Theorem cjne0d

Description: A number is nonzero iff its complex conjugate is nonzero. (Contributed by Mario Carneiro, 29-May-2016)

	Ref	Expression
Hypotheses	recld.1	⊢ ( 𝜑 → 𝐴 ∈ ℂ )
	cjne0d.2	⊢ ( 𝜑 → 𝐴 ≠ 0 )
Assertion	cjne0d	⊢ ( 𝜑 → ( ∗ ‘ 𝐴 ) ≠ 0 )

Proof

Step	Hyp	Ref	Expression
1		recld.1	⊢ ( 𝜑 → 𝐴 ∈ ℂ )
2		cjne0d.2	⊢ ( 𝜑 → 𝐴 ≠ 0 )
3		cjne0	⊢ ( 𝐴 ∈ ℂ → ( 𝐴 ≠ 0 ↔ ( ∗ ‘ 𝐴 ) ≠ 0 ) )
4	1 3	syl	⊢ ( 𝜑 → ( 𝐴 ≠ 0 ↔ ( ∗ ‘ 𝐴 ) ≠ 0 ) )
5	2 4	mpbid	⊢ ( 𝜑 → ( ∗ ‘ 𝐴 ) ≠ 0 )