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Theorem blof 21797
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 2366 . . 3  |-  ( U 
LnOp  W )  =  ( U  LnOp  W )
2 blof.5 . . 3  |-  B  =  ( U  BLnOp  W )
31, 2bloln 21796 . 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 21767 . 2  |-  ( ( U  e.  NrmCVec  /\  W  e.  NrmCVec  /\  T  e.  ( U  LnOp  W ) )  ->  T : X
--> Y )
73, 6syld3an3 1228 1  |-  ( ( U  e.  NrmCVec  /\  W  e.  NrmCVec  /\  T  e.  B )  ->  T : X --> Y )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ w3a 935    = wceq 1647    e. wcel 1715   -->wf 5354   ` cfv 5358  (class class class)co 5981   NrmCVeccnv 21574   BaseSetcba 21576    LnOp clno 21752    BLnOp cblo 21754
This theorem is referenced by:  nmblore  21798  nmblolbii  21811  blometi  21815  ubthlem3  21885  htthlem  21931
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1551  ax-5 1562  ax-17 1621  ax-9 1659  ax-8 1680  ax-13 1717  ax-14 1719  ax-6 1734  ax-7 1739  ax-11 1751  ax-12 1937  ax-ext 2347  ax-sep 4243  ax-nul 4251  ax-pow 4290  ax-pr 4316  ax-un 4615
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 937  df-tru 1324  df-ex 1547  df-nf 1550  df-sb 1654  df-eu 2221  df-mo 2222  df-clab 2353  df-cleq 2359  df-clel 2362  df-nfc 2491  df-ne 2531  df-ral 2633  df-rex 2634  df-rab 2637  df-v 2875  df-sbc 3078  df-dif 3241  df-un 3243  df-in 3245  df-ss 3252  df-nul 3544  df-if 3655  df-pw 3716  df-sn 3735  df-pr 3736  df-op 3738  df-uni 3930  df-br 4126  df-opab 4180  df-id 4412  df-xp 4798  df-rel 4799  df-cnv 4800  df-co 4801  df-dm 4802  df-rn 4803  df-iota 5322  df-fun 5360  df-fn 5361  df-f 5362  df-fv 5366  df-ov 5984  df-oprab 5985  df-mpt2 5986  df-map 6917  df-lno 21756  df-blo 21758
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