Metamath Proof Explorer
Description: Absolute value of positive integer exponentiation. (Contributed by Mario Carneiro, 29-May-2016)
|
|
Ref |
Expression |
|
Hypotheses |
abscld.1 |
⊢ ( 𝜑 → 𝐴 ∈ ℂ ) |
|
|
absexpd.2 |
⊢ ( 𝜑 → 𝑁 ∈ ℕ0 ) |
|
Assertion |
absexpd |
⊢ ( 𝜑 → ( abs ‘ ( 𝐴 ↑ 𝑁 ) ) = ( ( abs ‘ 𝐴 ) ↑ 𝑁 ) ) |
Proof
Step |
Hyp |
Ref |
Expression |
1 |
|
abscld.1 |
⊢ ( 𝜑 → 𝐴 ∈ ℂ ) |
2 |
|
absexpd.2 |
⊢ ( 𝜑 → 𝑁 ∈ ℕ0 ) |
3 |
|
absexp |
⊢ ( ( 𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ0 ) → ( abs ‘ ( 𝐴 ↑ 𝑁 ) ) = ( ( abs ‘ 𝐴 ) ↑ 𝑁 ) ) |
4 |
1 2 3
|
syl2anc |
⊢ ( 𝜑 → ( abs ‘ ( 𝐴 ↑ 𝑁 ) ) = ( ( abs ‘ 𝐴 ) ↑ 𝑁 ) ) |