Metamath Proof Explorer
Description: The ratio of nonzero numbers is nonzero. (Contributed by Mario
Carneiro, 27-May-2016)
|
|
Ref |
Expression |
|
Hypotheses |
div1d.1 |
⊢ ( 𝜑 → 𝐴 ∈ ℂ ) |
|
|
divcld.2 |
⊢ ( 𝜑 → 𝐵 ∈ ℂ ) |
|
|
divne0d.3 |
⊢ ( 𝜑 → 𝐴 ≠ 0 ) |
|
|
divne0d.4 |
⊢ ( 𝜑 → 𝐵 ≠ 0 ) |
|
Assertion |
divne0d |
⊢ ( 𝜑 → ( 𝐴 / 𝐵 ) ≠ 0 ) |
Proof
Step |
Hyp |
Ref |
Expression |
1 |
|
div1d.1 |
⊢ ( 𝜑 → 𝐴 ∈ ℂ ) |
2 |
|
divcld.2 |
⊢ ( 𝜑 → 𝐵 ∈ ℂ ) |
3 |
|
divne0d.3 |
⊢ ( 𝜑 → 𝐴 ≠ 0 ) |
4 |
|
divne0d.4 |
⊢ ( 𝜑 → 𝐵 ≠ 0 ) |
5 |
|
divne0 |
⊢ ( ( ( 𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ) ∧ ( 𝐵 ∈ ℂ ∧ 𝐵 ≠ 0 ) ) → ( 𝐴 / 𝐵 ) ≠ 0 ) |
6 |
1 3 2 4 5
|
syl22anc |
⊢ ( 𝜑 → ( 𝐴 / 𝐵 ) ≠ 0 ) |