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Metamath Proof Explorer

Theorem immul2d

Description: Imaginary part of a product. (Contributed by Mario Carneiro, 29-May-2016)

Ref Expression
Hypotheses crred.1 ⊢ ( 𝜑 → 𝐴 ∈ ℝ )
remul2d.2 ⊢ ( 𝜑 → 𝐵 ∈ ℂ )
Assertion immul2d ( 𝜑 → ( ℑ ‘ ( 𝐴 · 𝐵 ) ) = ( 𝐴 · ( ℑ ‘ 𝐵 ) ) )

Proof

Step Hyp Ref Expression
1 crred.1 ⊢ ( 𝜑 → 𝐴 ∈ ℝ )
2 remul2d.2 ⊢ ( 𝜑 → 𝐵 ∈ ℂ )
3 immul2 ⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℂ ) → ( ℑ ‘ ( 𝐴 · 𝐵 ) ) = ( 𝐴 · ( ℑ ‘ 𝐵 ) ) )
4 1 2 3 syl2anc ⊢ ( 𝜑 → ( ℑ ‘ ( 𝐴 · 𝐵 ) ) = ( 𝐴 · ( ℑ ‘ 𝐵 ) ) )