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Theorem nmorepnf 24347
Description: The norm of an operator is either real or plus infinity. (Contributed by NM, 8-Dec-2007.) (New usage is discouraged.)
Hypotheses
Ref Expression
nmoxr.1  |-  X  =  ( BaseSet `  U )
nmoxr.2  |-  Y  =  ( BaseSet `  W )
nmoxr.3  |-  N  =  ( U normOpOLD W
)
Assertion
Ref Expression
nmorepnf  |-  ( ( U  e.  NrmCVec  /\  W  e.  NrmCVec  /\  T : X
--> Y )  ->  (
( N `  T
)  e.  RR  <->  ( N `  T )  =/= +oo ) )

Proof of Theorem nmorepnf
Dummy variables  x  z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 nmoxr.2 . . . . 5  |-  Y  =  ( BaseSet `  W )
2 eqid 2454 . . . . 5  |-  ( normCV `  W )  =  (
normCV
`  W )
31, 2nmosetre 24343 . . . 4  |-  ( ( W  e.  NrmCVec  /\  T : X --> Y )  ->  { x  |  E. z  e.  X  (
( ( normCV `  U
) `  z )  <_  1  /\  x  =  ( ( normCV `  W
) `  ( T `  z ) ) ) }  C_  RR )
4 nmoxr.1 . . . . . 6  |-  X  =  ( BaseSet `  U )
5 eqid 2454 . . . . . 6  |-  ( 0vec `  U )  =  (
0vec `  U )
6 eqid 2454 . . . . . 6  |-  ( normCV `  U )  =  (
normCV
`  U )
74, 5, 6nmosetn0 24344 . . . . 5  |-  ( U  e.  NrmCVec  ->  ( ( normCV `  W ) `  ( T `  ( 0vec `  U ) ) )  e.  { x  |  E. z  e.  X  ( ( ( normCV `  U ) `  z
)  <_  1  /\  x  =  ( ( normCV `  W ) `  ( T `  z )
) ) } )
8 ne0i 3754 . . . . 5  |-  ( ( ( normCV `  W ) `  ( T `  ( 0vec `  U ) ) )  e.  { x  |  E. z  e.  X  ( ( ( normCV `  U ) `  z
)  <_  1  /\  x  =  ( ( normCV `  W ) `  ( T `  z )
) ) }  ->  { x  |  E. z  e.  X  ( (
( normCV `  U ) `  z )  <_  1  /\  x  =  (
( normCV `  W ) `  ( T `  z ) ) ) }  =/=  (/) )
97, 8syl 16 . . . 4  |-  ( U  e.  NrmCVec  ->  { x  |  E. z  e.  X  ( ( ( normCV `  U ) `  z
)  <_  1  /\  x  =  ( ( normCV `  W ) `  ( T `  z )
) ) }  =/=  (/) )
10 supxrre2 11409 . . . 4  |-  ( ( { x  |  E. z  e.  X  (
( ( normCV `  U
) `  z )  <_  1  /\  x  =  ( ( normCV `  W
) `  ( T `  z ) ) ) }  C_  RR  /\  {
x  |  E. z  e.  X  ( (
( normCV `  U ) `  z )  <_  1  /\  x  =  (
( normCV `  W ) `  ( T `  z ) ) ) }  =/=  (/) )  ->  ( sup ( { x  |  E. z  e.  X  (
( ( normCV `  U
) `  z )  <_  1  /\  x  =  ( ( normCV `  W
) `  ( T `  z ) ) ) } ,  RR* ,  <  )  e.  RR  <->  sup ( { x  |  E. z  e.  X  (
( ( normCV `  U
) `  z )  <_  1  /\  x  =  ( ( normCV `  W
) `  ( T `  z ) ) ) } ,  RR* ,  <  )  =/= +oo ) )
113, 9, 10syl2anr 478 . . 3  |-  ( ( U  e.  NrmCVec  /\  ( W  e.  NrmCVec  /\  T : X --> Y ) )  ->  ( sup ( { x  |  E. z  e.  X  (
( ( normCV `  U
) `  z )  <_  1  /\  x  =  ( ( normCV `  W
) `  ( T `  z ) ) ) } ,  RR* ,  <  )  e.  RR  <->  sup ( { x  |  E. z  e.  X  (
( ( normCV `  U
) `  z )  <_  1  /\  x  =  ( ( normCV `  W
) `  ( T `  z ) ) ) } ,  RR* ,  <  )  =/= +oo ) )
12113impb 1184 . 2  |-  ( ( U  e.  NrmCVec  /\  W  e.  NrmCVec  /\  T : X
--> Y )  ->  ( sup ( { x  |  E. z  e.  X  ( ( ( normCV `  U ) `  z
)  <_  1  /\  x  =  ( ( normCV `  W ) `  ( T `  z )
) ) } ,  RR* ,  <  )  e.  RR  <->  sup ( { x  |  E. z  e.  X  ( ( ( normCV `  U ) `  z
)  <_  1  /\  x  =  ( ( normCV `  W ) `  ( T `  z )
) ) } ,  RR* ,  <  )  =/= +oo ) )
13 nmoxr.3 . . . 4  |-  N  =  ( U normOpOLD W
)
144, 1, 6, 2, 13nmooval 24342 . . 3  |-  ( ( U  e.  NrmCVec  /\  W  e.  NrmCVec  /\  T : X
--> Y )  ->  ( N `  T )  =  sup ( { x  |  E. z  e.  X  ( ( ( normCV `  U ) `  z
)  <_  1  /\  x  =  ( ( normCV `  W ) `  ( T `  z )
) ) } ,  RR* ,  <  ) )
1514eleq1d 2523 . 2  |-  ( ( U  e.  NrmCVec  /\  W  e.  NrmCVec  /\  T : X
--> Y )  ->  (
( N `  T
)  e.  RR  <->  sup ( { x  |  E. z  e.  X  (
( ( normCV `  U
) `  z )  <_  1  /\  x  =  ( ( normCV `  W
) `  ( T `  z ) ) ) } ,  RR* ,  <  )  e.  RR ) )
1614neeq1d 2729 . 2  |-  ( ( U  e.  NrmCVec  /\  W  e.  NrmCVec  /\  T : X
--> Y )  ->  (
( N `  T
)  =/= +oo  <->  sup ( { x  |  E. z  e.  X  (
( ( normCV `  U
) `  z )  <_  1  /\  x  =  ( ( normCV `  W
) `  ( T `  z ) ) ) } ,  RR* ,  <  )  =/= +oo ) )
1712, 15, 163bitr4d 285 1  |-  ( ( U  e.  NrmCVec  /\  W  e.  NrmCVec  /\  T : X
--> Y )  ->  (
( N `  T
)  e.  RR  <->  ( N `  T )  =/= +oo ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 965    = wceq 1370    e. wcel 1758   {cab 2439    =/= wne 2648   E.wrex 2800    C_ wss 3439   (/)c0 3748   class class class wbr 4403   -->wf 5525   ` cfv 5529  (class class class)co 6203   supcsup 7805   RRcr 9396   1c1 9398   +oocpnf 9530   RR*cxr 9532    < clt 9533    <_ cle 9534   NrmCVeccnv 24141   BaseSetcba 24143   0veccn0v 24145   normCVcnmcv 24147   normOpOLDcnmoo 24320
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 1955  ax-ext 2432  ax-rep 4514  ax-sep 4524  ax-nul 4532  ax-pow 4581  ax-pr 4642  ax-un 6485  ax-cnex 9453  ax-resscn 9454  ax-1cn 9455  ax-icn 9456  ax-addcl 9457  ax-addrcl 9458  ax-mulcl 9459  ax-mulrcl 9460  ax-mulcom 9461  ax-addass 9462  ax-mulass 9463  ax-distr 9464  ax-i2m1 9465  ax-1ne0 9466  ax-1rid 9467  ax-rnegex 9468  ax-rrecex 9469  ax-cnre 9470  ax-pre-lttri 9471  ax-pre-lttrn 9472  ax-pre-ltadd 9473  ax-pre-mulgt0 9474  ax-pre-sup 9475
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-nel 2651  df-ral 2804  df-rex 2805  df-reu 2806  df-rmo 2807  df-rab 2808  df-v 3080  df-sbc 3295  df-csb 3399  df-dif 3442  df-un 3444  df-in 3446  df-ss 3453  df-pss 3455  df-nul 3749  df-if 3903  df-pw 3973  df-sn 3989  df-pr 3991  df-tp 3993  df-op 3995  df-uni 4203  df-iun 4284  df-br 4404  df-opab 4462  df-mpt 4463  df-tr 4497  df-eprel 4743  df-id 4747  df-po 4752  df-so 4753  df-fr 4790  df-we 4792  df-ord 4833  df-on 4834  df-lim 4835  df-suc 4836  df-xp 4957  df-rel 4958  df-cnv 4959  df-co 4960  df-dm 4961  df-rn 4962  df-res 4963  df-ima 4964  df-iota 5492  df-fun 5531  df-fn 5532  df-f 5533  df-f1 5534  df-fo 5535  df-f1o 5536  df-fv 5537  df-riota 6164  df-ov 6206  df-oprab 6207  df-mpt2 6208  df-om 6590  df-1st 6690  df-2nd 6691  df-recs 6945  df-rdg 6979  df-er 7214  df-map 7329  df-en 7424  df-dom 7425  df-sdom 7426  df-sup 7806  df-pnf 9535  df-mnf 9536  df-xr 9537  df-ltxr 9538  df-le 9539  df-sub 9712  df-neg 9713  df-div 10109  df-nn 10438  df-2 10495  df-3 10496  df-n0 10695  df-z 10762  df-uz 10977  df-rp 11107  df-seq 11928  df-exp 11987  df-cj 12710  df-re 12711  df-im 12712  df-sqr 12846  df-abs 12847  df-grpo 23857  df-gid 23858  df-ginv 23859  df-ablo 23948  df-vc 24103  df-nv 24149  df-va 24152  df-ba 24153  df-sm 24154  df-0v 24155  df-nmcv 24157  df-nmoo 24324
This theorem is referenced by:  nmoreltpnf  24348  nmogtmnf  24349  nmounbi  24355
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