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| Mirrors > Home > MPE Home > Th. List > mulne0d | Structured version Visualization version GIF version | ||
| Description: The product of two nonzero numbers is nonzero. (Contributed by Mario Carneiro, 27-May-2016.) |
| Ref | Expression |
|---|---|
| msq0d.1 | ⊢ (𝜑 → 𝐴 ∈ ℂ) |
| mul0ord.2 | ⊢ (𝜑 → 𝐵 ∈ ℂ) |
| mulne0d.3 | ⊢ (𝜑 → 𝐴 ≠ 0) |
| mulne0d.4 | ⊢ (𝜑 → 𝐵 ≠ 0) |
| Ref | Expression |
|---|---|
| mulne0d | ⊢ (𝜑 → (𝐴 · 𝐵) ≠ 0) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | mulne0d.3 | . 2 ⊢ (𝜑 → 𝐴 ≠ 0) | |
| 2 | mulne0d.4 | . 2 ⊢ (𝜑 → 𝐵 ≠ 0) | |
| 3 | msq0d.1 | . . 3 ⊢ (𝜑 → 𝐴 ∈ ℂ) | |
| 4 | mul0ord.2 | . . 3 ⊢ (𝜑 → 𝐵 ∈ ℂ) | |
| 5 | 3, 4 | mulne0bd 11914 | . 2 ⊢ (𝜑 → ((𝐴 ≠ 0 ∧ 𝐵 ≠ 0) ↔ (𝐴 · 𝐵) ≠ 0)) |
| 6 | 1, 2, 5 | mpbi2and 712 | 1 ⊢ (𝜑 → (𝐴 · 𝐵) ≠ 0) |
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