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Metamath Proof Explorer

Theorem diveq0

Description: A ratio is zero iff the numerator is zero. (Contributed by NM, 20-Apr-2006) (Revised by Mario Carneiro, 27-May-2016)

Ref Expression
Assertion diveq0 ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐵 ≠ 0 ) → ( ( 𝐴 / 𝐵 ) = 0 ↔ 𝐴 = 0 ) )

Proof

Step Hyp Ref Expression
1 0cn ⊢ 0 ∈ ℂ
2 divmul2 ⊢ ( ( 𝐴 ∈ ℂ ∧ 0 ∈ ℂ ∧ ( 𝐵 ∈ ℂ ∧ 𝐵 ≠ 0 ) ) → ( ( 𝐴 / 𝐵 ) = 0 ↔ 𝐴 = ( 𝐵 · 0 ) ) )
3 1 2 mp3an2 ⊢ ( ( 𝐴 ∈ ℂ ∧ ( 𝐵 ∈ ℂ ∧ 𝐵 ≠ 0 ) ) → ( ( 𝐴 / 𝐵 ) = 0 ↔ 𝐴 = ( 𝐵 · 0 ) ) )
4 3 3impb ⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐵 ≠ 0 ) → ( ( 𝐴 / 𝐵 ) = 0 ↔ 𝐴 = ( 𝐵 · 0 ) ) )
5 simp2 ⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐵 ≠ 0 ) → 𝐵 ∈ ℂ )
6 5 mul01d ⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐵 ≠ 0 ) → ( 𝐵 · 0 ) = 0 )
7 6 eqeq2d ⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐵 ≠ 0 ) → ( 𝐴 = ( 𝐵 · 0 ) ↔ 𝐴 = 0 ) )
8 4 7 bitrd ⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐵 ≠ 0 ) → ( ( 𝐴 / 𝐵 ) = 0 ↔ 𝐴 = 0 ) )