Metamath Proof Explorer


Theorem oddm1d2

Description: An integer is odd iff its predecessor divided by 2 is an integer. This is another representation of odd numbers without using the divides relation. (Contributed by AV, 18-Jun-2021) (Proof shortened by AV, 22-Jun-2021)

Ref Expression
Assertion oddm1d2 ( 𝑁 ∈ ℤ → ( ¬ 2 ∥ 𝑁 ↔ ( ( 𝑁 − 1 ) / 2 ) ∈ ℤ ) )

Proof

Step Hyp Ref Expression
1 oddp1d2 ( 𝑁 ∈ ℤ → ( ¬ 2 ∥ 𝑁 ↔ ( ( 𝑁 + 1 ) / 2 ) ∈ ℤ ) )
2 zob ( 𝑁 ∈ ℤ → ( ( ( 𝑁 + 1 ) / 2 ) ∈ ℤ ↔ ( ( 𝑁 − 1 ) / 2 ) ∈ ℤ ) )
3 1 2 bitrd ( 𝑁 ∈ ℤ → ( ¬ 2 ∥ 𝑁 ↔ ( ( 𝑁 − 1 ) / 2 ) ∈ ℤ ) )