Theorem ewlkprop

Description: Properties of an s-walk of edges. (Contributed by AV, 4-Jan-2021)

	Ref	Expression
Hypothesis	ewlksfval.i	⊢ 𝐼 = ( iEdg ‘ 𝐺 )
Assertion	ewlkprop	⊢ ( 𝐹 ∈ ( 𝐺 EdgWalks 𝑆 ) → ( ( 𝐺 ∈ V ∧ 𝑆 ∈ ℕ0* ) ∧ 𝐹 ∈ Word dom 𝐼 ∧ ∀ 𝑘 ∈ ( 1 ..^ ( ♯ ‘ 𝐹 ) ) 𝑆 ≤ ( ♯ ‘ ( ( 𝐼 ‘ ( 𝐹 ‘ ( 𝑘 − 1 ) ) ) ∩ ( 𝐼 ‘ ( 𝐹 ‘ 𝑘 ) ) ) ) ) )

Proof

Step	Hyp	Ref	Expression
1		ewlksfval.i	⊢ 𝐼 = ( iEdg ‘ 𝐺 )
2		df-ewlks	⊢ EdgWalks = ( 𝑔 ∈ V , 𝑠 ∈ ℕ0* ↦ { 𝑓 ∣ [ ( iEdg ‘ 𝑔 ) / 𝑖 ] ( 𝑓 ∈ Word dom 𝑖 ∧ ∀ 𝑘 ∈ ( 1 ..^ ( ♯ ‘ 𝑓 ) ) 𝑠 ≤ ( ♯ ‘ ( ( 𝑖 ‘ ( 𝑓 ‘ ( 𝑘 − 1 ) ) ) ∩ ( 𝑖 ‘ ( 𝑓 ‘ 𝑘 ) ) ) ) ) } )
3	2	elmpocl	⊢ ( 𝐹 ∈ ( 𝐺 EdgWalks 𝑆 ) → ( 𝐺 ∈ V ∧ 𝑆 ∈ ℕ0* ) )
4		simpr	⊢ ( ( 𝐹 ∈ ( 𝐺 EdgWalks 𝑆 ) ∧ ( 𝐺 ∈ V ∧ 𝑆 ∈ ℕ0* ) ) → ( 𝐺 ∈ V ∧ 𝑆 ∈ ℕ0* ) )
5	1	isewlk	⊢ ( ( 𝐺 ∈ V ∧ 𝑆 ∈ ℕ0* ∧ 𝐹 ∈ ( 𝐺 EdgWalks 𝑆 ) ) → ( 𝐹 ∈ ( 𝐺 EdgWalks 𝑆 ) ↔ ( 𝐹 ∈ Word dom 𝐼 ∧ ∀ 𝑘 ∈ ( 1 ..^ ( ♯ ‘ 𝐹 ) ) 𝑆 ≤ ( ♯ ‘ ( ( 𝐼 ‘ ( 𝐹 ‘ ( 𝑘 − 1 ) ) ) ∩ ( 𝐼 ‘ ( 𝐹 ‘ 𝑘 ) ) ) ) ) ) )
6	5	3expa	⊢ ( ( ( 𝐺 ∈ V ∧ 𝑆 ∈ ℕ0* ) ∧ 𝐹 ∈ ( 𝐺 EdgWalks 𝑆 ) ) → ( 𝐹 ∈ ( 𝐺 EdgWalks 𝑆 ) ↔ ( 𝐹 ∈ Word dom 𝐼 ∧ ∀ 𝑘 ∈ ( 1 ..^ ( ♯ ‘ 𝐹 ) ) 𝑆 ≤ ( ♯ ‘ ( ( 𝐼 ‘ ( 𝐹 ‘ ( 𝑘 − 1 ) ) ) ∩ ( 𝐼 ‘ ( 𝐹 ‘ 𝑘 ) ) ) ) ) ) )
7	6	biimpd	⊢ ( ( ( 𝐺 ∈ V ∧ 𝑆 ∈ ℕ0* ) ∧ 𝐹 ∈ ( 𝐺 EdgWalks 𝑆 ) ) → ( 𝐹 ∈ ( 𝐺 EdgWalks 𝑆 ) → ( 𝐹 ∈ Word dom 𝐼 ∧ ∀ 𝑘 ∈ ( 1 ..^ ( ♯ ‘ 𝐹 ) ) 𝑆 ≤ ( ♯ ‘ ( ( 𝐼 ‘ ( 𝐹 ‘ ( 𝑘 − 1 ) ) ) ∩ ( 𝐼 ‘ ( 𝐹 ‘ 𝑘 ) ) ) ) ) ) )
8	7	expcom	⊢ ( 𝐹 ∈ ( 𝐺 EdgWalks 𝑆 ) → ( ( 𝐺 ∈ V ∧ 𝑆 ∈ ℕ0* ) → ( 𝐹 ∈ ( 𝐺 EdgWalks 𝑆 ) → ( 𝐹 ∈ Word dom 𝐼 ∧ ∀ 𝑘 ∈ ( 1 ..^ ( ♯ ‘ 𝐹 ) ) 𝑆 ≤ ( ♯ ‘ ( ( 𝐼 ‘ ( 𝐹 ‘ ( 𝑘 − 1 ) ) ) ∩ ( 𝐼 ‘ ( 𝐹 ‘ 𝑘 ) ) ) ) ) ) ) )
9	8	pm2.43a	⊢ ( 𝐹 ∈ ( 𝐺 EdgWalks 𝑆 ) → ( ( 𝐺 ∈ V ∧ 𝑆 ∈ ℕ0* ) → ( 𝐹 ∈ Word dom 𝐼 ∧ ∀ 𝑘 ∈ ( 1 ..^ ( ♯ ‘ 𝐹 ) ) 𝑆 ≤ ( ♯ ‘ ( ( 𝐼 ‘ ( 𝐹 ‘ ( 𝑘 − 1 ) ) ) ∩ ( 𝐼 ‘ ( 𝐹 ‘ 𝑘 ) ) ) ) ) ) )
10	9	imp	⊢ ( ( 𝐹 ∈ ( 𝐺 EdgWalks 𝑆 ) ∧ ( 𝐺 ∈ V ∧ 𝑆 ∈ ℕ0* ) ) → ( 𝐹 ∈ Word dom 𝐼 ∧ ∀ 𝑘 ∈ ( 1 ..^ ( ♯ ‘ 𝐹 ) ) 𝑆 ≤ ( ♯ ‘ ( ( 𝐼 ‘ ( 𝐹 ‘ ( 𝑘 − 1 ) ) ) ∩ ( 𝐼 ‘ ( 𝐹 ‘ 𝑘 ) ) ) ) ) )
11		3anass	⊢ ( ( ( 𝐺 ∈ V ∧ 𝑆 ∈ ℕ0* ) ∧ 𝐹 ∈ Word dom 𝐼 ∧ ∀ 𝑘 ∈ ( 1 ..^ ( ♯ ‘ 𝐹 ) ) 𝑆 ≤ ( ♯ ‘ ( ( 𝐼 ‘ ( 𝐹 ‘ ( 𝑘 − 1 ) ) ) ∩ ( 𝐼 ‘ ( 𝐹 ‘ 𝑘 ) ) ) ) ) ↔ ( ( 𝐺 ∈ V ∧ 𝑆 ∈ ℕ0* ) ∧ ( 𝐹 ∈ Word dom 𝐼 ∧ ∀ 𝑘 ∈ ( 1 ..^ ( ♯ ‘ 𝐹 ) ) 𝑆 ≤ ( ♯ ‘ ( ( 𝐼 ‘ ( 𝐹 ‘ ( 𝑘 − 1 ) ) ) ∩ ( 𝐼 ‘ ( 𝐹 ‘ 𝑘 ) ) ) ) ) ) )
12	4 10 11	sylanbrc	⊢ ( ( 𝐹 ∈ ( 𝐺 EdgWalks 𝑆 ) ∧ ( 𝐺 ∈ V ∧ 𝑆 ∈ ℕ0* ) ) → ( ( 𝐺 ∈ V ∧ 𝑆 ∈ ℕ0* ) ∧ 𝐹 ∈ Word dom 𝐼 ∧ ∀ 𝑘 ∈ ( 1 ..^ ( ♯ ‘ 𝐹 ) ) 𝑆 ≤ ( ♯ ‘ ( ( 𝐼 ‘ ( 𝐹 ‘ ( 𝑘 − 1 ) ) ) ∩ ( 𝐼 ‘ ( 𝐹 ‘ 𝑘 ) ) ) ) ) )
13	3 12	mpdan	⊢ ( 𝐹 ∈ ( 𝐺 EdgWalks 𝑆 ) → ( ( 𝐺 ∈ V ∧ 𝑆 ∈ ℕ0* ) ∧ 𝐹 ∈ Word dom 𝐼 ∧ ∀ 𝑘 ∈ ( 1 ..^ ( ♯ ‘ 𝐹 ) ) 𝑆 ≤ ( ♯ ‘ ( ( 𝐼 ‘ ( 𝐹 ‘ ( 𝑘 − 1 ) ) ) ∩ ( 𝐼 ‘ ( 𝐹 ‘ 𝑘 ) ) ) ) ) )