Definition df-cpmat2mat 21315

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 21367, 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 21312	. 2 class cPolyMatToMat
2		vn	. . 3 setvar 𝑛
3		vr	. . 3 setvar 𝑟
4		cfn 8508	. . 3 class Fin
5		cvv 3494	. . 3 class V
6		vm	. . . 4 setvar 𝑚
7	2	cv 1532	. . . . 5 class 𝑛
8	3	cv 1532	. . . . 5 class 𝑟
9		ccpmat 21310	. . . . 5 class ConstPolyMat
10	7, 8, 9	co 7155	. . . 4 class (𝑛 ConstPolyMat 𝑟)
11		vx	. . . . 5 setvar 𝑥
12		vy	. . . . 5 setvar 𝑦
13		cc0 10536	. . . . . 6 class 0
14	11	cv 1532	. . . . . . . 8 class 𝑥
15	12	cv 1532	. . . . . . . 8 class 𝑦
16	6	cv 1532	. . . . . . . 8 class 𝑚
17	14, 15, 16	co 7155	. . . . . . 7 class (𝑥𝑚𝑦)
18		cco1 20345	. . . . . . 7 class coe1
19	17, 18	cfv 6354	. . . . . 6 class (coe1‘(𝑥𝑚𝑦))
20	13, 19	cfv 6354	. . . . 5 class ((coe1‘(𝑥𝑚𝑦))‘0)
21	11, 12, 7, 7, 20	cmpo 7157	. . . 4 class (𝑥 ∈ 𝑛, 𝑦 ∈ 𝑛 ↦ ((coe1‘(𝑥𝑚𝑦))‘0))
22	6, 10, 21	cmpt 5145	. . 3 class (𝑚 ∈ (𝑛 ConstPolyMat 𝑟) ↦ (𝑥 ∈ 𝑛, 𝑦 ∈ 𝑛 ↦ ((coe1‘(𝑥𝑚𝑦))‘0)))
23	2, 3, 4, 5, 22	cmpo 7157	. 2 class (𝑛 ∈ Fin, 𝑟 ∈ V ↦ (𝑚 ∈ (𝑛 ConstPolyMat 𝑟) ↦ (𝑥 ∈ 𝑛, 𝑦 ∈ 𝑛 ↦ ((coe1‘(𝑥𝑚𝑦))‘0))))
24	1, 23	wceq 1533	1 wff cPolyMatToMat = (𝑛 ∈ Fin, 𝑟 ∈ V ↦ (𝑚 ∈ (𝑛 ConstPolyMat 𝑟) ↦ (𝑥 ∈ 𝑛, 𝑦 ∈ 𝑛 ↦ ((coe1‘(𝑥𝑚𝑦))‘0))))

This definition is referenced by:  cpm2mfval  21356