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Theorem bdopf 26604
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 26603 . 2  |-  ( T  e.  BndLinOp  ->  T  e.  LinOp )
2 lnopf 26601 . 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 1767   -->wf 5590   ~Hchil 25659   LinOpclo 25687   BndLinOpcbo 25688
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4574  ax-nul 4582  ax-pow 4631  ax-pr 4692  ax-un 6587  ax-hilex 25739
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2822  df-rex 2823  df-rab 2826  df-v 3120  df-sbc 3337  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-nul 3791  df-if 3946  df-pw 4018  df-sn 4034  df-pr 4036  df-op 4040  df-uni 4252  df-br 4454  df-opab 4512  df-id 4801  df-xp 5011  df-rel 5012  df-cnv 5013  df-co 5014  df-dm 5015  df-rn 5016  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-fv 5602  df-ov 6298  df-oprab 6299  df-mpt2 6300  df-map 7434  df-lnop 26583  df-bdop 26584
This theorem is referenced by:  nmopre  26612  nmophmi  26773  adjbdln  26825  nmopadjlem  26831  nmoptrii  26836  nmopcoi  26837  bdophsi  26838  bdophdi  26839  nmoptri2i  26841  adjcoi  26842  nmopcoadji  26843  unierri  26846
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