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 )