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Theorem ishmo 21389
Description: The predicate "is a hermitian operator." (Contributed by NM, 26-Jan-2008.) (New usage is discouraged.)
Hypotheses
Ref Expression
hmoval.8  |-  H  =  ( HmOp `  U
)
hmoval.9  |-  A  =  ( U adj U
)
Assertion
Ref Expression
ishmo  |-  ( U  e.  NrmCVec  ->  ( T  e.  H  <->  ( T  e. 
dom  A  /\  ( A `  T )  =  T ) ) )

Proof of Theorem ishmo
Dummy variable  t is distinct from all other variables.
StepHypRef Expression
1 hmoval.8 . . . 4  |-  H  =  ( HmOp `  U
)
2 hmoval.9 . . . 4  |-  A  =  ( U adj U
)
31, 2hmoval 21388 . . 3  |-  ( U  e.  NrmCVec  ->  H  =  {
t  e.  dom  A  |  ( A `  t )  =  t } )
43eleq2d 2350 . 2  |-  ( U  e.  NrmCVec  ->  ( T  e.  H  <->  T  e.  { t  e.  dom  A  | 
( A `  t
)  =  t } ) )
5 fveq2 5525 . . . 4  |-  ( t  =  T  ->  ( A `  t )  =  ( A `  T ) )
6 id 19 . . . 4  |-  ( t  =  T  ->  t  =  T )
75, 6eqeq12d 2297 . . 3  |-  ( t  =  T  ->  (
( A `  t
)  =  t  <->  ( A `  T )  =  T ) )
87elrab 2923 . 2  |-  ( T  e.  { t  e. 
dom  A  |  ( A `  t )  =  t }  <->  ( T  e.  dom  A  /\  ( A `  T )  =  T ) )
94, 8syl6bb 252 1  |-  ( U  e.  NrmCVec  ->  ( T  e.  H  <->  ( T  e. 
dom  A  /\  ( A `  T )  =  T ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    = wceq 1623    e. wcel 1684   {crab 2547   dom cdm 4689   ` cfv 5255  (class class class)co 5858   NrmCVeccnv 21140   adjcaj 21326   HmOpchmo 21327
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-iota 5219  df-fun 5257  df-fv 5263  df-ov 5861  df-hmo 21329
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