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Definition df-cpmat2mat 20838
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 20890, 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 20835 . 2 class cPolyMatToMat
2 vn . . 3 setvar 𝑛
3 vr . . 3 setvar 𝑟
4 cfn 8194 . . 3 class Fin
5 cvv 3384 . . 3 class V
6 vm . . . 4 setvar 𝑚
72cv 1652 . . . . 5 class 𝑛
83cv 1652 . . . . 5 class 𝑟
9 ccpmat 20833 . . . . 5 class ConstPolyMat
107, 8, 9co 6877 . . . 4 class (𝑛 ConstPolyMat 𝑟)
11 vx . . . . 5 setvar 𝑥
12 vy . . . . 5 setvar 𝑦
13 cc0 10223 . . . . . 6 class 0
1411cv 1652 . . . . . . . 8 class 𝑥
1512cv 1652 . . . . . . . 8 class 𝑦
166cv 1652 . . . . . . . 8 class 𝑚
1714, 15, 16co 6877 . . . . . . 7 class (𝑥𝑚𝑦)
18 cco1 19867 . . . . . . 7 class coe1
1917, 18cfv 6100 . . . . . 6 class (coe1‘(𝑥𝑚𝑦))
2013, 19cfv 6100 . . . . 5 class ((coe1‘(𝑥𝑚𝑦))‘0)
2111, 12, 7, 7, 20cmpt2 6879 . . . 4 class (𝑥𝑛, 𝑦𝑛 ↦ ((coe1‘(𝑥𝑚𝑦))‘0))
226, 10, 21cmpt 4921 . . 3 class (𝑚 ∈ (𝑛 ConstPolyMat 𝑟) ↦ (𝑥𝑛, 𝑦𝑛 ↦ ((coe1‘(𝑥𝑚𝑦))‘0)))
232, 3, 4, 5, 22cmpt2 6879 . 2 class (𝑛 ∈ Fin, 𝑟 ∈ V ↦ (𝑚 ∈ (𝑛 ConstPolyMat 𝑟) ↦ (𝑥𝑛, 𝑦𝑛 ↦ ((coe1‘(𝑥𝑚𝑦))‘0))))
241, 23wceq 1653 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  20879
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