Theorem znegcl

Description: Closure law for negative integers. (Contributed by NM, 9-May-2004)

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

Proof

Step	Hyp	Ref	Expression
1		elz	⊢ ( 𝑁 ∈ ℤ ↔ ( 𝑁 ∈ ℝ ∧ ( 𝑁 = 0 ∨ 𝑁 ∈ ℕ ∨ - 𝑁 ∈ ℕ ) ) )
2		negeq	⊢ ( 𝑁 = 0 → - 𝑁 = - 0 )
3		neg0	⊢ - 0 = 0
4	2 3	syl6eq	⊢ ( 𝑁 = 0 → - 𝑁 = 0 )
5		0z	⊢ 0 ∈ ℤ
6	4 5	syl6eqel	⊢ ( 𝑁 = 0 → - 𝑁 ∈ ℤ )
7		nnnegz	⊢ ( 𝑁 ∈ ℕ → - 𝑁 ∈ ℤ )
8		nnz	⊢ ( - 𝑁 ∈ ℕ → - 𝑁 ∈ ℤ )
9	6 7 8	3jaoi	⊢ ( ( 𝑁 = 0 ∨ 𝑁 ∈ ℕ ∨ - 𝑁 ∈ ℕ ) → - 𝑁 ∈ ℤ )
10	1 9	simplbiim	⊢ ( 𝑁 ∈ ℤ → - 𝑁 ∈ ℤ )