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Theorem bloval 25400
Description: The class of bounded linear operators between two normed complex vector spaces. (Contributed by NM, 6-Nov-2007.) (Revised by Mario Carneiro, 16-Nov-2013.) (New usage is discouraged.)
Hypotheses
Ref Expression
bloval.3  |-  N  =  ( U normOpOLD W
)
bloval.4  |-  L  =  ( U  LnOp  W
)
bloval.5  |-  B  =  ( U  BLnOp  W )
Assertion
Ref Expression
bloval  |-  ( ( U  e.  NrmCVec  /\  W  e.  NrmCVec )  ->  B  =  { t  e.  L  |  ( N `  t )  < +oo } )
Distinct variable groups:    t, L    t, N    t, U    t, W
Allowed substitution hint:    B( t)

Proof of Theorem bloval
Dummy variables  u  w are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 bloval.5 . 2  |-  B  =  ( U  BLnOp  W )
2 oveq1 6291 . . . 4  |-  ( u  =  U  ->  (
u  LnOp  w )  =  ( U  LnOp  w ) )
3 oveq1 6291 . . . . . 6  |-  ( u  =  U  ->  (
u normOpOLD w )  =  ( U normOpOLD w
) )
43fveq1d 5868 . . . . 5  |-  ( u  =  U  ->  (
( u normOpOLD w
) `  t )  =  ( ( U
normOpOLD w ) `  t ) )
54breq1d 4457 . . . 4  |-  ( u  =  U  ->  (
( ( u normOpOLD w ) `  t
)  < +oo  <->  ( ( U normOpOLD w ) `  t )  < +oo ) )
62, 5rabeqbidv 3108 . . 3  |-  ( u  =  U  ->  { t  e.  ( u  LnOp  w )  |  ( ( u normOpOLD w ) `  t )  < +oo }  =  { t  e.  ( U  LnOp  w
)  |  ( ( U normOpOLD w ) `  t )  < +oo } )
7 oveq2 6292 . . . . 5  |-  ( w  =  W  ->  ( U  LnOp  w )  =  ( U  LnOp  W
) )
8 bloval.4 . . . . 5  |-  L  =  ( U  LnOp  W
)
97, 8syl6eqr 2526 . . . 4  |-  ( w  =  W  ->  ( U  LnOp  w )  =  L )
10 oveq2 6292 . . . . . . 7  |-  ( w  =  W  ->  ( U normOpOLD w )  =  ( U normOpOLD W
) )
11 bloval.3 . . . . . . 7  |-  N  =  ( U normOpOLD W
)
1210, 11syl6eqr 2526 . . . . . 6  |-  ( w  =  W  ->  ( U normOpOLD w )  =  N )
1312fveq1d 5868 . . . . 5  |-  ( w  =  W  ->  (
( U normOpOLD w
) `  t )  =  ( N `  t ) )
1413breq1d 4457 . . . 4  |-  ( w  =  W  ->  (
( ( U normOpOLD w ) `  t
)  < +oo  <->  ( N `  t )  < +oo ) )
159, 14rabeqbidv 3108 . . 3  |-  ( w  =  W  ->  { t  e.  ( U  LnOp  w )  |  ( ( U normOpOLD w ) `  t )  < +oo }  =  { t  e.  L  |  ( N `
 t )  < +oo } )
16 df-blo 25365 . . 3  |-  BLnOp  =  ( u  e.  NrmCVec ,  w  e.  NrmCVec  |->  { t  e.  ( u  LnOp  w
)  |  ( ( u normOpOLD w ) `  t )  < +oo } )
17 ovex 6309 . . . . 5  |-  ( U 
LnOp  W )  e.  _V
188, 17eqeltri 2551 . . . 4  |-  L  e. 
_V
1918rabex 4598 . . 3  |-  { t  e.  L  |  ( N `  t )  < +oo }  e.  _V
206, 15, 16, 19ovmpt2 6422 . 2  |-  ( ( U  e.  NrmCVec  /\  W  e.  NrmCVec )  ->  ( U  BLnOp  W )  =  { t  e.  L  |  ( N `  t )  < +oo } )
211, 20syl5eq 2520 1  |-  ( ( U  e.  NrmCVec  /\  W  e.  NrmCVec )  ->  B  =  { t  e.  L  |  ( N `  t )  < +oo } )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1379    e. wcel 1767   {crab 2818   _Vcvv 3113   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:  isblo  25401  hhbloi  26525
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