Theorem upgrimwlklen

Description: Graph isomorphisms between simple pseudographs map walks onto walks of the same length. (Contributed by AV, 6-Nov-2025)

	Ref	Expression
Hypotheses	upgrimwlk.i	⊢ 𝐼 = ( iEdg ‘ 𝐺 )
	upgrimwlk.j	⊢ 𝐽 = ( iEdg ‘ 𝐻 )
	upgrimwlk.g	⊢ ( 𝜑 → 𝐺 ∈ USPGraph )
	upgrimwlk.h	⊢ ( 𝜑 → 𝐻 ∈ USPGraph )
	upgrimwlk.n	⊢ ( 𝜑 → 𝑁 ∈ ( 𝐺 GraphIso 𝐻 ) )
	upgrimwlk.e	⊢ 𝐸 = ( 𝑥 ∈ dom 𝐹 ↦ ( ◡ 𝐽 ‘ ( 𝑁 “ ( 𝐼 ‘ ( 𝐹 ‘ 𝑥 ) ) ) ) )
	upgrimwlk.w	⊢ ( 𝜑 → 𝐹 ( Walks ‘ 𝐺 ) 𝑃 )
Assertion	upgrimwlklen	⊢ ( 𝜑 → ( 𝐸 ( Walks ‘ 𝐻 ) ( 𝑁 ∘ 𝑃 ) ∧ ( ♯ ‘ 𝐸 ) = ( ♯ ‘ 𝐹 ) ) )

Proof

Step	Hyp	Ref	Expression
1		upgrimwlk.i	⊢ 𝐼 = ( iEdg ‘ 𝐺 )
2		upgrimwlk.j	⊢ 𝐽 = ( iEdg ‘ 𝐻 )
3		upgrimwlk.g	⊢ ( 𝜑 → 𝐺 ∈ USPGraph )
4		upgrimwlk.h	⊢ ( 𝜑 → 𝐻 ∈ USPGraph )
5		upgrimwlk.n	⊢ ( 𝜑 → 𝑁 ∈ ( 𝐺 GraphIso 𝐻 ) )
6		upgrimwlk.e	⊢ 𝐸 = ( 𝑥 ∈ dom 𝐹 ↦ ( ◡ 𝐽 ‘ ( 𝑁 “ ( 𝐼 ‘ ( 𝐹 ‘ 𝑥 ) ) ) ) )
7		upgrimwlk.w	⊢ ( 𝜑 → 𝐹 ( Walks ‘ 𝐺 ) 𝑃 )
8	1 2 3 4 5 6 7	upgrimwlk	⊢ ( 𝜑 → 𝐸 ( Walks ‘ 𝐻 ) ( 𝑁 ∘ 𝑃 ) )
9	1	wlkf	⊢ ( 𝐹 ( Walks ‘ 𝐺 ) 𝑃 → 𝐹 ∈ Word dom 𝐼 )
10	7 9	syl	⊢ ( 𝜑 → 𝐹 ∈ Word dom 𝐼 )
11	1 2 3 4 5 6 10	upgrimwlklem1	⊢ ( 𝜑 → ( ♯ ‘ 𝐸 ) = ( ♯ ‘ 𝐹 ) )
12	8 11	jca	⊢ ( 𝜑 → ( 𝐸 ( Walks ‘ 𝐻 ) ( 𝑁 ∘ 𝑃 ) ∧ ( ♯ ‘ 𝐸 ) = ( ♯ ‘ 𝐹 ) ) )