Step |
Hyp |
Ref |
Expression |
0 |
|
csmat |
⊢ subMat1 |
1 |
|
vm |
⊢ 𝑚 |
2 |
|
cvv |
⊢ V |
3 |
|
vk |
⊢ 𝑘 |
4 |
|
cn |
⊢ ℕ |
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 ) ) 〉 ) ) ) ) |