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Theorem hlphl 23976
 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 (𝑊 ∈ ℂHil → 𝑊 ∈ PreHil)

Proof of Theorem hlphl
StepHypRef Expression
1 hlcph 23975 . 2 (𝑊 ∈ ℂHil → 𝑊 ∈ ℂPreHil)
2 cphphl 23783 . 2 (𝑊 ∈ ℂPreHil → 𝑊 ∈ PreHil)
31, 2syl 17 1 (𝑊 ∈ ℂHil → 𝑊 ∈ PreHil)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∈ wcel 2111  PreHilcphl 20317  ℂPreHilccph 23778  ℂHilchl 23945 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-nul 5174 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ral 3111  df-rex 3112  df-rab 3115  df-v 3443  df-sbc 3721  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-nul 4244  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4801  df-br 5031  df-opab 5093  df-mpt 5111  df-xp 5525  df-cnv 5527  df-dm 5529  df-rn 5530  df-res 5531  df-ima 5532  df-iota 6283  df-fv 6332  df-ov 7138  df-cph 23780  df-hl 23948 This theorem is referenced by:  chlcsschl  23989  pjthlem1  24048  pjth  24050  pjth2  24051  cldcss  24052  hlhil  24054
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