Definition df-mat 21305

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

Step	Hyp	Ref	Expression
1		cmat 21304	. 2 class Mat
2		vn	. . 3 setvar 𝑛
3		vr	. . 3 setvar 𝑟
4		cfn 8626	. . 3 class Fin
5		cvv 3408	. . 3 class V
6	3	cv 1542	. . . . 5 class 𝑟
7	2	cv 1542	. . . . . 6 class 𝑛
8	7, 7	cxp 5549	. . . . 5 class (𝑛 × 𝑛)
9		cfrlm 20708	. . . . 5 class freeLMod
10	6, 8, 9	co 7213	. . . 4 class (𝑟 freeLMod (𝑛 × 𝑛))
11		cnx 16744	. . . . . 6 class ndx
12		cmulr 16803	. . . . . 6 class .r
13	11, 12	cfv 6380	. . . . 5 class (.r‘ndx)
14	7, 7, 7	cotp 4549	. . . . . 6 class ⟨𝑛, 𝑛, 𝑛⟩
15		cmmul 21282	. . . . . 6 class maMul
16	6, 14, 15	co 7213	. . . . 5 class (𝑟 maMul ⟨𝑛, 𝑛, 𝑛⟩)
17	13, 16	cop 4547	. . . 4 class ⟨(.r‘ndx), (𝑟 maMul ⟨𝑛, 𝑛, 𝑛⟩)⟩
18		csts 16716	. . . 4 class sSet
19	10, 17, 18	co 7213	. . 3 class ((𝑟 freeLMod (𝑛 × 𝑛)) sSet ⟨(.r‘ndx), (𝑟 maMul ⟨𝑛, 𝑛, 𝑛⟩)⟩)
20	2, 3, 4, 5, 19	cmpo 7215	. 2 class (𝑛 ∈ Fin, 𝑟 ∈ V ↦ ((𝑟 freeLMod (𝑛 × 𝑛)) sSet ⟨(.r‘ndx), (𝑟 maMul ⟨𝑛, 𝑛, 𝑛⟩)⟩))
21	1, 20	wceq 1543	1 wff Mat = (𝑛 ∈ Fin, 𝑟 ∈ V ↦ ((𝑟 freeLMod (𝑛 × 𝑛)) sSet ⟨(.r‘ndx), (𝑟 maMul ⟨𝑛, 𝑛, 𝑛⟩)⟩))

This definition is referenced by:  matbas0pc  21306  matbas0  21307  matval  21308  matrcl  21309  mdetfval  21483  madufval  21534