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 )