Definition df-cpmat2mat 22643

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 22695, 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 22640	. 2 class cPolyMatToMat
2		vn	. . 3 setvar 𝑛
3		vr	. . 3 setvar 𝑟
4		cfn 8879	. . 3 class Fin
5		cvv 3437	. . 3 class V
6		vm	. . . 4 setvar 𝑚
7	2	cv 1540	. . . . 5 class 𝑛
8	3	cv 1540	. . . . 5 class 𝑟
9		ccpmat 22638	. . . . 5 class ConstPolyMat
10	7, 8, 9	co 7355	. . . 4 class (𝑛 ConstPolyMat 𝑟)
11		vx	. . . . 5 setvar 𝑥
12		vy	. . . . 5 setvar 𝑦
13		cc0 11017	. . . . . 6 class 0
14	11	cv 1540	. . . . . . . 8 class 𝑥
15	12	cv 1540	. . . . . . . 8 class 𝑦
16	6	cv 1540	. . . . . . . 8 class 𝑚
17	14, 15, 16	co 7355	. . . . . . 7 class (𝑥𝑚𝑦)
18		cco1 22109	. . . . . . 7 class coe1
19	17, 18	cfv 6489	. . . . . 6 class (coe1‘(𝑥𝑚𝑦))
20	13, 19	cfv 6489	. . . . 5 class ((coe1‘(𝑥𝑚𝑦))‘0)
21	11, 12, 7, 7, 20	cmpo 7357	. . . 4 class (𝑥 ∈ 𝑛, 𝑦 ∈ 𝑛 ↦ ((coe1‘(𝑥𝑚𝑦))‘0))
22	6, 10, 21	cmpt 5176	. . 3 class (𝑚 ∈ (𝑛 ConstPolyMat 𝑟) ↦ (𝑥 ∈ 𝑛, 𝑦 ∈ 𝑛 ↦ ((coe1‘(𝑥𝑚𝑦))‘0)))
23	2, 3, 4, 5, 22	cmpo 7357	. 2 class (𝑛 ∈ Fin, 𝑟 ∈ V ↦ (𝑚 ∈ (𝑛 ConstPolyMat 𝑟) ↦ (𝑥 ∈ 𝑛, 𝑦 ∈ 𝑛 ↦ ((coe1‘(𝑥𝑚𝑦))‘0))))
24	1, 23	wceq 1541	1 wff cPolyMatToMat = (𝑛 ∈ Fin, 𝑟 ∈ V ↦ (𝑚 ∈ (𝑛 ConstPolyMat 𝑟) ↦ (𝑥 ∈ 𝑛, 𝑦 ∈ 𝑛 ↦ ((coe1‘(𝑥𝑚𝑦))‘0))))

This definition is referenced by:  cpm2mfval  22684