Metamath Proof Explorer
Description: Relate the complex power function to the integer power function.
(Contributed by Mario Carneiro, 30-May-2016)
|
|
Ref |
Expression |
|
Hypotheses |
cxp0d.1 |
⊢ ( 𝜑 → 𝐴 ∈ ℂ ) |
|
|
cxpefd.2 |
⊢ ( 𝜑 → 𝐴 ≠ 0 ) |
|
|
cxpexpzd.3 |
⊢ ( 𝜑 → 𝐵 ∈ ℤ ) |
|
Assertion |
cxpexpzd |
⊢ ( 𝜑 → ( 𝐴 ↑𝑐 𝐵 ) = ( 𝐴 ↑ 𝐵 ) ) |
Proof
Step |
Hyp |
Ref |
Expression |
1 |
|
cxp0d.1 |
⊢ ( 𝜑 → 𝐴 ∈ ℂ ) |
2 |
|
cxpefd.2 |
⊢ ( 𝜑 → 𝐴 ≠ 0 ) |
3 |
|
cxpexpzd.3 |
⊢ ( 𝜑 → 𝐵 ∈ ℤ ) |
4 |
|
cxpexpz |
⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 𝐵 ∈ ℤ ) → ( 𝐴 ↑𝑐 𝐵 ) = ( 𝐴 ↑ 𝐵 ) ) |
5 |
1 2 3 4
|
syl3anc |
⊢ ( 𝜑 → ( 𝐴 ↑𝑐 𝐵 ) = ( 𝐴 ↑ 𝐵 ) ) |