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Theorem hmoval 24209
Description: The set of Hermitian (self-adjoint) operators on a normed complex vector space. (Contributed by NM, 26-Jan-2008.) (Revised by Mario Carneiro, 16-Nov-2013.) (New usage is discouraged.)
Hypotheses
Ref Expression
hmoval.8  |-  H  =  ( HmOp `  U
)
hmoval.9  |-  A  =  ( U adj U
)
Assertion
Ref Expression
hmoval  |-  ( U  e.  NrmCVec  ->  H  =  {
t  e.  dom  A  |  ( A `  t )  =  t } )
Distinct variable groups:    t, A    t, U
Allowed substitution hint:    H( t)

Proof of Theorem hmoval
Dummy variable  u is distinct from all other variables.
StepHypRef Expression
1 hmoval.8 . 2  |-  H  =  ( HmOp `  U
)
2 oveq12 6099 . . . . . . 7  |-  ( ( u  =  U  /\  u  =  U )  ->  ( u adj u
)  =  ( U adj U ) )
32anidms 645 . . . . . 6  |-  ( u  =  U  ->  (
u adj u )  =  ( U adj U ) )
4 hmoval.9 . . . . . 6  |-  A  =  ( U adj U
)
53, 4syl6eqr 2492 . . . . 5  |-  ( u  =  U  ->  (
u adj u )  =  A )
65dmeqd 5041 . . . 4  |-  ( u  =  U  ->  dom  ( u adj u
)  =  dom  A
)
75fveq1d 5692 . . . . 5  |-  ( u  =  U  ->  (
( u adj u
) `  t )  =  ( A `  t ) )
87eqeq1d 2450 . . . 4  |-  ( u  =  U  ->  (
( ( u adj u ) `  t
)  =  t  <->  ( A `  t )  =  t ) )
96, 8rabeqbidv 2966 . . 3  |-  ( u  =  U  ->  { t  e.  dom  ( u adj u )  |  ( ( u adj u ) `  t
)  =  t }  =  { t  e. 
dom  A  |  ( A `  t )  =  t } )
10 df-hmo 24150 . . 3  |-  HmOp  =  ( u  e.  NrmCVec  |->  { t  e.  dom  ( u adj u )  |  ( ( u adj u ) `  t
)  =  t } )
11 ovex 6115 . . . . . 6  |-  ( U adj U )  e. 
_V
124, 11eqeltri 2512 . . . . 5  |-  A  e. 
_V
1312dmex 6510 . . . 4  |-  dom  A  e.  _V
1413rabex 4442 . . 3  |-  { t  e.  dom  A  | 
( A `  t
)  =  t }  e.  _V
159, 10, 14fvmpt 5773 . 2  |-  ( U  e.  NrmCVec  ->  ( HmOp `  U
)  =  { t  e.  dom  A  | 
( A `  t
)  =  t } )
161, 15syl5eq 2486 1  |-  ( U  e.  NrmCVec  ->  H  =  {
t  e.  dom  A  |  ( A `  t )  =  t } )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1369    e. wcel 1756   {crab 2718   _Vcvv 2971   dom cdm 4839   ` cfv 5417  (class class class)co 6090   NrmCVeccnv 23961   adjcaj 24147   HmOpchmo 24148
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-sep 4412  ax-nul 4420  ax-pr 4530  ax-un 6371
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2567  df-ne 2607  df-ral 2719  df-rex 2720  df-rab 2723  df-v 2973  df-sbc 3186  df-dif 3330  df-un 3332  df-in 3334  df-ss 3341  df-nul 3637  df-if 3791  df-sn 3877  df-pr 3879  df-op 3883  df-uni 4091  df-br 4292  df-opab 4350  df-mpt 4351  df-id 4635  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-iota 5380  df-fun 5419  df-fv 5425  df-ov 6093  df-hmo 24150
This theorem is referenced by:  ishmo  24210
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