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Theorem nmosetre 21342
Description: The set in the supremum of the operator norm definition df-nmoo 21323 is a set of reals. (Contributed by NM, 13-Nov-2007.) (New usage is discouraged.)
Hypotheses
Ref Expression
nmosetre.2  |-  Y  =  ( BaseSet `  W )
nmosetre.4  |-  N  =  ( normCV `  W )
Assertion
Ref Expression
nmosetre  |-  ( ( W  e.  NrmCVec  /\  T : X --> Y )  ->  { x  |  E. z  e.  X  (
( M `  z
)  <_  1  /\  x  =  ( N `  ( T `  z
) ) ) } 
C_  RR )
Distinct variable groups:    x, z, T    x, W, z    x, X, z    x, Y, z
Allowed substitution hints:    M( x, z)    N( x, z)

Proof of Theorem nmosetre
StepHypRef Expression
1 ffvelrn 5663 . . . . . . . . 9  |-  ( ( T : X --> Y  /\  z  e.  X )  ->  ( T `  z
)  e.  Y )
2 nmosetre.2 . . . . . . . . . 10  |-  Y  =  ( BaseSet `  W )
3 nmosetre.4 . . . . . . . . . 10  |-  N  =  ( normCV `  W )
42, 3nvcl 21225 . . . . . . . . 9  |-  ( ( W  e.  NrmCVec  /\  ( T `  z )  e.  Y )  ->  ( N `  ( T `  z ) )  e.  RR )
51, 4sylan2 460 . . . . . . . 8  |-  ( ( W  e.  NrmCVec  /\  ( T : X --> Y  /\  z  e.  X )
)  ->  ( N `  ( T `  z
) )  e.  RR )
65anassrs 629 . . . . . . 7  |-  ( ( ( W  e.  NrmCVec  /\  T : X --> Y )  /\  z  e.  X
)  ->  ( N `  ( T `  z
) )  e.  RR )
7 eleq1 2343 . . . . . . 7  |-  ( x  =  ( N `  ( T `  z ) )  ->  ( x  e.  RR  <->  ( N `  ( T `  z ) )  e.  RR ) )
86, 7syl5ibr 212 . . . . . 6  |-  ( x  =  ( N `  ( T `  z ) )  ->  ( (
( W  e.  NrmCVec  /\  T : X --> Y )  /\  z  e.  X
)  ->  x  e.  RR ) )
98impcom 419 . . . . 5  |-  ( ( ( ( W  e.  NrmCVec 
/\  T : X --> Y )  /\  z  e.  X )  /\  x  =  ( N `  ( T `  z ) ) )  ->  x  e.  RR )
109adantrl 696 . . . 4  |-  ( ( ( ( W  e.  NrmCVec 
/\  T : X --> Y )  /\  z  e.  X )  /\  (
( M `  z
)  <_  1  /\  x  =  ( N `  ( T `  z
) ) ) )  ->  x  e.  RR )
1110exp31 587 . . 3  |-  ( ( W  e.  NrmCVec  /\  T : X --> Y )  -> 
( z  e.  X  ->  ( ( ( M `
 z )  <_ 
1  /\  x  =  ( N `  ( T `
 z ) ) )  ->  x  e.  RR ) ) )
1211rexlimdv 2666 . 2  |-  ( ( W  e.  NrmCVec  /\  T : X --> Y )  -> 
( E. z  e.  X  ( ( M `
 z )  <_ 
1  /\  x  =  ( N `  ( T `
 z ) ) )  ->  x  e.  RR ) )
1312abssdv 3247 1  |-  ( ( W  e.  NrmCVec  /\  T : X --> Y )  ->  { x  |  E. z  e.  X  (
( M `  z
)  <_  1  /\  x  =  ( N `  ( T `  z
) ) ) } 
C_  RR )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1623    e. wcel 1684   {cab 2269   E.wrex 2544    C_ wss 3152   class class class wbr 4023   -->wf 5251   ` cfv 5255   RRcr 8736   1c1 8738    <_ cle 8868   NrmCVeccnv 21140   BaseSetcba 21142   normCVcnmcv 21146
This theorem is referenced by:  nmoxr  21344  nmooge0  21345  nmorepnf  21346  nmoolb  21349  nmoubi  21350  nmlno0lem  21371  nmopsetretHIL  22444
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-13 1686  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-pow 4188  ax-pr 4214  ax-un 4512
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-1st 6122  df-2nd 6123  df-vc 21102  df-nv 21148  df-va 21151  df-ba 21152  df-sm 21153  df-0v 21154  df-nmcv 21156
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