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Theorem nmooval 22217
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 5701 . . . . 5  |-  ( BaseSet `  W )  e.  _V
31, 2eqeltri 2474 . . . 4  |-  Y  e. 
_V
4 nmoofval.1 . . . . 5  |-  X  =  ( BaseSet `  U )
5 fvex 5701 . . . . 5  |-  ( BaseSet `  U )  e.  _V
64, 5eqeltri 2474 . . . 4  |-  X  e. 
_V
73, 6elmap 7001 . . 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 22216 . . . . 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 5689 . . . 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 5686 . . . . . . . . . . 11  |-  ( t  =  T  ->  (
t `  z )  =  ( T `  z ) )
1413fveq2d 5691 . . . . . . . . . 10  |-  ( t  =  T  ->  ( M `  ( t `  z ) )  =  ( M `  ( T `  z )
) )
1514eqeq2d 2415 . . . . . . . . 9  |-  ( t  =  T  ->  (
x  =  ( M `
 ( t `  z ) )  <->  x  =  ( M `  ( T `
 z ) ) ) )
1615anbi2d 685 . . . . . . . 8  |-  ( t  =  T  ->  (
( ( L `  z )  <_  1  /\  x  =  ( M `  ( t `  z ) ) )  <-> 
( ( L `  z )  <_  1  /\  x  =  ( M `  ( T `  z ) ) ) ) )
1716rexbidv 2687 . . . . . . 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 2518 . . . . . 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 7409 . . . . 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 2404 . . . . 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 10690 . . . . . 6  |-  <  Or  RR*
2221supex 7424 . . . . 5  |-  sup ( { x  |  E. z  e.  X  (
( L `  z
)  <_  1  /\  x  =  ( M `  ( T `  z
) ) ) } ,  RR* ,  <  )  e.  _V
2319, 20, 22fvmpt 5765 . . . 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 2456 . . 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 463 . 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 1148 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 359    /\ w3a 936    = wceq 1649    e. wcel 1721   {cab 2390   E.wrex 2667   _Vcvv 2916   class class class wbr 4172    e. cmpt 4226   -->wf 5409   ` cfv 5413  (class class class)co 6040    ^m cmap 6977   supcsup 7403   1c1 8947   RR*cxr 9075    < clt 9076    <_ cle 9077   NrmCVeccnv 22016   BaseSetcba 22018   normCVcnmcv 22022   normOp OLDcnmoo 22195
This theorem is referenced by:  nmoxr  22220  nmooge0  22221  nmorepnf  22222  nmoolb  22225  nmoubi  22226  nmoo0  22245  nmlno0lem  22247
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-rep 4280  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660  ax-cnex 9002  ax-resscn 9003  ax-pre-lttri 9020  ax-pre-lttrn 9021
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-nel 2570  df-ral 2671  df-rex 2672  df-reu 2673  df-rmo 2674  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-op 3783  df-uni 3976  df-iun 4055  df-br 4173  df-opab 4227  df-mpt 4228  df-id 4458  df-po 4463  df-so 4464  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-ov 6043  df-oprab 6044  df-mpt2 6045  df-er 6864  df-map 6979  df-en 7069  df-dom 7070  df-sdom 7071  df-sup 7404  df-pnf 9078  df-mnf 9079  df-xr 9080  df-ltxr 9081  df-nmoo 22199
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