Step |
Hyp |
Ref |
Expression |
1 |
|
simpr |
⊢ ( ( ( 𝑊 ∈ Word 𝑉 ∧ 𝑀 ∈ ( 0 ... 𝑁 ) ∧ 𝑁 ∈ ( 0 ... ( ♯ ‘ 𝑊 ) ) ) ∧ 𝑖 ∈ ( 0 ..^ ( 𝑁 − 𝑀 ) ) ) → 𝑖 ∈ ( 0 ..^ ( 𝑁 − 𝑀 ) ) ) |
2 |
|
fzssz |
⊢ ( 0 ... ( ♯ ‘ 𝑊 ) ) ⊆ ℤ |
3 |
|
simpl3 |
⊢ ( ( ( 𝑊 ∈ Word 𝑉 ∧ 𝑀 ∈ ( 0 ... 𝑁 ) ∧ 𝑁 ∈ ( 0 ... ( ♯ ‘ 𝑊 ) ) ) ∧ 𝑖 ∈ ( 0 ..^ ( 𝑁 − 𝑀 ) ) ) → 𝑁 ∈ ( 0 ... ( ♯ ‘ 𝑊 ) ) ) |
4 |
2 3
|
sseldi |
⊢ ( ( ( 𝑊 ∈ Word 𝑉 ∧ 𝑀 ∈ ( 0 ... 𝑁 ) ∧ 𝑁 ∈ ( 0 ... ( ♯ ‘ 𝑊 ) ) ) ∧ 𝑖 ∈ ( 0 ..^ ( 𝑁 − 𝑀 ) ) ) → 𝑁 ∈ ℤ ) |
5 |
|
fzssz |
⊢ ( 0 ... 𝑁 ) ⊆ ℤ |
6 |
|
simpl2 |
⊢ ( ( ( 𝑊 ∈ Word 𝑉 ∧ 𝑀 ∈ ( 0 ... 𝑁 ) ∧ 𝑁 ∈ ( 0 ... ( ♯ ‘ 𝑊 ) ) ) ∧ 𝑖 ∈ ( 0 ..^ ( 𝑁 − 𝑀 ) ) ) → 𝑀 ∈ ( 0 ... 𝑁 ) ) |
7 |
5 6
|
sseldi |
⊢ ( ( ( 𝑊 ∈ Word 𝑉 ∧ 𝑀 ∈ ( 0 ... 𝑁 ) ∧ 𝑁 ∈ ( 0 ... ( ♯ ‘ 𝑊 ) ) ) ∧ 𝑖 ∈ ( 0 ..^ ( 𝑁 − 𝑀 ) ) ) → 𝑀 ∈ ℤ ) |
8 |
|
fzoaddel2 |
⊢ ( ( 𝑖 ∈ ( 0 ..^ ( 𝑁 − 𝑀 ) ) ∧ 𝑁 ∈ ℤ ∧ 𝑀 ∈ ℤ ) → ( 𝑖 + 𝑀 ) ∈ ( 𝑀 ..^ 𝑁 ) ) |
9 |
1 4 7 8
|
syl3anc |
⊢ ( ( ( 𝑊 ∈ Word 𝑉 ∧ 𝑀 ∈ ( 0 ... 𝑁 ) ∧ 𝑁 ∈ ( 0 ... ( ♯ ‘ 𝑊 ) ) ) ∧ 𝑖 ∈ ( 0 ..^ ( 𝑁 − 𝑀 ) ) ) → ( 𝑖 + 𝑀 ) ∈ ( 𝑀 ..^ 𝑁 ) ) |
10 |
|
simpr |
⊢ ( ( ( 𝑊 ∈ Word 𝑉 ∧ 𝑀 ∈ ( 0 ... 𝑁 ) ∧ 𝑁 ∈ ( 0 ... ( ♯ ‘ 𝑊 ) ) ) ∧ 𝑗 ∈ ( 𝑀 ..^ 𝑁 ) ) → 𝑗 ∈ ( 𝑀 ..^ 𝑁 ) ) |
11 |
|
simpl2 |
⊢ ( ( ( 𝑊 ∈ Word 𝑉 ∧ 𝑀 ∈ ( 0 ... 𝑁 ) ∧ 𝑁 ∈ ( 0 ... ( ♯ ‘ 𝑊 ) ) ) ∧ 𝑗 ∈ ( 𝑀 ..^ 𝑁 ) ) → 𝑀 ∈ ( 0 ... 𝑁 ) ) |
12 |
5 11
|
sseldi |
⊢ ( ( ( 𝑊 ∈ Word 𝑉 ∧ 𝑀 ∈ ( 0 ... 𝑁 ) ∧ 𝑁 ∈ ( 0 ... ( ♯ ‘ 𝑊 ) ) ) ∧ 𝑗 ∈ ( 𝑀 ..^ 𝑁 ) ) → 𝑀 ∈ ℤ ) |
13 |
12
|
zcnd |
⊢ ( ( ( 𝑊 ∈ Word 𝑉 ∧ 𝑀 ∈ ( 0 ... 𝑁 ) ∧ 𝑁 ∈ ( 0 ... ( ♯ ‘ 𝑊 ) ) ) ∧ 𝑗 ∈ ( 𝑀 ..^ 𝑁 ) ) → 𝑀 ∈ ℂ ) |
14 |
|
simpl3 |
⊢ ( ( ( 𝑊 ∈ Word 𝑉 ∧ 𝑀 ∈ ( 0 ... 𝑁 ) ∧ 𝑁 ∈ ( 0 ... ( ♯ ‘ 𝑊 ) ) ) ∧ 𝑗 ∈ ( 𝑀 ..^ 𝑁 ) ) → 𝑁 ∈ ( 0 ... ( ♯ ‘ 𝑊 ) ) ) |
15 |
2 14
|
sseldi |
⊢ ( ( ( 𝑊 ∈ Word 𝑉 ∧ 𝑀 ∈ ( 0 ... 𝑁 ) ∧ 𝑁 ∈ ( 0 ... ( ♯ ‘ 𝑊 ) ) ) ∧ 𝑗 ∈ ( 𝑀 ..^ 𝑁 ) ) → 𝑁 ∈ ℤ ) |
16 |
15
|
zcnd |
⊢ ( ( ( 𝑊 ∈ Word 𝑉 ∧ 𝑀 ∈ ( 0 ... 𝑁 ) ∧ 𝑁 ∈ ( 0 ... ( ♯ ‘ 𝑊 ) ) ) ∧ 𝑗 ∈ ( 𝑀 ..^ 𝑁 ) ) → 𝑁 ∈ ℂ ) |
17 |
13 16
|
pncan3d |
⊢ ( ( ( 𝑊 ∈ Word 𝑉 ∧ 𝑀 ∈ ( 0 ... 𝑁 ) ∧ 𝑁 ∈ ( 0 ... ( ♯ ‘ 𝑊 ) ) ) ∧ 𝑗 ∈ ( 𝑀 ..^ 𝑁 ) ) → ( 𝑀 + ( 𝑁 − 𝑀 ) ) = 𝑁 ) |
18 |
17
|
oveq2d |
⊢ ( ( ( 𝑊 ∈ Word 𝑉 ∧ 𝑀 ∈ ( 0 ... 𝑁 ) ∧ 𝑁 ∈ ( 0 ... ( ♯ ‘ 𝑊 ) ) ) ∧ 𝑗 ∈ ( 𝑀 ..^ 𝑁 ) ) → ( 𝑀 ..^ ( 𝑀 + ( 𝑁 − 𝑀 ) ) ) = ( 𝑀 ..^ 𝑁 ) ) |
19 |
10 18
|
eleqtrrd |
⊢ ( ( ( 𝑊 ∈ Word 𝑉 ∧ 𝑀 ∈ ( 0 ... 𝑁 ) ∧ 𝑁 ∈ ( 0 ... ( ♯ ‘ 𝑊 ) ) ) ∧ 𝑗 ∈ ( 𝑀 ..^ 𝑁 ) ) → 𝑗 ∈ ( 𝑀 ..^ ( 𝑀 + ( 𝑁 − 𝑀 ) ) ) ) |
20 |
15 12
|
zsubcld |
⊢ ( ( ( 𝑊 ∈ Word 𝑉 ∧ 𝑀 ∈ ( 0 ... 𝑁 ) ∧ 𝑁 ∈ ( 0 ... ( ♯ ‘ 𝑊 ) ) ) ∧ 𝑗 ∈ ( 𝑀 ..^ 𝑁 ) ) → ( 𝑁 − 𝑀 ) ∈ ℤ ) |
21 |
|
fzosubel3 |
⊢ ( ( 𝑗 ∈ ( 𝑀 ..^ ( 𝑀 + ( 𝑁 − 𝑀 ) ) ) ∧ ( 𝑁 − 𝑀 ) ∈ ℤ ) → ( 𝑗 − 𝑀 ) ∈ ( 0 ..^ ( 𝑁 − 𝑀 ) ) ) |
22 |
19 20 21
|
syl2anc |
⊢ ( ( ( 𝑊 ∈ Word 𝑉 ∧ 𝑀 ∈ ( 0 ... 𝑁 ) ∧ 𝑁 ∈ ( 0 ... ( ♯ ‘ 𝑊 ) ) ) ∧ 𝑗 ∈ ( 𝑀 ..^ 𝑁 ) ) → ( 𝑗 − 𝑀 ) ∈ ( 0 ..^ ( 𝑁 − 𝑀 ) ) ) |
23 |
|
simpr |
⊢ ( ( ( ( 𝑊 ∈ Word 𝑉 ∧ 𝑀 ∈ ( 0 ... 𝑁 ) ∧ 𝑁 ∈ ( 0 ... ( ♯ ‘ 𝑊 ) ) ) ∧ 𝑗 ∈ ( 𝑀 ..^ 𝑁 ) ) ∧ 𝑖 = ( 𝑗 − 𝑀 ) ) → 𝑖 = ( 𝑗 − 𝑀 ) ) |
24 |
23
|
oveq1d |
⊢ ( ( ( ( 𝑊 ∈ Word 𝑉 ∧ 𝑀 ∈ ( 0 ... 𝑁 ) ∧ 𝑁 ∈ ( 0 ... ( ♯ ‘ 𝑊 ) ) ) ∧ 𝑗 ∈ ( 𝑀 ..^ 𝑁 ) ) ∧ 𝑖 = ( 𝑗 − 𝑀 ) ) → ( 𝑖 + 𝑀 ) = ( ( 𝑗 − 𝑀 ) + 𝑀 ) ) |
25 |
24
|
eqeq2d |
⊢ ( ( ( ( 𝑊 ∈ Word 𝑉 ∧ 𝑀 ∈ ( 0 ... 𝑁 ) ∧ 𝑁 ∈ ( 0 ... ( ♯ ‘ 𝑊 ) ) ) ∧ 𝑗 ∈ ( 𝑀 ..^ 𝑁 ) ) ∧ 𝑖 = ( 𝑗 − 𝑀 ) ) → ( 𝑗 = ( 𝑖 + 𝑀 ) ↔ 𝑗 = ( ( 𝑗 − 𝑀 ) + 𝑀 ) ) ) |
26 |
|
fzossz |
⊢ ( 𝑀 ..^ 𝑁 ) ⊆ ℤ |
27 |
26 10
|
sseldi |
⊢ ( ( ( 𝑊 ∈ Word 𝑉 ∧ 𝑀 ∈ ( 0 ... 𝑁 ) ∧ 𝑁 ∈ ( 0 ... ( ♯ ‘ 𝑊 ) ) ) ∧ 𝑗 ∈ ( 𝑀 ..^ 𝑁 ) ) → 𝑗 ∈ ℤ ) |
28 |
27
|
zcnd |
⊢ ( ( ( 𝑊 ∈ Word 𝑉 ∧ 𝑀 ∈ ( 0 ... 𝑁 ) ∧ 𝑁 ∈ ( 0 ... ( ♯ ‘ 𝑊 ) ) ) ∧ 𝑗 ∈ ( 𝑀 ..^ 𝑁 ) ) → 𝑗 ∈ ℂ ) |
29 |
28 13
|
npcand |
⊢ ( ( ( 𝑊 ∈ Word 𝑉 ∧ 𝑀 ∈ ( 0 ... 𝑁 ) ∧ 𝑁 ∈ ( 0 ... ( ♯ ‘ 𝑊 ) ) ) ∧ 𝑗 ∈ ( 𝑀 ..^ 𝑁 ) ) → ( ( 𝑗 − 𝑀 ) + 𝑀 ) = 𝑗 ) |
30 |
29
|
eqcomd |
⊢ ( ( ( 𝑊 ∈ Word 𝑉 ∧ 𝑀 ∈ ( 0 ... 𝑁 ) ∧ 𝑁 ∈ ( 0 ... ( ♯ ‘ 𝑊 ) ) ) ∧ 𝑗 ∈ ( 𝑀 ..^ 𝑁 ) ) → 𝑗 = ( ( 𝑗 − 𝑀 ) + 𝑀 ) ) |
31 |
22 25 30
|
rspcedvd |
⊢ ( ( ( 𝑊 ∈ Word 𝑉 ∧ 𝑀 ∈ ( 0 ... 𝑁 ) ∧ 𝑁 ∈ ( 0 ... ( ♯ ‘ 𝑊 ) ) ) ∧ 𝑗 ∈ ( 𝑀 ..^ 𝑁 ) ) → ∃ 𝑖 ∈ ( 0 ..^ ( 𝑁 − 𝑀 ) ) 𝑗 = ( 𝑖 + 𝑀 ) ) |
32 |
|
eqcom |
⊢ ( 𝑦 = ( 𝑊 ‘ 𝑗 ) ↔ ( 𝑊 ‘ 𝑗 ) = 𝑦 ) |
33 |
|
simpr |
⊢ ( ( ( 𝑊 ∈ Word 𝑉 ∧ 𝑀 ∈ ( 0 ... 𝑁 ) ∧ 𝑁 ∈ ( 0 ... ( ♯ ‘ 𝑊 ) ) ) ∧ 𝑗 = ( 𝑖 + 𝑀 ) ) → 𝑗 = ( 𝑖 + 𝑀 ) ) |
34 |
33
|
fveq2d |
⊢ ( ( ( 𝑊 ∈ Word 𝑉 ∧ 𝑀 ∈ ( 0 ... 𝑁 ) ∧ 𝑁 ∈ ( 0 ... ( ♯ ‘ 𝑊 ) ) ) ∧ 𝑗 = ( 𝑖 + 𝑀 ) ) → ( 𝑊 ‘ 𝑗 ) = ( 𝑊 ‘ ( 𝑖 + 𝑀 ) ) ) |
35 |
34
|
eqeq2d |
⊢ ( ( ( 𝑊 ∈ Word 𝑉 ∧ 𝑀 ∈ ( 0 ... 𝑁 ) ∧ 𝑁 ∈ ( 0 ... ( ♯ ‘ 𝑊 ) ) ) ∧ 𝑗 = ( 𝑖 + 𝑀 ) ) → ( 𝑦 = ( 𝑊 ‘ 𝑗 ) ↔ 𝑦 = ( 𝑊 ‘ ( 𝑖 + 𝑀 ) ) ) ) |
36 |
32 35
|
bitr3id |
⊢ ( ( ( 𝑊 ∈ Word 𝑉 ∧ 𝑀 ∈ ( 0 ... 𝑁 ) ∧ 𝑁 ∈ ( 0 ... ( ♯ ‘ 𝑊 ) ) ) ∧ 𝑗 = ( 𝑖 + 𝑀 ) ) → ( ( 𝑊 ‘ 𝑗 ) = 𝑦 ↔ 𝑦 = ( 𝑊 ‘ ( 𝑖 + 𝑀 ) ) ) ) |
37 |
9 31 36
|
rexxfrd |
⊢ ( ( 𝑊 ∈ Word 𝑉 ∧ 𝑀 ∈ ( 0 ... 𝑁 ) ∧ 𝑁 ∈ ( 0 ... ( ♯ ‘ 𝑊 ) ) ) → ( ∃ 𝑗 ∈ ( 𝑀 ..^ 𝑁 ) ( 𝑊 ‘ 𝑗 ) = 𝑦 ↔ ∃ 𝑖 ∈ ( 0 ..^ ( 𝑁 − 𝑀 ) ) 𝑦 = ( 𝑊 ‘ ( 𝑖 + 𝑀 ) ) ) ) |
38 |
|
eqid |
⊢ ( 𝑖 ∈ ( 0 ..^ ( 𝑁 − 𝑀 ) ) ↦ ( 𝑊 ‘ ( 𝑖 + 𝑀 ) ) ) = ( 𝑖 ∈ ( 0 ..^ ( 𝑁 − 𝑀 ) ) ↦ ( 𝑊 ‘ ( 𝑖 + 𝑀 ) ) ) |
39 |
|
fvex |
⊢ ( 𝑊 ‘ ( 𝑖 + 𝑀 ) ) ∈ V |
40 |
38 39
|
elrnmpti |
⊢ ( 𝑦 ∈ ran ( 𝑖 ∈ ( 0 ..^ ( 𝑁 − 𝑀 ) ) ↦ ( 𝑊 ‘ ( 𝑖 + 𝑀 ) ) ) ↔ ∃ 𝑖 ∈ ( 0 ..^ ( 𝑁 − 𝑀 ) ) 𝑦 = ( 𝑊 ‘ ( 𝑖 + 𝑀 ) ) ) |
41 |
37 40
|
bitr4di |
⊢ ( ( 𝑊 ∈ Word 𝑉 ∧ 𝑀 ∈ ( 0 ... 𝑁 ) ∧ 𝑁 ∈ ( 0 ... ( ♯ ‘ 𝑊 ) ) ) → ( ∃ 𝑗 ∈ ( 𝑀 ..^ 𝑁 ) ( 𝑊 ‘ 𝑗 ) = 𝑦 ↔ 𝑦 ∈ ran ( 𝑖 ∈ ( 0 ..^ ( 𝑁 − 𝑀 ) ) ↦ ( 𝑊 ‘ ( 𝑖 + 𝑀 ) ) ) ) ) |
42 |
|
wrdf |
⊢ ( 𝑊 ∈ Word 𝑉 → 𝑊 : ( 0 ..^ ( ♯ ‘ 𝑊 ) ) ⟶ 𝑉 ) |
43 |
42
|
3ad2ant1 |
⊢ ( ( 𝑊 ∈ Word 𝑉 ∧ 𝑀 ∈ ( 0 ... 𝑁 ) ∧ 𝑁 ∈ ( 0 ... ( ♯ ‘ 𝑊 ) ) ) → 𝑊 : ( 0 ..^ ( ♯ ‘ 𝑊 ) ) ⟶ 𝑉 ) |
44 |
43
|
ffnd |
⊢ ( ( 𝑊 ∈ Word 𝑉 ∧ 𝑀 ∈ ( 0 ... 𝑁 ) ∧ 𝑁 ∈ ( 0 ... ( ♯ ‘ 𝑊 ) ) ) → 𝑊 Fn ( 0 ..^ ( ♯ ‘ 𝑊 ) ) ) |
45 |
|
elfzuz |
⊢ ( 𝑀 ∈ ( 0 ... 𝑁 ) → 𝑀 ∈ ( ℤ≥ ‘ 0 ) ) |
46 |
45
|
3ad2ant2 |
⊢ ( ( 𝑊 ∈ Word 𝑉 ∧ 𝑀 ∈ ( 0 ... 𝑁 ) ∧ 𝑁 ∈ ( 0 ... ( ♯ ‘ 𝑊 ) ) ) → 𝑀 ∈ ( ℤ≥ ‘ 0 ) ) |
47 |
|
fzoss1 |
⊢ ( 𝑀 ∈ ( ℤ≥ ‘ 0 ) → ( 𝑀 ..^ 𝑁 ) ⊆ ( 0 ..^ 𝑁 ) ) |
48 |
46 47
|
syl |
⊢ ( ( 𝑊 ∈ Word 𝑉 ∧ 𝑀 ∈ ( 0 ... 𝑁 ) ∧ 𝑁 ∈ ( 0 ... ( ♯ ‘ 𝑊 ) ) ) → ( 𝑀 ..^ 𝑁 ) ⊆ ( 0 ..^ 𝑁 ) ) |
49 |
|
elfzuz3 |
⊢ ( 𝑁 ∈ ( 0 ... ( ♯ ‘ 𝑊 ) ) → ( ♯ ‘ 𝑊 ) ∈ ( ℤ≥ ‘ 𝑁 ) ) |
50 |
49
|
3ad2ant3 |
⊢ ( ( 𝑊 ∈ Word 𝑉 ∧ 𝑀 ∈ ( 0 ... 𝑁 ) ∧ 𝑁 ∈ ( 0 ... ( ♯ ‘ 𝑊 ) ) ) → ( ♯ ‘ 𝑊 ) ∈ ( ℤ≥ ‘ 𝑁 ) ) |
51 |
|
fzoss2 |
⊢ ( ( ♯ ‘ 𝑊 ) ∈ ( ℤ≥ ‘ 𝑁 ) → ( 0 ..^ 𝑁 ) ⊆ ( 0 ..^ ( ♯ ‘ 𝑊 ) ) ) |
52 |
50 51
|
syl |
⊢ ( ( 𝑊 ∈ Word 𝑉 ∧ 𝑀 ∈ ( 0 ... 𝑁 ) ∧ 𝑁 ∈ ( 0 ... ( ♯ ‘ 𝑊 ) ) ) → ( 0 ..^ 𝑁 ) ⊆ ( 0 ..^ ( ♯ ‘ 𝑊 ) ) ) |
53 |
48 52
|
sstrd |
⊢ ( ( 𝑊 ∈ Word 𝑉 ∧ 𝑀 ∈ ( 0 ... 𝑁 ) ∧ 𝑁 ∈ ( 0 ... ( ♯ ‘ 𝑊 ) ) ) → ( 𝑀 ..^ 𝑁 ) ⊆ ( 0 ..^ ( ♯ ‘ 𝑊 ) ) ) |
54 |
44 53
|
fvelimabd |
⊢ ( ( 𝑊 ∈ Word 𝑉 ∧ 𝑀 ∈ ( 0 ... 𝑁 ) ∧ 𝑁 ∈ ( 0 ... ( ♯ ‘ 𝑊 ) ) ) → ( 𝑦 ∈ ( 𝑊 “ ( 𝑀 ..^ 𝑁 ) ) ↔ ∃ 𝑗 ∈ ( 𝑀 ..^ 𝑁 ) ( 𝑊 ‘ 𝑗 ) = 𝑦 ) ) |
55 |
|
swrdval2 |
⊢ ( ( 𝑊 ∈ Word 𝑉 ∧ 𝑀 ∈ ( 0 ... 𝑁 ) ∧ 𝑁 ∈ ( 0 ... ( ♯ ‘ 𝑊 ) ) ) → ( 𝑊 substr 〈 𝑀 , 𝑁 〉 ) = ( 𝑖 ∈ ( 0 ..^ ( 𝑁 − 𝑀 ) ) ↦ ( 𝑊 ‘ ( 𝑖 + 𝑀 ) ) ) ) |
56 |
55
|
rneqd |
⊢ ( ( 𝑊 ∈ Word 𝑉 ∧ 𝑀 ∈ ( 0 ... 𝑁 ) ∧ 𝑁 ∈ ( 0 ... ( ♯ ‘ 𝑊 ) ) ) → ran ( 𝑊 substr 〈 𝑀 , 𝑁 〉 ) = ran ( 𝑖 ∈ ( 0 ..^ ( 𝑁 − 𝑀 ) ) ↦ ( 𝑊 ‘ ( 𝑖 + 𝑀 ) ) ) ) |
57 |
56
|
eleq2d |
⊢ ( ( 𝑊 ∈ Word 𝑉 ∧ 𝑀 ∈ ( 0 ... 𝑁 ) ∧ 𝑁 ∈ ( 0 ... ( ♯ ‘ 𝑊 ) ) ) → ( 𝑦 ∈ ran ( 𝑊 substr 〈 𝑀 , 𝑁 〉 ) ↔ 𝑦 ∈ ran ( 𝑖 ∈ ( 0 ..^ ( 𝑁 − 𝑀 ) ) ↦ ( 𝑊 ‘ ( 𝑖 + 𝑀 ) ) ) ) ) |
58 |
41 54 57
|
3bitr4rd |
⊢ ( ( 𝑊 ∈ Word 𝑉 ∧ 𝑀 ∈ ( 0 ... 𝑁 ) ∧ 𝑁 ∈ ( 0 ... ( ♯ ‘ 𝑊 ) ) ) → ( 𝑦 ∈ ran ( 𝑊 substr 〈 𝑀 , 𝑁 〉 ) ↔ 𝑦 ∈ ( 𝑊 “ ( 𝑀 ..^ 𝑁 ) ) ) ) |
59 |
58
|
eqrdv |
⊢ ( ( 𝑊 ∈ Word 𝑉 ∧ 𝑀 ∈ ( 0 ... 𝑁 ) ∧ 𝑁 ∈ ( 0 ... ( ♯ ‘ 𝑊 ) ) ) → ran ( 𝑊 substr 〈 𝑀 , 𝑁 〉 ) = ( 𝑊 “ ( 𝑀 ..^ 𝑁 ) ) ) |