Description: The product of two nonzero numbers is nonzero. (Contributed by NM, 1-Aug-2004) (Proof shortened by Andrew Salmon, 19-Nov-2011)
Ref | Expression | ||
---|---|---|---|
Assertion | mulne0b | ⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ) → ( ( 𝐴 ≠ 0 ∧ 𝐵 ≠ 0 ) ↔ ( 𝐴 · 𝐵 ) ≠ 0 ) ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | neanior | ⊢ ( ( 𝐴 ≠ 0 ∧ 𝐵 ≠ 0 ) ↔ ¬ ( 𝐴 = 0 ∨ 𝐵 = 0 ) ) | |
2 | mul0or | ⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ) → ( ( 𝐴 · 𝐵 ) = 0 ↔ ( 𝐴 = 0 ∨ 𝐵 = 0 ) ) ) | |
3 | 2 | necon3abid | ⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ) → ( ( 𝐴 · 𝐵 ) ≠ 0 ↔ ¬ ( 𝐴 = 0 ∨ 𝐵 = 0 ) ) ) |
4 | 1 3 | bitr4id | ⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ) → ( ( 𝐴 ≠ 0 ∧ 𝐵 ≠ 0 ) ↔ ( 𝐴 · 𝐵 ) ≠ 0 ) ) |