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Theorem shss 28407
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 28405 . . 3 (𝐻S ↔ ((𝐻 ⊆ ℋ ∧ 0𝐻) ∧ (( + “ (𝐻 × 𝐻)) ⊆ 𝐻 ∧ ( · “ (ℂ × 𝐻)) ⊆ 𝐻)))
21simplbi 485 . 2 (𝐻S → (𝐻 ⊆ ℋ ∧ 0𝐻))
32simpld 482 1 (𝐻S𝐻 ⊆ ℋ)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 382  wcel 2145  wss 3723   × cxp 5247  cima 5252  cc 10136  chil 28116   + cva 28117   · csm 28118  0c0v 28121   S csh 28125
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-sep 4915  ax-hilex 28196
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-rab 3070  df-v 3353  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4226  df-pw 4299  df-sn 4317  df-pr 4319  df-op 4323  df-br 4787  df-opab 4847  df-xp 5255  df-cnv 5257  df-dm 5259  df-rn 5260  df-res 5261  df-ima 5262  df-sh 28404
This theorem is referenced by:  shel  28408  shex  28409  shssii  28410  shsubcl  28417  chss  28426  shsspwh  28443  hhsssh  28466  shocel  28481  shocsh  28483  ocss  28484  shocss  28485  shocorth  28491  shococss  28493  shorth  28494  shoccl  28504  shsel  28513  shintcli  28528  spanid  28546  shjval  28550  shjcl  28555  shlej1  28559  shlub  28613  chscllem2  28837  chscllem4  28839
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