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Metamath Proof Explorer

Theorem cjmulval

Description: A complex number times its conjugate. (Contributed by NM, 1-Feb-2007) (Revised by Mario Carneiro, 14-Jul-2014)

Ref Expression
Assertion cjmulval ( 𝐴 ∈ ℂ → ( 𝐴 · ( ∗ ‘ 𝐴 ) ) = ( ( ( ℜ ‘ 𝐴 ) ↑ 2 ) + ( ( ℑ ‘ 𝐴 ) ↑ 2 ) ) )

Proof

Step Hyp Ref Expression
1 recl ⊢ ( 𝐴 ∈ ℂ → ( ℜ ‘ 𝐴 ) ∈ ℝ )
2 1 recnd ⊢ ( 𝐴 ∈ ℂ → ( ℜ ‘ 𝐴 ) ∈ ℂ )
3 2 sqvald ⊢ ( 𝐴 ∈ ℂ → ( ( ℜ ‘ 𝐴 ) ↑ 2 ) = ( ( ℜ ‘ 𝐴 ) · ( ℜ ‘ 𝐴 ) ) )
4 imcl ⊢ ( 𝐴 ∈ ℂ → ( ℑ ‘ 𝐴 ) ∈ ℝ )
5 4 recnd ⊢ ( 𝐴 ∈ ℂ → ( ℑ ‘ 𝐴 ) ∈ ℂ )
6 5 sqvald ⊢ ( 𝐴 ∈ ℂ → ( ( ℑ ‘ 𝐴 ) ↑ 2 ) = ( ( ℑ ‘ 𝐴 ) · ( ℑ ‘ 𝐴 ) ) )
7 3 6 oveq12d ⊢ ( 𝐴 ∈ ℂ → ( ( ( ℜ ‘ 𝐴 ) ↑ 2 ) + ( ( ℑ ‘ 𝐴 ) ↑ 2 ) ) = ( ( ( ℜ ‘ 𝐴 ) · ( ℜ ‘ 𝐴 ) ) + ( ( ℑ ‘ 𝐴 ) · ( ℑ ‘ 𝐴 ) ) ) )
8 ipcnval ⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐴 ∈ ℂ ) → ( ℜ ‘ ( 𝐴 · ( ∗ ‘ 𝐴 ) ) ) = ( ( ( ℜ ‘ 𝐴 ) · ( ℜ ‘ 𝐴 ) ) + ( ( ℑ ‘ 𝐴 ) · ( ℑ ‘ 𝐴 ) ) ) )
9 8 anidms ⊢ ( 𝐴 ∈ ℂ → ( ℜ ‘ ( 𝐴 · ( ∗ ‘ 𝐴 ) ) ) = ( ( ( ℜ ‘ 𝐴 ) · ( ℜ ‘ 𝐴 ) ) + ( ( ℑ ‘ 𝐴 ) · ( ℑ ‘ 𝐴 ) ) ) )
10 cjmulrcl ⊢ ( 𝐴 ∈ ℂ → ( 𝐴 · ( ∗ ‘ 𝐴 ) ) ∈ ℝ )
11 rere ⊢ ( ( 𝐴 · ( ∗ ‘ 𝐴 ) ) ∈ ℝ → ( ℜ ‘ ( 𝐴 · ( ∗ ‘ 𝐴 ) ) ) = ( 𝐴 · ( ∗ ‘ 𝐴 ) ) )
12 10 11 syl ⊢ ( 𝐴 ∈ ℂ → ( ℜ ‘ ( 𝐴 · ( ∗ ‘ 𝐴 ) ) ) = ( 𝐴 · ( ∗ ‘ 𝐴 ) ) )
13 7 9 12 3eqtr2rd ⊢ ( 𝐴 ∈ ℂ → ( 𝐴 · ( ∗ ‘ 𝐴 ) ) = ( ( ( ℜ ‘ 𝐴 ) ↑ 2 ) + ( ( ℑ ‘ 𝐴 ) ↑ 2 ) ) )