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

Theorem nmooval 22269
Description: The operator norm function. (Contributed by NM, 27-Nov-2007.) (Revised by Mario Carneiro, 16-Nov-2013.) (New usage is discouraged.)
Hypotheses
Ref Expression
nmoofval.1  |-  X  =  ( BaseSet `  U )
nmoofval.2  |-  Y  =  ( BaseSet `  W )
nmoofval.3  |-  L  =  ( normCV `  U )
nmoofval.4  |-  M  =  ( normCV `  W )
nmoofval.6  |-  N  =  ( U normOp OLD W
)
Assertion
Ref Expression
nmooval  |-  ( ( U  e.  NrmCVec  /\  W  e.  NrmCVec  /\  T : X
--> Y )  ->  ( N `  T )  =  sup ( { x  |  E. z  e.  X  ( ( L `  z )  <_  1  /\  x  =  ( M `  ( T `  z ) ) ) } ,  RR* ,  <  ) )
Distinct variable groups:    x, z, U    x, W, z    z, X    x, Y    x, T, z
Allowed substitution hints:    L( x, z)    M( x, z)    N( x, z)    X( x)    Y( z)

Proof of Theorem nmooval
Dummy variable  t is distinct from all other variables.
StepHypRef Expression
1 nmoofval.2 . . . . 5  |-  Y  =  ( BaseSet `  W )
2 fvex 5745 . . . . 5  |-  ( BaseSet `  W )  e.  _V
31, 2eqeltri 2508 . . . 4  |-  Y  e. 
_V
4 nmoofval.1 . . . . 5  |-  X  =  ( BaseSet `  U )
5 fvex 5745 . . . . 5  |-  ( BaseSet `  U )  e.  _V
64, 5eqeltri 2508 . . . 4  |-  X  e. 
_V
73, 6elmap 7045 . . 3  |-  ( T  e.  ( Y  ^m  X )  <->  T : X
--> Y )
8 nmoofval.3 . . . . . 6  |-  L  =  ( normCV `  U )
9 nmoofval.4 . . . . . 6  |-  M  =  ( normCV `  W )
10 nmoofval.6 . . . . . 6  |-  N  =  ( U normOp OLD W
)
114, 1, 8, 9, 10nmoofval 22268 . . . . 5  |-  ( ( U  e.  NrmCVec  /\  W  e.  NrmCVec )  ->  N  =  ( t  e.  ( Y  ^m  X
)  |->  sup ( { x  |  E. z  e.  X  ( ( L `  z )  <_  1  /\  x  =  ( M `  ( t `  z ) ) ) } ,  RR* ,  <  ) ) )
1211fveq1d 5733 . . . 4  |-  ( ( U  e.  NrmCVec  /\  W  e.  NrmCVec )  ->  ( N `  T )  =  ( ( t  e.  ( Y  ^m  X )  |->  sup ( { x  |  E. z  e.  X  (
( L `  z
)  <_  1  /\  x  =  ( M `  ( t `  z
) ) ) } ,  RR* ,  <  )
) `  T )
)
13 fveq1 5730 . . . . . . . . . . 11  |-  ( t  =  T  ->  (
t `  z )  =  ( T `  z ) )
1413fveq2d 5735 . . . . . . . . . 10  |-  ( t  =  T  ->  ( M `  ( t `  z ) )  =  ( M `  ( T `  z )
) )
1514eqeq2d 2449 . . . . . . . . 9  |-  ( t  =  T  ->  (
x  =  ( M `
 ( t `  z ) )  <->  x  =  ( M `  ( T `
 z ) ) ) )
1615anbi2d 686 . . . . . . . 8  |-  ( t  =  T  ->  (
( ( L `  z )  <_  1  /\  x  =  ( M `  ( t `  z ) ) )  <-> 
( ( L `  z )  <_  1  /\  x  =  ( M `  ( T `  z ) ) ) ) )
1716rexbidv 2728 . . . . . . 7  |-  ( t  =  T  ->  ( E. z  e.  X  ( ( L `  z )  <_  1  /\  x  =  ( M `  ( t `  z ) ) )  <->  E. z  e.  X  ( ( L `  z )  <_  1  /\  x  =  ( M `  ( T `  z ) ) ) ) )
1817abbidv 2552 . . . . . 6  |-  ( t  =  T  ->  { x  |  E. z  e.  X  ( ( L `  z )  <_  1  /\  x  =  ( M `  ( t `  z ) ) ) }  =  { x  |  E. z  e.  X  ( ( L `  z )  <_  1  /\  x  =  ( M `  ( T `  z ) ) ) } )
1918supeq1d 7454 . . . . 5  |-  ( t  =  T  ->  sup ( { x  |  E. z  e.  X  (
( L `  z
)  <_  1  /\  x  =  ( M `  ( t `  z
) ) ) } ,  RR* ,  <  )  =  sup ( { x  |  E. z  e.  X  ( ( L `  z )  <_  1  /\  x  =  ( M `  ( T `  z ) ) ) } ,  RR* ,  <  ) )
20 eqid 2438 . . . . 5  |-  ( t  e.  ( Y  ^m  X )  |->  sup ( { x  |  E. z  e.  X  (
( L `  z
)  <_  1  /\  x  =  ( M `  ( t `  z
) ) ) } ,  RR* ,  <  )
)  =  ( t  e.  ( Y  ^m  X )  |->  sup ( { x  |  E. z  e.  X  (
( L `  z
)  <_  1  /\  x  =  ( M `  ( t `  z
) ) ) } ,  RR* ,  <  )
)
21 xrltso 10739 . . . . . 6  |-  <  Or  RR*
2221supex 7471 . . . . 5  |-  sup ( { x  |  E. z  e.  X  (
( L `  z
)  <_  1  /\  x  =  ( M `  ( T `  z
) ) ) } ,  RR* ,  <  )  e.  _V
2319, 20, 22fvmpt 5809 . . . 4  |-  ( T  e.  ( Y  ^m  X )  ->  (
( t  e.  ( Y  ^m  X ) 
|->  sup ( { x  |  E. z  e.  X  ( ( L `  z )  <_  1  /\  x  =  ( M `  ( t `  z ) ) ) } ,  RR* ,  <  ) ) `  T )  =  sup ( { x  |  E. z  e.  X  ( ( L `  z )  <_  1  /\  x  =  ( M `  ( T `  z )
) ) } ,  RR* ,  <  ) )
2412, 23sylan9eq 2490 . . 3  |-  ( ( ( U  e.  NrmCVec  /\  W  e.  NrmCVec )  /\  T  e.  ( Y  ^m  X ) )  -> 
( N `  T
)  =  sup ( { x  |  E. z  e.  X  (
( L `  z
)  <_  1  /\  x  =  ( M `  ( T `  z
) ) ) } ,  RR* ,  <  )
)
257, 24sylan2br 464 . 2  |-  ( ( ( U  e.  NrmCVec  /\  W  e.  NrmCVec )  /\  T : X --> Y )  ->  ( N `  T )  =  sup ( { x  |  E. z  e.  X  (
( L `  z
)  <_  1  /\  x  =  ( M `  ( T `  z
) ) ) } ,  RR* ,  <  )
)
26253impa 1149 1  |-  ( ( U  e.  NrmCVec  /\  W  e.  NrmCVec  /\  T : X
--> Y )  ->  ( N `  T )  =  sup ( { x  |  E. z  e.  X  ( ( L `  z )  <_  1  /\  x  =  ( M `  ( T `  z ) ) ) } ,  RR* ,  <  ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 360    /\ w3a 937    = wceq 1653    e. wcel 1726   {cab 2424   E.wrex 2708   _Vcvv 2958   class class class wbr 4215    e. cmpt 4269   -->wf 5453   ` cfv 5457  (class class class)co 6084    ^m cmap 7021   supcsup 7448   1c1 8996   RR*cxr 9124    < clt 9125    <_ cle 9126   NrmCVeccnv 22068   BaseSetcba 22070   normCVcnmcv 22074   normOp OLDcnmoo 22247
This theorem is referenced by:  nmoxr  22272  nmooge0  22273  nmorepnf  22274  nmoolb  22277  nmoubi  22278  nmoo0  22297  nmlno0lem  22299
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-rep 4323  ax-sep 4333  ax-nul 4341  ax-pow 4380  ax-pr 4406  ax-un 4704  ax-cnex 9051  ax-resscn 9052  ax-pre-lttri 9069  ax-pre-lttrn 9070
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-nel 2604  df-ral 2712  df-rex 2713  df-reu 2714  df-rmo 2715  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-iun 4097  df-br 4216  df-opab 4270  df-mpt 4271  df-id 4501  df-po 4506  df-so 4507  df-xp 4887  df-rel 4888  df-cnv 4889  df-co 4890  df-dm 4891  df-rn 4892  df-res 4893  df-ima 4894  df-iota 5421  df-fun 5459  df-fn 5460  df-f 5461  df-f1 5462  df-fo 5463  df-f1o 5464  df-fv 5465  df-ov 6087  df-oprab 6088  df-mpt2 6089  df-er 6908  df-map 7023  df-en 7113  df-dom 7114  df-sdom 7115  df-sup 7449  df-pnf 9127  df-mnf 9128  df-xr 9129  df-ltxr 9130  df-nmoo 22251
  Copyright terms: Public domain W3C validator