Theorem clwlkcomp

Description: A closed walk expressed by properties of its components. (Contributed by Alexander van der Vekens, 24-Jun-2018) (Revised by AV, 17-Feb-2021)

	Ref	Expression
Hypotheses	isclwlke.v	⊢ 𝑉 = ( Vtx ‘ 𝐺 )
	isclwlke.i	⊢ 𝐼 = ( iEdg ‘ 𝐺 )
	clwlkcomp.1	⊢ 𝐹 = ( 1st ‘ 𝑊 )
	clwlkcomp.2	⊢ 𝑃 = ( 2nd ‘ 𝑊 )
Assertion	clwlkcomp	⊢ ( ( 𝐺 ∈ 𝑋 ∧ 𝑊 ∈ ( 𝑆 × 𝑇 ) ) → ( 𝑊 ∈ ( ClWalks ‘ 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 ) ) ) ∧ ( 𝑃 ‘ 0 ) = ( 𝑃 ‘ ( ♯ ‘ 𝐹 ) ) ) ) ) )

Proof

Step	Hyp	Ref	Expression
1		isclwlke.v	⊢ 𝑉 = ( Vtx ‘ 𝐺 )
2		isclwlke.i	⊢ 𝐼 = ( iEdg ‘ 𝐺 )
3		clwlkcomp.1	⊢ 𝐹 = ( 1st ‘ 𝑊 )
4		clwlkcomp.2	⊢ 𝑃 = ( 2nd ‘ 𝑊 )
5	3	eqcomi	⊢ ( 1st ‘ 𝑊 ) = 𝐹
6	4	eqcomi	⊢ ( 2nd ‘ 𝑊 ) = 𝑃
7	5 6	pm3.2i	⊢ ( ( 1st ‘ 𝑊 ) = 𝐹 ∧ ( 2nd ‘ 𝑊 ) = 𝑃 )
8		eqop	⊢ ( 𝑊 ∈ ( 𝑆 × 𝑇 ) → ( 𝑊 = ⟨ 𝐹 , 𝑃 ⟩ ↔ ( ( 1st ‘ 𝑊 ) = 𝐹 ∧ ( 2nd ‘ 𝑊 ) = 𝑃 ) ) )
9	7 8	mpbiri	⊢ ( 𝑊 ∈ ( 𝑆 × 𝑇 ) → 𝑊 = ⟨ 𝐹 , 𝑃 ⟩ )
10	9	eleq1d	⊢ ( 𝑊 ∈ ( 𝑆 × 𝑇 ) → ( 𝑊 ∈ ( ClWalks ‘ 𝐺 ) ↔ ⟨ 𝐹 , 𝑃 ⟩ ∈ ( ClWalks ‘ 𝐺 ) ) )
11		df-br	⊢ ( 𝐹 ( ClWalks ‘ 𝐺 ) 𝑃 ↔ ⟨ 𝐹 , 𝑃 ⟩ ∈ ( ClWalks ‘ 𝐺 ) )
12	10 11	bitr4di	⊢ ( 𝑊 ∈ ( 𝑆 × 𝑇 ) → ( 𝑊 ∈ ( ClWalks ‘ 𝐺 ) ↔ 𝐹 ( ClWalks ‘ 𝐺 ) 𝑃 ) )
13	1 2	isclwlke	⊢ ( 𝐺 ∈ 𝑋 → ( 𝐹 ( ClWalks ‘ 𝐺 ) 𝑃 ↔ ( ( 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 ) ) ) ∧ ( 𝑃 ‘ 0 ) = ( 𝑃 ‘ ( ♯ ‘ 𝐹 ) ) ) ) ) )
14	12 13	sylan9bbr	⊢ ( ( 𝐺 ∈ 𝑋 ∧ 𝑊 ∈ ( 𝑆 × 𝑇 ) ) → ( 𝑊 ∈ ( ClWalks ‘ 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 ) ) ) ∧ ( 𝑃 ‘ 0 ) = ( 𝑃 ‘ ( ♯ ‘ 𝐹 ) ) ) ) ) )