Theorem wlkcompim

Description: Implications for the properties of the components of a walk. (Contributed by Alexander van der Vekens, 23-Jun-2018) (Revised by AV, 2-Jan-2021)

	Ref	Expression
Hypotheses	wlkcomp.v	⊢ 𝑉 = ( Vtx ‘ 𝐺 )
	wlkcomp.i	⊢ 𝐼 = ( iEdg ‘ 𝐺 )
	wlkcomp.1	⊢ 𝐹 = ( 1st ‘ 𝑊 )
	wlkcomp.2	⊢ 𝑃 = ( 2nd ‘ 𝑊 )
Assertion	wlkcompim	⊢ ( 𝑊 ∈ ( Walks ‘ 𝐺 ) → ( 𝐹 ∈ Word dom 𝐼 ∧ 𝑃 : ( 0 ... ( ♯ ‘ 𝐹 ) ) ⟶ 𝑉 ∧ ∀ 𝑘 ∈ ( 0 ..^ ( ♯ ‘ 𝐹 ) ) if- ( ( 𝑃 ‘ 𝑘 ) = ( 𝑃 ‘ ( 𝑘 + 1 ) ) , ( 𝐼 ‘ ( 𝐹 ‘ 𝑘 ) ) = { ( 𝑃 ‘ 𝑘 ) } , { ( 𝑃 ‘ 𝑘 ) , ( 𝑃 ‘ ( 𝑘 + 1 ) ) } ⊆ ( 𝐼 ‘ ( 𝐹 ‘ 𝑘 ) ) ) ) )

Proof

Step	Hyp	Ref	Expression
1		wlkcomp.v	⊢ 𝑉 = ( Vtx ‘ 𝐺 )
2		wlkcomp.i	⊢ 𝐼 = ( iEdg ‘ 𝐺 )
3		wlkcomp.1	⊢ 𝐹 = ( 1st ‘ 𝑊 )
4		wlkcomp.2	⊢ 𝑃 = ( 2nd ‘ 𝑊 )
5		elfvex	⊢ ( 𝑊 ∈ ( Walks ‘ 𝐺 ) → 𝐺 ∈ V )
6		wlkcpr	⊢ ( 𝑊 ∈ ( Walks ‘ 𝐺 ) ↔ ( 1st ‘ 𝑊 ) ( Walks ‘ 𝐺 ) ( 2nd ‘ 𝑊 ) )
7		wlkvv	⊢ ( ( 1st ‘ 𝑊 ) ( Walks ‘ 𝐺 ) ( 2nd ‘ 𝑊 ) → 𝑊 ∈ ( V × V ) )
8	6 7	sylbi	⊢ ( 𝑊 ∈ ( Walks ‘ 𝐺 ) → 𝑊 ∈ ( V × V ) )
9	1 2 3 4	wlkcomp	⊢ ( ( 𝐺 ∈ V ∧ 𝑊 ∈ ( V × V ) ) → ( 𝑊 ∈ ( Walks ‘ 𝐺 ) ↔ ( 𝐹 ∈ Word dom 𝐼 ∧ 𝑃 : ( 0 ... ( ♯ ‘ 𝐹 ) ) ⟶ 𝑉 ∧ ∀ 𝑘 ∈ ( 0 ..^ ( ♯ ‘ 𝐹 ) ) if- ( ( 𝑃 ‘ 𝑘 ) = ( 𝑃 ‘ ( 𝑘 + 1 ) ) , ( 𝐼 ‘ ( 𝐹 ‘ 𝑘 ) ) = { ( 𝑃 ‘ 𝑘 ) } , { ( 𝑃 ‘ 𝑘 ) , ( 𝑃 ‘ ( 𝑘 + 1 ) ) } ⊆ ( 𝐼 ‘ ( 𝐹 ‘ 𝑘 ) ) ) ) ) )
10	9	biimpcd	⊢ ( 𝑊 ∈ ( Walks ‘ 𝐺 ) → ( ( 𝐺 ∈ V ∧ 𝑊 ∈ ( V × V ) ) → ( 𝐹 ∈ Word dom 𝐼 ∧ 𝑃 : ( 0 ... ( ♯ ‘ 𝐹 ) ) ⟶ 𝑉 ∧ ∀ 𝑘 ∈ ( 0 ..^ ( ♯ ‘ 𝐹 ) ) if- ( ( 𝑃 ‘ 𝑘 ) = ( 𝑃 ‘ ( 𝑘 + 1 ) ) , ( 𝐼 ‘ ( 𝐹 ‘ 𝑘 ) ) = { ( 𝑃 ‘ 𝑘 ) } , { ( 𝑃 ‘ 𝑘 ) , ( 𝑃 ‘ ( 𝑘 + 1 ) ) } ⊆ ( 𝐼 ‘ ( 𝐹 ‘ 𝑘 ) ) ) ) ) )
11	5 8 10	mp2and	⊢ ( 𝑊 ∈ ( Walks ‘ 𝐺 ) → ( 𝐹 ∈ Word dom 𝐼 ∧ 𝑃 : ( 0 ... ( ♯ ‘ 𝐹 ) ) ⟶ 𝑉 ∧ ∀ 𝑘 ∈ ( 0 ..^ ( ♯ ‘ 𝐹 ) ) if- ( ( 𝑃 ‘ 𝑘 ) = ( 𝑃 ‘ ( 𝑘 + 1 ) ) , ( 𝐼 ‘ ( 𝐹 ‘ 𝑘 ) ) = { ( 𝑃 ‘ 𝑘 ) } , { ( 𝑃 ‘ 𝑘 ) , ( 𝑃 ‘ ( 𝑘 + 1 ) ) } ⊆ ( 𝐼 ‘ ( 𝐹 ‘ 𝑘 ) ) ) ) )