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Theorem nmooge0 21361
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 8894 . . 3  |-  0  e.  RR*
21a1i 10 . 2  |-  ( ( U  e.  NrmCVec  /\  W  e.  NrmCVec  /\  T : X
--> Y )  ->  0  e.  RR* )
3 simp2 956 . . . 4  |-  ( ( U  e.  NrmCVec  /\  W  e.  NrmCVec  /\  T : X
--> Y )  ->  W  e.  NrmCVec )
4 nmoxr.1 . . . . . . . 8  |-  X  =  ( BaseSet `  U )
5 eqid 2296 . . . . . . . 8  |-  ( 0vec `  U )  =  (
0vec `  U )
64, 5nvzcl 21208 . . . . . . 7  |-  ( U  e.  NrmCVec  ->  ( 0vec `  U
)  e.  X )
7 ffvelrn 5679 . . . . . . 7  |-  ( ( T : X --> Y  /\  ( 0vec `  U )  e.  X )  ->  ( T `  ( 0vec `  U ) )  e.  Y )
86, 7sylan2 460 . . . . . 6  |-  ( ( T : X --> Y  /\  U  e.  NrmCVec )  -> 
( T `  ( 0vec `  U ) )  e.  Y )
98ancoms 439 . . . . 5  |-  ( ( U  e.  NrmCVec  /\  T : X --> Y )  -> 
( T `  ( 0vec `  U ) )  e.  Y )
1093adant2 974 . . . 4  |-  ( ( U  e.  NrmCVec  /\  W  e.  NrmCVec  /\  T : X
--> Y )  ->  ( T `  ( 0vec `  U ) )  e.  Y )
11 nmoxr.2 . . . . 5  |-  Y  =  ( BaseSet `  W )
12 eqid 2296 . . . . 5  |-  ( normCV `  W )  =  (
normCV
`  W )
1311, 12nvcl 21241 . . . 4  |-  ( ( W  e.  NrmCVec  /\  ( T `  ( 0vec `  U ) )  e.  Y )  ->  (
( normCV `  W ) `  ( T `  ( 0vec `  U ) ) )  e.  RR )
143, 10, 13syl2anc 642 . . 3  |-  ( ( U  e.  NrmCVec  /\  W  e.  NrmCVec  /\  T : X
--> Y )  ->  (
( normCV `  W ) `  ( T `  ( 0vec `  U ) ) )  e.  RR )
1514rexrd 8897 . 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 21360 . 2  |-  ( ( U  e.  NrmCVec  /\  W  e.  NrmCVec  /\  T : X
--> Y )  ->  ( N `  T )  e.  RR* )
1811, 12nvge0 21256 . . 3  |-  ( ( W  e.  NrmCVec  /\  ( T `  ( 0vec `  U ) )  e.  Y )  ->  0  <_  ( ( normCV `  W
) `  ( T `  ( 0vec `  U
) ) ) )
193, 10, 18syl2anc 642 . 2  |-  ( ( U  e.  NrmCVec  /\  W  e.  NrmCVec  /\  T : X
--> Y )  ->  0  <_  ( ( normCV `  W
) `  ( T `  ( 0vec `  U
) ) ) )
2011, 12nmosetre 21358 . . . . . . 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 8892 . . . . . . 7  |-  RR  C_  RR*
2220, 21syl6ss 3204 . . . . . 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 2296 . . . . . . 7  |-  ( normCV `  U )  =  (
normCV
`  U )
244, 5, 23nmosetn0 21359 . . . . . 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 10659 . . . . . 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 463 . . . . 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 1146 . . . 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 1159 . . 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 21357 . . 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 4065 . 2  |-  ( ( U  e.  NrmCVec  /\  W  e.  NrmCVec  /\  T : X
--> Y )  ->  (
( normCV `  W ) `  ( T `  ( 0vec `  U ) ) )  <_  ( N `  T ) )
312, 15, 17, 19, 30xrletrd 10509 1  |-  ( ( U  e.  NrmCVec  /\  W  e.  NrmCVec  /\  T : X
--> Y )  ->  0  <_  ( N `  T
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    /\ w3a 934    = wceq 1632    e. wcel 1696   {cab 2282   E.wrex 2557    C_ wss 3165   class class class wbr 4039   -->wf 5267   ` cfv 5271  (class class class)co 5874   supcsup 7209   RRcr 8752   0cc0 8753   1c1 8754   RR*cxr 8882    < clt 8883    <_ cle 8884   NrmCVeccnv 21156   BaseSetcba 21158   0veccn0v 21160   normCVcnmcv 21162   normOp OLDcnmoo 21335
This theorem is referenced by:  nmlnogt0  21391  htthlem  21513
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-cnex 8809  ax-resscn 8810  ax-1cn 8811  ax-icn 8812  ax-addcl 8813  ax-addrcl 8814  ax-mulcl 8815  ax-mulrcl 8816  ax-mulcom 8817  ax-addass 8818  ax-mulass 8819  ax-distr 8820  ax-i2m1 8821  ax-1ne0 8822  ax-1rid 8823  ax-rnegex 8824  ax-rrecex 8825  ax-cnre 8826  ax-pre-lttri 8827  ax-pre-lttrn 8828  ax-pre-ltadd 8829  ax-pre-mulgt0 8830  ax-pre-sup 8831
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rmo 2564  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-1st 6138  df-2nd 6139  df-riota 6320  df-recs 6404  df-rdg 6439  df-er 6676  df-map 6790  df-en 6880  df-dom 6881  df-sdom 6882  df-sup 7210  df-pnf 8885  df-mnf 8886  df-xr 8887  df-ltxr 8888  df-le 8889  df-sub 9055  df-neg 9056  df-div 9440  df-nn 9763  df-2 9820  df-3 9821  df-n0 9982  df-z 10041  df-uz 10247  df-rp 10371  df-seq 11063  df-exp 11121  df-cj 11600  df-re 11601  df-im 11602  df-sqr 11736  df-abs 11737  df-grpo 20874  df-gid 20875  df-ginv 20876  df-ablo 20965  df-vc 21118  df-nv 21164  df-va 21167  df-ba 21168  df-sm 21169  df-0v 21170  df-nmcv 21172  df-nmoo 21339
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