MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  hlcmet Structured version   Visualization version   GIF version

Theorem hlcmet 30652
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 = (BaseSetβ€˜π‘ˆ)
hlcmet.8 𝐷 = (IndMetβ€˜π‘ˆ)
Assertion
Ref Expression
hlcmet (π‘ˆ ∈ CHilOLD β†’ 𝐷 ∈ (CMetβ€˜π‘‹))

Proof of Theorem hlcmet
StepHypRef Expression
1 hlobn 30646 . 2 (π‘ˆ ∈ CHilOLD β†’ π‘ˆ ∈ CBan)
2 hlcmet.x . . 3 𝑋 = (BaseSetβ€˜π‘ˆ)
3 hlcmet.8 . . 3 𝐷 = (IndMetβ€˜π‘ˆ)
42, 3cbncms 30623 . 2 (π‘ˆ ∈ CBan β†’ 𝐷 ∈ (CMetβ€˜π‘‹))
51, 4syl 17 1 (π‘ˆ ∈ CHilOLD β†’ 𝐷 ∈ (CMetβ€˜π‘‹))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   = wceq 1533   ∈ wcel 2098  β€˜cfv 6536  CMetccmet 25133  BaseSetcba 30344  IndMetcims 30349  CBanccbn 30620  CHilOLDchlo 30643
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2697
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-sb 2060  df-clab 2704  df-cleq 2718  df-clel 2804  df-rab 3427  df-v 3470  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4524  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-br 5142  df-iota 6488  df-fv 6544  df-cbn 30621  df-hlo 30644
This theorem is referenced by:  hlmet  30653  hlcompl  30673
  Copyright terms: Public domain W3C validator