Metamath Proof Explorer
Description: Generalize cxpmul2 to negative integers. (Contributed by Mario
Carneiro, 30-May-2016)
|
|
Ref |
Expression |
|
Hypotheses |
cxp0d.1 |
⊢ ( 𝜑 → 𝐴 ∈ ℂ ) |
|
|
cxpefd.2 |
⊢ ( 𝜑 → 𝐴 ≠ 0 ) |
|
|
cxpefd.3 |
⊢ ( 𝜑 → 𝐵 ∈ ℂ ) |
|
|
cxpmul2zd.4 |
⊢ ( 𝜑 → 𝐶 ∈ ℤ ) |
|
Assertion |
cxpmul2zd |
⊢ ( 𝜑 → ( 𝐴 ↑𝑐 ( 𝐵 · 𝐶 ) ) = ( ( 𝐴 ↑𝑐 𝐵 ) ↑ 𝐶 ) ) |
Proof
Step |
Hyp |
Ref |
Expression |
1 |
|
cxp0d.1 |
⊢ ( 𝜑 → 𝐴 ∈ ℂ ) |
2 |
|
cxpefd.2 |
⊢ ( 𝜑 → 𝐴 ≠ 0 ) |
3 |
|
cxpefd.3 |
⊢ ( 𝜑 → 𝐵 ∈ ℂ ) |
4 |
|
cxpmul2zd.4 |
⊢ ( 𝜑 → 𝐶 ∈ ℤ ) |
5 |
|
cxpmul2z |
⊢ ( ( ( 𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ) ∧ ( 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℤ ) ) → ( 𝐴 ↑𝑐 ( 𝐵 · 𝐶 ) ) = ( ( 𝐴 ↑𝑐 𝐵 ) ↑ 𝐶 ) ) |
6 |
1 2 3 4 5
|
syl22anc |
⊢ ( 𝜑 → ( 𝐴 ↑𝑐 ( 𝐵 · 𝐶 ) ) = ( ( 𝐴 ↑𝑐 𝐵 ) ↑ 𝐶 ) ) |