Step |
Hyp |
Ref |
Expression |
1 |
|
oveq2 |
⊢ ( 𝑙 = 𝐿 → ( 0 ..^ 𝑙 ) = ( 0 ..^ 𝐿 ) ) |
2 |
1
|
feq2d |
⊢ ( 𝑙 = 𝐿 → ( 𝑊 : ( 0 ..^ 𝑙 ) ⟶ 𝑆 ↔ 𝑊 : ( 0 ..^ 𝐿 ) ⟶ 𝑆 ) ) |
3 |
2
|
rspcev |
⊢ ( ( 𝐿 ∈ ℕ0 ∧ 𝑊 : ( 0 ..^ 𝐿 ) ⟶ 𝑆 ) → ∃ 𝑙 ∈ ℕ0 𝑊 : ( 0 ..^ 𝑙 ) ⟶ 𝑆 ) |
4 |
|
0nn0 |
⊢ 0 ∈ ℕ0 |
5 |
|
fzo0n0 |
⊢ ( ( 0 ..^ 𝐿 ) ≠ ∅ ↔ 𝐿 ∈ ℕ ) |
6 |
|
nnnn0 |
⊢ ( 𝐿 ∈ ℕ → 𝐿 ∈ ℕ0 ) |
7 |
5 6
|
sylbi |
⊢ ( ( 0 ..^ 𝐿 ) ≠ ∅ → 𝐿 ∈ ℕ0 ) |
8 |
7
|
necon1bi |
⊢ ( ¬ 𝐿 ∈ ℕ0 → ( 0 ..^ 𝐿 ) = ∅ ) |
9 |
|
fzo0 |
⊢ ( 0 ..^ 0 ) = ∅ |
10 |
8 9
|
eqtr4di |
⊢ ( ¬ 𝐿 ∈ ℕ0 → ( 0 ..^ 𝐿 ) = ( 0 ..^ 0 ) ) |
11 |
10
|
feq2d |
⊢ ( ¬ 𝐿 ∈ ℕ0 → ( 𝑊 : ( 0 ..^ 𝐿 ) ⟶ 𝑆 ↔ 𝑊 : ( 0 ..^ 0 ) ⟶ 𝑆 ) ) |
12 |
11
|
biimpa |
⊢ ( ( ¬ 𝐿 ∈ ℕ0 ∧ 𝑊 : ( 0 ..^ 𝐿 ) ⟶ 𝑆 ) → 𝑊 : ( 0 ..^ 0 ) ⟶ 𝑆 ) |
13 |
|
oveq2 |
⊢ ( 𝑙 = 0 → ( 0 ..^ 𝑙 ) = ( 0 ..^ 0 ) ) |
14 |
13
|
feq2d |
⊢ ( 𝑙 = 0 → ( 𝑊 : ( 0 ..^ 𝑙 ) ⟶ 𝑆 ↔ 𝑊 : ( 0 ..^ 0 ) ⟶ 𝑆 ) ) |
15 |
14
|
rspcev |
⊢ ( ( 0 ∈ ℕ0 ∧ 𝑊 : ( 0 ..^ 0 ) ⟶ 𝑆 ) → ∃ 𝑙 ∈ ℕ0 𝑊 : ( 0 ..^ 𝑙 ) ⟶ 𝑆 ) |
16 |
4 12 15
|
sylancr |
⊢ ( ( ¬ 𝐿 ∈ ℕ0 ∧ 𝑊 : ( 0 ..^ 𝐿 ) ⟶ 𝑆 ) → ∃ 𝑙 ∈ ℕ0 𝑊 : ( 0 ..^ 𝑙 ) ⟶ 𝑆 ) |
17 |
3 16
|
pm2.61ian |
⊢ ( 𝑊 : ( 0 ..^ 𝐿 ) ⟶ 𝑆 → ∃ 𝑙 ∈ ℕ0 𝑊 : ( 0 ..^ 𝑙 ) ⟶ 𝑆 ) |
18 |
|
iswrd |
⊢ ( 𝑊 ∈ Word 𝑆 ↔ ∃ 𝑙 ∈ ℕ0 𝑊 : ( 0 ..^ 𝑙 ) ⟶ 𝑆 ) |
19 |
17 18
|
sylibr |
⊢ ( 𝑊 : ( 0 ..^ 𝐿 ) ⟶ 𝑆 → 𝑊 ∈ Word 𝑆 ) |