Theorem wwlknbp

Description: Basic properties of a walk of a fixed length (in an undirected graph) as word. (Contributed by Alexander van der Vekens, 16-Jul-2018) (Revised by AV, 9-Apr-2021) (Proof shortened by AV, 20-May-2021)

	Ref	Expression
Hypothesis	wwlkbp.v	⊢ 𝑉 = ( Vtx ‘ 𝐺 )
Assertion	wwlknbp	⊢ ( 𝑊 ∈ ( 𝑁 WWalksN 𝐺 ) → ( 𝐺 ∈ V ∧ 𝑁 ∈ ℕ0 ∧ 𝑊 ∈ Word 𝑉 ) )

Proof

Step	Hyp	Ref	Expression
1		wwlkbp.v	⊢ 𝑉 = ( Vtx ‘ 𝐺 )
2		df-wwlksn	⊢ WWalksN = ( 𝑛 ∈ ℕ0 , 𝑔 ∈ V ↦ { 𝑤 ∈ ( WWalks ‘ 𝑔 ) ∣ ( ♯ ‘ 𝑤 ) = ( 𝑛 + 1 ) } )
3	2	elmpocl	⊢ ( 𝑊 ∈ ( 𝑁 WWalksN 𝐺 ) → ( 𝑁 ∈ ℕ0 ∧ 𝐺 ∈ V ) )
4		simpl	⊢ ( ( ( 𝑁 ∈ ℕ0 ∧ 𝐺 ∈ V ) ∧ 𝑊 ∈ ( 𝑁 WWalksN 𝐺 ) ) → ( 𝑁 ∈ ℕ0 ∧ 𝐺 ∈ V ) )
5	4	ancomd	⊢ ( ( ( 𝑁 ∈ ℕ0 ∧ 𝐺 ∈ V ) ∧ 𝑊 ∈ ( 𝑁 WWalksN 𝐺 ) ) → ( 𝐺 ∈ V ∧ 𝑁 ∈ ℕ0 ) )
6		iswwlksn	⊢ ( 𝑁 ∈ ℕ0 → ( 𝑊 ∈ ( 𝑁 WWalksN 𝐺 ) ↔ ( 𝑊 ∈ ( WWalks ‘ 𝐺 ) ∧ ( ♯ ‘ 𝑊 ) = ( 𝑁 + 1 ) ) ) )
7	6	adantr	⊢ ( ( 𝑁 ∈ ℕ0 ∧ 𝐺 ∈ V ) → ( 𝑊 ∈ ( 𝑁 WWalksN 𝐺 ) ↔ ( 𝑊 ∈ ( WWalks ‘ 𝐺 ) ∧ ( ♯ ‘ 𝑊 ) = ( 𝑁 + 1 ) ) ) )
8	1	wwlkbp	⊢ ( 𝑊 ∈ ( WWalks ‘ 𝐺 ) → ( 𝐺 ∈ V ∧ 𝑊 ∈ Word 𝑉 ) )
9	8	simprd	⊢ ( 𝑊 ∈ ( WWalks ‘ 𝐺 ) → 𝑊 ∈ Word 𝑉 )
10	9	adantr	⊢ ( ( 𝑊 ∈ ( WWalks ‘ 𝐺 ) ∧ ( ♯ ‘ 𝑊 ) = ( 𝑁 + 1 ) ) → 𝑊 ∈ Word 𝑉 )
11	7 10	biimtrdi	⊢ ( ( 𝑁 ∈ ℕ0 ∧ 𝐺 ∈ V ) → ( 𝑊 ∈ ( 𝑁 WWalksN 𝐺 ) → 𝑊 ∈ Word 𝑉 ) )
12	11	imp	⊢ ( ( ( 𝑁 ∈ ℕ0 ∧ 𝐺 ∈ V ) ∧ 𝑊 ∈ ( 𝑁 WWalksN 𝐺 ) ) → 𝑊 ∈ Word 𝑉 )
13		df-3an	⊢ ( ( 𝐺 ∈ V ∧ 𝑁 ∈ ℕ0 ∧ 𝑊 ∈ Word 𝑉 ) ↔ ( ( 𝐺 ∈ V ∧ 𝑁 ∈ ℕ0 ) ∧ 𝑊 ∈ Word 𝑉 ) )
14	5 12 13	sylanbrc	⊢ ( ( ( 𝑁 ∈ ℕ0 ∧ 𝐺 ∈ V ) ∧ 𝑊 ∈ ( 𝑁 WWalksN 𝐺 ) ) → ( 𝐺 ∈ V ∧ 𝑁 ∈ ℕ0 ∧ 𝑊 ∈ Word 𝑉 ) )
15	3 14	mpancom	⊢ ( 𝑊 ∈ ( 𝑁 WWalksN 𝐺 ) → ( 𝐺 ∈ V ∧ 𝑁 ∈ ℕ0 ∧ 𝑊 ∈ Word 𝑉 ) )