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Metamath Proof Explorer
Description: Square of absolute value of sum. Proposition 10-3.7(g) of Gleason
p. 133. (Contributed by NM, 2-Oct-1999)
|
|
Ref |
Expression |
|
Hypotheses |
absvalsqi.1 |
⊢ 𝐴 ∈ ℂ |
|
|
abssub.2 |
⊢ 𝐵 ∈ ℂ |
|
Assertion |
sqabsaddi |
⊢ ( ( abs ‘ ( 𝐴 + 𝐵 ) ) ↑ 2 ) = ( ( ( ( abs ‘ 𝐴 ) ↑ 2 ) + ( ( abs ‘ 𝐵 ) ↑ 2 ) ) + ( 2 · ( ℜ ‘ ( 𝐴 · ( ∗ ‘ 𝐵 ) ) ) ) ) |
Proof
| Step |
Hyp |
Ref |
Expression |
| 1 |
|
absvalsqi.1 |
⊢ 𝐴 ∈ ℂ |
| 2 |
|
abssub.2 |
⊢ 𝐵 ∈ ℂ |
| 3 |
|
sqabsadd |
⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ) → ( ( abs ‘ ( 𝐴 + 𝐵 ) ) ↑ 2 ) = ( ( ( ( abs ‘ 𝐴 ) ↑ 2 ) + ( ( abs ‘ 𝐵 ) ↑ 2 ) ) + ( 2 · ( ℜ ‘ ( 𝐴 · ( ∗ ‘ 𝐵 ) ) ) ) ) ) |
| 4 |
1 2 3
|
mp2an |
⊢ ( ( abs ‘ ( 𝐴 + 𝐵 ) ) ↑ 2 ) = ( ( ( ( abs ‘ 𝐴 ) ↑ 2 ) + ( ( abs ‘ 𝐵 ) ↑ 2 ) ) + ( 2 · ( ℜ ‘ ( 𝐴 · ( ∗ ‘ 𝐵 ) ) ) ) ) |