Definition df-cpmat2mat 21313

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 21365, 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 21310	. 2 class cPolyMatToMat
2		vn	. . 3 setvar 𝑛
3		vr	. . 3 setvar 𝑟
4		cfn 8492	. . 3 class Fin
5		cvv 3441	. . 3 class V
6		vm	. . . 4 setvar 𝑚
7	2	cv 1537	. . . . 5 class 𝑛
8	3	cv 1537	. . . . 5 class 𝑟
9		ccpmat 21308	. . . . 5 class ConstPolyMat
10	7, 8, 9	co 7135	. . . 4 class (𝑛 ConstPolyMat 𝑟)
11		vx	. . . . 5 setvar 𝑥
12		vy	. . . . 5 setvar 𝑦
13		cc0 10526	. . . . . 6 class 0
14	11	cv 1537	. . . . . . . 8 class 𝑥
15	12	cv 1537	. . . . . . . 8 class 𝑦
16	6	cv 1537	. . . . . . . 8 class 𝑚
17	14, 15, 16	co 7135	. . . . . . 7 class (𝑥𝑚𝑦)
18		cco1 20807	. . . . . . 7 class coe1
19	17, 18	cfv 6324	. . . . . 6 class (coe1‘(𝑥𝑚𝑦))
20	13, 19	cfv 6324	. . . . 5 class ((coe1‘(𝑥𝑚𝑦))‘0)
21	11, 12, 7, 7, 20	cmpo 7137	. . . 4 class (𝑥 ∈ 𝑛, 𝑦 ∈ 𝑛 ↦ ((coe1‘(𝑥𝑚𝑦))‘0))
22	6, 10, 21	cmpt 5110	. . 3 class (𝑚 ∈ (𝑛 ConstPolyMat 𝑟) ↦ (𝑥 ∈ 𝑛, 𝑦 ∈ 𝑛 ↦ ((coe1‘(𝑥𝑚𝑦))‘0)))
23	2, 3, 4, 5, 22	cmpo 7137	. 2 class (𝑛 ∈ Fin, 𝑟 ∈ V ↦ (𝑚 ∈ (𝑛 ConstPolyMat 𝑟) ↦ (𝑥 ∈ 𝑛, 𝑦 ∈ 𝑛 ↦ ((coe1‘(𝑥𝑚𝑦))‘0))))
24	1, 23	wceq 1538	1 wff cPolyMatToMat = (𝑛 ∈ Fin, 𝑟 ∈ V ↦ (𝑚 ∈ (𝑛 ConstPolyMat 𝑟) ↦ (𝑥 ∈ 𝑛, 𝑦 ∈ 𝑛 ↦ ((coe1‘(𝑥𝑚𝑦))‘0))))

This definition is referenced by:  cpm2mfval  21354