Step |
Hyp |
Ref |
Expression |
1 |
|
smat.s |
⊢ 𝑆 = ( 𝐾 ( subMat1 ‘ 𝐴 ) 𝐿 ) |
2 |
|
smat.m |
⊢ ( 𝜑 → 𝑀 ∈ ℕ ) |
3 |
|
smat.n |
⊢ ( 𝜑 → 𝑁 ∈ ℕ ) |
4 |
|
smat.k |
⊢ ( 𝜑 → 𝐾 ∈ ( 1 ... 𝑀 ) ) |
5 |
|
smat.l |
⊢ ( 𝜑 → 𝐿 ∈ ( 1 ... 𝑁 ) ) |
6 |
|
smat.a |
⊢ ( 𝜑 → 𝐴 ∈ ( 𝐵 ↑m ( ( 1 ... 𝑀 ) × ( 1 ... 𝑁 ) ) ) ) |
7 |
|
smatbl.i |
⊢ ( 𝜑 → 𝐼 ∈ ( 1 ..^ 𝐾 ) ) |
8 |
|
smatbl.j |
⊢ ( 𝜑 → 𝐽 ∈ ( 𝐿 ... 𝑁 ) ) |
9 |
|
fzossnn |
⊢ ( 1 ..^ 𝐾 ) ⊆ ℕ |
10 |
9 7
|
sselid |
⊢ ( 𝜑 → 𝐼 ∈ ℕ ) |
11 |
|
fz1ssnn |
⊢ ( 1 ... 𝑁 ) ⊆ ℕ |
12 |
11 5
|
sselid |
⊢ ( 𝜑 → 𝐿 ∈ ℕ ) |
13 |
|
fzssnn |
⊢ ( 𝐿 ∈ ℕ → ( 𝐿 ... 𝑁 ) ⊆ ℕ ) |
14 |
12 13
|
syl |
⊢ ( 𝜑 → ( 𝐿 ... 𝑁 ) ⊆ ℕ ) |
15 |
14 8
|
sseldd |
⊢ ( 𝜑 → 𝐽 ∈ ℕ ) |
16 |
|
elfzolt2 |
⊢ ( 𝐼 ∈ ( 1 ..^ 𝐾 ) → 𝐼 < 𝐾 ) |
17 |
7 16
|
syl |
⊢ ( 𝜑 → 𝐼 < 𝐾 ) |
18 |
17
|
iftrued |
⊢ ( 𝜑 → if ( 𝐼 < 𝐾 , 𝐼 , ( 𝐼 + 1 ) ) = 𝐼 ) |
19 |
|
elfzle1 |
⊢ ( 𝐽 ∈ ( 𝐿 ... 𝑁 ) → 𝐿 ≤ 𝐽 ) |
20 |
8 19
|
syl |
⊢ ( 𝜑 → 𝐿 ≤ 𝐽 ) |
21 |
12
|
nnred |
⊢ ( 𝜑 → 𝐿 ∈ ℝ ) |
22 |
15
|
nnred |
⊢ ( 𝜑 → 𝐽 ∈ ℝ ) |
23 |
21 22
|
lenltd |
⊢ ( 𝜑 → ( 𝐿 ≤ 𝐽 ↔ ¬ 𝐽 < 𝐿 ) ) |
24 |
20 23
|
mpbid |
⊢ ( 𝜑 → ¬ 𝐽 < 𝐿 ) |
25 |
24
|
iffalsed |
⊢ ( 𝜑 → if ( 𝐽 < 𝐿 , 𝐽 , ( 𝐽 + 1 ) ) = ( 𝐽 + 1 ) ) |
26 |
1 2 3 4 5 6 10 15 18 25
|
smatlem |
⊢ ( 𝜑 → ( 𝐼 𝑆 𝐽 ) = ( 𝐼 𝐴 ( 𝐽 + 1 ) ) ) |