Theorem nnpw2even

Description: 2 to the power of a positive integer is even. (Contributed by AV, 2-Jun-2020)

	Ref	Expression
Assertion	nnpw2even	⊢ ( 𝑁 ∈ ℕ → ( ( 2 ↑ 𝑁 ) / 2 ) ∈ ℕ )

Proof

Step	Hyp	Ref	Expression
1		2cnd	⊢ ( 𝑁 ∈ ℕ → 2 ∈ ℂ )
2		2ne0	⊢ 2 ≠ 0
3	2	a1i	⊢ ( 𝑁 ∈ ℕ → 2 ≠ 0 )
4		nnz	⊢ ( 𝑁 ∈ ℕ → 𝑁 ∈ ℤ )
5	1 3 4	expm1d	⊢ ( 𝑁 ∈ ℕ → ( 2 ↑ ( 𝑁 − 1 ) ) = ( ( 2 ↑ 𝑁 ) / 2 ) )
6		2nn	⊢ 2 ∈ ℕ
7	6	a1i	⊢ ( 𝑁 ∈ ℕ → 2 ∈ ℕ )
8		nnm1nn0	⊢ ( 𝑁 ∈ ℕ → ( 𝑁 − 1 ) ∈ ℕ0 )
9	7 8	nnexpcld	⊢ ( 𝑁 ∈ ℕ → ( 2 ↑ ( 𝑁 − 1 ) ) ∈ ℕ )
10	5 9	eqeltrrd	⊢ ( 𝑁 ∈ ℕ → ( ( 2 ↑ 𝑁 ) / 2 ) ∈ ℕ )