Theorem abshicom

Description: Commuted inner products have the same absolute values. (Contributed by NM, 26-May-2006) (New usage is discouraged.)

	Ref	Expression
Assertion	abshicom	⊢ ( ( 𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ ) → ( abs ‘ ( 𝐴 ·ih 𝐵 ) ) = ( abs ‘ ( 𝐵 ·ih 𝐴 ) ) )

Proof

Step	Hyp	Ref	Expression
1		ax-his1	⊢ ( ( 𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ ) → ( 𝐴 ·ih 𝐵 ) = ( ∗ ‘ ( 𝐵 ·ih 𝐴 ) ) )
2	1	fveq2d	⊢ ( ( 𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ ) → ( abs ‘ ( 𝐴 ·ih 𝐵 ) ) = ( abs ‘ ( ∗ ‘ ( 𝐵 ·ih 𝐴 ) ) ) )
3		hicl	⊢ ( ( 𝐵 ∈ ℋ ∧ 𝐴 ∈ ℋ ) → ( 𝐵 ·ih 𝐴 ) ∈ ℂ )
4	3	ancoms	⊢ ( ( 𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ ) → ( 𝐵 ·ih 𝐴 ) ∈ ℂ )
5	4	abscjd	⊢ ( ( 𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ ) → ( abs ‘ ( ∗ ‘ ( 𝐵 ·ih 𝐴 ) ) ) = ( abs ‘ ( 𝐵 ·ih 𝐴 ) ) )
6	2 5	eqtrd	⊢ ( ( 𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ ) → ( abs ‘ ( 𝐴 ·ih 𝐵 ) ) = ( abs ‘ ( 𝐵 ·ih 𝐴 ) ) )