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Theorem bloval 21375
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 5881 . . . 4  |-  ( u  =  U  ->  (
u  LnOp  w )  =  ( U  LnOp  w ) )
3 oveq1 5881 . . . . . 6  |-  ( u  =  U  ->  (
u normOp OLD w )  =  ( U normOp OLD w
) )
43fveq1d 5543 . . . . 5  |-  ( u  =  U  ->  (
( u normOp OLD w
) `  t )  =  ( ( U
normOp OLD w ) `  t ) )
54breq1d 4049 . . . 4  |-  ( u  =  U  ->  (
( ( u normOp OLD w ) `  t
)  <  +oo  <->  ( ( U normOp OLD w ) `  t )  <  +oo ) )
62, 5rabeqbidv 2796 . . 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 5882 . . . . 5  |-  ( w  =  W  ->  ( U  LnOp  w )  =  ( U  LnOp  W
) )
8 bloval.4 . . . . 5  |-  L  =  ( U  LnOp  W
)
97, 8syl6eqr 2346 . . . 4  |-  ( w  =  W  ->  ( U  LnOp  w )  =  L )
10 oveq2 5882 . . . . . . 7  |-  ( w  =  W  ->  ( U normOp OLD w )  =  ( U normOp OLD W
) )
11 bloval.3 . . . . . . 7  |-  N  =  ( U normOp OLD W
)
1210, 11syl6eqr 2346 . . . . . 6  |-  ( w  =  W  ->  ( U normOp OLD w )  =  N )
1312fveq1d 5543 . . . . 5  |-  ( w  =  W  ->  (
( U normOp OLD w
) `  t )  =  ( N `  t ) )
1413breq1d 4049 . . . 4  |-  ( w  =  W  ->  (
( ( U normOp OLD w ) `  t
)  <  +oo  <->  ( N `  t )  <  +oo ) )
159, 14rabeqbidv 2796 . . 3  |-  ( w  =  W  ->  { t  e.  ( U  LnOp  w )  |  ( ( U normOp OLD w ) `  t )  <  +oo }  =  { t  e.  L  |  ( N `
 t )  <  +oo } )
16 df-blo 21340 . . 3  |-  BLnOp  =  ( u  e.  NrmCVec ,  w  e.  NrmCVec  |->  { t  e.  ( u  LnOp  w
)  |  ( ( u normOp OLD w ) `  t )  <  +oo } )
17 ovex 5899 . . . . 5  |-  ( U 
LnOp  W )  e.  _V
188, 17eqeltri 2366 . . . 4  |-  L  e. 
_V
1918rabex 4181 . . 3  |-  { t  e.  L  |  ( N `  t )  <  +oo }  e.  _V
206, 15, 16, 19ovmpt2 5999 . 2  |-  ( ( U  e.  NrmCVec  /\  W  e.  NrmCVec )  ->  ( U  BLnOp  W )  =  { t  e.  L  |  ( N `  t )  <  +oo } )
211, 20syl5eq 2340 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 1632    e. wcel 1696   {crab 2560   _Vcvv 2801   class class class wbr 4039   ` cfv 5271  (class class class)co 5874    +oocpnf 8880    < clt 8883   NrmCVeccnv 21156    LnOp clno 21334   normOp OLDcnmoo 21335    BLnOp cblo 21336
This theorem is referenced by:  isblo  21376  hhbloi  22498
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pr 4230
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-sbc 3005  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-br 4040  df-opab 4094  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-iota 5235  df-fun 5273  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-blo 21340
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