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Theorem hlmet 22358
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 22357 . 2  |-  ( U  e.  CHil OLD  ->  D  e.  ( CMet `  X
) )
4 cmetmet 19200 . 2  |-  ( D  e.  ( CMet `  X
)  ->  D  e.  ( Met `  X ) )
53, 4syl 16 1  |-  ( U  e.  CHil OLD  ->  D  e.  ( Met `  X
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1649    e. wcel 1721   ` cfv 5421   Metcme 16650   CMetcms 19168   BaseSetcba 22026   IndMetcims 22031   CHil
OLDchlo 22348
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2393  ax-sep 4298  ax-nul 4306  ax-pow 4345  ax-pr 4371
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2266  df-mo 2267  df-clab 2399  df-cleq 2405  df-clel 2408  df-nfc 2537  df-ne 2577  df-ral 2679  df-rex 2680  df-rab 2683  df-v 2926  df-sbc 3130  df-dif 3291  df-un 3293  df-in 3295  df-ss 3302  df-nul 3597  df-if 3708  df-sn 3788  df-pr 3789  df-op 3791  df-uni 3984  df-br 4181  df-opab 4235  df-mpt 4236  df-id 4466  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-iota 5385  df-fun 5423  df-fv 5429  df-ov 6051  df-cmet 19171  df-cbn 22326  df-hlo 22349
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