Metamath Proof Explorer


Theorem cjmuld

Description: Complex conjugate distributes over multiplication. Proposition 10-3.4(c) of Gleason p. 133. (Contributed by Mario Carneiro, 29-May-2016)

Ref Expression
Hypotheses recld.1 ( 𝜑𝐴 ∈ ℂ )
readdd.2 ( 𝜑𝐵 ∈ ℂ )
Assertion cjmuld ( 𝜑 → ( ∗ ‘ ( 𝐴 · 𝐵 ) ) = ( ( ∗ ‘ 𝐴 ) · ( ∗ ‘ 𝐵 ) ) )

Proof

Step Hyp Ref Expression
1 recld.1 ( 𝜑𝐴 ∈ ℂ )
2 readdd.2 ( 𝜑𝐵 ∈ ℂ )
3 cjmul ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ) → ( ∗ ‘ ( 𝐴 · 𝐵 ) ) = ( ( ∗ ‘ 𝐴 ) · ( ∗ ‘ 𝐵 ) ) )
4 1 2 3 syl2anc ( 𝜑 → ( ∗ ‘ ( 𝐴 · 𝐵 ) ) = ( ( ∗ ‘ 𝐴 ) · ( ∗ ‘ 𝐵 ) ) )