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Theorem nmorepnf 22119
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 normOp OLD 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 2389 . . . . 5  |-  ( normCV `  W )  =  (
normCV
`  W )
31, 2nmosetre 22115 . . . 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 2389 . . . . . 6  |-  ( 0vec `  U )  =  (
0vec `  U )
6 eqid 2389 . . . . . 6  |-  ( normCV `  U )  =  (
normCV
`  U )
74, 5, 6nmosetn0 22116 . . . . 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 3579 . . . . 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 10844 . . . 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 465 . . 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 1149 . 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 normOp OLD W
)
144, 1, 6, 2, 13nmooval 22114 . . 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 2455 . 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 2565 . 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 277 1  |-  ( ( U  e.  NrmCVec  /\  W  e.  NrmCVec  /\  T : X
--> Y )  ->  (
( N `  T
)  e.  RR  <->  ( N `  T )  =/=  +oo ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    /\ w3a 936    = wceq 1649    e. wcel 1717   {cab 2375    =/= wne 2552   E.wrex 2652    C_ wss 3265   (/)c0 3573   class class class wbr 4155   -->wf 5392   ` cfv 5396  (class class class)co 6022   supcsup 7382   RRcr 8924   1c1 8926    +oocpnf 9052   RR*cxr 9054    < clt 9055    <_ cle 9056   NrmCVeccnv 21913   BaseSetcba 21915   0veccn0v 21917   normCVcnmcv 21919   normOp OLDcnmoo 22092
This theorem is referenced by:  nmoreltpnf  22120  nmogtmnf  22121  nmounbi  22127
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2370  ax-rep 4263  ax-sep 4273  ax-nul 4281  ax-pow 4320  ax-pr 4346  ax-un 4643  ax-cnex 8981  ax-resscn 8982  ax-1cn 8983  ax-icn 8984  ax-addcl 8985  ax-addrcl 8986  ax-mulcl 8987  ax-mulrcl 8988  ax-mulcom 8989  ax-addass 8990  ax-mulass 8991  ax-distr 8992  ax-i2m1 8993  ax-1ne0 8994  ax-1rid 8995  ax-rnegex 8996  ax-rrecex 8997  ax-cnre 8998  ax-pre-lttri 8999  ax-pre-lttrn 9000  ax-pre-ltadd 9001  ax-pre-mulgt0 9002  ax-pre-sup 9003
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2244  df-mo 2245  df-clab 2376  df-cleq 2382  df-clel 2385  df-nfc 2514  df-ne 2554  df-nel 2555  df-ral 2656  df-rex 2657  df-reu 2658  df-rmo 2659  df-rab 2660  df-v 2903  df-sbc 3107  df-csb 3197  df-dif 3268  df-un 3270  df-in 3272  df-ss 3279  df-pss 3281  df-nul 3574  df-if 3685  df-pw 3746  df-sn 3765  df-pr 3766  df-tp 3767  df-op 3768  df-uni 3960  df-iun 4039  df-br 4156  df-opab 4210  df-mpt 4211  df-tr 4246  df-eprel 4437  df-id 4441  df-po 4446  df-so 4447  df-fr 4484  df-we 4486  df-ord 4527  df-on 4528  df-lim 4529  df-suc 4530  df-om 4788  df-xp 4826  df-rel 4827  df-cnv 4828  df-co 4829  df-dm 4830  df-rn 4831  df-res 4832  df-ima 4833  df-iota 5360  df-fun 5398  df-fn 5399  df-f 5400  df-f1 5401  df-fo 5402  df-f1o 5403  df-fv 5404  df-ov 6025  df-oprab 6026  df-mpt2 6027  df-1st 6290  df-2nd 6291  df-riota 6487  df-recs 6571  df-rdg 6606  df-er 6843  df-map 6958  df-en 7048  df-dom 7049  df-sdom 7050  df-sup 7383  df-pnf 9057  df-mnf 9058  df-xr 9059  df-ltxr 9060  df-le 9061  df-sub 9227  df-neg 9228  df-div 9612  df-nn 9935  df-2 9992  df-3 9993  df-n0 10156  df-z 10217  df-uz 10423  df-rp 10547  df-seq 11253  df-exp 11312  df-cj 11833  df-re 11834  df-im 11835  df-sqr 11969  df-abs 11970  df-grpo 21629  df-gid 21630  df-ginv 21631  df-ablo 21720  df-vc 21875  df-nv 21921  df-va 21924  df-ba 21925  df-sm 21926  df-0v 21927  df-nmcv 21929  df-nmoo 22096
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