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Theorem nmoofval 24167
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 normOpOLD 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 normOpOLD W
)
2 fveq2 5696 . . . . . 6  |-  ( u  =  U  ->  ( BaseSet
`  u )  =  ( BaseSet `  U )
)
3 nmoofval.1 . . . . . 6  |-  X  =  ( BaseSet `  U )
42, 3syl6eqr 2493 . . . . 5  |-  ( u  =  U  ->  ( BaseSet
`  u )  =  X )
54oveq2d 6112 . . . 4  |-  ( u  =  U  ->  (
( BaseSet `  w )  ^m  ( BaseSet `  u )
)  =  ( (
BaseSet `  w )  ^m  X ) )
6 fveq2 5696 . . . . . . . . . . 11  |-  ( u  =  U  ->  ( normCV `  u )  =  (
normCV
`  U ) )
7 nmoofval.3 . . . . . . . . . . 11  |-  L  =  ( normCV `  U )
86, 7syl6eqr 2493 . . . . . . . . . 10  |-  ( u  =  U  ->  ( normCV `  u )  =  L )
98fveq1d 5698 . . . . . . . . 9  |-  ( u  =  U  ->  (
( normCV `  u ) `  z )  =  ( L `  z ) )
109breq1d 4307 . . . . . . . 8  |-  ( u  =  U  ->  (
( ( normCV `  u
) `  z )  <_  1  <->  ( L `  z )  <_  1
) )
1110anbi1d 704 . . . . . . 7  |-  ( u  =  U  ->  (
( ( ( normCV `  u ) `  z
)  <_  1  /\  x  =  ( ( normCV `  w ) `  (
t `  z )
) )  <->  ( ( L `  z )  <_  1  /\  x  =  ( ( normCV `  w
) `  ( t `  z ) ) ) ) )
124, 11rexeqbidv 2937 . . . . . 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 2562 . . . . 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 7701 . . . 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 4375 . . 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 5696 . . . . . 6  |-  ( w  =  W  ->  ( BaseSet
`  w )  =  ( BaseSet `  W )
)
17 nmoofval.2 . . . . . 6  |-  Y  =  ( BaseSet `  W )
1816, 17syl6eqr 2493 . . . . 5  |-  ( w  =  W  ->  ( BaseSet
`  w )  =  Y )
1918oveq1d 6111 . . . 4  |-  ( w  =  W  ->  (
( BaseSet `  w )  ^m  X )  =  ( Y  ^m  X ) )
20 fveq2 5696 . . . . . . . . . . 11  |-  ( w  =  W  ->  ( normCV `  w )  =  (
normCV
`  W ) )
21 nmoofval.4 . . . . . . . . . . 11  |-  M  =  ( normCV `  W )
2220, 21syl6eqr 2493 . . . . . . . . . 10  |-  ( w  =  W  ->  ( normCV `  w )  =  M )
2322fveq1d 5698 . . . . . . . . 9  |-  ( w  =  W  ->  (
( normCV `  w ) `  ( t `  z
) )  =  ( M `  ( t `
 z ) ) )
2423eqeq2d 2454 . . . . . . . 8  |-  ( w  =  W  ->  (
x  =  ( (
normCV
`  w ) `  ( t `  z
) )  <->  x  =  ( M `  ( t `
 z ) ) ) )
2524anbi2d 703 . . . . . . 7  |-  ( w  =  W  ->  (
( ( L `  z )  <_  1  /\  x  =  (
( normCV `  w ) `  ( t `  z
) ) )  <->  ( ( L `  z )  <_  1  /\  x  =  ( M `  (
t `  z )
) ) ) )
2625rexbidv 2741 . . . . . 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 2562 . . . . 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 7701 . . . 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 4375 . . 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 24150 . . 3  |-  normOpOLD  =  ( 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 6121 . . . 4  |-  ( Y  ^m  X )  e. 
_V
3231mptex 5953 . . 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 6231 . 2  |-  ( ( U  e.  NrmCVec  /\  W  e.  NrmCVec )  ->  ( U normOpOLD W )  =  ( t  e.  ( Y  ^m  X ) 
|->  sup ( { x  |  E. z  e.  X  ( ( L `  z )  <_  1  /\  x  =  ( M `  ( t `  z ) ) ) } ,  RR* ,  <  ) ) )
341, 33syl5eq 2487 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 setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1369    e. wcel 1756   {cab 2429   E.wrex 2721   class class class wbr 4297    e. cmpt 4355   ` 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-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-pr 4536
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  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-ral 2725  df-rex 2726  df-reu 2727  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-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-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-sup 7696  df-nmoo 24150
This theorem is referenced by:  nmooval  24168  hhnmoi  25310
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