MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  ishmo Structured version   Unicode version

Theorem ishmo 25549
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 25548 . . 3  |-  ( U  e.  NrmCVec  ->  H  =  {
t  e.  dom  A  |  ( A `  t )  =  t } )
43eleq2d 2537 . 2  |-  ( U  e.  NrmCVec  ->  ( T  e.  H  <->  T  e.  { t  e.  dom  A  | 
( A `  t
)  =  t } ) )
5 fveq2 5872 . . . 4  |-  ( t  =  T  ->  ( A `  t )  =  ( A `  T ) )
6 id 22 . . . 4  |-  ( t  =  T  ->  t  =  T )
75, 6eqeq12d 2489 . . 3  |-  ( t  =  T  ->  (
( A `  t
)  =  t  <->  ( A `  T )  =  T ) )
87elrab 3266 . 2  |-  ( T  e.  { t  e. 
dom  A  |  ( A `  t )  =  t }  <->  ( T  e.  dom  A  /\  ( A `  T )  =  T ) )
94, 8syl6bb 261 1  |-  ( U  e.  NrmCVec  ->  ( T  e.  H  <->  ( T  e. 
dom  A  /\  ( A `  T )  =  T ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1379    e. wcel 1767   {crab 2821   dom cdm 5005   ` cfv 5594  (class class class)co 6295   NrmCVeccnv 25300   adjcaj 25486   HmOpchmo 25487
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4574  ax-nul 4582  ax-pr 4692  ax-un 6587
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2822  df-rex 2823  df-rab 2826  df-v 3120  df-sbc 3337  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-nul 3791  df-if 3946  df-sn 4034  df-pr 4036  df-op 4040  df-uni 4252  df-br 4454  df-opab 4512  df-mpt 4513  df-id 4801  df-xp 5011  df-rel 5012  df-cnv 5013  df-co 5014  df-dm 5015  df-rn 5016  df-iota 5557  df-fun 5596  df-fv 5602  df-ov 6298  df-hmo 25489
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator