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Theorem hmopf 22509
Description: A Hermitian operator is a Hilbert space operator (mapping). (Contributed by NM, 19-Mar-2006.) (New usage is discouraged.)
Assertion
Ref Expression
hmopf  |-  ( T  e.  HrmOp  ->  T : ~H
--> ~H )

Proof of Theorem hmopf
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 elhmop 22508 . 2  |-  ( T  e.  HrmOp 
<->  ( T : ~H --> ~H  /\  A. x  e. 
~H  A. y  e.  ~H  ( x  .ih  ( T `
 y ) )  =  ( ( T `
 x )  .ih  y ) ) )
21simplbi 446 1  |-  ( T  e.  HrmOp  ->  T : ~H
--> ~H )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1633    e. wcel 1701   A.wral 2577   -->wf 5288   ` cfv 5292  (class class class)co 5900   ~Hchil 21554    .ih csp 21557   HrmOpcho 21585
This theorem is referenced by:  hmopex  22510  hmopre  22558  hmopadj  22574  hmdmadj  22575  hmoplin  22577  eighmre  22598  eighmorth  22599  hmops  22655  hmopm  22656  hmopd  22657  hmopco  22658  leop2  22759  leoppos  22761  leoprf  22763  leopsq  22764  leopadd  22767  leopmuli  22768  leopmul  22769  leopmul2i  22770  leopnmid  22773  nmopleid  22774  opsqrlem1  22775  opsqrlem6  22780  elpjrn  22825
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1537  ax-5 1548  ax-17 1607  ax-9 1645  ax-8 1666  ax-13 1703  ax-14 1705  ax-6 1720  ax-7 1725  ax-11 1732  ax-12 1897  ax-ext 2297  ax-sep 4178  ax-nul 4186  ax-pow 4225  ax-pr 4251  ax-un 4549  ax-hilex 21634
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1533  df-nf 1536  df-sb 1640  df-eu 2180  df-mo 2181  df-clab 2303  df-cleq 2309  df-clel 2312  df-nfc 2441  df-ne 2481  df-ral 2582  df-rex 2583  df-rab 2586  df-v 2824  df-sbc 3026  df-dif 3189  df-un 3191  df-in 3193  df-ss 3200  df-nul 3490  df-if 3600  df-pw 3661  df-sn 3680  df-pr 3681  df-op 3683  df-uni 3865  df-br 4061  df-opab 4115  df-id 4346  df-xp 4732  df-rel 4733  df-cnv 4734  df-co 4735  df-dm 4736  df-rn 4737  df-iota 5256  df-fun 5294  df-fn 5295  df-f 5296  df-fv 5300  df-ov 5903  df-oprab 5904  df-mpt2 5905  df-map 6817  df-hmop 22479
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