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 ) ) ) ) |
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 ) ) ) ) |