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Theorem nmosetre 25973
Description: The set in the supremum of the operator norm definition df-nmoo 25954 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 5963 . . . . . . . . 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 25856 . . . . . . . . 9  |-  ( ( W  e.  NrmCVec  /\  ( T `  z )  e.  Y )  ->  ( N `  ( T `  z ) )  e.  RR )
51, 4sylan2 472 . . . . . . . 8  |-  ( ( W  e.  NrmCVec  /\  ( T : X --> Y  /\  z  e.  X )
)  ->  ( N `  ( T `  z
) )  e.  RR )
65anassrs 646 . . . . . . 7  |-  ( ( ( W  e.  NrmCVec  /\  T : X --> Y )  /\  z  e.  X
)  ->  ( N `  ( T `  z
) )  e.  RR )
7 eleq1 2474 . . . . . . 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 428 . . . . 5  |-  ( ( ( ( W  e.  NrmCVec 
/\  T : X --> Y )  /\  z  e.  X )  /\  x  =  ( N `  ( T `  z ) ) )  ->  x  e.  RR )
109adantrl 714 . . . 4  |-  ( ( ( ( W  e.  NrmCVec 
/\  T : X --> Y )  /\  z  e.  X )  /\  (
( M `  z
)  <_  1  /\  x  =  ( N `  ( T `  z
) ) ) )  ->  x  e.  RR )
1110exp31 602 . . 3  |-  ( ( W  e.  NrmCVec  /\  T : X --> Y )  -> 
( z  e.  X  ->  ( ( ( M `
 z )  <_ 
1  /\  x  =  ( N `  ( T `
 z ) ) )  ->  x  e.  RR ) ) )
1211rexlimdv 2893 . 2  |-  ( ( W  e.  NrmCVec  /\  T : X --> Y )  -> 
( E. z  e.  X  ( ( M `
 z )  <_ 
1  /\  x  =  ( N `  ( T `
 z ) ) )  ->  x  e.  RR ) )
1312abssdv 3512 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 367    = wceq 1405    e. wcel 1842   {cab 2387   E.wrex 2754    C_ wss 3413   class class class wbr 4394   -->wf 5521   ` cfv 5525   RRcr 9441   1c1 9443    <_ cle 9579   NrmCVeccnv 25771   BaseSetcba 25773   normCVcnmcv 25777
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-8 1844  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-rep 4506  ax-sep 4516  ax-nul 4524  ax-pow 4571  ax-pr 4629  ax-un 6530
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 976  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-ral 2758  df-rex 2759  df-reu 2760  df-rab 2762  df-v 3060  df-sbc 3277  df-csb 3373  df-dif 3416  df-un 3418  df-in 3420  df-ss 3427  df-nul 3738  df-if 3885  df-sn 3972  df-pr 3974  df-op 3978  df-uni 4191  df-iun 4272  df-br 4395  df-opab 4453  df-mpt 4454  df-id 4737  df-xp 4948  df-rel 4949  df-cnv 4950  df-co 4951  df-dm 4952  df-rn 4953  df-res 4954  df-ima 4955  df-iota 5489  df-fun 5527  df-fn 5528  df-f 5529  df-f1 5530  df-fo 5531  df-f1o 5532  df-fv 5533  df-ov 6237  df-oprab 6238  df-1st 6738  df-2nd 6739  df-vc 25733  df-nv 25779  df-va 25782  df-ba 25783  df-sm 25784  df-0v 25785  df-nmcv 25787
This theorem is referenced by:  nmoxr  25975  nmooge0  25976  nmorepnf  25977  nmoolb  25980  nmoubi  25981  nmlno0lem  26002  nmopsetretHIL  27076
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