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Metamath Proof Explorer

Theorem cyclispthon

Description: A cycle is a path starting and ending at its first vertex. (Contributed by Alexander van der Vekens, 8-Nov-2017) (Revised by AV, 31-Jan-2021) (Proof shortened by AV, 30-Oct-2021)

Ref Expression
Assertion cyclispthon ( 𝐹 ( Cycles ‘ 𝐺 ) 𝑃 → 𝐹 ( ( 𝑃 ‘ 0 ) ( PathsOn ‘ 𝐺 ) ( 𝑃 ‘ 0 ) ) 𝑃 )

Proof

Step Hyp Ref Expression
1 cyclispth ⊢ ( 𝐹 ( Cycles ‘ 𝐺 ) 𝑃 → 𝐹 ( Paths ‘ 𝐺 ) 𝑃 )
2 pthonpth ⊢ ( 𝐹 ( Paths ‘ 𝐺 ) 𝑃 → 𝐹 ( ( 𝑃 ‘ 0 ) ( PathsOn ‘ 𝐺 ) ( 𝑃 ‘ ( ♯ ‘ 𝐹 ) ) ) 𝑃 )
3 1 2 syl ⊢ ( 𝐹 ( Cycles ‘ 𝐺 ) 𝑃 → 𝐹 ( ( 𝑃 ‘ 0 ) ( PathsOn ‘ 𝐺 ) ( 𝑃 ‘ ( ♯ ‘ 𝐹 ) ) ) 𝑃 )
4 iscycl ⊢ ( 𝐹 ( Cycles ‘ 𝐺 ) 𝑃 ↔ ( 𝐹 ( Paths ‘ 𝐺 ) 𝑃 ∧ ( 𝑃 ‘ 0 ) = ( 𝑃 ‘ ( ♯ ‘ 𝐹 ) ) ) )
5 4 simprbi ⊢ ( 𝐹 ( Cycles ‘ 𝐺 ) 𝑃 → ( 𝑃 ‘ 0 ) = ( 𝑃 ‘ ( ♯ ‘ 𝐹 ) ) )
6 5 oveq2d ⊢ ( 𝐹 ( Cycles ‘ 𝐺 ) 𝑃 → ( ( 𝑃 ‘ 0 ) ( PathsOn ‘ 𝐺 ) ( 𝑃 ‘ 0 ) ) = ( ( 𝑃 ‘ 0 ) ( PathsOn ‘ 𝐺 ) ( 𝑃 ‘ ( ♯ ‘ 𝐹 ) ) ) )
7 6 breqd ⊢ ( 𝐹 ( Cycles ‘ 𝐺 ) 𝑃 → ( 𝐹 ( ( 𝑃 ‘ 0 ) ( PathsOn ‘ 𝐺 ) ( 𝑃 ‘ 0 ) ) 𝑃 ↔ 𝐹 ( ( 𝑃 ‘ 0 ) ( PathsOn ‘ 𝐺 ) ( 𝑃 ‘ ( ♯ ‘ 𝐹 ) ) ) 𝑃 ) )
8 3 7 mpbird ⊢ ( 𝐹 ( Cycles ‘ 𝐺 ) 𝑃 → 𝐹 ( ( 𝑃 ‘ 0 ) ( PathsOn ‘ 𝐺 ) ( 𝑃 ‘ 0 ) ) 𝑃 )