Theorem sh0 31568

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 31560	. . 3 ⊢ (𝐻 ∈ Sℋ ↔ ((𝐻 ⊆ ℋ ∧ 0ℎ ∈ 𝐻) ∧ (( +ℎ “ (𝐻 × 𝐻)) ⊆ 𝐻 ∧ ( ·ℎ “ (ℂ × 𝐻)) ⊆ 𝐻)))
2	1	simplbi 501	. 2 ⊢ (𝐻 ∈ Sℋ → (𝐻 ⊆ ℋ ∧ 0ℎ ∈ 𝐻))
3	2	simprd 500	1 ⊢ (𝐻 ∈ Sℋ → 0ℎ ∈ 𝐻)

Syntax hints:   → wi 4   ∧ wa 400   ∈ wcel 2143   ⊆ wss 3905   × cxp 5659   “ cima 5664  ℂcc 11093   ℋchba 31271   +_ℎ cva 31272   ·_ℎ csm 31273  0_ℎc0v 31276   S_ℋ csh 31280

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-ext 2735  ax-sep 5257  ax-hilex 31351

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-sb 2097  df-clab 2742  df-cleq 2755  df-clel 2838  df-rab 3417  df-v 3457  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-nul 4287  df-if 4488  df-pw 4564  df-sn 4590  df-pr 4592  df-op 4596  df-br 5110  df-opab 5174  df-xp 5667  df-cnv 5669  df-dm 5671  df-rn 5672  df-res 5673  df-ima 5674  df-sh 31559

This theorem is referenced by:  ch0  31580  hhssabloilem  31613  hhssnv  31616  oc0  31642  ocin  31648  shscli  31669  shsel1  31673  shintcli  31681  shunssi  31720  omlsii  31755  sh0le  31792  imaelshi  32410  shatomistici  32713