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Theorem hlvc 28664
 Description: Every complex Hilbert space is a complex vector space. (Contributed by NM, 7-Sep-2007.) (New usage is discouraged.)
Hypothesis
Ref Expression
hlvc.1 𝑊 = (1st𝑈)
Assertion
Ref Expression
hlvc (𝑈 ∈ CHilOLD𝑊 ∈ CVecOLD)

Proof of Theorem hlvc
StepHypRef Expression
1 hlnv 28662 . 2 (𝑈 ∈ CHilOLD𝑈 ∈ NrmCVec)
2 hlvc.1 . . 3 𝑊 = (1st𝑈)
32nvvc 28386 . 2 (𝑈 ∈ NrmCVec → 𝑊 ∈ CVecOLD)
41, 3syl 17 1 (𝑈 ∈ CHilOLD𝑊 ∈ CVecOLD)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   = wceq 1533   ∈ wcel 2110  ‘cfv 6349  1st c1st 7681  CVecOLDcvc 28329  NrmCVeccnv 28355  CHilOLDchlo 28656 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2157  ax-12 2173  ax-ext 2793  ax-rep 5182  ax-sep 5195  ax-nul 5202  ax-pow 5258  ax-pr 5321  ax-un 7455 This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-mo 2618  df-eu 2650  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ne 3017  df-ral 3143  df-rex 3144  df-reu 3145  df-rab 3147  df-v 3496  df-sbc 3772  df-csb 3883  df-dif 3938  df-un 3940  df-in 3942  df-ss 3951  df-nul 4291  df-if 4467  df-sn 4561  df-pr 4563  df-op 4567  df-uni 4832  df-iun 4913  df-br 5059  df-opab 5121  df-mpt 5139  df-id 5454  df-xp 5555  df-rel 5556  df-cnv 5557  df-co 5558  df-dm 5559  df-rn 5560  df-res 5561  df-ima 5562  df-iota 6308  df-fun 6351  df-fn 6352  df-f 6353  df-f1 6354  df-fo 6355  df-f1o 6356  df-fv 6357  df-ov 7153  df-oprab 7154  df-1st 7683  df-2nd 7684  df-vc 28330  df-nv 28363  df-va 28366  df-ba 28367  df-sm 28368  df-0v 28369  df-nmcv 28371  df-cbn 28634  df-hlo 28657 This theorem is referenced by: (None)
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