Step |
Hyp |
Ref |
Expression |
1 |
|
nv0.1 |
⊢ 𝑋 = ( BaseSet ‘ 𝑈 ) |
2 |
|
nv0.4 |
⊢ 𝑆 = ( ·𝑠OLD ‘ 𝑈 ) |
3 |
|
nv0.6 |
⊢ 𝑍 = ( 0vec ‘ 𝑈 ) |
4 |
|
eqid |
⊢ ( 1st ‘ 𝑈 ) = ( 1st ‘ 𝑈 ) |
5 |
4
|
nvvc |
⊢ ( 𝑈 ∈ NrmCVec → ( 1st ‘ 𝑈 ) ∈ CVecOLD ) |
6 |
|
eqid |
⊢ ( +𝑣 ‘ 𝑈 ) = ( +𝑣 ‘ 𝑈 ) |
7 |
6
|
vafval |
⊢ ( +𝑣 ‘ 𝑈 ) = ( 1st ‘ ( 1st ‘ 𝑈 ) ) |
8 |
2
|
smfval |
⊢ 𝑆 = ( 2nd ‘ ( 1st ‘ 𝑈 ) ) |
9 |
1 6
|
bafval |
⊢ 𝑋 = ran ( +𝑣 ‘ 𝑈 ) |
10 |
|
eqid |
⊢ ( GId ‘ ( +𝑣 ‘ 𝑈 ) ) = ( GId ‘ ( +𝑣 ‘ 𝑈 ) ) |
11 |
7 8 9 10
|
vc0 |
⊢ ( ( ( 1st ‘ 𝑈 ) ∈ CVecOLD ∧ 𝐴 ∈ 𝑋 ) → ( 0 𝑆 𝐴 ) = ( GId ‘ ( +𝑣 ‘ 𝑈 ) ) ) |
12 |
5 11
|
sylan |
⊢ ( ( 𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋 ) → ( 0 𝑆 𝐴 ) = ( GId ‘ ( +𝑣 ‘ 𝑈 ) ) ) |
13 |
6 3
|
0vfval |
⊢ ( 𝑈 ∈ NrmCVec → 𝑍 = ( GId ‘ ( +𝑣 ‘ 𝑈 ) ) ) |
14 |
13
|
adantr |
⊢ ( ( 𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋 ) → 𝑍 = ( GId ‘ ( +𝑣 ‘ 𝑈 ) ) ) |
15 |
12 14
|
eqtr4d |
⊢ ( ( 𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋 ) → ( 0 𝑆 𝐴 ) = 𝑍 ) |