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Metamath Proof Explorer

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 ) } )