Definition df-subma 20758

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 20757	. 2 class subMat
2		vn	. . 3 setvar 𝑛
3		vr	. . 3 setvar 𝑟
4		cvv 3414	. . 3 class V
5		vm	. . . 4 setvar 𝑚
6	2	cv 1655	. . . . . 6 class 𝑛
7	3	cv 1655	. . . . . 6 class 𝑟
8		cmat 20587	. . . . . 6 class Mat
9	6, 7, 8	co 6910	. . . . 5 class (𝑛 Mat 𝑟)
10		cbs 16229	. . . . 5 class Base
11	9, 10	cfv 6127	. . . 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 1655	. . . . . . . 8 class 𝑘
17	16	csn 4399	. . . . . . 7 class {𝑘}
18	6, 17	cdif 3795	. . . . . 6 class (𝑛 ∖ {𝑘})
19	13	cv 1655	. . . . . . . 8 class 𝑙
20	19	csn 4399	. . . . . . 7 class {𝑙}
21	6, 20	cdif 3795	. . . . . 6 class (𝑛 ∖ {𝑙})
22	14	cv 1655	. . . . . . 7 class 𝑖
23	15	cv 1655	. . . . . . 7 class 𝑗
24	5	cv 1655	. . . . . . 7 class 𝑚
25	22, 23, 24	co 6910	. . . . . 6 class (𝑖𝑚𝑗)
26	14, 15, 18, 21, 25	cmpt2 6912	. . . . 5 class (𝑖 ∈ (𝑛 ∖ {𝑘}), 𝑗 ∈ (𝑛 ∖ {𝑙}) ↦ (𝑖𝑚𝑗))
27	12, 13, 6, 6, 26	cmpt2 6912	. . . 4 class (𝑘 ∈ 𝑛, 𝑙 ∈ 𝑛 ↦ (𝑖 ∈ (𝑛 ∖ {𝑘}), 𝑗 ∈ (𝑛 ∖ {𝑙}) ↦ (𝑖𝑚𝑗)))
28	5, 11, 27	cmpt 4954	. . 3 class (𝑚 ∈ (Base‘(𝑛 Mat 𝑟)) ↦ (𝑘 ∈ 𝑛, 𝑙 ∈ 𝑛 ↦ (𝑖 ∈ (𝑛 ∖ {𝑘}), 𝑗 ∈ (𝑛 ∖ {𝑙}) ↦ (𝑖𝑚𝑗))))
29	2, 3, 4, 4, 28	cmpt2 6912	. 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 1656	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  20760