Step |
Hyp |
Ref |
Expression |
1 |
|
ral0 |
⊢ ∀ 𝑖 ∈ ∅ ( 𝑊 ‘ 𝑖 ) = ( 𝑊 ‘ 0 ) |
2 |
|
oveq2 |
⊢ ( ( ♯ ‘ 𝑊 ) = 0 → ( 0 ..^ ( ♯ ‘ 𝑊 ) ) = ( 0 ..^ 0 ) ) |
3 |
|
fzo0 |
⊢ ( 0 ..^ 0 ) = ∅ |
4 |
2 3
|
eqtrdi |
⊢ ( ( ♯ ‘ 𝑊 ) = 0 → ( 0 ..^ ( ♯ ‘ 𝑊 ) ) = ∅ ) |
5 |
4
|
raleqdv |
⊢ ( ( ♯ ‘ 𝑊 ) = 0 → ( ∀ 𝑖 ∈ ( 0 ..^ ( ♯ ‘ 𝑊 ) ) ( 𝑊 ‘ 𝑖 ) = ( 𝑊 ‘ 0 ) ↔ ∀ 𝑖 ∈ ∅ ( 𝑊 ‘ 𝑖 ) = ( 𝑊 ‘ 0 ) ) ) |
6 |
1 5
|
mpbiri |
⊢ ( ( ♯ ‘ 𝑊 ) = 0 → ∀ 𝑖 ∈ ( 0 ..^ ( ♯ ‘ 𝑊 ) ) ( 𝑊 ‘ 𝑖 ) = ( 𝑊 ‘ 0 ) ) |
7 |
6
|
a1d |
⊢ ( ( ♯ ‘ 𝑊 ) = 0 → ( ( 𝑊 ∈ Word 𝑉 ∧ ( 𝑊 cyclShift 1 ) = 𝑊 ) → ∀ 𝑖 ∈ ( 0 ..^ ( ♯ ‘ 𝑊 ) ) ( 𝑊 ‘ 𝑖 ) = ( 𝑊 ‘ 0 ) ) ) |
8 |
|
simprl |
⊢ ( ( ( ¬ ( ♯ ‘ 𝑊 ) = 0 ∧ ¬ ( ♯ ‘ 𝑊 ) = 1 ) ∧ ( 𝑊 ∈ Word 𝑉 ∧ ( 𝑊 cyclShift 1 ) = 𝑊 ) ) → 𝑊 ∈ Word 𝑉 ) |
9 |
|
lencl |
⊢ ( 𝑊 ∈ Word 𝑉 → ( ♯ ‘ 𝑊 ) ∈ ℕ0 ) |
10 |
|
1nn0 |
⊢ 1 ∈ ℕ0 |
11 |
10
|
a1i |
⊢ ( ( ( ♯ ‘ 𝑊 ) ∈ ℕ0 ∧ ( ¬ ( ♯ ‘ 𝑊 ) = 0 ∧ ¬ ( ♯ ‘ 𝑊 ) = 1 ) ) → 1 ∈ ℕ0 ) |
12 |
|
df-ne |
⊢ ( ( ♯ ‘ 𝑊 ) ≠ 0 ↔ ¬ ( ♯ ‘ 𝑊 ) = 0 ) |
13 |
|
elnnne0 |
⊢ ( ( ♯ ‘ 𝑊 ) ∈ ℕ ↔ ( ( ♯ ‘ 𝑊 ) ∈ ℕ0 ∧ ( ♯ ‘ 𝑊 ) ≠ 0 ) ) |
14 |
13
|
simplbi2com |
⊢ ( ( ♯ ‘ 𝑊 ) ≠ 0 → ( ( ♯ ‘ 𝑊 ) ∈ ℕ0 → ( ♯ ‘ 𝑊 ) ∈ ℕ ) ) |
15 |
12 14
|
sylbir |
⊢ ( ¬ ( ♯ ‘ 𝑊 ) = 0 → ( ( ♯ ‘ 𝑊 ) ∈ ℕ0 → ( ♯ ‘ 𝑊 ) ∈ ℕ ) ) |
16 |
15
|
adantr |
⊢ ( ( ¬ ( ♯ ‘ 𝑊 ) = 0 ∧ ¬ ( ♯ ‘ 𝑊 ) = 1 ) → ( ( ♯ ‘ 𝑊 ) ∈ ℕ0 → ( ♯ ‘ 𝑊 ) ∈ ℕ ) ) |
17 |
16
|
impcom |
⊢ ( ( ( ♯ ‘ 𝑊 ) ∈ ℕ0 ∧ ( ¬ ( ♯ ‘ 𝑊 ) = 0 ∧ ¬ ( ♯ ‘ 𝑊 ) = 1 ) ) → ( ♯ ‘ 𝑊 ) ∈ ℕ ) |
18 |
|
neqne |
⊢ ( ¬ ( ♯ ‘ 𝑊 ) = 1 → ( ♯ ‘ 𝑊 ) ≠ 1 ) |
19 |
18
|
ad2antll |
⊢ ( ( ( ♯ ‘ 𝑊 ) ∈ ℕ0 ∧ ( ¬ ( ♯ ‘ 𝑊 ) = 0 ∧ ¬ ( ♯ ‘ 𝑊 ) = 1 ) ) → ( ♯ ‘ 𝑊 ) ≠ 1 ) |
20 |
|
nngt1ne1 |
⊢ ( ( ♯ ‘ 𝑊 ) ∈ ℕ → ( 1 < ( ♯ ‘ 𝑊 ) ↔ ( ♯ ‘ 𝑊 ) ≠ 1 ) ) |
21 |
17 20
|
syl |
⊢ ( ( ( ♯ ‘ 𝑊 ) ∈ ℕ0 ∧ ( ¬ ( ♯ ‘ 𝑊 ) = 0 ∧ ¬ ( ♯ ‘ 𝑊 ) = 1 ) ) → ( 1 < ( ♯ ‘ 𝑊 ) ↔ ( ♯ ‘ 𝑊 ) ≠ 1 ) ) |
22 |
19 21
|
mpbird |
⊢ ( ( ( ♯ ‘ 𝑊 ) ∈ ℕ0 ∧ ( ¬ ( ♯ ‘ 𝑊 ) = 0 ∧ ¬ ( ♯ ‘ 𝑊 ) = 1 ) ) → 1 < ( ♯ ‘ 𝑊 ) ) |
23 |
|
elfzo0 |
⊢ ( 1 ∈ ( 0 ..^ ( ♯ ‘ 𝑊 ) ) ↔ ( 1 ∈ ℕ0 ∧ ( ♯ ‘ 𝑊 ) ∈ ℕ ∧ 1 < ( ♯ ‘ 𝑊 ) ) ) |
24 |
11 17 22 23
|
syl3anbrc |
⊢ ( ( ( ♯ ‘ 𝑊 ) ∈ ℕ0 ∧ ( ¬ ( ♯ ‘ 𝑊 ) = 0 ∧ ¬ ( ♯ ‘ 𝑊 ) = 1 ) ) → 1 ∈ ( 0 ..^ ( ♯ ‘ 𝑊 ) ) ) |
25 |
24
|
ex |
⊢ ( ( ♯ ‘ 𝑊 ) ∈ ℕ0 → ( ( ¬ ( ♯ ‘ 𝑊 ) = 0 ∧ ¬ ( ♯ ‘ 𝑊 ) = 1 ) → 1 ∈ ( 0 ..^ ( ♯ ‘ 𝑊 ) ) ) ) |
26 |
9 25
|
syl |
⊢ ( 𝑊 ∈ Word 𝑉 → ( ( ¬ ( ♯ ‘ 𝑊 ) = 0 ∧ ¬ ( ♯ ‘ 𝑊 ) = 1 ) → 1 ∈ ( 0 ..^ ( ♯ ‘ 𝑊 ) ) ) ) |
27 |
26
|
adantr |
⊢ ( ( 𝑊 ∈ Word 𝑉 ∧ ( 𝑊 cyclShift 1 ) = 𝑊 ) → ( ( ¬ ( ♯ ‘ 𝑊 ) = 0 ∧ ¬ ( ♯ ‘ 𝑊 ) = 1 ) → 1 ∈ ( 0 ..^ ( ♯ ‘ 𝑊 ) ) ) ) |
28 |
27
|
impcom |
⊢ ( ( ( ¬ ( ♯ ‘ 𝑊 ) = 0 ∧ ¬ ( ♯ ‘ 𝑊 ) = 1 ) ∧ ( 𝑊 ∈ Word 𝑉 ∧ ( 𝑊 cyclShift 1 ) = 𝑊 ) ) → 1 ∈ ( 0 ..^ ( ♯ ‘ 𝑊 ) ) ) |
29 |
|
simprr |
⊢ ( ( ( ¬ ( ♯ ‘ 𝑊 ) = 0 ∧ ¬ ( ♯ ‘ 𝑊 ) = 1 ) ∧ ( 𝑊 ∈ Word 𝑉 ∧ ( 𝑊 cyclShift 1 ) = 𝑊 ) ) → ( 𝑊 cyclShift 1 ) = 𝑊 ) |
30 |
|
lbfzo0 |
⊢ ( 0 ∈ ( 0 ..^ ( ♯ ‘ 𝑊 ) ) ↔ ( ♯ ‘ 𝑊 ) ∈ ℕ ) |
31 |
30 13
|
sylbbr |
⊢ ( ( ( ♯ ‘ 𝑊 ) ∈ ℕ0 ∧ ( ♯ ‘ 𝑊 ) ≠ 0 ) → 0 ∈ ( 0 ..^ ( ♯ ‘ 𝑊 ) ) ) |
32 |
31
|
ex |
⊢ ( ( ♯ ‘ 𝑊 ) ∈ ℕ0 → ( ( ♯ ‘ 𝑊 ) ≠ 0 → 0 ∈ ( 0 ..^ ( ♯ ‘ 𝑊 ) ) ) ) |
33 |
12 32
|
syl5bir |
⊢ ( ( ♯ ‘ 𝑊 ) ∈ ℕ0 → ( ¬ ( ♯ ‘ 𝑊 ) = 0 → 0 ∈ ( 0 ..^ ( ♯ ‘ 𝑊 ) ) ) ) |
34 |
9 33
|
syl |
⊢ ( 𝑊 ∈ Word 𝑉 → ( ¬ ( ♯ ‘ 𝑊 ) = 0 → 0 ∈ ( 0 ..^ ( ♯ ‘ 𝑊 ) ) ) ) |
35 |
34
|
adantr |
⊢ ( ( 𝑊 ∈ Word 𝑉 ∧ ( 𝑊 cyclShift 1 ) = 𝑊 ) → ( ¬ ( ♯ ‘ 𝑊 ) = 0 → 0 ∈ ( 0 ..^ ( ♯ ‘ 𝑊 ) ) ) ) |
36 |
35
|
com12 |
⊢ ( ¬ ( ♯ ‘ 𝑊 ) = 0 → ( ( 𝑊 ∈ Word 𝑉 ∧ ( 𝑊 cyclShift 1 ) = 𝑊 ) → 0 ∈ ( 0 ..^ ( ♯ ‘ 𝑊 ) ) ) ) |
37 |
36
|
adantr |
⊢ ( ( ¬ ( ♯ ‘ 𝑊 ) = 0 ∧ ¬ ( ♯ ‘ 𝑊 ) = 1 ) → ( ( 𝑊 ∈ Word 𝑉 ∧ ( 𝑊 cyclShift 1 ) = 𝑊 ) → 0 ∈ ( 0 ..^ ( ♯ ‘ 𝑊 ) ) ) ) |
38 |
37
|
imp |
⊢ ( ( ( ¬ ( ♯ ‘ 𝑊 ) = 0 ∧ ¬ ( ♯ ‘ 𝑊 ) = 1 ) ∧ ( 𝑊 ∈ Word 𝑉 ∧ ( 𝑊 cyclShift 1 ) = 𝑊 ) ) → 0 ∈ ( 0 ..^ ( ♯ ‘ 𝑊 ) ) ) |
39 |
|
elfzoelz |
⊢ ( 1 ∈ ( 0 ..^ ( ♯ ‘ 𝑊 ) ) → 1 ∈ ℤ ) |
40 |
|
cshweqrep |
⊢ ( ( 𝑊 ∈ Word 𝑉 ∧ 1 ∈ ℤ ) → ( ( ( 𝑊 cyclShift 1 ) = 𝑊 ∧ 0 ∈ ( 0 ..^ ( ♯ ‘ 𝑊 ) ) ) → ∀ 𝑖 ∈ ℕ0 ( 𝑊 ‘ 0 ) = ( 𝑊 ‘ ( ( 0 + ( 𝑖 · 1 ) ) mod ( ♯ ‘ 𝑊 ) ) ) ) ) |
41 |
39 40
|
sylan2 |
⊢ ( ( 𝑊 ∈ Word 𝑉 ∧ 1 ∈ ( 0 ..^ ( ♯ ‘ 𝑊 ) ) ) → ( ( ( 𝑊 cyclShift 1 ) = 𝑊 ∧ 0 ∈ ( 0 ..^ ( ♯ ‘ 𝑊 ) ) ) → ∀ 𝑖 ∈ ℕ0 ( 𝑊 ‘ 0 ) = ( 𝑊 ‘ ( ( 0 + ( 𝑖 · 1 ) ) mod ( ♯ ‘ 𝑊 ) ) ) ) ) |
42 |
41
|
imp |
⊢ ( ( ( 𝑊 ∈ Word 𝑉 ∧ 1 ∈ ( 0 ..^ ( ♯ ‘ 𝑊 ) ) ) ∧ ( ( 𝑊 cyclShift 1 ) = 𝑊 ∧ 0 ∈ ( 0 ..^ ( ♯ ‘ 𝑊 ) ) ) ) → ∀ 𝑖 ∈ ℕ0 ( 𝑊 ‘ 0 ) = ( 𝑊 ‘ ( ( 0 + ( 𝑖 · 1 ) ) mod ( ♯ ‘ 𝑊 ) ) ) ) |
43 |
8 28 29 38 42
|
syl22anc |
⊢ ( ( ( ¬ ( ♯ ‘ 𝑊 ) = 0 ∧ ¬ ( ♯ ‘ 𝑊 ) = 1 ) ∧ ( 𝑊 ∈ Word 𝑉 ∧ ( 𝑊 cyclShift 1 ) = 𝑊 ) ) → ∀ 𝑖 ∈ ℕ0 ( 𝑊 ‘ 0 ) = ( 𝑊 ‘ ( ( 0 + ( 𝑖 · 1 ) ) mod ( ♯ ‘ 𝑊 ) ) ) ) |
44 |
|
0nn0 |
⊢ 0 ∈ ℕ0 |
45 |
|
fzossnn0 |
⊢ ( 0 ∈ ℕ0 → ( 0 ..^ ( ♯ ‘ 𝑊 ) ) ⊆ ℕ0 ) |
46 |
|
ssralv |
⊢ ( ( 0 ..^ ( ♯ ‘ 𝑊 ) ) ⊆ ℕ0 → ( ∀ 𝑖 ∈ ℕ0 ( 𝑊 ‘ 0 ) = ( 𝑊 ‘ ( ( 0 + ( 𝑖 · 1 ) ) mod ( ♯ ‘ 𝑊 ) ) ) → ∀ 𝑖 ∈ ( 0 ..^ ( ♯ ‘ 𝑊 ) ) ( 𝑊 ‘ 0 ) = ( 𝑊 ‘ ( ( 0 + ( 𝑖 · 1 ) ) mod ( ♯ ‘ 𝑊 ) ) ) ) ) |
47 |
44 45 46
|
mp2b |
⊢ ( ∀ 𝑖 ∈ ℕ0 ( 𝑊 ‘ 0 ) = ( 𝑊 ‘ ( ( 0 + ( 𝑖 · 1 ) ) mod ( ♯ ‘ 𝑊 ) ) ) → ∀ 𝑖 ∈ ( 0 ..^ ( ♯ ‘ 𝑊 ) ) ( 𝑊 ‘ 0 ) = ( 𝑊 ‘ ( ( 0 + ( 𝑖 · 1 ) ) mod ( ♯ ‘ 𝑊 ) ) ) ) |
48 |
|
eqcom |
⊢ ( ( 𝑊 ‘ 0 ) = ( 𝑊 ‘ ( ( 0 + ( 𝑖 · 1 ) ) mod ( ♯ ‘ 𝑊 ) ) ) ↔ ( 𝑊 ‘ ( ( 0 + ( 𝑖 · 1 ) ) mod ( ♯ ‘ 𝑊 ) ) ) = ( 𝑊 ‘ 0 ) ) |
49 |
|
elfzoelz |
⊢ ( 𝑖 ∈ ( 0 ..^ ( ♯ ‘ 𝑊 ) ) → 𝑖 ∈ ℤ ) |
50 |
|
zre |
⊢ ( 𝑖 ∈ ℤ → 𝑖 ∈ ℝ ) |
51 |
|
ax-1rid |
⊢ ( 𝑖 ∈ ℝ → ( 𝑖 · 1 ) = 𝑖 ) |
52 |
50 51
|
syl |
⊢ ( 𝑖 ∈ ℤ → ( 𝑖 · 1 ) = 𝑖 ) |
53 |
52
|
oveq2d |
⊢ ( 𝑖 ∈ ℤ → ( 0 + ( 𝑖 · 1 ) ) = ( 0 + 𝑖 ) ) |
54 |
|
zcn |
⊢ ( 𝑖 ∈ ℤ → 𝑖 ∈ ℂ ) |
55 |
54
|
addid2d |
⊢ ( 𝑖 ∈ ℤ → ( 0 + 𝑖 ) = 𝑖 ) |
56 |
53 55
|
eqtrd |
⊢ ( 𝑖 ∈ ℤ → ( 0 + ( 𝑖 · 1 ) ) = 𝑖 ) |
57 |
49 56
|
syl |
⊢ ( 𝑖 ∈ ( 0 ..^ ( ♯ ‘ 𝑊 ) ) → ( 0 + ( 𝑖 · 1 ) ) = 𝑖 ) |
58 |
57
|
oveq1d |
⊢ ( 𝑖 ∈ ( 0 ..^ ( ♯ ‘ 𝑊 ) ) → ( ( 0 + ( 𝑖 · 1 ) ) mod ( ♯ ‘ 𝑊 ) ) = ( 𝑖 mod ( ♯ ‘ 𝑊 ) ) ) |
59 |
|
zmodidfzoimp |
⊢ ( 𝑖 ∈ ( 0 ..^ ( ♯ ‘ 𝑊 ) ) → ( 𝑖 mod ( ♯ ‘ 𝑊 ) ) = 𝑖 ) |
60 |
58 59
|
eqtrd |
⊢ ( 𝑖 ∈ ( 0 ..^ ( ♯ ‘ 𝑊 ) ) → ( ( 0 + ( 𝑖 · 1 ) ) mod ( ♯ ‘ 𝑊 ) ) = 𝑖 ) |
61 |
60
|
fveqeq2d |
⊢ ( 𝑖 ∈ ( 0 ..^ ( ♯ ‘ 𝑊 ) ) → ( ( 𝑊 ‘ ( ( 0 + ( 𝑖 · 1 ) ) mod ( ♯ ‘ 𝑊 ) ) ) = ( 𝑊 ‘ 0 ) ↔ ( 𝑊 ‘ 𝑖 ) = ( 𝑊 ‘ 0 ) ) ) |
62 |
61
|
biimpd |
⊢ ( 𝑖 ∈ ( 0 ..^ ( ♯ ‘ 𝑊 ) ) → ( ( 𝑊 ‘ ( ( 0 + ( 𝑖 · 1 ) ) mod ( ♯ ‘ 𝑊 ) ) ) = ( 𝑊 ‘ 0 ) → ( 𝑊 ‘ 𝑖 ) = ( 𝑊 ‘ 0 ) ) ) |
63 |
48 62
|
syl5bi |
⊢ ( 𝑖 ∈ ( 0 ..^ ( ♯ ‘ 𝑊 ) ) → ( ( 𝑊 ‘ 0 ) = ( 𝑊 ‘ ( ( 0 + ( 𝑖 · 1 ) ) mod ( ♯ ‘ 𝑊 ) ) ) → ( 𝑊 ‘ 𝑖 ) = ( 𝑊 ‘ 0 ) ) ) |
64 |
63
|
ralimia |
⊢ ( ∀ 𝑖 ∈ ( 0 ..^ ( ♯ ‘ 𝑊 ) ) ( 𝑊 ‘ 0 ) = ( 𝑊 ‘ ( ( 0 + ( 𝑖 · 1 ) ) mod ( ♯ ‘ 𝑊 ) ) ) → ∀ 𝑖 ∈ ( 0 ..^ ( ♯ ‘ 𝑊 ) ) ( 𝑊 ‘ 𝑖 ) = ( 𝑊 ‘ 0 ) ) |
65 |
47 64
|
syl |
⊢ ( ∀ 𝑖 ∈ ℕ0 ( 𝑊 ‘ 0 ) = ( 𝑊 ‘ ( ( 0 + ( 𝑖 · 1 ) ) mod ( ♯ ‘ 𝑊 ) ) ) → ∀ 𝑖 ∈ ( 0 ..^ ( ♯ ‘ 𝑊 ) ) ( 𝑊 ‘ 𝑖 ) = ( 𝑊 ‘ 0 ) ) |
66 |
43 65
|
syl |
⊢ ( ( ( ¬ ( ♯ ‘ 𝑊 ) = 0 ∧ ¬ ( ♯ ‘ 𝑊 ) = 1 ) ∧ ( 𝑊 ∈ Word 𝑉 ∧ ( 𝑊 cyclShift 1 ) = 𝑊 ) ) → ∀ 𝑖 ∈ ( 0 ..^ ( ♯ ‘ 𝑊 ) ) ( 𝑊 ‘ 𝑖 ) = ( 𝑊 ‘ 0 ) ) |
67 |
66
|
ex |
⊢ ( ( ¬ ( ♯ ‘ 𝑊 ) = 0 ∧ ¬ ( ♯ ‘ 𝑊 ) = 1 ) → ( ( 𝑊 ∈ Word 𝑉 ∧ ( 𝑊 cyclShift 1 ) = 𝑊 ) → ∀ 𝑖 ∈ ( 0 ..^ ( ♯ ‘ 𝑊 ) ) ( 𝑊 ‘ 𝑖 ) = ( 𝑊 ‘ 0 ) ) ) |
68 |
67
|
impancom |
⊢ ( ( ¬ ( ♯ ‘ 𝑊 ) = 0 ∧ ( 𝑊 ∈ Word 𝑉 ∧ ( 𝑊 cyclShift 1 ) = 𝑊 ) ) → ( ¬ ( ♯ ‘ 𝑊 ) = 1 → ∀ 𝑖 ∈ ( 0 ..^ ( ♯ ‘ 𝑊 ) ) ( 𝑊 ‘ 𝑖 ) = ( 𝑊 ‘ 0 ) ) ) |
69 |
|
eqid |
⊢ ( 𝑊 ‘ 0 ) = ( 𝑊 ‘ 0 ) |
70 |
|
c0ex |
⊢ 0 ∈ V |
71 |
|
fveqeq2 |
⊢ ( 𝑖 = 0 → ( ( 𝑊 ‘ 𝑖 ) = ( 𝑊 ‘ 0 ) ↔ ( 𝑊 ‘ 0 ) = ( 𝑊 ‘ 0 ) ) ) |
72 |
70 71
|
ralsn |
⊢ ( ∀ 𝑖 ∈ { 0 } ( 𝑊 ‘ 𝑖 ) = ( 𝑊 ‘ 0 ) ↔ ( 𝑊 ‘ 0 ) = ( 𝑊 ‘ 0 ) ) |
73 |
69 72
|
mpbir |
⊢ ∀ 𝑖 ∈ { 0 } ( 𝑊 ‘ 𝑖 ) = ( 𝑊 ‘ 0 ) |
74 |
|
oveq2 |
⊢ ( ( ♯ ‘ 𝑊 ) = 1 → ( 0 ..^ ( ♯ ‘ 𝑊 ) ) = ( 0 ..^ 1 ) ) |
75 |
|
fzo01 |
⊢ ( 0 ..^ 1 ) = { 0 } |
76 |
74 75
|
eqtrdi |
⊢ ( ( ♯ ‘ 𝑊 ) = 1 → ( 0 ..^ ( ♯ ‘ 𝑊 ) ) = { 0 } ) |
77 |
76
|
raleqdv |
⊢ ( ( ♯ ‘ 𝑊 ) = 1 → ( ∀ 𝑖 ∈ ( 0 ..^ ( ♯ ‘ 𝑊 ) ) ( 𝑊 ‘ 𝑖 ) = ( 𝑊 ‘ 0 ) ↔ ∀ 𝑖 ∈ { 0 } ( 𝑊 ‘ 𝑖 ) = ( 𝑊 ‘ 0 ) ) ) |
78 |
73 77
|
mpbiri |
⊢ ( ( ♯ ‘ 𝑊 ) = 1 → ∀ 𝑖 ∈ ( 0 ..^ ( ♯ ‘ 𝑊 ) ) ( 𝑊 ‘ 𝑖 ) = ( 𝑊 ‘ 0 ) ) |
79 |
68 78
|
pm2.61d2 |
⊢ ( ( ¬ ( ♯ ‘ 𝑊 ) = 0 ∧ ( 𝑊 ∈ Word 𝑉 ∧ ( 𝑊 cyclShift 1 ) = 𝑊 ) ) → ∀ 𝑖 ∈ ( 0 ..^ ( ♯ ‘ 𝑊 ) ) ( 𝑊 ‘ 𝑖 ) = ( 𝑊 ‘ 0 ) ) |
80 |
79
|
ex |
⊢ ( ¬ ( ♯ ‘ 𝑊 ) = 0 → ( ( 𝑊 ∈ Word 𝑉 ∧ ( 𝑊 cyclShift 1 ) = 𝑊 ) → ∀ 𝑖 ∈ ( 0 ..^ ( ♯ ‘ 𝑊 ) ) ( 𝑊 ‘ 𝑖 ) = ( 𝑊 ‘ 0 ) ) ) |
81 |
7 80
|
pm2.61i |
⊢ ( ( 𝑊 ∈ Word 𝑉 ∧ ( 𝑊 cyclShift 1 ) = 𝑊 ) → ∀ 𝑖 ∈ ( 0 ..^ ( ♯ ‘ 𝑊 ) ) ( 𝑊 ‘ 𝑖 ) = ( 𝑊 ‘ 0 ) ) |