Metamath Proof Explorer

Theorem clwlkcompim

Description: Implications for the properties of the components of a closed walk. (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 = ( 1^st ‘ 𝑊 )
		clwlkcomp.2	⊢ 𝑃 = ( 2^nd ‘ 𝑊 )
	Assertion	clwlkcompim	⊢ ( 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 = ( 1^st ‘ 𝑊 )
4		clwlkcomp.2	⊢ 𝑃 = ( 2^nd ‘ 𝑊 )
5		elfvex	⊢ ( 𝑊 ∈ ( ClWalks ‘ 𝐺 ) → 𝐺 ∈ V )
6		clwlks	⊢ ( ClWalks ‘ 𝐺 ) = { ⟨ 𝑓 , 𝑔 ⟩ ∣ ( 𝑓 ( Walks ‘ 𝐺 ) 𝑔 ∧ ( 𝑔 ‘ 0 ) = ( 𝑔 ‘ ( ♯ ‘ 𝑓 ) ) ) }
7	6	a1i	⊢ ( 𝐺 ∈ V → ( ClWalks ‘ 𝐺 ) = { ⟨ 𝑓 , 𝑔 ⟩ ∣ ( 𝑓 ( Walks ‘ 𝐺 ) 𝑔 ∧ ( 𝑔 ‘ 0 ) = ( 𝑔 ‘ ( ♯ ‘ 𝑓 ) ) ) } )
8	7	eleq2d	⊢ ( 𝐺 ∈ V → ( 𝑊 ∈ ( ClWalks ‘ 𝐺 ) ↔ 𝑊 ∈ { ⟨ 𝑓 , 𝑔 ⟩ ∣ ( 𝑓 ( Walks ‘ 𝐺 ) 𝑔 ∧ ( 𝑔 ‘ 0 ) = ( 𝑔 ‘ ( ♯ ‘ 𝑓 ) ) ) } ) )
9		elopaelxp	⊢ ( 𝑊 ∈ { ⟨ 𝑓 , 𝑔 ⟩ ∣ ( 𝑓 ( Walks ‘ 𝐺 ) 𝑔 ∧ ( 𝑔 ‘ 0 ) = ( 𝑔 ‘ ( ♯ ‘ 𝑓 ) ) ) } → 𝑊 ∈ ( V × V ) )
10	9	anim2i	⊢ ( ( 𝐺 ∈ V ∧ 𝑊 ∈ { ⟨ 𝑓 , 𝑔 ⟩ ∣ ( 𝑓 ( Walks ‘ 𝐺 ) 𝑔 ∧ ( 𝑔 ‘ 0 ) = ( 𝑔 ‘ ( ♯ ‘ 𝑓 ) ) ) } ) → ( 𝐺 ∈ V ∧ 𝑊 ∈ ( V × V ) ) )
11	10	ex	⊢ ( 𝐺 ∈ V → ( 𝑊 ∈ { ⟨ 𝑓 , 𝑔 ⟩ ∣ ( 𝑓 ( Walks ‘ 𝐺 ) 𝑔 ∧ ( 𝑔 ‘ 0 ) = ( 𝑔 ‘ ( ♯ ‘ 𝑓 ) ) ) } → ( 𝐺 ∈ V ∧ 𝑊 ∈ ( V × V ) ) ) )
12	8 11	sylbid	⊢ ( 𝐺 ∈ V → ( 𝑊 ∈ ( ClWalks ‘ 𝐺 ) → ( 𝐺 ∈ V ∧ 𝑊 ∈ ( V × V ) ) ) )
13	5 12	mpcom	⊢ ( 𝑊 ∈ ( ClWalks ‘ 𝐺 ) → ( 𝐺 ∈ V ∧ 𝑊 ∈ ( V × V ) ) )
14	1 2 3 4	clwlkcomp	⊢ ( ( 𝐺 ∈ V ∧ 𝑊 ∈ ( V × V ) ) → ( 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 ) ) ) ) ) )
15	13 14	syl	⊢ ( 𝑊 ∈ ( ClWalks ‘ 𝐺 ) → ( 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 ) ) ) ) ) )
16	15	ibi	⊢ ( 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 ) ) ) ) )