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Metamath Proof Explorer

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 )