Definition df-subma 22078

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 22077	. 2 class subMat
2		vn	. . 3 setvar 𝑛
3		vr	. . 3 setvar 𝑟
4		cvv 3474	. . 3 class V
5		vm	. . . 4 setvar 𝑚
6	2	cv 1540	. . . . . 6 class 𝑛
7	3	cv 1540	. . . . . 6 class 𝑟
8		cmat 21906	. . . . . 6 class Mat
9	6, 7, 8	co 7408	. . . . 5 class (𝑛 Mat 𝑟)
10		cbs 17143	. . . . 5 class Base
11	9, 10	cfv 6543	. . . 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 4628	. . . . . . 7 class {𝑘}
18	6, 17	cdif 3945	. . . . . 6 class (𝑛 ∖ {𝑘})
19	13	cv 1540	. . . . . . . 8 class 𝑙
20	19	csn 4628	. . . . . . 7 class {𝑙}
21	6, 20	cdif 3945	. . . . . 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 7408	. . . . . 6 class (𝑖𝑚𝑗)
26	14, 15, 18, 21, 25	cmpo 7410	. . . . 5 class (𝑖 ∈ (𝑛 ∖ {𝑘}), 𝑗 ∈ (𝑛 ∖ {𝑙}) ↦ (𝑖𝑚𝑗))
27	12, 13, 6, 6, 26	cmpo 7410	. . . 4 class (𝑘 ∈ 𝑛, 𝑙 ∈ 𝑛 ↦ (𝑖 ∈ (𝑛 ∖ {𝑘}), 𝑗 ∈ (𝑛 ∖ {𝑙}) ↦ (𝑖𝑚𝑗)))
28	5, 11, 27	cmpt 5231	. . 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 7410	. 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  22080