Theorem sh0 30734

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 30726	. . 3 ⊢ (𝐻 ∈ Sℋ ↔ ((𝐻 ⊆ ℋ ∧ 0ℎ ∈ 𝐻) ∧ (( +ℎ “ (𝐻 × 𝐻)) ⊆ 𝐻 ∧ ( ·ℎ “ (ℂ × 𝐻)) ⊆ 𝐻)))
2	1	simplbi 496	. 2 ⊢ (𝐻 ∈ Sℋ → (𝐻 ⊆ ℋ ∧ 0ℎ ∈ 𝐻))
3	2	simprd 494	1 ⊢ (𝐻 ∈ Sℋ → 0ℎ ∈ 𝐻)

Syntax hints:   → wi 4   ∧ wa 394   ∈ wcel 2104   ⊆ wss 3949   × cxp 5675   “ cima 5680  ℂcc 11112   ℋchba 30437   +_ℎ cva 30438   ·_ℎ csm 30439  0_ℎc0v 30442   S_ℋ csh 30446

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 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-ext 2701  ax-sep 5300  ax-hilex 30517

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1780  df-sb 2066  df-clab 2708  df-cleq 2722  df-clel 2808  df-rab 3431  df-v 3474  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-br 5150  df-opab 5212  df-xp 5683  df-cnv 5685  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-sh 30725

This theorem is referenced by:  ch0  30746  hhssabloilem  30779  hhssnv  30782  oc0  30808  ocin  30814  shscli  30835  shsel1  30839  shintcli  30847  shunssi  30886  omlsii  30921  sh0le  30958  imaelshi  31576  shatomistici  31879