Theorem ch0 31561

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 31557	. 2 ⊢ (𝐻 ∈ Cℋ → 𝐻 ∈ Sℋ )
2		sh0 31549	. 2 ⊢ (𝐻 ∈ Sℋ → 0ℎ ∈ 𝐻)
3	1, 2	syl 18	1 ⊢ (𝐻 ∈ Cℋ → 0ℎ ∈ 𝐻)

Syntax hints:   → wi 4   ∈ wcel 2143  0_ℎc0v 31257   S_ℋ csh 31261   C_ℋ cch 31262

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-ext 2735  ax-sep 5258  ax-hilex 31332

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-sb 2097  df-clab 2742  df-cleq 2755  df-clel 2838  df-rab 3417  df-v 3457  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-nul 4288  df-if 4489  df-pw 4565  df-sn 4591  df-pr 4593  df-op 4597  df-uni 4874  df-br 5111  df-opab 5175  df-xp 5669  df-cnv 5671  df-dm 5673  df-rn 5674  df-res 5675  df-ima 5676  df-iota 6494  df-fv 6546  df-ov 7415  df-sh 31540  df-ch 31554

This theorem is referenced by:  omlsii  31736  nonbooli  31984  strlem1  32583