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Metamath Proof Explorer

Theorem mulne0bd

Description: The product of two nonzero numbers is nonzero. (Contributed by Mario Carneiro, 27-May-2016)

Ref Expression
Hypotheses msq0d.1 ⊢ ( 𝜑 → 𝐴 ∈ ℂ )
mulne0bd.2 ⊢ ( 𝜑 → 𝐵 ∈ ℂ )
Assertion mulne0bd ( 𝜑 → ( ( 𝐴 ≠ 0 ∧ 𝐵 ≠ 0 ) ↔ ( 𝐴 · 𝐵 ) ≠ 0 ) )

Proof

Step Hyp Ref Expression
1 msq0d.1 ⊢ ( 𝜑 → 𝐴 ∈ ℂ )
2 mulne0bd.2 ⊢ ( 𝜑 → 𝐵 ∈ ℂ )
3 mulne0b ⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ) → ( ( 𝐴 ≠ 0 ∧ 𝐵 ≠ 0 ) ↔ ( 𝐴 · 𝐵 ) ≠ 0 ) )
4 1 2 3 syl2anc ⊢ ( 𝜑 → ( ( 𝐴 ≠ 0 ∧ 𝐵 ≠ 0 ) ↔ ( 𝐴 · 𝐵 ) ≠ 0 ) )