Theorem sh0 31179

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 31171	. . 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 2109   ⊆ wss 3905   × cxp 5621   “ cima 5626  ℂcc 11026   ℋchba 30882   +_ℎ cva 30883   ·_ℎ csm 30884  0_ℎc0v 30887   S_ℋ csh 30891

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 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-ext 2701  ax-sep 5238  ax-hilex 30962

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2066  df-clab 2708  df-cleq 2721  df-clel 2803  df-rab 3397  df-v 3440  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-nul 4287  df-if 4479  df-pw 4555  df-sn 4580  df-pr 4582  df-op 4586  df-br 5096  df-opab 5158  df-xp 5629  df-cnv 5631  df-dm 5633  df-rn 5634  df-res 5635  df-ima 5636  df-sh 31170

This theorem is referenced by:  ch0  31191  hhssabloilem  31224  hhssnv  31227  oc0  31253  ocin  31259  shscli  31280  shsel1  31284  shintcli  31292  shunssi  31331  omlsii  31366  sh0le  31403  imaelshi  32021  shatomistici  32324