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Theorem hmoval 22303
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 6082 . . . . . . 7  |-  ( ( u  =  U  /\  u  =  U )  ->  ( u adj u
)  =  ( U adj U ) )
32anidms 627 . . . . . 6  |-  ( u  =  U  ->  (
u adj u )  =  ( U adj U ) )
4 hmoval.9 . . . . . 6  |-  A  =  ( U adj U
)
53, 4syl6eqr 2485 . . . . 5  |-  ( u  =  U  ->  (
u adj u )  =  A )
65dmeqd 5064 . . . 4  |-  ( u  =  U  ->  dom  ( u adj u
)  =  dom  A
)
75fveq1d 5722 . . . . 5  |-  ( u  =  U  ->  (
( u adj u
) `  t )  =  ( A `  t ) )
87eqeq1d 2443 . . . 4  |-  ( u  =  U  ->  (
( ( u adj u ) `  t
)  =  t  <->  ( A `  t )  =  t ) )
96, 8rabeqbidv 2943 . . 3  |-  ( u  =  U  ->  { t  e.  dom  ( u adj u )  |  ( ( u adj u ) `  t
)  =  t }  =  { t  e. 
dom  A  |  ( A `  t )  =  t } )
10 df-hmo 22244 . . 3  |-  HmOp  =  ( u  e.  NrmCVec  |->  { t  e.  dom  ( u adj u )  |  ( ( u adj u ) `  t
)  =  t } )
11 ovex 6098 . . . . . 6  |-  ( U adj U )  e. 
_V
124, 11eqeltri 2505 . . . . 5  |-  A  e. 
_V
1312dmex 5124 . . . 4  |-  dom  A  e.  _V
1413rabex 4346 . . 3  |-  { t  e.  dom  A  | 
( A `  t
)  =  t }  e.  _V
159, 10, 14fvmpt 5798 . 2  |-  ( U  e.  NrmCVec  ->  ( HmOp `  U
)  =  { t  e.  dom  A  | 
( A `  t
)  =  t } )
161, 15syl5eq 2479 1  |-  ( U  e.  NrmCVec  ->  H  =  {
t  e.  dom  A  |  ( A `  t )  =  t } )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1652    e. wcel 1725   {crab 2701   _Vcvv 2948   dom cdm 4870   ` cfv 5446  (class class class)co 6073   NrmCVeccnv 22055   adjcaj 22241   HmOpchmo 22242
This theorem is referenced by:  ishmo  22304
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pr 4395  ax-un 4693
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-rab 2706  df-v 2950  df-sbc 3154  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-br 4205  df-opab 4259  df-mpt 4260  df-id 4490  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-iota 5410  df-fun 5448  df-fv 5454  df-ov 6076  df-hmo 22244
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