Step |
Hyp |
Ref |
Expression |
1 |
|
cshw1 |
⊢ ( ( 𝑊 ∈ Word 𝑉 ∧ ( 𝑊 cyclShift 1 ) = 𝑊 ) → ∀ 𝑖 ∈ ( 0 ..^ ( ♯ ‘ 𝑊 ) ) ( 𝑊 ‘ 𝑖 ) = ( 𝑊 ‘ 0 ) ) |
2 |
|
repswsymballbi |
⊢ ( 𝑊 ∈ Word 𝑉 → ( 𝑊 = ( ( 𝑊 ‘ 0 ) repeatS ( ♯ ‘ 𝑊 ) ) ↔ ∀ 𝑖 ∈ ( 0 ..^ ( ♯ ‘ 𝑊 ) ) ( 𝑊 ‘ 𝑖 ) = ( 𝑊 ‘ 0 ) ) ) |
3 |
2
|
bicomd |
⊢ ( 𝑊 ∈ Word 𝑉 → ( ∀ 𝑖 ∈ ( 0 ..^ ( ♯ ‘ 𝑊 ) ) ( 𝑊 ‘ 𝑖 ) = ( 𝑊 ‘ 0 ) ↔ 𝑊 = ( ( 𝑊 ‘ 0 ) repeatS ( ♯ ‘ 𝑊 ) ) ) ) |
4 |
3
|
adantr |
⊢ ( ( 𝑊 ∈ Word 𝑉 ∧ ( 𝑊 cyclShift 1 ) = 𝑊 ) → ( ∀ 𝑖 ∈ ( 0 ..^ ( ♯ ‘ 𝑊 ) ) ( 𝑊 ‘ 𝑖 ) = ( 𝑊 ‘ 0 ) ↔ 𝑊 = ( ( 𝑊 ‘ 0 ) repeatS ( ♯ ‘ 𝑊 ) ) ) ) |
5 |
1 4
|
mpbid |
⊢ ( ( 𝑊 ∈ Word 𝑉 ∧ ( 𝑊 cyclShift 1 ) = 𝑊 ) → 𝑊 = ( ( 𝑊 ‘ 0 ) repeatS ( ♯ ‘ 𝑊 ) ) ) |