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

Theorem bdopf 25271
Description: A bounded linear Hilbert space operator is a Hilbert space operator. (Contributed by NM, 2-Feb-2006.) (New usage is discouraged.)
Assertion
Ref Expression
bdopf  |-  ( T  e.  BndLinOp  ->  T : ~H --> ~H )

Proof of Theorem bdopf
StepHypRef Expression
1 bdopln 25270 . 2  |-  ( T  e.  BndLinOp  ->  T  e.  LinOp )
2 lnopf 25268 . 2  |-  ( T  e.  LinOp  ->  T : ~H
--> ~H )
31, 2syl 16 1  |-  ( T  e.  BndLinOp  ->  T : ~H --> ~H )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    e. wcel 1756   -->wf 5419   ~Hchil 24326   LinOpclo 24354   BndLinOpcbo 24355
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-sep 4418  ax-nul 4426  ax-pow 4475  ax-pr 4536  ax-un 6377  ax-hilex 24406
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2573  df-ne 2613  df-ral 2725  df-rex 2726  df-rab 2729  df-v 2979  df-sbc 3192  df-dif 3336  df-un 3338  df-in 3340  df-ss 3347  df-nul 3643  df-if 3797  df-pw 3867  df-sn 3883  df-pr 3885  df-op 3889  df-uni 4097  df-br 4298  df-opab 4356  df-id 4641  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-iota 5386  df-fun 5425  df-fn 5426  df-f 5427  df-fv 5431  df-ov 6099  df-oprab 6100  df-mpt2 6101  df-map 7221  df-lnop 25250  df-bdop 25251
This theorem is referenced by:  nmopre  25279  nmophmi  25440  adjbdln  25492  nmopadjlem  25498  nmoptrii  25503  nmopcoi  25504  bdophsi  25505  bdophdi  25506  nmoptri2i  25508  adjcoi  25509  nmopcoadji  25510  unierri  25513
  Copyright terms: Public domain W3C validator