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Theorem hlmet 27591
Description: The induced metric on a complex Hilbert space. (Contributed by NM, 7-Sep-2007.) (New usage is discouraged.)
Hypotheses
Ref Expression
hlcmet.x 𝑋 = (BaseSet‘𝑈)
hlcmet.8 𝐷 = (IndMet‘𝑈)
Assertion
Ref Expression
hlmet (𝑈 ∈ CHilOLD𝐷 ∈ (Met‘𝑋))

Proof of Theorem hlmet
StepHypRef Expression
1 hlcmet.x . . 3 𝑋 = (BaseSet‘𝑈)
2 hlcmet.8 . . 3 𝐷 = (IndMet‘𝑈)
31, 2hlcmet 27590 . 2 (𝑈 ∈ CHilOLD𝐷 ∈ (CMet‘𝑋))
4 cmetmet 22987 . 2 (𝐷 ∈ (CMet‘𝑋) → 𝐷 ∈ (Met‘𝑋))
53, 4syl 17 1 (𝑈 ∈ CHilOLD𝐷 ∈ (Met‘𝑋))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1480  wcel 1992  cfv 5850  Metcme 19646  CMetcms 22955  BaseSetcba 27281  IndMetcims 27286  CHilOLDchlo 27581
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1841  ax-6 1890  ax-7 1937  ax-8 1994  ax-9 2001  ax-10 2021  ax-11 2036  ax-12 2049  ax-13 2250  ax-ext 2606  ax-sep 4746  ax-nul 4754  ax-pow 4808  ax-pr 4872
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1883  df-eu 2478  df-mo 2479  df-clab 2613  df-cleq 2619  df-clel 2622  df-nfc 2756  df-ne 2797  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3193  df-sbc 3423  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-nul 3897  df-if 4064  df-sn 4154  df-pr 4156  df-op 4160  df-uni 4408  df-br 4619  df-opab 4679  df-mpt 4680  df-id 4994  df-xp 5085  df-rel 5086  df-cnv 5087  df-co 5088  df-dm 5089  df-iota 5813  df-fun 5852  df-fv 5858  df-ov 6608  df-cmet 22958  df-cbn 27559  df-hlo 27582
This theorem is referenced by: (None)
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