Metamath Proof Explorer


Theorem divne0b

Description: The ratio of nonzero numbers is nonzero. (Contributed by NM, 2-Aug-2004) (Proof shortened by Mario Carneiro, 27-May-2016)

Ref Expression
Assertion divne0b ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐵 ≠ 0 ) → ( 𝐴 ≠ 0 ↔ ( 𝐴 / 𝐵 ) ≠ 0 ) )

Proof

Step Hyp Ref Expression
1 diveq0 ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐵 ≠ 0 ) → ( ( 𝐴 / 𝐵 ) = 0 ↔ 𝐴 = 0 ) )
2 1 bicomd ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐵 ≠ 0 ) → ( 𝐴 = 0 ↔ ( 𝐴 / 𝐵 ) = 0 ) )
3 2 necon3bid ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐵 ≠ 0 ) → ( 𝐴 ≠ 0 ↔ ( 𝐴 / 𝐵 ) ≠ 0 ) )