Theorem gzsubcl

Description: The gaussian integers are closed under subtraction. (Contributed by Mario Carneiro, 14-Jul-2014)

	Ref	Expression
Assertion	gzsubcl	⊢ ( ( 𝐴 ∈ ℤ[i] ∧ 𝐵 ∈ ℤ[i] ) → ( 𝐴 − 𝐵 ) ∈ ℤ[i] )

Proof

Step	Hyp	Ref	Expression
1		gzcn	⊢ ( 𝐴 ∈ ℤ[i] → 𝐴 ∈ ℂ )
2		gzcn	⊢ ( 𝐵 ∈ ℤ[i] → 𝐵 ∈ ℂ )
3		negsub	⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ) → ( 𝐴 + - 𝐵 ) = ( 𝐴 − 𝐵 ) )
4	1 2 3	syl2an	⊢ ( ( 𝐴 ∈ ℤ[i] ∧ 𝐵 ∈ ℤ[i] ) → ( 𝐴 + - 𝐵 ) = ( 𝐴 − 𝐵 ) )
5		gznegcl	⊢ ( 𝐵 ∈ ℤ[i] → - 𝐵 ∈ ℤ[i] )
6		gzaddcl	⊢ ( ( 𝐴 ∈ ℤ[i] ∧ - 𝐵 ∈ ℤ[i] ) → ( 𝐴 + - 𝐵 ) ∈ ℤ[i] )
7	5 6	sylan2	⊢ ( ( 𝐴 ∈ ℤ[i] ∧ 𝐵 ∈ ℤ[i] ) → ( 𝐴 + - 𝐵 ) ∈ ℤ[i] )
8	4 7	eqeltrrd	⊢ ( ( 𝐴 ∈ ℤ[i] ∧ 𝐵 ∈ ℤ[i] ) → ( 𝐴 − 𝐵 ) ∈ ℤ[i] )