Step |
Hyp |
Ref |
Expression |
1 |
|
pfxmpt |
⊢ ( ( 𝑊 ∈ Word 𝑉 ∧ 𝐿 ∈ ( 0 ... ( ♯ ‘ 𝑊 ) ) ) → ( 𝑊 prefix 𝐿 ) = ( 𝑥 ∈ ( 0 ..^ 𝐿 ) ↦ ( 𝑊 ‘ 𝑥 ) ) ) |
2 |
|
simpll |
⊢ ( ( ( 𝑊 ∈ Word 𝑉 ∧ 𝐿 ∈ ( 0 ... ( ♯ ‘ 𝑊 ) ) ) ∧ 𝑥 ∈ ( 0 ..^ 𝐿 ) ) → 𝑊 ∈ Word 𝑉 ) |
3 |
|
elfzuz3 |
⊢ ( 𝐿 ∈ ( 0 ... ( ♯ ‘ 𝑊 ) ) → ( ♯ ‘ 𝑊 ) ∈ ( ℤ≥ ‘ 𝐿 ) ) |
4 |
3
|
adantl |
⊢ ( ( 𝑊 ∈ Word 𝑉 ∧ 𝐿 ∈ ( 0 ... ( ♯ ‘ 𝑊 ) ) ) → ( ♯ ‘ 𝑊 ) ∈ ( ℤ≥ ‘ 𝐿 ) ) |
5 |
|
fzoss2 |
⊢ ( ( ♯ ‘ 𝑊 ) ∈ ( ℤ≥ ‘ 𝐿 ) → ( 0 ..^ 𝐿 ) ⊆ ( 0 ..^ ( ♯ ‘ 𝑊 ) ) ) |
6 |
4 5
|
syl |
⊢ ( ( 𝑊 ∈ Word 𝑉 ∧ 𝐿 ∈ ( 0 ... ( ♯ ‘ 𝑊 ) ) ) → ( 0 ..^ 𝐿 ) ⊆ ( 0 ..^ ( ♯ ‘ 𝑊 ) ) ) |
7 |
6
|
sselda |
⊢ ( ( ( 𝑊 ∈ Word 𝑉 ∧ 𝐿 ∈ ( 0 ... ( ♯ ‘ 𝑊 ) ) ) ∧ 𝑥 ∈ ( 0 ..^ 𝐿 ) ) → 𝑥 ∈ ( 0 ..^ ( ♯ ‘ 𝑊 ) ) ) |
8 |
|
wrdsymbcl |
⊢ ( ( 𝑊 ∈ Word 𝑉 ∧ 𝑥 ∈ ( 0 ..^ ( ♯ ‘ 𝑊 ) ) ) → ( 𝑊 ‘ 𝑥 ) ∈ 𝑉 ) |
9 |
2 7 8
|
syl2anc |
⊢ ( ( ( 𝑊 ∈ Word 𝑉 ∧ 𝐿 ∈ ( 0 ... ( ♯ ‘ 𝑊 ) ) ) ∧ 𝑥 ∈ ( 0 ..^ 𝐿 ) ) → ( 𝑊 ‘ 𝑥 ) ∈ 𝑉 ) |
10 |
1 9
|
fmpt3d |
⊢ ( ( 𝑊 ∈ Word 𝑉 ∧ 𝐿 ∈ ( 0 ... ( ♯ ‘ 𝑊 ) ) ) → ( 𝑊 prefix 𝐿 ) : ( 0 ..^ 𝐿 ) ⟶ 𝑉 ) |