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Metamath Proof Explorer

Theorem divne0i

Description: The ratio of nonzero numbers is nonzero. (Contributed by NM, 9-Feb-1995)

Ref Expression
Hypotheses divclz.1 ⊢ A ∈ ℂ
divclz.2 ⊢ B ∈ ℂ
divneq0.3 ⊢ A ≠ 0
divneq0.4 ⊢ B ≠ 0
Assertion divne0i ⊢ A B ≠ 0

Proof

Step Hyp Ref Expression
1 divclz.1 ⊢ A ∈ ℂ
2 divclz.2 ⊢ B ∈ ℂ
3 divneq0.3 ⊢ A ≠ 0
4 divneq0.4 ⊢ B ≠ 0
5 divne0 ⊢ A ∈ ℂ ∧ A ≠ 0 ∧ B ∈ ℂ ∧ B ≠ 0 → A B ≠ 0
6 1 3 2 4 5 mp4an ⊢ A B ≠ 0