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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 = ( ·𝑖OLD ‘ 𝑈 )
Assertion ipipcj ( ( 𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ) → ( ( 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 = ( ·𝑖OLD ‘ 𝑈 )
3 1 2 dipcl ⊢ ( ( 𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ) → ( 𝐴 𝑃 𝐵 ) ∈ ℂ )
4 3 absvalsqd ⊢ ( ( 𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ) → ( ( abs ‘ ( 𝐴 𝑃 𝐵 ) ) ↑ 2 ) = ( ( 𝐴 𝑃 𝐵 ) · ( ∗ ‘ ( 𝐴 𝑃 𝐵 ) ) ) )
5 1 2 dipcj ⊢ ( ( 𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ) → ( ∗ ‘ ( 𝐴 𝑃 𝐵 ) ) = ( 𝐵 𝑃 𝐴 ) )
6 5 oveq2d ⊢ ( ( 𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ) → ( ( 𝐴 𝑃 𝐵 ) · ( ∗ ‘ ( 𝐴 𝑃 𝐵 ) ) ) = ( ( 𝐴 𝑃 𝐵 ) · ( 𝐵 𝑃 𝐴 ) ) )
7 4 6 eqtr2d ⊢ ( ( 𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ) → ( ( 𝐴 𝑃 𝐵 ) · ( 𝐵 𝑃 𝐴 ) ) = ( ( abs ‘ ( 𝐴 𝑃 𝐵 ) ) ↑ 2 ) )