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 ) ) )