Theorem isodd7

Description: The predicate "is an odd number". An odd number and 2 have 1 as greatest common divisor. (Contributed by AV, 1-Jul-2020)

	Ref	Expression
Assertion	isodd7	⊢ ( 𝑍 ∈ Odd ↔ ( 𝑍 ∈ ℤ ∧ ( 2 gcd 𝑍 ) = 1 ) )

Proof

Step	Hyp	Ref	Expression
1		isodd3	⊢ ( 𝑍 ∈ Odd ↔ ( 𝑍 ∈ ℤ ∧ ¬ 2 ∥ 𝑍 ) )
2		2prm	⊢ 2 ∈ ℙ
3		coprm	⊢ ( ( 2 ∈ ℙ ∧ 𝑍 ∈ ℤ ) → ( ¬ 2 ∥ 𝑍 ↔ ( 2 gcd 𝑍 ) = 1 ) )
4	2 3	mpan	⊢ ( 𝑍 ∈ ℤ → ( ¬ 2 ∥ 𝑍 ↔ ( 2 gcd 𝑍 ) = 1 ) )
5	4	pm5.32i	⊢ ( ( 𝑍 ∈ ℤ ∧ ¬ 2 ∥ 𝑍 ) ↔ ( 𝑍 ∈ ℤ ∧ ( 2 gcd 𝑍 ) = 1 ) )
6	1 5	bitri	⊢ ( 𝑍 ∈ Odd ↔ ( 𝑍 ∈ ℤ ∧ ( 2 gcd 𝑍 ) = 1 ) )