Theorem ch0 29491

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 29487	. 2 ⊢ (𝐻 ∈ Cℋ → 𝐻 ∈ Sℋ )
2		sh0 29479	. 2 ⊢ (𝐻 ∈ Sℋ → 0ℎ ∈ 𝐻)
3	1, 2	syl 17	1 ⊢ (𝐻 ∈ Cℋ → 0ℎ ∈ 𝐻)

Syntax hints:   → wi 4   ∈ wcel 2108  0_ℎc0v 29187   S_ℋ csh 29191   C_ℋ cch 29192

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-ext 2709  ax-sep 5218  ax-hilex 29262

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-sb 2069  df-clab 2716  df-cleq 2730  df-clel 2817  df-rab 3072  df-v 3424  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4254  df-if 4457  df-pw 4532  df-sn 4559  df-pr 4561  df-op 4565  df-uni 4837  df-br 5071  df-opab 5133  df-xp 5586  df-cnv 5588  df-dm 5590  df-rn 5591  df-res 5592  df-ima 5593  df-iota 6376  df-fv 6426  df-ov 7258  df-sh 29470  df-ch 29484

This theorem is referenced by:  omlsii  29666  nonbooli  29914  strlem1  30513