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Metamath Proof Explorer

Theorem recne0d

Description: The reciprocal of a nonzero number is nonzero. (Contributed by Mario Carneiro, 27-May-2016)

Ref Expression
Hypotheses div1d.1 ⊢ ( 𝜑 → 𝐴 ∈ ℂ )
reccld.2 ⊢ ( 𝜑 → 𝐴 ≠ 0 )
Assertion recne0d ( 𝜑 → ( 1 / 𝐴 ) ≠ 0 )

Proof

Step Hyp Ref Expression
1 div1d.1 ⊢ ( 𝜑 → 𝐴 ∈ ℂ )
2 reccld.2 ⊢ ( 𝜑 → 𝐴 ≠ 0 )
3 recne0 ⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ) → ( 1 / 𝐴 ) ≠ 0 )
4 1 2 3 syl2anc ⊢ ( 𝜑 → ( 1 / 𝐴 ) ≠ 0 )