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

Theorem hlphl 24732
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 24731 . 2 (π‘Š ∈ β„‚Hil β†’ π‘Š ∈ β„‚PreHil)
2 cphphl 24538 . 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 2107  PreHilcphl 21031  β„‚PreHilccph 24533  β„‚Hilchl 24701
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 2708  ax-nul 5264
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 2715  df-cleq 2729  df-clel 2815  df-ne 2945  df-rab 3409  df-v 3448  df-sbc 3741  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4284  df-if 4488  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4867  df-br 5107  df-opab 5169  df-mpt 5190  df-xp 5640  df-cnv 5642  df-dm 5644  df-rn 5645  df-res 5646  df-ima 5647  df-iota 6449  df-fv 6505  df-ov 7361  df-cph 24535  df-hl 24704
This theorem is referenced by:  chlcsschl  24745  pjthlem1  24804  pjth  24806  pjth2  24807  cldcss  24808  hlhil  24810
  Copyright terms: Public domain W3C validator