Theorem ch0 31248

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 31244	. 2 ⊢ (𝐻 ∈ Cℋ → 𝐻 ∈ Sℋ )
2		sh0 31236	. 2 ⊢ (𝐻 ∈ Sℋ → 0ℎ ∈ 𝐻)
3	1, 2	syl 17	1 ⊢ (𝐻 ∈ Cℋ → 0ℎ ∈ 𝐻)

Syntax hints:   → wi 4   ∈ wcel 2107  0_ℎc0v 30944   S_ℋ csh 30948   C_ℋ cch 30949

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-ext 2707  ax-sep 5295  ax-hilex 31019

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-sb 2064  df-clab 2714  df-cleq 2728  df-clel 2815  df-rab 3436  df-v 3481  df-dif 3953  df-un 3955  df-in 3957  df-ss 3967  df-nul 4333  df-if 4525  df-pw 4601  df-sn 4626  df-pr 4628  df-op 4632  df-uni 4907  df-br 5143  df-opab 5205  df-xp 5690  df-cnv 5692  df-dm 5694  df-rn 5695  df-res 5696  df-ima 5697  df-iota 6513  df-fv 6568  df-ov 7435  df-sh 31227  df-ch 31241

This theorem is referenced by:  omlsii  31423  nonbooli  31671  strlem1  32270