Step |
Hyp |
Ref |
Expression |
1 |
|
clwlkclwwlklem2.f |
⊢ 𝐹 = ( 𝑥 ∈ ( 0 ..^ ( ( ♯ ‘ 𝑃 ) − 1 ) ) ↦ if ( 𝑥 < ( ( ♯ ‘ 𝑃 ) − 2 ) , ( ◡ 𝐸 ‘ { ( 𝑃 ‘ 𝑥 ) , ( 𝑃 ‘ ( 𝑥 + 1 ) ) } ) , ( ◡ 𝐸 ‘ { ( 𝑃 ‘ 𝑥 ) , ( 𝑃 ‘ 0 ) } ) ) ) |
2 |
|
lsw |
⊢ ( 𝑃 ∈ Word 𝑉 → ( lastS ‘ 𝑃 ) = ( 𝑃 ‘ ( ( ♯ ‘ 𝑃 ) − 1 ) ) ) |
3 |
2
|
adantr |
⊢ ( ( 𝑃 ∈ Word 𝑉 ∧ 2 ≤ ( ♯ ‘ 𝑃 ) ) → ( lastS ‘ 𝑃 ) = ( 𝑃 ‘ ( ( ♯ ‘ 𝑃 ) − 1 ) ) ) |
4 |
1
|
clwlkclwwlklem2a2 |
⊢ ( ( 𝑃 ∈ Word 𝑉 ∧ 2 ≤ ( ♯ ‘ 𝑃 ) ) → ( ♯ ‘ 𝐹 ) = ( ( ♯ ‘ 𝑃 ) − 1 ) ) |
5 |
4
|
eqcomd |
⊢ ( ( 𝑃 ∈ Word 𝑉 ∧ 2 ≤ ( ♯ ‘ 𝑃 ) ) → ( ( ♯ ‘ 𝑃 ) − 1 ) = ( ♯ ‘ 𝐹 ) ) |
6 |
5
|
fveq2d |
⊢ ( ( 𝑃 ∈ Word 𝑉 ∧ 2 ≤ ( ♯ ‘ 𝑃 ) ) → ( 𝑃 ‘ ( ( ♯ ‘ 𝑃 ) − 1 ) ) = ( 𝑃 ‘ ( ♯ ‘ 𝐹 ) ) ) |
7 |
3 6
|
eqtr2d |
⊢ ( ( 𝑃 ∈ Word 𝑉 ∧ 2 ≤ ( ♯ ‘ 𝑃 ) ) → ( 𝑃 ‘ ( ♯ ‘ 𝐹 ) ) = ( lastS ‘ 𝑃 ) ) |