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Theorem hmopf 26455
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 26454 . 2  |-  ( T  e.  HrmOp 
<->  ( T : ~H --> ~H  /\  A. x  e. 
~H  A. y  e.  ~H  ( x  .ih  ( T `
 y ) )  =  ( ( T `
 x )  .ih  y ) ) )
21simplbi 460 1  |-  ( T  e.  HrmOp  ->  T : ~H
--> ~H )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1374    e. wcel 1762   A.wral 2807   -->wf 5575   ` cfv 5579  (class class class)co 6275   ~Hchil 25498    .ih csp 25501   HrmOpcho 25529
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-8 1764  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1961  ax-ext 2438  ax-sep 4561  ax-nul 4569  ax-pow 4618  ax-pr 4679  ax-un 6567  ax-hilex 25578
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2272  df-mo 2273  df-clab 2446  df-cleq 2452  df-clel 2455  df-nfc 2610  df-ne 2657  df-ral 2812  df-rex 2813  df-rab 2816  df-v 3108  df-sbc 3325  df-dif 3472  df-un 3474  df-in 3476  df-ss 3483  df-nul 3779  df-if 3933  df-pw 4005  df-sn 4021  df-pr 4023  df-op 4027  df-uni 4239  df-br 4441  df-opab 4499  df-id 4788  df-xp 4998  df-rel 4999  df-cnv 5000  df-co 5001  df-dm 5002  df-rn 5003  df-iota 5542  df-fun 5581  df-fn 5582  df-f 5583  df-fv 5587  df-ov 6278  df-oprab 6279  df-mpt2 6280  df-map 7412  df-hmop 26425
This theorem is referenced by:  hmopex  26456  hmopre  26504  hmopadj  26520  hmdmadj  26521  hmoplin  26523  eighmre  26544  eighmorth  26545  hmops  26601  hmopm  26602  hmopd  26603  hmopco  26604  leop2  26705  leoppos  26707  leoprf  26709  leopsq  26710  leopadd  26713  leopmuli  26714  leopmul  26715  leopmul2i  26716  leopnmid  26719  nmopleid  26720  opsqrlem1  26721  opsqrlem6  26726  elpjrn  26771
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