**Description:** The zero vector belongs to any closed subspace of a Hilbert space.
(Contributed by NM, 24-Aug-1999) (New usage is discouraged.)

Ref | Expression | ||
---|---|---|---|

Assertion | ch0 | $${\u22a2}{H}\in {\mathbf{C}}_{\mathscr{H}}\to {0}_{\u210e}\in {H}$$ |

Step | Hyp | Ref | Expression |
---|---|---|---|

1 | chsh | $${\u22a2}{H}\in {\mathbf{C}}_{\mathscr{H}}\to {H}\in {\mathbf{S}}_{\mathscr{H}}$$ | |

2 | sh0 | $${\u22a2}{H}\in {\mathbf{S}}_{\mathscr{H}}\to {0}_{\u210e}\in {H}$$ | |

3 | 1 2 | syl | $${\u22a2}{H}\in {\mathbf{C}}_{\mathscr{H}}\to {0}_{\u210e}\in {H}$$ |