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Metamath Proof Explorer

Theorem recclzi

Description: Closure law for reciprocal. (Contributed by NM, 30-Apr-2005)

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

Proof

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