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	$$⊢\left(N\in {\mathbb{N}}_0\wedge \frac{N+1}{2}\in {\mathbb{N}}_0\right)\to ⌊\frac{N}{2}⌋=⌊\frac{N-1}{2}⌋$$

Proof

Step	Hyp	Ref	Expression
1		nn0ofldiv2	$$⊢\left(N\in {\mathbb{N}}_0\wedge \frac{N+1}{2}\in {\mathbb{N}}_0\right)\to ⌊\frac{N}{2}⌋=\frac{N-1}{2}$$
2		nn0o	$$⊢\left(N\in {\mathbb{N}}_0\wedge \frac{N+1}{2}\in {\mathbb{N}}_0\right)\to \frac{N-1}{2}\in {\mathbb{N}}_0$$
3	2	nn0zd	$$⊢\left(N\in {\mathbb{N}}_0\wedge \frac{N+1}{2}\in {\mathbb{N}}_0\right)\to \frac{N-1}{2}\in \mathbb{Z}$$
4		flid	$$⊢\frac{N-1}{2}\in \mathbb{Z}\to ⌊\frac{N-1}{2}⌋=\frac{N-1}{2}$$
5	3 4	syl	$$⊢\left(N\in {\mathbb{N}}_0\wedge \frac{N+1}{2}\in {\mathbb{N}}_0\right)\to ⌊\frac{N-1}{2}⌋=\frac{N-1}{2}$$
6	1 5	eqtr4d	$$⊢\left(N\in {\mathbb{N}}_0\wedge \frac{N+1}{2}\in {\mathbb{N}}_0\right)\to ⌊\frac{N}{2}⌋=⌊\frac{N-1}{2}⌋$$