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Theorem hlmet 30643
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 = (BaseSetβ€˜π‘ˆ)
hlcmet.8 𝐷 = (IndMetβ€˜π‘ˆ)
Assertion
Ref Expression
hlmet (π‘ˆ ∈ CHilOLD β†’ 𝐷 ∈ (Metβ€˜π‘‹))

Proof of Theorem hlmet
StepHypRef Expression
1 hlcmet.x . . 3 𝑋 = (BaseSetβ€˜π‘ˆ)
2 hlcmet.8 . . 3 𝐷 = (IndMetβ€˜π‘ˆ)
31, 2hlcmet 30642 . 2 (π‘ˆ ∈ CHilOLD β†’ 𝐷 ∈ (CMetβ€˜π‘‹))
4 cmetmet 25158 . 2 (𝐷 ∈ (CMetβ€˜π‘‹) β†’ 𝐷 ∈ (Metβ€˜π‘‹))
53, 4syl 17 1 (π‘ˆ ∈ CHilOLD β†’ 𝐷 ∈ (Metβ€˜π‘‹))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   = wceq 1533   ∈ wcel 2098  β€˜cfv 6534  Metcmet 21220  CMetccmet 25126  BaseSetcba 30334  IndMetcims 30339  CHilOLDchlo 30633
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-10 2129  ax-11 2146  ax-12 2163  ax-ext 2695  ax-sep 5290  ax-nul 5297  ax-pr 5418
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-nf 1778  df-sb 2060  df-mo 2526  df-eu 2555  df-clab 2702  df-cleq 2716  df-clel 2802  df-nfc 2877  df-ne 2933  df-ral 3054  df-rex 3063  df-rab 3425  df-v 3468  df-dif 3944  df-un 3946  df-in 3948  df-ss 3958  df-nul 4316  df-if 4522  df-sn 4622  df-pr 4624  df-op 4628  df-uni 4901  df-br 5140  df-opab 5202  df-mpt 5223  df-id 5565  df-xp 5673  df-rel 5674  df-cnv 5675  df-co 5676  df-dm 5677  df-iota 6486  df-fun 6536  df-fv 6542  df-ov 7405  df-cmet 25129  df-cbn 30611  df-hlo 30634
This theorem is referenced by: (None)
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