Theorem nn0ofldiv2

Description: The floor of an odd nonnegative integer divided by 2 is equal to the integer first decreased by 1 and then divided by 2. (Contributed by AV, 1-Jun-2020) (Proof shortened by AV, 7-Jun-2020)

	Ref	Expression
Assertion	nn0ofldiv2	⊢ ( ( 𝑁 ∈ ℕ0 ∧ ( ( 𝑁 + 1 ) / 2 ) ∈ ℕ0 ) → ( ⌊ ‘ ( 𝑁 / 2 ) ) = ( ( 𝑁 − 1 ) / 2 ) )

Proof

Step	Hyp	Ref	Expression
1		nn0z	⊢ ( 𝑁 ∈ ℕ0 → 𝑁 ∈ ℤ )
2		nn0z	⊢ ( ( ( 𝑁 + 1 ) / 2 ) ∈ ℕ0 → ( ( 𝑁 + 1 ) / 2 ) ∈ ℤ )
3		zofldiv2	⊢ ( ( 𝑁 ∈ ℤ ∧ ( ( 𝑁 + 1 ) / 2 ) ∈ ℤ ) → ( ⌊ ‘ ( 𝑁 / 2 ) ) = ( ( 𝑁 − 1 ) / 2 ) )
4	1 2 3	syl2an	⊢ ( ( 𝑁 ∈ ℕ0 ∧ ( ( 𝑁 + 1 ) / 2 ) ∈ ℕ0 ) → ( ⌊ ‘ ( 𝑁 / 2 ) ) = ( ( 𝑁 − 1 ) / 2 ) )