Theorem cjsubd

Description: Complex conjugate distributes over subtraction. (Contributed by Thierry Arnoux, 1-Jul-2025)

	Ref	Expression
Hypotheses	cjsubd.1	⊢ ( 𝜑 → 𝐴 ∈ ℂ )
	cjsubd.2	⊢ ( 𝜑 → 𝐵 ∈ ℂ )
Assertion	cjsubd	⊢ ( 𝜑 → ( ∗ ‘ ( 𝐴 − 𝐵 ) ) = ( ( ∗ ‘ 𝐴 ) − ( ∗ ‘ 𝐵 ) ) )

Proof

Step	Hyp	Ref	Expression
1		cjsubd.1	⊢ ( 𝜑 → 𝐴 ∈ ℂ )
2		cjsubd.2	⊢ ( 𝜑 → 𝐵 ∈ ℂ )
3		cjsub	⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ) → ( ∗ ‘ ( 𝐴 − 𝐵 ) ) = ( ( ∗ ‘ 𝐴 ) − ( ∗ ‘ 𝐵 ) ) )
4	1 2 3	syl2anc	⊢ ( 𝜑 → ( ∗ ‘ ( 𝐴 − 𝐵 ) ) = ( ( ∗ ‘ 𝐴 ) − ( ∗ ‘ 𝐵 ) ) )