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Metamath Proof Explorer
Description: An integer power of a reciprocal is the reciprocal of the integer
power with same exponent. (Contributed by Mario Carneiro, 28-May-2016)
|
|
Ref |
Expression |
|
Hypotheses |
expcld.1 |
⊢ ( 𝜑 → 𝐴 ∈ ℂ ) |
|
|
sqrecd.1 |
⊢ ( 𝜑 → 𝐴 ≠ 0 ) |
|
|
expclzd.3 |
⊢ ( 𝜑 → 𝑁 ∈ ℤ ) |
|
Assertion |
exprecd |
⊢ ( 𝜑 → ( ( 1 / 𝐴 ) ↑ 𝑁 ) = ( 1 / ( 𝐴 ↑ 𝑁 ) ) ) |
Proof
| Step |
Hyp |
Ref |
Expression |
| 1 |
|
expcld.1 |
⊢ ( 𝜑 → 𝐴 ∈ ℂ ) |
| 2 |
|
sqrecd.1 |
⊢ ( 𝜑 → 𝐴 ≠ 0 ) |
| 3 |
|
expclzd.3 |
⊢ ( 𝜑 → 𝑁 ∈ ℤ ) |
| 4 |
|
exprec |
⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 𝑁 ∈ ℤ ) → ( ( 1 / 𝐴 ) ↑ 𝑁 ) = ( 1 / ( 𝐴 ↑ 𝑁 ) ) ) |
| 5 |
1 2 3 4
|
syl3anc |
⊢ ( 𝜑 → ( ( 1 / 𝐴 ) ↑ 𝑁 ) = ( 1 / ( 𝐴 ↑ 𝑁 ) ) ) |