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Theorem shss 28915
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 28913 . . 3 (𝐻S ↔ ((𝐻 ⊆ ℋ ∧ 0𝐻) ∧ (( + “ (𝐻 × 𝐻)) ⊆ 𝐻 ∧ ( · “ (ℂ × 𝐻)) ⊆ 𝐻)))
21simplbi 498 . 2 (𝐻S → (𝐻 ⊆ ℋ ∧ 0𝐻))
32simpld 495 1 (𝐻S𝐻 ⊆ ℋ)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396  wcel 2105  wss 3935   × cxp 5547  cima 5552  cc 10524  chba 28624   + cva 28625   · csm 28626  0c0v 28629   S csh 28633
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2793  ax-sep 5195  ax-hilex 28704
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-rab 3147  df-v 3497  df-dif 3938  df-un 3940  df-in 3942  df-ss 3951  df-nul 4291  df-if 4466  df-pw 4539  df-sn 4560  df-pr 4562  df-op 4566  df-br 5059  df-opab 5121  df-xp 5555  df-cnv 5557  df-dm 5559  df-rn 5560  df-res 5561  df-ima 5562  df-sh 28912
This theorem is referenced by:  shel  28916  shex  28917  shssii  28918  shsubcl  28925  chss  28934  shsspwh  28951  hhsssh  28974  shocel  28987  shocsh  28989  ocss  28990  shocss  28991  shocorth  28997  shococss  28999  shorth  29000  shoccl  29010  shsel  29019  shintcli  29034  spanid  29052  shjval  29056  shjcl  29061  shlej1  29065  shlub  29119  chscllem2  29343  chscllem4  29345
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