Theorem zsubcl

Description: Closure of subtraction of integers. (Contributed by NM, 11-May-2004)

	Ref	Expression
Assertion	zsubcl	⊢ ( ( 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ) → ( 𝑀 − 𝑁 ) ∈ ℤ )

Proof

Step	Hyp	Ref	Expression
1		zcn	⊢ ( 𝑀 ∈ ℤ → 𝑀 ∈ ℂ )
2		zcn	⊢ ( 𝑁 ∈ ℤ → 𝑁 ∈ ℂ )
3		negsub	⊢ ( ( 𝑀 ∈ ℂ ∧ 𝑁 ∈ ℂ ) → ( 𝑀 + - 𝑁 ) = ( 𝑀 − 𝑁 ) )
4	1 2 3	syl2an	⊢ ( ( 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ) → ( 𝑀 + - 𝑁 ) = ( 𝑀 − 𝑁 ) )
5		znegcl	⊢ ( 𝑁 ∈ ℤ → - 𝑁 ∈ ℤ )
6		zaddcl	⊢ ( ( 𝑀 ∈ ℤ ∧ - 𝑁 ∈ ℤ ) → ( 𝑀 + - 𝑁 ) ∈ ℤ )
7	5 6	sylan2	⊢ ( ( 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ) → ( 𝑀 + - 𝑁 ) ∈ ℤ )
8	4 7	eqeltrrd	⊢ ( ( 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ) → ( 𝑀 − 𝑁 ) ∈ ℤ )