This is an inofficial mirror of http://metamath.tirix.org for personal testing of a visualizer extension only.
Metamath Proof Explorer
Description: The Hilbert space norm determines a metric space. (Contributed by Mario
Carneiro, 10-Sep-2015) (New usage is discouraged.)
|
|
Ref |
Expression |
|
Hypothesis |
hilmet.1 |
⊢ 𝐷 = ( normℎ ∘ −ℎ ) |
|
Assertion |
hilxmet |
⊢ 𝐷 ∈ ( ∞Met ‘ ℋ ) |
Proof
| Step |
Hyp |
Ref |
Expression |
| 1 |
|
hilmet.1 |
⊢ 𝐷 = ( normℎ ∘ −ℎ ) |
| 2 |
1
|
hilmet |
⊢ 𝐷 ∈ ( Met ‘ ℋ ) |
| 3 |
|
metxmet |
⊢ ( 𝐷 ∈ ( Met ‘ ℋ ) → 𝐷 ∈ ( ∞Met ‘ ℋ ) ) |
| 4 |
2 3
|
ax-mp |
⊢ 𝐷 ∈ ( ∞Met ‘ ℋ ) |