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Theorem nmoofval 21340
Description: The operator norm function. (Contributed by NM, 6-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
nmoofval  |-  ( ( 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* ,  <  ) ) )
Distinct variable groups:    x, t,
z, U    t, W, x, z    t, X, z   
t, Y, x    t, L    t, M
Allowed substitution hints:    L( x, z)    M( x, z)    N( x, z, t)    X( x)    Y( z)

Proof of Theorem nmoofval
Dummy variables  u  w are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 nmoofval.6 . 2  |-  N  =  ( U normOp OLD W
)
2 fveq2 5525 . . . . . 6  |-  ( u  =  U  ->  ( BaseSet
`  u )  =  ( BaseSet `  U )
)
3 nmoofval.1 . . . . . 6  |-  X  =  ( BaseSet `  U )
42, 3syl6eqr 2333 . . . . 5  |-  ( u  =  U  ->  ( BaseSet
`  u )  =  X )
54oveq2d 5874 . . . 4  |-  ( u  =  U  ->  (
( BaseSet `  w )  ^m  ( BaseSet `  u )
)  =  ( (
BaseSet `  w )  ^m  X ) )
6 fveq2 5525 . . . . . . . . . . 11  |-  ( u  =  U  ->  ( normCV `  u )  =  (
normCV
`  U ) )
7 nmoofval.3 . . . . . . . . . . 11  |-  L  =  ( normCV `  U )
86, 7syl6eqr 2333 . . . . . . . . . 10  |-  ( u  =  U  ->  ( normCV `  u )  =  L )
98fveq1d 5527 . . . . . . . . 9  |-  ( u  =  U  ->  (
( normCV `  u ) `  z )  =  ( L `  z ) )
109breq1d 4033 . . . . . . . 8  |-  ( u  =  U  ->  (
( ( normCV `  u
) `  z )  <_  1  <->  ( L `  z )  <_  1
) )
1110anbi1d 685 . . . . . . 7  |-  ( u  =  U  ->  (
( ( ( normCV `  u ) `  z
)  <_  1  /\  x  =  ( ( normCV `  w ) `  (
t `  z )
) )  <->  ( ( L `  z )  <_  1  /\  x  =  ( ( normCV `  w
) `  ( t `  z ) ) ) ) )
124, 11rexeqbidv 2749 . . . . . 6  |-  ( u  =  U  ->  ( E. z  e.  ( BaseSet
`  u ) ( ( ( normCV `  u
) `  z )  <_  1  /\  x  =  ( ( normCV `  w
) `  ( t `  z ) ) )  <->  E. z  e.  X  ( ( L `  z )  <_  1  /\  x  =  (
( normCV `  w ) `  ( t `  z
) ) ) ) )
1312abbidv 2397 . . . . 5  |-  ( u  =  U  ->  { x  |  E. z  e.  (
BaseSet `  u ) ( ( ( normCV `  u
) `  z )  <_  1  /\  x  =  ( ( normCV `  w
) `  ( t `  z ) ) ) }  =  { x  |  E. z  e.  X  ( ( L `  z )  <_  1  /\  x  =  (
( normCV `  w ) `  ( t `  z
) ) ) } )
1413supeq1d 7199 . . . 4  |-  ( u  =  U  ->  sup ( { x  |  E. z  e.  ( BaseSet `  u ) ( ( ( normCV `  u ) `  z )  <_  1  /\  x  =  (
( normCV `  w ) `  ( t `  z
) ) ) } ,  RR* ,  <  )  =  sup ( { x  |  E. z  e.  X  ( ( L `  z )  <_  1  /\  x  =  (
( normCV `  w ) `  ( t `  z
) ) ) } ,  RR* ,  <  )
)
155, 14mpteq12dv 4098 . . 3  |-  ( u  =  U  ->  (
t  e.  ( (
BaseSet `  w )  ^m  ( BaseSet `  u )
)  |->  sup ( { x  |  E. z  e.  (
BaseSet `  u ) ( ( ( normCV `  u
) `  z )  <_  1  /\  x  =  ( ( normCV `  w
) `  ( t `  z ) ) ) } ,  RR* ,  <  ) )  =  ( t  e.  ( ( BaseSet `  w )  ^m  X
)  |->  sup ( { x  |  E. z  e.  X  ( ( L `  z )  <_  1  /\  x  =  (
( normCV `  w ) `  ( t `  z
) ) ) } ,  RR* ,  <  )
) )
16 fveq2 5525 . . . . . 6  |-  ( w  =  W  ->  ( BaseSet
`  w )  =  ( BaseSet `  W )
)
17 nmoofval.2 . . . . . 6  |-  Y  =  ( BaseSet `  W )
1816, 17syl6eqr 2333 . . . . 5  |-  ( w  =  W  ->  ( BaseSet
`  w )  =  Y )
1918oveq1d 5873 . . . 4  |-  ( w  =  W  ->  (
( BaseSet `  w )  ^m  X )  =  ( Y  ^m  X ) )
20 fveq2 5525 . . . . . . . . . . 11  |-  ( w  =  W  ->  ( normCV `  w )  =  (
normCV
`  W ) )
21 nmoofval.4 . . . . . . . . . . 11  |-  M  =  ( normCV `  W )
2220, 21syl6eqr 2333 . . . . . . . . . 10  |-  ( w  =  W  ->  ( normCV `  w )  =  M )
2322fveq1d 5527 . . . . . . . . 9  |-  ( w  =  W  ->  (
( normCV `  w ) `  ( t `  z
) )  =  ( M `  ( t `
 z ) ) )
2423eqeq2d 2294 . . . . . . . 8  |-  ( w  =  W  ->  (
x  =  ( (
normCV
`  w ) `  ( t `  z
) )  <->  x  =  ( M `  ( t `
 z ) ) ) )
2524anbi2d 684 . . . . . . 7  |-  ( w  =  W  ->  (
( ( L `  z )  <_  1  /\  x  =  (
( normCV `  w ) `  ( t `  z
) ) )  <->  ( ( L `  z )  <_  1  /\  x  =  ( M `  (
t `  z )
) ) ) )
2625rexbidv 2564 . . . . . 6  |-  ( w  =  W  ->  ( E. z  e.  X  ( ( L `  z )  <_  1  /\  x  =  (
( normCV `  w ) `  ( t `  z
) ) )  <->  E. z  e.  X  ( ( L `  z )  <_  1  /\  x  =  ( M `  (
t `  z )
) ) ) )
2726abbidv 2397 . . . . 5  |-  ( w  =  W  ->  { x  |  E. z  e.  X  ( ( L `  z )  <_  1  /\  x  =  (
( normCV `  w ) `  ( t `  z
) ) ) }  =  { x  |  E. z  e.  X  ( ( L `  z )  <_  1  /\  x  =  ( M `  ( t `  z ) ) ) } )
2827supeq1d 7199 . . . 4  |-  ( w  =  W  ->  sup ( { x  |  E. z  e.  X  (
( L `  z
)  <_  1  /\  x  =  ( ( normCV `  w ) `  (
t `  z )
) ) } ,  RR* ,  <  )  =  sup ( { x  |  E. z  e.  X  ( ( L `  z )  <_  1  /\  x  =  ( M `  ( t `  z ) ) ) } ,  RR* ,  <  ) )
2919, 28mpteq12dv 4098 . . 3  |-  ( w  =  W  ->  (
t  e.  ( (
BaseSet `  w )  ^m  X )  |->  sup ( { x  |  E. z  e.  X  (
( L `  z
)  <_  1  /\  x  =  ( ( normCV `  w ) `  (
t `  z )
) ) } ,  RR* ,  <  ) )  =  ( t  e.  ( Y  ^m  X
)  |->  sup ( { x  |  E. z  e.  X  ( ( L `  z )  <_  1  /\  x  =  ( M `  ( t `  z ) ) ) } ,  RR* ,  <  ) ) )
30 df-nmoo 21323 . . 3  |-  normOp OLD  =  ( u  e.  NrmCVec ,  w  e.  NrmCVec  |->  ( t  e.  ( ( BaseSet `  w
)  ^m  ( BaseSet `  u ) )  |->  sup ( { x  |  E. z  e.  (
BaseSet `  u ) ( ( ( normCV `  u
) `  z )  <_  1  /\  x  =  ( ( normCV `  w
) `  ( t `  z ) ) ) } ,  RR* ,  <  ) ) )
31 ovex 5883 . . . 4  |-  ( Y  ^m  X )  e. 
_V
3231mptex 5746 . . 3  |-  ( t  e.  ( Y  ^m  X )  |->  sup ( { x  |  E. z  e.  X  (
( L `  z
)  <_  1  /\  x  =  ( M `  ( t `  z
) ) ) } ,  RR* ,  <  )
)  e.  _V
3315, 29, 30, 32ovmpt2 5983 . 2  |-  ( ( U  e.  NrmCVec  /\  W  e.  NrmCVec )  ->  ( U normOp OLD W )  =  ( t  e.  ( Y  ^m  X ) 
|->  sup ( { x  |  E. z  e.  X  ( ( L `  z )  <_  1  /\  x  =  ( M `  ( t `  z ) ) ) } ,  RR* ,  <  ) ) )
341, 33syl5eq 2327 1  |-  ( ( 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* ,  <  ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1623    e. wcel 1684   {cab 2269   E.wrex 2544   class class class wbr 4023    e. cmpt 4077   ` cfv 5255  (class class class)co 5858    ^m cmap 6772   supcsup 7193   1c1 8738   RR*cxr 8866    < clt 8867    <_ cle 8868   NrmCVeccnv 21140   BaseSetcba 21142   normCVcnmcv 21146   normOp OLDcnmoo 21319
This theorem is referenced by:  nmooval  21341  hhnmoi  22481
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pr 4214
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-reu 2550  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-sup 7194  df-nmoo 21323
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