Theorem crimd

Description: The imaginary part of a complex number representation. Definition 10-3.1 of Gleason p. 132. (Contributed by Mario Carneiro, 29-May-2016)

	Ref	Expression
Hypotheses	crred.1	⊢ ( 𝜑 → 𝐴 ∈ ℝ )
	crred.2	⊢ ( 𝜑 → 𝐵 ∈ ℝ )
Assertion	crimd	⊢ ( 𝜑 → ( ℑ ‘ ( 𝐴 + ( i · 𝐵 ) ) ) = 𝐵 )

Proof

Step	Hyp	Ref	Expression
1		crred.1	⊢ ( 𝜑 → 𝐴 ∈ ℝ )
2		crred.2	⊢ ( 𝜑 → 𝐵 ∈ ℝ )
3		crim	⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ) → ( ℑ ‘ ( 𝐴 + ( i · 𝐵 ) ) ) = 𝐵 )
4	1 2 3	syl2anc	⊢ ( 𝜑 → ( ℑ ‘ ( 𝐴 + ( i · 𝐵 ) ) ) = 𝐵 )