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

Theorem lnopf 23367
Description: A linear Hilbert space operator is a Hilbert space operator. (Contributed by NM, 18-Jan-2006.) (New usage is discouraged.)
Assertion
Ref Expression
lnopf  |-  ( T  e.  LinOp  ->  T : ~H
--> ~H )

Proof of Theorem lnopf
Dummy variables  x  y  z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ellnop 23366 . 2  |-  ( T  e.  LinOp 
<->  ( T : ~H --> ~H  /\  A. x  e.  CC  A. y  e. 
~H  A. z  e.  ~H  ( T `  ( ( x  .h  y )  +h  z ) )  =  ( ( x  .h  ( T `  y ) )  +h  ( T `  z
) ) ) )
21simplbi 448 1  |-  ( T  e.  LinOp  ->  T : ~H
--> ~H )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1653    e. wcel 1726   A.wral 2707   -->wf 5453   ` cfv 5457  (class class class)co 6084   CCcc 8993   ~Hchil 22427    +h cva 22428    .h csm 22429   LinOpclo 22455
This theorem is referenced by:  bdopf  23370  elbdop2  23379  unopadj2  23446  lnop0  23474  lnopmul  23475  lnopfi  23477  homco2  23485  nmopun  23522  cnlnadjeui  23585  cnlnssadj  23588
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 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 4333  ax-nul 4341  ax-pow 4380  ax-pr 4406  ax-un 4704  ax-hilex 22507
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 4216  df-opab 4270  df-id 4501  df-xp 4887  df-rel 4888  df-cnv 4889  df-co 4890  df-dm 4891  df-rn 4892  df-iota 5421  df-fun 5459  df-fn 5460  df-f 5461  df-fv 5465  df-ov 6087  df-oprab 6088  df-mpt2 6089  df-map 7023  df-lnop 23349
  Copyright terms: Public domain W3C validator