Description: The zero vector belongs to any subspace of a Hilbert space. (Contributed by NM, 11-Oct-1999) (Revised by Mario Carneiro, 23-Dec-2013) (New usage is discouraged.)
| Ref | Expression | ||
|---|---|---|---|
| Assertion | sh0 | |- ( H e. SH -> 0h e. H ) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | issh | |- ( H e. SH <-> ( ( H C_ ~H /\ 0h e. H ) /\ ( ( +h " ( H X. H ) ) C_ H /\ ( .h " ( CC X. H ) ) C_ H ) ) ) |
|
| 2 | 1 | simplbi | |- ( H e. SH -> ( H C_ ~H /\ 0h e. H ) ) |
| 3 | 2 | simprd | |- ( H e. SH -> 0h e. H ) |