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Theorem hmopf 23379
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 23378 . 2  |-  ( T  e.  HrmOp 
<->  ( T : ~H --> ~H  /\  A. x  e. 
~H  A. y  e.  ~H  ( x  .ih  ( T `
 y ) )  =  ( ( T `
 x )  .ih  y ) ) )
21simplbi 448 1  |-  ( T  e.  HrmOp  ->  T : ~H
--> ~H )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1653    e. wcel 1726   A.wral 2707   -->wf 5452   ` cfv 5456  (class class class)co 6083   ~Hchil 22424    .ih csp 22427   HrmOpcho 22455
This theorem is referenced by:  hmopex  23380  hmopre  23428  hmopadj  23444  hmdmadj  23445  hmoplin  23447  eighmre  23468  eighmorth  23469  hmops  23525  hmopm  23526  hmopd  23527  hmopco  23528  leop2  23629  leoppos  23631  leoprf  23633  leopsq  23634  leopadd  23637  leopmuli  23638  leopmul  23639  leopmul2i  23640  leopnmid  23643  nmopleid  23644  opsqrlem1  23645  opsqrlem6  23650  elpjrn  23695
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-sep 4332  ax-nul 4340  ax-pow 4379  ax-pr 4405  ax-un 4703  ax-hilex 22504
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-rab 2716  df-v 2960  df-sbc 3164  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-br 4215  df-opab 4269  df-id 4500  df-xp 4886  df-rel 4887  df-cnv 4888  df-co 4889  df-dm 4890  df-rn 4891  df-iota 5420  df-fun 5458  df-fn 5459  df-f 5460  df-fv 5464  df-ov 6086  df-oprab 6087  df-mpt2 6088  df-map 7022  df-hmop 23349
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