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Theorem nmooge0 22229
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 9095 . . 3  |-  0  e.  RR*
21a1i 11 . 2  |-  ( ( U  e.  NrmCVec  /\  W  e.  NrmCVec  /\  T : X
--> Y )  ->  0  e.  RR* )
3 simp2 958 . . . 4  |-  ( ( U  e.  NrmCVec  /\  W  e.  NrmCVec  /\  T : X
--> Y )  ->  W  e.  NrmCVec )
4 nmoxr.1 . . . . . . . 8  |-  X  =  ( BaseSet `  U )
5 eqid 2412 . . . . . . . 8  |-  ( 0vec `  U )  =  (
0vec `  U )
64, 5nvzcl 22076 . . . . . . 7  |-  ( U  e.  NrmCVec  ->  ( 0vec `  U
)  e.  X )
7 ffvelrn 5835 . . . . . . 7  |-  ( ( T : X --> Y  /\  ( 0vec `  U )  e.  X )  ->  ( T `  ( 0vec `  U ) )  e.  Y )
86, 7sylan2 461 . . . . . 6  |-  ( ( T : X --> Y  /\  U  e.  NrmCVec )  -> 
( T `  ( 0vec `  U ) )  e.  Y )
98ancoms 440 . . . . 5  |-  ( ( U  e.  NrmCVec  /\  T : X --> Y )  -> 
( T `  ( 0vec `  U ) )  e.  Y )
1093adant2 976 . . . 4  |-  ( ( U  e.  NrmCVec  /\  W  e.  NrmCVec  /\  T : X
--> Y )  ->  ( T `  ( 0vec `  U ) )  e.  Y )
11 nmoxr.2 . . . . 5  |-  Y  =  ( BaseSet `  W )
12 eqid 2412 . . . . 5  |-  ( normCV `  W )  =  (
normCV
`  W )
1311, 12nvcl 22109 . . . 4  |-  ( ( W  e.  NrmCVec  /\  ( T `  ( 0vec `  U ) )  e.  Y )  ->  (
( normCV `  W ) `  ( T `  ( 0vec `  U ) ) )  e.  RR )
143, 10, 13syl2anc 643 . . 3  |-  ( ( U  e.  NrmCVec  /\  W  e.  NrmCVec  /\  T : X
--> Y )  ->  (
( normCV `  W ) `  ( T `  ( 0vec `  U ) ) )  e.  RR )
1514rexrd 9098 . 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 22228 . 2  |-  ( ( U  e.  NrmCVec  /\  W  e.  NrmCVec  /\  T : X
--> Y )  ->  ( N `  T )  e.  RR* )
1811, 12nvge0 22124 . . 3  |-  ( ( W  e.  NrmCVec  /\  ( T `  ( 0vec `  U ) )  e.  Y )  ->  0  <_  ( ( normCV `  W
) `  ( T `  ( 0vec `  U
) ) ) )
193, 10, 18syl2anc 643 . 2  |-  ( ( U  e.  NrmCVec  /\  W  e.  NrmCVec  /\  T : X
--> Y )  ->  0  <_  ( ( normCV `  W
) `  ( T `  ( 0vec `  U
) ) ) )
2011, 12nmosetre 22226 . . . . . . 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 9093 . . . . . . 7  |-  RR  C_  RR*
2220, 21syl6ss 3328 . . . . . 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 2412 . . . . . . 7  |-  ( normCV `  U )  =  (
normCV
`  U )
244, 5, 23nmosetn0 22227 . . . . . 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 10867 . . . . . 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 464 . . . . 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 1148 . . . 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 1161 . . 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 22225 . . 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 4206 . 2  |-  ( ( U  e.  NrmCVec  /\  W  e.  NrmCVec  /\  T : X
--> Y )  ->  (
( normCV `  W ) `  ( T `  ( 0vec `  U ) ) )  <_  ( N `  T ) )
312, 15, 17, 19, 30xrletrd 10716 1  |-  ( ( U  e.  NrmCVec  /\  W  e.  NrmCVec  /\  T : X
--> Y )  ->  0  <_  ( N `  T
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    /\ w3a 936    = wceq 1649    e. wcel 1721   {cab 2398   E.wrex 2675    C_ wss 3288   class class class wbr 4180   -->wf 5417   ` cfv 5421  (class class class)co 6048   supcsup 7411   RRcr 8953   0cc0 8954   1c1 8955   RR*cxr 9083    < clt 9084    <_ cle 9085   NrmCVeccnv 22024   BaseSetcba 22026   0veccn0v 22028   normCVcnmcv 22030   normOp OLDcnmoo 22203
This theorem is referenced by:  nmlnogt0  22259  htthlem  22381
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 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2393  ax-rep 4288  ax-sep 4298  ax-nul 4306  ax-pow 4345  ax-pr 4371  ax-un 4668  ax-cnex 9010  ax-resscn 9011  ax-1cn 9012  ax-icn 9013  ax-addcl 9014  ax-addrcl 9015  ax-mulcl 9016  ax-mulrcl 9017  ax-mulcom 9018  ax-addass 9019  ax-mulass 9020  ax-distr 9021  ax-i2m1 9022  ax-1ne0 9023  ax-1rid 9024  ax-rnegex 9025  ax-rrecex 9026  ax-cnre 9027  ax-pre-lttri 9028  ax-pre-lttrn 9029  ax-pre-ltadd 9030  ax-pre-mulgt0 9031  ax-pre-sup 9032
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 2266  df-mo 2267  df-clab 2399  df-cleq 2405  df-clel 2408  df-nfc 2537  df-ne 2577  df-nel 2578  df-ral 2679  df-rex 2680  df-reu 2681  df-rmo 2682  df-rab 2683  df-v 2926  df-sbc 3130  df-csb 3220  df-dif 3291  df-un 3293  df-in 3295  df-ss 3302  df-pss 3304  df-nul 3597  df-if 3708  df-pw 3769  df-sn 3788  df-pr 3789  df-tp 3790  df-op 3791  df-uni 3984  df-iun 4063  df-br 4181  df-opab 4235  df-mpt 4236  df-tr 4271  df-eprel 4462  df-id 4466  df-po 4471  df-so 4472  df-fr 4509  df-we 4511  df-ord 4552  df-on 4553  df-lim 4554  df-suc 4555  df-om 4813  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-iota 5385  df-fun 5423  df-fn 5424  df-f 5425  df-f1 5426  df-fo 5427  df-f1o 5428  df-fv 5429  df-ov 6051  df-oprab 6052  df-mpt2 6053  df-1st 6316  df-2nd 6317  df-riota 6516  df-recs 6600  df-rdg 6635  df-er 6872  df-map 6987  df-en 7077  df-dom 7078  df-sdom 7079  df-sup 7412  df-pnf 9086  df-mnf 9087  df-xr 9088  df-ltxr 9089  df-le 9090  df-sub 9257  df-neg 9258  df-div 9642  df-nn 9965  df-2 10022  df-3 10023  df-n0 10186  df-z 10247  df-uz 10453  df-rp 10577  df-seq 11287  df-exp 11346  df-cj 11867  df-re 11868  df-im 11869  df-sqr 12003  df-abs 12004  df-grpo 21740  df-gid 21741  df-ginv 21742  df-ablo 21831  df-vc 21986  df-nv 22032  df-va 22035  df-ba 22036  df-sm 22037  df-0v 22038  df-nmcv 22040  df-nmoo 22207
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