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

Theorem bloval 21359
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 normOp OLD 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 5865 . . . 4  |-  ( u  =  U  ->  (
u  LnOp  w )  =  ( U  LnOp  w ) )
3 oveq1 5865 . . . . . 6  |-  ( u  =  U  ->  (
u normOp OLD w )  =  ( U normOp OLD w
) )
43fveq1d 5527 . . . . 5  |-  ( u  =  U  ->  (
( u normOp OLD w
) `  t )  =  ( ( U
normOp OLD w ) `  t ) )
54breq1d 4033 . . . 4  |-  ( u  =  U  ->  (
( ( u normOp OLD w ) `  t
)  <  +oo  <->  ( ( U normOp OLD w ) `  t )  <  +oo ) )
62, 5rabeqbidv 2783 . . 3  |-  ( u  =  U  ->  { t  e.  ( u  LnOp  w )  |  ( ( u normOp OLD w ) `  t )  <  +oo }  =  { t  e.  ( U  LnOp  w
)  |  ( ( U normOp OLD w ) `  t )  <  +oo } )
7 oveq2 5866 . . . . 5  |-  ( w  =  W  ->  ( U  LnOp  w )  =  ( U  LnOp  W
) )
8 bloval.4 . . . . 5  |-  L  =  ( U  LnOp  W
)
97, 8syl6eqr 2333 . . . 4  |-  ( w  =  W  ->  ( U  LnOp  w )  =  L )
10 oveq2 5866 . . . . . . 7  |-  ( w  =  W  ->  ( U normOp OLD w )  =  ( U normOp OLD W
) )
11 bloval.3 . . . . . . 7  |-  N  =  ( U normOp OLD W
)
1210, 11syl6eqr 2333 . . . . . 6  |-  ( w  =  W  ->  ( U normOp OLD w )  =  N )
1312fveq1d 5527 . . . . 5  |-  ( w  =  W  ->  (
( U normOp OLD w
) `  t )  =  ( N `  t ) )
1413breq1d 4033 . . . 4  |-  ( w  =  W  ->  (
( ( U normOp OLD w ) `  t
)  <  +oo  <->  ( N `  t )  <  +oo ) )
159, 14rabeqbidv 2783 . . 3  |-  ( w  =  W  ->  { t  e.  ( U  LnOp  w )  |  ( ( U normOp OLD w ) `  t )  <  +oo }  =  { t  e.  L  |  ( N `
 t )  <  +oo } )
16 df-blo 21324 . . 3  |-  BLnOp  =  ( u  e.  NrmCVec ,  w  e.  NrmCVec  |->  { t  e.  ( u  LnOp  w
)  |  ( ( u normOp OLD w ) `  t )  <  +oo } )
17 ovex 5883 . . . . 5  |-  ( U 
LnOp  W )  e.  _V
188, 17eqeltri 2353 . . . 4  |-  L  e. 
_V
1918rabex 4165 . . 3  |-  { t  e.  L  |  ( N `  t )  <  +oo }  e.  _V
206, 15, 16, 19ovmpt2 5983 . 2  |-  ( ( U  e.  NrmCVec  /\  W  e.  NrmCVec )  ->  ( U  BLnOp  W )  =  { t  e.  L  |  ( N `  t )  <  +oo } )
211, 20syl5eq 2327 1  |-  ( ( U  e.  NrmCVec  /\  W  e.  NrmCVec )  ->  B  =  { t  e.  L  |  ( N `  t )  <  +oo } )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1623    e. wcel 1684   {crab 2547   _Vcvv 2788   class class class wbr 4023   ` cfv 5255  (class class class)co 5858    +oocpnf 8864    < clt 8867   NrmCVeccnv 21140    LnOp clno 21318   normOp OLDcnmoo 21319    BLnOp cblo 21320
This theorem is referenced by:  isblo  21360  hhbloi  22482
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pr 4214
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-iota 5219  df-fun 5257  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-blo 21324
  Copyright terms: Public domain W3C validator