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 ) ) |