Metamath Proof Explorer


Theorem remul

Description: Real part of a product. (Contributed by NM, 28-Jul-1999) (Revised by Mario Carneiro, 14-Jul-2014)

Ref Expression
Assertion remul ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ) → ( ℜ ‘ ( 𝐴 · 𝐵 ) ) = ( ( ( ℜ ‘ 𝐴 ) · ( ℜ ‘ 𝐵 ) ) − ( ( ℑ ‘ 𝐴 ) · ( ℑ ‘ 𝐵 ) ) ) )

Proof

Step Hyp Ref Expression
1 remullem ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ) → ( ( ℜ ‘ ( 𝐴 · 𝐵 ) ) = ( ( ( ℜ ‘ 𝐴 ) · ( ℜ ‘ 𝐵 ) ) − ( ( ℑ ‘ 𝐴 ) · ( ℑ ‘ 𝐵 ) ) ) ∧ ( ℑ ‘ ( 𝐴 · 𝐵 ) ) = ( ( ( ℜ ‘ 𝐴 ) · ( ℑ ‘ 𝐵 ) ) + ( ( ℑ ‘ 𝐴 ) · ( ℜ ‘ 𝐵 ) ) ) ∧ ( ∗ ‘ ( 𝐴 · 𝐵 ) ) = ( ( ∗ ‘ 𝐴 ) · ( ∗ ‘ 𝐵 ) ) ) )
2 1 simp1d ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ) → ( ℜ ‘ ( 𝐴 · 𝐵 ) ) = ( ( ( ℜ ‘ 𝐴 ) · ( ℜ ‘ 𝐵 ) ) − ( ( ℑ ‘ 𝐴 ) · ( ℑ ‘ 𝐵 ) ) ) )