Theorem nnnegz

Description: The negative of a positive integer is an integer. (Contributed by NM, 12-Jan-2002)

	Ref	Expression
Assertion	nnnegz	⊢ ( 𝑁 ∈ ℕ → - 𝑁 ∈ ℤ )

Proof

Step	Hyp	Ref	Expression
1		nnre	⊢ ( 𝑁 ∈ ℕ → 𝑁 ∈ ℝ )
2	1	renegcld	⊢ ( 𝑁 ∈ ℕ → - 𝑁 ∈ ℝ )
3		nncn	⊢ ( 𝑁 ∈ ℕ → 𝑁 ∈ ℂ )
4		negneg	⊢ ( 𝑁 ∈ ℂ → - - 𝑁 = 𝑁 )
5	4	eleq1d	⊢ ( 𝑁 ∈ ℂ → ( - - 𝑁 ∈ ℕ ↔ 𝑁 ∈ ℕ ) )
6	5	biimprd	⊢ ( 𝑁 ∈ ℂ → ( 𝑁 ∈ ℕ → - - 𝑁 ∈ ℕ ) )
7	3 6	mpcom	⊢ ( 𝑁 ∈ ℕ → - - 𝑁 ∈ ℕ )
8	7	3mix3d	⊢ ( 𝑁 ∈ ℕ → ( - 𝑁 = 0 ∨ - 𝑁 ∈ ℕ ∨ - - 𝑁 ∈ ℕ ) )
9		elz	⊢ ( - 𝑁 ∈ ℤ ↔ ( - 𝑁 ∈ ℝ ∧ ( - 𝑁 = 0 ∨ - 𝑁 ∈ ℕ ∨ - - 𝑁 ∈ ℕ ) ) )
10	2 8 9	sylanbrc	⊢ ( 𝑁 ∈ ℕ → - 𝑁 ∈ ℤ )