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Definition df-mat2pmat 20279
Description: Transformation of a matrix (over a ring) into a matrix over the corresponding polynomial ring. (Contributed by AV, 31-Jul-2019.)
Assertion
Ref Expression
df-mat2pmat matToPolyMat = (𝑛 ∈ Fin, 𝑟 ∈ V ↦ (𝑚 ∈ (Base‘(𝑛 Mat 𝑟)) ↦ (𝑥𝑛, 𝑦𝑛 ↦ ((algSc‘(Poly1𝑟))‘(𝑥𝑚𝑦)))))
Distinct variable group:   𝑚,𝑛,𝑟,𝑥,𝑦

Detailed syntax breakdown of Definition df-mat2pmat
StepHypRef Expression
1 cmat2pmat 20276 . 2 class matToPolyMat
2 vn . . 3 setvar 𝑛
3 vr . . 3 setvar 𝑟
4 cfn 7819 . . 3 class Fin
5 cvv 3173 . . 3 class V
6 vm . . . 4 setvar 𝑚
72cv 1474 . . . . . 6 class 𝑛
83cv 1474 . . . . . 6 class 𝑟
9 cmat 19980 . . . . . 6 class Mat
107, 8, 9co 6527 . . . . 5 class (𝑛 Mat 𝑟)
11 cbs 15644 . . . . 5 class Base
1210, 11cfv 5790 . . . 4 class (Base‘(𝑛 Mat 𝑟))
13 vx . . . . 5 setvar 𝑥
14 vy . . . . 5 setvar 𝑦
1513cv 1474 . . . . . . 7 class 𝑥
1614cv 1474 . . . . . . 7 class 𝑦
176cv 1474 . . . . . . 7 class 𝑚
1815, 16, 17co 6527 . . . . . 6 class (𝑥𝑚𝑦)
19 cpl1 19317 . . . . . . . 8 class Poly1
208, 19cfv 5790 . . . . . . 7 class (Poly1𝑟)
21 cascl 19081 . . . . . . 7 class algSc
2220, 21cfv 5790 . . . . . 6 class (algSc‘(Poly1𝑟))
2318, 22cfv 5790 . . . . 5 class ((algSc‘(Poly1𝑟))‘(𝑥𝑚𝑦))
2413, 14, 7, 7, 23cmpt2 6529 . . . 4 class (𝑥𝑛, 𝑦𝑛 ↦ ((algSc‘(Poly1𝑟))‘(𝑥𝑚𝑦)))
256, 12, 24cmpt 4638 . . 3 class (𝑚 ∈ (Base‘(𝑛 Mat 𝑟)) ↦ (𝑥𝑛, 𝑦𝑛 ↦ ((algSc‘(Poly1𝑟))‘(𝑥𝑚𝑦))))
262, 3, 4, 5, 25cmpt2 6529 . 2 class (𝑛 ∈ Fin, 𝑟 ∈ V ↦ (𝑚 ∈ (Base‘(𝑛 Mat 𝑟)) ↦ (𝑥𝑛, 𝑦𝑛 ↦ ((algSc‘(Poly1𝑟))‘(𝑥𝑚𝑦)))))
271, 26wceq 1475 1 wff matToPolyMat = (𝑛 ∈ Fin, 𝑟 ∈ V ↦ (𝑚 ∈ (Base‘(𝑛 Mat 𝑟)) ↦ (𝑥𝑛, 𝑦𝑛 ↦ ((algSc‘(Poly1𝑟))‘(𝑥𝑚𝑦)))))
Colors of variables: wff setvar class
This definition is referenced by:  mat2pmatfval  20295
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