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 ) ) ) )