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Theorem nmoofval 22255
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 5720 . . . . . 6  |-  ( u  =  U  ->  ( BaseSet
`  u )  =  ( BaseSet `  U )
)
3 nmoofval.1 . . . . . 6  |-  X  =  ( BaseSet `  U )
42, 3syl6eqr 2485 . . . . 5  |-  ( u  =  U  ->  ( BaseSet
`  u )  =  X )
54oveq2d 6089 . . . 4  |-  ( u  =  U  ->  (
( BaseSet `  w )  ^m  ( BaseSet `  u )
)  =  ( (
BaseSet `  w )  ^m  X ) )
6 fveq2 5720 . . . . . . . . . . 11  |-  ( u  =  U  ->  ( normCV `  u )  =  (
normCV
`  U ) )
7 nmoofval.3 . . . . . . . . . . 11  |-  L  =  ( normCV `  U )
86, 7syl6eqr 2485 . . . . . . . . . 10  |-  ( u  =  U  ->  ( normCV `  u )  =  L )
98fveq1d 5722 . . . . . . . . 9  |-  ( u  =  U  ->  (
( normCV `  u ) `  z )  =  ( L `  z ) )
109breq1d 4214 . . . . . . . 8  |-  ( u  =  U  ->  (
( ( normCV `  u
) `  z )  <_  1  <->  ( L `  z )  <_  1
) )
1110anbi1d 686 . . . . . . 7  |-  ( u  =  U  ->  (
( ( ( normCV `  u ) `  z
)  <_  1  /\  x  =  ( ( normCV `  w ) `  (
t `  z )
) )  <->  ( ( L `  z )  <_  1  /\  x  =  ( ( normCV `  w
) `  ( t `  z ) ) ) ) )
124, 11rexeqbidv 2909 . . . . . 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 2549 . . . . 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 7443 . . . 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 4279 . . 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 5720 . . . . . 6  |-  ( w  =  W  ->  ( BaseSet
`  w )  =  ( BaseSet `  W )
)
17 nmoofval.2 . . . . . 6  |-  Y  =  ( BaseSet `  W )
1816, 17syl6eqr 2485 . . . . 5  |-  ( w  =  W  ->  ( BaseSet
`  w )  =  Y )
1918oveq1d 6088 . . . 4  |-  ( w  =  W  ->  (
( BaseSet `  w )  ^m  X )  =  ( Y  ^m  X ) )
20 fveq2 5720 . . . . . . . . . . 11  |-  ( w  =  W  ->  ( normCV `  w )  =  (
normCV
`  W ) )
21 nmoofval.4 . . . . . . . . . . 11  |-  M  =  ( normCV `  W )
2220, 21syl6eqr 2485 . . . . . . . . . 10  |-  ( w  =  W  ->  ( normCV `  w )  =  M )
2322fveq1d 5722 . . . . . . . . 9  |-  ( w  =  W  ->  (
( normCV `  w ) `  ( t `  z
) )  =  ( M `  ( t `
 z ) ) )
2423eqeq2d 2446 . . . . . . . 8  |-  ( w  =  W  ->  (
x  =  ( (
normCV
`  w ) `  ( t `  z
) )  <->  x  =  ( M `  ( t `
 z ) ) ) )
2524anbi2d 685 . . . . . . 7  |-  ( w  =  W  ->  (
( ( L `  z )  <_  1  /\  x  =  (
( normCV `  w ) `  ( t `  z
) ) )  <->  ( ( L `  z )  <_  1  /\  x  =  ( M `  (
t `  z )
) ) ) )
2625rexbidv 2718 . . . . . 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 2549 . . . . 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 7443 . . . 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 4279 . . 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 22238 . . 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 6098 . . . 4  |-  ( Y  ^m  X )  e. 
_V
3231mptex 5958 . . 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 6201 . 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 2479 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 359    = wceq 1652    e. wcel 1725   {cab 2421   E.wrex 2698   class class class wbr 4204    e. cmpt 4258   ` cfv 5446  (class class class)co 6073    ^m cmap 7010   supcsup 7437   1c1 8983   RR*cxr 9111    < clt 9112    <_ cle 9113   NrmCVeccnv 22055   BaseSetcba 22057   normCVcnmcv 22061   normOp OLDcnmoo 22234
This theorem is referenced by:  nmooval  22256  hhnmoi  23396
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-rep 4312  ax-sep 4322  ax-nul 4330  ax-pr 4395
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-reu 2704  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  df-id 4490  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-ov 6076  df-oprab 6077  df-mpt2 6078  df-sup 7438  df-nmoo 22238
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