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Metamath Proof Explorer

Theorem crctisclwlk

Description: A circuit is a closed walk. (Contributed by AV, 17-Feb-2021) (Proof shortened by AV, 30-Oct-2021)

Ref Expression
Assertion crctisclwlk ( 𝐹 ( Circuits ‘ 𝐺 ) 𝑃 → 𝐹 ( ClWalks ‘ 𝐺 ) 𝑃 )

Proof

Step Hyp Ref Expression
1 crctprop ⊢ ( 𝐹 ( Circuits ‘ 𝐺 ) 𝑃 → ( 𝐹 ( Trails ‘ 𝐺 ) 𝑃 ∧ ( 𝑃 ‘ 0 ) = ( 𝑃 ‘ ( ♯ ‘ 𝐹 ) ) ) )
2 trliswlk ⊢ ( 𝐹 ( Trails ‘ 𝐺 ) 𝑃 → 𝐹 ( Walks ‘ 𝐺 ) 𝑃 )
3 isclwlk ⊢ ( 𝐹 ( ClWalks ‘ 𝐺 ) 𝑃 ↔ ( 𝐹 ( Walks ‘ 𝐺 ) 𝑃 ∧ ( 𝑃 ‘ 0 ) = ( 𝑃 ‘ ( ♯ ‘ 𝐹 ) ) ) )
4 3 biimpri ⊢ ( ( 𝐹 ( Walks ‘ 𝐺 ) 𝑃 ∧ ( 𝑃 ‘ 0 ) = ( 𝑃 ‘ ( ♯ ‘ 𝐹 ) ) ) → 𝐹 ( ClWalks ‘ 𝐺 ) 𝑃 )
5 2 4 sylan ⊢ ( ( 𝐹 ( Trails ‘ 𝐺 ) 𝑃 ∧ ( 𝑃 ‘ 0 ) = ( 𝑃 ‘ ( ♯ ‘ 𝐹 ) ) ) → 𝐹 ( ClWalks ‘ 𝐺 ) 𝑃 )
6 1 5 syl ⊢ ( 𝐹 ( Circuits ‘ 𝐺 ) 𝑃 → 𝐹 ( ClWalks ‘ 𝐺 ) 𝑃 )