Metamath Proof Explorer


Theorem recne0

Description: The reciprocal of a nonzero number is nonzero. (Contributed by NM, 9-Feb-2006) (Proof shortened by Mario Carneiro, 27-May-2016)

Ref Expression
Assertion recne0 ( ( 𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ) → ( 1 / 𝐴 ) ≠ 0 )

Proof

Step Hyp Ref Expression
1 ax-1cn 1 ∈ ℂ
2 ax-1ne0 1 ≠ 0
3 divne0 ( ( ( 1 ∈ ℂ ∧ 1 ≠ 0 ) ∧ ( 𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ) ) → ( 1 / 𝐴 ) ≠ 0 )
4 1 2 3 mpanl12 ( ( 𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ) → ( 1 / 𝐴 ) ≠ 0 )