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Metamath Proof Explorer

Theorem ipcni

Description: Standard inner product on complex numbers. (Contributed by NM, 2-Oct-1999)

Ref Expression
Hypotheses recl.1 ⊢ 𝐴 ∈ ℂ
readdi.2 ⊢ 𝐵 ∈ ℂ
Assertion ipcni ( ℜ ‘ ( 𝐴 · ( ∗ ‘ 𝐵 ) ) ) = ( ( ( ℜ ‘ 𝐴 ) · ( ℜ ‘ 𝐵 ) ) + ( ( ℑ ‘ 𝐴 ) · ( ℑ ‘ 𝐵 ) ) )

Proof

Step Hyp Ref Expression
1 recl.1 ⊢ 𝐴 ∈ ℂ
2 readdi.2 ⊢ 𝐵 ∈ ℂ
3 ipcnval ⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ) → ( ℜ ‘ ( 𝐴 · ( ∗ ‘ 𝐵 ) ) ) = ( ( ( ℜ ‘ 𝐴 ) · ( ℜ ‘ 𝐵 ) ) + ( ( ℑ ‘ 𝐴 ) · ( ℑ ‘ 𝐵 ) ) ) )
4 1 2 3 mp2an ⊢ ( ℜ ‘ ( 𝐴 · ( ∗ ‘ 𝐵 ) ) ) = ( ( ( ℜ ‘ 𝐴 ) · ( ℜ ‘ 𝐵 ) ) + ( ( ℑ ‘ 𝐴 ) · ( ℑ ‘ 𝐵 ) ) )