Metamath Proof Explorer


Theorem absreimsq

Description: Square of the absolute value of a number that has been decomposed into real and imaginary parts. (Contributed by NM, 1-Feb-2007)

Ref Expression
Assertion absreimsq ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ) → ( ( abs ‘ ( 𝐴 + ( i · 𝐵 ) ) ) ↑ 2 ) = ( ( 𝐴 ↑ 2 ) + ( 𝐵 ↑ 2 ) ) )

Proof

Step Hyp Ref Expression
1 recn ( 𝐴 ∈ ℝ → 𝐴 ∈ ℂ )
2 ax-icn i ∈ ℂ
3 recn ( 𝐵 ∈ ℝ → 𝐵 ∈ ℂ )
4 mulcl ( ( i ∈ ℂ ∧ 𝐵 ∈ ℂ ) → ( i · 𝐵 ) ∈ ℂ )
5 2 3 4 sylancr ( 𝐵 ∈ ℝ → ( i · 𝐵 ) ∈ ℂ )
6 addcl ( ( 𝐴 ∈ ℂ ∧ ( i · 𝐵 ) ∈ ℂ ) → ( 𝐴 + ( i · 𝐵 ) ) ∈ ℂ )
7 1 5 6 syl2an ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ) → ( 𝐴 + ( i · 𝐵 ) ) ∈ ℂ )
8 absvalsq2 ( ( 𝐴 + ( i · 𝐵 ) ) ∈ ℂ → ( ( abs ‘ ( 𝐴 + ( i · 𝐵 ) ) ) ↑ 2 ) = ( ( ( ℜ ‘ ( 𝐴 + ( i · 𝐵 ) ) ) ↑ 2 ) + ( ( ℑ ‘ ( 𝐴 + ( i · 𝐵 ) ) ) ↑ 2 ) ) )
9 7 8 syl ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ) → ( ( abs ‘ ( 𝐴 + ( i · 𝐵 ) ) ) ↑ 2 ) = ( ( ( ℜ ‘ ( 𝐴 + ( i · 𝐵 ) ) ) ↑ 2 ) + ( ( ℑ ‘ ( 𝐴 + ( i · 𝐵 ) ) ) ↑ 2 ) ) )
10 crre ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ) → ( ℜ ‘ ( 𝐴 + ( i · 𝐵 ) ) ) = 𝐴 )
11 10 oveq1d ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ) → ( ( ℜ ‘ ( 𝐴 + ( i · 𝐵 ) ) ) ↑ 2 ) = ( 𝐴 ↑ 2 ) )
12 crim ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ) → ( ℑ ‘ ( 𝐴 + ( i · 𝐵 ) ) ) = 𝐵 )
13 12 oveq1d ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ) → ( ( ℑ ‘ ( 𝐴 + ( i · 𝐵 ) ) ) ↑ 2 ) = ( 𝐵 ↑ 2 ) )
14 11 13 oveq12d ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ) → ( ( ( ℜ ‘ ( 𝐴 + ( i · 𝐵 ) ) ) ↑ 2 ) + ( ( ℑ ‘ ( 𝐴 + ( i · 𝐵 ) ) ) ↑ 2 ) ) = ( ( 𝐴 ↑ 2 ) + ( 𝐵 ↑ 2 ) ) )
15 9 14 eqtrd ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ) → ( ( abs ‘ ( 𝐴 + ( i · 𝐵 ) ) ) ↑ 2 ) = ( ( 𝐴 ↑ 2 ) + ( 𝐵 ↑ 2 ) ) )