Metamath Proof Explorer


Theorem ipipcj

Description: An inner product times its conjugate. (Contributed by NM, 23-Nov-2007) (New usage is discouraged.)

Ref Expression
Hypotheses ipcl.1 𝑋 = ( BaseSet ‘ 𝑈 )
ipcl.7 𝑃 = ( ·𝑖OLD𝑈 )
Assertion ipipcj ( ( 𝑈 ∈ NrmCVec ∧ 𝐴𝑋𝐵𝑋 ) → ( ( 𝐴 𝑃 𝐵 ) · ( 𝐵 𝑃 𝐴 ) ) = ( ( abs ‘ ( 𝐴 𝑃 𝐵 ) ) ↑ 2 ) )

Proof

Step Hyp Ref Expression
1 ipcl.1 𝑋 = ( BaseSet ‘ 𝑈 )
2 ipcl.7 𝑃 = ( ·𝑖OLD𝑈 )
3 1 2 dipcl ( ( 𝑈 ∈ NrmCVec ∧ 𝐴𝑋𝐵𝑋 ) → ( 𝐴 𝑃 𝐵 ) ∈ ℂ )
4 3 absvalsqd ( ( 𝑈 ∈ NrmCVec ∧ 𝐴𝑋𝐵𝑋 ) → ( ( abs ‘ ( 𝐴 𝑃 𝐵 ) ) ↑ 2 ) = ( ( 𝐴 𝑃 𝐵 ) · ( ∗ ‘ ( 𝐴 𝑃 𝐵 ) ) ) )
5 1 2 dipcj ( ( 𝑈 ∈ NrmCVec ∧ 𝐴𝑋𝐵𝑋 ) → ( ∗ ‘ ( 𝐴 𝑃 𝐵 ) ) = ( 𝐵 𝑃 𝐴 ) )
6 5 oveq2d ( ( 𝑈 ∈ NrmCVec ∧ 𝐴𝑋𝐵𝑋 ) → ( ( 𝐴 𝑃 𝐵 ) · ( ∗ ‘ ( 𝐴 𝑃 𝐵 ) ) ) = ( ( 𝐴 𝑃 𝐵 ) · ( 𝐵 𝑃 𝐴 ) ) )
7 4 6 eqtr2d ( ( 𝑈 ∈ NrmCVec ∧ 𝐴𝑋𝐵𝑋 ) → ( ( 𝐴 𝑃 𝐵 ) · ( 𝐵 𝑃 𝐴 ) ) = ( ( abs ‘ ( 𝐴 𝑃 𝐵 ) ) ↑ 2 ) )