Step | Hyp | Ref
| Expression |
1 | | simprl 790 |
. . . . . 6
⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑆 ∈ 𝐵 ∧ 𝑇 ∈ 𝐵)) → 𝑆 ∈ 𝐵) |
2 | | scmatscmide.1 |
. . . . . . . 8
⊢ 1 =
(1r‘𝐴) |
3 | | scmatscmide.a |
. . . . . . . . 9
⊢ 𝐴 = (𝑁 Mat 𝑅) |
4 | | eqid 2610 |
. . . . . . . . 9
⊢
(Base‘𝐴) =
(Base‘𝐴) |
5 | | scmatscmide.0 |
. . . . . . . . 9
⊢ 0 =
(0g‘𝑅) |
6 | | eqid 2610 |
. . . . . . . . 9
⊢ (𝑁 DMat 𝑅) = (𝑁 DMat 𝑅) |
7 | 3, 4, 5, 6 | dmatid 20120 |
. . . . . . . 8
⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) →
(1r‘𝐴)
∈ (𝑁 DMat 𝑅)) |
8 | 2, 7 | syl5eqel 2692 |
. . . . . . 7
⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 1 ∈ (𝑁 DMat 𝑅)) |
9 | 8 | adantr 480 |
. . . . . 6
⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑆 ∈ 𝐵 ∧ 𝑇 ∈ 𝐵)) → 1 ∈ (𝑁 DMat 𝑅)) |
10 | 1, 9 | jca 553 |
. . . . 5
⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑆 ∈ 𝐵 ∧ 𝑇 ∈ 𝐵)) → (𝑆 ∈ 𝐵 ∧ 1 ∈ (𝑁 DMat 𝑅))) |
11 | | scmatscmide.b |
. . . . . 6
⊢ 𝐵 = (Base‘𝑅) |
12 | | scmatscmide.m |
. . . . . 6
⊢ ∗ = (
·𝑠 ‘𝐴) |
13 | 11, 3, 4, 12, 6 | dmatscmcl 20128 |
. . . . 5
⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑆 ∈ 𝐵 ∧ 1 ∈ (𝑁 DMat 𝑅))) → (𝑆 ∗ 1 ) ∈ (𝑁 DMat 𝑅)) |
14 | 10, 13 | syldan 486 |
. . . 4
⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑆 ∈ 𝐵 ∧ 𝑇 ∈ 𝐵)) → (𝑆 ∗ 1 ) ∈ (𝑁 DMat 𝑅)) |
15 | | simprr 792 |
. . . . . 6
⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑆 ∈ 𝐵 ∧ 𝑇 ∈ 𝐵)) → 𝑇 ∈ 𝐵) |
16 | 15, 9 | jca 553 |
. . . . 5
⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑆 ∈ 𝐵 ∧ 𝑇 ∈ 𝐵)) → (𝑇 ∈ 𝐵 ∧ 1 ∈ (𝑁 DMat 𝑅))) |
17 | 11, 3, 4, 12, 6 | dmatscmcl 20128 |
. . . . 5
⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑇 ∈ 𝐵 ∧ 1 ∈ (𝑁 DMat 𝑅))) → (𝑇 ∗ 1 ) ∈ (𝑁 DMat 𝑅)) |
18 | 16, 17 | syldan 486 |
. . . 4
⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑆 ∈ 𝐵 ∧ 𝑇 ∈ 𝐵)) → (𝑇 ∗ 1 ) ∈ (𝑁 DMat 𝑅)) |
19 | 14, 18 | jca 553 |
. . 3
⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑆 ∈ 𝐵 ∧ 𝑇 ∈ 𝐵)) → ((𝑆 ∗ 1 ) ∈ (𝑁 DMat 𝑅) ∧ (𝑇 ∗ 1 ) ∈ (𝑁 DMat 𝑅))) |
20 | | scmatscmiddistr.m |
. . . . 5
⊢ × =
(.r‘𝐴) |
21 | 20 | oveqi 6562 |
. . . 4
⊢ ((𝑆 ∗ 1 ) × (𝑇 ∗ 1 )) = ((𝑆 ∗ 1
)(.r‘𝐴)(𝑇 ∗ 1 )) |
22 | 3, 4, 5, 6 | dmatmul 20122 |
. . . 4
⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ ((𝑆 ∗ 1 ) ∈ (𝑁 DMat 𝑅) ∧ (𝑇 ∗ 1 ) ∈ (𝑁 DMat 𝑅))) → ((𝑆 ∗ 1
)(.r‘𝐴)(𝑇 ∗ 1 )) = (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ if(𝑖 = 𝑗, ((𝑖(𝑆 ∗ 1 )𝑗)(.r‘𝑅)(𝑖(𝑇 ∗ 1 )𝑗)), 0 ))) |
23 | 21, 22 | syl5eq 2656 |
. . 3
⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ ((𝑆 ∗ 1 ) ∈ (𝑁 DMat 𝑅) ∧ (𝑇 ∗ 1 ) ∈ (𝑁 DMat 𝑅))) → ((𝑆 ∗ 1 ) × (𝑇 ∗ 1 )) = (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ if(𝑖 = 𝑗, ((𝑖(𝑆 ∗ 1 )𝑗)(.r‘𝑅)(𝑖(𝑇 ∗ 1 )𝑗)), 0 ))) |
24 | 19, 23 | syldan 486 |
. 2
⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑆 ∈ 𝐵 ∧ 𝑇 ∈ 𝐵)) → ((𝑆 ∗ 1 ) × (𝑇 ∗ 1 )) = (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ if(𝑖 = 𝑗, ((𝑖(𝑆 ∗ 1 )𝑗)(.r‘𝑅)(𝑖(𝑇 ∗ 1 )𝑗)), 0 ))) |
25 | | simpll 786 |
. . . . . . . . 9
⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑆 ∈ 𝐵 ∧ 𝑇 ∈ 𝐵)) → 𝑁 ∈ Fin) |
26 | | simplr 788 |
. . . . . . . . 9
⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑆 ∈ 𝐵 ∧ 𝑇 ∈ 𝐵)) → 𝑅 ∈ Ring) |
27 | 25, 26, 1 | 3jca 1235 |
. . . . . . . 8
⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑆 ∈ 𝐵 ∧ 𝑇 ∈ 𝐵)) → (𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑆 ∈ 𝐵)) |
28 | 27 | 3ad2ant1 1075 |
. . . . . . 7
⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑆 ∈ 𝐵 ∧ 𝑇 ∈ 𝐵)) ∧ 𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁) → (𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑆 ∈ 𝐵)) |
29 | | 3simpc 1053 |
. . . . . . 7
⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑆 ∈ 𝐵 ∧ 𝑇 ∈ 𝐵)) ∧ 𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁) → (𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁)) |
30 | 3, 11, 5, 2, 12 | scmatscmide 20132 |
. . . . . . 7
⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑆 ∈ 𝐵) ∧ (𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁)) → (𝑖(𝑆 ∗ 1 )𝑗) = if(𝑖 = 𝑗, 𝑆, 0 )) |
31 | 28, 29, 30 | syl2anc 691 |
. . . . . 6
⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑆 ∈ 𝐵 ∧ 𝑇 ∈ 𝐵)) ∧ 𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁) → (𝑖(𝑆 ∗ 1 )𝑗) = if(𝑖 = 𝑗, 𝑆, 0 )) |
32 | 25, 26, 15 | 3jca 1235 |
. . . . . . . 8
⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑆 ∈ 𝐵 ∧ 𝑇 ∈ 𝐵)) → (𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑇 ∈ 𝐵)) |
33 | 32 | 3ad2ant1 1075 |
. . . . . . 7
⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑆 ∈ 𝐵 ∧ 𝑇 ∈ 𝐵)) ∧ 𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁) → (𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑇 ∈ 𝐵)) |
34 | 3, 11, 5, 2, 12 | scmatscmide 20132 |
. . . . . . 7
⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑇 ∈ 𝐵) ∧ (𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁)) → (𝑖(𝑇 ∗ 1 )𝑗) = if(𝑖 = 𝑗, 𝑇, 0 )) |
35 | 33, 29, 34 | syl2anc 691 |
. . . . . 6
⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑆 ∈ 𝐵 ∧ 𝑇 ∈ 𝐵)) ∧ 𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁) → (𝑖(𝑇 ∗ 1 )𝑗) = if(𝑖 = 𝑗, 𝑇, 0 )) |
36 | 31, 35 | oveq12d 6567 |
. . . . 5
⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑆 ∈ 𝐵 ∧ 𝑇 ∈ 𝐵)) ∧ 𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁) → ((𝑖(𝑆 ∗ 1 )𝑗)(.r‘𝑅)(𝑖(𝑇 ∗ 1 )𝑗)) = (if(𝑖 = 𝑗, 𝑆, 0
)(.r‘𝑅)if(𝑖 = 𝑗, 𝑇, 0 ))) |
37 | 36 | ifeq1d 4054 |
. . . 4
⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑆 ∈ 𝐵 ∧ 𝑇 ∈ 𝐵)) ∧ 𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁) → if(𝑖 = 𝑗, ((𝑖(𝑆 ∗ 1 )𝑗)(.r‘𝑅)(𝑖(𝑇 ∗ 1 )𝑗)), 0 ) = if(𝑖 = 𝑗, (if(𝑖 = 𝑗, 𝑆, 0
)(.r‘𝑅)if(𝑖 = 𝑗, 𝑇, 0 )), 0 )) |
38 | 37 | mpt2eq3dva 6617 |
. . 3
⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑆 ∈ 𝐵 ∧ 𝑇 ∈ 𝐵)) → (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ if(𝑖 = 𝑗, ((𝑖(𝑆 ∗ 1 )𝑗)(.r‘𝑅)(𝑖(𝑇 ∗ 1 )𝑗)), 0 )) = (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ if(𝑖 = 𝑗, (if(𝑖 = 𝑗, 𝑆, 0
)(.r‘𝑅)if(𝑖 = 𝑗, 𝑇, 0 )), 0 ))) |
39 | | iftrue 4042 |
. . . . . . . 8
⊢ (𝑖 = 𝑗 → if(𝑖 = 𝑗, 𝑆, 0 ) = 𝑆) |
40 | | iftrue 4042 |
. . . . . . . 8
⊢ (𝑖 = 𝑗 → if(𝑖 = 𝑗, 𝑇, 0 ) = 𝑇) |
41 | 39, 40 | oveq12d 6567 |
. . . . . . 7
⊢ (𝑖 = 𝑗 → (if(𝑖 = 𝑗, 𝑆, 0
)(.r‘𝑅)if(𝑖 = 𝑗, 𝑇, 0 )) = (𝑆(.r‘𝑅)𝑇)) |
42 | 41 | adantl 481 |
. . . . . 6
⊢
(((((𝑁 ∈ Fin
∧ 𝑅 ∈ Ring) ∧
(𝑆 ∈ 𝐵 ∧ 𝑇 ∈ 𝐵)) ∧ 𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁) ∧ 𝑖 = 𝑗) → (if(𝑖 = 𝑗, 𝑆, 0
)(.r‘𝑅)if(𝑖 = 𝑗, 𝑇, 0 )) = (𝑆(.r‘𝑅)𝑇)) |
43 | 42 | ifeq1da 4066 |
. . . . 5
⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑆 ∈ 𝐵 ∧ 𝑇 ∈ 𝐵)) ∧ 𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁) → if(𝑖 = 𝑗, (if(𝑖 = 𝑗, 𝑆, 0
)(.r‘𝑅)if(𝑖 = 𝑗, 𝑇, 0 )), 0 ) = if(𝑖 = 𝑗, (𝑆(.r‘𝑅)𝑇), 0 )) |
44 | 43 | mpt2eq3dva 6617 |
. . . 4
⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑆 ∈ 𝐵 ∧ 𝑇 ∈ 𝐵)) → (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ if(𝑖 = 𝑗, (if(𝑖 = 𝑗, 𝑆, 0
)(.r‘𝑅)if(𝑖 = 𝑗, 𝑇, 0 )), 0 )) = (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ if(𝑖 = 𝑗, (𝑆(.r‘𝑅)𝑇), 0 ))) |
45 | | eqidd 2611 |
. . . . . . . 8
⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑆 ∈ 𝐵 ∧ 𝑇 ∈ 𝐵)) ∧ (𝑥 ∈ 𝑁 ∧ 𝑦 ∈ 𝑁)) → (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ if(𝑖 = 𝑗, (𝑆(.r‘𝑅)𝑇), 0 )) = (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ if(𝑖 = 𝑗, (𝑆(.r‘𝑅)𝑇), 0 ))) |
46 | | eqeq12 2623 |
. . . . . . . . . 10
⊢ ((𝑖 = 𝑥 ∧ 𝑗 = 𝑦) → (𝑖 = 𝑗 ↔ 𝑥 = 𝑦)) |
47 | | scmatscmiddistr.t |
. . . . . . . . . . . . 13
⊢ · =
(.r‘𝑅) |
48 | 47 | eqcomi 2619 |
. . . . . . . . . . . 12
⊢
(.r‘𝑅) = · |
49 | 48 | oveqi 6562 |
. . . . . . . . . . 11
⊢ (𝑆(.r‘𝑅)𝑇) = (𝑆 · 𝑇) |
50 | 49 | a1i 11 |
. . . . . . . . . 10
⊢ ((𝑖 = 𝑥 ∧ 𝑗 = 𝑦) → (𝑆(.r‘𝑅)𝑇) = (𝑆 · 𝑇)) |
51 | 46, 50 | ifbieq1d 4059 |
. . . . . . . . 9
⊢ ((𝑖 = 𝑥 ∧ 𝑗 = 𝑦) → if(𝑖 = 𝑗, (𝑆(.r‘𝑅)𝑇), 0 ) = if(𝑥 = 𝑦, (𝑆 · 𝑇), 0 )) |
52 | 51 | adantl 481 |
. . . . . . . 8
⊢
(((((𝑁 ∈ Fin
∧ 𝑅 ∈ Ring) ∧
(𝑆 ∈ 𝐵 ∧ 𝑇 ∈ 𝐵)) ∧ (𝑥 ∈ 𝑁 ∧ 𝑦 ∈ 𝑁)) ∧ (𝑖 = 𝑥 ∧ 𝑗 = 𝑦)) → if(𝑖 = 𝑗, (𝑆(.r‘𝑅)𝑇), 0 ) = if(𝑥 = 𝑦, (𝑆 · 𝑇), 0 )) |
53 | | simprl 790 |
. . . . . . . 8
⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑆 ∈ 𝐵 ∧ 𝑇 ∈ 𝐵)) ∧ (𝑥 ∈ 𝑁 ∧ 𝑦 ∈ 𝑁)) → 𝑥 ∈ 𝑁) |
54 | | simprr 792 |
. . . . . . . 8
⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑆 ∈ 𝐵 ∧ 𝑇 ∈ 𝐵)) ∧ (𝑥 ∈ 𝑁 ∧ 𝑦 ∈ 𝑁)) → 𝑦 ∈ 𝑁) |
55 | | ovex 6577 |
. . . . . . . . . 10
⊢ (𝑆 · 𝑇) ∈ V |
56 | | fvex 6113 |
. . . . . . . . . . 11
⊢
(0g‘𝑅) ∈ V |
57 | 5, 56 | eqeltri 2684 |
. . . . . . . . . 10
⊢ 0 ∈
V |
58 | 55, 57 | ifex 4106 |
. . . . . . . . 9
⊢ if(𝑥 = 𝑦, (𝑆 · 𝑇), 0 ) ∈
V |
59 | 58 | a1i 11 |
. . . . . . . 8
⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑆 ∈ 𝐵 ∧ 𝑇 ∈ 𝐵)) ∧ (𝑥 ∈ 𝑁 ∧ 𝑦 ∈ 𝑁)) → if(𝑥 = 𝑦, (𝑆 · 𝑇), 0 ) ∈
V) |
60 | 45, 52, 53, 54, 59 | ovmpt2d 6686 |
. . . . . . 7
⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑆 ∈ 𝐵 ∧ 𝑇 ∈ 𝐵)) ∧ (𝑥 ∈ 𝑁 ∧ 𝑦 ∈ 𝑁)) → (𝑥(𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ if(𝑖 = 𝑗, (𝑆(.r‘𝑅)𝑇), 0 ))𝑦) = if(𝑥 = 𝑦, (𝑆 · 𝑇), 0 )) |
61 | 26, 1, 15 | 3jca 1235 |
. . . . . . . . . 10
⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑆 ∈ 𝐵 ∧ 𝑇 ∈ 𝐵)) → (𝑅 ∈ Ring ∧ 𝑆 ∈ 𝐵 ∧ 𝑇 ∈ 𝐵)) |
62 | 11, 47 | ringcl 18384 |
. . . . . . . . . 10
⊢ ((𝑅 ∈ Ring ∧ 𝑆 ∈ 𝐵 ∧ 𝑇 ∈ 𝐵) → (𝑆 · 𝑇) ∈ 𝐵) |
63 | 61, 62 | syl 17 |
. . . . . . . . 9
⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑆 ∈ 𝐵 ∧ 𝑇 ∈ 𝐵)) → (𝑆 · 𝑇) ∈ 𝐵) |
64 | 25, 26, 63 | 3jca 1235 |
. . . . . . . 8
⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑆 ∈ 𝐵 ∧ 𝑇 ∈ 𝐵)) → (𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ (𝑆 · 𝑇) ∈ 𝐵)) |
65 | 3, 11, 5, 2, 12 | scmatscmide 20132 |
. . . . . . . 8
⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ (𝑆 · 𝑇) ∈ 𝐵) ∧ (𝑥 ∈ 𝑁 ∧ 𝑦 ∈ 𝑁)) → (𝑥((𝑆 · 𝑇) ∗ 1 )𝑦) = if(𝑥 = 𝑦, (𝑆 · 𝑇), 0 )) |
66 | 64, 65 | sylan 487 |
. . . . . . 7
⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑆 ∈ 𝐵 ∧ 𝑇 ∈ 𝐵)) ∧ (𝑥 ∈ 𝑁 ∧ 𝑦 ∈ 𝑁)) → (𝑥((𝑆 · 𝑇) ∗ 1 )𝑦) = if(𝑥 = 𝑦, (𝑆 · 𝑇), 0 )) |
67 | 60, 66 | eqtr4d 2647 |
. . . . . 6
⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑆 ∈ 𝐵 ∧ 𝑇 ∈ 𝐵)) ∧ (𝑥 ∈ 𝑁 ∧ 𝑦 ∈ 𝑁)) → (𝑥(𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ if(𝑖 = 𝑗, (𝑆(.r‘𝑅)𝑇), 0 ))𝑦) = (𝑥((𝑆 · 𝑇) ∗ 1 )𝑦)) |
68 | 67 | ralrimivva 2954 |
. . . . 5
⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑆 ∈ 𝐵 ∧ 𝑇 ∈ 𝐵)) → ∀𝑥 ∈ 𝑁 ∀𝑦 ∈ 𝑁 (𝑥(𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ if(𝑖 = 𝑗, (𝑆(.r‘𝑅)𝑇), 0 ))𝑦) = (𝑥((𝑆 · 𝑇) ∗ 1 )𝑦)) |
69 | | eqid 2610 |
. . . . . . . . . . 11
⊢
(.r‘𝑅) = (.r‘𝑅) |
70 | 11, 69 | ringcl 18384 |
. . . . . . . . . 10
⊢ ((𝑅 ∈ Ring ∧ 𝑆 ∈ 𝐵 ∧ 𝑇 ∈ 𝐵) → (𝑆(.r‘𝑅)𝑇) ∈ 𝐵) |
71 | 61, 70 | syl 17 |
. . . . . . . . 9
⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑆 ∈ 𝐵 ∧ 𝑇 ∈ 𝐵)) → (𝑆(.r‘𝑅)𝑇) ∈ 𝐵) |
72 | 11, 5 | ring0cl 18392 |
. . . . . . . . . . 11
⊢ (𝑅 ∈ Ring → 0 ∈ 𝐵) |
73 | 72 | adantl 481 |
. . . . . . . . . 10
⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 0 ∈ 𝐵) |
74 | 73 | adantr 480 |
. . . . . . . . 9
⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑆 ∈ 𝐵 ∧ 𝑇 ∈ 𝐵)) → 0 ∈ 𝐵) |
75 | 71, 74 | ifcld 4081 |
. . . . . . . 8
⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑆 ∈ 𝐵 ∧ 𝑇 ∈ 𝐵)) → if(𝑖 = 𝑗, (𝑆(.r‘𝑅)𝑇), 0 ) ∈ 𝐵) |
76 | 75 | 3ad2ant1 1075 |
. . . . . . 7
⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑆 ∈ 𝐵 ∧ 𝑇 ∈ 𝐵)) ∧ 𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁) → if(𝑖 = 𝑗, (𝑆(.r‘𝑅)𝑇), 0 ) ∈ 𝐵) |
77 | 3, 11, 4, 25, 26, 76 | matbas2d 20048 |
. . . . . 6
⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑆 ∈ 𝐵 ∧ 𝑇 ∈ 𝐵)) → (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ if(𝑖 = 𝑗, (𝑆(.r‘𝑅)𝑇), 0 )) ∈
(Base‘𝐴)) |
78 | 3 | matring 20068 |
. . . . . . . . . 10
⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝐴 ∈ Ring) |
79 | 4, 2 | ringidcl 18391 |
. . . . . . . . . 10
⊢ (𝐴 ∈ Ring → 1 ∈
(Base‘𝐴)) |
80 | 78, 79 | syl 17 |
. . . . . . . . 9
⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 1 ∈
(Base‘𝐴)) |
81 | 80 | adantr 480 |
. . . . . . . 8
⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑆 ∈ 𝐵 ∧ 𝑇 ∈ 𝐵)) → 1 ∈ (Base‘𝐴)) |
82 | 63, 81 | jca 553 |
. . . . . . 7
⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑆 ∈ 𝐵 ∧ 𝑇 ∈ 𝐵)) → ((𝑆 · 𝑇) ∈ 𝐵 ∧ 1 ∈ (Base‘𝐴))) |
83 | 11, 3, 4, 12 | matvscl 20056 |
. . . . . . 7
⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ ((𝑆 · 𝑇) ∈ 𝐵 ∧ 1 ∈ (Base‘𝐴))) → ((𝑆 · 𝑇) ∗ 1 ) ∈ (Base‘𝐴)) |
84 | 82, 83 | syldan 486 |
. . . . . 6
⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑆 ∈ 𝐵 ∧ 𝑇 ∈ 𝐵)) → ((𝑆 · 𝑇) ∗ 1 ) ∈ (Base‘𝐴)) |
85 | 3, 4 | eqmat 20049 |
. . . . . 6
⊢ (((𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ if(𝑖 = 𝑗, (𝑆(.r‘𝑅)𝑇), 0 )) ∈
(Base‘𝐴) ∧
((𝑆 · 𝑇) ∗ 1 ) ∈ (Base‘𝐴)) → ((𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ if(𝑖 = 𝑗, (𝑆(.r‘𝑅)𝑇), 0 )) = ((𝑆 · 𝑇) ∗ 1 ) ↔ ∀𝑥 ∈ 𝑁 ∀𝑦 ∈ 𝑁 (𝑥(𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ if(𝑖 = 𝑗, (𝑆(.r‘𝑅)𝑇), 0 ))𝑦) = (𝑥((𝑆 · 𝑇) ∗ 1 )𝑦))) |
86 | 77, 84, 85 | syl2anc 691 |
. . . . 5
⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑆 ∈ 𝐵 ∧ 𝑇 ∈ 𝐵)) → ((𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ if(𝑖 = 𝑗, (𝑆(.r‘𝑅)𝑇), 0 )) = ((𝑆 · 𝑇) ∗ 1 ) ↔ ∀𝑥 ∈ 𝑁 ∀𝑦 ∈ 𝑁 (𝑥(𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ if(𝑖 = 𝑗, (𝑆(.r‘𝑅)𝑇), 0 ))𝑦) = (𝑥((𝑆 · 𝑇) ∗ 1 )𝑦))) |
87 | 68, 86 | mpbird 246 |
. . . 4
⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑆 ∈ 𝐵 ∧ 𝑇 ∈ 𝐵)) → (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ if(𝑖 = 𝑗, (𝑆(.r‘𝑅)𝑇), 0 )) = ((𝑆 · 𝑇) ∗ 1 )) |
88 | 44, 87 | eqtrd 2644 |
. . 3
⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑆 ∈ 𝐵 ∧ 𝑇 ∈ 𝐵)) → (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ if(𝑖 = 𝑗, (if(𝑖 = 𝑗, 𝑆, 0
)(.r‘𝑅)if(𝑖 = 𝑗, 𝑇, 0 )), 0 )) = ((𝑆 · 𝑇) ∗ 1 )) |
89 | 38, 88 | eqtrd 2644 |
. 2
⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑆 ∈ 𝐵 ∧ 𝑇 ∈ 𝐵)) → (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ if(𝑖 = 𝑗, ((𝑖(𝑆 ∗ 1 )𝑗)(.r‘𝑅)(𝑖(𝑇 ∗ 1 )𝑗)), 0 )) = ((𝑆 · 𝑇) ∗ 1 )) |
90 | 24, 89 | eqtrd 2644 |
1
⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑆 ∈ 𝐵 ∧ 𝑇 ∈ 𝐵)) → ((𝑆 ∗ 1 ) × (𝑇 ∗ 1 )) = ((𝑆 · 𝑇) ∗ 1 )) |