Theorem gzreim

Description: Construct a gaussian integer from real and imaginary parts. (Contributed by Mario Carneiro, 16-Jul-2014)

	Ref	Expression
Assertion	gzreim	⊢ ( ( 𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ) → ( 𝐴 + ( i · 𝐵 ) ) ∈ ℤ[i] )

Proof

Step	Hyp	Ref	Expression
1		zgz	⊢ ( 𝐴 ∈ ℤ → 𝐴 ∈ ℤ[i] )
2		igz	⊢ i ∈ ℤ[i]
3		zgz	⊢ ( 𝐵 ∈ ℤ → 𝐵 ∈ ℤ[i] )
4		gzmulcl	⊢ ( ( i ∈ ℤ[i] ∧ 𝐵 ∈ ℤ[i] ) → ( i · 𝐵 ) ∈ ℤ[i] )
5	2 3 4	sylancr	⊢ ( 𝐵 ∈ ℤ → ( i · 𝐵 ) ∈ ℤ[i] )
6		gzaddcl	⊢ ( ( 𝐴 ∈ ℤ[i] ∧ ( i · 𝐵 ) ∈ ℤ[i] ) → ( 𝐴 + ( i · 𝐵 ) ) ∈ ℤ[i] )
7	1 5 6	syl2an	⊢ ( ( 𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ) → ( 𝐴 + ( i · 𝐵 ) ) ∈ ℤ[i] )