Definition df-subma 22502

Description: Define the submatrices of a square matrix. A submatrix is obtained by deleting a row and a column of the original matrix. Since the indices of a matrix need not to be sequential integers, it does not matter that there may be gaps in the numbering of the indices for the submatrix. The determinants of such submatrices are called the "minors" of the original matrix. (Contributed by AV, 27-Dec-2018.)

Assertion

Ref	Expression
df-subma	⊢ subMat = (𝑛 ∈ V, 𝑟 ∈ V ↦ (𝑚 ∈ (Base‘(𝑛 Mat 𝑟)) ↦ (𝑘 ∈ 𝑛, 𝑙 ∈ 𝑛 ↦ (𝑖 ∈ (𝑛 ∖ {𝑘}), 𝑗 ∈ (𝑛 ∖ {𝑙}) ↦ (𝑖𝑚𝑗)))))

Distinct variable group:   𝑛,𝑟,𝑚,𝑖,𝑗,𝑘,𝑙

Detailed syntax breakdown of Definition df-subma

Step	Hyp	Ref	Expression
1		csubma 22501	. 2 class subMat
2		vn	. . 3 setvar 𝑛
3		vr	. . 3 setvar 𝑟
4		cvv 3438	. . 3 class V
5		vm	. . . 4 setvar 𝑚
6	2	cv 1540	. . . . . 6 class 𝑛
7	3	cv 1540	. . . . . 6 class 𝑟
8		cmat 22332	. . . . . 6 class Mat
9	6, 7, 8	co 7355	. . . . 5 class (𝑛 Mat 𝑟)
10		cbs 17130	. . . . 5 class Base
11	9, 10	cfv 6489	. . . 4 class (Base‘(𝑛 Mat 𝑟))
12		vk	. . . . 5 setvar 𝑘
13		vl	. . . . 5 setvar 𝑙
14		vi	. . . . . 6 setvar 𝑖
15		vj	. . . . . 6 setvar 𝑗
16	12	cv 1540	. . . . . . . 8 class 𝑘
17	16	csn 4577	. . . . . . 7 class {𝑘}
18	6, 17	cdif 3896	. . . . . 6 class (𝑛 ∖ {𝑘})
19	13	cv 1540	. . . . . . . 8 class 𝑙
20	19	csn 4577	. . . . . . 7 class {𝑙}
21	6, 20	cdif 3896	. . . . . 6 class (𝑛 ∖ {𝑙})
22	14	cv 1540	. . . . . . 7 class 𝑖
23	15	cv 1540	. . . . . . 7 class 𝑗
24	5	cv 1540	. . . . . . 7 class 𝑚
25	22, 23, 24	co 7355	. . . . . 6 class (𝑖𝑚𝑗)
26	14, 15, 18, 21, 25	cmpo 7357	. . . . 5 class (𝑖 ∈ (𝑛 ∖ {𝑘}), 𝑗 ∈ (𝑛 ∖ {𝑙}) ↦ (𝑖𝑚𝑗))
27	12, 13, 6, 6, 26	cmpo 7357	. . . 4 class (𝑘 ∈ 𝑛, 𝑙 ∈ 𝑛 ↦ (𝑖 ∈ (𝑛 ∖ {𝑘}), 𝑗 ∈ (𝑛 ∖ {𝑙}) ↦ (𝑖𝑚𝑗)))
28	5, 11, 27	cmpt 5176	. . 3 class (𝑚 ∈ (Base‘(𝑛 Mat 𝑟)) ↦ (𝑘 ∈ 𝑛, 𝑙 ∈ 𝑛 ↦ (𝑖 ∈ (𝑛 ∖ {𝑘}), 𝑗 ∈ (𝑛 ∖ {𝑙}) ↦ (𝑖𝑚𝑗))))
29	2, 3, 4, 4, 28	cmpo 7357	. 2 class (𝑛 ∈ V, 𝑟 ∈ V ↦ (𝑚 ∈ (Base‘(𝑛 Mat 𝑟)) ↦ (𝑘 ∈ 𝑛, 𝑙 ∈ 𝑛 ↦ (𝑖 ∈ (𝑛 ∖ {𝑘}), 𝑗 ∈ (𝑛 ∖ {𝑙}) ↦ (𝑖𝑚𝑗)))))
30	1, 29	wceq 1541	1 wff subMat = (𝑛 ∈ V, 𝑟 ∈ V ↦ (𝑚 ∈ (Base‘(𝑛 Mat 𝑟)) ↦ (𝑘 ∈ 𝑛, 𝑙 ∈ 𝑛 ↦ (𝑖 ∈ (𝑛 ∖ {𝑘}), 𝑗 ∈ (𝑛 ∖ {𝑙}) ↦ (𝑖𝑚𝑗)))))

This definition is referenced by:  submafval  22504