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Metamath Proof Explorer

Theorem diveq0d

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

Ref Expression
Hypotheses div1d.1 ⊢ ( 𝜑 → 𝐴 ∈ ℂ )
divcld.2 ⊢ ( 𝜑 → 𝐵 ∈ ℂ )
divcld.3 ⊢ ( 𝜑 → 𝐵 ≠ 0 )
diveq0d.4 ⊢ ( 𝜑 → ( 𝐴 / 𝐵 ) = 0 )
Assertion diveq0d ( 𝜑 → 𝐴 = 0 )

Proof

Step Hyp Ref Expression
1 div1d.1 ⊢ ( 𝜑 → 𝐴 ∈ ℂ )
2 divcld.2 ⊢ ( 𝜑 → 𝐵 ∈ ℂ )
3 divcld.3 ⊢ ( 𝜑 → 𝐵 ≠ 0 )
4 diveq0d.4 ⊢ ( 𝜑 → ( 𝐴 / 𝐵 ) = 0 )
5 diveq0 ⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐵 ≠ 0 ) → ( ( 𝐴 / 𝐵 ) = 0 ↔ 𝐴 = 0 ) )
6 1 2 3 5 syl3anc ⊢ ( 𝜑 → ( ( 𝐴 / 𝐵 ) = 0 ↔ 𝐴 = 0 ) )
7 4 6 mpbid ⊢ ( 𝜑 → 𝐴 = 0 )