MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  bloln Structured version   Unicode version

Theorem bloln 25403
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 2467 . . . 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 25401 . . 3  |-  ( ( U  e.  NrmCVec  /\  W  e.  NrmCVec )  ->  ( T  e.  B  <->  ( T  e.  L  /\  (
( U normOpOLD W
) `  T )  < +oo ) ) )
54simprbda 623 . 2  |-  ( ( ( U  e.  NrmCVec  /\  W  e.  NrmCVec )  /\  T  e.  B )  ->  T  e.  L )
653impa 1191 1  |-  ( ( U  e.  NrmCVec  /\  W  e.  NrmCVec  /\  T  e.  B )  ->  T  e.  L )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    /\ w3a 973    = wceq 1379    e. wcel 1767   class class class wbr 4447   ` cfv 5588  (class class class)co 6284   +oocpnf 9625    < clt 9628   NrmCVeccnv 25181    LnOp clno 25359   normOpOLDcnmoo 25360    BLnOp cblo 25361
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-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pr 4686
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-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-iota 5551  df-fun 5590  df-fv 5596  df-ov 6287  df-oprab 6288  df-mpt2 6289  df-blo 25365
This theorem is referenced by:  blof  25404  nmblolbii  25418  isblo3i  25420  blometi  25422  blocn2  25427  ubthlem2  25491
  Copyright terms: Public domain W3C validator