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Theorem hlvc 30821
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 30819 . 2 (𝑈 ∈ CHilOLD𝑈 ∈ NrmCVec)
2 hlvc.1 . . 3 𝑊 = (1st𝑈)
32nvvc 30543 . 2 (𝑈 ∈ NrmCVec → 𝑊 ∈ CVecOLD)
41, 3syl 17 1 (𝑈 ∈ CHilOLD𝑊 ∈ CVecOLD)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1534  wcel 2099  cfv 6544  1st c1st 7991  CVecOLDcvc 30486  NrmCVeccnv 30512  CHilOLDchlo 30813
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-10 2130  ax-11 2147  ax-12 2167  ax-ext 2697  ax-rep 5281  ax-sep 5295  ax-nul 5302  ax-pr 5424  ax-un 7736
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2704  df-cleq 2718  df-clel 2803  df-nfc 2878  df-ne 2931  df-ral 3052  df-rex 3061  df-reu 3366  df-rab 3421  df-v 3465  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4324  df-if 4525  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4907  df-iun 4996  df-br 5145  df-opab 5207  df-mpt 5228  df-id 5571  df-xp 5679  df-rel 5680  df-cnv 5681  df-co 5682  df-dm 5683  df-rn 5684  df-res 5685  df-ima 5686  df-iota 6496  df-fun 6546  df-fn 6547  df-f 6548  df-f1 6549  df-fo 6550  df-f1o 6551  df-fv 6552  df-ov 7417  df-oprab 7418  df-1st 7993  df-2nd 7994  df-vc 30487  df-nv 30520  df-va 30523  df-ba 30524  df-sm 30525  df-0v 30526  df-nmcv 30528  df-cbn 30791  df-hlo 30814
This theorem is referenced by: (None)
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