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Theorem nmooge0 22269
Description: The norm of an operator is nonnegative. (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
nmooge0  |-  ( ( U  e.  NrmCVec  /\  W  e.  NrmCVec  /\  T : X
--> Y )  ->  0  <_  ( N `  T
) )

Proof of Theorem nmooge0
Dummy variables  x  z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 0xr 9132 . . 3  |-  0  e.  RR*
21a1i 11 . 2  |-  ( ( U  e.  NrmCVec  /\  W  e.  NrmCVec  /\  T : X
--> Y )  ->  0  e.  RR* )
3 simp2 959 . . . 4  |-  ( ( U  e.  NrmCVec  /\  W  e.  NrmCVec  /\  T : X
--> Y )  ->  W  e.  NrmCVec )
4 nmoxr.1 . . . . . . . 8  |-  X  =  ( BaseSet `  U )
5 eqid 2437 . . . . . . . 8  |-  ( 0vec `  U )  =  (
0vec `  U )
64, 5nvzcl 22116 . . . . . . 7  |-  ( U  e.  NrmCVec  ->  ( 0vec `  U
)  e.  X )
7 ffvelrn 5869 . . . . . . 7  |-  ( ( T : X --> Y  /\  ( 0vec `  U )  e.  X )  ->  ( T `  ( 0vec `  U ) )  e.  Y )
86, 7sylan2 462 . . . . . 6  |-  ( ( T : X --> Y  /\  U  e.  NrmCVec )  -> 
( T `  ( 0vec `  U ) )  e.  Y )
98ancoms 441 . . . . 5  |-  ( ( U  e.  NrmCVec  /\  T : X --> Y )  -> 
( T `  ( 0vec `  U ) )  e.  Y )
1093adant2 977 . . . 4  |-  ( ( U  e.  NrmCVec  /\  W  e.  NrmCVec  /\  T : X
--> Y )  ->  ( T `  ( 0vec `  U ) )  e.  Y )
11 nmoxr.2 . . . . 5  |-  Y  =  ( BaseSet `  W )
12 eqid 2437 . . . . 5  |-  ( normCV `  W )  =  (
normCV
`  W )
1311, 12nvcl 22149 . . . 4  |-  ( ( W  e.  NrmCVec  /\  ( T `  ( 0vec `  U ) )  e.  Y )  ->  (
( normCV `  W ) `  ( T `  ( 0vec `  U ) ) )  e.  RR )
143, 10, 13syl2anc 644 . . 3  |-  ( ( U  e.  NrmCVec  /\  W  e.  NrmCVec  /\  T : X
--> Y )  ->  (
( normCV `  W ) `  ( T `  ( 0vec `  U ) ) )  e.  RR )
1514rexrd 9135 . 2  |-  ( ( U  e.  NrmCVec  /\  W  e.  NrmCVec  /\  T : X
--> Y )  ->  (
( normCV `  W ) `  ( T `  ( 0vec `  U ) ) )  e.  RR* )
16 nmoxr.3 . . 3  |-  N  =  ( U normOp OLD W
)
174, 11, 16nmoxr 22268 . 2  |-  ( ( U  e.  NrmCVec  /\  W  e.  NrmCVec  /\  T : X
--> Y )  ->  ( N `  T )  e.  RR* )
1811, 12nvge0 22164 . . 3  |-  ( ( W  e.  NrmCVec  /\  ( T `  ( 0vec `  U ) )  e.  Y )  ->  0  <_  ( ( normCV `  W
) `  ( T `  ( 0vec `  U
) ) ) )
193, 10, 18syl2anc 644 . 2  |-  ( ( U  e.  NrmCVec  /\  W  e.  NrmCVec  /\  T : X
--> Y )  ->  0  <_  ( ( normCV `  W
) `  ( T `  ( 0vec `  U
) ) ) )
2011, 12nmosetre 22266 . . . . . . 7  |-  ( ( W  e.  NrmCVec  /\  T : X --> Y )  ->  { x  |  E. z  e.  X  (
( ( normCV `  U
) `  z )  <_  1  /\  x  =  ( ( normCV `  W
) `  ( T `  z ) ) ) }  C_  RR )
21 ressxr 9130 . . . . . . 7  |-  RR  C_  RR*
2220, 21syl6ss 3361 . . . . . 6  |-  ( ( W  e.  NrmCVec  /\  T : X --> Y )  ->  { x  |  E. z  e.  X  (
( ( normCV `  U
) `  z )  <_  1  /\  x  =  ( ( normCV `  W
) `  ( T `  z ) ) ) }  C_  RR* )
23 eqid 2437 . . . . . . 7  |-  ( normCV `  U )  =  (
normCV
`  U )
244, 5, 23nmosetn0 22267 . . . . . 6  |-  ( U  e.  NrmCVec  ->  ( ( normCV `  W ) `  ( T `  ( 0vec `  U ) ) )  e.  { x  |  E. z  e.  X  ( ( ( normCV `  U ) `  z
)  <_  1  /\  x  =  ( ( normCV `  W ) `  ( T `  z )
) ) } )
25 supxrub 10904 . . . . . 6  |-  ( ( { x  |  E. z  e.  X  (
( ( normCV `  U
) `  z )  <_  1  /\  x  =  ( ( normCV `  W
) `  ( T `  z ) ) ) }  C_  RR*  /\  (
( normCV `  W ) `  ( T `  ( 0vec `  U ) ) )  e.  { x  |  E. z  e.  X  ( ( ( normCV `  U ) `  z
)  <_  1  /\  x  =  ( ( normCV `  W ) `  ( T `  z )
) ) } )  ->  ( ( normCV `  W ) `  ( T `  ( 0vec `  U ) ) )  <_  sup ( { x  |  E. z  e.  X  ( ( ( normCV `  U ) `  z
)  <_  1  /\  x  =  ( ( normCV `  W ) `  ( T `  z )
) ) } ,  RR* ,  <  ) )
2622, 24, 25syl2an 465 . . . . 5  |-  ( ( ( W  e.  NrmCVec  /\  T : X --> Y )  /\  U  e.  NrmCVec )  ->  ( ( normCV `  W ) `  ( T `  ( 0vec `  U ) ) )  <_  sup ( { x  |  E. z  e.  X  ( ( ( normCV `  U ) `  z
)  <_  1  /\  x  =  ( ( normCV `  W ) `  ( T `  z )
) ) } ,  RR* ,  <  ) )
27263impa 1149 . . . 4  |-  ( ( W  e.  NrmCVec  /\  T : X --> Y  /\  U  e.  NrmCVec )  ->  (
( normCV `  W ) `  ( T `  ( 0vec `  U ) ) )  <_  sup ( { x  |  E. z  e.  X  ( ( ( normCV `  U ) `  z
)  <_  1  /\  x  =  ( ( normCV `  W ) `  ( T `  z )
) ) } ,  RR* ,  <  ) )
28273comr 1162 . . 3  |-  ( ( U  e.  NrmCVec  /\  W  e.  NrmCVec  /\  T : X
--> Y )  ->  (
( normCV `  W ) `  ( T `  ( 0vec `  U ) ) )  <_  sup ( { x  |  E. z  e.  X  ( ( ( normCV `  U ) `  z
)  <_  1  /\  x  =  ( ( normCV `  W ) `  ( T `  z )
) ) } ,  RR* ,  <  ) )
294, 11, 23, 12, 16nmooval 22265 . . 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* ,  <  ) )
3028, 29breqtrrd 4239 . 2  |-  ( ( U  e.  NrmCVec  /\  W  e.  NrmCVec  /\  T : X
--> Y )  ->  (
( normCV `  W ) `  ( T `  ( 0vec `  U ) ) )  <_  ( N `  T ) )
312, 15, 17, 19, 30xrletrd 10753 1  |-  ( ( U  e.  NrmCVec  /\  W  e.  NrmCVec  /\  T : X
--> Y )  ->  0  <_  ( N `  T
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 360    /\ w3a 937    = wceq 1653    e. wcel 1726   {cab 2423   E.wrex 2707    C_ wss 3321   class class class wbr 4213   -->wf 5451   ` cfv 5455  (class class class)co 6082   supcsup 7446   RRcr 8990   0cc0 8991   1c1 8992   RR*cxr 9120    < clt 9121    <_ cle 9122   NrmCVeccnv 22064   BaseSetcba 22066   0veccn0v 22068   normCVcnmcv 22070   normOp OLDcnmoo 22243
This theorem is referenced by:  nmlnogt0  22299  htthlem  22421
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  ax-cnex 9047  ax-resscn 9048  ax-1cn 9049  ax-icn 9050  ax-addcl 9051  ax-addrcl 9052  ax-mulcl 9053  ax-mulrcl 9054  ax-mulcom 9055  ax-addass 9056  ax-mulass 9057  ax-distr 9058  ax-i2m1 9059  ax-1ne0 9060  ax-1rid 9061  ax-rnegex 9062  ax-rrecex 9063  ax-cnre 9064  ax-pre-lttri 9065  ax-pre-lttrn 9066  ax-pre-ltadd 9067  ax-pre-mulgt0 9068  ax-pre-sup 9069
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  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-nel 2603  df-ral 2711  df-rex 2712  df-reu 2713  df-rmo 2714  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-pss 3337  df-nul 3630  df-if 3741  df-pw 3802  df-sn 3821  df-pr 3822  df-tp 3823  df-op 3824  df-uni 4017  df-iun 4096  df-br 4214  df-opab 4268  df-mpt 4269  df-tr 4304  df-eprel 4495  df-id 4499  df-po 4504  df-so 4505  df-fr 4542  df-we 4544  df-ord 4585  df-on 4586  df-lim 4587  df-suc 4588  df-om 4847  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-mpt2 6087  df-1st 6350  df-2nd 6351  df-riota 6550  df-recs 6634  df-rdg 6669  df-er 6906  df-map 7021  df-en 7111  df-dom 7112  df-sdom 7113  df-sup 7447  df-pnf 9123  df-mnf 9124  df-xr 9125  df-ltxr 9126  df-le 9127  df-sub 9294  df-neg 9295  df-div 9679  df-nn 10002  df-2 10059  df-3 10060  df-n0 10223  df-z 10284  df-uz 10490  df-rp 10614  df-seq 11325  df-exp 11384  df-cj 11905  df-re 11906  df-im 11907  df-sqr 12041  df-abs 12042  df-grpo 21780  df-gid 21781  df-ginv 21782  df-ablo 21871  df-vc 22026  df-nv 22072  df-va 22075  df-ba 22076  df-sm 22077  df-0v 22078  df-nmcv 22080  df-nmoo 22247
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