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 ) )