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Metamath Proof Explorer
Description: An integer power is nonzero if its base is nonzero. (Contributed by NM, 2-Aug-2006) (Revised by Mario Carneiro, 4-Jun-2014)
|
|
Ref |
Expression |
|
Assertion |
expne0i |
⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 𝑁 ∈ ℤ ) → ( 𝐴 ↑ 𝑁 ) ≠ 0 ) |
Proof
| Step |
Hyp |
Ref |
Expression |
| 1 |
|
expclzlem |
⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 𝑁 ∈ ℤ ) → ( 𝐴 ↑ 𝑁 ) ∈ ( ℂ ∖ { 0 } ) ) |
| 2 |
|
eldifsni |
⊢ ( ( 𝐴 ↑ 𝑁 ) ∈ ( ℂ ∖ { 0 } ) → ( 𝐴 ↑ 𝑁 ) ≠ 0 ) |
| 3 |
1 2
|
syl |
⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 𝑁 ∈ ℤ ) → ( 𝐴 ↑ 𝑁 ) ≠ 0 ) |