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

Theorem hmopf 25423
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 25422 . 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 1370    e. wcel 1758   A.wral 2795   -->wf 5515   ` cfv 5519  (class class class)co 6193   ~Hchil 24466    .ih csp 24469   HrmOpcho 24497
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-sep 4514  ax-nul 4522  ax-pow 4571  ax-pr 4632  ax-un 6475  ax-hilex 24546
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ne 2646  df-ral 2800  df-rex 2801  df-rab 2804  df-v 3073  df-sbc 3288  df-dif 3432  df-un 3434  df-in 3436  df-ss 3443  df-nul 3739  df-if 3893  df-pw 3963  df-sn 3979  df-pr 3981  df-op 3985  df-uni 4193  df-br 4394  df-opab 4452  df-id 4737  df-xp 4947  df-rel 4948  df-cnv 4949  df-co 4950  df-dm 4951  df-rn 4952  df-iota 5482  df-fun 5521  df-fn 5522  df-f 5523  df-fv 5527  df-ov 6196  df-oprab 6197  df-mpt2 6198  df-map 7319  df-hmop 25393
This theorem is referenced by:  hmopex  25424  hmopre  25472  hmopadj  25488  hmdmadj  25489  hmoplin  25491  eighmre  25512  eighmorth  25513  hmops  25569  hmopm  25570  hmopd  25571  hmopco  25572  leop2  25673  leoppos  25675  leoprf  25677  leopsq  25678  leopadd  25681  leopmuli  25682  leopmul  25683  leopmul2i  25684  leopnmid  25687  nmopleid  25688  opsqrlem1  25689  opsqrlem6  25694  elpjrn  25739
  Copyright terms: Public domain W3C validator