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Definition df-cpmat2mat 22701
Description: Transformation of a constant polynomial matrix (over a ring) into a matrix over the corresponding ring. Since this function is the inverse function of matToPolyMat, see m2cpminv 22753, it is also called "inverse matrix transformation" in the following. (Contributed by AV, 14-Dec-2019.)
Assertion
Ref Expression
df-cpmat2mat cPolyMatToMat = (𝑛 ∈ Fin, 𝑟 ∈ V ↦ (𝑚 ∈ (𝑛 ConstPolyMat 𝑟) ↦ (𝑥𝑛, 𝑦𝑛 ↦ ((coe1‘(𝑥𝑚𝑦))‘0))))
Distinct variable group:   𝑚,𝑛,𝑟,𝑥,𝑦

Detailed syntax breakdown of Definition df-cpmat2mat
StepHypRef Expression
1 ccpmat2mat 22698 . 2 class cPolyMatToMat
2 vn . . 3 setvar 𝑛
3 vr . . 3 setvar 𝑟
4 cfn 8974 . . 3 class Fin
5 cvv 3462 . . 3 class V
6 vm . . . 4 setvar 𝑚
72cv 1533 . . . . 5 class 𝑛
83cv 1533 . . . . 5 class 𝑟
9 ccpmat 22696 . . . . 5 class ConstPolyMat
107, 8, 9co 7424 . . . 4 class (𝑛 ConstPolyMat 𝑟)
11 vx . . . . 5 setvar 𝑥
12 vy . . . . 5 setvar 𝑦
13 cc0 11158 . . . . . 6 class 0
1411cv 1533 . . . . . . . 8 class 𝑥
1512cv 1533 . . . . . . . 8 class 𝑦
166cv 1533 . . . . . . . 8 class 𝑚
1714, 15, 16co 7424 . . . . . . 7 class (𝑥𝑚𝑦)
18 cco1 22167 . . . . . . 7 class coe1
1917, 18cfv 6554 . . . . . 6 class (coe1‘(𝑥𝑚𝑦))
2013, 19cfv 6554 . . . . 5 class ((coe1‘(𝑥𝑚𝑦))‘0)
2111, 12, 7, 7, 20cmpo 7426 . . . 4 class (𝑥𝑛, 𝑦𝑛 ↦ ((coe1‘(𝑥𝑚𝑦))‘0))
226, 10, 21cmpt 5236 . . 3 class (𝑚 ∈ (𝑛 ConstPolyMat 𝑟) ↦ (𝑥𝑛, 𝑦𝑛 ↦ ((coe1‘(𝑥𝑚𝑦))‘0)))
232, 3, 4, 5, 22cmpo 7426 . 2 class (𝑛 ∈ Fin, 𝑟 ∈ V ↦ (𝑚 ∈ (𝑛 ConstPolyMat 𝑟) ↦ (𝑥𝑛, 𝑦𝑛 ↦ ((coe1‘(𝑥𝑚𝑦))‘0))))
241, 23wceq 1534 1 wff cPolyMatToMat = (𝑛 ∈ Fin, 𝑟 ∈ V ↦ (𝑚 ∈ (𝑛 ConstPolyMat 𝑟) ↦ (𝑥𝑛, 𝑦𝑛 ↦ ((coe1‘(𝑥𝑚𝑦))‘0))))
Colors of variables: wff setvar class
This definition is referenced by:  cpm2mfval  22742
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