Definition df-cpmat2mat 22646

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 22698, 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 22643	. 2 class cPolyMatToMat
2		vn	. . 3 setvar 𝑛
3		vr	. . 3 setvar 𝑟
4		cfn 8959	. . 3 class Fin
5		cvv 3459	. . 3 class V
6		vm	. . . 4 setvar 𝑚
7	2	cv 1539	. . . . 5 class 𝑛
8	3	cv 1539	. . . . 5 class 𝑟
9		ccpmat 22641	. . . . 5 class ConstPolyMat
10	7, 8, 9	co 7405	. . . 4 class (𝑛 ConstPolyMat 𝑟)
11		vx	. . . . 5 setvar 𝑥
12		vy	. . . . 5 setvar 𝑦
13		cc0 11129	. . . . . 6 class 0
14	11	cv 1539	. . . . . . . 8 class 𝑥
15	12	cv 1539	. . . . . . . 8 class 𝑦
16	6	cv 1539	. . . . . . . 8 class 𝑚
17	14, 15, 16	co 7405	. . . . . . 7 class (𝑥𝑚𝑦)
18		cco1 22113	. . . . . . 7 class coe1
19	17, 18	cfv 6531	. . . . . 6 class (coe1‘(𝑥𝑚𝑦))
20	13, 19	cfv 6531	. . . . 5 class ((coe1‘(𝑥𝑚𝑦))‘0)
21	11, 12, 7, 7, 20	cmpo 7407	. . . 4 class (𝑥 ∈ 𝑛, 𝑦 ∈ 𝑛 ↦ ((coe1‘(𝑥𝑚𝑦))‘0))
22	6, 10, 21	cmpt 5201	. . 3 class (𝑚 ∈ (𝑛 ConstPolyMat 𝑟) ↦ (𝑥 ∈ 𝑛, 𝑦 ∈ 𝑛 ↦ ((coe1‘(𝑥𝑚𝑦))‘0)))
23	2, 3, 4, 5, 22	cmpo 7407	. 2 class (𝑛 ∈ Fin, 𝑟 ∈ V ↦ (𝑚 ∈ (𝑛 ConstPolyMat 𝑟) ↦ (𝑥 ∈ 𝑛, 𝑦 ∈ 𝑛 ↦ ((coe1‘(𝑥𝑚𝑦))‘0))))
24	1, 23	wceq 1540	1 wff cPolyMatToMat = (𝑛 ∈ Fin, 𝑟 ∈ V ↦ (𝑚 ∈ (𝑛 ConstPolyMat 𝑟) ↦ (𝑥 ∈ 𝑛, 𝑦 ∈ 𝑛 ↦ ((coe1‘(𝑥𝑚𝑦))‘0))))

This definition is referenced by:  cpm2mfval  22687