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Theorem bloln 26270
Description: A bounded operator is a linear operator. (Contributed by NM, 8-Dec-2007.) (New usage is discouraged.)
Hypotheses
Ref Expression
bloln.4  |-  L  =  ( U  LnOp  W
)
bloln.5  |-  B  =  ( U  BLnOp  W )
Assertion
Ref Expression
bloln  |-  ( ( U  e.  NrmCVec  /\  W  e.  NrmCVec  /\  T  e.  B )  ->  T  e.  L )

Proof of Theorem bloln
StepHypRef Expression
1 eqid 2429 . . . 4  |-  ( U
normOpOLD W )  =  ( U normOpOLD W
)
2 bloln.4 . . . 4  |-  L  =  ( U  LnOp  W
)
3 bloln.5 . . . 4  |-  B  =  ( U  BLnOp  W )
41, 2, 3isblo 26268 . . 3  |-  ( ( U  e.  NrmCVec  /\  W  e.  NrmCVec )  ->  ( T  e.  B  <->  ( T  e.  L  /\  (
( U normOpOLD W
) `  T )  < +oo ) ) )
54simprbda 627 . 2  |-  ( ( ( U  e.  NrmCVec  /\  W  e.  NrmCVec )  /\  T  e.  B )  ->  T  e.  L )
653impa 1200 1  |-  ( ( U  e.  NrmCVec  /\  W  e.  NrmCVec  /\  T  e.  B )  ->  T  e.  L )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 370    /\ w3a 982    = wceq 1437    e. wcel 1870   class class class wbr 4426   ` cfv 5601  (class class class)co 6305   +oocpnf 9671    < clt 9674   NrmCVeccnv 26048    LnOp clno 26226   normOpOLDcnmoo 26227    BLnOp cblo 26228
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1751  ax-6 1797  ax-7 1841  ax-9 1874  ax-10 1889  ax-11 1894  ax-12 1907  ax-13 2055  ax-ext 2407  ax-sep 4548  ax-nul 4556  ax-pr 4661
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1790  df-eu 2270  df-mo 2271  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2579  df-ne 2627  df-ral 2787  df-rex 2788  df-rab 2791  df-v 3089  df-sbc 3306  df-dif 3445  df-un 3447  df-in 3449  df-ss 3456  df-nul 3768  df-if 3916  df-sn 4003  df-pr 4005  df-op 4009  df-uni 4223  df-br 4427  df-opab 4485  df-id 4769  df-xp 4860  df-rel 4861  df-cnv 4862  df-co 4863  df-dm 4864  df-iota 5565  df-fun 5603  df-fv 5609  df-ov 6308  df-oprab 6309  df-mpt2 6310  df-blo 26232
This theorem is referenced by:  blof  26271  nmblolbii  26285  isblo3i  26287  blometi  26289  blocn2  26294  ubthlem2  26358
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