Step | Hyp | Ref
| Expression |
1 | | simprl 768 |
. . . 4
⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝐶 ∈ 𝐾 ∧ 𝑀 ∈ 𝐷)) → 𝐶 ∈ 𝐾) |
2 | | dmatscmcl.a |
. . . . . . . 8
⊢ 𝐴 = (𝑁 Mat 𝑅) |
3 | | dmatscmcl.b |
. . . . . . . 8
⊢ 𝐵 = (Base‘𝐴) |
4 | | eqid 2740 |
. . . . . . . 8
⊢
(0g‘𝑅) = (0g‘𝑅) |
5 | | dmatscmcl.d |
. . . . . . . 8
⊢ 𝐷 = (𝑁 DMat 𝑅) |
6 | 2, 3, 4, 5 | dmatmat 21639 |
. . . . . . 7
⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → (𝑀 ∈ 𝐷 → 𝑀 ∈ 𝐵)) |
7 | 6 | com12 32 |
. . . . . 6
⊢ (𝑀 ∈ 𝐷 → ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝑀 ∈ 𝐵)) |
8 | 7 | adantl 482 |
. . . . 5
⊢ ((𝐶 ∈ 𝐾 ∧ 𝑀 ∈ 𝐷) → ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝑀 ∈ 𝐵)) |
9 | 8 | impcom 408 |
. . . 4
⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝐶 ∈ 𝐾 ∧ 𝑀 ∈ 𝐷)) → 𝑀 ∈ 𝐵) |
10 | 1, 9 | jca 512 |
. . 3
⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝐶 ∈ 𝐾 ∧ 𝑀 ∈ 𝐷)) → (𝐶 ∈ 𝐾 ∧ 𝑀 ∈ 𝐵)) |
11 | | dmatscmcl.k |
. . . 4
⊢ 𝐾 = (Base‘𝑅) |
12 | | dmatscmcl.s |
. . . 4
⊢ ∗ = (
·𝑠 ‘𝐴) |
13 | 11, 2, 3, 12 | matvscl 21576 |
. . 3
⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝐶 ∈ 𝐾 ∧ 𝑀 ∈ 𝐵)) → (𝐶 ∗ 𝑀) ∈ 𝐵) |
14 | 10, 13 | syldan 591 |
. 2
⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝐶 ∈ 𝐾 ∧ 𝑀 ∈ 𝐷)) → (𝐶 ∗ 𝑀) ∈ 𝐵) |
15 | 2, 3, 4, 5 | dmatel 21638 |
. . . . 5
⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → (𝑀 ∈ 𝐷 ↔ (𝑀 ∈ 𝐵 ∧ ∀𝑖 ∈ 𝑁 ∀𝑗 ∈ 𝑁 (𝑖 ≠ 𝑗 → (𝑖𝑀𝑗) = (0g‘𝑅))))) |
16 | 15 | adantr 481 |
. . . 4
⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝐶 ∈ 𝐾) → (𝑀 ∈ 𝐷 ↔ (𝑀 ∈ 𝐵 ∧ ∀𝑖 ∈ 𝑁 ∀𝑗 ∈ 𝑁 (𝑖 ≠ 𝑗 → (𝑖𝑀𝑗) = (0g‘𝑅))))) |
17 | | simp-4r 781 |
. . . . . . . . . . . 12
⊢
(((((𝑁 ∈ Fin
∧ 𝑅 ∈ Ring) ∧
𝐶 ∈ 𝐾) ∧ 𝑀 ∈ 𝐵) ∧ (𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁)) → 𝑅 ∈ Ring) |
18 | | simpr 485 |
. . . . . . . . . . . . . 14
⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝐶 ∈ 𝐾) → 𝐶 ∈ 𝐾) |
19 | 18 | anim1i 615 |
. . . . . . . . . . . . 13
⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝐶 ∈ 𝐾) ∧ 𝑀 ∈ 𝐵) → (𝐶 ∈ 𝐾 ∧ 𝑀 ∈ 𝐵)) |
20 | 19 | adantr 481 |
. . . . . . . . . . . 12
⊢
(((((𝑁 ∈ Fin
∧ 𝑅 ∈ Ring) ∧
𝐶 ∈ 𝐾) ∧ 𝑀 ∈ 𝐵) ∧ (𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁)) → (𝐶 ∈ 𝐾 ∧ 𝑀 ∈ 𝐵)) |
21 | | simpr 485 |
. . . . . . . . . . . 12
⊢
(((((𝑁 ∈ Fin
∧ 𝑅 ∈ Ring) ∧
𝐶 ∈ 𝐾) ∧ 𝑀 ∈ 𝐵) ∧ (𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁)) → (𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁)) |
22 | 17, 20, 21 | 3jca 1127 |
. . . . . . . . . . 11
⊢
(((((𝑁 ∈ Fin
∧ 𝑅 ∈ Ring) ∧
𝐶 ∈ 𝐾) ∧ 𝑀 ∈ 𝐵) ∧ (𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁)) → (𝑅 ∈ Ring ∧ (𝐶 ∈ 𝐾 ∧ 𝑀 ∈ 𝐵) ∧ (𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁))) |
23 | 22 | adantr 481 |
. . . . . . . . . 10
⊢
((((((𝑁 ∈ Fin
∧ 𝑅 ∈ Ring) ∧
𝐶 ∈ 𝐾) ∧ 𝑀 ∈ 𝐵) ∧ (𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁)) ∧ (𝑖𝑀𝑗) = (0g‘𝑅)) → (𝑅 ∈ Ring ∧ (𝐶 ∈ 𝐾 ∧ 𝑀 ∈ 𝐵) ∧ (𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁))) |
24 | | eqid 2740 |
. . . . . . . . . . 11
⊢
(.r‘𝑅) = (.r‘𝑅) |
25 | 2, 3, 11, 12, 24 | matvscacell 21581 |
. . . . . . . . . 10
⊢ ((𝑅 ∈ Ring ∧ (𝐶 ∈ 𝐾 ∧ 𝑀 ∈ 𝐵) ∧ (𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁)) → (𝑖(𝐶 ∗ 𝑀)𝑗) = (𝐶(.r‘𝑅)(𝑖𝑀𝑗))) |
26 | 23, 25 | syl 17 |
. . . . . . . . 9
⊢
((((((𝑁 ∈ Fin
∧ 𝑅 ∈ Ring) ∧
𝐶 ∈ 𝐾) ∧ 𝑀 ∈ 𝐵) ∧ (𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁)) ∧ (𝑖𝑀𝑗) = (0g‘𝑅)) → (𝑖(𝐶 ∗ 𝑀)𝑗) = (𝐶(.r‘𝑅)(𝑖𝑀𝑗))) |
27 | | oveq2 7277 |
. . . . . . . . . 10
⊢ ((𝑖𝑀𝑗) = (0g‘𝑅) → (𝐶(.r‘𝑅)(𝑖𝑀𝑗)) = (𝐶(.r‘𝑅)(0g‘𝑅))) |
28 | 27 | adantl 482 |
. . . . . . . . 9
⊢
((((((𝑁 ∈ Fin
∧ 𝑅 ∈ Ring) ∧
𝐶 ∈ 𝐾) ∧ 𝑀 ∈ 𝐵) ∧ (𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁)) ∧ (𝑖𝑀𝑗) = (0g‘𝑅)) → (𝐶(.r‘𝑅)(𝑖𝑀𝑗)) = (𝐶(.r‘𝑅)(0g‘𝑅))) |
29 | 11, 24, 4 | ringrz 19823 |
. . . . . . . . . . 11
⊢ ((𝑅 ∈ Ring ∧ 𝐶 ∈ 𝐾) → (𝐶(.r‘𝑅)(0g‘𝑅)) = (0g‘𝑅)) |
30 | 29 | ad5ant23 757 |
. . . . . . . . . 10
⊢
(((((𝑁 ∈ Fin
∧ 𝑅 ∈ Ring) ∧
𝐶 ∈ 𝐾) ∧ 𝑀 ∈ 𝐵) ∧ (𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁)) → (𝐶(.r‘𝑅)(0g‘𝑅)) = (0g‘𝑅)) |
31 | 30 | adantr 481 |
. . . . . . . . 9
⊢
((((((𝑁 ∈ Fin
∧ 𝑅 ∈ Ring) ∧
𝐶 ∈ 𝐾) ∧ 𝑀 ∈ 𝐵) ∧ (𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁)) ∧ (𝑖𝑀𝑗) = (0g‘𝑅)) → (𝐶(.r‘𝑅)(0g‘𝑅)) = (0g‘𝑅)) |
32 | 26, 28, 31 | 3eqtrd 2784 |
. . . . . . . 8
⊢
((((((𝑁 ∈ Fin
∧ 𝑅 ∈ Ring) ∧
𝐶 ∈ 𝐾) ∧ 𝑀 ∈ 𝐵) ∧ (𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁)) ∧ (𝑖𝑀𝑗) = (0g‘𝑅)) → (𝑖(𝐶 ∗ 𝑀)𝑗) = (0g‘𝑅)) |
33 | 32 | ex 413 |
. . . . . . 7
⊢
(((((𝑁 ∈ Fin
∧ 𝑅 ∈ Ring) ∧
𝐶 ∈ 𝐾) ∧ 𝑀 ∈ 𝐵) ∧ (𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁)) → ((𝑖𝑀𝑗) = (0g‘𝑅) → (𝑖(𝐶 ∗ 𝑀)𝑗) = (0g‘𝑅))) |
34 | 33 | imim2d 57 |
. . . . . 6
⊢
(((((𝑁 ∈ Fin
∧ 𝑅 ∈ Ring) ∧
𝐶 ∈ 𝐾) ∧ 𝑀 ∈ 𝐵) ∧ (𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁)) → ((𝑖 ≠ 𝑗 → (𝑖𝑀𝑗) = (0g‘𝑅)) → (𝑖 ≠ 𝑗 → (𝑖(𝐶 ∗ 𝑀)𝑗) = (0g‘𝑅)))) |
35 | 34 | ralimdvva 3107 |
. . . . 5
⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝐶 ∈ 𝐾) ∧ 𝑀 ∈ 𝐵) → (∀𝑖 ∈ 𝑁 ∀𝑗 ∈ 𝑁 (𝑖 ≠ 𝑗 → (𝑖𝑀𝑗) = (0g‘𝑅)) → ∀𝑖 ∈ 𝑁 ∀𝑗 ∈ 𝑁 (𝑖 ≠ 𝑗 → (𝑖(𝐶 ∗ 𝑀)𝑗) = (0g‘𝑅)))) |
36 | 35 | expimpd 454 |
. . . 4
⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝐶 ∈ 𝐾) → ((𝑀 ∈ 𝐵 ∧ ∀𝑖 ∈ 𝑁 ∀𝑗 ∈ 𝑁 (𝑖 ≠ 𝑗 → (𝑖𝑀𝑗) = (0g‘𝑅))) → ∀𝑖 ∈ 𝑁 ∀𝑗 ∈ 𝑁 (𝑖 ≠ 𝑗 → (𝑖(𝐶 ∗ 𝑀)𝑗) = (0g‘𝑅)))) |
37 | 16, 36 | sylbid 239 |
. . 3
⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝐶 ∈ 𝐾) → (𝑀 ∈ 𝐷 → ∀𝑖 ∈ 𝑁 ∀𝑗 ∈ 𝑁 (𝑖 ≠ 𝑗 → (𝑖(𝐶 ∗ 𝑀)𝑗) = (0g‘𝑅)))) |
38 | 37 | impr 455 |
. 2
⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝐶 ∈ 𝐾 ∧ 𝑀 ∈ 𝐷)) → ∀𝑖 ∈ 𝑁 ∀𝑗 ∈ 𝑁 (𝑖 ≠ 𝑗 → (𝑖(𝐶 ∗ 𝑀)𝑗) = (0g‘𝑅))) |
39 | 2, 3, 4, 5 | dmatel 21638 |
. . 3
⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → ((𝐶 ∗ 𝑀) ∈ 𝐷 ↔ ((𝐶 ∗ 𝑀) ∈ 𝐵 ∧ ∀𝑖 ∈ 𝑁 ∀𝑗 ∈ 𝑁 (𝑖 ≠ 𝑗 → (𝑖(𝐶 ∗ 𝑀)𝑗) = (0g‘𝑅))))) |
40 | 39 | adantr 481 |
. 2
⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝐶 ∈ 𝐾 ∧ 𝑀 ∈ 𝐷)) → ((𝐶 ∗ 𝑀) ∈ 𝐷 ↔ ((𝐶 ∗ 𝑀) ∈ 𝐵 ∧ ∀𝑖 ∈ 𝑁 ∀𝑗 ∈ 𝑁 (𝑖 ≠ 𝑗 → (𝑖(𝐶 ∗ 𝑀)𝑗) = (0g‘𝑅))))) |
41 | 14, 38, 40 | mpbir2and 710 |
1
⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝐶 ∈ 𝐾 ∧ 𝑀 ∈ 𝐷)) → (𝐶 ∗ 𝑀) ∈ 𝐷) |