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Theorem hmoval 21404
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 5883 . . . . . . 7  |-  ( ( u  =  U  /\  u  =  U )  ->  ( u adj u
)  =  ( U adj U ) )
32anidms 626 . . . . . 6  |-  ( u  =  U  ->  (
u adj u )  =  ( U adj U ) )
4 hmoval.9 . . . . . 6  |-  A  =  ( U adj U
)
53, 4syl6eqr 2346 . . . . 5  |-  ( u  =  U  ->  (
u adj u )  =  A )
65dmeqd 4897 . . . 4  |-  ( u  =  U  ->  dom  ( u adj u
)  =  dom  A
)
75fveq1d 5543 . . . . 5  |-  ( u  =  U  ->  (
( u adj u
) `  t )  =  ( A `  t ) )
87eqeq1d 2304 . . . 4  |-  ( u  =  U  ->  (
( ( u adj u ) `  t
)  =  t  <->  ( A `  t )  =  t ) )
96, 8rabeqbidv 2796 . . 3  |-  ( u  =  U  ->  { t  e.  dom  ( u adj u )  |  ( ( u adj u ) `  t
)  =  t }  =  { t  e. 
dom  A  |  ( A `  t )  =  t } )
10 df-hmo 21345 . . 3  |-  HmOp  =  ( u  e.  NrmCVec  |->  { t  e.  dom  ( u adj u )  |  ( ( u adj u ) `  t
)  =  t } )
11 ovex 5899 . . . . . 6  |-  ( U adj U )  e. 
_V
124, 11eqeltri 2366 . . . . 5  |-  A  e. 
_V
1312dmex 4957 . . . 4  |-  dom  A  e.  _V
1413rabex 4181 . . 3  |-  { t  e.  dom  A  | 
( A `  t
)  =  t }  e.  _V
159, 10, 14fvmpt 5618 . 2  |-  ( U  e.  NrmCVec  ->  ( HmOp `  U
)  =  { t  e.  dom  A  | 
( A `  t
)  =  t } )
161, 15syl5eq 2340 1  |-  ( U  e.  NrmCVec  ->  H  =  {
t  e.  dom  A  |  ( A `  t )  =  t } )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1632    e. wcel 1696   {crab 2560   _Vcvv 2801   dom cdm 4705   ` cfv 5271  (class class class)co 5874   NrmCVeccnv 21156   adjcaj 21342   HmOpchmo 21343
This theorem is referenced by:  ishmo  21405
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pr 4230  ax-un 4528
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-sbc 3005  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-iota 5235  df-fun 5273  df-fv 5279  df-ov 5877  df-hmo 21345
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