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

Theorem bloval 24200
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 6117 . . . 4  |-  ( u  =  U  ->  (
u  LnOp  w )  =  ( U  LnOp  w ) )
3 oveq1 6117 . . . . . 6  |-  ( u  =  U  ->  (
u normOpOLD w )  =  ( U normOpOLD w
) )
43fveq1d 5712 . . . . 5  |-  ( u  =  U  ->  (
( u normOpOLD w
) `  t )  =  ( ( U
normOpOLD w ) `  t ) )
54breq1d 4321 . . . 4  |-  ( u  =  U  ->  (
( ( u normOpOLD w ) `  t
)  < +oo  <->  ( ( U normOpOLD w ) `  t )  < +oo ) )
62, 5rabeqbidv 2986 . . 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 6118 . . . . 5  |-  ( w  =  W  ->  ( U  LnOp  w )  =  ( U  LnOp  W
) )
8 bloval.4 . . . . 5  |-  L  =  ( U  LnOp  W
)
97, 8syl6eqr 2493 . . . 4  |-  ( w  =  W  ->  ( U  LnOp  w )  =  L )
10 oveq2 6118 . . . . . . 7  |-  ( w  =  W  ->  ( U normOpOLD w )  =  ( U normOpOLD W
) )
11 bloval.3 . . . . . . 7  |-  N  =  ( U normOpOLD W
)
1210, 11syl6eqr 2493 . . . . . 6  |-  ( w  =  W  ->  ( U normOpOLD w )  =  N )
1312fveq1d 5712 . . . . 5  |-  ( w  =  W  ->  (
( U normOpOLD w
) `  t )  =  ( N `  t ) )
1413breq1d 4321 . . . 4  |-  ( w  =  W  ->  (
( ( U normOpOLD w ) `  t
)  < +oo  <->  ( N `  t )  < +oo ) )
159, 14rabeqbidv 2986 . . 3  |-  ( w  =  W  ->  { t  e.  ( U  LnOp  w )  |  ( ( U normOpOLD w ) `  t )  < +oo }  =  { t  e.  L  |  ( N `
 t )  < +oo } )
16 df-blo 24165 . . 3  |-  BLnOp  =  ( u  e.  NrmCVec ,  w  e.  NrmCVec  |->  { t  e.  ( u  LnOp  w
)  |  ( ( u normOpOLD w ) `  t )  < +oo } )
17 ovex 6135 . . . . 5  |-  ( U 
LnOp  W )  e.  _V
188, 17eqeltri 2513 . . . 4  |-  L  e. 
_V
1918rabex 4462 . . 3  |-  { t  e.  L  |  ( N `  t )  < +oo }  e.  _V
206, 15, 16, 19ovmpt2 6245 . 2  |-  ( ( U  e.  NrmCVec  /\  W  e.  NrmCVec )  ->  ( U  BLnOp  W )  =  { t  e.  L  |  ( N `  t )  < +oo } )
211, 20syl5eq 2487 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 1369    e. wcel 1756   {crab 2738   _Vcvv 2991   class class class wbr 4311   ` cfv 5437  (class class class)co 6110   +oocpnf 9434    < clt 9437   NrmCVeccnv 23981    LnOp clno 24159   normOpOLDcnmoo 24160    BLnOp cblo 24161
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-sep 4432  ax-nul 4440  ax-pr 4550
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2577  df-ne 2622  df-ral 2739  df-rex 2740  df-rab 2743  df-v 2993  df-sbc 3206  df-dif 3350  df-un 3352  df-in 3354  df-ss 3361  df-nul 3657  df-if 3811  df-sn 3897  df-pr 3899  df-op 3903  df-uni 4111  df-br 4312  df-opab 4370  df-id 4655  df-xp 4865  df-rel 4866  df-cnv 4867  df-co 4868  df-dm 4869  df-iota 5400  df-fun 5439  df-fv 5445  df-ov 6113  df-oprab 6114  df-mpt2 6115  df-blo 24165
This theorem is referenced by:  isblo  24201  hhbloi  25325
  Copyright terms: Public domain W3C validator