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Theorem hmoval 25851
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 6305 . . . . . . 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 2516 . . . . 5  |-  ( u  =  U  ->  (
u adj u )  =  A )
65dmeqd 5215 . . . 4  |-  ( u  =  U  ->  dom  ( u adj u
)  =  dom  A
)
75fveq1d 5874 . . . . 5  |-  ( u  =  U  ->  (
( u adj u
) `  t )  =  ( A `  t ) )
87eqeq1d 2459 . . . 4  |-  ( u  =  U  ->  (
( ( u adj u ) `  t
)  =  t  <->  ( A `  t )  =  t ) )
96, 8rabeqbidv 3104 . . 3  |-  ( u  =  U  ->  { t  e.  dom  ( u adj u )  |  ( ( u adj u ) `  t
)  =  t }  =  { t  e. 
dom  A  |  ( A `  t )  =  t } )
10 df-hmo 25792 . . 3  |-  HmOp  =  ( u  e.  NrmCVec  |->  { t  e.  dom  ( u adj u )  |  ( ( u adj u ) `  t
)  =  t } )
11 ovex 6324 . . . . . 6  |-  ( U adj U )  e. 
_V
124, 11eqeltri 2541 . . . . 5  |-  A  e. 
_V
1312dmex 6732 . . . 4  |-  dom  A  e.  _V
1413rabex 4607 . . 3  |-  { t  e.  dom  A  | 
( A `  t
)  =  t }  e.  _V
159, 10, 14fvmpt 5956 . 2  |-  ( U  e.  NrmCVec  ->  ( HmOp `  U
)  =  { t  e.  dom  A  | 
( A `  t
)  =  t } )
161, 15syl5eq 2510 1  |-  ( U  e.  NrmCVec  ->  H  =  {
t  e.  dom  A  |  ( A `  t )  =  t } )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1395    e. wcel 1819   {crab 2811   _Vcvv 3109   dom cdm 5008   ` cfv 5594  (class class class)co 6296   NrmCVeccnv 25603   adjcaj 25789   HmOpchmo 25790
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-sep 4578  ax-nul 4586  ax-pr 4695  ax-un 6591
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-rex 2813  df-rab 2816  df-v 3111  df-sbc 3328  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3794  df-if 3945  df-sn 4033  df-pr 4035  df-op 4039  df-uni 4252  df-br 4457  df-opab 4516  df-mpt 4517  df-id 4804  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-iota 5557  df-fun 5596  df-fv 5602  df-ov 6299  df-hmo 25792
This theorem is referenced by:  ishmo  25852
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