Step |
Hyp |
Ref |
Expression |
1 |
|
simpl |
⊢ ( ( 𝑊 ∈ Word 𝑉 ∧ 𝑁 ∈ ( 1 ... ( ♯ ‘ 𝑊 ) ) ) → 𝑊 ∈ Word 𝑉 ) |
2 |
|
elfzelz |
⊢ ( 𝑁 ∈ ( 1 ... ( ♯ ‘ 𝑊 ) ) → 𝑁 ∈ ℤ ) |
3 |
2
|
adantl |
⊢ ( ( 𝑊 ∈ Word 𝑉 ∧ 𝑁 ∈ ( 1 ... ( ♯ ‘ 𝑊 ) ) ) → 𝑁 ∈ ℤ ) |
4 |
|
elfz1b |
⊢ ( 𝑁 ∈ ( 1 ... ( ♯ ‘ 𝑊 ) ) ↔ ( 𝑁 ∈ ℕ ∧ ( ♯ ‘ 𝑊 ) ∈ ℕ ∧ 𝑁 ≤ ( ♯ ‘ 𝑊 ) ) ) |
5 |
|
simp2 |
⊢ ( ( 𝑁 ∈ ℕ ∧ ( ♯ ‘ 𝑊 ) ∈ ℕ ∧ 𝑁 ≤ ( ♯ ‘ 𝑊 ) ) → ( ♯ ‘ 𝑊 ) ∈ ℕ ) |
6 |
4 5
|
sylbi |
⊢ ( 𝑁 ∈ ( 1 ... ( ♯ ‘ 𝑊 ) ) → ( ♯ ‘ 𝑊 ) ∈ ℕ ) |
7 |
6
|
adantl |
⊢ ( ( 𝑊 ∈ Word 𝑉 ∧ 𝑁 ∈ ( 1 ... ( ♯ ‘ 𝑊 ) ) ) → ( ♯ ‘ 𝑊 ) ∈ ℕ ) |
8 |
|
fzo0end |
⊢ ( ( ♯ ‘ 𝑊 ) ∈ ℕ → ( ( ♯ ‘ 𝑊 ) − 1 ) ∈ ( 0 ..^ ( ♯ ‘ 𝑊 ) ) ) |
9 |
7 8
|
syl |
⊢ ( ( 𝑊 ∈ Word 𝑉 ∧ 𝑁 ∈ ( 1 ... ( ♯ ‘ 𝑊 ) ) ) → ( ( ♯ ‘ 𝑊 ) − 1 ) ∈ ( 0 ..^ ( ♯ ‘ 𝑊 ) ) ) |
10 |
|
cshwidxmod |
⊢ ( ( 𝑊 ∈ Word 𝑉 ∧ 𝑁 ∈ ℤ ∧ ( ( ♯ ‘ 𝑊 ) − 1 ) ∈ ( 0 ..^ ( ♯ ‘ 𝑊 ) ) ) → ( ( 𝑊 cyclShift 𝑁 ) ‘ ( ( ♯ ‘ 𝑊 ) − 1 ) ) = ( 𝑊 ‘ ( ( ( ( ♯ ‘ 𝑊 ) − 1 ) + 𝑁 ) mod ( ♯ ‘ 𝑊 ) ) ) ) |
11 |
1 3 9 10
|
syl3anc |
⊢ ( ( 𝑊 ∈ Word 𝑉 ∧ 𝑁 ∈ ( 1 ... ( ♯ ‘ 𝑊 ) ) ) → ( ( 𝑊 cyclShift 𝑁 ) ‘ ( ( ♯ ‘ 𝑊 ) − 1 ) ) = ( 𝑊 ‘ ( ( ( ( ♯ ‘ 𝑊 ) − 1 ) + 𝑁 ) mod ( ♯ ‘ 𝑊 ) ) ) ) |
12 |
|
nncn |
⊢ ( ( ♯ ‘ 𝑊 ) ∈ ℕ → ( ♯ ‘ 𝑊 ) ∈ ℂ ) |
13 |
12
|
adantl |
⊢ ( ( 𝑁 ∈ ℕ ∧ ( ♯ ‘ 𝑊 ) ∈ ℕ ) → ( ♯ ‘ 𝑊 ) ∈ ℂ ) |
14 |
|
1cnd |
⊢ ( ( 𝑁 ∈ ℕ ∧ ( ♯ ‘ 𝑊 ) ∈ ℕ ) → 1 ∈ ℂ ) |
15 |
|
nncn |
⊢ ( 𝑁 ∈ ℕ → 𝑁 ∈ ℂ ) |
16 |
15
|
adantr |
⊢ ( ( 𝑁 ∈ ℕ ∧ ( ♯ ‘ 𝑊 ) ∈ ℕ ) → 𝑁 ∈ ℂ ) |
17 |
13 14 16
|
3jca |
⊢ ( ( 𝑁 ∈ ℕ ∧ ( ♯ ‘ 𝑊 ) ∈ ℕ ) → ( ( ♯ ‘ 𝑊 ) ∈ ℂ ∧ 1 ∈ ℂ ∧ 𝑁 ∈ ℂ ) ) |
18 |
17
|
3adant3 |
⊢ ( ( 𝑁 ∈ ℕ ∧ ( ♯ ‘ 𝑊 ) ∈ ℕ ∧ 𝑁 ≤ ( ♯ ‘ 𝑊 ) ) → ( ( ♯ ‘ 𝑊 ) ∈ ℂ ∧ 1 ∈ ℂ ∧ 𝑁 ∈ ℂ ) ) |
19 |
4 18
|
sylbi |
⊢ ( 𝑁 ∈ ( 1 ... ( ♯ ‘ 𝑊 ) ) → ( ( ♯ ‘ 𝑊 ) ∈ ℂ ∧ 1 ∈ ℂ ∧ 𝑁 ∈ ℂ ) ) |
20 |
|
subadd23 |
⊢ ( ( ( ♯ ‘ 𝑊 ) ∈ ℂ ∧ 1 ∈ ℂ ∧ 𝑁 ∈ ℂ ) → ( ( ( ♯ ‘ 𝑊 ) − 1 ) + 𝑁 ) = ( ( ♯ ‘ 𝑊 ) + ( 𝑁 − 1 ) ) ) |
21 |
19 20
|
syl |
⊢ ( 𝑁 ∈ ( 1 ... ( ♯ ‘ 𝑊 ) ) → ( ( ( ♯ ‘ 𝑊 ) − 1 ) + 𝑁 ) = ( ( ♯ ‘ 𝑊 ) + ( 𝑁 − 1 ) ) ) |
22 |
21
|
oveq1d |
⊢ ( 𝑁 ∈ ( 1 ... ( ♯ ‘ 𝑊 ) ) → ( ( ( ( ♯ ‘ 𝑊 ) − 1 ) + 𝑁 ) mod ( ♯ ‘ 𝑊 ) ) = ( ( ( ♯ ‘ 𝑊 ) + ( 𝑁 − 1 ) ) mod ( ♯ ‘ 𝑊 ) ) ) |
23 |
|
nnm1nn0 |
⊢ ( 𝑁 ∈ ℕ → ( 𝑁 − 1 ) ∈ ℕ0 ) |
24 |
23
|
3ad2ant1 |
⊢ ( ( 𝑁 ∈ ℕ ∧ ( ♯ ‘ 𝑊 ) ∈ ℕ ∧ 𝑁 ≤ ( ♯ ‘ 𝑊 ) ) → ( 𝑁 − 1 ) ∈ ℕ0 ) |
25 |
|
simp3 |
⊢ ( ( 𝑁 ∈ ℕ ∧ ( ♯ ‘ 𝑊 ) ∈ ℕ ∧ 𝑁 ≤ ( ♯ ‘ 𝑊 ) ) → 𝑁 ≤ ( ♯ ‘ 𝑊 ) ) |
26 |
|
nnz |
⊢ ( 𝑁 ∈ ℕ → 𝑁 ∈ ℤ ) |
27 |
|
nnz |
⊢ ( ( ♯ ‘ 𝑊 ) ∈ ℕ → ( ♯ ‘ 𝑊 ) ∈ ℤ ) |
28 |
26 27
|
anim12i |
⊢ ( ( 𝑁 ∈ ℕ ∧ ( ♯ ‘ 𝑊 ) ∈ ℕ ) → ( 𝑁 ∈ ℤ ∧ ( ♯ ‘ 𝑊 ) ∈ ℤ ) ) |
29 |
28
|
3adant3 |
⊢ ( ( 𝑁 ∈ ℕ ∧ ( ♯ ‘ 𝑊 ) ∈ ℕ ∧ 𝑁 ≤ ( ♯ ‘ 𝑊 ) ) → ( 𝑁 ∈ ℤ ∧ ( ♯ ‘ 𝑊 ) ∈ ℤ ) ) |
30 |
|
zlem1lt |
⊢ ( ( 𝑁 ∈ ℤ ∧ ( ♯ ‘ 𝑊 ) ∈ ℤ ) → ( 𝑁 ≤ ( ♯ ‘ 𝑊 ) ↔ ( 𝑁 − 1 ) < ( ♯ ‘ 𝑊 ) ) ) |
31 |
29 30
|
syl |
⊢ ( ( 𝑁 ∈ ℕ ∧ ( ♯ ‘ 𝑊 ) ∈ ℕ ∧ 𝑁 ≤ ( ♯ ‘ 𝑊 ) ) → ( 𝑁 ≤ ( ♯ ‘ 𝑊 ) ↔ ( 𝑁 − 1 ) < ( ♯ ‘ 𝑊 ) ) ) |
32 |
25 31
|
mpbid |
⊢ ( ( 𝑁 ∈ ℕ ∧ ( ♯ ‘ 𝑊 ) ∈ ℕ ∧ 𝑁 ≤ ( ♯ ‘ 𝑊 ) ) → ( 𝑁 − 1 ) < ( ♯ ‘ 𝑊 ) ) |
33 |
24 5 32
|
3jca |
⊢ ( ( 𝑁 ∈ ℕ ∧ ( ♯ ‘ 𝑊 ) ∈ ℕ ∧ 𝑁 ≤ ( ♯ ‘ 𝑊 ) ) → ( ( 𝑁 − 1 ) ∈ ℕ0 ∧ ( ♯ ‘ 𝑊 ) ∈ ℕ ∧ ( 𝑁 − 1 ) < ( ♯ ‘ 𝑊 ) ) ) |
34 |
4 33
|
sylbi |
⊢ ( 𝑁 ∈ ( 1 ... ( ♯ ‘ 𝑊 ) ) → ( ( 𝑁 − 1 ) ∈ ℕ0 ∧ ( ♯ ‘ 𝑊 ) ∈ ℕ ∧ ( 𝑁 − 1 ) < ( ♯ ‘ 𝑊 ) ) ) |
35 |
|
addmodid |
⊢ ( ( ( 𝑁 − 1 ) ∈ ℕ0 ∧ ( ♯ ‘ 𝑊 ) ∈ ℕ ∧ ( 𝑁 − 1 ) < ( ♯ ‘ 𝑊 ) ) → ( ( ( ♯ ‘ 𝑊 ) + ( 𝑁 − 1 ) ) mod ( ♯ ‘ 𝑊 ) ) = ( 𝑁 − 1 ) ) |
36 |
34 35
|
syl |
⊢ ( 𝑁 ∈ ( 1 ... ( ♯ ‘ 𝑊 ) ) → ( ( ( ♯ ‘ 𝑊 ) + ( 𝑁 − 1 ) ) mod ( ♯ ‘ 𝑊 ) ) = ( 𝑁 − 1 ) ) |
37 |
22 36
|
eqtrd |
⊢ ( 𝑁 ∈ ( 1 ... ( ♯ ‘ 𝑊 ) ) → ( ( ( ( ♯ ‘ 𝑊 ) − 1 ) + 𝑁 ) mod ( ♯ ‘ 𝑊 ) ) = ( 𝑁 − 1 ) ) |
38 |
37
|
fveq2d |
⊢ ( 𝑁 ∈ ( 1 ... ( ♯ ‘ 𝑊 ) ) → ( 𝑊 ‘ ( ( ( ( ♯ ‘ 𝑊 ) − 1 ) + 𝑁 ) mod ( ♯ ‘ 𝑊 ) ) ) = ( 𝑊 ‘ ( 𝑁 − 1 ) ) ) |
39 |
38
|
adantl |
⊢ ( ( 𝑊 ∈ Word 𝑉 ∧ 𝑁 ∈ ( 1 ... ( ♯ ‘ 𝑊 ) ) ) → ( 𝑊 ‘ ( ( ( ( ♯ ‘ 𝑊 ) − 1 ) + 𝑁 ) mod ( ♯ ‘ 𝑊 ) ) ) = ( 𝑊 ‘ ( 𝑁 − 1 ) ) ) |
40 |
11 39
|
eqtrd |
⊢ ( ( 𝑊 ∈ Word 𝑉 ∧ 𝑁 ∈ ( 1 ... ( ♯ ‘ 𝑊 ) ) ) → ( ( 𝑊 cyclShift 𝑁 ) ‘ ( ( ♯ ‘ 𝑊 ) − 1 ) ) = ( 𝑊 ‘ ( 𝑁 − 1 ) ) ) |