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Theorem nmosetre 24309
Description: The set in the supremum of the operator norm definition df-nmoo 24290 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 5943 . . . . . . . . 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 24192 . . . . . . . . 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 2523 . . . . . . 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 2939 . 2  |-  ( ( W  e.  NrmCVec  /\  T : X --> Y )  -> 
( E. z  e.  X  ( ( M `
 z )  <_ 
1  /\  x  =  ( N `  ( T `
 z ) ) )  ->  x  e.  RR ) )
1312abssdv 3527 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 1370    e. wcel 1758   {cab 2436   E.wrex 2796    C_ wss 3429   class class class wbr 4393   -->wf 5515   ` cfv 5519   RRcr 9385   1c1 9387    <_ cle 9523   NrmCVeccnv 24107   BaseSetcba 24109   normCVcnmcv 24113
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-rep 4504  ax-sep 4514  ax-nul 4522  ax-pow 4571  ax-pr 4632  ax-un 6475
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ne 2646  df-ral 2800  df-rex 2801  df-reu 2802  df-rab 2804  df-v 3073  df-sbc 3288  df-csb 3390  df-dif 3432  df-un 3434  df-in 3436  df-ss 3443  df-nul 3739  df-if 3893  df-sn 3979  df-pr 3981  df-op 3985  df-uni 4193  df-iun 4274  df-br 4394  df-opab 4452  df-mpt 4453  df-id 4737  df-xp 4947  df-rel 4948  df-cnv 4949  df-co 4950  df-dm 4951  df-rn 4952  df-res 4953  df-ima 4954  df-iota 5482  df-fun 5521  df-fn 5522  df-f 5523  df-f1 5524  df-fo 5525  df-f1o 5526  df-fv 5527  df-ov 6196  df-oprab 6197  df-1st 6680  df-2nd 6681  df-vc 24069  df-nv 24115  df-va 24118  df-ba 24119  df-sm 24120  df-0v 24121  df-nmcv 24123
This theorem is referenced by:  nmoxr  24311  nmooge0  24312  nmorepnf  24313  nmoolb  24316  nmoubi  24317  nmlno0lem  24338  nmopsetretHIL  25413
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