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Theorem blof 25362
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 2460 . . 3  |-  ( U 
LnOp  W )  =  ( U  LnOp  W )
2 blof.5 . . 3  |-  B  =  ( U  BLnOp  W )
31, 2bloln 25361 . 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 25332 . 2  |-  ( ( U  e.  NrmCVec  /\  W  e.  NrmCVec  /\  T  e.  ( U  LnOp  W ) )  ->  T : X
--> Y )
73, 6syld3an3 1268 1  |-  ( ( U  e.  NrmCVec  /\  W  e.  NrmCVec  /\  T  e.  B )  ->  T : X --> Y )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ w3a 968    = wceq 1374    e. wcel 1762   -->wf 5575   ` cfv 5579  (class class class)co 6275   NrmCVeccnv 25139   BaseSetcba 25141    LnOp clno 25317    BLnOp cblo 25319
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-8 1764  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1961  ax-ext 2438  ax-sep 4561  ax-nul 4569  ax-pow 4618  ax-pr 4679  ax-un 6567
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2272  df-mo 2273  df-clab 2446  df-cleq 2452  df-clel 2455  df-nfc 2610  df-ne 2657  df-ral 2812  df-rex 2813  df-rab 2816  df-v 3108  df-sbc 3325  df-dif 3472  df-un 3474  df-in 3476  df-ss 3483  df-nul 3779  df-if 3933  df-pw 4005  df-sn 4021  df-pr 4023  df-op 4027  df-uni 4239  df-br 4441  df-opab 4499  df-id 4788  df-xp 4998  df-rel 4999  df-cnv 5000  df-co 5001  df-dm 5002  df-rn 5003  df-iota 5542  df-fun 5581  df-fn 5582  df-f 5583  df-fv 5587  df-ov 6278  df-oprab 6279  df-mpt2 6280  df-map 7412  df-lno 25321  df-blo 25323
This theorem is referenced by:  nmblore  25363  nmblolbii  25376  blometi  25380  ubthlem3  25450  htthlem  25496
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