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Metamath Proof Explorer

Theorem wwlkseq

Description: Equality of two walks (as words). (Contributed by Alexander van der Vekens, 4-Aug-2018) (Revised by AV, 16-Apr-2021)

Ref Expression
Assertion wwlkseq ( ( 𝑊 ∈ ( WWalks ‘ 𝐺 ) ∧ 𝑇 ∈ ( WWalks ‘ 𝐺 ) ) → ( 𝑊 = 𝑇 ↔ ( ( ♯ ‘ 𝑊 ) = ( ♯ ‘ 𝑇 ) ∧ ∀ 𝑖 ∈ ( 0 ..^ ( ♯ ‘ 𝑊 ) ) ( 𝑊 ‘ 𝑖 ) = ( 𝑇 ‘ 𝑖 ) ) ) )

Proof

Step Hyp Ref Expression
1 eqid ⊢ ( Vtx ‘ 𝐺 ) = ( Vtx ‘ 𝐺 )
2 1 wwlkbp ⊢ ( 𝑊 ∈ ( WWalks ‘ 𝐺 ) → ( 𝐺 ∈ V ∧ 𝑊 ∈ Word ( Vtx ‘ 𝐺 ) ) )
3 2 simprd ⊢ ( 𝑊 ∈ ( WWalks ‘ 𝐺 ) → 𝑊 ∈ Word ( Vtx ‘ 𝐺 ) )
4 1 wwlkbp ⊢ ( 𝑇 ∈ ( WWalks ‘ 𝐺 ) → ( 𝐺 ∈ V ∧ 𝑇 ∈ Word ( Vtx ‘ 𝐺 ) ) )
5 4 simprd ⊢ ( 𝑇 ∈ ( WWalks ‘ 𝐺 ) → 𝑇 ∈ Word ( Vtx ‘ 𝐺 ) )
6 eqwrd ⊢ ( ( 𝑊 ∈ Word ( Vtx ‘ 𝐺 ) ∧ 𝑇 ∈ Word ( Vtx ‘ 𝐺 ) ) → ( 𝑊 = 𝑇 ↔ ( ( ♯ ‘ 𝑊 ) = ( ♯ ‘ 𝑇 ) ∧ ∀ 𝑖 ∈ ( 0 ..^ ( ♯ ‘ 𝑊 ) ) ( 𝑊 ‘ 𝑖 ) = ( 𝑇 ‘ 𝑖 ) ) ) )
7 3 5 6 syl2an ⊢ ( ( 𝑊 ∈ ( WWalks ‘ 𝐺 ) ∧ 𝑇 ∈ ( WWalks ‘ 𝐺 ) ) → ( 𝑊 = 𝑇 ↔ ( ( ♯ ‘ 𝑊 ) = ( ♯ ‘ 𝑇 ) ∧ ∀ 𝑖 ∈ ( 0 ..^ ( ♯ ‘ 𝑊 ) ) ( 𝑊 ‘ 𝑖 ) = ( 𝑇 ‘ 𝑖 ) ) ) )