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Metamath Proof Explorer

Theorem recne0zi

Description: The reciprocal of a nonzero number is nonzero. (Contributed by NM, 14-May-1999)

Ref Expression
Hypothesis divclz.1 ⊢ 𝐴 ∈ ℂ
Assertion recne0zi ( 𝐴 ≠ 0 → ( 1 / 𝐴 ) ≠ 0 )

Proof

Step Hyp Ref Expression
1 divclz.1 ⊢ 𝐴 ∈ ℂ
2 recne0 ⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ) → ( 1 / 𝐴 ) ≠ 0 )
3 1 2 mpan ⊢ ( 𝐴 ≠ 0 → ( 1 / 𝐴 ) ≠ 0 )