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Metamath Proof Explorer
Description: The reciprocal of a nonzero number is nonzero. (Contributed by NM, 9-Feb-2006) (Proof shortened by Mario Carneiro, 27-May-2016)
|
|
Ref |
Expression |
|
Assertion |
recne0 |
⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ) → ( 1 / 𝐴 ) ≠ 0 ) |
Proof
| Step |
Hyp |
Ref |
Expression |
| 1 |
|
ax-1cn |
⊢ 1 ∈ ℂ |
| 2 |
|
ax-1ne0 |
⊢ 1 ≠ 0 |
| 3 |
|
divne0 |
⊢ ( ( ( 1 ∈ ℂ ∧ 1 ≠ 0 ) ∧ ( 𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ) ) → ( 1 / 𝐴 ) ≠ 0 ) |
| 4 |
1 2 3
|
mpanl12 |
⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ) → ( 1 / 𝐴 ) ≠ 0 ) |