Step |
Hyp |
Ref |
Expression |
1 |
|
pfxmpt |
⊢ ( ( 𝑆 ∈ Word 𝐴 ∧ 𝐿 ∈ ( 0 ... ( ♯ ‘ 𝑆 ) ) ) → ( 𝑆 prefix 𝐿 ) = ( 𝑥 ∈ ( 0 ..^ 𝐿 ) ↦ ( 𝑆 ‘ 𝑥 ) ) ) |
2 |
|
wrdf |
⊢ ( 𝑆 ∈ Word 𝐴 → 𝑆 : ( 0 ..^ ( ♯ ‘ 𝑆 ) ) ⟶ 𝐴 ) |
3 |
2
|
adantr |
⊢ ( ( 𝑆 ∈ Word 𝐴 ∧ 𝐿 ∈ ( 0 ... ( ♯ ‘ 𝑆 ) ) ) → 𝑆 : ( 0 ..^ ( ♯ ‘ 𝑆 ) ) ⟶ 𝐴 ) |
4 |
|
elfzuz3 |
⊢ ( 𝐿 ∈ ( 0 ... ( ♯ ‘ 𝑆 ) ) → ( ♯ ‘ 𝑆 ) ∈ ( ℤ≥ ‘ 𝐿 ) ) |
5 |
4
|
adantl |
⊢ ( ( 𝑆 ∈ Word 𝐴 ∧ 𝐿 ∈ ( 0 ... ( ♯ ‘ 𝑆 ) ) ) → ( ♯ ‘ 𝑆 ) ∈ ( ℤ≥ ‘ 𝐿 ) ) |
6 |
|
fzoss2 |
⊢ ( ( ♯ ‘ 𝑆 ) ∈ ( ℤ≥ ‘ 𝐿 ) → ( 0 ..^ 𝐿 ) ⊆ ( 0 ..^ ( ♯ ‘ 𝑆 ) ) ) |
7 |
5 6
|
syl |
⊢ ( ( 𝑆 ∈ Word 𝐴 ∧ 𝐿 ∈ ( 0 ... ( ♯ ‘ 𝑆 ) ) ) → ( 0 ..^ 𝐿 ) ⊆ ( 0 ..^ ( ♯ ‘ 𝑆 ) ) ) |
8 |
3 7
|
feqresmpt |
⊢ ( ( 𝑆 ∈ Word 𝐴 ∧ 𝐿 ∈ ( 0 ... ( ♯ ‘ 𝑆 ) ) ) → ( 𝑆 ↾ ( 0 ..^ 𝐿 ) ) = ( 𝑥 ∈ ( 0 ..^ 𝐿 ) ↦ ( 𝑆 ‘ 𝑥 ) ) ) |
9 |
1 8
|
eqtr4d |
⊢ ( ( 𝑆 ∈ Word 𝐴 ∧ 𝐿 ∈ ( 0 ... ( ♯ ‘ 𝑆 ) ) ) → ( 𝑆 prefix 𝐿 ) = ( 𝑆 ↾ ( 0 ..^ 𝐿 ) ) ) |