Theorem oddm1div2z

Description: The result of dividing an odd number decreased by 1 and then divided by 2 is an integer. (Contributed by AV, 15-Jun-2020)

	Ref	Expression
Assertion	oddm1div2z	⊢ ( 𝑍 ∈ Odd → ( ( 𝑍 − 1 ) / 2 ) ∈ ℤ )

Proof

Step	Hyp	Ref	Expression
1		oddp1div2z	⊢ ( 𝑍 ∈ Odd → ( ( 𝑍 + 1 ) / 2 ) ∈ ℤ )
2		oddz	⊢ ( 𝑍 ∈ Odd → 𝑍 ∈ ℤ )
3		zob	⊢ ( 𝑍 ∈ ℤ → ( ( ( 𝑍 + 1 ) / 2 ) ∈ ℤ ↔ ( ( 𝑍 − 1 ) / 2 ) ∈ ℤ ) )
4	2 3	syl	⊢ ( 𝑍 ∈ Odd → ( ( ( 𝑍 + 1 ) / 2 ) ∈ ℤ ↔ ( ( 𝑍 − 1 ) / 2 ) ∈ ℤ ) )
5	1 4	mpbid	⊢ ( 𝑍 ∈ Odd → ( ( 𝑍 − 1 ) / 2 ) ∈ ℤ )