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Metamath Proof Explorer

Theorem ipcnd

Description: Standard inner product on complex numbers. (Contributed by Mario Carneiro, 29-May-2016)

Ref Expression
Hypotheses recld.1 ⊢ ( 𝜑 → 𝐴 ∈ ℂ )
readdd.2 ⊢ ( 𝜑 → 𝐵 ∈ ℂ )
Assertion ipcnd ( 𝜑 → ( ℜ ‘ ( 𝐴 · ( ∗ ‘ 𝐵 ) ) ) = ( ( ( ℜ ‘ 𝐴 ) · ( ℜ ‘ 𝐵 ) ) + ( ( ℑ ‘ 𝐴 ) · ( ℑ ‘ 𝐵 ) ) ) )

Proof

Step Hyp Ref Expression
1 recld.1 ⊢ ( 𝜑 → 𝐴 ∈ ℂ )
2 readdd.2 ⊢ ( 𝜑 → 𝐵 ∈ ℂ )
3 ipcnval ⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ) → ( ℜ ‘ ( 𝐴 · ( ∗ ‘ 𝐵 ) ) ) = ( ( ( ℜ ‘ 𝐴 ) · ( ℜ ‘ 𝐵 ) ) + ( ( ℑ ‘ 𝐴 ) · ( ℑ ‘ 𝐵 ) ) ) )
4 1 2 3 syl2anc ⊢ ( 𝜑 → ( ℜ ‘ ( 𝐴 · ( ∗ ‘ 𝐵 ) ) ) = ( ( ( ℜ ‘ 𝐴 ) · ( ℜ ‘ 𝐵 ) ) + ( ( ℑ ‘ 𝐴 ) · ( ℑ ‘ 𝐵 ) ) ) )