HSE Home Hilbert Space Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  HSE Home  >  Th. List  >  hmopadj2 Unicode version

Theorem hmopadj2 23397
Description: An operator is Hermitian iff it is self-adjoint. Definition of Hermitian in [Halmos] p. 41. (Contributed by NM, 9-Apr-2006.) (New usage is discouraged.)
Assertion
Ref Expression
hmopadj2  |-  ( T  e.  dom  adjh  ->  ( T  e.  HrmOp  <->  ( adjh `  T )  =  T ) )

Proof of Theorem hmopadj2
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 hmopadj 23395 . 2  |-  ( T  e.  HrmOp  ->  ( adjh `  T )  =  T )
2 dmadjop 23344 . . . . 5  |-  ( T  e.  dom  adjh  ->  T : ~H --> ~H )
32adantr 452 . . . 4  |-  ( ( T  e.  dom  adjh  /\  ( adjh `  T
)  =  T )  ->  T : ~H --> ~H )
4 adj1 23389 . . . . . . . 8  |-  ( ( T  e.  dom  adjh  /\  x  e.  ~H  /\  y  e.  ~H )  ->  ( x  .ih  ( T `  y )
)  =  ( ( ( adjh `  T
) `  x )  .ih  y ) )
543expb 1154 . . . . . . 7  |-  ( ( T  e.  dom  adjh  /\  ( x  e.  ~H  /\  y  e.  ~H )
)  ->  ( x  .ih  ( T `  y
) )  =  ( ( ( adjh `  T
) `  x )  .ih  y ) )
65adantlr 696 . . . . . 6  |-  ( ( ( T  e.  dom  adjh  /\  ( adjh `  T
)  =  T )  /\  ( x  e. 
~H  /\  y  e.  ~H ) )  ->  (
x  .ih  ( T `  y ) )  =  ( ( ( adjh `  T ) `  x
)  .ih  y )
)
7 fveq1 5686 . . . . . . . 8  |-  ( (
adjh `  T )  =  T  ->  ( (
adjh `  T ) `  x )  =  ( T `  x ) )
87oveq1d 6055 . . . . . . 7  |-  ( (
adjh `  T )  =  T  ->  ( ( ( adjh `  T
) `  x )  .ih  y )  =  ( ( T `  x
)  .ih  y )
)
98ad2antlr 708 . . . . . 6  |-  ( ( ( T  e.  dom  adjh  /\  ( adjh `  T
)  =  T )  /\  ( x  e. 
~H  /\  y  e.  ~H ) )  ->  (
( ( adjh `  T
) `  x )  .ih  y )  =  ( ( T `  x
)  .ih  y )
)
106, 9eqtrd 2436 . . . . 5  |-  ( ( ( T  e.  dom  adjh  /\  ( adjh `  T
)  =  T )  /\  ( x  e. 
~H  /\  y  e.  ~H ) )  ->  (
x  .ih  ( T `  y ) )  =  ( ( T `  x )  .ih  y
) )
1110ralrimivva 2758 . . . 4  |-  ( ( T  e.  dom  adjh  /\  ( adjh `  T
)  =  T )  ->  A. x  e.  ~H  A. y  e.  ~H  (
x  .ih  ( T `  y ) )  =  ( ( T `  x )  .ih  y
) )
12 elhmop 23329 . . . 4  |-  ( T  e.  HrmOp 
<->  ( T : ~H --> ~H  /\  A. x  e. 
~H  A. y  e.  ~H  ( x  .ih  ( T `
 y ) )  =  ( ( T `
 x )  .ih  y ) ) )
133, 11, 12sylanbrc 646 . . 3  |-  ( ( T  e.  dom  adjh  /\  ( adjh `  T
)  =  T )  ->  T  e.  HrmOp )
1413ex 424 . 2  |-  ( T  e.  dom  adjh  ->  ( ( adjh `  T
)  =  T  ->  T  e.  HrmOp ) )
151, 14impbid2 196 1  |-  ( T  e.  dom  adjh  ->  ( T  e.  HrmOp  <->  ( adjh `  T )  =  T ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    = wceq 1649    e. wcel 1721   A.wral 2666   dom cdm 4837   -->wf 5409   ` cfv 5413  (class class class)co 6040   ~Hchil 22375    .ih csp 22378   HrmOpcho 22406   adjhcado 22411
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-rep 4280  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660  ax-resscn 9003  ax-1cn 9004  ax-icn 9005  ax-addcl 9006  ax-addrcl 9007  ax-mulcl 9008  ax-mulrcl 9009  ax-mulcom 9010  ax-addass 9011  ax-mulass 9012  ax-distr 9013  ax-i2m1 9014  ax-1ne0 9015  ax-1rid 9016  ax-rnegex 9017  ax-rrecex 9018  ax-cnre 9019  ax-pre-lttri 9020  ax-pre-lttrn 9021  ax-pre-ltadd 9022  ax-pre-mulgt0 9023  ax-hilex 22455  ax-hfvadd 22456  ax-hvcom 22457  ax-hvass 22458  ax-hv0cl 22459  ax-hvaddid 22460  ax-hfvmul 22461  ax-hvmulid 22462  ax-hvdistr2 22465  ax-hvmul0 22466  ax-hfi 22534  ax-his1 22537  ax-his2 22538  ax-his3 22539  ax-his4 22540
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-nel 2570  df-ral 2671  df-rex 2672  df-reu 2673  df-rmo 2674  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-op 3783  df-uni 3976  df-iun 4055  df-br 4173  df-opab 4227  df-mpt 4228  df-id 4458  df-po 4463  df-so 4464  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-ov 6043  df-oprab 6044  df-mpt2 6045  df-riota 6508  df-er 6864  df-map 6979  df-en 7069  df-dom 7070  df-sdom 7071  df-pnf 9078  df-mnf 9079  df-xr 9080  df-ltxr 9081  df-le 9082  df-sub 9249  df-neg 9250  df-div 9634  df-2 10014  df-cj 11859  df-re 11860  df-im 11861  df-hvsub 22427  df-hmop 23300  df-adjh 23305
  Copyright terms: Public domain W3C validator