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Theorem nmooval 21654
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 5646 . . . . 5  |-  ( BaseSet `  W )  e.  _V
31, 2eqeltri 2436 . . . 4  |-  Y  e. 
_V
4 nmoofval.1 . . . . 5  |-  X  =  ( BaseSet `  U )
5 fvex 5646 . . . . 5  |-  ( BaseSet `  U )  e.  _V
64, 5eqeltri 2436 . . . 4  |-  X  e. 
_V
73, 6elmap 6939 . . 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 21653 . . . . 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 5634 . . . 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 5631 . . . . . . . . . . 11  |-  ( t  =  T  ->  (
t `  z )  =  ( T `  z ) )
1413fveq2d 5636 . . . . . . . . . 10  |-  ( t  =  T  ->  ( M `  ( t `  z ) )  =  ( M `  ( T `  z )
) )
1514eqeq2d 2377 . . . . . . . . 9  |-  ( t  =  T  ->  (
x  =  ( M `
 ( t `  z ) )  <->  x  =  ( M `  ( T `
 z ) ) ) )
1615anbi2d 684 . . . . . . . 8  |-  ( t  =  T  ->  (
( ( L `  z )  <_  1  /\  x  =  ( M `  ( t `  z ) ) )  <-> 
( ( L `  z )  <_  1  /\  x  =  ( M `  ( T `  z ) ) ) ) )
1716rexbidv 2649 . . . . . . 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 2480 . . . . . 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 7346 . . . . 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 2366 . . . . 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 10627 . . . . . 6  |-  <  Or  RR*
2221supex 7361 . . . . 5  |-  sup ( { x  |  E. z  e.  X  (
( L `  z
)  <_  1  /\  x  =  ( M `  ( T `  z
) ) ) } ,  RR* ,  <  )  e.  _V
2319, 20, 22fvmpt 5709 . . . 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 2418 . . 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 462 . 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 1147 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 358    /\ w3a 935    = wceq 1647    e. wcel 1715   {cab 2352   E.wrex 2629   _Vcvv 2873   class class class wbr 4125    e. cmpt 4179   -->wf 5354   ` cfv 5358  (class class class)co 5981    ^m cmap 6915   supcsup 7340   1c1 8885   RR*cxr 9013    < clt 9014    <_ cle 9015   NrmCVeccnv 21453   BaseSetcba 21455   normCVcnmcv 21459   normOp OLDcnmoo 21632
This theorem is referenced by:  nmoxr  21657  nmooge0  21658  nmorepnf  21659  nmoolb  21662  nmoubi  21663  nmoo0  21682  nmlno0lem  21684
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1551  ax-5 1562  ax-17 1621  ax-9 1659  ax-8 1680  ax-13 1717  ax-14 1719  ax-6 1734  ax-7 1739  ax-11 1751  ax-12 1937  ax-ext 2347  ax-rep 4233  ax-sep 4243  ax-nul 4251  ax-pow 4290  ax-pr 4316  ax-un 4615  ax-cnex 8940  ax-resscn 8941  ax-pre-lttri 8958  ax-pre-lttrn 8959
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 936  df-3an 937  df-tru 1324  df-ex 1547  df-nf 1550  df-sb 1654  df-eu 2221  df-mo 2222  df-clab 2353  df-cleq 2359  df-clel 2362  df-nfc 2491  df-ne 2531  df-nel 2532  df-ral 2633  df-rex 2634  df-reu 2635  df-rmo 2636  df-rab 2637  df-v 2875  df-sbc 3078  df-csb 3168  df-dif 3241  df-un 3243  df-in 3245  df-ss 3252  df-nul 3544  df-if 3655  df-pw 3716  df-sn 3735  df-pr 3736  df-op 3738  df-uni 3930  df-iun 4009  df-br 4126  df-opab 4180  df-mpt 4181  df-id 4412  df-po 4417  df-so 4418  df-xp 4798  df-rel 4799  df-cnv 4800  df-co 4801  df-dm 4802  df-rn 4803  df-res 4804  df-ima 4805  df-iota 5322  df-fun 5360  df-fn 5361  df-f 5362  df-f1 5363  df-fo 5364  df-f1o 5365  df-fv 5366  df-ov 5984  df-oprab 5985  df-mpt2 5986  df-er 6802  df-map 6917  df-en 7007  df-dom 7008  df-sdom 7009  df-sup 7341  df-pnf 9016  df-mnf 9017  df-xr 9018  df-ltxr 9019  df-nmoo 21636
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