Theorem eupthcl

Description: An Eulerian path has length # ( F ) , which is an integer. (Contributed by Mario Carneiro, 12-Mar-2015) (Revised by AV, 18-Feb-2021)

	Ref	Expression
Assertion	eupthcl	⊢ ( 𝐹 ( EulerPaths ‘ 𝐺 ) 𝑃 → ( ♯ ‘ 𝐹 ) ∈ ℕ0 )

Proof

Step	Hyp	Ref	Expression
1		eqid	⊢ ( iEdg ‘ 𝐺 ) = ( iEdg ‘ 𝐺 )
2	1	eupthi	⊢ ( 𝐹 ( EulerPaths ‘ 𝐺 ) 𝑃 → ( 𝐹 ( Walks ‘ 𝐺 ) 𝑃 ∧ 𝐹 : ( 0 ..^ ( ♯ ‘ 𝐹 ) ) –1-1-onto→ dom ( iEdg ‘ 𝐺 ) ) )
3		wlkcl	⊢ ( 𝐹 ( Walks ‘ 𝐺 ) 𝑃 → ( ♯ ‘ 𝐹 ) ∈ ℕ0 )
4	3	adantr	⊢ ( ( 𝐹 ( Walks ‘ 𝐺 ) 𝑃 ∧ 𝐹 : ( 0 ..^ ( ♯ ‘ 𝐹 ) ) –1-1-onto→ dom ( iEdg ‘ 𝐺 ) ) → ( ♯ ‘ 𝐹 ) ∈ ℕ0 )
5	2 4	syl	⊢ ( 𝐹 ( EulerPaths ‘ 𝐺 ) 𝑃 → ( ♯ ‘ 𝐹 ) ∈ ℕ0 )