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Metamath Proof Explorer
Description: Define the induced metric on a normed complex vector space.
(Contributed by NM, 11-Sep-2007) (New usage is discouraged.)
|
|
Ref |
Expression |
|
Assertion |
df-ims |
⊢ IndMet = ( 𝑢 ∈ NrmCVec ↦ ( ( normCV ‘ 𝑢 ) ∘ ( −𝑣 ‘ 𝑢 ) ) ) |
Detailed syntax breakdown
| Step |
Hyp |
Ref |
Expression |
| 0 |
|
cims |
⊢ IndMet |
| 1 |
|
vu |
⊢ 𝑢 |
| 2 |
|
cnv |
⊢ NrmCVec |
| 3 |
|
cnmcv |
⊢ normCV |
| 4 |
1
|
cv |
⊢ 𝑢 |
| 5 |
4 3
|
cfv |
⊢ ( normCV ‘ 𝑢 ) |
| 6 |
|
cnsb |
⊢ −𝑣 |
| 7 |
4 6
|
cfv |
⊢ ( −𝑣 ‘ 𝑢 ) |
| 8 |
5 7
|
ccom |
⊢ ( ( normCV ‘ 𝑢 ) ∘ ( −𝑣 ‘ 𝑢 ) ) |
| 9 |
1 2 8
|
cmpt |
⊢ ( 𝑢 ∈ NrmCVec ↦ ( ( normCV ‘ 𝑢 ) ∘ ( −𝑣 ‘ 𝑢 ) ) ) |
| 10 |
0 9
|
wceq |
⊢ IndMet = ( 𝑢 ∈ NrmCVec ↦ ( ( normCV ‘ 𝑢 ) ∘ ( −𝑣 ‘ 𝑢 ) ) ) |