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Theorem lnopfi 26561
Description: A linear Hilbert space operator is a Hilbert space operator. (Contributed by NM, 23-Jan-2006.) (New usage is discouraged.)
Hypothesis
Ref Expression
lnopl.1  |-  T  e. 
LinOp
Assertion
Ref Expression
lnopfi  |-  T : ~H
--> ~H

Proof of Theorem lnopfi
StepHypRef Expression
1 lnopl.1 . 2  |-  T  e. 
LinOp
2 lnopf 26451 . 2  |-  ( T  e.  LinOp  ->  T : ~H
--> ~H )
31, 2ax-mp 5 1  |-  T : ~H
--> ~H
Colors of variables: wff setvar class
Syntax hints:    e. wcel 1767   -->wf 5582   ~Hchil 25509   LinOpclo 25537
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 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6574  ax-hilex 25589
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 2819  df-rex 2820  df-rab 2823  df-v 3115  df-sbc 3332  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-br 4448  df-opab 4506  df-id 4795  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-iota 5549  df-fun 5588  df-fn 5589  df-f 5590  df-fv 5594  df-ov 6285  df-oprab 6286  df-mpt2 6287  df-map 7419  df-lnop 26433
This theorem is referenced by:  lnopaddi  26563  lnopsubi  26566  hoddii  26581  nmlnop0iALT  26587  nmlnopgt0i  26589  lnopmi  26592  lnophsi  26593  lnophdi  26594  lnopcoi  26595  lnopco0i  26596  lnopeq0lem1  26597  lnopeq0i  26599  lnopeqi  26600  lnopunilem1  26602  lnopunilem2  26603  lnophmlem2  26609  lnophmi  26610  nmbdoplbi  26616  nmcopexi  26619  nmcoplbi  26620  lnopconi  26626  imaelshi  26650  rnelshi  26651  cnlnadjlem2  26660  cnlnadjlem6  26664  cnlnadjlem7  26665  cnlnadjeui  26669  nmopcoi  26687  bdopcoi  26690  hmopidmchi  26743  hmopidmpji  26744
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