MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  df-mat Structured version   Visualization version   GIF version

Definition df-mat 20262
Description: Define the algebra of n x n matrices over a ring r. (Contributed by Stefan O'Rear, 31-Aug-2015.)
Assertion
Ref Expression
df-mat Mat = (𝑛 ∈ Fin, 𝑟 ∈ V ↦ ((𝑟 freeLMod (𝑛 × 𝑛)) sSet ⟨(.r‘ndx), (𝑟 maMul ⟨𝑛, 𝑛, 𝑛⟩)⟩))
Distinct variable group:   𝑛,𝑟

Detailed syntax breakdown of Definition df-mat
StepHypRef Expression
1 cmat 20261 . 2 class Mat
2 vn . . 3 setvar 𝑛
3 vr . . 3 setvar 𝑟
4 cfn 7997 . . 3 class Fin
5 cvv 3231 . . 3 class V
63cv 1522 . . . . 5 class 𝑟
72cv 1522 . . . . . 6 class 𝑛
87, 7cxp 5141 . . . . 5 class (𝑛 × 𝑛)
9 cfrlm 20138 . . . . 5 class freeLMod
106, 8, 9co 6690 . . . 4 class (𝑟 freeLMod (𝑛 × 𝑛))
11 cnx 15901 . . . . . 6 class ndx
12 cmulr 15989 . . . . . 6 class .r
1311, 12cfv 5926 . . . . 5 class (.r‘ndx)
147, 7, 7cotp 4218 . . . . . 6 class 𝑛, 𝑛, 𝑛
15 cmmul 20237 . . . . . 6 class maMul
166, 14, 15co 6690 . . . . 5 class (𝑟 maMul ⟨𝑛, 𝑛, 𝑛⟩)
1713, 16cop 4216 . . . 4 class ⟨(.r‘ndx), (𝑟 maMul ⟨𝑛, 𝑛, 𝑛⟩)⟩
18 csts 15902 . . . 4 class sSet
1910, 17, 18co 6690 . . 3 class ((𝑟 freeLMod (𝑛 × 𝑛)) sSet ⟨(.r‘ndx), (𝑟 maMul ⟨𝑛, 𝑛, 𝑛⟩)⟩)
202, 3, 4, 5, 19cmpt2 6692 . 2 class (𝑛 ∈ Fin, 𝑟 ∈ V ↦ ((𝑟 freeLMod (𝑛 × 𝑛)) sSet ⟨(.r‘ndx), (𝑟 maMul ⟨𝑛, 𝑛, 𝑛⟩)⟩))
211, 20wceq 1523 1 wff Mat = (𝑛 ∈ Fin, 𝑟 ∈ V ↦ ((𝑟 freeLMod (𝑛 × 𝑛)) sSet ⟨(.r‘ndx), (𝑟 maMul ⟨𝑛, 𝑛, 𝑛⟩)⟩))
Colors of variables: wff setvar class
This definition is referenced by:  matbas0pc  20263  matbas0  20264  matval  20265  matrcl  20266  mdetfval  20440  madufval  20491
  Copyright terms: Public domain W3C validator