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Theorem shss 27928
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 27926 . . 3 (𝐻S ↔ ((𝐻 ⊆ ℋ ∧ 0𝐻) ∧ (( + “ (𝐻 × 𝐻)) ⊆ 𝐻 ∧ ( · “ (ℂ × 𝐻)) ⊆ 𝐻)))
21simplbi 476 . 2 (𝐻S → (𝐻 ⊆ ℋ ∧ 0𝐻))
32simpld 475 1 (𝐻S𝐻 ⊆ ℋ)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384  wcel 1987  wss 3556   × cxp 5074  cima 5079  cc 9881  chil 27637   + cva 27638   · csm 27639  0c0v 27642   S csh 27646
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4743  ax-hilex 27717
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-rab 2916  df-v 3188  df-dif 3559  df-un 3561  df-in 3563  df-ss 3570  df-nul 3894  df-if 4061  df-pw 4134  df-sn 4151  df-pr 4153  df-op 4157  df-br 4616  df-opab 4676  df-xp 5082  df-cnv 5084  df-dm 5086  df-rn 5087  df-res 5088  df-ima 5089  df-sh 27925
This theorem is referenced by:  shel  27929  shex  27930  shssii  27931  shsubcl  27938  chss  27947  shsspwh  27964  hhsssh  27987  shocel  28002  shocsh  28004  ocss  28005  shocss  28006  shocorth  28012  shococss  28014  shorth  28015  shoccl  28025  shsel  28034  shintcli  28049  spanid  28067  shjval  28071  shjcl  28076  shlej1  28080  shlub  28134  chscllem2  28358  chscllem4  28360
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