Step |
Hyp |
Ref |
Expression |
1 |
|
nvsz.4 |
⊢ 𝑆 = ( ·𝑠OLD ‘ 𝑈 ) |
2 |
|
nvsz.6 |
⊢ 𝑍 = ( 0vec ‘ 𝑈 ) |
3 |
|
eqid |
⊢ ( 1st ‘ 𝑈 ) = ( 1st ‘ 𝑈 ) |
4 |
3
|
nvvc |
⊢ ( 𝑈 ∈ NrmCVec → ( 1st ‘ 𝑈 ) ∈ CVecOLD ) |
5 |
|
eqid |
⊢ ( +𝑣 ‘ 𝑈 ) = ( +𝑣 ‘ 𝑈 ) |
6 |
5
|
vafval |
⊢ ( +𝑣 ‘ 𝑈 ) = ( 1st ‘ ( 1st ‘ 𝑈 ) ) |
7 |
1
|
smfval |
⊢ 𝑆 = ( 2nd ‘ ( 1st ‘ 𝑈 ) ) |
8 |
|
eqid |
⊢ ( BaseSet ‘ 𝑈 ) = ( BaseSet ‘ 𝑈 ) |
9 |
8 5
|
bafval |
⊢ ( BaseSet ‘ 𝑈 ) = ran ( +𝑣 ‘ 𝑈 ) |
10 |
|
eqid |
⊢ ( GId ‘ ( +𝑣 ‘ 𝑈 ) ) = ( GId ‘ ( +𝑣 ‘ 𝑈 ) ) |
11 |
6 7 9 10
|
vcz |
⊢ ( ( ( 1st ‘ 𝑈 ) ∈ CVecOLD ∧ 𝐴 ∈ ℂ ) → ( 𝐴 𝑆 ( GId ‘ ( +𝑣 ‘ 𝑈 ) ) ) = ( GId ‘ ( +𝑣 ‘ 𝑈 ) ) ) |
12 |
4 11
|
sylan |
⊢ ( ( 𝑈 ∈ NrmCVec ∧ 𝐴 ∈ ℂ ) → ( 𝐴 𝑆 ( GId ‘ ( +𝑣 ‘ 𝑈 ) ) ) = ( GId ‘ ( +𝑣 ‘ 𝑈 ) ) ) |
13 |
5 2
|
0vfval |
⊢ ( 𝑈 ∈ NrmCVec → 𝑍 = ( GId ‘ ( +𝑣 ‘ 𝑈 ) ) ) |
14 |
13
|
adantr |
⊢ ( ( 𝑈 ∈ NrmCVec ∧ 𝐴 ∈ ℂ ) → 𝑍 = ( GId ‘ ( +𝑣 ‘ 𝑈 ) ) ) |
15 |
14
|
oveq2d |
⊢ ( ( 𝑈 ∈ NrmCVec ∧ 𝐴 ∈ ℂ ) → ( 𝐴 𝑆 𝑍 ) = ( 𝐴 𝑆 ( GId ‘ ( +𝑣 ‘ 𝑈 ) ) ) ) |
16 |
12 15 14
|
3eqtr4d |
⊢ ( ( 𝑈 ∈ NrmCVec ∧ 𝐴 ∈ ℂ ) → ( 𝐴 𝑆 𝑍 ) = 𝑍 ) |