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Definition df-cpmat2mat 20733
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 20785, 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 20730 . 2 class cPolyMatToMat
2 vn . . 3 setvar 𝑛
3 vr . . 3 setvar 𝑟
4 cfn 8113 . . 3 class Fin
5 cvv 3351 . . 3 class V
6 vm . . . 4 setvar 𝑚
72cv 1630 . . . . 5 class 𝑛
83cv 1630 . . . . 5 class 𝑟
9 ccpmat 20728 . . . . 5 class ConstPolyMat
107, 8, 9co 6796 . . . 4 class (𝑛 ConstPolyMat 𝑟)
11 vx . . . . 5 setvar 𝑥
12 vy . . . . 5 setvar 𝑦
13 cc0 10142 . . . . . 6 class 0
1411cv 1630 . . . . . . . 8 class 𝑥
1512cv 1630 . . . . . . . 8 class 𝑦
166cv 1630 . . . . . . . 8 class 𝑚
1714, 15, 16co 6796 . . . . . . 7 class (𝑥𝑚𝑦)
18 cco1 19763 . . . . . . 7 class coe1
1917, 18cfv 6030 . . . . . 6 class (coe1‘(𝑥𝑚𝑦))
2013, 19cfv 6030 . . . . 5 class ((coe1‘(𝑥𝑚𝑦))‘0)
2111, 12, 7, 7, 20cmpt2 6798 . . . 4 class (𝑥𝑛, 𝑦𝑛 ↦ ((coe1‘(𝑥𝑚𝑦))‘0))
226, 10, 21cmpt 4864 . . 3 class (𝑚 ∈ (𝑛 ConstPolyMat 𝑟) ↦ (𝑥𝑛, 𝑦𝑛 ↦ ((coe1‘(𝑥𝑚𝑦))‘0)))
232, 3, 4, 5, 22cmpt2 6798 . 2 class (𝑛 ∈ Fin, 𝑟 ∈ V ↦ (𝑚 ∈ (𝑛 ConstPolyMat 𝑟) ↦ (𝑥𝑛, 𝑦𝑛 ↦ ((coe1‘(𝑥𝑚𝑦))‘0))))
241, 23wceq 1631 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  20774
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