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Metamath Proof Explorer

Theorem cjmuli

Description: Complex conjugate distributes over multiplication. Proposition 10-3.4(c) of Gleason p. 133. (Contributed by NM, 28-Jul-1999)

Ref Expression
Hypotheses recl.1 ⊢ 𝐴 ∈ ℂ
readdi.2 ⊢ 𝐵 ∈ ℂ
Assertion cjmuli ( ∗ ‘ ( 𝐴 · 𝐵 ) ) = ( ( ∗ ‘ 𝐴 ) · ( ∗ ‘ 𝐵 ) )

Proof

Step Hyp Ref Expression
1 recl.1 ⊢ 𝐴 ∈ ℂ
2 readdi.2 ⊢ 𝐵 ∈ ℂ
3 cjmul ⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ) → ( ∗ ‘ ( 𝐴 · 𝐵 ) ) = ( ( ∗ ‘ 𝐴 ) · ( ∗ ‘ 𝐵 ) ) )
4 1 2 3 mp2an ⊢ ( ∗ ‘ ( 𝐴 · 𝐵 ) ) = ( ( ∗ ‘ 𝐴 ) · ( ∗ ‘ 𝐵 ) )