Theorem iseven5

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

	Ref	Expression
Assertion	iseven5	⊢ ( 𝑍 ∈ Even ↔ ( 𝑍 ∈ ℤ ∧ ( 2 gcd 𝑍 ) = 2 ) )

Proof

Step	Hyp	Ref	Expression
1		iseven2	⊢ ( 𝑍 ∈ Even ↔ ( 𝑍 ∈ ℤ ∧ 2 ∥ 𝑍 ) )
2		2nn	⊢ 2 ∈ ℕ
3		gcdzeq	⊢ ( ( 2 ∈ ℕ ∧ 𝑍 ∈ ℤ ) → ( ( 2 gcd 𝑍 ) = 2 ↔ 2 ∥ 𝑍 ) )
4	2 3	mpan	⊢ ( 𝑍 ∈ ℤ → ( ( 2 gcd 𝑍 ) = 2 ↔ 2 ∥ 𝑍 ) )
5	4	bicomd	⊢ ( 𝑍 ∈ ℤ → ( 2 ∥ 𝑍 ↔ ( 2 gcd 𝑍 ) = 2 ) )
6	5	pm5.32i	⊢ ( ( 𝑍 ∈ ℤ ∧ 2 ∥ 𝑍 ) ↔ ( 𝑍 ∈ ℤ ∧ ( 2 gcd 𝑍 ) = 2 ) )
7	1 6	bitri	⊢ ( 𝑍 ∈ Even ↔ ( 𝑍 ∈ ℤ ∧ ( 2 gcd 𝑍 ) = 2 ) )