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Theorem hmopadj2 22514
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 ) )
Dummy variables  x  y are mutually distinct and distinct from all other variables.

Proof of Theorem hmopadj2
StepHypRef Expression
1 hmopadj 22512 . 2  |-  ( T  e.  HrmOp  ->  ( adjh `  T )  =  T )
2 dmadjop 22461 . . . . 5  |-  ( T  e.  dom  adjh  ->  T : ~H --> ~H )
32adantr 453 . . . 4  |-  ( ( T  e.  dom  adjh  /\  ( adjh `  T
)  =  T )  ->  T : ~H --> ~H )
4 adj1 22506 . . . . . . . 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 697 . . . . . 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 5485 . . . . . . . 8  |-  ( (
adjh `  T )  =  T  ->  ( (
adjh `  T ) `  x )  =  ( T `  x ) )
87oveq1d 5835 . . . . . . 7  |-  ( (
adjh `  T )  =  T  ->  ( ( ( adjh `  T
) `  x )  .ih  y )  =  ( ( T `  x
)  .ih  y )
)
98ad2antlr 709 . . . . . 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 2317 . . . . 5  |-  ( ( ( T  e.  dom  adjh  /\  ( adjh `  T
)  =  T )  /\  ( x  e. 
~H  /\  y  e.  ~H ) )  ->  (
x  .ih  ( T `  y ) )  =  ( ( T `  x )  .ih  y
) )
1110ralrimivva 2637 . . . 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 22446 . . . 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 647 . . 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 1624    e. wcel 1685   A.wral 2545   dom cdm 4689   -->wf 5218   ` cfv 5222  (class class class)co 5820   ~Hchil 21492    .ih csp 21495   HrmOpcho 21523   adjhcado 21528
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-gen 1534  ax-5 1545  ax-17 1604  ax-9 1637  ax-8 1645  ax-13 1687  ax-14 1689  ax-6 1704  ax-7 1709  ax-11 1716  ax-12 1868  ax-ext 2266  ax-rep 4133  ax-sep 4143  ax-nul 4151  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-resscn 8790  ax-1cn 8791  ax-icn 8792  ax-addcl 8793  ax-addrcl 8794  ax-mulcl 8795  ax-mulrcl 8796  ax-mulcom 8797  ax-addass 8798  ax-mulass 8799  ax-distr 8800  ax-i2m1 8801  ax-1ne0 8802  ax-1rid 8803  ax-rnegex 8804  ax-rrecex 8805  ax-cnre 8806  ax-pre-lttri 8807  ax-pre-lttrn 8808  ax-pre-ltadd 8809  ax-pre-mulgt0 8810  ax-hilex 21572  ax-hfvadd 21573  ax-hvcom 21574  ax-hvass 21575  ax-hv0cl 21576  ax-hvaddid 21577  ax-hfvmul 21578  ax-hvmulid 21579  ax-hvdistr2 21582  ax-hvmul0 21583  ax-hfi 21651  ax-his1 21654  ax-his2 21655  ax-his3 21656  ax-his4 21657
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 937  df-3an 938  df-tru 1312  df-ex 1530  df-nf 1533  df-sb 1632  df-eu 2149  df-mo 2150  df-clab 2272  df-cleq 2278  df-clel 2281  df-nfc 2410  df-ne 2450  df-nel 2451  df-ral 2550  df-rex 2551  df-reu 2552  df-rmo 2553  df-rab 2554  df-v 2792  df-sbc 2994  df-csb 3084  df-dif 3157  df-un 3159  df-in 3161  df-ss 3168  df-nul 3458  df-if 3568  df-pw 3629  df-sn 3648  df-pr 3649  df-op 3651  df-uni 3830  df-iun 3909  df-br 4026  df-opab 4080  df-mpt 4081  df-id 4309  df-po 4314  df-so 4315  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-fun 5224  df-fn 5225  df-f 5226  df-f1 5227  df-fo 5228  df-f1o 5229  df-fv 5230  df-ov 5823  df-oprab 5824  df-mpt2 5825  df-iota 6253  df-riota 6300  df-er 6656  df-map 6770  df-en 6860  df-dom 6861  df-sdom 6862  df-pnf 8865  df-mnf 8866  df-xr 8867  df-ltxr 8868  df-le 8869  df-sub 9035  df-neg 9036  df-div 9420  df-2 9800  df-cj 11579  df-re 11580  df-im 11581  df-hvsub 21544  df-hmop 22417  df-adjh 22422
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