| Step | Hyp | Ref
| Expression |
| 1 | | crngring 20242 |
. . . 4
⊢ (𝑅 ∈ CRing → 𝑅 ∈ Ring) |
| 2 | | scmatid.a |
. . . . 5
⊢ 𝐴 = (𝑁 Mat 𝑅) |
| 3 | | scmatid.b |
. . . . 5
⊢ 𝐵 = (Base‘𝐴) |
| 4 | | scmatid.e |
. . . . 5
⊢ 𝐸 = (Base‘𝑅) |
| 5 | | scmatid.0 |
. . . . 5
⊢ 0 =
(0g‘𝑅) |
| 6 | | scmatid.s |
. . . . 5
⊢ 𝑆 = (𝑁 ScMat 𝑅) |
| 7 | 2, 3, 4, 5, 6 | scmatsrng 22526 |
. . . 4
⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝑆 ∈ (SubRing‘𝐴)) |
| 8 | 1, 7 | sylan2 593 |
. . 3
⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) → 𝑆 ∈ (SubRing‘𝐴)) |
| 9 | | scmatcrng.c |
. . . 4
⊢ 𝐶 = (𝐴 ↾s 𝑆) |
| 10 | 9 | subrgring 20574 |
. . 3
⊢ (𝑆 ∈ (SubRing‘𝐴) → 𝐶 ∈ Ring) |
| 11 | 8, 10 | syl 17 |
. 2
⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) → 𝐶 ∈ Ring) |
| 12 | | simp1lr 1238 |
. . . . . . . 8
⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) ∧ 𝑎 ∈ 𝑁 ∧ 𝑏 ∈ 𝑁) → 𝑅 ∈ CRing) |
| 13 | | eqid 2737 |
. . . . . . . . 9
⊢
(Base‘𝐴) =
(Base‘𝐴) |
| 14 | | simp2 1138 |
. . . . . . . . 9
⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) ∧ 𝑎 ∈ 𝑁 ∧ 𝑏 ∈ 𝑁) → 𝑎 ∈ 𝑁) |
| 15 | | simp3 1139 |
. . . . . . . . 9
⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) ∧ 𝑎 ∈ 𝑁 ∧ 𝑏 ∈ 𝑁) → 𝑏 ∈ 𝑁) |
| 16 | 2, 13, 6 | scmatmat 22515 |
. . . . . . . . . . . 12
⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) → (𝑥 ∈ 𝑆 → 𝑥 ∈ (Base‘𝐴))) |
| 17 | 16 | imp 406 |
. . . . . . . . . . 11
⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ 𝑥 ∈ 𝑆) → 𝑥 ∈ (Base‘𝐴)) |
| 18 | 17 | adantrr 717 |
. . . . . . . . . 10
⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → 𝑥 ∈ (Base‘𝐴)) |
| 19 | 18 | 3ad2ant1 1134 |
. . . . . . . . 9
⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) ∧ 𝑎 ∈ 𝑁 ∧ 𝑏 ∈ 𝑁) → 𝑥 ∈ (Base‘𝐴)) |
| 20 | 2, 4, 13, 14, 15, 19 | matecld 22432 |
. . . . . . . 8
⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) ∧ 𝑎 ∈ 𝑁 ∧ 𝑏 ∈ 𝑁) → (𝑎𝑥𝑏) ∈ 𝐸) |
| 21 | 2, 13, 6 | scmatmat 22515 |
. . . . . . . . . . . 12
⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) → (𝑦 ∈ 𝑆 → 𝑦 ∈ (Base‘𝐴))) |
| 22 | 21 | imp 406 |
. . . . . . . . . . 11
⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ 𝑦 ∈ 𝑆) → 𝑦 ∈ (Base‘𝐴)) |
| 23 | 22 | adantrl 716 |
. . . . . . . . . 10
⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → 𝑦 ∈ (Base‘𝐴)) |
| 24 | 23 | 3ad2ant1 1134 |
. . . . . . . . 9
⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) ∧ 𝑎 ∈ 𝑁 ∧ 𝑏 ∈ 𝑁) → 𝑦 ∈ (Base‘𝐴)) |
| 25 | 2, 4, 13, 14, 15, 24 | matecld 22432 |
. . . . . . . 8
⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) ∧ 𝑎 ∈ 𝑁 ∧ 𝑏 ∈ 𝑁) → (𝑎𝑦𝑏) ∈ 𝐸) |
| 26 | | eqid 2737 |
. . . . . . . . 9
⊢
(.r‘𝑅) = (.r‘𝑅) |
| 27 | 4, 26 | crngcom 20248 |
. . . . . . . 8
⊢ ((𝑅 ∈ CRing ∧ (𝑎𝑥𝑏) ∈ 𝐸 ∧ (𝑎𝑦𝑏) ∈ 𝐸) → ((𝑎𝑥𝑏)(.r‘𝑅)(𝑎𝑦𝑏)) = ((𝑎𝑦𝑏)(.r‘𝑅)(𝑎𝑥𝑏))) |
| 28 | 12, 20, 25, 27 | syl3anc 1373 |
. . . . . . 7
⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) ∧ 𝑎 ∈ 𝑁 ∧ 𝑏 ∈ 𝑁) → ((𝑎𝑥𝑏)(.r‘𝑅)(𝑎𝑦𝑏)) = ((𝑎𝑦𝑏)(.r‘𝑅)(𝑎𝑥𝑏))) |
| 29 | 28 | ifeq1d 4545 |
. . . . . 6
⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) ∧ 𝑎 ∈ 𝑁 ∧ 𝑏 ∈ 𝑁) → if(𝑎 = 𝑏, ((𝑎𝑥𝑏)(.r‘𝑅)(𝑎𝑦𝑏)), 0 ) = if(𝑎 = 𝑏, ((𝑎𝑦𝑏)(.r‘𝑅)(𝑎𝑥𝑏)), 0 )) |
| 30 | 29 | mpoeq3dva 7510 |
. . . . 5
⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑎 ∈ 𝑁, 𝑏 ∈ 𝑁 ↦ if(𝑎 = 𝑏, ((𝑎𝑥𝑏)(.r‘𝑅)(𝑎𝑦𝑏)), 0 )) = (𝑎 ∈ 𝑁, 𝑏 ∈ 𝑁 ↦ if(𝑎 = 𝑏, ((𝑎𝑦𝑏)(.r‘𝑅)(𝑎𝑥𝑏)), 0 ))) |
| 31 | 1 | anim2i 617 |
. . . . . 6
⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) → (𝑁 ∈ Fin ∧ 𝑅 ∈ Ring)) |
| 32 | | eqid 2737 |
. . . . . . . . . 10
⊢ (𝑁 DMat 𝑅) = (𝑁 DMat 𝑅) |
| 33 | 2, 3, 4, 5, 6, 32 | scmatdmat 22521 |
. . . . . . . . 9
⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → (𝑥 ∈ 𝑆 → 𝑥 ∈ (𝑁 DMat 𝑅))) |
| 34 | 1, 33 | sylan2 593 |
. . . . . . . 8
⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) → (𝑥 ∈ 𝑆 → 𝑥 ∈ (𝑁 DMat 𝑅))) |
| 35 | 2, 3, 4, 5, 6, 32 | scmatdmat 22521 |
. . . . . . . . 9
⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → (𝑦 ∈ 𝑆 → 𝑦 ∈ (𝑁 DMat 𝑅))) |
| 36 | 1, 35 | sylan2 593 |
. . . . . . . 8
⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) → (𝑦 ∈ 𝑆 → 𝑦 ∈ (𝑁 DMat 𝑅))) |
| 37 | 34, 36 | anim12d 609 |
. . . . . . 7
⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) → ((𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆) → (𝑥 ∈ (𝑁 DMat 𝑅) ∧ 𝑦 ∈ (𝑁 DMat 𝑅)))) |
| 38 | 37 | imp 406 |
. . . . . 6
⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥 ∈ (𝑁 DMat 𝑅) ∧ 𝑦 ∈ (𝑁 DMat 𝑅))) |
| 39 | 2, 3, 5, 32 | dmatmul 22503 |
. . . . . 6
⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ (𝑁 DMat 𝑅) ∧ 𝑦 ∈ (𝑁 DMat 𝑅))) → (𝑥(.r‘𝐴)𝑦) = (𝑎 ∈ 𝑁, 𝑏 ∈ 𝑁 ↦ if(𝑎 = 𝑏, ((𝑎𝑥𝑏)(.r‘𝑅)(𝑎𝑦𝑏)), 0 ))) |
| 40 | 31, 38, 39 | syl2an2r 685 |
. . . . 5
⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥(.r‘𝐴)𝑦) = (𝑎 ∈ 𝑁, 𝑏 ∈ 𝑁 ↦ if(𝑎 = 𝑏, ((𝑎𝑥𝑏)(.r‘𝑅)(𝑎𝑦𝑏)), 0 ))) |
| 41 | 38 | ancomd 461 |
. . . . . 6
⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑦 ∈ (𝑁 DMat 𝑅) ∧ 𝑥 ∈ (𝑁 DMat 𝑅))) |
| 42 | 2, 3, 5, 32 | dmatmul 22503 |
. . . . . 6
⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑦 ∈ (𝑁 DMat 𝑅) ∧ 𝑥 ∈ (𝑁 DMat 𝑅))) → (𝑦(.r‘𝐴)𝑥) = (𝑎 ∈ 𝑁, 𝑏 ∈ 𝑁 ↦ if(𝑎 = 𝑏, ((𝑎𝑦𝑏)(.r‘𝑅)(𝑎𝑥𝑏)), 0 ))) |
| 43 | 31, 41, 42 | syl2an2r 685 |
. . . . 5
⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑦(.r‘𝐴)𝑥) = (𝑎 ∈ 𝑁, 𝑏 ∈ 𝑁 ↦ if(𝑎 = 𝑏, ((𝑎𝑦𝑏)(.r‘𝑅)(𝑎𝑥𝑏)), 0 ))) |
| 44 | 30, 40, 43 | 3eqtr4d 2787 |
. . . 4
⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥(.r‘𝐴)𝑦) = (𝑦(.r‘𝐴)𝑥)) |
| 45 | 44 | ralrimivva 3202 |
. . 3
⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) →
∀𝑥 ∈ 𝑆 ∀𝑦 ∈ 𝑆 (𝑥(.r‘𝐴)𝑦) = (𝑦(.r‘𝐴)𝑥)) |
| 46 | 9 | subrgbas 20581 |
. . . . . 6
⊢ (𝑆 ∈ (SubRing‘𝐴) → 𝑆 = (Base‘𝐶)) |
| 47 | 46 | eqcomd 2743 |
. . . . 5
⊢ (𝑆 ∈ (SubRing‘𝐴) → (Base‘𝐶) = 𝑆) |
| 48 | | eqid 2737 |
. . . . . . . . . 10
⊢
(.r‘𝐴) = (.r‘𝐴) |
| 49 | 9, 48 | ressmulr 17351 |
. . . . . . . . 9
⊢ (𝑆 ∈ (SubRing‘𝐴) →
(.r‘𝐴) =
(.r‘𝐶)) |
| 50 | 49 | eqcomd 2743 |
. . . . . . . 8
⊢ (𝑆 ∈ (SubRing‘𝐴) →
(.r‘𝐶) =
(.r‘𝐴)) |
| 51 | 50 | oveqd 7448 |
. . . . . . 7
⊢ (𝑆 ∈ (SubRing‘𝐴) → (𝑥(.r‘𝐶)𝑦) = (𝑥(.r‘𝐴)𝑦)) |
| 52 | 50 | oveqd 7448 |
. . . . . . 7
⊢ (𝑆 ∈ (SubRing‘𝐴) → (𝑦(.r‘𝐶)𝑥) = (𝑦(.r‘𝐴)𝑥)) |
| 53 | 51, 52 | eqeq12d 2753 |
. . . . . 6
⊢ (𝑆 ∈ (SubRing‘𝐴) → ((𝑥(.r‘𝐶)𝑦) = (𝑦(.r‘𝐶)𝑥) ↔ (𝑥(.r‘𝐴)𝑦) = (𝑦(.r‘𝐴)𝑥))) |
| 54 | 47, 53 | raleqbidv 3346 |
. . . . 5
⊢ (𝑆 ∈ (SubRing‘𝐴) → (∀𝑦 ∈ (Base‘𝐶)(𝑥(.r‘𝐶)𝑦) = (𝑦(.r‘𝐶)𝑥) ↔ ∀𝑦 ∈ 𝑆 (𝑥(.r‘𝐴)𝑦) = (𝑦(.r‘𝐴)𝑥))) |
| 55 | 47, 54 | raleqbidv 3346 |
. . . 4
⊢ (𝑆 ∈ (SubRing‘𝐴) → (∀𝑥 ∈ (Base‘𝐶)∀𝑦 ∈ (Base‘𝐶)(𝑥(.r‘𝐶)𝑦) = (𝑦(.r‘𝐶)𝑥) ↔ ∀𝑥 ∈ 𝑆 ∀𝑦 ∈ 𝑆 (𝑥(.r‘𝐴)𝑦) = (𝑦(.r‘𝐴)𝑥))) |
| 56 | 8, 55 | syl 17 |
. . 3
⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) →
(∀𝑥 ∈
(Base‘𝐶)∀𝑦 ∈ (Base‘𝐶)(𝑥(.r‘𝐶)𝑦) = (𝑦(.r‘𝐶)𝑥) ↔ ∀𝑥 ∈ 𝑆 ∀𝑦 ∈ 𝑆 (𝑥(.r‘𝐴)𝑦) = (𝑦(.r‘𝐴)𝑥))) |
| 57 | 45, 56 | mpbird 257 |
. 2
⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) →
∀𝑥 ∈
(Base‘𝐶)∀𝑦 ∈ (Base‘𝐶)(𝑥(.r‘𝐶)𝑦) = (𝑦(.r‘𝐶)𝑥)) |
| 58 | | eqid 2737 |
. . 3
⊢
(Base‘𝐶) =
(Base‘𝐶) |
| 59 | | eqid 2737 |
. . 3
⊢
(.r‘𝐶) = (.r‘𝐶) |
| 60 | 58, 59 | iscrng2 20249 |
. 2
⊢ (𝐶 ∈ CRing ↔ (𝐶 ∈ Ring ∧ ∀𝑥 ∈ (Base‘𝐶)∀𝑦 ∈ (Base‘𝐶)(𝑥(.r‘𝐶)𝑦) = (𝑦(.r‘𝐶)𝑥))) |
| 61 | 11, 57, 60 | sylanbrc 583 |
1
⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) → 𝐶 ∈ CRing) |