Step |
Hyp |
Ref |
Expression |
1 |
|
swrdsb0eq |
⊢ ( ( ( 𝑊 ∈ Word 𝑉 ∧ 𝑈 ∈ Word 𝑉 ) ∧ ( 𝑀 ∈ ℕ0 ∧ 𝑁 ∈ ℕ0 ) ∧ 𝑁 ≤ 𝑀 ) → ( 𝑊 substr 〈 𝑀 , 𝑁 〉 ) = ( 𝑈 substr 〈 𝑀 , 𝑁 〉 ) ) |
2 |
1
|
3expa |
⊢ ( ( ( ( 𝑊 ∈ Word 𝑉 ∧ 𝑈 ∈ Word 𝑉 ) ∧ ( 𝑀 ∈ ℕ0 ∧ 𝑁 ∈ ℕ0 ) ) ∧ 𝑁 ≤ 𝑀 ) → ( 𝑊 substr 〈 𝑀 , 𝑁 〉 ) = ( 𝑈 substr 〈 𝑀 , 𝑁 〉 ) ) |
3 |
2
|
ancoms |
⊢ ( ( 𝑁 ≤ 𝑀 ∧ ( ( 𝑊 ∈ Word 𝑉 ∧ 𝑈 ∈ Word 𝑉 ) ∧ ( 𝑀 ∈ ℕ0 ∧ 𝑁 ∈ ℕ0 ) ) ) → ( 𝑊 substr 〈 𝑀 , 𝑁 〉 ) = ( 𝑈 substr 〈 𝑀 , 𝑁 〉 ) ) |
4 |
3
|
3adantr3 |
⊢ ( ( 𝑁 ≤ 𝑀 ∧ ( ( 𝑊 ∈ Word 𝑉 ∧ 𝑈 ∈ Word 𝑉 ) ∧ ( 𝑀 ∈ ℕ0 ∧ 𝑁 ∈ ℕ0 ) ∧ ( 𝑁 ≤ ( ♯ ‘ 𝑊 ) ∧ 𝑁 ≤ ( ♯ ‘ 𝑈 ) ) ) ) → ( 𝑊 substr 〈 𝑀 , 𝑁 〉 ) = ( 𝑈 substr 〈 𝑀 , 𝑁 〉 ) ) |
5 |
|
ral0 |
⊢ ∀ 𝑖 ∈ ∅ ( 𝑊 ‘ 𝑖 ) = ( 𝑈 ‘ 𝑖 ) |
6 |
|
nn0z |
⊢ ( 𝑀 ∈ ℕ0 → 𝑀 ∈ ℤ ) |
7 |
|
nn0z |
⊢ ( 𝑁 ∈ ℕ0 → 𝑁 ∈ ℤ ) |
8 |
|
fzon |
⊢ ( ( 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ) → ( 𝑁 ≤ 𝑀 ↔ ( 𝑀 ..^ 𝑁 ) = ∅ ) ) |
9 |
6 7 8
|
syl2an |
⊢ ( ( 𝑀 ∈ ℕ0 ∧ 𝑁 ∈ ℕ0 ) → ( 𝑁 ≤ 𝑀 ↔ ( 𝑀 ..^ 𝑁 ) = ∅ ) ) |
10 |
9
|
biimpa |
⊢ ( ( ( 𝑀 ∈ ℕ0 ∧ 𝑁 ∈ ℕ0 ) ∧ 𝑁 ≤ 𝑀 ) → ( 𝑀 ..^ 𝑁 ) = ∅ ) |
11 |
10
|
raleqdv |
⊢ ( ( ( 𝑀 ∈ ℕ0 ∧ 𝑁 ∈ ℕ0 ) ∧ 𝑁 ≤ 𝑀 ) → ( ∀ 𝑖 ∈ ( 𝑀 ..^ 𝑁 ) ( 𝑊 ‘ 𝑖 ) = ( 𝑈 ‘ 𝑖 ) ↔ ∀ 𝑖 ∈ ∅ ( 𝑊 ‘ 𝑖 ) = ( 𝑈 ‘ 𝑖 ) ) ) |
12 |
5 11
|
mpbiri |
⊢ ( ( ( 𝑀 ∈ ℕ0 ∧ 𝑁 ∈ ℕ0 ) ∧ 𝑁 ≤ 𝑀 ) → ∀ 𝑖 ∈ ( 𝑀 ..^ 𝑁 ) ( 𝑊 ‘ 𝑖 ) = ( 𝑈 ‘ 𝑖 ) ) |
13 |
12
|
ex |
⊢ ( ( 𝑀 ∈ ℕ0 ∧ 𝑁 ∈ ℕ0 ) → ( 𝑁 ≤ 𝑀 → ∀ 𝑖 ∈ ( 𝑀 ..^ 𝑁 ) ( 𝑊 ‘ 𝑖 ) = ( 𝑈 ‘ 𝑖 ) ) ) |
14 |
13
|
3ad2ant2 |
⊢ ( ( ( 𝑊 ∈ Word 𝑉 ∧ 𝑈 ∈ Word 𝑉 ) ∧ ( 𝑀 ∈ ℕ0 ∧ 𝑁 ∈ ℕ0 ) ∧ ( 𝑁 ≤ ( ♯ ‘ 𝑊 ) ∧ 𝑁 ≤ ( ♯ ‘ 𝑈 ) ) ) → ( 𝑁 ≤ 𝑀 → ∀ 𝑖 ∈ ( 𝑀 ..^ 𝑁 ) ( 𝑊 ‘ 𝑖 ) = ( 𝑈 ‘ 𝑖 ) ) ) |
15 |
14
|
impcom |
⊢ ( ( 𝑁 ≤ 𝑀 ∧ ( ( 𝑊 ∈ Word 𝑉 ∧ 𝑈 ∈ Word 𝑉 ) ∧ ( 𝑀 ∈ ℕ0 ∧ 𝑁 ∈ ℕ0 ) ∧ ( 𝑁 ≤ ( ♯ ‘ 𝑊 ) ∧ 𝑁 ≤ ( ♯ ‘ 𝑈 ) ) ) ) → ∀ 𝑖 ∈ ( 𝑀 ..^ 𝑁 ) ( 𝑊 ‘ 𝑖 ) = ( 𝑈 ‘ 𝑖 ) ) |
16 |
4 15
|
2thd |
⊢ ( ( 𝑁 ≤ 𝑀 ∧ ( ( 𝑊 ∈ Word 𝑉 ∧ 𝑈 ∈ Word 𝑉 ) ∧ ( 𝑀 ∈ ℕ0 ∧ 𝑁 ∈ ℕ0 ) ∧ ( 𝑁 ≤ ( ♯ ‘ 𝑊 ) ∧ 𝑁 ≤ ( ♯ ‘ 𝑈 ) ) ) ) → ( ( 𝑊 substr 〈 𝑀 , 𝑁 〉 ) = ( 𝑈 substr 〈 𝑀 , 𝑁 〉 ) ↔ ∀ 𝑖 ∈ ( 𝑀 ..^ 𝑁 ) ( 𝑊 ‘ 𝑖 ) = ( 𝑈 ‘ 𝑖 ) ) ) |
17 |
|
swrdcl |
⊢ ( 𝑊 ∈ Word 𝑉 → ( 𝑊 substr 〈 𝑀 , 𝑁 〉 ) ∈ Word 𝑉 ) |
18 |
|
swrdcl |
⊢ ( 𝑈 ∈ Word 𝑉 → ( 𝑈 substr 〈 𝑀 , 𝑁 〉 ) ∈ Word 𝑉 ) |
19 |
|
eqwrd |
⊢ ( ( ( 𝑊 substr 〈 𝑀 , 𝑁 〉 ) ∈ Word 𝑉 ∧ ( 𝑈 substr 〈 𝑀 , 𝑁 〉 ) ∈ Word 𝑉 ) → ( ( 𝑊 substr 〈 𝑀 , 𝑁 〉 ) = ( 𝑈 substr 〈 𝑀 , 𝑁 〉 ) ↔ ( ( ♯ ‘ ( 𝑊 substr 〈 𝑀 , 𝑁 〉 ) ) = ( ♯ ‘ ( 𝑈 substr 〈 𝑀 , 𝑁 〉 ) ) ∧ ∀ 𝑗 ∈ ( 0 ..^ ( ♯ ‘ ( 𝑊 substr 〈 𝑀 , 𝑁 〉 ) ) ) ( ( 𝑊 substr 〈 𝑀 , 𝑁 〉 ) ‘ 𝑗 ) = ( ( 𝑈 substr 〈 𝑀 , 𝑁 〉 ) ‘ 𝑗 ) ) ) ) |
20 |
17 18 19
|
syl2an |
⊢ ( ( 𝑊 ∈ Word 𝑉 ∧ 𝑈 ∈ Word 𝑉 ) → ( ( 𝑊 substr 〈 𝑀 , 𝑁 〉 ) = ( 𝑈 substr 〈 𝑀 , 𝑁 〉 ) ↔ ( ( ♯ ‘ ( 𝑊 substr 〈 𝑀 , 𝑁 〉 ) ) = ( ♯ ‘ ( 𝑈 substr 〈 𝑀 , 𝑁 〉 ) ) ∧ ∀ 𝑗 ∈ ( 0 ..^ ( ♯ ‘ ( 𝑊 substr 〈 𝑀 , 𝑁 〉 ) ) ) ( ( 𝑊 substr 〈 𝑀 , 𝑁 〉 ) ‘ 𝑗 ) = ( ( 𝑈 substr 〈 𝑀 , 𝑁 〉 ) ‘ 𝑗 ) ) ) ) |
21 |
20
|
3ad2ant1 |
⊢ ( ( ( 𝑊 ∈ Word 𝑉 ∧ 𝑈 ∈ Word 𝑉 ) ∧ ( 𝑀 ∈ ℕ0 ∧ 𝑁 ∈ ℕ0 ) ∧ ( 𝑁 ≤ ( ♯ ‘ 𝑊 ) ∧ 𝑁 ≤ ( ♯ ‘ 𝑈 ) ) ) → ( ( 𝑊 substr 〈 𝑀 , 𝑁 〉 ) = ( 𝑈 substr 〈 𝑀 , 𝑁 〉 ) ↔ ( ( ♯ ‘ ( 𝑊 substr 〈 𝑀 , 𝑁 〉 ) ) = ( ♯ ‘ ( 𝑈 substr 〈 𝑀 , 𝑁 〉 ) ) ∧ ∀ 𝑗 ∈ ( 0 ..^ ( ♯ ‘ ( 𝑊 substr 〈 𝑀 , 𝑁 〉 ) ) ) ( ( 𝑊 substr 〈 𝑀 , 𝑁 〉 ) ‘ 𝑗 ) = ( ( 𝑈 substr 〈 𝑀 , 𝑁 〉 ) ‘ 𝑗 ) ) ) ) |
22 |
21
|
adantl |
⊢ ( ( ¬ 𝑁 ≤ 𝑀 ∧ ( ( 𝑊 ∈ Word 𝑉 ∧ 𝑈 ∈ Word 𝑉 ) ∧ ( 𝑀 ∈ ℕ0 ∧ 𝑁 ∈ ℕ0 ) ∧ ( 𝑁 ≤ ( ♯ ‘ 𝑊 ) ∧ 𝑁 ≤ ( ♯ ‘ 𝑈 ) ) ) ) → ( ( 𝑊 substr 〈 𝑀 , 𝑁 〉 ) = ( 𝑈 substr 〈 𝑀 , 𝑁 〉 ) ↔ ( ( ♯ ‘ ( 𝑊 substr 〈 𝑀 , 𝑁 〉 ) ) = ( ♯ ‘ ( 𝑈 substr 〈 𝑀 , 𝑁 〉 ) ) ∧ ∀ 𝑗 ∈ ( 0 ..^ ( ♯ ‘ ( 𝑊 substr 〈 𝑀 , 𝑁 〉 ) ) ) ( ( 𝑊 substr 〈 𝑀 , 𝑁 〉 ) ‘ 𝑗 ) = ( ( 𝑈 substr 〈 𝑀 , 𝑁 〉 ) ‘ 𝑗 ) ) ) ) |
23 |
|
swrdsbslen |
⊢ ( ( ( 𝑊 ∈ Word 𝑉 ∧ 𝑈 ∈ Word 𝑉 ) ∧ ( 𝑀 ∈ ℕ0 ∧ 𝑁 ∈ ℕ0 ) ∧ ( 𝑁 ≤ ( ♯ ‘ 𝑊 ) ∧ 𝑁 ≤ ( ♯ ‘ 𝑈 ) ) ) → ( ♯ ‘ ( 𝑊 substr 〈 𝑀 , 𝑁 〉 ) ) = ( ♯ ‘ ( 𝑈 substr 〈 𝑀 , 𝑁 〉 ) ) ) |
24 |
23
|
adantl |
⊢ ( ( ¬ 𝑁 ≤ 𝑀 ∧ ( ( 𝑊 ∈ Word 𝑉 ∧ 𝑈 ∈ Word 𝑉 ) ∧ ( 𝑀 ∈ ℕ0 ∧ 𝑁 ∈ ℕ0 ) ∧ ( 𝑁 ≤ ( ♯ ‘ 𝑊 ) ∧ 𝑁 ≤ ( ♯ ‘ 𝑈 ) ) ) ) → ( ♯ ‘ ( 𝑊 substr 〈 𝑀 , 𝑁 〉 ) ) = ( ♯ ‘ ( 𝑈 substr 〈 𝑀 , 𝑁 〉 ) ) ) |
25 |
24
|
biantrurd |
⊢ ( ( ¬ 𝑁 ≤ 𝑀 ∧ ( ( 𝑊 ∈ Word 𝑉 ∧ 𝑈 ∈ Word 𝑉 ) ∧ ( 𝑀 ∈ ℕ0 ∧ 𝑁 ∈ ℕ0 ) ∧ ( 𝑁 ≤ ( ♯ ‘ 𝑊 ) ∧ 𝑁 ≤ ( ♯ ‘ 𝑈 ) ) ) ) → ( ∀ 𝑗 ∈ ( 0 ..^ ( ♯ ‘ ( 𝑊 substr 〈 𝑀 , 𝑁 〉 ) ) ) ( ( 𝑊 substr 〈 𝑀 , 𝑁 〉 ) ‘ 𝑗 ) = ( ( 𝑈 substr 〈 𝑀 , 𝑁 〉 ) ‘ 𝑗 ) ↔ ( ( ♯ ‘ ( 𝑊 substr 〈 𝑀 , 𝑁 〉 ) ) = ( ♯ ‘ ( 𝑈 substr 〈 𝑀 , 𝑁 〉 ) ) ∧ ∀ 𝑗 ∈ ( 0 ..^ ( ♯ ‘ ( 𝑊 substr 〈 𝑀 , 𝑁 〉 ) ) ) ( ( 𝑊 substr 〈 𝑀 , 𝑁 〉 ) ‘ 𝑗 ) = ( ( 𝑈 substr 〈 𝑀 , 𝑁 〉 ) ‘ 𝑗 ) ) ) ) |
26 |
|
nn0re |
⊢ ( 𝑀 ∈ ℕ0 → 𝑀 ∈ ℝ ) |
27 |
|
nn0re |
⊢ ( 𝑁 ∈ ℕ0 → 𝑁 ∈ ℝ ) |
28 |
|
ltnle |
⊢ ( ( 𝑀 ∈ ℝ ∧ 𝑁 ∈ ℝ ) → ( 𝑀 < 𝑁 ↔ ¬ 𝑁 ≤ 𝑀 ) ) |
29 |
|
ltle |
⊢ ( ( 𝑀 ∈ ℝ ∧ 𝑁 ∈ ℝ ) → ( 𝑀 < 𝑁 → 𝑀 ≤ 𝑁 ) ) |
30 |
28 29
|
sylbird |
⊢ ( ( 𝑀 ∈ ℝ ∧ 𝑁 ∈ ℝ ) → ( ¬ 𝑁 ≤ 𝑀 → 𝑀 ≤ 𝑁 ) ) |
31 |
26 27 30
|
syl2an |
⊢ ( ( 𝑀 ∈ ℕ0 ∧ 𝑁 ∈ ℕ0 ) → ( ¬ 𝑁 ≤ 𝑀 → 𝑀 ≤ 𝑁 ) ) |
32 |
31
|
3ad2ant2 |
⊢ ( ( ( 𝑊 ∈ Word 𝑉 ∧ 𝑈 ∈ Word 𝑉 ) ∧ ( 𝑀 ∈ ℕ0 ∧ 𝑁 ∈ ℕ0 ) ∧ ( 𝑁 ≤ ( ♯ ‘ 𝑊 ) ∧ 𝑁 ≤ ( ♯ ‘ 𝑈 ) ) ) → ( ¬ 𝑁 ≤ 𝑀 → 𝑀 ≤ 𝑁 ) ) |
33 |
|
simpl1l |
⊢ ( ( ( ( 𝑊 ∈ Word 𝑉 ∧ 𝑈 ∈ Word 𝑉 ) ∧ ( 𝑀 ∈ ℕ0 ∧ 𝑁 ∈ ℕ0 ) ∧ ( 𝑁 ≤ ( ♯ ‘ 𝑊 ) ∧ 𝑁 ≤ ( ♯ ‘ 𝑈 ) ) ) ∧ 𝑀 ≤ 𝑁 ) → 𝑊 ∈ Word 𝑉 ) |
34 |
|
simpl2l |
⊢ ( ( ( ( 𝑊 ∈ Word 𝑉 ∧ 𝑈 ∈ Word 𝑉 ) ∧ ( 𝑀 ∈ ℕ0 ∧ 𝑁 ∈ ℕ0 ) ∧ ( 𝑁 ≤ ( ♯ ‘ 𝑊 ) ∧ 𝑁 ≤ ( ♯ ‘ 𝑈 ) ) ) ∧ 𝑀 ≤ 𝑁 ) → 𝑀 ∈ ℕ0 ) |
35 |
6 7
|
anim12i |
⊢ ( ( 𝑀 ∈ ℕ0 ∧ 𝑁 ∈ ℕ0 ) → ( 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ) ) |
36 |
35
|
3ad2ant2 |
⊢ ( ( ( 𝑊 ∈ Word 𝑉 ∧ 𝑈 ∈ Word 𝑉 ) ∧ ( 𝑀 ∈ ℕ0 ∧ 𝑁 ∈ ℕ0 ) ∧ ( 𝑁 ≤ ( ♯ ‘ 𝑊 ) ∧ 𝑁 ≤ ( ♯ ‘ 𝑈 ) ) ) → ( 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ) ) |
37 |
36
|
anim1i |
⊢ ( ( ( ( 𝑊 ∈ Word 𝑉 ∧ 𝑈 ∈ Word 𝑉 ) ∧ ( 𝑀 ∈ ℕ0 ∧ 𝑁 ∈ ℕ0 ) ∧ ( 𝑁 ≤ ( ♯ ‘ 𝑊 ) ∧ 𝑁 ≤ ( ♯ ‘ 𝑈 ) ) ) ∧ 𝑀 ≤ 𝑁 ) → ( ( 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ) ∧ 𝑀 ≤ 𝑁 ) ) |
38 |
|
df-3an |
⊢ ( ( 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝑀 ≤ 𝑁 ) ↔ ( ( 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ) ∧ 𝑀 ≤ 𝑁 ) ) |
39 |
37 38
|
sylibr |
⊢ ( ( ( ( 𝑊 ∈ Word 𝑉 ∧ 𝑈 ∈ Word 𝑉 ) ∧ ( 𝑀 ∈ ℕ0 ∧ 𝑁 ∈ ℕ0 ) ∧ ( 𝑁 ≤ ( ♯ ‘ 𝑊 ) ∧ 𝑁 ≤ ( ♯ ‘ 𝑈 ) ) ) ∧ 𝑀 ≤ 𝑁 ) → ( 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝑀 ≤ 𝑁 ) ) |
40 |
|
eluz2 |
⊢ ( 𝑁 ∈ ( ℤ≥ ‘ 𝑀 ) ↔ ( 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝑀 ≤ 𝑁 ) ) |
41 |
39 40
|
sylibr |
⊢ ( ( ( ( 𝑊 ∈ Word 𝑉 ∧ 𝑈 ∈ Word 𝑉 ) ∧ ( 𝑀 ∈ ℕ0 ∧ 𝑁 ∈ ℕ0 ) ∧ ( 𝑁 ≤ ( ♯ ‘ 𝑊 ) ∧ 𝑁 ≤ ( ♯ ‘ 𝑈 ) ) ) ∧ 𝑀 ≤ 𝑁 ) → 𝑁 ∈ ( ℤ≥ ‘ 𝑀 ) ) |
42 |
34 41
|
jca |
⊢ ( ( ( ( 𝑊 ∈ Word 𝑉 ∧ 𝑈 ∈ Word 𝑉 ) ∧ ( 𝑀 ∈ ℕ0 ∧ 𝑁 ∈ ℕ0 ) ∧ ( 𝑁 ≤ ( ♯ ‘ 𝑊 ) ∧ 𝑁 ≤ ( ♯ ‘ 𝑈 ) ) ) ∧ 𝑀 ≤ 𝑁 ) → ( 𝑀 ∈ ℕ0 ∧ 𝑁 ∈ ( ℤ≥ ‘ 𝑀 ) ) ) |
43 |
|
simpl3l |
⊢ ( ( ( ( 𝑊 ∈ Word 𝑉 ∧ 𝑈 ∈ Word 𝑉 ) ∧ ( 𝑀 ∈ ℕ0 ∧ 𝑁 ∈ ℕ0 ) ∧ ( 𝑁 ≤ ( ♯ ‘ 𝑊 ) ∧ 𝑁 ≤ ( ♯ ‘ 𝑈 ) ) ) ∧ 𝑀 ≤ 𝑁 ) → 𝑁 ≤ ( ♯ ‘ 𝑊 ) ) |
44 |
|
swrdlen2 |
⊢ ( ( 𝑊 ∈ Word 𝑉 ∧ ( 𝑀 ∈ ℕ0 ∧ 𝑁 ∈ ( ℤ≥ ‘ 𝑀 ) ) ∧ 𝑁 ≤ ( ♯ ‘ 𝑊 ) ) → ( ♯ ‘ ( 𝑊 substr 〈 𝑀 , 𝑁 〉 ) ) = ( 𝑁 − 𝑀 ) ) |
45 |
33 42 43 44
|
syl3anc |
⊢ ( ( ( ( 𝑊 ∈ Word 𝑉 ∧ 𝑈 ∈ Word 𝑉 ) ∧ ( 𝑀 ∈ ℕ0 ∧ 𝑁 ∈ ℕ0 ) ∧ ( 𝑁 ≤ ( ♯ ‘ 𝑊 ) ∧ 𝑁 ≤ ( ♯ ‘ 𝑈 ) ) ) ∧ 𝑀 ≤ 𝑁 ) → ( ♯ ‘ ( 𝑊 substr 〈 𝑀 , 𝑁 〉 ) ) = ( 𝑁 − 𝑀 ) ) |
46 |
45
|
oveq2d |
⊢ ( ( ( ( 𝑊 ∈ Word 𝑉 ∧ 𝑈 ∈ Word 𝑉 ) ∧ ( 𝑀 ∈ ℕ0 ∧ 𝑁 ∈ ℕ0 ) ∧ ( 𝑁 ≤ ( ♯ ‘ 𝑊 ) ∧ 𝑁 ≤ ( ♯ ‘ 𝑈 ) ) ) ∧ 𝑀 ≤ 𝑁 ) → ( 0 ..^ ( ♯ ‘ ( 𝑊 substr 〈 𝑀 , 𝑁 〉 ) ) ) = ( 0 ..^ ( 𝑁 − 𝑀 ) ) ) |
47 |
46
|
raleqdv |
⊢ ( ( ( ( 𝑊 ∈ Word 𝑉 ∧ 𝑈 ∈ Word 𝑉 ) ∧ ( 𝑀 ∈ ℕ0 ∧ 𝑁 ∈ ℕ0 ) ∧ ( 𝑁 ≤ ( ♯ ‘ 𝑊 ) ∧ 𝑁 ≤ ( ♯ ‘ 𝑈 ) ) ) ∧ 𝑀 ≤ 𝑁 ) → ( ∀ 𝑗 ∈ ( 0 ..^ ( ♯ ‘ ( 𝑊 substr 〈 𝑀 , 𝑁 〉 ) ) ) ( ( 𝑊 substr 〈 𝑀 , 𝑁 〉 ) ‘ 𝑗 ) = ( ( 𝑈 substr 〈 𝑀 , 𝑁 〉 ) ‘ 𝑗 ) ↔ ∀ 𝑗 ∈ ( 0 ..^ ( 𝑁 − 𝑀 ) ) ( ( 𝑊 substr 〈 𝑀 , 𝑁 〉 ) ‘ 𝑗 ) = ( ( 𝑈 substr 〈 𝑀 , 𝑁 〉 ) ‘ 𝑗 ) ) ) |
48 |
|
0zd |
⊢ ( ( ( 𝑊 ∈ Word 𝑉 ∧ 𝑈 ∈ Word 𝑉 ) ∧ ( 𝑀 ∈ ℕ0 ∧ 𝑁 ∈ ℕ0 ) ∧ ( 𝑁 ≤ ( ♯ ‘ 𝑊 ) ∧ 𝑁 ≤ ( ♯ ‘ 𝑈 ) ) ) → 0 ∈ ℤ ) |
49 |
|
zsubcl |
⊢ ( ( 𝑁 ∈ ℤ ∧ 𝑀 ∈ ℤ ) → ( 𝑁 − 𝑀 ) ∈ ℤ ) |
50 |
7 6 49
|
syl2anr |
⊢ ( ( 𝑀 ∈ ℕ0 ∧ 𝑁 ∈ ℕ0 ) → ( 𝑁 − 𝑀 ) ∈ ℤ ) |
51 |
50
|
3ad2ant2 |
⊢ ( ( ( 𝑊 ∈ Word 𝑉 ∧ 𝑈 ∈ Word 𝑉 ) ∧ ( 𝑀 ∈ ℕ0 ∧ 𝑁 ∈ ℕ0 ) ∧ ( 𝑁 ≤ ( ♯ ‘ 𝑊 ) ∧ 𝑁 ≤ ( ♯ ‘ 𝑈 ) ) ) → ( 𝑁 − 𝑀 ) ∈ ℤ ) |
52 |
6
|
adantr |
⊢ ( ( 𝑀 ∈ ℕ0 ∧ 𝑁 ∈ ℕ0 ) → 𝑀 ∈ ℤ ) |
53 |
52
|
3ad2ant2 |
⊢ ( ( ( 𝑊 ∈ Word 𝑉 ∧ 𝑈 ∈ Word 𝑉 ) ∧ ( 𝑀 ∈ ℕ0 ∧ 𝑁 ∈ ℕ0 ) ∧ ( 𝑁 ≤ ( ♯ ‘ 𝑊 ) ∧ 𝑁 ≤ ( ♯ ‘ 𝑈 ) ) ) → 𝑀 ∈ ℤ ) |
54 |
|
fzoshftral |
⊢ ( ( 0 ∈ ℤ ∧ ( 𝑁 − 𝑀 ) ∈ ℤ ∧ 𝑀 ∈ ℤ ) → ( ∀ 𝑗 ∈ ( 0 ..^ ( 𝑁 − 𝑀 ) ) ( ( 𝑊 substr 〈 𝑀 , 𝑁 〉 ) ‘ 𝑗 ) = ( ( 𝑈 substr 〈 𝑀 , 𝑁 〉 ) ‘ 𝑗 ) ↔ ∀ 𝑖 ∈ ( ( 0 + 𝑀 ) ..^ ( ( 𝑁 − 𝑀 ) + 𝑀 ) ) [ ( 𝑖 − 𝑀 ) / 𝑗 ] ( ( 𝑊 substr 〈 𝑀 , 𝑁 〉 ) ‘ 𝑗 ) = ( ( 𝑈 substr 〈 𝑀 , 𝑁 〉 ) ‘ 𝑗 ) ) ) |
55 |
48 51 53 54
|
syl3anc |
⊢ ( ( ( 𝑊 ∈ Word 𝑉 ∧ 𝑈 ∈ Word 𝑉 ) ∧ ( 𝑀 ∈ ℕ0 ∧ 𝑁 ∈ ℕ0 ) ∧ ( 𝑁 ≤ ( ♯ ‘ 𝑊 ) ∧ 𝑁 ≤ ( ♯ ‘ 𝑈 ) ) ) → ( ∀ 𝑗 ∈ ( 0 ..^ ( 𝑁 − 𝑀 ) ) ( ( 𝑊 substr 〈 𝑀 , 𝑁 〉 ) ‘ 𝑗 ) = ( ( 𝑈 substr 〈 𝑀 , 𝑁 〉 ) ‘ 𝑗 ) ↔ ∀ 𝑖 ∈ ( ( 0 + 𝑀 ) ..^ ( ( 𝑁 − 𝑀 ) + 𝑀 ) ) [ ( 𝑖 − 𝑀 ) / 𝑗 ] ( ( 𝑊 substr 〈 𝑀 , 𝑁 〉 ) ‘ 𝑗 ) = ( ( 𝑈 substr 〈 𝑀 , 𝑁 〉 ) ‘ 𝑗 ) ) ) |
56 |
55
|
adantr |
⊢ ( ( ( ( 𝑊 ∈ Word 𝑉 ∧ 𝑈 ∈ Word 𝑉 ) ∧ ( 𝑀 ∈ ℕ0 ∧ 𝑁 ∈ ℕ0 ) ∧ ( 𝑁 ≤ ( ♯ ‘ 𝑊 ) ∧ 𝑁 ≤ ( ♯ ‘ 𝑈 ) ) ) ∧ 𝑀 ≤ 𝑁 ) → ( ∀ 𝑗 ∈ ( 0 ..^ ( 𝑁 − 𝑀 ) ) ( ( 𝑊 substr 〈 𝑀 , 𝑁 〉 ) ‘ 𝑗 ) = ( ( 𝑈 substr 〈 𝑀 , 𝑁 〉 ) ‘ 𝑗 ) ↔ ∀ 𝑖 ∈ ( ( 0 + 𝑀 ) ..^ ( ( 𝑁 − 𝑀 ) + 𝑀 ) ) [ ( 𝑖 − 𝑀 ) / 𝑗 ] ( ( 𝑊 substr 〈 𝑀 , 𝑁 〉 ) ‘ 𝑗 ) = ( ( 𝑈 substr 〈 𝑀 , 𝑁 〉 ) ‘ 𝑗 ) ) ) |
57 |
|
nn0cn |
⊢ ( 𝑁 ∈ ℕ0 → 𝑁 ∈ ℂ ) |
58 |
|
nn0cn |
⊢ ( 𝑀 ∈ ℕ0 → 𝑀 ∈ ℂ ) |
59 |
|
addid2 |
⊢ ( 𝑀 ∈ ℂ → ( 0 + 𝑀 ) = 𝑀 ) |
60 |
59
|
adantl |
⊢ ( ( 𝑁 ∈ ℂ ∧ 𝑀 ∈ ℂ ) → ( 0 + 𝑀 ) = 𝑀 ) |
61 |
|
npcan |
⊢ ( ( 𝑁 ∈ ℂ ∧ 𝑀 ∈ ℂ ) → ( ( 𝑁 − 𝑀 ) + 𝑀 ) = 𝑁 ) |
62 |
60 61
|
oveq12d |
⊢ ( ( 𝑁 ∈ ℂ ∧ 𝑀 ∈ ℂ ) → ( ( 0 + 𝑀 ) ..^ ( ( 𝑁 − 𝑀 ) + 𝑀 ) ) = ( 𝑀 ..^ 𝑁 ) ) |
63 |
57 58 62
|
syl2anr |
⊢ ( ( 𝑀 ∈ ℕ0 ∧ 𝑁 ∈ ℕ0 ) → ( ( 0 + 𝑀 ) ..^ ( ( 𝑁 − 𝑀 ) + 𝑀 ) ) = ( 𝑀 ..^ 𝑁 ) ) |
64 |
63
|
3ad2ant2 |
⊢ ( ( ( 𝑊 ∈ Word 𝑉 ∧ 𝑈 ∈ Word 𝑉 ) ∧ ( 𝑀 ∈ ℕ0 ∧ 𝑁 ∈ ℕ0 ) ∧ ( 𝑁 ≤ ( ♯ ‘ 𝑊 ) ∧ 𝑁 ≤ ( ♯ ‘ 𝑈 ) ) ) → ( ( 0 + 𝑀 ) ..^ ( ( 𝑁 − 𝑀 ) + 𝑀 ) ) = ( 𝑀 ..^ 𝑁 ) ) |
65 |
64
|
adantr |
⊢ ( ( ( ( 𝑊 ∈ Word 𝑉 ∧ 𝑈 ∈ Word 𝑉 ) ∧ ( 𝑀 ∈ ℕ0 ∧ 𝑁 ∈ ℕ0 ) ∧ ( 𝑁 ≤ ( ♯ ‘ 𝑊 ) ∧ 𝑁 ≤ ( ♯ ‘ 𝑈 ) ) ) ∧ 𝑀 ≤ 𝑁 ) → ( ( 0 + 𝑀 ) ..^ ( ( 𝑁 − 𝑀 ) + 𝑀 ) ) = ( 𝑀 ..^ 𝑁 ) ) |
66 |
65
|
raleqdv |
⊢ ( ( ( ( 𝑊 ∈ Word 𝑉 ∧ 𝑈 ∈ Word 𝑉 ) ∧ ( 𝑀 ∈ ℕ0 ∧ 𝑁 ∈ ℕ0 ) ∧ ( 𝑁 ≤ ( ♯ ‘ 𝑊 ) ∧ 𝑁 ≤ ( ♯ ‘ 𝑈 ) ) ) ∧ 𝑀 ≤ 𝑁 ) → ( ∀ 𝑖 ∈ ( ( 0 + 𝑀 ) ..^ ( ( 𝑁 − 𝑀 ) + 𝑀 ) ) [ ( 𝑖 − 𝑀 ) / 𝑗 ] ( ( 𝑊 substr 〈 𝑀 , 𝑁 〉 ) ‘ 𝑗 ) = ( ( 𝑈 substr 〈 𝑀 , 𝑁 〉 ) ‘ 𝑗 ) ↔ ∀ 𝑖 ∈ ( 𝑀 ..^ 𝑁 ) [ ( 𝑖 − 𝑀 ) / 𝑗 ] ( ( 𝑊 substr 〈 𝑀 , 𝑁 〉 ) ‘ 𝑗 ) = ( ( 𝑈 substr 〈 𝑀 , 𝑁 〉 ) ‘ 𝑗 ) ) ) |
67 |
|
ovex |
⊢ ( 𝑖 − 𝑀 ) ∈ V |
68 |
|
sbceqg |
⊢ ( ( 𝑖 − 𝑀 ) ∈ V → ( [ ( 𝑖 − 𝑀 ) / 𝑗 ] ( ( 𝑊 substr 〈 𝑀 , 𝑁 〉 ) ‘ 𝑗 ) = ( ( 𝑈 substr 〈 𝑀 , 𝑁 〉 ) ‘ 𝑗 ) ↔ ⦋ ( 𝑖 − 𝑀 ) / 𝑗 ⦌ ( ( 𝑊 substr 〈 𝑀 , 𝑁 〉 ) ‘ 𝑗 ) = ⦋ ( 𝑖 − 𝑀 ) / 𝑗 ⦌ ( ( 𝑈 substr 〈 𝑀 , 𝑁 〉 ) ‘ 𝑗 ) ) ) |
69 |
|
csbfv2g |
⊢ ( ( 𝑖 − 𝑀 ) ∈ V → ⦋ ( 𝑖 − 𝑀 ) / 𝑗 ⦌ ( ( 𝑊 substr 〈 𝑀 , 𝑁 〉 ) ‘ 𝑗 ) = ( ( 𝑊 substr 〈 𝑀 , 𝑁 〉 ) ‘ ⦋ ( 𝑖 − 𝑀 ) / 𝑗 ⦌ 𝑗 ) ) |
70 |
|
csbvarg |
⊢ ( ( 𝑖 − 𝑀 ) ∈ V → ⦋ ( 𝑖 − 𝑀 ) / 𝑗 ⦌ 𝑗 = ( 𝑖 − 𝑀 ) ) |
71 |
70
|
fveq2d |
⊢ ( ( 𝑖 − 𝑀 ) ∈ V → ( ( 𝑊 substr 〈 𝑀 , 𝑁 〉 ) ‘ ⦋ ( 𝑖 − 𝑀 ) / 𝑗 ⦌ 𝑗 ) = ( ( 𝑊 substr 〈 𝑀 , 𝑁 〉 ) ‘ ( 𝑖 − 𝑀 ) ) ) |
72 |
69 71
|
eqtrd |
⊢ ( ( 𝑖 − 𝑀 ) ∈ V → ⦋ ( 𝑖 − 𝑀 ) / 𝑗 ⦌ ( ( 𝑊 substr 〈 𝑀 , 𝑁 〉 ) ‘ 𝑗 ) = ( ( 𝑊 substr 〈 𝑀 , 𝑁 〉 ) ‘ ( 𝑖 − 𝑀 ) ) ) |
73 |
|
csbfv2g |
⊢ ( ( 𝑖 − 𝑀 ) ∈ V → ⦋ ( 𝑖 − 𝑀 ) / 𝑗 ⦌ ( ( 𝑈 substr 〈 𝑀 , 𝑁 〉 ) ‘ 𝑗 ) = ( ( 𝑈 substr 〈 𝑀 , 𝑁 〉 ) ‘ ⦋ ( 𝑖 − 𝑀 ) / 𝑗 ⦌ 𝑗 ) ) |
74 |
70
|
fveq2d |
⊢ ( ( 𝑖 − 𝑀 ) ∈ V → ( ( 𝑈 substr 〈 𝑀 , 𝑁 〉 ) ‘ ⦋ ( 𝑖 − 𝑀 ) / 𝑗 ⦌ 𝑗 ) = ( ( 𝑈 substr 〈 𝑀 , 𝑁 〉 ) ‘ ( 𝑖 − 𝑀 ) ) ) |
75 |
73 74
|
eqtrd |
⊢ ( ( 𝑖 − 𝑀 ) ∈ V → ⦋ ( 𝑖 − 𝑀 ) / 𝑗 ⦌ ( ( 𝑈 substr 〈 𝑀 , 𝑁 〉 ) ‘ 𝑗 ) = ( ( 𝑈 substr 〈 𝑀 , 𝑁 〉 ) ‘ ( 𝑖 − 𝑀 ) ) ) |
76 |
72 75
|
eqeq12d |
⊢ ( ( 𝑖 − 𝑀 ) ∈ V → ( ⦋ ( 𝑖 − 𝑀 ) / 𝑗 ⦌ ( ( 𝑊 substr 〈 𝑀 , 𝑁 〉 ) ‘ 𝑗 ) = ⦋ ( 𝑖 − 𝑀 ) / 𝑗 ⦌ ( ( 𝑈 substr 〈 𝑀 , 𝑁 〉 ) ‘ 𝑗 ) ↔ ( ( 𝑊 substr 〈 𝑀 , 𝑁 〉 ) ‘ ( 𝑖 − 𝑀 ) ) = ( ( 𝑈 substr 〈 𝑀 , 𝑁 〉 ) ‘ ( 𝑖 − 𝑀 ) ) ) ) |
77 |
68 76
|
bitrd |
⊢ ( ( 𝑖 − 𝑀 ) ∈ V → ( [ ( 𝑖 − 𝑀 ) / 𝑗 ] ( ( 𝑊 substr 〈 𝑀 , 𝑁 〉 ) ‘ 𝑗 ) = ( ( 𝑈 substr 〈 𝑀 , 𝑁 〉 ) ‘ 𝑗 ) ↔ ( ( 𝑊 substr 〈 𝑀 , 𝑁 〉 ) ‘ ( 𝑖 − 𝑀 ) ) = ( ( 𝑈 substr 〈 𝑀 , 𝑁 〉 ) ‘ ( 𝑖 − 𝑀 ) ) ) ) |
78 |
67 77
|
mp1i |
⊢ ( ( ( ( ( 𝑊 ∈ Word 𝑉 ∧ 𝑈 ∈ Word 𝑉 ) ∧ ( 𝑀 ∈ ℕ0 ∧ 𝑁 ∈ ℕ0 ) ∧ ( 𝑁 ≤ ( ♯ ‘ 𝑊 ) ∧ 𝑁 ≤ ( ♯ ‘ 𝑈 ) ) ) ∧ 𝑀 ≤ 𝑁 ) ∧ 𝑖 ∈ ( 𝑀 ..^ 𝑁 ) ) → ( [ ( 𝑖 − 𝑀 ) / 𝑗 ] ( ( 𝑊 substr 〈 𝑀 , 𝑁 〉 ) ‘ 𝑗 ) = ( ( 𝑈 substr 〈 𝑀 , 𝑁 〉 ) ‘ 𝑗 ) ↔ ( ( 𝑊 substr 〈 𝑀 , 𝑁 〉 ) ‘ ( 𝑖 − 𝑀 ) ) = ( ( 𝑈 substr 〈 𝑀 , 𝑁 〉 ) ‘ ( 𝑖 − 𝑀 ) ) ) ) |
79 |
33 42 43
|
3jca |
⊢ ( ( ( ( 𝑊 ∈ Word 𝑉 ∧ 𝑈 ∈ Word 𝑉 ) ∧ ( 𝑀 ∈ ℕ0 ∧ 𝑁 ∈ ℕ0 ) ∧ ( 𝑁 ≤ ( ♯ ‘ 𝑊 ) ∧ 𝑁 ≤ ( ♯ ‘ 𝑈 ) ) ) ∧ 𝑀 ≤ 𝑁 ) → ( 𝑊 ∈ Word 𝑉 ∧ ( 𝑀 ∈ ℕ0 ∧ 𝑁 ∈ ( ℤ≥ ‘ 𝑀 ) ) ∧ 𝑁 ≤ ( ♯ ‘ 𝑊 ) ) ) |
80 |
|
swrdfv2 |
⊢ ( ( ( 𝑊 ∈ Word 𝑉 ∧ ( 𝑀 ∈ ℕ0 ∧ 𝑁 ∈ ( ℤ≥ ‘ 𝑀 ) ) ∧ 𝑁 ≤ ( ♯ ‘ 𝑊 ) ) ∧ 𝑖 ∈ ( 𝑀 ..^ 𝑁 ) ) → ( ( 𝑊 substr 〈 𝑀 , 𝑁 〉 ) ‘ ( 𝑖 − 𝑀 ) ) = ( 𝑊 ‘ 𝑖 ) ) |
81 |
79 80
|
sylan |
⊢ ( ( ( ( ( 𝑊 ∈ Word 𝑉 ∧ 𝑈 ∈ Word 𝑉 ) ∧ ( 𝑀 ∈ ℕ0 ∧ 𝑁 ∈ ℕ0 ) ∧ ( 𝑁 ≤ ( ♯ ‘ 𝑊 ) ∧ 𝑁 ≤ ( ♯ ‘ 𝑈 ) ) ) ∧ 𝑀 ≤ 𝑁 ) ∧ 𝑖 ∈ ( 𝑀 ..^ 𝑁 ) ) → ( ( 𝑊 substr 〈 𝑀 , 𝑁 〉 ) ‘ ( 𝑖 − 𝑀 ) ) = ( 𝑊 ‘ 𝑖 ) ) |
82 |
|
simpl1r |
⊢ ( ( ( ( 𝑊 ∈ Word 𝑉 ∧ 𝑈 ∈ Word 𝑉 ) ∧ ( 𝑀 ∈ ℕ0 ∧ 𝑁 ∈ ℕ0 ) ∧ ( 𝑁 ≤ ( ♯ ‘ 𝑊 ) ∧ 𝑁 ≤ ( ♯ ‘ 𝑈 ) ) ) ∧ 𝑀 ≤ 𝑁 ) → 𝑈 ∈ Word 𝑉 ) |
83 |
|
simpl3r |
⊢ ( ( ( ( 𝑊 ∈ Word 𝑉 ∧ 𝑈 ∈ Word 𝑉 ) ∧ ( 𝑀 ∈ ℕ0 ∧ 𝑁 ∈ ℕ0 ) ∧ ( 𝑁 ≤ ( ♯ ‘ 𝑊 ) ∧ 𝑁 ≤ ( ♯ ‘ 𝑈 ) ) ) ∧ 𝑀 ≤ 𝑁 ) → 𝑁 ≤ ( ♯ ‘ 𝑈 ) ) |
84 |
82 42 83
|
3jca |
⊢ ( ( ( ( 𝑊 ∈ Word 𝑉 ∧ 𝑈 ∈ Word 𝑉 ) ∧ ( 𝑀 ∈ ℕ0 ∧ 𝑁 ∈ ℕ0 ) ∧ ( 𝑁 ≤ ( ♯ ‘ 𝑊 ) ∧ 𝑁 ≤ ( ♯ ‘ 𝑈 ) ) ) ∧ 𝑀 ≤ 𝑁 ) → ( 𝑈 ∈ Word 𝑉 ∧ ( 𝑀 ∈ ℕ0 ∧ 𝑁 ∈ ( ℤ≥ ‘ 𝑀 ) ) ∧ 𝑁 ≤ ( ♯ ‘ 𝑈 ) ) ) |
85 |
|
swrdfv2 |
⊢ ( ( ( 𝑈 ∈ Word 𝑉 ∧ ( 𝑀 ∈ ℕ0 ∧ 𝑁 ∈ ( ℤ≥ ‘ 𝑀 ) ) ∧ 𝑁 ≤ ( ♯ ‘ 𝑈 ) ) ∧ 𝑖 ∈ ( 𝑀 ..^ 𝑁 ) ) → ( ( 𝑈 substr 〈 𝑀 , 𝑁 〉 ) ‘ ( 𝑖 − 𝑀 ) ) = ( 𝑈 ‘ 𝑖 ) ) |
86 |
84 85
|
sylan |
⊢ ( ( ( ( ( 𝑊 ∈ Word 𝑉 ∧ 𝑈 ∈ Word 𝑉 ) ∧ ( 𝑀 ∈ ℕ0 ∧ 𝑁 ∈ ℕ0 ) ∧ ( 𝑁 ≤ ( ♯ ‘ 𝑊 ) ∧ 𝑁 ≤ ( ♯ ‘ 𝑈 ) ) ) ∧ 𝑀 ≤ 𝑁 ) ∧ 𝑖 ∈ ( 𝑀 ..^ 𝑁 ) ) → ( ( 𝑈 substr 〈 𝑀 , 𝑁 〉 ) ‘ ( 𝑖 − 𝑀 ) ) = ( 𝑈 ‘ 𝑖 ) ) |
87 |
81 86
|
eqeq12d |
⊢ ( ( ( ( ( 𝑊 ∈ Word 𝑉 ∧ 𝑈 ∈ Word 𝑉 ) ∧ ( 𝑀 ∈ ℕ0 ∧ 𝑁 ∈ ℕ0 ) ∧ ( 𝑁 ≤ ( ♯ ‘ 𝑊 ) ∧ 𝑁 ≤ ( ♯ ‘ 𝑈 ) ) ) ∧ 𝑀 ≤ 𝑁 ) ∧ 𝑖 ∈ ( 𝑀 ..^ 𝑁 ) ) → ( ( ( 𝑊 substr 〈 𝑀 , 𝑁 〉 ) ‘ ( 𝑖 − 𝑀 ) ) = ( ( 𝑈 substr 〈 𝑀 , 𝑁 〉 ) ‘ ( 𝑖 − 𝑀 ) ) ↔ ( 𝑊 ‘ 𝑖 ) = ( 𝑈 ‘ 𝑖 ) ) ) |
88 |
78 87
|
bitrd |
⊢ ( ( ( ( ( 𝑊 ∈ Word 𝑉 ∧ 𝑈 ∈ Word 𝑉 ) ∧ ( 𝑀 ∈ ℕ0 ∧ 𝑁 ∈ ℕ0 ) ∧ ( 𝑁 ≤ ( ♯ ‘ 𝑊 ) ∧ 𝑁 ≤ ( ♯ ‘ 𝑈 ) ) ) ∧ 𝑀 ≤ 𝑁 ) ∧ 𝑖 ∈ ( 𝑀 ..^ 𝑁 ) ) → ( [ ( 𝑖 − 𝑀 ) / 𝑗 ] ( ( 𝑊 substr 〈 𝑀 , 𝑁 〉 ) ‘ 𝑗 ) = ( ( 𝑈 substr 〈 𝑀 , 𝑁 〉 ) ‘ 𝑗 ) ↔ ( 𝑊 ‘ 𝑖 ) = ( 𝑈 ‘ 𝑖 ) ) ) |
89 |
88
|
ralbidva |
⊢ ( ( ( ( 𝑊 ∈ Word 𝑉 ∧ 𝑈 ∈ Word 𝑉 ) ∧ ( 𝑀 ∈ ℕ0 ∧ 𝑁 ∈ ℕ0 ) ∧ ( 𝑁 ≤ ( ♯ ‘ 𝑊 ) ∧ 𝑁 ≤ ( ♯ ‘ 𝑈 ) ) ) ∧ 𝑀 ≤ 𝑁 ) → ( ∀ 𝑖 ∈ ( 𝑀 ..^ 𝑁 ) [ ( 𝑖 − 𝑀 ) / 𝑗 ] ( ( 𝑊 substr 〈 𝑀 , 𝑁 〉 ) ‘ 𝑗 ) = ( ( 𝑈 substr 〈 𝑀 , 𝑁 〉 ) ‘ 𝑗 ) ↔ ∀ 𝑖 ∈ ( 𝑀 ..^ 𝑁 ) ( 𝑊 ‘ 𝑖 ) = ( 𝑈 ‘ 𝑖 ) ) ) |
90 |
66 89
|
bitrd |
⊢ ( ( ( ( 𝑊 ∈ Word 𝑉 ∧ 𝑈 ∈ Word 𝑉 ) ∧ ( 𝑀 ∈ ℕ0 ∧ 𝑁 ∈ ℕ0 ) ∧ ( 𝑁 ≤ ( ♯ ‘ 𝑊 ) ∧ 𝑁 ≤ ( ♯ ‘ 𝑈 ) ) ) ∧ 𝑀 ≤ 𝑁 ) → ( ∀ 𝑖 ∈ ( ( 0 + 𝑀 ) ..^ ( ( 𝑁 − 𝑀 ) + 𝑀 ) ) [ ( 𝑖 − 𝑀 ) / 𝑗 ] ( ( 𝑊 substr 〈 𝑀 , 𝑁 〉 ) ‘ 𝑗 ) = ( ( 𝑈 substr 〈 𝑀 , 𝑁 〉 ) ‘ 𝑗 ) ↔ ∀ 𝑖 ∈ ( 𝑀 ..^ 𝑁 ) ( 𝑊 ‘ 𝑖 ) = ( 𝑈 ‘ 𝑖 ) ) ) |
91 |
47 56 90
|
3bitrd |
⊢ ( ( ( ( 𝑊 ∈ Word 𝑉 ∧ 𝑈 ∈ Word 𝑉 ) ∧ ( 𝑀 ∈ ℕ0 ∧ 𝑁 ∈ ℕ0 ) ∧ ( 𝑁 ≤ ( ♯ ‘ 𝑊 ) ∧ 𝑁 ≤ ( ♯ ‘ 𝑈 ) ) ) ∧ 𝑀 ≤ 𝑁 ) → ( ∀ 𝑗 ∈ ( 0 ..^ ( ♯ ‘ ( 𝑊 substr 〈 𝑀 , 𝑁 〉 ) ) ) ( ( 𝑊 substr 〈 𝑀 , 𝑁 〉 ) ‘ 𝑗 ) = ( ( 𝑈 substr 〈 𝑀 , 𝑁 〉 ) ‘ 𝑗 ) ↔ ∀ 𝑖 ∈ ( 𝑀 ..^ 𝑁 ) ( 𝑊 ‘ 𝑖 ) = ( 𝑈 ‘ 𝑖 ) ) ) |
92 |
91
|
ex |
⊢ ( ( ( 𝑊 ∈ Word 𝑉 ∧ 𝑈 ∈ Word 𝑉 ) ∧ ( 𝑀 ∈ ℕ0 ∧ 𝑁 ∈ ℕ0 ) ∧ ( 𝑁 ≤ ( ♯ ‘ 𝑊 ) ∧ 𝑁 ≤ ( ♯ ‘ 𝑈 ) ) ) → ( 𝑀 ≤ 𝑁 → ( ∀ 𝑗 ∈ ( 0 ..^ ( ♯ ‘ ( 𝑊 substr 〈 𝑀 , 𝑁 〉 ) ) ) ( ( 𝑊 substr 〈 𝑀 , 𝑁 〉 ) ‘ 𝑗 ) = ( ( 𝑈 substr 〈 𝑀 , 𝑁 〉 ) ‘ 𝑗 ) ↔ ∀ 𝑖 ∈ ( 𝑀 ..^ 𝑁 ) ( 𝑊 ‘ 𝑖 ) = ( 𝑈 ‘ 𝑖 ) ) ) ) |
93 |
32 92
|
syld |
⊢ ( ( ( 𝑊 ∈ Word 𝑉 ∧ 𝑈 ∈ Word 𝑉 ) ∧ ( 𝑀 ∈ ℕ0 ∧ 𝑁 ∈ ℕ0 ) ∧ ( 𝑁 ≤ ( ♯ ‘ 𝑊 ) ∧ 𝑁 ≤ ( ♯ ‘ 𝑈 ) ) ) → ( ¬ 𝑁 ≤ 𝑀 → ( ∀ 𝑗 ∈ ( 0 ..^ ( ♯ ‘ ( 𝑊 substr 〈 𝑀 , 𝑁 〉 ) ) ) ( ( 𝑊 substr 〈 𝑀 , 𝑁 〉 ) ‘ 𝑗 ) = ( ( 𝑈 substr 〈 𝑀 , 𝑁 〉 ) ‘ 𝑗 ) ↔ ∀ 𝑖 ∈ ( 𝑀 ..^ 𝑁 ) ( 𝑊 ‘ 𝑖 ) = ( 𝑈 ‘ 𝑖 ) ) ) ) |
94 |
93
|
impcom |
⊢ ( ( ¬ 𝑁 ≤ 𝑀 ∧ ( ( 𝑊 ∈ Word 𝑉 ∧ 𝑈 ∈ Word 𝑉 ) ∧ ( 𝑀 ∈ ℕ0 ∧ 𝑁 ∈ ℕ0 ) ∧ ( 𝑁 ≤ ( ♯ ‘ 𝑊 ) ∧ 𝑁 ≤ ( ♯ ‘ 𝑈 ) ) ) ) → ( ∀ 𝑗 ∈ ( 0 ..^ ( ♯ ‘ ( 𝑊 substr 〈 𝑀 , 𝑁 〉 ) ) ) ( ( 𝑊 substr 〈 𝑀 , 𝑁 〉 ) ‘ 𝑗 ) = ( ( 𝑈 substr 〈 𝑀 , 𝑁 〉 ) ‘ 𝑗 ) ↔ ∀ 𝑖 ∈ ( 𝑀 ..^ 𝑁 ) ( 𝑊 ‘ 𝑖 ) = ( 𝑈 ‘ 𝑖 ) ) ) |
95 |
22 25 94
|
3bitr2d |
⊢ ( ( ¬ 𝑁 ≤ 𝑀 ∧ ( ( 𝑊 ∈ Word 𝑉 ∧ 𝑈 ∈ Word 𝑉 ) ∧ ( 𝑀 ∈ ℕ0 ∧ 𝑁 ∈ ℕ0 ) ∧ ( 𝑁 ≤ ( ♯ ‘ 𝑊 ) ∧ 𝑁 ≤ ( ♯ ‘ 𝑈 ) ) ) ) → ( ( 𝑊 substr 〈 𝑀 , 𝑁 〉 ) = ( 𝑈 substr 〈 𝑀 , 𝑁 〉 ) ↔ ∀ 𝑖 ∈ ( 𝑀 ..^ 𝑁 ) ( 𝑊 ‘ 𝑖 ) = ( 𝑈 ‘ 𝑖 ) ) ) |
96 |
16 95
|
pm2.61ian |
⊢ ( ( ( 𝑊 ∈ Word 𝑉 ∧ 𝑈 ∈ Word 𝑉 ) ∧ ( 𝑀 ∈ ℕ0 ∧ 𝑁 ∈ ℕ0 ) ∧ ( 𝑁 ≤ ( ♯ ‘ 𝑊 ) ∧ 𝑁 ≤ ( ♯ ‘ 𝑈 ) ) ) → ( ( 𝑊 substr 〈 𝑀 , 𝑁 〉 ) = ( 𝑈 substr 〈 𝑀 , 𝑁 〉 ) ↔ ∀ 𝑖 ∈ ( 𝑀 ..^ 𝑁 ) ( 𝑊 ‘ 𝑖 ) = ( 𝑈 ‘ 𝑖 ) ) ) |