Metamath Proof Explorer


Definition df-smat

Description: Define a function generating submatrices of an integer-indexed matrix. The function maps an index in ( ( 1 ... M ) X. ( 1 ... N ) ) into a new index in ( ( 1 ... ( M - 1 ) ) X. ( 1 ... ( N - 1 ) ) ) . A submatrix is obtained by deleting a row and a column of the original matrix. Because this function re-indexes the matrix, the resulting submatrix still has the same index set for rows and columns, and its determinent is defined, unlike the current df-subma . (Contributed by Thierry Arnoux, 18-Aug-2020)

Ref Expression
Assertion df-smat subMat1 = ( 𝑚 ∈ V ↦ ( 𝑘 ∈ ℕ , 𝑙 ∈ ℕ ↦ ( 𝑚 ∘ ( 𝑖 ∈ ℕ , 𝑗 ∈ ℕ ↦ ⟨ if ( 𝑖 < 𝑘 , 𝑖 , ( 𝑖 + 1 ) ) , if ( 𝑗 < 𝑙 , 𝑗 , ( 𝑗 + 1 ) ) ⟩ ) ) ) )

Detailed syntax breakdown

Step Hyp Ref Expression
0 csmat subMat1
1 vm 𝑚
2 cvv V
3 vk 𝑘
4 cn
5 vl 𝑙
6 1 cv 𝑚
7 vi 𝑖
8 vj 𝑗
9 7 cv 𝑖
10 clt <
11 3 cv 𝑘
12 9 11 10 wbr 𝑖 < 𝑘
13 caddc +
14 c1 1
15 9 14 13 co ( 𝑖 + 1 )
16 12 9 15 cif if ( 𝑖 < 𝑘 , 𝑖 , ( 𝑖 + 1 ) )
17 8 cv 𝑗
18 5 cv 𝑙
19 17 18 10 wbr 𝑗 < 𝑙
20 17 14 13 co ( 𝑗 + 1 )
21 19 17 20 cif if ( 𝑗 < 𝑙 , 𝑗 , ( 𝑗 + 1 ) )
22 16 21 cop ⟨ if ( 𝑖 < 𝑘 , 𝑖 , ( 𝑖 + 1 ) ) , if ( 𝑗 < 𝑙 , 𝑗 , ( 𝑗 + 1 ) ) ⟩
23 7 8 4 4 22 cmpo ( 𝑖 ∈ ℕ , 𝑗 ∈ ℕ ↦ ⟨ if ( 𝑖 < 𝑘 , 𝑖 , ( 𝑖 + 1 ) ) , if ( 𝑗 < 𝑙 , 𝑗 , ( 𝑗 + 1 ) ) ⟩ )
24 6 23 ccom ( 𝑚 ∘ ( 𝑖 ∈ ℕ , 𝑗 ∈ ℕ ↦ ⟨ if ( 𝑖 < 𝑘 , 𝑖 , ( 𝑖 + 1 ) ) , if ( 𝑗 < 𝑙 , 𝑗 , ( 𝑗 + 1 ) ) ⟩ ) )
25 3 5 4 4 24 cmpo ( 𝑘 ∈ ℕ , 𝑙 ∈ ℕ ↦ ( 𝑚 ∘ ( 𝑖 ∈ ℕ , 𝑗 ∈ ℕ ↦ ⟨ if ( 𝑖 < 𝑘 , 𝑖 , ( 𝑖 + 1 ) ) , if ( 𝑗 < 𝑙 , 𝑗 , ( 𝑗 + 1 ) ) ⟩ ) ) )
26 1 2 25 cmpt ( 𝑚 ∈ V ↦ ( 𝑘 ∈ ℕ , 𝑙 ∈ ℕ ↦ ( 𝑚 ∘ ( 𝑖 ∈ ℕ , 𝑗 ∈ ℕ ↦ ⟨ if ( 𝑖 < 𝑘 , 𝑖 , ( 𝑖 + 1 ) ) , if ( 𝑗 < 𝑙 , 𝑗 , ( 𝑗 + 1 ) ) ⟩ ) ) ) )
27 0 26 wceq subMat1 = ( 𝑚 ∈ V ↦ ( 𝑘 ∈ ℕ , 𝑙 ∈ ℕ ↦ ( 𝑚 ∘ ( 𝑖 ∈ ℕ , 𝑗 ∈ ℕ ↦ ⟨ if ( 𝑖 < 𝑘 , 𝑖 , ( 𝑖 + 1 ) ) , if ( 𝑗 < 𝑙 , 𝑗 , ( 𝑗 + 1 ) ) ⟩ ) ) ) )