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Theorem nmosetre 25355
Description: The set in the supremum of the operator norm definition df-nmoo 25336 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 6017 . . . . . . . . 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 25238 . . . . . . . . 9  |-  ( ( W  e.  NrmCVec  /\  ( T `  z )  e.  Y )  ->  ( N `  ( T `  z ) )  e.  RR )
51, 4sylan2 474 . . . . . . . 8  |-  ( ( W  e.  NrmCVec  /\  ( T : X --> Y  /\  z  e.  X )
)  ->  ( N `  ( T `  z
) )  e.  RR )
65anassrs 648 . . . . . . 7  |-  ( ( ( W  e.  NrmCVec  /\  T : X --> Y )  /\  z  e.  X
)  ->  ( N `  ( T `  z
) )  e.  RR )
7 eleq1 2539 . . . . . . 7  |-  ( x  =  ( N `  ( T `  z ) )  ->  ( x  e.  RR  <->  ( N `  ( T `  z ) )  e.  RR ) )
86, 7syl5ibr 221 . . . . . 6  |-  ( x  =  ( N `  ( T `  z ) )  ->  ( (
( W  e.  NrmCVec  /\  T : X --> Y )  /\  z  e.  X
)  ->  x  e.  RR ) )
98impcom 430 . . . . 5  |-  ( ( ( ( W  e.  NrmCVec 
/\  T : X --> Y )  /\  z  e.  X )  /\  x  =  ( N `  ( T `  z ) ) )  ->  x  e.  RR )
109adantrl 715 . . . 4  |-  ( ( ( ( W  e.  NrmCVec 
/\  T : X --> Y )  /\  z  e.  X )  /\  (
( M `  z
)  <_  1  /\  x  =  ( N `  ( T `  z
) ) ) )  ->  x  e.  RR )
1110exp31 604 . . 3  |-  ( ( W  e.  NrmCVec  /\  T : X --> Y )  -> 
( z  e.  X  ->  ( ( ( M `
 z )  <_ 
1  /\  x  =  ( N `  ( T `
 z ) ) )  ->  x  e.  RR ) ) )
1211rexlimdv 2953 . 2  |-  ( ( W  e.  NrmCVec  /\  T : X --> Y )  -> 
( E. z  e.  X  ( ( M `
 z )  <_ 
1  /\  x  =  ( N `  ( T `
 z ) ) )  ->  x  e.  RR ) )
1312abssdv 3574 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 setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1379    e. wcel 1767   {cab 2452   E.wrex 2815    C_ wss 3476   class class class wbr 4447   -->wf 5582   ` cfv 5586   RRcr 9487   1c1 9489    <_ cle 9625   NrmCVeccnv 25153   BaseSetcba 25155   normCVcnmcv 25159
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6574
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-reu 2821  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-id 4795  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5549  df-fun 5588  df-fn 5589  df-f 5590  df-f1 5591  df-fo 5592  df-f1o 5593  df-fv 5594  df-ov 6285  df-oprab 6286  df-1st 6781  df-2nd 6782  df-vc 25115  df-nv 25161  df-va 25164  df-ba 25165  df-sm 25166  df-0v 25167  df-nmcv 25169
This theorem is referenced by:  nmoxr  25357  nmooge0  25358  nmorepnf  25359  nmoolb  25362  nmoubi  25363  nmlno0lem  25384  nmopsetretHIL  26459
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