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Theorem nmoofval 26274
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 5872 . . . . . 6  |-  ( u  =  U  ->  ( BaseSet
`  u )  =  ( BaseSet `  U )
)
3 nmoofval.1 . . . . . 6  |-  X  =  ( BaseSet `  U )
42, 3syl6eqr 2479 . . . . 5  |-  ( u  =  U  ->  ( BaseSet
`  u )  =  X )
54oveq2d 6312 . . . 4  |-  ( u  =  U  ->  (
( BaseSet `  w )  ^m  ( BaseSet `  u )
)  =  ( (
BaseSet `  w )  ^m  X ) )
6 fveq2 5872 . . . . . . . . . . 11  |-  ( u  =  U  ->  ( normCV `  u )  =  (
normCV
`  U ) )
7 nmoofval.3 . . . . . . . . . . 11  |-  L  =  ( normCV `  U )
86, 7syl6eqr 2479 . . . . . . . . . 10  |-  ( u  =  U  ->  ( normCV `  u )  =  L )
98fveq1d 5874 . . . . . . . . 9  |-  ( u  =  U  ->  (
( normCV `  u ) `  z )  =  ( L `  z ) )
109breq1d 4427 . . . . . . . 8  |-  ( u  =  U  ->  (
( ( normCV `  u
) `  z )  <_  1  <->  ( L `  z )  <_  1
) )
1110anbi1d 709 . . . . . . 7  |-  ( u  =  U  ->  (
( ( ( normCV `  u ) `  z
)  <_  1  /\  x  =  ( ( normCV `  w ) `  (
t `  z )
) )  <->  ( ( L `  z )  <_  1  /\  x  =  ( ( normCV `  w
) `  ( t `  z ) ) ) ) )
124, 11rexeqbidv 3038 . . . . . 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 2556 . . . . 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 7957 . . . 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 4495 . . 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 5872 . . . . . 6  |-  ( w  =  W  ->  ( BaseSet
`  w )  =  ( BaseSet `  W )
)
17 nmoofval.2 . . . . . 6  |-  Y  =  ( BaseSet `  W )
1816, 17syl6eqr 2479 . . . . 5  |-  ( w  =  W  ->  ( BaseSet
`  w )  =  Y )
1918oveq1d 6311 . . . 4  |-  ( w  =  W  ->  (
( BaseSet `  w )  ^m  X )  =  ( Y  ^m  X ) )
20 fveq2 5872 . . . . . . . . . . 11  |-  ( w  =  W  ->  ( normCV `  w )  =  (
normCV
`  W ) )
21 nmoofval.4 . . . . . . . . . . 11  |-  M  =  ( normCV `  W )
2220, 21syl6eqr 2479 . . . . . . . . . 10  |-  ( w  =  W  ->  ( normCV `  w )  =  M )
2322fveq1d 5874 . . . . . . . . 9  |-  ( w  =  W  ->  (
( normCV `  w ) `  ( t `  z
) )  =  ( M `  ( t `
 z ) ) )
2423eqeq2d 2434 . . . . . . . 8  |-  ( w  =  W  ->  (
x  =  ( (
normCV
`  w ) `  ( t `  z
) )  <->  x  =  ( M `  ( t `
 z ) ) ) )
2524anbi2d 708 . . . . . . 7  |-  ( w  =  W  ->  (
( ( L `  z )  <_  1  /\  x  =  (
( normCV `  w ) `  ( t `  z
) ) )  <->  ( ( L `  z )  <_  1  /\  x  =  ( M `  (
t `  z )
) ) ) )
2625rexbidv 2937 . . . . . 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 2556 . . . . 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 7957 . . . 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 4495 . . 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 26257 . . 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 6324 . . . 4  |-  ( Y  ^m  X )  e. 
_V
3231mptex 6142 . . 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 6437 . 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 2473 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 370    = wceq 1437    e. wcel 1867   {cab 2405   E.wrex 2774   class class class wbr 4417    |-> cmpt 4475   ` cfv 5592  (class class class)co 6296    ^m cmap 7471   supcsup 7951   1c1 9529   RR*cxr 9663    < clt 9664    <_ cle 9665   NrmCVeccnv 26074   BaseSetcba 26076   normCVcnmcv 26080   normOpOLDcnmoo 26253
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1748  ax-6 1794  ax-7 1838  ax-9 1871  ax-10 1886  ax-11 1891  ax-12 1904  ax-13 2052  ax-ext 2398  ax-rep 4529  ax-sep 4539  ax-nul 4547  ax-pr 4652
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1787  df-eu 2267  df-mo 2268  df-clab 2406  df-cleq 2412  df-clel 2415  df-nfc 2570  df-ne 2618  df-ral 2778  df-rex 2779  df-reu 2780  df-rab 2782  df-v 3080  df-sbc 3297  df-csb 3393  df-dif 3436  df-un 3438  df-in 3440  df-ss 3447  df-nul 3759  df-if 3907  df-sn 3994  df-pr 3996  df-op 4000  df-uni 4214  df-iun 4295  df-br 4418  df-opab 4476  df-mpt 4477  df-id 4760  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 5556  df-fun 5594  df-fn 5595  df-f 5596  df-f1 5597  df-fo 5598  df-f1o 5599  df-fv 5600  df-ov 6299  df-oprab 6300  df-mpt2 6301  df-sup 7953  df-nmoo 26257
This theorem is referenced by:  nmooval  26275  hhnmoi  27415
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