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Theorem hlmet 29879
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 29878 . 2 (π‘ˆ ∈ CHilOLD β†’ 𝐷 ∈ (CMetβ€˜π‘‹))
4 cmetmet 24666 . 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 1542   ∈ wcel 2107  β€˜cfv 6497  Metcmet 20798  CMetccmet 24634  BaseSetcba 29570  IndMetcims 29575  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-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5257  ax-nul 5264  ax-pr 5385
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-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2941  df-ral 3062  df-rex 3071  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-opab 5169  df-mpt 5190  df-id 5532  df-xp 5640  df-rel 5641  df-cnv 5642  df-co 5643  df-dm 5644  df-iota 6449  df-fun 6499  df-fv 6505  df-ov 7361  df-cmet 24637  df-cbn 29847  df-hlo 29870
This theorem is referenced by: (None)
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