Step |
Hyp |
Ref |
Expression |
1 |
|
simpl |
⊢ ( ( 𝑊 ∈ Word 𝑉 ∧ 2 ≤ ( ♯ ‘ 𝑊 ) ) → 𝑊 ∈ Word 𝑉 ) |
2 |
|
wrdlenge2n0 |
⊢ ( ( 𝑊 ∈ Word 𝑉 ∧ 2 ≤ ( ♯ ‘ 𝑊 ) ) → 𝑊 ≠ ∅ ) |
3 |
|
2z |
⊢ 2 ∈ ℤ |
4 |
3
|
a1i |
⊢ ( ( 𝑊 ∈ Word 𝑉 ∧ 2 ≤ ( ♯ ‘ 𝑊 ) ) → 2 ∈ ℤ ) |
5 |
|
lencl |
⊢ ( 𝑊 ∈ Word 𝑉 → ( ♯ ‘ 𝑊 ) ∈ ℕ0 ) |
6 |
5
|
nn0zd |
⊢ ( 𝑊 ∈ Word 𝑉 → ( ♯ ‘ 𝑊 ) ∈ ℤ ) |
7 |
6
|
adantr |
⊢ ( ( 𝑊 ∈ Word 𝑉 ∧ 2 ≤ ( ♯ ‘ 𝑊 ) ) → ( ♯ ‘ 𝑊 ) ∈ ℤ ) |
8 |
|
simpr |
⊢ ( ( 𝑊 ∈ Word 𝑉 ∧ 2 ≤ ( ♯ ‘ 𝑊 ) ) → 2 ≤ ( ♯ ‘ 𝑊 ) ) |
9 |
|
eluz2 |
⊢ ( ( ♯ ‘ 𝑊 ) ∈ ( ℤ≥ ‘ 2 ) ↔ ( 2 ∈ ℤ ∧ ( ♯ ‘ 𝑊 ) ∈ ℤ ∧ 2 ≤ ( ♯ ‘ 𝑊 ) ) ) |
10 |
4 7 8 9
|
syl3anbrc |
⊢ ( ( 𝑊 ∈ Word 𝑉 ∧ 2 ≤ ( ♯ ‘ 𝑊 ) ) → ( ♯ ‘ 𝑊 ) ∈ ( ℤ≥ ‘ 2 ) ) |
11 |
|
uz2m1nn |
⊢ ( ( ♯ ‘ 𝑊 ) ∈ ( ℤ≥ ‘ 2 ) → ( ( ♯ ‘ 𝑊 ) − 1 ) ∈ ℕ ) |
12 |
10 11
|
syl |
⊢ ( ( 𝑊 ∈ Word 𝑉 ∧ 2 ≤ ( ♯ ‘ 𝑊 ) ) → ( ( ♯ ‘ 𝑊 ) − 1 ) ∈ ℕ ) |
13 |
|
lbfzo0 |
⊢ ( 0 ∈ ( 0 ..^ ( ( ♯ ‘ 𝑊 ) − 1 ) ) ↔ ( ( ♯ ‘ 𝑊 ) − 1 ) ∈ ℕ ) |
14 |
12 13
|
sylibr |
⊢ ( ( 𝑊 ∈ Word 𝑉 ∧ 2 ≤ ( ♯ ‘ 𝑊 ) ) → 0 ∈ ( 0 ..^ ( ( ♯ ‘ 𝑊 ) − 1 ) ) ) |
15 |
|
pfxtrcfv |
⊢ ( ( 𝑊 ∈ Word 𝑉 ∧ 𝑊 ≠ ∅ ∧ 0 ∈ ( 0 ..^ ( ( ♯ ‘ 𝑊 ) − 1 ) ) ) → ( ( 𝑊 prefix ( ( ♯ ‘ 𝑊 ) − 1 ) ) ‘ 0 ) = ( 𝑊 ‘ 0 ) ) |
16 |
1 2 14 15
|
syl3anc |
⊢ ( ( 𝑊 ∈ Word 𝑉 ∧ 2 ≤ ( ♯ ‘ 𝑊 ) ) → ( ( 𝑊 prefix ( ( ♯ ‘ 𝑊 ) − 1 ) ) ‘ 0 ) = ( 𝑊 ‘ 0 ) ) |