Step |
Hyp |
Ref |
Expression |
1 |
|
nvdi.1 |
⊢ 𝑋 = ( BaseSet ‘ 𝑈 ) |
2 |
|
nvdi.2 |
⊢ 𝐺 = ( +𝑣 ‘ 𝑈 ) |
3 |
|
nvdi.4 |
⊢ 𝑆 = ( ·𝑠OLD ‘ 𝑈 ) |
4 |
|
eqid |
⊢ ( 1st ‘ 𝑈 ) = ( 1st ‘ 𝑈 ) |
5 |
4
|
nvvc |
⊢ ( 𝑈 ∈ NrmCVec → ( 1st ‘ 𝑈 ) ∈ CVecOLD ) |
6 |
2
|
vafval |
⊢ 𝐺 = ( 1st ‘ ( 1st ‘ 𝑈 ) ) |
7 |
3
|
smfval |
⊢ 𝑆 = ( 2nd ‘ ( 1st ‘ 𝑈 ) ) |
8 |
1 2
|
bafval |
⊢ 𝑋 = ran 𝐺 |
9 |
6 7 8
|
vcdi |
⊢ ( ( ( 1st ‘ 𝑈 ) ∈ CVecOLD ∧ ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋 ) ) → ( 𝐴 𝑆 ( 𝐵 𝐺 𝐶 ) ) = ( ( 𝐴 𝑆 𝐵 ) 𝐺 ( 𝐴 𝑆 𝐶 ) ) ) |
10 |
5 9
|
sylan |
⊢ ( ( 𝑈 ∈ NrmCVec ∧ ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋 ) ) → ( 𝐴 𝑆 ( 𝐵 𝐺 𝐶 ) ) = ( ( 𝐴 𝑆 𝐵 ) 𝐺 ( 𝐴 𝑆 𝐶 ) ) ) |