Theorem clwwlknwwlkncl

Description: Obtaining a closed walk (as word) by appending the first symbol to the word representing a walk. (Contributed by Alexander van der Vekens, 29-Sep-2018) (Revised by AV, 26-Apr-2021) (Revised by AV, 22-Mar-2022)

	Ref	Expression
Assertion	clwwlknwwlkncl	⊢ ( 𝑊 ∈ ( 𝑁 ClWWalksN 𝐺 ) → ( 𝑊 ++ ⟨“ ( 𝑊 ‘ 0 ) ”⟩ ) ∈ { 𝑤 ∈ ( 𝑁 WWalksN 𝐺 ) ∣ ( lastS ‘ 𝑤 ) = ( 𝑤 ‘ 0 ) } )

Proof

Step	Hyp	Ref	Expression
1		clwwlknnn	⊢ ( 𝑊 ∈ ( 𝑁 ClWWalksN 𝐺 ) → 𝑁 ∈ ℕ )
2		eqid	⊢ ( Vtx ‘ 𝐺 ) = ( Vtx ‘ 𝐺 )
3	2	clwwlknbp	⊢ ( 𝑊 ∈ ( 𝑁 ClWWalksN 𝐺 ) → ( 𝑊 ∈ Word ( Vtx ‘ 𝐺 ) ∧ ( ♯ ‘ 𝑊 ) = 𝑁 ) )
4		eqid	⊢ ( Edg ‘ 𝐺 ) = ( Edg ‘ 𝐺 )
5	2 4	clwwlknp	⊢ ( 𝑊 ∈ ( 𝑁 ClWWalksN 𝐺 ) → ( ( 𝑊 ∈ Word ( Vtx ‘ 𝐺 ) ∧ ( ♯ ‘ 𝑊 ) = 𝑁 ) ∧ ∀ 𝑖 ∈ ( 0 ..^ ( 𝑁 − 1 ) ) { ( 𝑊 ‘ 𝑖 ) , ( 𝑊 ‘ ( 𝑖 + 1 ) ) } ∈ ( Edg ‘ 𝐺 ) ∧ { ( lastS ‘ 𝑊 ) , ( 𝑊 ‘ 0 ) } ∈ ( Edg ‘ 𝐺 ) ) )
6		3simpc	⊢ ( ( ( 𝑊 ∈ Word ( Vtx ‘ 𝐺 ) ∧ ( ♯ ‘ 𝑊 ) = 𝑁 ) ∧ ∀ 𝑖 ∈ ( 0 ..^ ( 𝑁 − 1 ) ) { ( 𝑊 ‘ 𝑖 ) , ( 𝑊 ‘ ( 𝑖 + 1 ) ) } ∈ ( Edg ‘ 𝐺 ) ∧ { ( lastS ‘ 𝑊 ) , ( 𝑊 ‘ 0 ) } ∈ ( Edg ‘ 𝐺 ) ) → ( ∀ 𝑖 ∈ ( 0 ..^ ( 𝑁 − 1 ) ) { ( 𝑊 ‘ 𝑖 ) , ( 𝑊 ‘ ( 𝑖 + 1 ) ) } ∈ ( Edg ‘ 𝐺 ) ∧ { ( lastS ‘ 𝑊 ) , ( 𝑊 ‘ 0 ) } ∈ ( Edg ‘ 𝐺 ) ) )
7	5 6	syl	⊢ ( 𝑊 ∈ ( 𝑁 ClWWalksN 𝐺 ) → ( ∀ 𝑖 ∈ ( 0 ..^ ( 𝑁 − 1 ) ) { ( 𝑊 ‘ 𝑖 ) , ( 𝑊 ‘ ( 𝑖 + 1 ) ) } ∈ ( Edg ‘ 𝐺 ) ∧ { ( lastS ‘ 𝑊 ) , ( 𝑊 ‘ 0 ) } ∈ ( Edg ‘ 𝐺 ) ) )
8		eqid	⊢ { 𝑤 ∈ ( 𝑁 WWalksN 𝐺 ) ∣ ( lastS ‘ 𝑤 ) = ( 𝑤 ‘ 0 ) } = { 𝑤 ∈ ( 𝑁 WWalksN 𝐺 ) ∣ ( lastS ‘ 𝑤 ) = ( 𝑤 ‘ 0 ) }
9	8	clwwlkel	⊢ ( ( 𝑁 ∈ ℕ ∧ ( 𝑊 ∈ Word ( Vtx ‘ 𝐺 ) ∧ ( ♯ ‘ 𝑊 ) = 𝑁 ) ∧ ( ∀ 𝑖 ∈ ( 0 ..^ ( 𝑁 − 1 ) ) { ( 𝑊 ‘ 𝑖 ) , ( 𝑊 ‘ ( 𝑖 + 1 ) ) } ∈ ( Edg ‘ 𝐺 ) ∧ { ( lastS ‘ 𝑊 ) , ( 𝑊 ‘ 0 ) } ∈ ( Edg ‘ 𝐺 ) ) ) → ( 𝑊 ++ ⟨“ ( 𝑊 ‘ 0 ) ”⟩ ) ∈ { 𝑤 ∈ ( 𝑁 WWalksN 𝐺 ) ∣ ( lastS ‘ 𝑤 ) = ( 𝑤 ‘ 0 ) } )
10	1 3 7 9	syl3anc	⊢ ( 𝑊 ∈ ( 𝑁 ClWWalksN 𝐺 ) → ( 𝑊 ++ ⟨“ ( 𝑊 ‘ 0 ) ”⟩ ) ∈ { 𝑤 ∈ ( 𝑁 WWalksN 𝐺 ) ∣ ( lastS ‘ 𝑤 ) = ( 𝑤 ‘ 0 ) } )