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Theorem nmooval 24098
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 normOpOLD 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 5698 . . . . 5  |-  ( BaseSet `  W )  e.  _V
31, 2eqeltri 2511 . . . 4  |-  Y  e. 
_V
4 nmoofval.1 . . . . 5  |-  X  =  ( BaseSet `  U )
5 fvex 5698 . . . . 5  |-  ( BaseSet `  U )  e.  _V
64, 5eqeltri 2511 . . . 4  |-  X  e. 
_V
73, 6elmap 7237 . . 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 normOpOLD W
)
114, 1, 8, 9, 10nmoofval 24097 . . . . 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 5690 . . . 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 5687 . . . . . . . . . . 11  |-  ( t  =  T  ->  (
t `  z )  =  ( T `  z ) )
1413fveq2d 5692 . . . . . . . . . 10  |-  ( t  =  T  ->  ( M `  ( t `  z ) )  =  ( M `  ( T `  z )
) )
1514eqeq2d 2452 . . . . . . . . 9  |-  ( t  =  T  ->  (
x  =  ( M `
 ( t `  z ) )  <->  x  =  ( M `  ( T `
 z ) ) ) )
1615anbi2d 698 . . . . . . . 8  |-  ( t  =  T  ->  (
( ( L `  z )  <_  1  /\  x  =  ( M `  ( t `  z ) ) )  <-> 
( ( L `  z )  <_  1  /\  x  =  ( M `  ( T `  z ) ) ) ) )
1716rexbidv 2734 . . . . . . 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 2555 . . . . . 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 7692 . . . . 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 2441 . . . . 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 11114 . . . . . 6  |-  <  Or  RR*
2221supex 7709 . . . . 5  |-  sup ( { x  |  E. z  e.  X  (
( L `  z
)  <_  1  /\  x  =  ( M `  ( T `  z
) ) ) } ,  RR* ,  <  )  e.  _V
2319, 20, 22fvmpt 5771 . . . 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 2493 . . 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 473 . 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 1177 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 setvar class
Syntax hints:    -> wi 4    /\ wa 369    /\ w3a 960    = wceq 1364    e. wcel 1761   {cab 2427   E.wrex 2714   _Vcvv 2970   class class class wbr 4289    e. cmpt 4347   -->wf 5411   ` cfv 5415  (class class class)co 6090    ^m cmap 7210   supcsup 7686   1c1 9279   RR*cxr 9413    < clt 9414    <_ cle 9415   NrmCVeccnv 23897   BaseSetcba 23899   normCVcnmcv 23903   normOpOLDcnmoo 24076
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1713  ax-7 1733  ax-8 1763  ax-9 1765  ax-10 1780  ax-11 1785  ax-12 1797  ax-13 1948  ax-ext 2422  ax-rep 4400  ax-sep 4410  ax-nul 4418  ax-pow 4467  ax-pr 4528  ax-un 6371  ax-cnex 9334  ax-resscn 9335  ax-pre-lttri 9352  ax-pre-lttrn 9353
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 961  df-3an 962  df-tru 1367  df-ex 1592  df-nf 1595  df-sb 1706  df-eu 2261  df-mo 2262  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ne 2606  df-nel 2607  df-ral 2718  df-rex 2719  df-reu 2720  df-rmo 2721  df-rab 2722  df-v 2972  df-sbc 3184  df-csb 3286  df-dif 3328  df-un 3330  df-in 3332  df-ss 3339  df-nul 3635  df-if 3789  df-pw 3859  df-sn 3875  df-pr 3877  df-op 3881  df-uni 4089  df-iun 4170  df-br 4290  df-opab 4348  df-mpt 4349  df-id 4632  df-po 4637  df-so 4638  df-xp 4842  df-rel 4843  df-cnv 4844  df-co 4845  df-dm 4846  df-rn 4847  df-res 4848  df-ima 4849  df-iota 5378  df-fun 5417  df-fn 5418  df-f 5419  df-f1 5420  df-fo 5421  df-f1o 5422  df-fv 5423  df-ov 6093  df-oprab 6094  df-mpt2 6095  df-er 7097  df-map 7212  df-en 7307  df-dom 7308  df-sdom 7309  df-sup 7687  df-pnf 9416  df-mnf 9417  df-xr 9418  df-ltxr 9419  df-nmoo 24080
This theorem is referenced by:  nmoxr  24101  nmooge0  24102  nmorepnf  24103  nmoolb  24106  nmoubi  24107  nmoo0  24126  nmlno0lem  24128
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