| Step |
Hyp |
Ref |
Expression |
| 1 |
|
swrdcl |
⊢ ( 𝑆 ∈ Word 𝐴 → ( 𝑆 substr ⟨ 𝑋 , 𝑌 ⟩ ) ∈ Word 𝐴 ) |
| 2 |
1
|
adantr |
⊢ ( ( 𝑆 ∈ Word 𝐴 ∧ ( 𝑋 ∈ ( 0 ... 𝑌 ) ∧ 𝑌 ∈ ( 0 ... 𝑍 ) ∧ 𝑍 ∈ ( 0 ... ( ♯ ‘ 𝑆 ) ) ) ) → ( 𝑆 substr ⟨ 𝑋 , 𝑌 ⟩ ) ∈ Word 𝐴 ) |
| 3 |
|
swrdcl |
⊢ ( 𝑆 ∈ Word 𝐴 → ( 𝑆 substr ⟨ 𝑌 , 𝑍 ⟩ ) ∈ Word 𝐴 ) |
| 4 |
3
|
adantr |
⊢ ( ( 𝑆 ∈ Word 𝐴 ∧ ( 𝑋 ∈ ( 0 ... 𝑌 ) ∧ 𝑌 ∈ ( 0 ... 𝑍 ) ∧ 𝑍 ∈ ( 0 ... ( ♯ ‘ 𝑆 ) ) ) ) → ( 𝑆 substr ⟨ 𝑌 , 𝑍 ⟩ ) ∈ Word 𝐴 ) |
| 5 |
|
ccatcl |
⊢ ( ( ( 𝑆 substr ⟨ 𝑋 , 𝑌 ⟩ ) ∈ Word 𝐴 ∧ ( 𝑆 substr ⟨ 𝑌 , 𝑍 ⟩ ) ∈ Word 𝐴 ) → ( ( 𝑆 substr ⟨ 𝑋 , 𝑌 ⟩ ) ++ ( 𝑆 substr ⟨ 𝑌 , 𝑍 ⟩ ) ) ∈ Word 𝐴 ) |
| 6 |
2 4 5
|
syl2anc |
⊢ ( ( 𝑆 ∈ Word 𝐴 ∧ ( 𝑋 ∈ ( 0 ... 𝑌 ) ∧ 𝑌 ∈ ( 0 ... 𝑍 ) ∧ 𝑍 ∈ ( 0 ... ( ♯ ‘ 𝑆 ) ) ) ) → ( ( 𝑆 substr ⟨ 𝑋 , 𝑌 ⟩ ) ++ ( 𝑆 substr ⟨ 𝑌 , 𝑍 ⟩ ) ) ∈ Word 𝐴 ) |
| 7 |
|
wrdfn |
⊢ ( ( ( 𝑆 substr ⟨ 𝑋 , 𝑌 ⟩ ) ++ ( 𝑆 substr ⟨ 𝑌 , 𝑍 ⟩ ) ) ∈ Word 𝐴 → ( ( 𝑆 substr ⟨ 𝑋 , 𝑌 ⟩ ) ++ ( 𝑆 substr ⟨ 𝑌 , 𝑍 ⟩ ) ) Fn ( 0 ..^ ( ♯ ‘ ( ( 𝑆 substr ⟨ 𝑋 , 𝑌 ⟩ ) ++ ( 𝑆 substr ⟨ 𝑌 , 𝑍 ⟩ ) ) ) ) ) |
| 8 |
6 7
|
syl |
⊢ ( ( 𝑆 ∈ Word 𝐴 ∧ ( 𝑋 ∈ ( 0 ... 𝑌 ) ∧ 𝑌 ∈ ( 0 ... 𝑍 ) ∧ 𝑍 ∈ ( 0 ... ( ♯ ‘ 𝑆 ) ) ) ) → ( ( 𝑆 substr ⟨ 𝑋 , 𝑌 ⟩ ) ++ ( 𝑆 substr ⟨ 𝑌 , 𝑍 ⟩ ) ) Fn ( 0 ..^ ( ♯ ‘ ( ( 𝑆 substr ⟨ 𝑋 , 𝑌 ⟩ ) ++ ( 𝑆 substr ⟨ 𝑌 , 𝑍 ⟩ ) ) ) ) ) |
| 9 |
|
ccatlen |
⊢ ( ( ( 𝑆 substr ⟨ 𝑋 , 𝑌 ⟩ ) ∈ Word 𝐴 ∧ ( 𝑆 substr ⟨ 𝑌 , 𝑍 ⟩ ) ∈ Word 𝐴 ) → ( ♯ ‘ ( ( 𝑆 substr ⟨ 𝑋 , 𝑌 ⟩ ) ++ ( 𝑆 substr ⟨ 𝑌 , 𝑍 ⟩ ) ) ) = ( ( ♯ ‘ ( 𝑆 substr ⟨ 𝑋 , 𝑌 ⟩ ) ) + ( ♯ ‘ ( 𝑆 substr ⟨ 𝑌 , 𝑍 ⟩ ) ) ) ) |
| 10 |
2 4 9
|
syl2anc |
⊢ ( ( 𝑆 ∈ Word 𝐴 ∧ ( 𝑋 ∈ ( 0 ... 𝑌 ) ∧ 𝑌 ∈ ( 0 ... 𝑍 ) ∧ 𝑍 ∈ ( 0 ... ( ♯ ‘ 𝑆 ) ) ) ) → ( ♯ ‘ ( ( 𝑆 substr ⟨ 𝑋 , 𝑌 ⟩ ) ++ ( 𝑆 substr ⟨ 𝑌 , 𝑍 ⟩ ) ) ) = ( ( ♯ ‘ ( 𝑆 substr ⟨ 𝑋 , 𝑌 ⟩ ) ) + ( ♯ ‘ ( 𝑆 substr ⟨ 𝑌 , 𝑍 ⟩ ) ) ) ) |
| 11 |
|
simpl |
⊢ ( ( 𝑆 ∈ Word 𝐴 ∧ ( 𝑋 ∈ ( 0 ... 𝑌 ) ∧ 𝑌 ∈ ( 0 ... 𝑍 ) ∧ 𝑍 ∈ ( 0 ... ( ♯ ‘ 𝑆 ) ) ) ) → 𝑆 ∈ Word 𝐴 ) |
| 12 |
|
simpr1 |
⊢ ( ( 𝑆 ∈ Word 𝐴 ∧ ( 𝑋 ∈ ( 0 ... 𝑌 ) ∧ 𝑌 ∈ ( 0 ... 𝑍 ) ∧ 𝑍 ∈ ( 0 ... ( ♯ ‘ 𝑆 ) ) ) ) → 𝑋 ∈ ( 0 ... 𝑌 ) ) |
| 13 |
|
simpr2 |
⊢ ( ( 𝑆 ∈ Word 𝐴 ∧ ( 𝑋 ∈ ( 0 ... 𝑌 ) ∧ 𝑌 ∈ ( 0 ... 𝑍 ) ∧ 𝑍 ∈ ( 0 ... ( ♯ ‘ 𝑆 ) ) ) ) → 𝑌 ∈ ( 0 ... 𝑍 ) ) |
| 14 |
|
simpr3 |
⊢ ( ( 𝑆 ∈ Word 𝐴 ∧ ( 𝑋 ∈ ( 0 ... 𝑌 ) ∧ 𝑌 ∈ ( 0 ... 𝑍 ) ∧ 𝑍 ∈ ( 0 ... ( ♯ ‘ 𝑆 ) ) ) ) → 𝑍 ∈ ( 0 ... ( ♯ ‘ 𝑆 ) ) ) |
| 15 |
|
fzass4 |
⊢ ( ( 𝑌 ∈ ( 0 ... ( ♯ ‘ 𝑆 ) ) ∧ 𝑍 ∈ ( 𝑌 ... ( ♯ ‘ 𝑆 ) ) ) ↔ ( 𝑌 ∈ ( 0 ... 𝑍 ) ∧ 𝑍 ∈ ( 0 ... ( ♯ ‘ 𝑆 ) ) ) ) |
| 16 |
15
|
biimpri |
⊢ ( ( 𝑌 ∈ ( 0 ... 𝑍 ) ∧ 𝑍 ∈ ( 0 ... ( ♯ ‘ 𝑆 ) ) ) → ( 𝑌 ∈ ( 0 ... ( ♯ ‘ 𝑆 ) ) ∧ 𝑍 ∈ ( 𝑌 ... ( ♯ ‘ 𝑆 ) ) ) ) |
| 17 |
16
|
simpld |
⊢ ( ( 𝑌 ∈ ( 0 ... 𝑍 ) ∧ 𝑍 ∈ ( 0 ... ( ♯ ‘ 𝑆 ) ) ) → 𝑌 ∈ ( 0 ... ( ♯ ‘ 𝑆 ) ) ) |
| 18 |
13 14 17
|
syl2anc |
⊢ ( ( 𝑆 ∈ Word 𝐴 ∧ ( 𝑋 ∈ ( 0 ... 𝑌 ) ∧ 𝑌 ∈ ( 0 ... 𝑍 ) ∧ 𝑍 ∈ ( 0 ... ( ♯ ‘ 𝑆 ) ) ) ) → 𝑌 ∈ ( 0 ... ( ♯ ‘ 𝑆 ) ) ) |
| 19 |
|
swrdlen |
⊢ ( ( 𝑆 ∈ Word 𝐴 ∧ 𝑋 ∈ ( 0 ... 𝑌 ) ∧ 𝑌 ∈ ( 0 ... ( ♯ ‘ 𝑆 ) ) ) → ( ♯ ‘ ( 𝑆 substr ⟨ 𝑋 , 𝑌 ⟩ ) ) = ( 𝑌 − 𝑋 ) ) |
| 20 |
11 12 18 19
|
syl3anc |
⊢ ( ( 𝑆 ∈ Word 𝐴 ∧ ( 𝑋 ∈ ( 0 ... 𝑌 ) ∧ 𝑌 ∈ ( 0 ... 𝑍 ) ∧ 𝑍 ∈ ( 0 ... ( ♯ ‘ 𝑆 ) ) ) ) → ( ♯ ‘ ( 𝑆 substr ⟨ 𝑋 , 𝑌 ⟩ ) ) = ( 𝑌 − 𝑋 ) ) |
| 21 |
|
swrdlen |
⊢ ( ( 𝑆 ∈ Word 𝐴 ∧ 𝑌 ∈ ( 0 ... 𝑍 ) ∧ 𝑍 ∈ ( 0 ... ( ♯ ‘ 𝑆 ) ) ) → ( ♯ ‘ ( 𝑆 substr ⟨ 𝑌 , 𝑍 ⟩ ) ) = ( 𝑍 − 𝑌 ) ) |
| 22 |
21
|
3adant3r1 |
⊢ ( ( 𝑆 ∈ Word 𝐴 ∧ ( 𝑋 ∈ ( 0 ... 𝑌 ) ∧ 𝑌 ∈ ( 0 ... 𝑍 ) ∧ 𝑍 ∈ ( 0 ... ( ♯ ‘ 𝑆 ) ) ) ) → ( ♯ ‘ ( 𝑆 substr ⟨ 𝑌 , 𝑍 ⟩ ) ) = ( 𝑍 − 𝑌 ) ) |
| 23 |
20 22
|
oveq12d |
⊢ ( ( 𝑆 ∈ Word 𝐴 ∧ ( 𝑋 ∈ ( 0 ... 𝑌 ) ∧ 𝑌 ∈ ( 0 ... 𝑍 ) ∧ 𝑍 ∈ ( 0 ... ( ♯ ‘ 𝑆 ) ) ) ) → ( ( ♯ ‘ ( 𝑆 substr ⟨ 𝑋 , 𝑌 ⟩ ) ) + ( ♯ ‘ ( 𝑆 substr ⟨ 𝑌 , 𝑍 ⟩ ) ) ) = ( ( 𝑌 − 𝑋 ) + ( 𝑍 − 𝑌 ) ) ) |
| 24 |
13
|
elfzelzd |
⊢ ( ( 𝑆 ∈ Word 𝐴 ∧ ( 𝑋 ∈ ( 0 ... 𝑌 ) ∧ 𝑌 ∈ ( 0 ... 𝑍 ) ∧ 𝑍 ∈ ( 0 ... ( ♯ ‘ 𝑆 ) ) ) ) → 𝑌 ∈ ℤ ) |
| 25 |
24
|
zcnd |
⊢ ( ( 𝑆 ∈ Word 𝐴 ∧ ( 𝑋 ∈ ( 0 ... 𝑌 ) ∧ 𝑌 ∈ ( 0 ... 𝑍 ) ∧ 𝑍 ∈ ( 0 ... ( ♯ ‘ 𝑆 ) ) ) ) → 𝑌 ∈ ℂ ) |
| 26 |
12
|
elfzelzd |
⊢ ( ( 𝑆 ∈ Word 𝐴 ∧ ( 𝑋 ∈ ( 0 ... 𝑌 ) ∧ 𝑌 ∈ ( 0 ... 𝑍 ) ∧ 𝑍 ∈ ( 0 ... ( ♯ ‘ 𝑆 ) ) ) ) → 𝑋 ∈ ℤ ) |
| 27 |
26
|
zcnd |
⊢ ( ( 𝑆 ∈ Word 𝐴 ∧ ( 𝑋 ∈ ( 0 ... 𝑌 ) ∧ 𝑌 ∈ ( 0 ... 𝑍 ) ∧ 𝑍 ∈ ( 0 ... ( ♯ ‘ 𝑆 ) ) ) ) → 𝑋 ∈ ℂ ) |
| 28 |
14
|
elfzelzd |
⊢ ( ( 𝑆 ∈ Word 𝐴 ∧ ( 𝑋 ∈ ( 0 ... 𝑌 ) ∧ 𝑌 ∈ ( 0 ... 𝑍 ) ∧ 𝑍 ∈ ( 0 ... ( ♯ ‘ 𝑆 ) ) ) ) → 𝑍 ∈ ℤ ) |
| 29 |
28
|
zcnd |
⊢ ( ( 𝑆 ∈ Word 𝐴 ∧ ( 𝑋 ∈ ( 0 ... 𝑌 ) ∧ 𝑌 ∈ ( 0 ... 𝑍 ) ∧ 𝑍 ∈ ( 0 ... ( ♯ ‘ 𝑆 ) ) ) ) → 𝑍 ∈ ℂ ) |
| 30 |
25 27 29
|
npncan3d |
⊢ ( ( 𝑆 ∈ Word 𝐴 ∧ ( 𝑋 ∈ ( 0 ... 𝑌 ) ∧ 𝑌 ∈ ( 0 ... 𝑍 ) ∧ 𝑍 ∈ ( 0 ... ( ♯ ‘ 𝑆 ) ) ) ) → ( ( 𝑌 − 𝑋 ) + ( 𝑍 − 𝑌 ) ) = ( 𝑍 − 𝑋 ) ) |
| 31 |
10 23 30
|
3eqtrd |
⊢ ( ( 𝑆 ∈ Word 𝐴 ∧ ( 𝑋 ∈ ( 0 ... 𝑌 ) ∧ 𝑌 ∈ ( 0 ... 𝑍 ) ∧ 𝑍 ∈ ( 0 ... ( ♯ ‘ 𝑆 ) ) ) ) → ( ♯ ‘ ( ( 𝑆 substr ⟨ 𝑋 , 𝑌 ⟩ ) ++ ( 𝑆 substr ⟨ 𝑌 , 𝑍 ⟩ ) ) ) = ( 𝑍 − 𝑋 ) ) |
| 32 |
31
|
oveq2d |
⊢ ( ( 𝑆 ∈ Word 𝐴 ∧ ( 𝑋 ∈ ( 0 ... 𝑌 ) ∧ 𝑌 ∈ ( 0 ... 𝑍 ) ∧ 𝑍 ∈ ( 0 ... ( ♯ ‘ 𝑆 ) ) ) ) → ( 0 ..^ ( ♯ ‘ ( ( 𝑆 substr ⟨ 𝑋 , 𝑌 ⟩ ) ++ ( 𝑆 substr ⟨ 𝑌 , 𝑍 ⟩ ) ) ) ) = ( 0 ..^ ( 𝑍 − 𝑋 ) ) ) |
| 33 |
32
|
fneq2d |
⊢ ( ( 𝑆 ∈ Word 𝐴 ∧ ( 𝑋 ∈ ( 0 ... 𝑌 ) ∧ 𝑌 ∈ ( 0 ... 𝑍 ) ∧ 𝑍 ∈ ( 0 ... ( ♯ ‘ 𝑆 ) ) ) ) → ( ( ( 𝑆 substr ⟨ 𝑋 , 𝑌 ⟩ ) ++ ( 𝑆 substr ⟨ 𝑌 , 𝑍 ⟩ ) ) Fn ( 0 ..^ ( ♯ ‘ ( ( 𝑆 substr ⟨ 𝑋 , 𝑌 ⟩ ) ++ ( 𝑆 substr ⟨ 𝑌 , 𝑍 ⟩ ) ) ) ) ↔ ( ( 𝑆 substr ⟨ 𝑋 , 𝑌 ⟩ ) ++ ( 𝑆 substr ⟨ 𝑌 , 𝑍 ⟩ ) ) Fn ( 0 ..^ ( 𝑍 − 𝑋 ) ) ) ) |
| 34 |
8 33
|
mpbid |
⊢ ( ( 𝑆 ∈ Word 𝐴 ∧ ( 𝑋 ∈ ( 0 ... 𝑌 ) ∧ 𝑌 ∈ ( 0 ... 𝑍 ) ∧ 𝑍 ∈ ( 0 ... ( ♯ ‘ 𝑆 ) ) ) ) → ( ( 𝑆 substr ⟨ 𝑋 , 𝑌 ⟩ ) ++ ( 𝑆 substr ⟨ 𝑌 , 𝑍 ⟩ ) ) Fn ( 0 ..^ ( 𝑍 − 𝑋 ) ) ) |
| 35 |
|
swrdcl |
⊢ ( 𝑆 ∈ Word 𝐴 → ( 𝑆 substr ⟨ 𝑋 , 𝑍 ⟩ ) ∈ Word 𝐴 ) |
| 36 |
35
|
adantr |
⊢ ( ( 𝑆 ∈ Word 𝐴 ∧ ( 𝑋 ∈ ( 0 ... 𝑌 ) ∧ 𝑌 ∈ ( 0 ... 𝑍 ) ∧ 𝑍 ∈ ( 0 ... ( ♯ ‘ 𝑆 ) ) ) ) → ( 𝑆 substr ⟨ 𝑋 , 𝑍 ⟩ ) ∈ Word 𝐴 ) |
| 37 |
|
wrdfn |
⊢ ( ( 𝑆 substr ⟨ 𝑋 , 𝑍 ⟩ ) ∈ Word 𝐴 → ( 𝑆 substr ⟨ 𝑋 , 𝑍 ⟩ ) Fn ( 0 ..^ ( ♯ ‘ ( 𝑆 substr ⟨ 𝑋 , 𝑍 ⟩ ) ) ) ) |
| 38 |
36 37
|
syl |
⊢ ( ( 𝑆 ∈ Word 𝐴 ∧ ( 𝑋 ∈ ( 0 ... 𝑌 ) ∧ 𝑌 ∈ ( 0 ... 𝑍 ) ∧ 𝑍 ∈ ( 0 ... ( ♯ ‘ 𝑆 ) ) ) ) → ( 𝑆 substr ⟨ 𝑋 , 𝑍 ⟩ ) Fn ( 0 ..^ ( ♯ ‘ ( 𝑆 substr ⟨ 𝑋 , 𝑍 ⟩ ) ) ) ) |
| 39 |
|
fzass4 |
⊢ ( ( 𝑋 ∈ ( 0 ... 𝑍 ) ∧ 𝑌 ∈ ( 𝑋 ... 𝑍 ) ) ↔ ( 𝑋 ∈ ( 0 ... 𝑌 ) ∧ 𝑌 ∈ ( 0 ... 𝑍 ) ) ) |
| 40 |
39
|
biimpri |
⊢ ( ( 𝑋 ∈ ( 0 ... 𝑌 ) ∧ 𝑌 ∈ ( 0 ... 𝑍 ) ) → ( 𝑋 ∈ ( 0 ... 𝑍 ) ∧ 𝑌 ∈ ( 𝑋 ... 𝑍 ) ) ) |
| 41 |
40
|
simpld |
⊢ ( ( 𝑋 ∈ ( 0 ... 𝑌 ) ∧ 𝑌 ∈ ( 0 ... 𝑍 ) ) → 𝑋 ∈ ( 0 ... 𝑍 ) ) |
| 42 |
12 13 41
|
syl2anc |
⊢ ( ( 𝑆 ∈ Word 𝐴 ∧ ( 𝑋 ∈ ( 0 ... 𝑌 ) ∧ 𝑌 ∈ ( 0 ... 𝑍 ) ∧ 𝑍 ∈ ( 0 ... ( ♯ ‘ 𝑆 ) ) ) ) → 𝑋 ∈ ( 0 ... 𝑍 ) ) |
| 43 |
|
swrdlen |
⊢ ( ( 𝑆 ∈ Word 𝐴 ∧ 𝑋 ∈ ( 0 ... 𝑍 ) ∧ 𝑍 ∈ ( 0 ... ( ♯ ‘ 𝑆 ) ) ) → ( ♯ ‘ ( 𝑆 substr ⟨ 𝑋 , 𝑍 ⟩ ) ) = ( 𝑍 − 𝑋 ) ) |
| 44 |
11 42 14 43
|
syl3anc |
⊢ ( ( 𝑆 ∈ Word 𝐴 ∧ ( 𝑋 ∈ ( 0 ... 𝑌 ) ∧ 𝑌 ∈ ( 0 ... 𝑍 ) ∧ 𝑍 ∈ ( 0 ... ( ♯ ‘ 𝑆 ) ) ) ) → ( ♯ ‘ ( 𝑆 substr ⟨ 𝑋 , 𝑍 ⟩ ) ) = ( 𝑍 − 𝑋 ) ) |
| 45 |
44
|
oveq2d |
⊢ ( ( 𝑆 ∈ Word 𝐴 ∧ ( 𝑋 ∈ ( 0 ... 𝑌 ) ∧ 𝑌 ∈ ( 0 ... 𝑍 ) ∧ 𝑍 ∈ ( 0 ... ( ♯ ‘ 𝑆 ) ) ) ) → ( 0 ..^ ( ♯ ‘ ( 𝑆 substr ⟨ 𝑋 , 𝑍 ⟩ ) ) ) = ( 0 ..^ ( 𝑍 − 𝑋 ) ) ) |
| 46 |
45
|
fneq2d |
⊢ ( ( 𝑆 ∈ Word 𝐴 ∧ ( 𝑋 ∈ ( 0 ... 𝑌 ) ∧ 𝑌 ∈ ( 0 ... 𝑍 ) ∧ 𝑍 ∈ ( 0 ... ( ♯ ‘ 𝑆 ) ) ) ) → ( ( 𝑆 substr ⟨ 𝑋 , 𝑍 ⟩ ) Fn ( 0 ..^ ( ♯ ‘ ( 𝑆 substr ⟨ 𝑋 , 𝑍 ⟩ ) ) ) ↔ ( 𝑆 substr ⟨ 𝑋 , 𝑍 ⟩ ) Fn ( 0 ..^ ( 𝑍 − 𝑋 ) ) ) ) |
| 47 |
38 46
|
mpbid |
⊢ ( ( 𝑆 ∈ Word 𝐴 ∧ ( 𝑋 ∈ ( 0 ... 𝑌 ) ∧ 𝑌 ∈ ( 0 ... 𝑍 ) ∧ 𝑍 ∈ ( 0 ... ( ♯ ‘ 𝑆 ) ) ) ) → ( 𝑆 substr ⟨ 𝑋 , 𝑍 ⟩ ) Fn ( 0 ..^ ( 𝑍 − 𝑋 ) ) ) |
| 48 |
24 26
|
zsubcld |
⊢ ( ( 𝑆 ∈ Word 𝐴 ∧ ( 𝑋 ∈ ( 0 ... 𝑌 ) ∧ 𝑌 ∈ ( 0 ... 𝑍 ) ∧ 𝑍 ∈ ( 0 ... ( ♯ ‘ 𝑆 ) ) ) ) → ( 𝑌 − 𝑋 ) ∈ ℤ ) |
| 49 |
48
|
anim1ci |
⊢ ( ( ( 𝑆 ∈ Word 𝐴 ∧ ( 𝑋 ∈ ( 0 ... 𝑌 ) ∧ 𝑌 ∈ ( 0 ... 𝑍 ) ∧ 𝑍 ∈ ( 0 ... ( ♯ ‘ 𝑆 ) ) ) ) ∧ 𝑥 ∈ ( 0 ..^ ( 𝑍 − 𝑋 ) ) ) → ( 𝑥 ∈ ( 0 ..^ ( 𝑍 − 𝑋 ) ) ∧ ( 𝑌 − 𝑋 ) ∈ ℤ ) ) |
| 50 |
|
fzospliti |
⊢ ( ( 𝑥 ∈ ( 0 ..^ ( 𝑍 − 𝑋 ) ) ∧ ( 𝑌 − 𝑋 ) ∈ ℤ ) → ( 𝑥 ∈ ( 0 ..^ ( 𝑌 − 𝑋 ) ) ∨ 𝑥 ∈ ( ( 𝑌 − 𝑋 ) ..^ ( 𝑍 − 𝑋 ) ) ) ) |
| 51 |
49 50
|
syl |
⊢ ( ( ( 𝑆 ∈ Word 𝐴 ∧ ( 𝑋 ∈ ( 0 ... 𝑌 ) ∧ 𝑌 ∈ ( 0 ... 𝑍 ) ∧ 𝑍 ∈ ( 0 ... ( ♯ ‘ 𝑆 ) ) ) ) ∧ 𝑥 ∈ ( 0 ..^ ( 𝑍 − 𝑋 ) ) ) → ( 𝑥 ∈ ( 0 ..^ ( 𝑌 − 𝑋 ) ) ∨ 𝑥 ∈ ( ( 𝑌 − 𝑋 ) ..^ ( 𝑍 − 𝑋 ) ) ) ) |
| 52 |
1
|
ad2antrr |
⊢ ( ( ( 𝑆 ∈ Word 𝐴 ∧ ( 𝑋 ∈ ( 0 ... 𝑌 ) ∧ 𝑌 ∈ ( 0 ... 𝑍 ) ∧ 𝑍 ∈ ( 0 ... ( ♯ ‘ 𝑆 ) ) ) ) ∧ 𝑥 ∈ ( 0 ..^ ( 𝑌 − 𝑋 ) ) ) → ( 𝑆 substr ⟨ 𝑋 , 𝑌 ⟩ ) ∈ Word 𝐴 ) |
| 53 |
3
|
ad2antrr |
⊢ ( ( ( 𝑆 ∈ Word 𝐴 ∧ ( 𝑋 ∈ ( 0 ... 𝑌 ) ∧ 𝑌 ∈ ( 0 ... 𝑍 ) ∧ 𝑍 ∈ ( 0 ... ( ♯ ‘ 𝑆 ) ) ) ) ∧ 𝑥 ∈ ( 0 ..^ ( 𝑌 − 𝑋 ) ) ) → ( 𝑆 substr ⟨ 𝑌 , 𝑍 ⟩ ) ∈ Word 𝐴 ) |
| 54 |
20
|
oveq2d |
⊢ ( ( 𝑆 ∈ Word 𝐴 ∧ ( 𝑋 ∈ ( 0 ... 𝑌 ) ∧ 𝑌 ∈ ( 0 ... 𝑍 ) ∧ 𝑍 ∈ ( 0 ... ( ♯ ‘ 𝑆 ) ) ) ) → ( 0 ..^ ( ♯ ‘ ( 𝑆 substr ⟨ 𝑋 , 𝑌 ⟩ ) ) ) = ( 0 ..^ ( 𝑌 − 𝑋 ) ) ) |
| 55 |
54
|
eleq2d |
⊢ ( ( 𝑆 ∈ Word 𝐴 ∧ ( 𝑋 ∈ ( 0 ... 𝑌 ) ∧ 𝑌 ∈ ( 0 ... 𝑍 ) ∧ 𝑍 ∈ ( 0 ... ( ♯ ‘ 𝑆 ) ) ) ) → ( 𝑥 ∈ ( 0 ..^ ( ♯ ‘ ( 𝑆 substr ⟨ 𝑋 , 𝑌 ⟩ ) ) ) ↔ 𝑥 ∈ ( 0 ..^ ( 𝑌 − 𝑋 ) ) ) ) |
| 56 |
55
|
biimpar |
⊢ ( ( ( 𝑆 ∈ Word 𝐴 ∧ ( 𝑋 ∈ ( 0 ... 𝑌 ) ∧ 𝑌 ∈ ( 0 ... 𝑍 ) ∧ 𝑍 ∈ ( 0 ... ( ♯ ‘ 𝑆 ) ) ) ) ∧ 𝑥 ∈ ( 0 ..^ ( 𝑌 − 𝑋 ) ) ) → 𝑥 ∈ ( 0 ..^ ( ♯ ‘ ( 𝑆 substr ⟨ 𝑋 , 𝑌 ⟩ ) ) ) ) |
| 57 |
|
ccatval1 |
⊢ ( ( ( 𝑆 substr ⟨ 𝑋 , 𝑌 ⟩ ) ∈ Word 𝐴 ∧ ( 𝑆 substr ⟨ 𝑌 , 𝑍 ⟩ ) ∈ Word 𝐴 ∧ 𝑥 ∈ ( 0 ..^ ( ♯ ‘ ( 𝑆 substr ⟨ 𝑋 , 𝑌 ⟩ ) ) ) ) → ( ( ( 𝑆 substr ⟨ 𝑋 , 𝑌 ⟩ ) ++ ( 𝑆 substr ⟨ 𝑌 , 𝑍 ⟩ ) ) ‘ 𝑥 ) = ( ( 𝑆 substr ⟨ 𝑋 , 𝑌 ⟩ ) ‘ 𝑥 ) ) |
| 58 |
52 53 56 57
|
syl3anc |
⊢ ( ( ( 𝑆 ∈ Word 𝐴 ∧ ( 𝑋 ∈ ( 0 ... 𝑌 ) ∧ 𝑌 ∈ ( 0 ... 𝑍 ) ∧ 𝑍 ∈ ( 0 ... ( ♯ ‘ 𝑆 ) ) ) ) ∧ 𝑥 ∈ ( 0 ..^ ( 𝑌 − 𝑋 ) ) ) → ( ( ( 𝑆 substr ⟨ 𝑋 , 𝑌 ⟩ ) ++ ( 𝑆 substr ⟨ 𝑌 , 𝑍 ⟩ ) ) ‘ 𝑥 ) = ( ( 𝑆 substr ⟨ 𝑋 , 𝑌 ⟩ ) ‘ 𝑥 ) ) |
| 59 |
|
simpll |
⊢ ( ( ( 𝑆 ∈ Word 𝐴 ∧ ( 𝑋 ∈ ( 0 ... 𝑌 ) ∧ 𝑌 ∈ ( 0 ... 𝑍 ) ∧ 𝑍 ∈ ( 0 ... ( ♯ ‘ 𝑆 ) ) ) ) ∧ 𝑥 ∈ ( 0 ..^ ( 𝑌 − 𝑋 ) ) ) → 𝑆 ∈ Word 𝐴 ) |
| 60 |
|
simplr1 |
⊢ ( ( ( 𝑆 ∈ Word 𝐴 ∧ ( 𝑋 ∈ ( 0 ... 𝑌 ) ∧ 𝑌 ∈ ( 0 ... 𝑍 ) ∧ 𝑍 ∈ ( 0 ... ( ♯ ‘ 𝑆 ) ) ) ) ∧ 𝑥 ∈ ( 0 ..^ ( 𝑌 − 𝑋 ) ) ) → 𝑋 ∈ ( 0 ... 𝑌 ) ) |
| 61 |
18
|
adantr |
⊢ ( ( ( 𝑆 ∈ Word 𝐴 ∧ ( 𝑋 ∈ ( 0 ... 𝑌 ) ∧ 𝑌 ∈ ( 0 ... 𝑍 ) ∧ 𝑍 ∈ ( 0 ... ( ♯ ‘ 𝑆 ) ) ) ) ∧ 𝑥 ∈ ( 0 ..^ ( 𝑌 − 𝑋 ) ) ) → 𝑌 ∈ ( 0 ... ( ♯ ‘ 𝑆 ) ) ) |
| 62 |
|
simpr |
⊢ ( ( ( 𝑆 ∈ Word 𝐴 ∧ ( 𝑋 ∈ ( 0 ... 𝑌 ) ∧ 𝑌 ∈ ( 0 ... 𝑍 ) ∧ 𝑍 ∈ ( 0 ... ( ♯ ‘ 𝑆 ) ) ) ) ∧ 𝑥 ∈ ( 0 ..^ ( 𝑌 − 𝑋 ) ) ) → 𝑥 ∈ ( 0 ..^ ( 𝑌 − 𝑋 ) ) ) |
| 63 |
|
swrdfv |
⊢ ( ( ( 𝑆 ∈ Word 𝐴 ∧ 𝑋 ∈ ( 0 ... 𝑌 ) ∧ 𝑌 ∈ ( 0 ... ( ♯ ‘ 𝑆 ) ) ) ∧ 𝑥 ∈ ( 0 ..^ ( 𝑌 − 𝑋 ) ) ) → ( ( 𝑆 substr ⟨ 𝑋 , 𝑌 ⟩ ) ‘ 𝑥 ) = ( 𝑆 ‘ ( 𝑥 + 𝑋 ) ) ) |
| 64 |
59 60 61 62 63
|
syl31anc |
⊢ ( ( ( 𝑆 ∈ Word 𝐴 ∧ ( 𝑋 ∈ ( 0 ... 𝑌 ) ∧ 𝑌 ∈ ( 0 ... 𝑍 ) ∧ 𝑍 ∈ ( 0 ... ( ♯ ‘ 𝑆 ) ) ) ) ∧ 𝑥 ∈ ( 0 ..^ ( 𝑌 − 𝑋 ) ) ) → ( ( 𝑆 substr ⟨ 𝑋 , 𝑌 ⟩ ) ‘ 𝑥 ) = ( 𝑆 ‘ ( 𝑥 + 𝑋 ) ) ) |
| 65 |
58 64
|
eqtrd |
⊢ ( ( ( 𝑆 ∈ Word 𝐴 ∧ ( 𝑋 ∈ ( 0 ... 𝑌 ) ∧ 𝑌 ∈ ( 0 ... 𝑍 ) ∧ 𝑍 ∈ ( 0 ... ( ♯ ‘ 𝑆 ) ) ) ) ∧ 𝑥 ∈ ( 0 ..^ ( 𝑌 − 𝑋 ) ) ) → ( ( ( 𝑆 substr ⟨ 𝑋 , 𝑌 ⟩ ) ++ ( 𝑆 substr ⟨ 𝑌 , 𝑍 ⟩ ) ) ‘ 𝑥 ) = ( 𝑆 ‘ ( 𝑥 + 𝑋 ) ) ) |
| 66 |
1
|
ad2antrr |
⊢ ( ( ( 𝑆 ∈ Word 𝐴 ∧ ( 𝑋 ∈ ( 0 ... 𝑌 ) ∧ 𝑌 ∈ ( 0 ... 𝑍 ) ∧ 𝑍 ∈ ( 0 ... ( ♯ ‘ 𝑆 ) ) ) ) ∧ 𝑥 ∈ ( ( 𝑌 − 𝑋 ) ..^ ( 𝑍 − 𝑋 ) ) ) → ( 𝑆 substr ⟨ 𝑋 , 𝑌 ⟩ ) ∈ Word 𝐴 ) |
| 67 |
3
|
ad2antrr |
⊢ ( ( ( 𝑆 ∈ Word 𝐴 ∧ ( 𝑋 ∈ ( 0 ... 𝑌 ) ∧ 𝑌 ∈ ( 0 ... 𝑍 ) ∧ 𝑍 ∈ ( 0 ... ( ♯ ‘ 𝑆 ) ) ) ) ∧ 𝑥 ∈ ( ( 𝑌 − 𝑋 ) ..^ ( 𝑍 − 𝑋 ) ) ) → ( 𝑆 substr ⟨ 𝑌 , 𝑍 ⟩ ) ∈ Word 𝐴 ) |
| 68 |
23 30
|
eqtrd |
⊢ ( ( 𝑆 ∈ Word 𝐴 ∧ ( 𝑋 ∈ ( 0 ... 𝑌 ) ∧ 𝑌 ∈ ( 0 ... 𝑍 ) ∧ 𝑍 ∈ ( 0 ... ( ♯ ‘ 𝑆 ) ) ) ) → ( ( ♯ ‘ ( 𝑆 substr ⟨ 𝑋 , 𝑌 ⟩ ) ) + ( ♯ ‘ ( 𝑆 substr ⟨ 𝑌 , 𝑍 ⟩ ) ) ) = ( 𝑍 − 𝑋 ) ) |
| 69 |
20 68
|
oveq12d |
⊢ ( ( 𝑆 ∈ Word 𝐴 ∧ ( 𝑋 ∈ ( 0 ... 𝑌 ) ∧ 𝑌 ∈ ( 0 ... 𝑍 ) ∧ 𝑍 ∈ ( 0 ... ( ♯ ‘ 𝑆 ) ) ) ) → ( ( ♯ ‘ ( 𝑆 substr ⟨ 𝑋 , 𝑌 ⟩ ) ) ..^ ( ( ♯ ‘ ( 𝑆 substr ⟨ 𝑋 , 𝑌 ⟩ ) ) + ( ♯ ‘ ( 𝑆 substr ⟨ 𝑌 , 𝑍 ⟩ ) ) ) ) = ( ( 𝑌 − 𝑋 ) ..^ ( 𝑍 − 𝑋 ) ) ) |
| 70 |
69
|
eleq2d |
⊢ ( ( 𝑆 ∈ Word 𝐴 ∧ ( 𝑋 ∈ ( 0 ... 𝑌 ) ∧ 𝑌 ∈ ( 0 ... 𝑍 ) ∧ 𝑍 ∈ ( 0 ... ( ♯ ‘ 𝑆 ) ) ) ) → ( 𝑥 ∈ ( ( ♯ ‘ ( 𝑆 substr ⟨ 𝑋 , 𝑌 ⟩ ) ) ..^ ( ( ♯ ‘ ( 𝑆 substr ⟨ 𝑋 , 𝑌 ⟩ ) ) + ( ♯ ‘ ( 𝑆 substr ⟨ 𝑌 , 𝑍 ⟩ ) ) ) ) ↔ 𝑥 ∈ ( ( 𝑌 − 𝑋 ) ..^ ( 𝑍 − 𝑋 ) ) ) ) |
| 71 |
70
|
biimpar |
⊢ ( ( ( 𝑆 ∈ Word 𝐴 ∧ ( 𝑋 ∈ ( 0 ... 𝑌 ) ∧ 𝑌 ∈ ( 0 ... 𝑍 ) ∧ 𝑍 ∈ ( 0 ... ( ♯ ‘ 𝑆 ) ) ) ) ∧ 𝑥 ∈ ( ( 𝑌 − 𝑋 ) ..^ ( 𝑍 − 𝑋 ) ) ) → 𝑥 ∈ ( ( ♯ ‘ ( 𝑆 substr ⟨ 𝑋 , 𝑌 ⟩ ) ) ..^ ( ( ♯ ‘ ( 𝑆 substr ⟨ 𝑋 , 𝑌 ⟩ ) ) + ( ♯ ‘ ( 𝑆 substr ⟨ 𝑌 , 𝑍 ⟩ ) ) ) ) ) |
| 72 |
|
ccatval2 |
⊢ ( ( ( 𝑆 substr ⟨ 𝑋 , 𝑌 ⟩ ) ∈ Word 𝐴 ∧ ( 𝑆 substr ⟨ 𝑌 , 𝑍 ⟩ ) ∈ Word 𝐴 ∧ 𝑥 ∈ ( ( ♯ ‘ ( 𝑆 substr ⟨ 𝑋 , 𝑌 ⟩ ) ) ..^ ( ( ♯ ‘ ( 𝑆 substr ⟨ 𝑋 , 𝑌 ⟩ ) ) + ( ♯ ‘ ( 𝑆 substr ⟨ 𝑌 , 𝑍 ⟩ ) ) ) ) ) → ( ( ( 𝑆 substr ⟨ 𝑋 , 𝑌 ⟩ ) ++ ( 𝑆 substr ⟨ 𝑌 , 𝑍 ⟩ ) ) ‘ 𝑥 ) = ( ( 𝑆 substr ⟨ 𝑌 , 𝑍 ⟩ ) ‘ ( 𝑥 − ( ♯ ‘ ( 𝑆 substr ⟨ 𝑋 , 𝑌 ⟩ ) ) ) ) ) |
| 73 |
66 67 71 72
|
syl3anc |
⊢ ( ( ( 𝑆 ∈ Word 𝐴 ∧ ( 𝑋 ∈ ( 0 ... 𝑌 ) ∧ 𝑌 ∈ ( 0 ... 𝑍 ) ∧ 𝑍 ∈ ( 0 ... ( ♯ ‘ 𝑆 ) ) ) ) ∧ 𝑥 ∈ ( ( 𝑌 − 𝑋 ) ..^ ( 𝑍 − 𝑋 ) ) ) → ( ( ( 𝑆 substr ⟨ 𝑋 , 𝑌 ⟩ ) ++ ( 𝑆 substr ⟨ 𝑌 , 𝑍 ⟩ ) ) ‘ 𝑥 ) = ( ( 𝑆 substr ⟨ 𝑌 , 𝑍 ⟩ ) ‘ ( 𝑥 − ( ♯ ‘ ( 𝑆 substr ⟨ 𝑋 , 𝑌 ⟩ ) ) ) ) ) |
| 74 |
|
simpll |
⊢ ( ( ( 𝑆 ∈ Word 𝐴 ∧ ( 𝑋 ∈ ( 0 ... 𝑌 ) ∧ 𝑌 ∈ ( 0 ... 𝑍 ) ∧ 𝑍 ∈ ( 0 ... ( ♯ ‘ 𝑆 ) ) ) ) ∧ 𝑥 ∈ ( ( 𝑌 − 𝑋 ) ..^ ( 𝑍 − 𝑋 ) ) ) → 𝑆 ∈ Word 𝐴 ) |
| 75 |
|
simplr2 |
⊢ ( ( ( 𝑆 ∈ Word 𝐴 ∧ ( 𝑋 ∈ ( 0 ... 𝑌 ) ∧ 𝑌 ∈ ( 0 ... 𝑍 ) ∧ 𝑍 ∈ ( 0 ... ( ♯ ‘ 𝑆 ) ) ) ) ∧ 𝑥 ∈ ( ( 𝑌 − 𝑋 ) ..^ ( 𝑍 − 𝑋 ) ) ) → 𝑌 ∈ ( 0 ... 𝑍 ) ) |
| 76 |
|
simplr3 |
⊢ ( ( ( 𝑆 ∈ Word 𝐴 ∧ ( 𝑋 ∈ ( 0 ... 𝑌 ) ∧ 𝑌 ∈ ( 0 ... 𝑍 ) ∧ 𝑍 ∈ ( 0 ... ( ♯ ‘ 𝑆 ) ) ) ) ∧ 𝑥 ∈ ( ( 𝑌 − 𝑋 ) ..^ ( 𝑍 − 𝑋 ) ) ) → 𝑍 ∈ ( 0 ... ( ♯ ‘ 𝑆 ) ) ) |
| 77 |
20
|
oveq2d |
⊢ ( ( 𝑆 ∈ Word 𝐴 ∧ ( 𝑋 ∈ ( 0 ... 𝑌 ) ∧ 𝑌 ∈ ( 0 ... 𝑍 ) ∧ 𝑍 ∈ ( 0 ... ( ♯ ‘ 𝑆 ) ) ) ) → ( 𝑥 − ( ♯ ‘ ( 𝑆 substr ⟨ 𝑋 , 𝑌 ⟩ ) ) ) = ( 𝑥 − ( 𝑌 − 𝑋 ) ) ) |
| 78 |
77
|
adantr |
⊢ ( ( ( 𝑆 ∈ Word 𝐴 ∧ ( 𝑋 ∈ ( 0 ... 𝑌 ) ∧ 𝑌 ∈ ( 0 ... 𝑍 ) ∧ 𝑍 ∈ ( 0 ... ( ♯ ‘ 𝑆 ) ) ) ) ∧ 𝑥 ∈ ( ( 𝑌 − 𝑋 ) ..^ ( 𝑍 − 𝑋 ) ) ) → ( 𝑥 − ( ♯ ‘ ( 𝑆 substr ⟨ 𝑋 , 𝑌 ⟩ ) ) ) = ( 𝑥 − ( 𝑌 − 𝑋 ) ) ) |
| 79 |
30
|
oveq2d |
⊢ ( ( 𝑆 ∈ Word 𝐴 ∧ ( 𝑋 ∈ ( 0 ... 𝑌 ) ∧ 𝑌 ∈ ( 0 ... 𝑍 ) ∧ 𝑍 ∈ ( 0 ... ( ♯ ‘ 𝑆 ) ) ) ) → ( ( 𝑌 − 𝑋 ) ..^ ( ( 𝑌 − 𝑋 ) + ( 𝑍 − 𝑌 ) ) ) = ( ( 𝑌 − 𝑋 ) ..^ ( 𝑍 − 𝑋 ) ) ) |
| 80 |
79
|
eleq2d |
⊢ ( ( 𝑆 ∈ Word 𝐴 ∧ ( 𝑋 ∈ ( 0 ... 𝑌 ) ∧ 𝑌 ∈ ( 0 ... 𝑍 ) ∧ 𝑍 ∈ ( 0 ... ( ♯ ‘ 𝑆 ) ) ) ) → ( 𝑥 ∈ ( ( 𝑌 − 𝑋 ) ..^ ( ( 𝑌 − 𝑋 ) + ( 𝑍 − 𝑌 ) ) ) ↔ 𝑥 ∈ ( ( 𝑌 − 𝑋 ) ..^ ( 𝑍 − 𝑋 ) ) ) ) |
| 81 |
80
|
biimpar |
⊢ ( ( ( 𝑆 ∈ Word 𝐴 ∧ ( 𝑋 ∈ ( 0 ... 𝑌 ) ∧ 𝑌 ∈ ( 0 ... 𝑍 ) ∧ 𝑍 ∈ ( 0 ... ( ♯ ‘ 𝑆 ) ) ) ) ∧ 𝑥 ∈ ( ( 𝑌 − 𝑋 ) ..^ ( 𝑍 − 𝑋 ) ) ) → 𝑥 ∈ ( ( 𝑌 − 𝑋 ) ..^ ( ( 𝑌 − 𝑋 ) + ( 𝑍 − 𝑌 ) ) ) ) |
| 82 |
28 24
|
zsubcld |
⊢ ( ( 𝑆 ∈ Word 𝐴 ∧ ( 𝑋 ∈ ( 0 ... 𝑌 ) ∧ 𝑌 ∈ ( 0 ... 𝑍 ) ∧ 𝑍 ∈ ( 0 ... ( ♯ ‘ 𝑆 ) ) ) ) → ( 𝑍 − 𝑌 ) ∈ ℤ ) |
| 83 |
82
|
adantr |
⊢ ( ( ( 𝑆 ∈ Word 𝐴 ∧ ( 𝑋 ∈ ( 0 ... 𝑌 ) ∧ 𝑌 ∈ ( 0 ... 𝑍 ) ∧ 𝑍 ∈ ( 0 ... ( ♯ ‘ 𝑆 ) ) ) ) ∧ 𝑥 ∈ ( ( 𝑌 − 𝑋 ) ..^ ( 𝑍 − 𝑋 ) ) ) → ( 𝑍 − 𝑌 ) ∈ ℤ ) |
| 84 |
|
fzosubel3 |
⊢ ( ( 𝑥 ∈ ( ( 𝑌 − 𝑋 ) ..^ ( ( 𝑌 − 𝑋 ) + ( 𝑍 − 𝑌 ) ) ) ∧ ( 𝑍 − 𝑌 ) ∈ ℤ ) → ( 𝑥 − ( 𝑌 − 𝑋 ) ) ∈ ( 0 ..^ ( 𝑍 − 𝑌 ) ) ) |
| 85 |
81 83 84
|
syl2anc |
⊢ ( ( ( 𝑆 ∈ Word 𝐴 ∧ ( 𝑋 ∈ ( 0 ... 𝑌 ) ∧ 𝑌 ∈ ( 0 ... 𝑍 ) ∧ 𝑍 ∈ ( 0 ... ( ♯ ‘ 𝑆 ) ) ) ) ∧ 𝑥 ∈ ( ( 𝑌 − 𝑋 ) ..^ ( 𝑍 − 𝑋 ) ) ) → ( 𝑥 − ( 𝑌 − 𝑋 ) ) ∈ ( 0 ..^ ( 𝑍 − 𝑌 ) ) ) |
| 86 |
78 85
|
eqeltrd |
⊢ ( ( ( 𝑆 ∈ Word 𝐴 ∧ ( 𝑋 ∈ ( 0 ... 𝑌 ) ∧ 𝑌 ∈ ( 0 ... 𝑍 ) ∧ 𝑍 ∈ ( 0 ... ( ♯ ‘ 𝑆 ) ) ) ) ∧ 𝑥 ∈ ( ( 𝑌 − 𝑋 ) ..^ ( 𝑍 − 𝑋 ) ) ) → ( 𝑥 − ( ♯ ‘ ( 𝑆 substr ⟨ 𝑋 , 𝑌 ⟩ ) ) ) ∈ ( 0 ..^ ( 𝑍 − 𝑌 ) ) ) |
| 87 |
|
swrdfv |
⊢ ( ( ( 𝑆 ∈ Word 𝐴 ∧ 𝑌 ∈ ( 0 ... 𝑍 ) ∧ 𝑍 ∈ ( 0 ... ( ♯ ‘ 𝑆 ) ) ) ∧ ( 𝑥 − ( ♯ ‘ ( 𝑆 substr ⟨ 𝑋 , 𝑌 ⟩ ) ) ) ∈ ( 0 ..^ ( 𝑍 − 𝑌 ) ) ) → ( ( 𝑆 substr ⟨ 𝑌 , 𝑍 ⟩ ) ‘ ( 𝑥 − ( ♯ ‘ ( 𝑆 substr ⟨ 𝑋 , 𝑌 ⟩ ) ) ) ) = ( 𝑆 ‘ ( ( 𝑥 − ( ♯ ‘ ( 𝑆 substr ⟨ 𝑋 , 𝑌 ⟩ ) ) ) + 𝑌 ) ) ) |
| 88 |
74 75 76 86 87
|
syl31anc |
⊢ ( ( ( 𝑆 ∈ Word 𝐴 ∧ ( 𝑋 ∈ ( 0 ... 𝑌 ) ∧ 𝑌 ∈ ( 0 ... 𝑍 ) ∧ 𝑍 ∈ ( 0 ... ( ♯ ‘ 𝑆 ) ) ) ) ∧ 𝑥 ∈ ( ( 𝑌 − 𝑋 ) ..^ ( 𝑍 − 𝑋 ) ) ) → ( ( 𝑆 substr ⟨ 𝑌 , 𝑍 ⟩ ) ‘ ( 𝑥 − ( ♯ ‘ ( 𝑆 substr ⟨ 𝑋 , 𝑌 ⟩ ) ) ) ) = ( 𝑆 ‘ ( ( 𝑥 − ( ♯ ‘ ( 𝑆 substr ⟨ 𝑋 , 𝑌 ⟩ ) ) ) + 𝑌 ) ) ) |
| 89 |
77
|
oveq1d |
⊢ ( ( 𝑆 ∈ Word 𝐴 ∧ ( 𝑋 ∈ ( 0 ... 𝑌 ) ∧ 𝑌 ∈ ( 0 ... 𝑍 ) ∧ 𝑍 ∈ ( 0 ... ( ♯ ‘ 𝑆 ) ) ) ) → ( ( 𝑥 − ( ♯ ‘ ( 𝑆 substr ⟨ 𝑋 , 𝑌 ⟩ ) ) ) + 𝑌 ) = ( ( 𝑥 − ( 𝑌 − 𝑋 ) ) + 𝑌 ) ) |
| 90 |
89
|
adantr |
⊢ ( ( ( 𝑆 ∈ Word 𝐴 ∧ ( 𝑋 ∈ ( 0 ... 𝑌 ) ∧ 𝑌 ∈ ( 0 ... 𝑍 ) ∧ 𝑍 ∈ ( 0 ... ( ♯ ‘ 𝑆 ) ) ) ) ∧ 𝑥 ∈ ( ( 𝑌 − 𝑋 ) ..^ ( 𝑍 − 𝑋 ) ) ) → ( ( 𝑥 − ( ♯ ‘ ( 𝑆 substr ⟨ 𝑋 , 𝑌 ⟩ ) ) ) + 𝑌 ) = ( ( 𝑥 − ( 𝑌 − 𝑋 ) ) + 𝑌 ) ) |
| 91 |
|
elfzoelz |
⊢ ( 𝑥 ∈ ( ( 𝑌 − 𝑋 ) ..^ ( 𝑍 − 𝑋 ) ) → 𝑥 ∈ ℤ ) |
| 92 |
91
|
zcnd |
⊢ ( 𝑥 ∈ ( ( 𝑌 − 𝑋 ) ..^ ( 𝑍 − 𝑋 ) ) → 𝑥 ∈ ℂ ) |
| 93 |
92
|
adantl |
⊢ ( ( ( 𝑆 ∈ Word 𝐴 ∧ ( 𝑋 ∈ ( 0 ... 𝑌 ) ∧ 𝑌 ∈ ( 0 ... 𝑍 ) ∧ 𝑍 ∈ ( 0 ... ( ♯ ‘ 𝑆 ) ) ) ) ∧ 𝑥 ∈ ( ( 𝑌 − 𝑋 ) ..^ ( 𝑍 − 𝑋 ) ) ) → 𝑥 ∈ ℂ ) |
| 94 |
25 27
|
subcld |
⊢ ( ( 𝑆 ∈ Word 𝐴 ∧ ( 𝑋 ∈ ( 0 ... 𝑌 ) ∧ 𝑌 ∈ ( 0 ... 𝑍 ) ∧ 𝑍 ∈ ( 0 ... ( ♯ ‘ 𝑆 ) ) ) ) → ( 𝑌 − 𝑋 ) ∈ ℂ ) |
| 95 |
94
|
adantr |
⊢ ( ( ( 𝑆 ∈ Word 𝐴 ∧ ( 𝑋 ∈ ( 0 ... 𝑌 ) ∧ 𝑌 ∈ ( 0 ... 𝑍 ) ∧ 𝑍 ∈ ( 0 ... ( ♯ ‘ 𝑆 ) ) ) ) ∧ 𝑥 ∈ ( ( 𝑌 − 𝑋 ) ..^ ( 𝑍 − 𝑋 ) ) ) → ( 𝑌 − 𝑋 ) ∈ ℂ ) |
| 96 |
25
|
adantr |
⊢ ( ( ( 𝑆 ∈ Word 𝐴 ∧ ( 𝑋 ∈ ( 0 ... 𝑌 ) ∧ 𝑌 ∈ ( 0 ... 𝑍 ) ∧ 𝑍 ∈ ( 0 ... ( ♯ ‘ 𝑆 ) ) ) ) ∧ 𝑥 ∈ ( ( 𝑌 − 𝑋 ) ..^ ( 𝑍 − 𝑋 ) ) ) → 𝑌 ∈ ℂ ) |
| 97 |
93 95 96
|
subadd23d |
⊢ ( ( ( 𝑆 ∈ Word 𝐴 ∧ ( 𝑋 ∈ ( 0 ... 𝑌 ) ∧ 𝑌 ∈ ( 0 ... 𝑍 ) ∧ 𝑍 ∈ ( 0 ... ( ♯ ‘ 𝑆 ) ) ) ) ∧ 𝑥 ∈ ( ( 𝑌 − 𝑋 ) ..^ ( 𝑍 − 𝑋 ) ) ) → ( ( 𝑥 − ( 𝑌 − 𝑋 ) ) + 𝑌 ) = ( 𝑥 + ( 𝑌 − ( 𝑌 − 𝑋 ) ) ) ) |
| 98 |
25 27
|
nncand |
⊢ ( ( 𝑆 ∈ Word 𝐴 ∧ ( 𝑋 ∈ ( 0 ... 𝑌 ) ∧ 𝑌 ∈ ( 0 ... 𝑍 ) ∧ 𝑍 ∈ ( 0 ... ( ♯ ‘ 𝑆 ) ) ) ) → ( 𝑌 − ( 𝑌 − 𝑋 ) ) = 𝑋 ) |
| 99 |
98
|
oveq2d |
⊢ ( ( 𝑆 ∈ Word 𝐴 ∧ ( 𝑋 ∈ ( 0 ... 𝑌 ) ∧ 𝑌 ∈ ( 0 ... 𝑍 ) ∧ 𝑍 ∈ ( 0 ... ( ♯ ‘ 𝑆 ) ) ) ) → ( 𝑥 + ( 𝑌 − ( 𝑌 − 𝑋 ) ) ) = ( 𝑥 + 𝑋 ) ) |
| 100 |
99
|
adantr |
⊢ ( ( ( 𝑆 ∈ Word 𝐴 ∧ ( 𝑋 ∈ ( 0 ... 𝑌 ) ∧ 𝑌 ∈ ( 0 ... 𝑍 ) ∧ 𝑍 ∈ ( 0 ... ( ♯ ‘ 𝑆 ) ) ) ) ∧ 𝑥 ∈ ( ( 𝑌 − 𝑋 ) ..^ ( 𝑍 − 𝑋 ) ) ) → ( 𝑥 + ( 𝑌 − ( 𝑌 − 𝑋 ) ) ) = ( 𝑥 + 𝑋 ) ) |
| 101 |
90 97 100
|
3eqtrd |
⊢ ( ( ( 𝑆 ∈ Word 𝐴 ∧ ( 𝑋 ∈ ( 0 ... 𝑌 ) ∧ 𝑌 ∈ ( 0 ... 𝑍 ) ∧ 𝑍 ∈ ( 0 ... ( ♯ ‘ 𝑆 ) ) ) ) ∧ 𝑥 ∈ ( ( 𝑌 − 𝑋 ) ..^ ( 𝑍 − 𝑋 ) ) ) → ( ( 𝑥 − ( ♯ ‘ ( 𝑆 substr ⟨ 𝑋 , 𝑌 ⟩ ) ) ) + 𝑌 ) = ( 𝑥 + 𝑋 ) ) |
| 102 |
101
|
fveq2d |
⊢ ( ( ( 𝑆 ∈ Word 𝐴 ∧ ( 𝑋 ∈ ( 0 ... 𝑌 ) ∧ 𝑌 ∈ ( 0 ... 𝑍 ) ∧ 𝑍 ∈ ( 0 ... ( ♯ ‘ 𝑆 ) ) ) ) ∧ 𝑥 ∈ ( ( 𝑌 − 𝑋 ) ..^ ( 𝑍 − 𝑋 ) ) ) → ( 𝑆 ‘ ( ( 𝑥 − ( ♯ ‘ ( 𝑆 substr ⟨ 𝑋 , 𝑌 ⟩ ) ) ) + 𝑌 ) ) = ( 𝑆 ‘ ( 𝑥 + 𝑋 ) ) ) |
| 103 |
73 88 102
|
3eqtrd |
⊢ ( ( ( 𝑆 ∈ Word 𝐴 ∧ ( 𝑋 ∈ ( 0 ... 𝑌 ) ∧ 𝑌 ∈ ( 0 ... 𝑍 ) ∧ 𝑍 ∈ ( 0 ... ( ♯ ‘ 𝑆 ) ) ) ) ∧ 𝑥 ∈ ( ( 𝑌 − 𝑋 ) ..^ ( 𝑍 − 𝑋 ) ) ) → ( ( ( 𝑆 substr ⟨ 𝑋 , 𝑌 ⟩ ) ++ ( 𝑆 substr ⟨ 𝑌 , 𝑍 ⟩ ) ) ‘ 𝑥 ) = ( 𝑆 ‘ ( 𝑥 + 𝑋 ) ) ) |
| 104 |
65 103
|
jaodan |
⊢ ( ( ( 𝑆 ∈ Word 𝐴 ∧ ( 𝑋 ∈ ( 0 ... 𝑌 ) ∧ 𝑌 ∈ ( 0 ... 𝑍 ) ∧ 𝑍 ∈ ( 0 ... ( ♯ ‘ 𝑆 ) ) ) ) ∧ ( 𝑥 ∈ ( 0 ..^ ( 𝑌 − 𝑋 ) ) ∨ 𝑥 ∈ ( ( 𝑌 − 𝑋 ) ..^ ( 𝑍 − 𝑋 ) ) ) ) → ( ( ( 𝑆 substr ⟨ 𝑋 , 𝑌 ⟩ ) ++ ( 𝑆 substr ⟨ 𝑌 , 𝑍 ⟩ ) ) ‘ 𝑥 ) = ( 𝑆 ‘ ( 𝑥 + 𝑋 ) ) ) |
| 105 |
51 104
|
syldan |
⊢ ( ( ( 𝑆 ∈ Word 𝐴 ∧ ( 𝑋 ∈ ( 0 ... 𝑌 ) ∧ 𝑌 ∈ ( 0 ... 𝑍 ) ∧ 𝑍 ∈ ( 0 ... ( ♯ ‘ 𝑆 ) ) ) ) ∧ 𝑥 ∈ ( 0 ..^ ( 𝑍 − 𝑋 ) ) ) → ( ( ( 𝑆 substr ⟨ 𝑋 , 𝑌 ⟩ ) ++ ( 𝑆 substr ⟨ 𝑌 , 𝑍 ⟩ ) ) ‘ 𝑥 ) = ( 𝑆 ‘ ( 𝑥 + 𝑋 ) ) ) |
| 106 |
|
simpll |
⊢ ( ( ( 𝑆 ∈ Word 𝐴 ∧ ( 𝑋 ∈ ( 0 ... 𝑌 ) ∧ 𝑌 ∈ ( 0 ... 𝑍 ) ∧ 𝑍 ∈ ( 0 ... ( ♯ ‘ 𝑆 ) ) ) ) ∧ 𝑥 ∈ ( 0 ..^ ( 𝑍 − 𝑋 ) ) ) → 𝑆 ∈ Word 𝐴 ) |
| 107 |
42
|
adantr |
⊢ ( ( ( 𝑆 ∈ Word 𝐴 ∧ ( 𝑋 ∈ ( 0 ... 𝑌 ) ∧ 𝑌 ∈ ( 0 ... 𝑍 ) ∧ 𝑍 ∈ ( 0 ... ( ♯ ‘ 𝑆 ) ) ) ) ∧ 𝑥 ∈ ( 0 ..^ ( 𝑍 − 𝑋 ) ) ) → 𝑋 ∈ ( 0 ... 𝑍 ) ) |
| 108 |
|
simplr3 |
⊢ ( ( ( 𝑆 ∈ Word 𝐴 ∧ ( 𝑋 ∈ ( 0 ... 𝑌 ) ∧ 𝑌 ∈ ( 0 ... 𝑍 ) ∧ 𝑍 ∈ ( 0 ... ( ♯ ‘ 𝑆 ) ) ) ) ∧ 𝑥 ∈ ( 0 ..^ ( 𝑍 − 𝑋 ) ) ) → 𝑍 ∈ ( 0 ... ( ♯ ‘ 𝑆 ) ) ) |
| 109 |
|
simpr |
⊢ ( ( ( 𝑆 ∈ Word 𝐴 ∧ ( 𝑋 ∈ ( 0 ... 𝑌 ) ∧ 𝑌 ∈ ( 0 ... 𝑍 ) ∧ 𝑍 ∈ ( 0 ... ( ♯ ‘ 𝑆 ) ) ) ) ∧ 𝑥 ∈ ( 0 ..^ ( 𝑍 − 𝑋 ) ) ) → 𝑥 ∈ ( 0 ..^ ( 𝑍 − 𝑋 ) ) ) |
| 110 |
|
swrdfv |
⊢ ( ( ( 𝑆 ∈ Word 𝐴 ∧ 𝑋 ∈ ( 0 ... 𝑍 ) ∧ 𝑍 ∈ ( 0 ... ( ♯ ‘ 𝑆 ) ) ) ∧ 𝑥 ∈ ( 0 ..^ ( 𝑍 − 𝑋 ) ) ) → ( ( 𝑆 substr ⟨ 𝑋 , 𝑍 ⟩ ) ‘ 𝑥 ) = ( 𝑆 ‘ ( 𝑥 + 𝑋 ) ) ) |
| 111 |
106 107 108 109 110
|
syl31anc |
⊢ ( ( ( 𝑆 ∈ Word 𝐴 ∧ ( 𝑋 ∈ ( 0 ... 𝑌 ) ∧ 𝑌 ∈ ( 0 ... 𝑍 ) ∧ 𝑍 ∈ ( 0 ... ( ♯ ‘ 𝑆 ) ) ) ) ∧ 𝑥 ∈ ( 0 ..^ ( 𝑍 − 𝑋 ) ) ) → ( ( 𝑆 substr ⟨ 𝑋 , 𝑍 ⟩ ) ‘ 𝑥 ) = ( 𝑆 ‘ ( 𝑥 + 𝑋 ) ) ) |
| 112 |
105 111
|
eqtr4d |
⊢ ( ( ( 𝑆 ∈ Word 𝐴 ∧ ( 𝑋 ∈ ( 0 ... 𝑌 ) ∧ 𝑌 ∈ ( 0 ... 𝑍 ) ∧ 𝑍 ∈ ( 0 ... ( ♯ ‘ 𝑆 ) ) ) ) ∧ 𝑥 ∈ ( 0 ..^ ( 𝑍 − 𝑋 ) ) ) → ( ( ( 𝑆 substr ⟨ 𝑋 , 𝑌 ⟩ ) ++ ( 𝑆 substr ⟨ 𝑌 , 𝑍 ⟩ ) ) ‘ 𝑥 ) = ( ( 𝑆 substr ⟨ 𝑋 , 𝑍 ⟩ ) ‘ 𝑥 ) ) |
| 113 |
34 47 112
|
eqfnfvd |
⊢ ( ( 𝑆 ∈ Word 𝐴 ∧ ( 𝑋 ∈ ( 0 ... 𝑌 ) ∧ 𝑌 ∈ ( 0 ... 𝑍 ) ∧ 𝑍 ∈ ( 0 ... ( ♯ ‘ 𝑆 ) ) ) ) → ( ( 𝑆 substr ⟨ 𝑋 , 𝑌 ⟩ ) ++ ( 𝑆 substr ⟨ 𝑌 , 𝑍 ⟩ ) ) = ( 𝑆 substr ⟨ 𝑋 , 𝑍 ⟩ ) ) |