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Theorem nmosetre 22266
Description: The set in the supremum of the operator norm definition df-nmoo 22247 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 5869 . . . . . . . . 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 22149 . . . . . . . . 9  |-  ( ( W  e.  NrmCVec  /\  ( T `  z )  e.  Y )  ->  ( N `  ( T `  z ) )  e.  RR )
51, 4sylan2 462 . . . . . . . 8  |-  ( ( W  e.  NrmCVec  /\  ( T : X --> Y  /\  z  e.  X )
)  ->  ( N `  ( T `  z
) )  e.  RR )
65anassrs 631 . . . . . . 7  |-  ( ( ( W  e.  NrmCVec  /\  T : X --> Y )  /\  z  e.  X
)  ->  ( N `  ( T `  z
) )  e.  RR )
7 eleq1 2497 . . . . . . 7  |-  ( x  =  ( N `  ( T `  z ) )  ->  ( x  e.  RR  <->  ( N `  ( T `  z ) )  e.  RR ) )
86, 7syl5ibr 214 . . . . . 6  |-  ( x  =  ( N `  ( T `  z ) )  ->  ( (
( W  e.  NrmCVec  /\  T : X --> Y )  /\  z  e.  X
)  ->  x  e.  RR ) )
98impcom 421 . . . . 5  |-  ( ( ( ( W  e.  NrmCVec 
/\  T : X --> Y )  /\  z  e.  X )  /\  x  =  ( N `  ( T `  z ) ) )  ->  x  e.  RR )
109adantrl 698 . . . 4  |-  ( ( ( ( W  e.  NrmCVec 
/\  T : X --> Y )  /\  z  e.  X )  /\  (
( M `  z
)  <_  1  /\  x  =  ( N `  ( T `  z
) ) ) )  ->  x  e.  RR )
1110exp31 589 . . 3  |-  ( ( W  e.  NrmCVec  /\  T : X --> Y )  -> 
( z  e.  X  ->  ( ( ( M `
 z )  <_ 
1  /\  x  =  ( N `  ( T `
 z ) ) )  ->  x  e.  RR ) ) )
1211rexlimdv 2830 . 2  |-  ( ( W  e.  NrmCVec  /\  T : X --> Y )  -> 
( E. z  e.  X  ( ( M `
 z )  <_ 
1  /\  x  =  ( N `  ( T `
 z ) ) )  ->  x  e.  RR ) )
1312abssdv 3418 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 360    = wceq 1653    e. wcel 1726   {cab 2423   E.wrex 2707    C_ wss 3321   class class class wbr 4213   -->wf 5451   ` cfv 5455   RRcr 8990   1c1 8992    <_ cle 9122   NrmCVeccnv 22064   BaseSetcba 22066   normCVcnmcv 22070
This theorem is referenced by:  nmoxr  22268  nmooge0  22269  nmorepnf  22270  nmoolb  22273  nmoubi  22274  nmlno0lem  22295  nmopsetretHIL  23368
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2418  ax-rep 4321  ax-sep 4331  ax-nul 4339  ax-pow 4378  ax-pr 4404  ax-un 4702
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2286  df-mo 2287  df-clab 2424  df-cleq 2430  df-clel 2433  df-nfc 2562  df-ne 2602  df-ral 2711  df-rex 2712  df-reu 2713  df-rab 2715  df-v 2959  df-sbc 3163  df-csb 3253  df-dif 3324  df-un 3326  df-in 3328  df-ss 3335  df-nul 3630  df-if 3741  df-sn 3821  df-pr 3822  df-op 3824  df-uni 4017  df-iun 4096  df-br 4214  df-opab 4268  df-mpt 4269  df-id 4499  df-xp 4885  df-rel 4886  df-cnv 4887  df-co 4888  df-dm 4889  df-rn 4890  df-res 4891  df-ima 4892  df-iota 5419  df-fun 5457  df-fn 5458  df-f 5459  df-f1 5460  df-fo 5461  df-f1o 5462  df-fv 5463  df-ov 6085  df-oprab 6086  df-1st 6350  df-2nd 6351  df-vc 22026  df-nv 22072  df-va 22075  df-ba 22076  df-sm 22077  df-0v 22078  df-nmcv 22080
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