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Theorem blof 26418
Description: A bounded operator is an operator. (Contributed by NM, 8-Dec-2007.) (New usage is discouraged.)
Hypotheses
Ref Expression
blof.1  |-  X  =  ( BaseSet `  U )
blof.2  |-  Y  =  ( BaseSet `  W )
blof.5  |-  B  =  ( U  BLnOp  W )
Assertion
Ref Expression
blof  |-  ( ( U  e.  NrmCVec  /\  W  e.  NrmCVec  /\  T  e.  B )  ->  T : X --> Y )

Proof of Theorem blof
StepHypRef Expression
1 eqid 2423 . . 3  |-  ( U 
LnOp  W )  =  ( U  LnOp  W )
2 blof.5 . . 3  |-  B  =  ( U  BLnOp  W )
31, 2bloln 26417 . 2  |-  ( ( U  e.  NrmCVec  /\  W  e.  NrmCVec  /\  T  e.  B )  ->  T  e.  ( U  LnOp  W
) )
4 blof.1 . . 3  |-  X  =  ( BaseSet `  U )
5 blof.2 . . 3  |-  Y  =  ( BaseSet `  W )
64, 5, 1lnof 26388 . 2  |-  ( ( U  e.  NrmCVec  /\  W  e.  NrmCVec  /\  T  e.  ( U  LnOp  W ) )  ->  T : X
--> Y )
73, 6syld3an3 1310 1  |-  ( ( U  e.  NrmCVec  /\  W  e.  NrmCVec  /\  T  e.  B )  ->  T : X --> Y )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ w3a 983    = wceq 1438    e. wcel 1869   -->wf 5595   ` cfv 5599  (class class class)co 6303   NrmCVeccnv 26195   BaseSetcba 26197    LnOp clno 26373    BLnOp cblo 26375
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1666  ax-4 1679  ax-5 1749  ax-6 1795  ax-7 1840  ax-8 1871  ax-9 1873  ax-10 1888  ax-11 1893  ax-12 1906  ax-13 2054  ax-ext 2401  ax-sep 4544  ax-nul 4553  ax-pow 4600  ax-pr 4658  ax-un 6595
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3an 985  df-tru 1441  df-ex 1661  df-nf 1665  df-sb 1788  df-eu 2270  df-mo 2271  df-clab 2409  df-cleq 2415  df-clel 2418  df-nfc 2573  df-ne 2621  df-ral 2781  df-rex 2782  df-rab 2785  df-v 3084  df-sbc 3301  df-dif 3440  df-un 3442  df-in 3444  df-ss 3451  df-nul 3763  df-if 3911  df-pw 3982  df-sn 3998  df-pr 4000  df-op 4004  df-uni 4218  df-br 4422  df-opab 4481  df-id 4766  df-xp 4857  df-rel 4858  df-cnv 4859  df-co 4860  df-dm 4861  df-rn 4862  df-iota 5563  df-fun 5601  df-fn 5602  df-f 5603  df-fv 5607  df-ov 6306  df-oprab 6307  df-mpt2 6308  df-map 7480  df-lno 26377  df-blo 26379
This theorem is referenced by:  nmblore  26419  nmblolbii  26432  blometi  26436  ubthlem3  26506  htthlem  26562
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