MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  hlphl Structured version   Visualization version   GIF version

Theorem hlphl 25237
Description: Every subcomplex Hilbert space is an inner product space (also called a pre-Hilbert space). (Contributed by NM, 28-Apr-2007.) (Revised by Mario Carneiro, 15-Oct-2015.)
Assertion
Ref Expression
hlphl (π‘Š ∈ β„‚Hil β†’ π‘Š ∈ PreHil)

Proof of Theorem hlphl
StepHypRef Expression
1 hlcph 25236 . 2 (π‘Š ∈ β„‚Hil β†’ π‘Š ∈ β„‚PreHil)
2 cphphl 25043 . 2 (π‘Š ∈ β„‚PreHil β†’ π‘Š ∈ PreHil)
31, 2syl 17 1 (π‘Š ∈ β„‚Hil β†’ π‘Š ∈ PreHil)
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∈ wcel 2098  PreHilcphl 21506  β„‚PreHilccph 25038  β„‚Hilchl 25206
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2695  ax-nul 5297
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-sb 2060  df-clab 2702  df-cleq 2716  df-clel 2802  df-ne 2933  df-rab 3425  df-v 3468  df-sbc 3771  df-dif 3944  df-un 3946  df-in 3948  df-ss 3958  df-nul 4316  df-if 4522  df-sn 4622  df-pr 4624  df-op 4628  df-uni 4901  df-br 5140  df-opab 5202  df-mpt 5223  df-xp 5673  df-cnv 5675  df-dm 5677  df-rn 5678  df-res 5679  df-ima 5680  df-iota 6486  df-fv 6542  df-ov 7405  df-cph 25040  df-hl 25209
This theorem is referenced by:  chlcsschl  25250  pjthlem1  25309  pjth  25311  pjth2  25312  cldcss  25313  hlhil  25315
  Copyright terms: Public domain W3C validator