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	⊢ ( 𝜑 → 𝐴 ∈ ℂ )
	readdd.2	⊢ ( 𝜑 → 𝐵 ∈ ℂ )
Assertion	cjmuld	⊢ ( 𝜑 → ( ∗ ‘ ( 𝐴 · 𝐵 ) ) = ( ( ∗ ‘ 𝐴 ) · ( ∗ ‘ 𝐵 ) ) )

Proof

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