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Definition df-subma 20377
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)) ↦ (𝑘𝑛, 𝑙𝑛 ↦ (𝑖 ∈ (𝑛 ∖ {𝑘}), 𝑗 ∈ (𝑛 ∖ {𝑙}) ↦ (𝑖𝑚𝑗)))))
Distinct variable group:   𝑛,𝑟,𝑚,𝑖,𝑗,𝑘,𝑙

Detailed syntax breakdown of Definition df-subma
StepHypRef Expression
1 csubma 20376 . 2 class subMat
2 vn . . 3 setvar 𝑛
3 vr . . 3 setvar 𝑟
4 cvv 3198 . . 3 class V
5 vm . . . 4 setvar 𝑚
62cv 1481 . . . . . 6 class 𝑛
73cv 1481 . . . . . 6 class 𝑟
8 cmat 20207 . . . . . 6 class Mat
96, 7, 8co 6647 . . . . 5 class (𝑛 Mat 𝑟)
10 cbs 15851 . . . . 5 class Base
119, 10cfv 5886 . . . 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
1612cv 1481 . . . . . . . 8 class 𝑘
1716csn 4175 . . . . . . 7 class {𝑘}
186, 17cdif 3569 . . . . . 6 class (𝑛 ∖ {𝑘})
1913cv 1481 . . . . . . . 8 class 𝑙
2019csn 4175 . . . . . . 7 class {𝑙}
216, 20cdif 3569 . . . . . 6 class (𝑛 ∖ {𝑙})
2214cv 1481 . . . . . . 7 class 𝑖
2315cv 1481 . . . . . . 7 class 𝑗
245cv 1481 . . . . . . 7 class 𝑚
2522, 23, 24co 6647 . . . . . 6 class (𝑖𝑚𝑗)
2614, 15, 18, 21, 25cmpt2 6649 . . . . 5 class (𝑖 ∈ (𝑛 ∖ {𝑘}), 𝑗 ∈ (𝑛 ∖ {𝑙}) ↦ (𝑖𝑚𝑗))
2712, 13, 6, 6, 26cmpt2 6649 . . . 4 class (𝑘𝑛, 𝑙𝑛 ↦ (𝑖 ∈ (𝑛 ∖ {𝑘}), 𝑗 ∈ (𝑛 ∖ {𝑙}) ↦ (𝑖𝑚𝑗)))
285, 11, 27cmpt 4727 . . 3 class (𝑚 ∈ (Base‘(𝑛 Mat 𝑟)) ↦ (𝑘𝑛, 𝑙𝑛 ↦ (𝑖 ∈ (𝑛 ∖ {𝑘}), 𝑗 ∈ (𝑛 ∖ {𝑙}) ↦ (𝑖𝑚𝑗))))
292, 3, 4, 4, 28cmpt2 6649 . 2 class (𝑛 ∈ V, 𝑟 ∈ V ↦ (𝑚 ∈ (Base‘(𝑛 Mat 𝑟)) ↦ (𝑘𝑛, 𝑙𝑛 ↦ (𝑖 ∈ (𝑛 ∖ {𝑘}), 𝑗 ∈ (𝑛 ∖ {𝑙}) ↦ (𝑖𝑚𝑗)))))
301, 29wceq 1482 1 wff subMat = (𝑛 ∈ V, 𝑟 ∈ V ↦ (𝑚 ∈ (Base‘(𝑛 Mat 𝑟)) ↦ (𝑘𝑛, 𝑙𝑛 ↦ (𝑖 ∈ (𝑛 ∖ {𝑘}), 𝑗 ∈ (𝑛 ∖ {𝑙}) ↦ (𝑖𝑚𝑗)))))
Colors of variables: wff setvar class
This definition is referenced by:  submafval  20379
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