Theorem ch0 31214

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

Assertion

Ref	Expression
ch0	⊢ (𝐻 ∈ Cℋ → 0ℎ ∈ 𝐻)

Proof of Theorem ch0

Step	Hyp	Ref	Expression
1		chsh 31210	. 2 ⊢ (𝐻 ∈ Cℋ → 𝐻 ∈ Sℋ )
2		sh0 31202	. 2 ⊢ (𝐻 ∈ Sℋ → 0ℎ ∈ 𝐻)
3	1, 2	syl 17	1 ⊢ (𝐻 ∈ Cℋ → 0ℎ ∈ 𝐻)

Syntax hints:   → wi 4   ∈ wcel 2109  0_ℎc0v 30910   S_ℋ csh 30914   C_ℋ cch 30915

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-ext 2708  ax-sep 5271  ax-hilex 30985

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2066  df-clab 2715  df-cleq 2728  df-clel 2810  df-rab 3421  df-v 3466  df-dif 3934  df-un 3936  df-in 3938  df-ss 3948  df-nul 4314  df-if 4506  df-pw 4582  df-sn 4607  df-pr 4609  df-op 4613  df-uni 4889  df-br 5125  df-opab 5187  df-xp 5665  df-cnv 5667  df-dm 5669  df-rn 5670  df-res 5671  df-ima 5672  df-iota 6489  df-fv 6544  df-ov 7413  df-sh 31193  df-ch 31207

This theorem is referenced by:  omlsii  31389  nonbooli  31637  strlem1  32236