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Theorem ishmo 22314
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 22313 . . 3  |-  ( U  e.  NrmCVec  ->  H  =  {
t  e.  dom  A  |  ( A `  t )  =  t } )
43eleq2d 2505 . 2  |-  ( U  e.  NrmCVec  ->  ( T  e.  H  <->  T  e.  { t  e.  dom  A  | 
( A `  t
)  =  t } ) )
5 fveq2 5730 . . . 4  |-  ( t  =  T  ->  ( A `  t )  =  ( A `  T ) )
6 id 21 . . . 4  |-  ( t  =  T  ->  t  =  T )
75, 6eqeq12d 2452 . . 3  |-  ( t  =  T  ->  (
( A `  t
)  =  t  <->  ( A `  T )  =  T ) )
87elrab 3094 . 2  |-  ( T  e.  { t  e. 
dom  A  |  ( A `  t )  =  t }  <->  ( T  e.  dom  A  /\  ( A `  T )  =  T ) )
94, 8syl6bb 254 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 178    /\ wa 360    = wceq 1653    e. wcel 1726   {crab 2711   dom cdm 4880   ` cfv 5456  (class class class)co 6083   NrmCVeccnv 22065   adjcaj 22251   HmOpchmo 22252
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-sep 4332  ax-nul 4340  ax-pr 4405  ax-un 4703
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-rab 2716  df-v 2960  df-sbc 3164  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-br 4215  df-opab 4269  df-mpt 4270  df-id 4500  df-xp 4886  df-rel 4887  df-cnv 4888  df-co 4889  df-dm 4890  df-rn 4891  df-iota 5420  df-fun 5458  df-fv 5464  df-ov 6086  df-hmo 22254
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