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Theorem hlmet 21588
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  e.  CHil OLD  ->  D  e.  ( Met `  X
) )

Proof of Theorem hlmet
StepHypRef Expression
1 hlcmet.x . . 3  |-  X  =  ( BaseSet `  U )
2 hlcmet.8 . . 3  |-  D  =  ( IndMet `  U )
31, 2hlcmet 21587 . 2  |-  ( U  e.  CHil OLD  ->  D  e.  ( CMet `  X
) )
4 cmetmet 18816 . 2  |-  ( D  e.  ( CMet `  X
)  ->  D  e.  ( Met `  X ) )
53, 4syl 15 1  |-  ( U  e.  CHil OLD  ->  D  e.  ( Met `  X
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1642    e. wcel 1710   ` cfv 5337   Metcme 16469   CMetcms 18784   BaseSetcba 21256   IndMetcims 21261   CHil
OLDchlo 21578
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-13 1712  ax-14 1714  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1930  ax-ext 2339  ax-sep 4222  ax-nul 4230  ax-pow 4269  ax-pr 4295
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2213  df-mo 2214  df-clab 2345  df-cleq 2351  df-clel 2354  df-nfc 2483  df-ne 2523  df-ral 2624  df-rex 2625  df-rab 2628  df-v 2866  df-sbc 3068  df-dif 3231  df-un 3233  df-in 3235  df-ss 3242  df-nul 3532  df-if 3642  df-sn 3722  df-pr 3723  df-op 3725  df-uni 3909  df-br 4105  df-opab 4159  df-mpt 4160  df-id 4391  df-xp 4777  df-rel 4778  df-cnv 4779  df-co 4780  df-dm 4781  df-iota 5301  df-fun 5339  df-fv 5345  df-ov 5948  df-cmet 18787  df-cbn 21556  df-hlo 21579
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