This is an inofficial mirror of http://metamath.tirix.org for personal testing of a visualizer extension only.
Metamath Proof Explorer
Description: Distribution of squaring over multiplication. (Contributed by NM, 21-Mar-2008)
|
|
Ref |
Expression |
|
Assertion |
sqmul |
⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ) → ( ( 𝐴 · 𝐵 ) ↑ 2 ) = ( ( 𝐴 ↑ 2 ) · ( 𝐵 ↑ 2 ) ) ) |
Proof
| Step |
Hyp |
Ref |
Expression |
| 1 |
|
2nn0 |
⊢ 2 ∈ ℕ0 |
| 2 |
|
mulexp |
⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 2 ∈ ℕ0 ) → ( ( 𝐴 · 𝐵 ) ↑ 2 ) = ( ( 𝐴 ↑ 2 ) · ( 𝐵 ↑ 2 ) ) ) |
| 3 |
1 2
|
mp3an3 |
⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ) → ( ( 𝐴 · 𝐵 ) ↑ 2 ) = ( ( 𝐴 ↑ 2 ) · ( 𝐵 ↑ 2 ) ) ) |