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		simp1	⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐵 ≠ 0 ) → 𝐴 ∈ ℂ )
2		0cnd	⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐵 ≠ 0 ) → 0 ∈ ℂ )
3		3simpc	⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐵 ≠ 0 ) → ( 𝐵 ∈ ℂ ∧ 𝐵 ≠ 0 ) )
4		divmul2	⊢ ( ( 𝐴 ∈ ℂ ∧ 0 ∈ ℂ ∧ ( 𝐵 ∈ ℂ ∧ 𝐵 ≠ 0 ) ) → ( ( 𝐴 / 𝐵 ) = 0 ↔ 𝐴 = ( 𝐵 · 0 ) ) )
5	1 2 3 4	syl3anc	⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐵 ≠ 0 ) → ( ( 𝐴 / 𝐵 ) = 0 ↔ 𝐴 = ( 𝐵 · 0 ) ) )
6		simp2	⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐵 ≠ 0 ) → 𝐵 ∈ ℂ )
7	6	mul01d	⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐵 ≠ 0 ) → ( 𝐵 · 0 ) = 0 )
8	7	eqeq2d	⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐵 ≠ 0 ) → ( 𝐴 = ( 𝐵 · 0 ) ↔ 𝐴 = 0 ) )
9	5 8	bitrd	⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐵 ≠ 0 ) → ( ( 𝐴 / 𝐵 ) = 0 ↔ 𝐴 = 0 ) )