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

Theorem hmopadj2 22481
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
StepHypRef Expression
1 hmopadj 22479 . 2  |-  ( T  e.  HrmOp  ->  ( adjh `  T )  =  T )
2 dmadjop 22428 . . . . 5  |-  ( T  e.  dom  adjh  ->  T : ~H --> ~H )
32adantr 453 . . . 4  |-  ( ( T  e.  dom  adjh  /\  ( adjh `  T
)  =  T )  ->  T : ~H --> ~H )
4 adj1 22473 . . . . . . . 8  |-  ( ( T  e.  dom  adjh  /\  x  e.  ~H  /\  y  e.  ~H )  ->  ( x  .ih  ( T `  y )
)  =  ( ( ( adjh `  T
) `  x )  .ih  y ) )
543expb 1157 . . . . . . 7  |-  ( ( T  e.  dom  adjh  /\  ( x  e.  ~H  /\  y  e.  ~H )
)  ->  ( x  .ih  ( T `  y
) )  =  ( ( ( adjh `  T
) `  x )  .ih  y ) )
65adantlr 698 . . . . . 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 5457 . . . . . . . 8  |-  ( (
adjh `  T )  =  T  ->  ( (
adjh `  T ) `  x )  =  ( T `  x ) )
87oveq1d 5807 . . . . . . 7  |-  ( (
adjh `  T )  =  T  ->  ( ( ( adjh `  T
) `  x )  .ih  y )  =  ( ( T `  x
)  .ih  y )
)
98ad2antlr 710 . . . . . 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 2290 . . . . 5  |-  ( ( ( T  e.  dom  adjh  /\  ( adjh `  T
)  =  T )  /\  ( x  e. 
~H  /\  y  e.  ~H ) )  ->  (
x  .ih  ( T `  y ) )  =  ( ( T `  x )  .ih  y
) )
1110ralrimivva 2610 . . . 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 22413 . . . 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 648 . . 3  |-  ( ( T  e.  dom  adjh  /\  ( adjh `  T
)  =  T )  ->  T  e.  HrmOp )
1413ex 425 . 2  |-  ( T  e.  dom  adjh  ->  ( ( adjh `  T
)  =  T  ->  T  e.  HrmOp ) )
151, 14impbid2 197 1  |-  ( T  e.  dom  adjh  ->  ( T  e.  HrmOp  <->  ( adjh `  T )  =  T ) )
Colors of variables: wff set class
Syntax hints:    -> wi 6    <-> wb 178    /\ wa 360    = wceq 1619    e. wcel 1621   A.wral 2518   dom cdm 4661   -->wf 4669   ` cfv 4673  (class class class)co 5792   ~Hchil 21459    .ih csp 21462   HrmOpcho 21490   adjhcado 21495
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-13 1625  ax-14 1626  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1927  ax-ext 2239  ax-rep 4105  ax-sep 4115  ax-nul 4123  ax-pow 4160  ax-pr 4186  ax-un 4484  ax-resscn 8762  ax-1cn 8763  ax-icn 8764  ax-addcl 8765  ax-addrcl 8766  ax-mulcl 8767  ax-mulrcl 8768  ax-mulcom 8769  ax-addass 8770  ax-mulass 8771  ax-distr 8772  ax-i2m1 8773  ax-1ne0 8774  ax-1rid 8775  ax-rnegex 8776  ax-rrecex 8777  ax-cnre 8778  ax-pre-lttri 8779  ax-pre-lttrn 8780  ax-pre-ltadd 8781  ax-pre-mulgt0 8782  ax-hilex 21539  ax-hfvadd 21540  ax-hvcom 21541  ax-hvass 21542  ax-hv0cl 21543  ax-hvaddid 21544  ax-hfvmul 21545  ax-hvmulid 21546  ax-hvdistr2 21549  ax-hvmul0 21550  ax-hfi 21618  ax-his1 21621  ax-his2 21622  ax-his3 21623  ax-his4 21624
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 940  df-3an 941  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1884  df-eu 2122  df-mo 2123  df-clab 2245  df-cleq 2251  df-clel 2254  df-nfc 2383  df-ne 2423  df-nel 2424  df-ral 2523  df-rex 2524  df-reu 2525  df-rmo 2526  df-rab 2527  df-v 2765  df-sbc 2967  df-csb 3057  df-dif 3130  df-un 3132  df-in 3134  df-ss 3141  df-nul 3431  df-if 3540  df-pw 3601  df-sn 3620  df-pr 3621  df-op 3623  df-uni 3802  df-iun 3881  df-br 3998  df-opab 4052  df-mpt 4053  df-id 4281  df-po 4286  df-so 4287  df-xp 4675  df-rel 4676  df-cnv 4677  df-co 4678  df-dm 4679  df-rn 4680  df-res 4681  df-ima 4682  df-fun 4683  df-fn 4684  df-f 4685  df-f1 4686  df-fo 4687  df-f1o 4688  df-fv 4689  df-ov 5795  df-oprab 5796  df-mpt2 5797  df-iota 6225  df-riota 6272  df-er 6628  df-map 6742  df-en 6832  df-dom 6833  df-sdom 6834  df-pnf 8837  df-mnf 8838  df-xr 8839  df-ltxr 8840  df-le 8841  df-sub 9007  df-neg 9008  df-div 9392  df-2 9772  df-cj 11549  df-re 11550  df-im 11551  df-hvsub 21511  df-hmop 22384  df-adjh 22389
  Copyright terms: Public domain W3C validator