Theorem sh0 31217

Description: The zero vector belongs to any subspace of a Hilbert space. (Contributed by NM, 11-Oct-1999.) (Revised by Mario Carneiro, 23-Dec-2013.) (New usage is discouraged.)

Assertion

Ref	Expression
sh0	⊢ (𝐻 ∈ Sℋ → 0ℎ ∈ 𝐻)

Proof of Theorem sh0

Step	Hyp	Ref	Expression
1		issh 31209	. . 3 ⊢ (𝐻 ∈ Sℋ ↔ ((𝐻 ⊆ ℋ ∧ 0ℎ ∈ 𝐻) ∧ (( +ℎ “ (𝐻 × 𝐻)) ⊆ 𝐻 ∧ ( ·ℎ “ (ℂ × 𝐻)) ⊆ 𝐻)))
2	1	simplbi 497	. 2 ⊢ (𝐻 ∈ Sℋ → (𝐻 ⊆ ℋ ∧ 0ℎ ∈ 𝐻))
3	2	simprd 495	1 ⊢ (𝐻 ∈ Sℋ → 0ℎ ∈ 𝐻)

Syntax hints:   → wi 4   ∧ wa 395   ∈ wcel 2113   ⊆ wss 3898   × cxp 5619   “ cima 5624  ℂcc 11015   ℋchba 30920   +_ℎ cva 30921   ·_ℎ csm 30922  0_ℎc0v 30925   S_ℋ csh 30929

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-ext 2705  ax-sep 5238  ax-hilex 31000

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-sb 2068  df-clab 2712  df-cleq 2725  df-clel 2808  df-rab 3397  df-v 3439  df-dif 3901  df-un 3903  df-in 3905  df-ss 3915  df-nul 4283  df-if 4477  df-pw 4553  df-sn 4578  df-pr 4580  df-op 4584  df-br 5096  df-opab 5158  df-xp 5627  df-cnv 5629  df-dm 5631  df-rn 5632  df-res 5633  df-ima 5634  df-sh 31208

This theorem is referenced by:  ch0  31229  hhssabloilem  31262  hhssnv  31265  oc0  31291  ocin  31297  shscli  31318  shsel1  31322  shintcli  31330  shunssi  31369  omlsii  31404  sh0le  31441  imaelshi  32059  shatomistici  32362