Theorem clwwlknp

Description: Properties of a set being a closed walk (represented by a word). (Contributed by Alexander van der Vekens, 17-Jun-2018) (Revised by AV, 24-Apr-2021) (Proof shortened by AV, 23-Mar-2022)

	Ref	Expression
Hypotheses	isclwwlknx.v	⊢ 𝑉 = ( Vtx ‘ 𝐺 )
	isclwwlknx.e	⊢ 𝐸 = ( Edg ‘ 𝐺 )
Assertion	clwwlknp	⊢ ( 𝑊 ∈ ( 𝑁 ClWWalksN 𝐺 ) → ( ( 𝑊 ∈ Word 𝑉 ∧ ( ♯ ‘ 𝑊 ) = 𝑁 ) ∧ ∀ 𝑖 ∈ ( 0 ..^ ( 𝑁 − 1 ) ) { ( 𝑊 ‘ 𝑖 ) , ( 𝑊 ‘ ( 𝑖 + 1 ) ) } ∈ 𝐸 ∧ { ( lastS ‘ 𝑊 ) , ( 𝑊 ‘ 0 ) } ∈ 𝐸 ) )

Proof

Step	Hyp	Ref	Expression
1		isclwwlknx.v	⊢ 𝑉 = ( Vtx ‘ 𝐺 )
2		isclwwlknx.e	⊢ 𝐸 = ( Edg ‘ 𝐺 )
3	1	clwwlknbp	⊢ ( 𝑊 ∈ ( 𝑁 ClWWalksN 𝐺 ) → ( 𝑊 ∈ Word 𝑉 ∧ ( ♯ ‘ 𝑊 ) = 𝑁 ) )
4		simpr	⊢ ( ( 𝑊 ∈ ( 𝑁 ClWWalksN 𝐺 ) ∧ ( 𝑊 ∈ Word 𝑉 ∧ ( ♯ ‘ 𝑊 ) = 𝑁 ) ) → ( 𝑊 ∈ Word 𝑉 ∧ ( ♯ ‘ 𝑊 ) = 𝑁 ) )
5		clwwlknnn	⊢ ( 𝑊 ∈ ( 𝑁 ClWWalksN 𝐺 ) → 𝑁 ∈ ℕ )
6	1 2	isclwwlknx	⊢ ( 𝑁 ∈ ℕ → ( 𝑊 ∈ ( 𝑁 ClWWalksN 𝐺 ) ↔ ( ( 𝑊 ∈ Word 𝑉 ∧ ∀ 𝑖 ∈ ( 0 ..^ ( ( ♯ ‘ 𝑊 ) − 1 ) ) { ( 𝑊 ‘ 𝑖 ) , ( 𝑊 ‘ ( 𝑖 + 1 ) ) } ∈ 𝐸 ∧ { ( lastS ‘ 𝑊 ) , ( 𝑊 ‘ 0 ) } ∈ 𝐸 ) ∧ ( ♯ ‘ 𝑊 ) = 𝑁 ) ) )
7		3simpc	⊢ ( ( 𝑊 ∈ Word 𝑉 ∧ ∀ 𝑖 ∈ ( 0 ..^ ( ( ♯ ‘ 𝑊 ) − 1 ) ) { ( 𝑊 ‘ 𝑖 ) , ( 𝑊 ‘ ( 𝑖 + 1 ) ) } ∈ 𝐸 ∧ { ( lastS ‘ 𝑊 ) , ( 𝑊 ‘ 0 ) } ∈ 𝐸 ) → ( ∀ 𝑖 ∈ ( 0 ..^ ( ( ♯ ‘ 𝑊 ) − 1 ) ) { ( 𝑊 ‘ 𝑖 ) , ( 𝑊 ‘ ( 𝑖 + 1 ) ) } ∈ 𝐸 ∧ { ( lastS ‘ 𝑊 ) , ( 𝑊 ‘ 0 ) } ∈ 𝐸 ) )
8	7	adantr	⊢ ( ( ( 𝑊 ∈ Word 𝑉 ∧ ∀ 𝑖 ∈ ( 0 ..^ ( ( ♯ ‘ 𝑊 ) − 1 ) ) { ( 𝑊 ‘ 𝑖 ) , ( 𝑊 ‘ ( 𝑖 + 1 ) ) } ∈ 𝐸 ∧ { ( lastS ‘ 𝑊 ) , ( 𝑊 ‘ 0 ) } ∈ 𝐸 ) ∧ ( ♯ ‘ 𝑊 ) = 𝑁 ) → ( ∀ 𝑖 ∈ ( 0 ..^ ( ( ♯ ‘ 𝑊 ) − 1 ) ) { ( 𝑊 ‘ 𝑖 ) , ( 𝑊 ‘ ( 𝑖 + 1 ) ) } ∈ 𝐸 ∧ { ( lastS ‘ 𝑊 ) , ( 𝑊 ‘ 0 ) } ∈ 𝐸 ) )
9	6 8	biimtrdi	⊢ ( 𝑁 ∈ ℕ → ( 𝑊 ∈ ( 𝑁 ClWWalksN 𝐺 ) → ( ∀ 𝑖 ∈ ( 0 ..^ ( ( ♯ ‘ 𝑊 ) − 1 ) ) { ( 𝑊 ‘ 𝑖 ) , ( 𝑊 ‘ ( 𝑖 + 1 ) ) } ∈ 𝐸 ∧ { ( lastS ‘ 𝑊 ) , ( 𝑊 ‘ 0 ) } ∈ 𝐸 ) ) )
10	5 9	mpcom	⊢ ( 𝑊 ∈ ( 𝑁 ClWWalksN 𝐺 ) → ( ∀ 𝑖 ∈ ( 0 ..^ ( ( ♯ ‘ 𝑊 ) − 1 ) ) { ( 𝑊 ‘ 𝑖 ) , ( 𝑊 ‘ ( 𝑖 + 1 ) ) } ∈ 𝐸 ∧ { ( lastS ‘ 𝑊 ) , ( 𝑊 ‘ 0 ) } ∈ 𝐸 ) )
11	10	adantr	⊢ ( ( 𝑊 ∈ ( 𝑁 ClWWalksN 𝐺 ) ∧ ( 𝑊 ∈ Word 𝑉 ∧ ( ♯ ‘ 𝑊 ) = 𝑁 ) ) → ( ∀ 𝑖 ∈ ( 0 ..^ ( ( ♯ ‘ 𝑊 ) − 1 ) ) { ( 𝑊 ‘ 𝑖 ) , ( 𝑊 ‘ ( 𝑖 + 1 ) ) } ∈ 𝐸 ∧ { ( lastS ‘ 𝑊 ) , ( 𝑊 ‘ 0 ) } ∈ 𝐸 ) )
12		oveq1	⊢ ( ( ♯ ‘ 𝑊 ) = 𝑁 → ( ( ♯ ‘ 𝑊 ) − 1 ) = ( 𝑁 − 1 ) )
13	12	oveq2d	⊢ ( ( ♯ ‘ 𝑊 ) = 𝑁 → ( 0 ..^ ( ( ♯ ‘ 𝑊 ) − 1 ) ) = ( 0 ..^ ( 𝑁 − 1 ) ) )
14	13	raleqdv	⊢ ( ( ♯ ‘ 𝑊 ) = 𝑁 → ( ∀ 𝑖 ∈ ( 0 ..^ ( ( ♯ ‘ 𝑊 ) − 1 ) ) { ( 𝑊 ‘ 𝑖 ) , ( 𝑊 ‘ ( 𝑖 + 1 ) ) } ∈ 𝐸 ↔ ∀ 𝑖 ∈ ( 0 ..^ ( 𝑁 − 1 ) ) { ( 𝑊 ‘ 𝑖 ) , ( 𝑊 ‘ ( 𝑖 + 1 ) ) } ∈ 𝐸 ) )
15	14	anbi1d	⊢ ( ( ♯ ‘ 𝑊 ) = 𝑁 → ( ( ∀ 𝑖 ∈ ( 0 ..^ ( ( ♯ ‘ 𝑊 ) − 1 ) ) { ( 𝑊 ‘ 𝑖 ) , ( 𝑊 ‘ ( 𝑖 + 1 ) ) } ∈ 𝐸 ∧ { ( lastS ‘ 𝑊 ) , ( 𝑊 ‘ 0 ) } ∈ 𝐸 ) ↔ ( ∀ 𝑖 ∈ ( 0 ..^ ( 𝑁 − 1 ) ) { ( 𝑊 ‘ 𝑖 ) , ( 𝑊 ‘ ( 𝑖 + 1 ) ) } ∈ 𝐸 ∧ { ( lastS ‘ 𝑊 ) , ( 𝑊 ‘ 0 ) } ∈ 𝐸 ) ) )
16	15	ad2antll	⊢ ( ( 𝑊 ∈ ( 𝑁 ClWWalksN 𝐺 ) ∧ ( 𝑊 ∈ Word 𝑉 ∧ ( ♯ ‘ 𝑊 ) = 𝑁 ) ) → ( ( ∀ 𝑖 ∈ ( 0 ..^ ( ( ♯ ‘ 𝑊 ) − 1 ) ) { ( 𝑊 ‘ 𝑖 ) , ( 𝑊 ‘ ( 𝑖 + 1 ) ) } ∈ 𝐸 ∧ { ( lastS ‘ 𝑊 ) , ( 𝑊 ‘ 0 ) } ∈ 𝐸 ) ↔ ( ∀ 𝑖 ∈ ( 0 ..^ ( 𝑁 − 1 ) ) { ( 𝑊 ‘ 𝑖 ) , ( 𝑊 ‘ ( 𝑖 + 1 ) ) } ∈ 𝐸 ∧ { ( lastS ‘ 𝑊 ) , ( 𝑊 ‘ 0 ) } ∈ 𝐸 ) ) )
17	11 16	mpbid	⊢ ( ( 𝑊 ∈ ( 𝑁 ClWWalksN 𝐺 ) ∧ ( 𝑊 ∈ Word 𝑉 ∧ ( ♯ ‘ 𝑊 ) = 𝑁 ) ) → ( ∀ 𝑖 ∈ ( 0 ..^ ( 𝑁 − 1 ) ) { ( 𝑊 ‘ 𝑖 ) , ( 𝑊 ‘ ( 𝑖 + 1 ) ) } ∈ 𝐸 ∧ { ( lastS ‘ 𝑊 ) , ( 𝑊 ‘ 0 ) } ∈ 𝐸 ) )
18	4 17	jca	⊢ ( ( 𝑊 ∈ ( 𝑁 ClWWalksN 𝐺 ) ∧ ( 𝑊 ∈ Word 𝑉 ∧ ( ♯ ‘ 𝑊 ) = 𝑁 ) ) → ( ( 𝑊 ∈ Word 𝑉 ∧ ( ♯ ‘ 𝑊 ) = 𝑁 ) ∧ ( ∀ 𝑖 ∈ ( 0 ..^ ( 𝑁 − 1 ) ) { ( 𝑊 ‘ 𝑖 ) , ( 𝑊 ‘ ( 𝑖 + 1 ) ) } ∈ 𝐸 ∧ { ( lastS ‘ 𝑊 ) , ( 𝑊 ‘ 0 ) } ∈ 𝐸 ) ) )
19	3 18	mpdan	⊢ ( 𝑊 ∈ ( 𝑁 ClWWalksN 𝐺 ) → ( ( 𝑊 ∈ Word 𝑉 ∧ ( ♯ ‘ 𝑊 ) = 𝑁 ) ∧ ( ∀ 𝑖 ∈ ( 0 ..^ ( 𝑁 − 1 ) ) { ( 𝑊 ‘ 𝑖 ) , ( 𝑊 ‘ ( 𝑖 + 1 ) ) } ∈ 𝐸 ∧ { ( lastS ‘ 𝑊 ) , ( 𝑊 ‘ 0 ) } ∈ 𝐸 ) ) )
20		3anass	⊢ ( ( ( 𝑊 ∈ Word 𝑉 ∧ ( ♯ ‘ 𝑊 ) = 𝑁 ) ∧ ∀ 𝑖 ∈ ( 0 ..^ ( 𝑁 − 1 ) ) { ( 𝑊 ‘ 𝑖 ) , ( 𝑊 ‘ ( 𝑖 + 1 ) ) } ∈ 𝐸 ∧ { ( lastS ‘ 𝑊 ) , ( 𝑊 ‘ 0 ) } ∈ 𝐸 ) ↔ ( ( 𝑊 ∈ Word 𝑉 ∧ ( ♯ ‘ 𝑊 ) = 𝑁 ) ∧ ( ∀ 𝑖 ∈ ( 0 ..^ ( 𝑁 − 1 ) ) { ( 𝑊 ‘ 𝑖 ) , ( 𝑊 ‘ ( 𝑖 + 1 ) ) } ∈ 𝐸 ∧ { ( lastS ‘ 𝑊 ) , ( 𝑊 ‘ 0 ) } ∈ 𝐸 ) ) )
21	19 20	sylibr	⊢ ( 𝑊 ∈ ( 𝑁 ClWWalksN 𝐺 ) → ( ( 𝑊 ∈ Word 𝑉 ∧ ( ♯ ‘ 𝑊 ) = 𝑁 ) ∧ ∀ 𝑖 ∈ ( 0 ..^ ( 𝑁 − 1 ) ) { ( 𝑊 ‘ 𝑖 ) , ( 𝑊 ‘ ( 𝑖 + 1 ) ) } ∈ 𝐸 ∧ { ( lastS ‘ 𝑊 ) , ( 𝑊 ‘ 0 ) } ∈ 𝐸 ) )