Theorem numclwwlkovh0

Description: Value of operation H , mapping a vertex v and an integer n greater than 1 to the "closed n-walks v(0) ... v(n-2) v(n-1) v(n) from v = v(0) = v(n) ... with v(n-2) =/= v" according to definition 7 in Huneke p. 2. (Contributed by AV, 1-May-2022)

	Ref	Expression
Hypothesis	numclwwlkovh.h	⊢ 𝐻 = ( 𝑣 ∈ 𝑉 , 𝑛 ∈ ( ℤ≥ ‘ 2 ) ↦ { 𝑤 ∈ ( 𝑣 ( ClWWalksNOn ‘ 𝐺 ) 𝑛 ) ∣ ( 𝑤 ‘ ( 𝑛 − 2 ) ) ≠ 𝑣 } )
Assertion	numclwwlkovh0	⊢ ( ( 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ ( ℤ≥ ‘ 2 ) ) → ( 𝑋 𝐻 𝑁 ) = { 𝑤 ∈ ( 𝑋 ( ClWWalksNOn ‘ 𝐺 ) 𝑁 ) ∣ ( 𝑤 ‘ ( 𝑁 − 2 ) ) ≠ 𝑋 } )

Proof

Step	Hyp	Ref	Expression
1		numclwwlkovh.h	⊢ 𝐻 = ( 𝑣 ∈ 𝑉 , 𝑛 ∈ ( ℤ≥ ‘ 2 ) ↦ { 𝑤 ∈ ( 𝑣 ( ClWWalksNOn ‘ 𝐺 ) 𝑛 ) ∣ ( 𝑤 ‘ ( 𝑛 − 2 ) ) ≠ 𝑣 } )
2		oveq12	⊢ ( ( 𝑣 = 𝑋 ∧ 𝑛 = 𝑁 ) → ( 𝑣 ( ClWWalksNOn ‘ 𝐺 ) 𝑛 ) = ( 𝑋 ( ClWWalksNOn ‘ 𝐺 ) 𝑁 ) )
3		oveq1	⊢ ( 𝑛 = 𝑁 → ( 𝑛 − 2 ) = ( 𝑁 − 2 ) )
4	3	adantl	⊢ ( ( 𝑣 = 𝑋 ∧ 𝑛 = 𝑁 ) → ( 𝑛 − 2 ) = ( 𝑁 − 2 ) )
5	4	fveq2d	⊢ ( ( 𝑣 = 𝑋 ∧ 𝑛 = 𝑁 ) → ( 𝑤 ‘ ( 𝑛 − 2 ) ) = ( 𝑤 ‘ ( 𝑁 − 2 ) ) )
6		simpl	⊢ ( ( 𝑣 = 𝑋 ∧ 𝑛 = 𝑁 ) → 𝑣 = 𝑋 )
7	5 6	neeq12d	⊢ ( ( 𝑣 = 𝑋 ∧ 𝑛 = 𝑁 ) → ( ( 𝑤 ‘ ( 𝑛 − 2 ) ) ≠ 𝑣 ↔ ( 𝑤 ‘ ( 𝑁 − 2 ) ) ≠ 𝑋 ) )
8	2 7	rabeqbidv	⊢ ( ( 𝑣 = 𝑋 ∧ 𝑛 = 𝑁 ) → { 𝑤 ∈ ( 𝑣 ( ClWWalksNOn ‘ 𝐺 ) 𝑛 ) ∣ ( 𝑤 ‘ ( 𝑛 − 2 ) ) ≠ 𝑣 } = { 𝑤 ∈ ( 𝑋 ( ClWWalksNOn ‘ 𝐺 ) 𝑁 ) ∣ ( 𝑤 ‘ ( 𝑁 − 2 ) ) ≠ 𝑋 } )
9		ovex	⊢ ( 𝑋 ( ClWWalksNOn ‘ 𝐺 ) 𝑁 ) ∈ V
10	9	rabex	⊢ { 𝑤 ∈ ( 𝑋 ( ClWWalksNOn ‘ 𝐺 ) 𝑁 ) ∣ ( 𝑤 ‘ ( 𝑁 − 2 ) ) ≠ 𝑋 } ∈ V
11	8 1 10	ovmpoa	⊢ ( ( 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ ( ℤ≥ ‘ 2 ) ) → ( 𝑋 𝐻 𝑁 ) = { 𝑤 ∈ ( 𝑋 ( ClWWalksNOn ‘ 𝐺 ) 𝑁 ) ∣ ( 𝑤 ‘ ( 𝑁 − 2 ) ) ≠ 𝑋 } )