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Theorem bloval 26293
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 6303 . . . 4  |-  ( u  =  U  ->  (
u  LnOp  w )  =  ( U  LnOp  w ) )
3 oveq1 6303 . . . . . 6  |-  ( u  =  U  ->  (
u normOpOLD w )  =  ( U normOpOLD w
) )
43fveq1d 5874 . . . . 5  |-  ( u  =  U  ->  (
( u normOpOLD w
) `  t )  =  ( ( U
normOpOLD w ) `  t ) )
54breq1d 4427 . . . 4  |-  ( u  =  U  ->  (
( ( u normOpOLD w ) `  t
)  < +oo  <->  ( ( U normOpOLD w ) `  t )  < +oo ) )
62, 5rabeqbidv 3073 . . 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 6304 . . . . 5  |-  ( w  =  W  ->  ( U  LnOp  w )  =  ( U  LnOp  W
) )
8 bloval.4 . . . . 5  |-  L  =  ( U  LnOp  W
)
97, 8syl6eqr 2479 . . . 4  |-  ( w  =  W  ->  ( U  LnOp  w )  =  L )
10 oveq2 6304 . . . . . . 7  |-  ( w  =  W  ->  ( U normOpOLD w )  =  ( U normOpOLD W
) )
11 bloval.3 . . . . . . 7  |-  N  =  ( U normOpOLD W
)
1210, 11syl6eqr 2479 . . . . . 6  |-  ( w  =  W  ->  ( U normOpOLD w )  =  N )
1312fveq1d 5874 . . . . 5  |-  ( w  =  W  ->  (
( U normOpOLD w
) `  t )  =  ( N `  t ) )
1413breq1d 4427 . . . 4  |-  ( w  =  W  ->  (
( ( U normOpOLD w ) `  t
)  < +oo  <->  ( N `  t )  < +oo ) )
159, 14rabeqbidv 3073 . . 3  |-  ( w  =  W  ->  { t  e.  ( U  LnOp  w )  |  ( ( U normOpOLD w ) `  t )  < +oo }  =  { t  e.  L  |  ( N `
 t )  < +oo } )
16 df-blo 26258 . . 3  |-  BLnOp  =  ( u  e.  NrmCVec ,  w  e.  NrmCVec  |->  { t  e.  ( u  LnOp  w
)  |  ( ( u normOpOLD w ) `  t )  < +oo } )
17 ovex 6324 . . . . 5  |-  ( U 
LnOp  W )  e.  _V
188, 17eqeltri 2504 . . . 4  |-  L  e. 
_V
1918rabex 4567 . . 3  |-  { t  e.  L  |  ( N `  t )  < +oo }  e.  _V
206, 15, 16, 19ovmpt2 6437 . 2  |-  ( ( U  e.  NrmCVec  /\  W  e.  NrmCVec )  ->  ( U  BLnOp  W )  =  { t  e.  L  |  ( N `  t )  < +oo } )
211, 20syl5eq 2473 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 370    = wceq 1437    e. wcel 1867   {crab 2777   _Vcvv 3078   class class class wbr 4417   ` cfv 5592  (class class class)co 6296   +oocpnf 9661    < clt 9664   NrmCVeccnv 26074    LnOp clno 26252   normOpOLDcnmoo 26253    BLnOp cblo 26254
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1748  ax-6 1794  ax-7 1838  ax-9 1871  ax-10 1886  ax-11 1891  ax-12 1904  ax-13 2052  ax-ext 2398  ax-sep 4539  ax-nul 4547  ax-pr 4652
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1787  df-eu 2267  df-mo 2268  df-clab 2406  df-cleq 2412  df-clel 2415  df-nfc 2570  df-ne 2618  df-ral 2778  df-rex 2779  df-rab 2782  df-v 3080  df-sbc 3297  df-dif 3436  df-un 3438  df-in 3440  df-ss 3447  df-nul 3759  df-if 3907  df-sn 3994  df-pr 3996  df-op 4000  df-uni 4214  df-br 4418  df-opab 4476  df-id 4760  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-iota 5556  df-fun 5594  df-fv 5600  df-ov 6299  df-oprab 6300  df-mpt2 6301  df-blo 26258
This theorem is referenced by:  isblo  26294  hhbloi  27416
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