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Theorem hlcmet 29878
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 29872 . 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 29849 . 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 1542   ∈ wcel 2107  β€˜cfv 6497  CMetccmet 24634  BaseSetcba 29570  IndMetcims 29575  CBanccbn 29846  CHilOLDchlo 29869
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2704
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-rab 3407  df-v 3446  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4284  df-if 4488  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4867  df-br 5107  df-iota 6449  df-fv 6505  df-cbn 29847  df-hlo 29870
This theorem is referenced by:  hlmet  29879  hlcompl  29899
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