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Theorem hlcmet 30722
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 30716 . 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 30693 . 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 6551  CMetccmet 25200  BaseSetcba 30414  IndMetcims 30419  CBanccbn 30690  CHilOLDchlo 30713
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 2698
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-sb 2060  df-clab 2705  df-cleq 2719  df-clel 2805  df-rab 3429  df-v 3473  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4325  df-if 4531  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4911  df-br 5151  df-iota 6503  df-fv 6559  df-cbn 30691  df-hlo 30714
This theorem is referenced by:  hlmet  30723  hlcompl  30743
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