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Theorem bloln 21378
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 2296 . . . 4  |-  ( U
normOp OLD W )  =  ( U normOp OLD W
)
2 bloln.4 . . . 4  |-  L  =  ( U  LnOp  W
)
3 bloln.5 . . . 4  |-  B  =  ( U  BLnOp  W )
41, 2, 3isblo 21376 . . 3  |-  ( ( U  e.  NrmCVec  /\  W  e.  NrmCVec )  ->  ( T  e.  B  <->  ( T  e.  L  /\  (
( U normOp OLD W
) `  T )  <  +oo ) ) )
54simprbda 606 . 2  |-  ( ( ( U  e.  NrmCVec  /\  W  e.  NrmCVec )  /\  T  e.  B )  ->  T  e.  L )
653impa 1146 1  |-  ( ( U  e.  NrmCVec  /\  W  e.  NrmCVec  /\  T  e.  B )  ->  T  e.  L )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    /\ w3a 934    = wceq 1632    e. wcel 1696   class class class wbr 4039   ` cfv 5271  (class class class)co 5874    +oocpnf 8880    < clt 8883   NrmCVeccnv 21156    LnOp clno 21334   normOp OLDcnmoo 21335    BLnOp cblo 21336
This theorem is referenced by:  blof  21379  nmblolbii  21393  isblo3i  21395  blometi  21397  blocn2  21402  ubthlem2  21466
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pr 4230
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-sbc 3005  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-br 4040  df-opab 4094  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-iota 5235  df-fun 5273  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-blo 21340
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