Theorem cjne0

Description: A number is nonzero iff its complex conjugate is nonzero. (Contributed by NM, 29-Apr-2005)

	Ref	Expression
Assertion	cjne0	⊢ ( 𝐴 ∈ ℂ → ( 𝐴 ≠ 0 ↔ ( ∗ ‘ 𝐴 ) ≠ 0 ) )

Proof

Step	Hyp	Ref	Expression
1		0cn	⊢ 0 ∈ ℂ
2		cj11	⊢ ( ( 𝐴 ∈ ℂ ∧ 0 ∈ ℂ ) → ( ( ∗ ‘ 𝐴 ) = ( ∗ ‘ 0 ) ↔ 𝐴 = 0 ) )
3	1 2	mpan2	⊢ ( 𝐴 ∈ ℂ → ( ( ∗ ‘ 𝐴 ) = ( ∗ ‘ 0 ) ↔ 𝐴 = 0 ) )
4		cj0	⊢ ( ∗ ‘ 0 ) = 0
5	4	eqeq2i	⊢ ( ( ∗ ‘ 𝐴 ) = ( ∗ ‘ 0 ) ↔ ( ∗ ‘ 𝐴 ) = 0 )
6	3 5	bitr3di	⊢ ( 𝐴 ∈ ℂ → ( 𝐴 = 0 ↔ ( ∗ ‘ 𝐴 ) = 0 ) )
7	6	necon3bid	⊢ ( 𝐴 ∈ ℂ → ( 𝐴 ≠ 0 ↔ ( ∗ ‘ 𝐴 ) ≠ 0 ) )