Definition df-cpmat2mat 22683

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 22735, 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) ↦ (𝑥 ∈ 𝑛, 𝑦 ∈ 𝑛 ↦ ((coe1‘(𝑥𝑚𝑦))‘0))))

Distinct variable group:   𝑚,𝑛,𝑟,𝑥,𝑦

Detailed syntax breakdown of Definition df-cpmat2mat

Step	Hyp	Ref	Expression
1		ccpmat2mat 22680	. 2 class cPolyMatToMat
2		vn	. . 3 setvar 𝑛
3		vr	. . 3 setvar 𝑟
4		cfn 8886	. . 3 class Fin
5		cvv 3430	. . 3 class V
6		vm	. . . 4 setvar 𝑚
7	2	cv 1541	. . . . 5 class 𝑛
8	3	cv 1541	. . . . 5 class 𝑟
9		ccpmat 22678	. . . . 5 class ConstPolyMat
10	7, 8, 9	co 7360	. . . 4 class (𝑛 ConstPolyMat 𝑟)
11		vx	. . . . 5 setvar 𝑥
12		vy	. . . . 5 setvar 𝑦
13		cc0 11029	. . . . . 6 class 0
14	11	cv 1541	. . . . . . . 8 class 𝑥
15	12	cv 1541	. . . . . . . 8 class 𝑦
16	6	cv 1541	. . . . . . . 8 class 𝑚
17	14, 15, 16	co 7360	. . . . . . 7 class (𝑥𝑚𝑦)
18		cco1 22151	. . . . . . 7 class coe1
19	17, 18	cfv 6492	. . . . . 6 class (coe1‘(𝑥𝑚𝑦))
20	13, 19	cfv 6492	. . . . 5 class ((coe1‘(𝑥𝑚𝑦))‘0)
21	11, 12, 7, 7, 20	cmpo 7362	. . . 4 class (𝑥 ∈ 𝑛, 𝑦 ∈ 𝑛 ↦ ((coe1‘(𝑥𝑚𝑦))‘0))
22	6, 10, 21	cmpt 5167	. . 3 class (𝑚 ∈ (𝑛 ConstPolyMat 𝑟) ↦ (𝑥 ∈ 𝑛, 𝑦 ∈ 𝑛 ↦ ((coe1‘(𝑥𝑚𝑦))‘0)))
23	2, 3, 4, 5, 22	cmpo 7362	. 2 class (𝑛 ∈ Fin, 𝑟 ∈ V ↦ (𝑚 ∈ (𝑛 ConstPolyMat 𝑟) ↦ (𝑥 ∈ 𝑛, 𝑦 ∈ 𝑛 ↦ ((coe1‘(𝑥𝑚𝑦))‘0))))
24	1, 23	wceq 1542	1 wff cPolyMatToMat = (𝑛 ∈ Fin, 𝑟 ∈ V ↦ (𝑚 ∈ (𝑛 ConstPolyMat 𝑟) ↦ (𝑥 ∈ 𝑛, 𝑦 ∈ 𝑛 ↦ ((coe1‘(𝑥𝑚𝑦))‘0))))

This definition is referenced by:  cpm2mfval  22724