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 ) ) ) |
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 ) ) ) |