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Definition df-subma 22509
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 22508 . 2 class subMat
2 vn . . 3 setvar 𝑛
3 vr . . 3 setvar 𝑟
4 cvv 3463 . . 3 class V
5 vm . . . 4 setvar 𝑚
62cv 1532 . . . . . 6 class 𝑛
73cv 1532 . . . . . 6 class 𝑟
8 cmat 22337 . . . . . 6 class Mat
96, 7, 8co 7417 . . . . 5 class (𝑛 Mat 𝑟)
10 cbs 17179 . . . . 5 class Base
119, 10cfv 6547 . . . 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 1532 . . . . . . . 8 class 𝑘
1716csn 4629 . . . . . . 7 class {𝑘}
186, 17cdif 3942 . . . . . 6 class (𝑛 ∖ {𝑘})
1913cv 1532 . . . . . . . 8 class 𝑙
2019csn 4629 . . . . . . 7 class {𝑙}
216, 20cdif 3942 . . . . . 6 class (𝑛 ∖ {𝑙})
2214cv 1532 . . . . . . 7 class 𝑖
2315cv 1532 . . . . . . 7 class 𝑗
245cv 1532 . . . . . . 7 class 𝑚
2522, 23, 24co 7417 . . . . . 6 class (𝑖𝑚𝑗)
2614, 15, 18, 21, 25cmpo 7419 . . . . 5 class (𝑖 ∈ (𝑛 ∖ {𝑘}), 𝑗 ∈ (𝑛 ∖ {𝑙}) ↦ (𝑖𝑚𝑗))
2712, 13, 6, 6, 26cmpo 7419 . . . 4 class (𝑘𝑛, 𝑙𝑛 ↦ (𝑖 ∈ (𝑛 ∖ {𝑘}), 𝑗 ∈ (𝑛 ∖ {𝑙}) ↦ (𝑖𝑚𝑗)))
285, 11, 27cmpt 5231 . . 3 class (𝑚 ∈ (Base‘(𝑛 Mat 𝑟)) ↦ (𝑘𝑛, 𝑙𝑛 ↦ (𝑖 ∈ (𝑛 ∖ {𝑘}), 𝑗 ∈ (𝑛 ∖ {𝑙}) ↦ (𝑖𝑚𝑗))))
292, 3, 4, 4, 28cmpo 7419 . 2 class (𝑛 ∈ V, 𝑟 ∈ V ↦ (𝑚 ∈ (Base‘(𝑛 Mat 𝑟)) ↦ (𝑘𝑛, 𝑙𝑛 ↦ (𝑖 ∈ (𝑛 ∖ {𝑘}), 𝑗 ∈ (𝑛 ∖ {𝑙}) ↦ (𝑖𝑚𝑗)))))
301, 29wceq 1533 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  22511
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