Step |
Hyp |
Ref |
Expression |
1 |
|
imsdval2.1 |
⊢ 𝑋 = ( BaseSet ‘ 𝑈 ) |
2 |
|
imsdval2.2 |
⊢ 𝐺 = ( +𝑣 ‘ 𝑈 ) |
3 |
|
imsdval2.4 |
⊢ 𝑆 = ( ·𝑠OLD ‘ 𝑈 ) |
4 |
|
imsdval2.6 |
⊢ 𝑁 = ( normCV ‘ 𝑈 ) |
5 |
|
imsdval2.8 |
⊢ 𝐷 = ( IndMet ‘ 𝑈 ) |
6 |
|
eqid |
⊢ ( −𝑣 ‘ 𝑈 ) = ( −𝑣 ‘ 𝑈 ) |
7 |
1 6 4 5
|
imsdval |
⊢ ( ( 𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ) → ( 𝐴 𝐷 𝐵 ) = ( 𝑁 ‘ ( 𝐴 ( −𝑣 ‘ 𝑈 ) 𝐵 ) ) ) |
8 |
1 2 3 6
|
nvmval |
⊢ ( ( 𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ) → ( 𝐴 ( −𝑣 ‘ 𝑈 ) 𝐵 ) = ( 𝐴 𝐺 ( - 1 𝑆 𝐵 ) ) ) |
9 |
8
|
fveq2d |
⊢ ( ( 𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ) → ( 𝑁 ‘ ( 𝐴 ( −𝑣 ‘ 𝑈 ) 𝐵 ) ) = ( 𝑁 ‘ ( 𝐴 𝐺 ( - 1 𝑆 𝐵 ) ) ) ) |
10 |
7 9
|
eqtrd |
⊢ ( ( 𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ) → ( 𝐴 𝐷 𝐵 ) = ( 𝑁 ‘ ( 𝐴 𝐺 ( - 1 𝑆 𝐵 ) ) ) ) |