) ∧ 𝑑 = 𝑎) → (𝑑𝑀𝑐) = (𝑎𝑀𝑐)) |
74 | 73 | ifeq1da 4496 |
. . . . . . . . . . 11
⊢ ((((𝑀 ∈ 𝐵 ∧ 𝑅 ∈ CRing) ∧ 𝑎 ∈ 𝑁 ∧ 𝑏 ∈ 𝑁) ∧ ¬ 𝑎 = 𝑏) → if(𝑑 = 𝑎, (𝑑𝑀𝑐), (𝑑𝑀𝑐)) = if(𝑑 = 𝑎, (𝑎𝑀𝑐), (𝑑𝑀𝑐))) |
75 | 71, 74 | eqtr3id 2794 |
. . . . . . . . . 10
⊢ ((((𝑀 ∈ 𝐵 ∧ 𝑅 ∈ CRing) ∧ 𝑎 ∈ 𝑁 ∧ 𝑏 ∈ 𝑁) ∧ ¬ 𝑎 = 𝑏) → (𝑑𝑀𝑐) = if(𝑑 = 𝑎, (𝑎𝑀𝑐), (𝑑𝑀𝑐))) |
76 | 75 | ifeq2d 4485 |
. . . . . . . . 9
⊢ ((((𝑀 ∈ 𝐵 ∧ 𝑅 ∈ CRing) ∧ 𝑎 ∈ 𝑁 ∧ 𝑏 ∈ 𝑁) ∧ ¬ 𝑎 = 𝑏) → if(𝑑 = 𝑏, (𝑎𝑀𝑐), (𝑑𝑀𝑐)) = if(𝑑 = 𝑏, (𝑎𝑀𝑐), if(𝑑 = 𝑎, (𝑎𝑀𝑐), (𝑑𝑀𝑐)))) |
77 | 76 | mpoeq3dv 7346 |
. . . . . . . 8
⊢ ((((𝑀 ∈ 𝐵 ∧ 𝑅 ∈ CRing) ∧ 𝑎 ∈ 𝑁 ∧ 𝑏 ∈ 𝑁) ∧ ¬ 𝑎 = 𝑏) → (𝑑 ∈ 𝑁, 𝑐 ∈ 𝑁 ↦ if(𝑑 = 𝑏, (𝑎𝑀𝑐), (𝑑𝑀𝑐))) = (𝑑 ∈ 𝑁, 𝑐 ∈ 𝑁 ↦ if(𝑑 = 𝑏, (𝑎𝑀𝑐), if(𝑑 = 𝑎, (𝑎𝑀𝑐), (𝑑𝑀𝑐))))) |
78 | 77 | fveq2d 6773 |
. . . . . . 7
⊢ ((((𝑀 ∈ 𝐵 ∧ 𝑅 ∈ CRing) ∧ 𝑎 ∈ 𝑁 ∧ 𝑏 ∈ 𝑁) ∧ ¬ 𝑎 = 𝑏) → (𝐷‘(𝑑 ∈ 𝑁, 𝑐 ∈ 𝑁 ↦ if(𝑑 = 𝑏, (𝑎𝑀𝑐), (𝑑𝑀𝑐)))) = (𝐷‘(𝑑 ∈ 𝑁, 𝑐 ∈ 𝑁 ↦ if(𝑑 = 𝑏, (𝑎𝑀𝑐), if(𝑑 = 𝑎, (𝑎𝑀𝑐), (𝑑𝑀𝑐)))))) |
79 | | iffalse 4474 |
. . . . . . . 8
⊢ (¬
𝑎 = 𝑏 → if(𝑎 = 𝑏, (𝐷‘𝑀), (0g‘𝑅)) = (0g‘𝑅)) |
80 | 79 | adantl 482 |
. . . . . . 7
⊢ ((((𝑀 ∈ 𝐵 ∧ 𝑅 ∈ CRing) ∧ 𝑎 ∈ 𝑁 ∧ 𝑏 ∈ 𝑁) ∧ ¬ 𝑎 = 𝑏) → if(𝑎 = 𝑏, (𝐷‘𝑀), (0g‘𝑅)) = (0g‘𝑅)) |
81 | 70, 78, 80 | 3eqtr4d 2790 |
. . . . . 6
⊢ ((((𝑀 ∈ 𝐵 ∧ 𝑅 ∈ CRing) ∧ 𝑎 ∈ 𝑁 ∧ 𝑏 ∈ 𝑁) ∧ ¬ 𝑎 = 𝑏) → (𝐷‘(𝑑 ∈ 𝑁, 𝑐 ∈ 𝑁 ↦ if(𝑑 = 𝑏, (𝑎𝑀𝑐), (𝑑𝑀𝑐)))) = if(𝑎 = 𝑏, (𝐷‘𝑀), (0g‘𝑅))) |
82 | 53, 81 | pm2.61dan 810 |
. . . . 5
⊢ (((𝑀 ∈ 𝐵 ∧ 𝑅 ∈ CRing) ∧ 𝑎 ∈ 𝑁 ∧ 𝑏 ∈ 𝑁) → (𝐷‘(𝑑 ∈ 𝑁, 𝑐 ∈ 𝑁 ↦ if(𝑑 = 𝑏, (𝑎𝑀𝑐), (𝑑𝑀𝑐)))) = if(𝑎 = 𝑏, (𝐷‘𝑀), (0g‘𝑅))) |
83 | 36, 82 | eqtrd 2780 |
. . . 4
⊢ (((𝑀 ∈ 𝐵 ∧ 𝑅 ∈ CRing) ∧ 𝑎 ∈ 𝑁 ∧ 𝑏 ∈ 𝑁) → (𝑅 Σg (𝑐 ∈ 𝑁 ↦ ((𝑎𝑀𝑐)(.r‘𝑅)(𝑐(𝐽‘𝑀)𝑏)))) = if(𝑎 = 𝑏, (𝐷‘𝑀), (0g‘𝑅))) |
84 | 83 | mpoeq3dva 7344 |
. . 3
⊢ ((𝑀 ∈ 𝐵 ∧ 𝑅 ∈ CRing) → (𝑎 ∈ 𝑁, 𝑏 ∈ 𝑁 ↦ (𝑅 Σg (𝑐 ∈ 𝑁 ↦ ((𝑎𝑀𝑐)(.r‘𝑅)(𝑐(𝐽‘𝑀)𝑏))))) = (𝑎 ∈ 𝑁, 𝑏 ∈ 𝑁 ↦ if(𝑎 = 𝑏, (𝐷‘𝑀), (0g‘𝑅)))) |
85 | | madurid.i |
. . . . 5
⊢ 1 =
(1r‘𝐴) |
86 | 85 | oveq2i 7280 |
. . . 4
⊢ ((𝐷‘𝑀) ∙ 1 ) = ((𝐷‘𝑀) ∙
(1r‘𝐴)) |
87 | | crngring 19791 |
. . . . . 6
⊢ (𝑅 ∈ CRing → 𝑅 ∈ Ring) |
88 | 87 | adantl 482 |
. . . . 5
⊢ ((𝑀 ∈ 𝐵 ∧ 𝑅 ∈ CRing) → 𝑅 ∈ Ring) |
89 | 25, 5, 6, 2 | mdetf 21740 |
. . . . . . 7
⊢ (𝑅 ∈ CRing → 𝐷:𝐵⟶(Base‘𝑅)) |
90 | 89 | adantl 482 |
. . . . . 6
⊢ ((𝑀 ∈ 𝐵 ∧ 𝑅 ∈ CRing) → 𝐷:𝐵⟶(Base‘𝑅)) |
91 | 90, 15 | ffvelrnd 6957 |
. . . . 5
⊢ ((𝑀 ∈ 𝐵 ∧ 𝑅 ∈ CRing) → (𝐷‘𝑀) ∈ (Base‘𝑅)) |
92 | | madurid.s |
. . . . . 6
⊢ ∙ = (
·𝑠 ‘𝐴) |
93 | 5, 2, 92, 54 | matsc 21595 |
. . . . 5
⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ (𝐷‘𝑀) ∈ (Base‘𝑅)) → ((𝐷‘𝑀) ∙
(1r‘𝐴)) =
(𝑎 ∈ 𝑁, 𝑏 ∈ 𝑁 ↦ if(𝑎 = 𝑏, (𝐷‘𝑀), (0g‘𝑅)))) |
94 | 9, 88, 91, 93 | syl3anc 1370 |
. . . 4
⊢ ((𝑀 ∈ 𝐵 ∧ 𝑅 ∈ CRing) → ((𝐷‘𝑀) ∙
(1r‘𝐴)) =
(𝑎 ∈ 𝑁, 𝑏 ∈ 𝑁 ↦ if(𝑎 = 𝑏, (𝐷‘𝑀), (0g‘𝑅)))) |
95 | 86, 94 | eqtrid 2792 |
. . 3
⊢ ((𝑀 ∈ 𝐵 ∧ 𝑅 ∈ CRing) → ((𝐷‘𝑀) ∙ 1 ) = (𝑎 ∈ 𝑁, 𝑏 ∈ 𝑁 ↦ if(𝑎 = 𝑏, (𝐷‘𝑀), (0g‘𝑅)))) |
96 | 84, 95 | eqtr4d 2783 |
. 2
⊢ ((𝑀 ∈ 𝐵 ∧ 𝑅 ∈ CRing) → (𝑎 ∈ 𝑁, 𝑏 ∈ 𝑁 ↦ (𝑅 Σg (𝑐 ∈ 𝑁 ↦ ((𝑎𝑀𝑐)(.r‘𝑅)(𝑐(𝐽‘𝑀)𝑏))))) = ((𝐷‘𝑀) ∙ 1 )) |
97 | 19, 24, 96 | 3eqtr3d 2788 |
1
⊢ ((𝑀 ∈ 𝐵 ∧ 𝑅 ∈ CRing) → (𝑀 · (𝐽‘𝑀)) = ((𝐷‘𝑀) ∙ 1 )) |