Step |
Hyp |
Ref |
Expression |
1 |
|
simp1 |
⊢ ( ( 𝑆 ∈ Word 𝑉 ∧ ( 𝐹 ∈ ℕ0 ∧ 𝐿 ∈ ( ℤ≥ ‘ 𝐹 ) ) ∧ 𝐿 ≤ ( ♯ ‘ 𝑆 ) ) → 𝑆 ∈ Word 𝑉 ) |
2 |
|
simpl |
⊢ ( ( 𝐹 ∈ ℕ0 ∧ 𝐿 ∈ ( ℤ≥ ‘ 𝐹 ) ) → 𝐹 ∈ ℕ0 ) |
3 |
|
eluznn0 |
⊢ ( ( 𝐹 ∈ ℕ0 ∧ 𝐿 ∈ ( ℤ≥ ‘ 𝐹 ) ) → 𝐿 ∈ ℕ0 ) |
4 |
|
eluzle |
⊢ ( 𝐿 ∈ ( ℤ≥ ‘ 𝐹 ) → 𝐹 ≤ 𝐿 ) |
5 |
4
|
adantl |
⊢ ( ( 𝐹 ∈ ℕ0 ∧ 𝐿 ∈ ( ℤ≥ ‘ 𝐹 ) ) → 𝐹 ≤ 𝐿 ) |
6 |
2 3 5
|
3jca |
⊢ ( ( 𝐹 ∈ ℕ0 ∧ 𝐿 ∈ ( ℤ≥ ‘ 𝐹 ) ) → ( 𝐹 ∈ ℕ0 ∧ 𝐿 ∈ ℕ0 ∧ 𝐹 ≤ 𝐿 ) ) |
7 |
6
|
3ad2ant2 |
⊢ ( ( 𝑆 ∈ Word 𝑉 ∧ ( 𝐹 ∈ ℕ0 ∧ 𝐿 ∈ ( ℤ≥ ‘ 𝐹 ) ) ∧ 𝐿 ≤ ( ♯ ‘ 𝑆 ) ) → ( 𝐹 ∈ ℕ0 ∧ 𝐿 ∈ ℕ0 ∧ 𝐹 ≤ 𝐿 ) ) |
8 |
|
elfz2nn0 |
⊢ ( 𝐹 ∈ ( 0 ... 𝐿 ) ↔ ( 𝐹 ∈ ℕ0 ∧ 𝐿 ∈ ℕ0 ∧ 𝐹 ≤ 𝐿 ) ) |
9 |
7 8
|
sylibr |
⊢ ( ( 𝑆 ∈ Word 𝑉 ∧ ( 𝐹 ∈ ℕ0 ∧ 𝐿 ∈ ( ℤ≥ ‘ 𝐹 ) ) ∧ 𝐿 ≤ ( ♯ ‘ 𝑆 ) ) → 𝐹 ∈ ( 0 ... 𝐿 ) ) |
10 |
3
|
3ad2ant2 |
⊢ ( ( 𝑆 ∈ Word 𝑉 ∧ ( 𝐹 ∈ ℕ0 ∧ 𝐿 ∈ ( ℤ≥ ‘ 𝐹 ) ) ∧ 𝐿 ≤ ( ♯ ‘ 𝑆 ) ) → 𝐿 ∈ ℕ0 ) |
11 |
|
lencl |
⊢ ( 𝑆 ∈ Word 𝑉 → ( ♯ ‘ 𝑆 ) ∈ ℕ0 ) |
12 |
11
|
3ad2ant1 |
⊢ ( ( 𝑆 ∈ Word 𝑉 ∧ ( 𝐹 ∈ ℕ0 ∧ 𝐿 ∈ ( ℤ≥ ‘ 𝐹 ) ) ∧ 𝐿 ≤ ( ♯ ‘ 𝑆 ) ) → ( ♯ ‘ 𝑆 ) ∈ ℕ0 ) |
13 |
|
simp3 |
⊢ ( ( 𝑆 ∈ Word 𝑉 ∧ ( 𝐹 ∈ ℕ0 ∧ 𝐿 ∈ ( ℤ≥ ‘ 𝐹 ) ) ∧ 𝐿 ≤ ( ♯ ‘ 𝑆 ) ) → 𝐿 ≤ ( ♯ ‘ 𝑆 ) ) |
14 |
10 12 13
|
3jca |
⊢ ( ( 𝑆 ∈ Word 𝑉 ∧ ( 𝐹 ∈ ℕ0 ∧ 𝐿 ∈ ( ℤ≥ ‘ 𝐹 ) ) ∧ 𝐿 ≤ ( ♯ ‘ 𝑆 ) ) → ( 𝐿 ∈ ℕ0 ∧ ( ♯ ‘ 𝑆 ) ∈ ℕ0 ∧ 𝐿 ≤ ( ♯ ‘ 𝑆 ) ) ) |
15 |
|
elfz2nn0 |
⊢ ( 𝐿 ∈ ( 0 ... ( ♯ ‘ 𝑆 ) ) ↔ ( 𝐿 ∈ ℕ0 ∧ ( ♯ ‘ 𝑆 ) ∈ ℕ0 ∧ 𝐿 ≤ ( ♯ ‘ 𝑆 ) ) ) |
16 |
14 15
|
sylibr |
⊢ ( ( 𝑆 ∈ Word 𝑉 ∧ ( 𝐹 ∈ ℕ0 ∧ 𝐿 ∈ ( ℤ≥ ‘ 𝐹 ) ) ∧ 𝐿 ≤ ( ♯ ‘ 𝑆 ) ) → 𝐿 ∈ ( 0 ... ( ♯ ‘ 𝑆 ) ) ) |
17 |
|
swrdlen |
⊢ ( ( 𝑆 ∈ Word 𝑉 ∧ 𝐹 ∈ ( 0 ... 𝐿 ) ∧ 𝐿 ∈ ( 0 ... ( ♯ ‘ 𝑆 ) ) ) → ( ♯ ‘ ( 𝑆 substr 〈 𝐹 , 𝐿 〉 ) ) = ( 𝐿 − 𝐹 ) ) |
18 |
1 9 16 17
|
syl3anc |
⊢ ( ( 𝑆 ∈ Word 𝑉 ∧ ( 𝐹 ∈ ℕ0 ∧ 𝐿 ∈ ( ℤ≥ ‘ 𝐹 ) ) ∧ 𝐿 ≤ ( ♯ ‘ 𝑆 ) ) → ( ♯ ‘ ( 𝑆 substr 〈 𝐹 , 𝐿 〉 ) ) = ( 𝐿 − 𝐹 ) ) |