Step |
Hyp |
Ref |
Expression |
1 |
|
simpll |
⊢ ( ( ( 𝑊 ∈ Word 𝑉 ∧ 𝑈 ∈ Word 𝑉 ) ∧ ( 𝑀 ∈ ℕ0 ∧ 𝑁 ∈ ℕ0 ) ) → 𝑊 ∈ Word 𝑉 ) |
2 |
|
nn0z |
⊢ ( 𝑀 ∈ ℕ0 → 𝑀 ∈ ℤ ) |
3 |
2
|
ad2antrl |
⊢ ( ( ( 𝑊 ∈ Word 𝑉 ∧ 𝑈 ∈ Word 𝑉 ) ∧ ( 𝑀 ∈ ℕ0 ∧ 𝑁 ∈ ℕ0 ) ) → 𝑀 ∈ ℤ ) |
4 |
|
nn0z |
⊢ ( 𝑁 ∈ ℕ0 → 𝑁 ∈ ℤ ) |
5 |
4
|
ad2antll |
⊢ ( ( ( 𝑊 ∈ Word 𝑉 ∧ 𝑈 ∈ Word 𝑉 ) ∧ ( 𝑀 ∈ ℕ0 ∧ 𝑁 ∈ ℕ0 ) ) → 𝑁 ∈ ℤ ) |
6 |
|
swrdlend |
⊢ ( ( 𝑊 ∈ Word 𝑉 ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ) → ( 𝑁 ≤ 𝑀 → ( 𝑊 substr 〈 𝑀 , 𝑁 〉 ) = ∅ ) ) |
7 |
1 3 5 6
|
syl3anc |
⊢ ( ( ( 𝑊 ∈ Word 𝑉 ∧ 𝑈 ∈ Word 𝑉 ) ∧ ( 𝑀 ∈ ℕ0 ∧ 𝑁 ∈ ℕ0 ) ) → ( 𝑁 ≤ 𝑀 → ( 𝑊 substr 〈 𝑀 , 𝑁 〉 ) = ∅ ) ) |
8 |
7
|
3impia |
⊢ ( ( ( 𝑊 ∈ Word 𝑉 ∧ 𝑈 ∈ Word 𝑉 ) ∧ ( 𝑀 ∈ ℕ0 ∧ 𝑁 ∈ ℕ0 ) ∧ 𝑁 ≤ 𝑀 ) → ( 𝑊 substr 〈 𝑀 , 𝑁 〉 ) = ∅ ) |
9 |
|
simplr |
⊢ ( ( ( 𝑊 ∈ Word 𝑉 ∧ 𝑈 ∈ Word 𝑉 ) ∧ ( 𝑀 ∈ ℕ0 ∧ 𝑁 ∈ ℕ0 ) ) → 𝑈 ∈ Word 𝑉 ) |
10 |
|
swrdlend |
⊢ ( ( 𝑈 ∈ Word 𝑉 ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ) → ( 𝑁 ≤ 𝑀 → ( 𝑈 substr 〈 𝑀 , 𝑁 〉 ) = ∅ ) ) |
11 |
9 3 5 10
|
syl3anc |
⊢ ( ( ( 𝑊 ∈ Word 𝑉 ∧ 𝑈 ∈ Word 𝑉 ) ∧ ( 𝑀 ∈ ℕ0 ∧ 𝑁 ∈ ℕ0 ) ) → ( 𝑁 ≤ 𝑀 → ( 𝑈 substr 〈 𝑀 , 𝑁 〉 ) = ∅ ) ) |
12 |
11
|
3impia |
⊢ ( ( ( 𝑊 ∈ Word 𝑉 ∧ 𝑈 ∈ Word 𝑉 ) ∧ ( 𝑀 ∈ ℕ0 ∧ 𝑁 ∈ ℕ0 ) ∧ 𝑁 ≤ 𝑀 ) → ( 𝑈 substr 〈 𝑀 , 𝑁 〉 ) = ∅ ) |
13 |
8 12
|
eqtr4d |
⊢ ( ( ( 𝑊 ∈ Word 𝑉 ∧ 𝑈 ∈ Word 𝑉 ) ∧ ( 𝑀 ∈ ℕ0 ∧ 𝑁 ∈ ℕ0 ) ∧ 𝑁 ≤ 𝑀 ) → ( 𝑊 substr 〈 𝑀 , 𝑁 〉 ) = ( 𝑈 substr 〈 𝑀 , 𝑁 〉 ) ) |