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Definition df-cpmat2mat 20332
 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 20384, 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 20329 . 2 class cPolyMatToMat
2 vn . . 3 setvar 𝑛
3 vr . . 3 setvar 𝑟
4 cfn 7841 . . 3 class Fin
5 cvv 3173 . . 3 class V
6 vm . . . 4 setvar 𝑚
72cv 1474 . . . . 5 class 𝑛
83cv 1474 . . . . 5 class 𝑟
9 ccpmat 20327 . . . . 5 class ConstPolyMat
107, 8, 9co 6549 . . . 4 class (𝑛 ConstPolyMat 𝑟)
11 vx . . . . 5 setvar 𝑥
12 vy . . . . 5 setvar 𝑦
13 cc0 9815 . . . . . 6 class 0
1411cv 1474 . . . . . . . 8 class 𝑥
1512cv 1474 . . . . . . . 8 class 𝑦
166cv 1474 . . . . . . . 8 class 𝑚
1714, 15, 16co 6549 . . . . . . 7 class (𝑥𝑚𝑦)
18 cco1 19369 . . . . . . 7 class coe1
1917, 18cfv 5804 . . . . . 6 class (coe1‘(𝑥𝑚𝑦))
2013, 19cfv 5804 . . . . 5 class ((coe1‘(𝑥𝑚𝑦))‘0)
2111, 12, 7, 7, 20cmpt2 6551 . . . 4 class (𝑥𝑛, 𝑦𝑛 ↦ ((coe1‘(𝑥𝑚𝑦))‘0))
226, 10, 21cmpt 4643 . . 3 class (𝑚 ∈ (𝑛 ConstPolyMat 𝑟) ↦ (𝑥𝑛, 𝑦𝑛 ↦ ((coe1‘(𝑥𝑚𝑦))‘0)))
232, 3, 4, 5, 22cmpt2 6551 . 2 class (𝑛 ∈ Fin, 𝑟 ∈ V ↦ (𝑚 ∈ (𝑛 ConstPolyMat 𝑟) ↦ (𝑥𝑛, 𝑦𝑛 ↦ ((coe1‘(𝑥𝑚𝑦))‘0))))
241, 23wceq 1475 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  20373
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