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Metamath Proof Explorer

Theorem absreim

Description: Absolute value of a number that has been decomposed into real and imaginary parts. (Contributed by NM, 14-Jan-2006)

Ref Expression
Assertion absreim ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ) → ( abs ‘ ( 𝐴 + ( i · 𝐵 ) ) ) = ( √ ‘ ( ( 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 abscl ⊢ ( ( 𝐴 + ( i · 𝐵 ) ) ∈ ℂ → ( abs ‘ ( 𝐴 + ( i · 𝐵 ) ) ) ∈ ℝ )
9 7 8 syl ⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ) → ( abs ‘ ( 𝐴 + ( i · 𝐵 ) ) ) ∈ ℝ )
10 absge0 ⊢ ( ( 𝐴 + ( i · 𝐵 ) ) ∈ ℂ → 0 ≤ ( abs ‘ ( 𝐴 + ( i · 𝐵 ) ) ) )
11 7 10 syl ⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ) → 0 ≤ ( abs ‘ ( 𝐴 + ( i · 𝐵 ) ) ) )
12 sqrtsq ⊢ ( ( ( abs ‘ ( 𝐴 + ( i · 𝐵 ) ) ) ∈ ℝ ∧ 0 ≤ ( abs ‘ ( 𝐴 + ( i · 𝐵 ) ) ) ) → ( √ ‘ ( ( abs ‘ ( 𝐴 + ( i · 𝐵 ) ) ) ↑ 2 ) ) = ( abs ‘ ( 𝐴 + ( i · 𝐵 ) ) ) )
13 9 11 12 syl2anc ⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ) → ( √ ‘ ( ( abs ‘ ( 𝐴 + ( i · 𝐵 ) ) ) ↑ 2 ) ) = ( abs ‘ ( 𝐴 + ( i · 𝐵 ) ) ) )
14 absreimsq ⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ) → ( ( abs ‘ ( 𝐴 + ( i · 𝐵 ) ) ) ↑ 2 ) = ( ( 𝐴 ↑ 2 ) + ( 𝐵 ↑ 2 ) ) )
15 14 fveq2d ⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ) → ( √ ‘ ( ( abs ‘ ( 𝐴 + ( i · 𝐵 ) ) ) ↑ 2 ) ) = ( √ ‘ ( ( 𝐴 ↑ 2 ) + ( 𝐵 ↑ 2 ) ) ) )
16 13 15 eqtr3d ⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ) → ( abs ‘ ( 𝐴 + ( i · 𝐵 ) ) ) = ( √ ‘ ( ( 𝐴 ↑ 2 ) + ( 𝐵 ↑ 2 ) ) ) )