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Theorem hlphl 25306
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 25305 . 2 (π‘Š ∈ β„‚Hil β†’ π‘Š ∈ β„‚PreHil)
2 cphphl 25112 . 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 2099  PreHilcphl 21556  β„‚PreHilccph 25107  β„‚Hilchl 25275
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-ext 2699  ax-nul 5306
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-sb 2061  df-clab 2706  df-cleq 2720  df-clel 2806  df-ne 2938  df-rab 3430  df-v 3473  df-sbc 3777  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4909  df-br 5149  df-opab 5211  df-mpt 5232  df-xp 5684  df-cnv 5686  df-dm 5688  df-rn 5689  df-res 5690  df-ima 5691  df-iota 6500  df-fv 6556  df-ov 7423  df-cph 25109  df-hl 25278
This theorem is referenced by:  chlcsschl  25319  pjthlem1  25378  pjth  25380  pjth2  25381  cldcss  25382  hlhil  25384
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