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Theorem nmooval 24168
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 5706 . . . . 5  |-  ( BaseSet `  W )  e.  _V
31, 2eqeltri 2513 . . . 4  |-  Y  e. 
_V
4 nmoofval.1 . . . . 5  |-  X  =  ( BaseSet `  U )
5 fvex 5706 . . . . 5  |-  ( BaseSet `  U )  e.  _V
64, 5eqeltri 2513 . . . 4  |-  X  e. 
_V
73, 6elmap 7246 . . 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 24167 . . . . 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 5698 . . . 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 5695 . . . . . . . . . . 11  |-  ( t  =  T  ->  (
t `  z )  =  ( T `  z ) )
1413fveq2d 5700 . . . . . . . . . 10  |-  ( t  =  T  ->  ( M `  ( t `  z ) )  =  ( M `  ( T `  z )
) )
1514eqeq2d 2454 . . . . . . . . 9  |-  ( t  =  T  ->  (
x  =  ( M `
 ( t `  z ) )  <->  x  =  ( M `  ( T `
 z ) ) ) )
1615anbi2d 703 . . . . . . . 8  |-  ( t  =  T  ->  (
( ( L `  z )  <_  1  /\  x  =  ( M `  ( t `  z ) ) )  <-> 
( ( L `  z )  <_  1  /\  x  =  ( M `  ( T `  z ) ) ) ) )
1716rexbidv 2741 . . . . . . 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 2562 . . . . . 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 7701 . . . . 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 2443 . . . . 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 11123 . . . . . 6  |-  <  Or  RR*
2221supex 7718 . . . . 5  |-  sup ( { x  |  E. z  e.  X  (
( L `  z
)  <_  1  /\  x  =  ( M `  ( T `  z
) ) ) } ,  RR* ,  <  )  e.  _V
2319, 20, 22fvmpt 5779 . . . 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 2495 . . 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 476 . 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 1182 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 965    = wceq 1369    e. wcel 1756   {cab 2429   E.wrex 2721   _Vcvv 2977   class class class wbr 4297    e. cmpt 4355   -->wf 5419   ` cfv 5423  (class class class)co 6096    ^m cmap 7219   supcsup 7695   1c1 9288   RR*cxr 9422    < clt 9423    <_ cle 9424   NrmCVeccnv 23967   BaseSetcba 23969   normCVcnmcv 23973   normOpOLDcnmoo 24146
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4408  ax-sep 4418  ax-nul 4426  ax-pow 4475  ax-pr 4536  ax-un 6377  ax-cnex 9343  ax-resscn 9344  ax-pre-lttri 9361  ax-pre-lttrn 9362
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2573  df-ne 2613  df-nel 2614  df-ral 2725  df-rex 2726  df-reu 2727  df-rmo 2728  df-rab 2729  df-v 2979  df-sbc 3192  df-csb 3294  df-dif 3336  df-un 3338  df-in 3340  df-ss 3347  df-nul 3643  df-if 3797  df-pw 3867  df-sn 3883  df-pr 3885  df-op 3889  df-uni 4097  df-iun 4178  df-br 4298  df-opab 4356  df-mpt 4357  df-id 4641  df-po 4646  df-so 4647  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-iota 5386  df-fun 5425  df-fn 5426  df-f 5427  df-f1 5428  df-fo 5429  df-f1o 5430  df-fv 5431  df-ov 6099  df-oprab 6100  df-mpt2 6101  df-er 7106  df-map 7221  df-en 7316  df-dom 7317  df-sdom 7318  df-sup 7696  df-pnf 9425  df-mnf 9426  df-xr 9427  df-ltxr 9428  df-nmoo 24150
This theorem is referenced by:  nmoxr  24171  nmooge0  24172  nmorepnf  24173  nmoolb  24176  nmoubi  24177  nmoo0  24196  nmlno0lem  24198
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