Theorem flnn0ohalf

Description: The floor of the half of an odd positive integer is equal to the floor of the half of the integer decreased by 1. (Contributed by AV, 5-Jun-2012)

	Ref	Expression
Assertion	flnn0ohalf	⊢ ( ( 𝑁 ∈ ℕ0 ∧ ( ( 𝑁 + 1 ) / 2 ) ∈ ℕ0 ) → ( ⌊ ‘ ( 𝑁 / 2 ) ) = ( ⌊ ‘ ( ( 𝑁 − 1 ) / 2 ) ) )

Proof

Step	Hyp	Ref	Expression
1		nn0ofldiv2	⊢ ( ( 𝑁 ∈ ℕ0 ∧ ( ( 𝑁 + 1 ) / 2 ) ∈ ℕ0 ) → ( ⌊ ‘ ( 𝑁 / 2 ) ) = ( ( 𝑁 − 1 ) / 2 ) )
2		nn0o	⊢ ( ( 𝑁 ∈ ℕ0 ∧ ( ( 𝑁 + 1 ) / 2 ) ∈ ℕ0 ) → ( ( 𝑁 − 1 ) / 2 ) ∈ ℕ0 )
3	2	nn0zd	⊢ ( ( 𝑁 ∈ ℕ0 ∧ ( ( 𝑁 + 1 ) / 2 ) ∈ ℕ0 ) → ( ( 𝑁 − 1 ) / 2 ) ∈ ℤ )
4		flid	⊢ ( ( ( 𝑁 − 1 ) / 2 ) ∈ ℤ → ( ⌊ ‘ ( ( 𝑁 − 1 ) / 2 ) ) = ( ( 𝑁 − 1 ) / 2 ) )
5	3 4	syl	⊢ ( ( 𝑁 ∈ ℕ0 ∧ ( ( 𝑁 + 1 ) / 2 ) ∈ ℕ0 ) → ( ⌊ ‘ ( ( 𝑁 − 1 ) / 2 ) ) = ( ( 𝑁 − 1 ) / 2 ) )
6	1 5	eqtr4d	⊢ ( ( 𝑁 ∈ ℕ0 ∧ ( ( 𝑁 + 1 ) / 2 ) ∈ ℕ0 ) → ( ⌊ ‘ ( 𝑁 / 2 ) ) = ( ⌊ ‘ ( ( 𝑁 − 1 ) / 2 ) ) )