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Definition df-mat 21908
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, π‘Ÿ ∈ V ↦ ((π‘Ÿ freeLMod (𝑛 Γ— 𝑛)) sSet ⟨(.rβ€˜ndx), (π‘Ÿ maMul βŸ¨π‘›, 𝑛, π‘›βŸ©)⟩))
Distinct variable group:   𝑛,π‘Ÿ

Detailed syntax breakdown of Definition df-mat
StepHypRef Expression
1 cmat 21907 . 2 class Mat
2 vn . . 3 setvar 𝑛
3 vr . . 3 setvar π‘Ÿ
4 cfn 8939 . . 3 class Fin
5 cvv 3475 . . 3 class V
63cv 1541 . . . . 5 class π‘Ÿ
72cv 1541 . . . . . 6 class 𝑛
87, 7cxp 5675 . . . . 5 class (𝑛 Γ— 𝑛)
9 cfrlm 21301 . . . . 5 class freeLMod
106, 8, 9co 7409 . . . 4 class (π‘Ÿ freeLMod (𝑛 Γ— 𝑛))
11 cnx 17126 . . . . . 6 class ndx
12 cmulr 17198 . . . . . 6 class .r
1311, 12cfv 6544 . . . . 5 class (.rβ€˜ndx)
147, 7, 7cotp 4637 . . . . . 6 class βŸ¨π‘›, 𝑛, π‘›βŸ©
15 cmmul 21885 . . . . . 6 class maMul
166, 14, 15co 7409 . . . . 5 class (π‘Ÿ maMul βŸ¨π‘›, 𝑛, π‘›βŸ©)
1713, 16cop 4635 . . . 4 class ⟨(.rβ€˜ndx), (π‘Ÿ maMul βŸ¨π‘›, 𝑛, π‘›βŸ©)⟩
18 csts 17096 . . . 4 class sSet
1910, 17, 18co 7409 . . 3 class ((π‘Ÿ freeLMod (𝑛 Γ— 𝑛)) sSet ⟨(.rβ€˜ndx), (π‘Ÿ maMul βŸ¨π‘›, 𝑛, π‘›βŸ©)⟩)
202, 3, 4, 5, 19cmpo 7411 . 2 class (𝑛 ∈ Fin, π‘Ÿ ∈ V ↦ ((π‘Ÿ freeLMod (𝑛 Γ— 𝑛)) sSet ⟨(.rβ€˜ndx), (π‘Ÿ maMul βŸ¨π‘›, 𝑛, π‘›βŸ©)⟩))
211, 20wceq 1542 1 wff Mat = (𝑛 ∈ Fin, π‘Ÿ ∈ V ↦ ((π‘Ÿ freeLMod (𝑛 Γ— 𝑛)) sSet ⟨(.rβ€˜ndx), (π‘Ÿ maMul βŸ¨π‘›, 𝑛, π‘›βŸ©)⟩))
Colors of variables: wff setvar class
This definition is referenced by:  matbas0pc  21909  matbas0  21910  matval  21911  matrcl  21912  mdetfval  22088  madufval  22139
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