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Theorem hlcmet 28999
Description: The induced metric on a complex Hilbert space is complete. (Contributed by NM, 8-Sep-2007.) (New usage is discouraged.)
Hypotheses
Ref Expression
hlcmet.x 𝑋 = (BaseSet‘𝑈)
hlcmet.8 𝐷 = (IndMet‘𝑈)
Assertion
Ref Expression
hlcmet (𝑈 ∈ CHilOLD𝐷 ∈ (CMet‘𝑋))

Proof of Theorem hlcmet
StepHypRef Expression
1 hlobn 28993 . 2 (𝑈 ∈ CHilOLD𝑈 ∈ CBan)
2 hlcmet.x . . 3 𝑋 = (BaseSet‘𝑈)
3 hlcmet.8 . . 3 𝐷 = (IndMet‘𝑈)
42, 3cbncms 28970 . 2 (𝑈 ∈ CBan → 𝐷 ∈ (CMet‘𝑋))
51, 4syl 17 1 (𝑈 ∈ CHilOLD𝐷 ∈ (CMet‘𝑋))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1543  wcel 2111  cfv 6398  CMetccmet 24175  BaseSetcba 28691  IndMetcims 28696  CBanccbn 28967  CHilOLDchlo 28990
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2113  ax-9 2121  ax-ext 2709
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-sb 2072  df-clab 2716  df-cleq 2730  df-clel 2817  df-rab 3071  df-v 3423  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-nul 4253  df-if 4455  df-sn 4557  df-pr 4559  df-op 4563  df-uni 4835  df-br 5069  df-iota 6356  df-fv 6406  df-cbn 28968  df-hlo 28991
This theorem is referenced by:  hlmet  29000  hlcompl  29020
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