Metamath Proof Explorer
Description: Value of absolute value function. Definition 10.36 of Gleason p. 133.
(Contributed by NM, 2-Oct-1999)
|
|
Ref |
Expression |
|
Hypothesis |
absvalsqi.1 |
⊢ 𝐴 ∈ ℂ |
|
Assertion |
absval2i |
⊢ ( abs ‘ 𝐴 ) = ( √ ‘ ( ( ( ℜ ‘ 𝐴 ) ↑ 2 ) + ( ( ℑ ‘ 𝐴 ) ↑ 2 ) ) ) |
Proof
Step |
Hyp |
Ref |
Expression |
1 |
|
absvalsqi.1 |
⊢ 𝐴 ∈ ℂ |
2 |
|
absval2 |
⊢ ( 𝐴 ∈ ℂ → ( abs ‘ 𝐴 ) = ( √ ‘ ( ( ( ℜ ‘ 𝐴 ) ↑ 2 ) + ( ( ℑ ‘ 𝐴 ) ↑ 2 ) ) ) ) |
3 |
1 2
|
ax-mp |
⊢ ( abs ‘ 𝐴 ) = ( √ ‘ ( ( ( ℜ ‘ 𝐴 ) ↑ 2 ) + ( ( ℑ ‘ 𝐴 ) ↑ 2 ) ) ) |