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Theorem shss 30441
Description: A subspace is a subset of Hilbert space. (Contributed by NM, 9-Oct-1999.) (Revised by Mario Carneiro, 23-Dec-2013.) (New usage is discouraged.)
Assertion
Ref Expression
shss (𝐻S𝐻 ⊆ ℋ)

Proof of Theorem shss
StepHypRef Expression
1 issh 30439 . . 3 (𝐻S ↔ ((𝐻 ⊆ ℋ ∧ 0𝐻) ∧ (( + “ (𝐻 × 𝐻)) ⊆ 𝐻 ∧ ( · “ (ℂ × 𝐻)) ⊆ 𝐻)))
21simplbi 499 . 2 (𝐻S → (𝐻 ⊆ ℋ ∧ 0𝐻))
32simpld 496 1 (𝐻S𝐻 ⊆ ℋ)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 397  wcel 2107  wss 3947   × cxp 5673  cima 5678  cc 11104  chba 30150   + cva 30151   · csm 30152  0c0v 30155   S csh 30159
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2704  ax-sep 5298  ax-hilex 30230
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-rab 3434  df-v 3477  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4322  df-if 4528  df-pw 4603  df-sn 4628  df-pr 4630  df-op 4634  df-br 5148  df-opab 5210  df-xp 5681  df-cnv 5683  df-dm 5685  df-rn 5686  df-res 5687  df-ima 5688  df-sh 30438
This theorem is referenced by:  shel  30442  shex  30443  shssii  30444  shsubcl  30451  chss  30460  shsspwh  30477  hhsssh  30500  shocel  30513  shocsh  30515  ocss  30516  shocss  30517  shocorth  30523  shococss  30525  shorth  30526  shoccl  30536  shsel  30545  shintcli  30560  spanid  30578  shjval  30582  shjcl  30587  shlej1  30591  shlub  30645  chscllem2  30869  chscllem4  30871
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