Metamath Proof Explorer
Description: Value of the complex power function when the first argument is zero.
(Contributed by Mario Carneiro, 30-May-2016)
|
|
Ref |
Expression |
|
Hypotheses |
cxp0d.1 |
⊢ ( 𝜑 → 𝐴 ∈ ℂ ) |
|
|
cxpefd.2 |
⊢ ( 𝜑 → 𝐴 ≠ 0 ) |
|
Assertion |
0cxpd |
⊢ ( 𝜑 → ( 0 ↑𝑐 𝐴 ) = 0 ) |
Proof
Step |
Hyp |
Ref |
Expression |
1 |
|
cxp0d.1 |
⊢ ( 𝜑 → 𝐴 ∈ ℂ ) |
2 |
|
cxpefd.2 |
⊢ ( 𝜑 → 𝐴 ≠ 0 ) |
3 |
|
0cxp |
⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ) → ( 0 ↑𝑐 𝐴 ) = 0 ) |
4 |
1 2 3
|
syl2anc |
⊢ ( 𝜑 → ( 0 ↑𝑐 𝐴 ) = 0 ) |