Definition df-subma 22584

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 22583	. 2 class subMat
2		vn	. . 3 setvar 𝑛
3		vr	. . 3 setvar 𝑟
4		cvv 3479	. . 3 class V
5		vm	. . . 4 setvar 𝑚
6	2	cv 1538	. . . . . 6 class 𝑛
7	3	cv 1538	. . . . . 6 class 𝑟
8		cmat 22412	. . . . . 6 class Mat
9	6, 7, 8	co 7432	. . . . 5 class (𝑛 Mat 𝑟)
10		cbs 17248	. . . . 5 class Base
11	9, 10	cfv 6560	. . . 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 1538	. . . . . . . 8 class 𝑘
17	16	csn 4625	. . . . . . 7 class {𝑘}
18	6, 17	cdif 3947	. . . . . 6 class (𝑛 ∖ {𝑘})
19	13	cv 1538	. . . . . . . 8 class 𝑙
20	19	csn 4625	. . . . . . 7 class {𝑙}
21	6, 20	cdif 3947	. . . . . 6 class (𝑛 ∖ {𝑙})
22	14	cv 1538	. . . . . . 7 class 𝑖
23	15	cv 1538	. . . . . . 7 class 𝑗
24	5	cv 1538	. . . . . . 7 class 𝑚
25	22, 23, 24	co 7432	. . . . . 6 class (𝑖𝑚𝑗)
26	14, 15, 18, 21, 25	cmpo 7434	. . . . 5 class (𝑖 ∈ (𝑛 ∖ {𝑘}), 𝑗 ∈ (𝑛 ∖ {𝑙}) ↦ (𝑖𝑚𝑗))
27	12, 13, 6, 6, 26	cmpo 7434	. . . 4 class (𝑘 ∈ 𝑛, 𝑙 ∈ 𝑛 ↦ (𝑖 ∈ (𝑛 ∖ {𝑘}), 𝑗 ∈ (𝑛 ∖ {𝑙}) ↦ (𝑖𝑚𝑗)))
28	5, 11, 27	cmpt 5224	. . 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 7434	. 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 1539	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  22586