Step |
Hyp |
Ref |
Expression |
1 |
|
dip0r.1 |
⊢ 𝑋 = ( BaseSet ‘ 𝑈 ) |
2 |
|
dip0r.5 |
⊢ 𝑍 = ( 0vec ‘ 𝑈 ) |
3 |
|
dip0r.7 |
⊢ 𝑃 = ( ·𝑖OLD ‘ 𝑈 ) |
4 |
1 2
|
nvzcl |
⊢ ( 𝑈 ∈ NrmCVec → 𝑍 ∈ 𝑋 ) |
5 |
4
|
adantr |
⊢ ( ( 𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋 ) → 𝑍 ∈ 𝑋 ) |
6 |
1 3
|
dipcj |
⊢ ( ( 𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋 ∧ 𝑍 ∈ 𝑋 ) → ( ∗ ‘ ( 𝐴 𝑃 𝑍 ) ) = ( 𝑍 𝑃 𝐴 ) ) |
7 |
5 6
|
mpd3an3 |
⊢ ( ( 𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋 ) → ( ∗ ‘ ( 𝐴 𝑃 𝑍 ) ) = ( 𝑍 𝑃 𝐴 ) ) |
8 |
1 2 3
|
dip0r |
⊢ ( ( 𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋 ) → ( 𝐴 𝑃 𝑍 ) = 0 ) |
9 |
8
|
fveq2d |
⊢ ( ( 𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋 ) → ( ∗ ‘ ( 𝐴 𝑃 𝑍 ) ) = ( ∗ ‘ 0 ) ) |
10 |
|
cj0 |
⊢ ( ∗ ‘ 0 ) = 0 |
11 |
9 10
|
eqtrdi |
⊢ ( ( 𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋 ) → ( ∗ ‘ ( 𝐴 𝑃 𝑍 ) ) = 0 ) |
12 |
7 11
|
eqtr3d |
⊢ ( ( 𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋 ) → ( 𝑍 𝑃 𝐴 ) = 0 ) |