Theorem sh0 31287

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 31279	. . 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 495	1 ⊢ (𝐻 ∈ Sℋ → 0ℎ ∈ 𝐻)

Syntax hints:   → wi 4   ∧ wa 395   ∈ wcel 2114   ⊆ wss 3889   × cxp 5629   “ cima 5634  ℂcc 11036   ℋchba 30990   +_ℎ cva 30991   ·_ℎ csm 30992  0_ℎc0v 30995   S_ℋ csh 30999

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2708  ax-sep 5231  ax-hilex 31070

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-sb 2069  df-clab 2715  df-cleq 2728  df-clel 2811  df-rab 3390  df-v 3431  df-dif 3892  df-un 3894  df-in 3896  df-ss 3906  df-nul 4274  df-if 4467  df-pw 4543  df-sn 4568  df-pr 4570  df-op 4574  df-br 5086  df-opab 5148  df-xp 5637  df-cnv 5639  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-sh 31278

This theorem is referenced by:  ch0  31299  hhssabloilem  31332  hhssnv  31335  oc0  31361  ocin  31367  shscli  31388  shsel1  31392  shintcli  31400  shunssi  31439  omlsii  31474  sh0le  31511  imaelshi  32129  shatomistici  32432