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Theorem nmooge0 24302
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 normOpOLD 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 9531 . . 3  |-  0  e.  RR*
21a1i 11 . 2  |-  ( ( U  e.  NrmCVec  /\  W  e.  NrmCVec  /\  T : X
--> Y )  ->  0  e.  RR* )
3 simp2 989 . . . 4  |-  ( ( U  e.  NrmCVec  /\  W  e.  NrmCVec  /\  T : X
--> Y )  ->  W  e.  NrmCVec )
4 nmoxr.1 . . . . . . . 8  |-  X  =  ( BaseSet `  U )
5 eqid 2451 . . . . . . . 8  |-  ( 0vec `  U )  =  (
0vec `  U )
64, 5nvzcl 24149 . . . . . . 7  |-  ( U  e.  NrmCVec  ->  ( 0vec `  U
)  e.  X )
7 ffvelrn 5940 . . . . . . 7  |-  ( ( T : X --> Y  /\  ( 0vec `  U )  e.  X )  ->  ( T `  ( 0vec `  U ) )  e.  Y )
86, 7sylan2 474 . . . . . 6  |-  ( ( T : X --> Y  /\  U  e.  NrmCVec )  -> 
( T `  ( 0vec `  U ) )  e.  Y )
98ancoms 453 . . . . 5  |-  ( ( U  e.  NrmCVec  /\  T : X --> Y )  -> 
( T `  ( 0vec `  U ) )  e.  Y )
1093adant2 1007 . . . 4  |-  ( ( U  e.  NrmCVec  /\  W  e.  NrmCVec  /\  T : X
--> Y )  ->  ( T `  ( 0vec `  U ) )  e.  Y )
11 nmoxr.2 . . . . 5  |-  Y  =  ( BaseSet `  W )
12 eqid 2451 . . . . 5  |-  ( normCV `  W )  =  (
normCV
`  W )
1311, 12nvcl 24182 . . . 4  |-  ( ( W  e.  NrmCVec  /\  ( T `  ( 0vec `  U ) )  e.  Y )  ->  (
( normCV `  W ) `  ( T `  ( 0vec `  U ) ) )  e.  RR )
143, 10, 13syl2anc 661 . . 3  |-  ( ( U  e.  NrmCVec  /\  W  e.  NrmCVec  /\  T : X
--> Y )  ->  (
( normCV `  W ) `  ( T `  ( 0vec `  U ) ) )  e.  RR )
1514rexrd 9534 . 2  |-  ( ( U  e.  NrmCVec  /\  W  e.  NrmCVec  /\  T : X
--> Y )  ->  (
( normCV `  W ) `  ( T `  ( 0vec `  U ) ) )  e.  RR* )
16 nmoxr.3 . . 3  |-  N  =  ( U normOpOLD W
)
174, 11, 16nmoxr 24301 . 2  |-  ( ( U  e.  NrmCVec  /\  W  e.  NrmCVec  /\  T : X
--> Y )  ->  ( N `  T )  e.  RR* )
1811, 12nvge0 24197 . . 3  |-  ( ( W  e.  NrmCVec  /\  ( T `  ( 0vec `  U ) )  e.  Y )  ->  0  <_  ( ( normCV `  W
) `  ( T `  ( 0vec `  U
) ) ) )
193, 10, 18syl2anc 661 . 2  |-  ( ( U  e.  NrmCVec  /\  W  e.  NrmCVec  /\  T : X
--> Y )  ->  0  <_  ( ( normCV `  W
) `  ( T `  ( 0vec `  U
) ) ) )
2011, 12nmosetre 24299 . . . . . . 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 9528 . . . . . . 7  |-  RR  C_  RR*
2220, 21syl6ss 3466 . . . . . 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 2451 . . . . . . 7  |-  ( normCV `  U )  =  (
normCV
`  U )
244, 5, 23nmosetn0 24300 . . . . . 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 11388 . . . . . 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 477 . . . . 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 1183 . . . 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 1196 . . 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 24298 . . 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 4416 . 2  |-  ( ( U  e.  NrmCVec  /\  W  e.  NrmCVec  /\  T : X
--> Y )  ->  (
( normCV `  W ) `  ( T `  ( 0vec `  U ) ) )  <_  ( N `  T ) )
312, 15, 17, 19, 30xrletrd 11237 1  |-  ( ( U  e.  NrmCVec  /\  W  e.  NrmCVec  /\  T : X
--> Y )  ->  0  <_  ( N `  T
) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    /\ w3a 965    = wceq 1370    e. wcel 1758   {cab 2436   E.wrex 2796    C_ wss 3426   class class class wbr 4390   -->wf 5512   ` cfv 5516  (class class class)co 6190   supcsup 7791   RRcr 9382   0cc0 9383   1c1 9384   RR*cxr 9518    < clt 9519    <_ cle 9520   NrmCVeccnv 24097   BaseSetcba 24099   0veccn0v 24101   normCVcnmcv 24103   normOpOLDcnmoo 24276
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 1952  ax-ext 2430  ax-rep 4501  ax-sep 4511  ax-nul 4519  ax-pow 4568  ax-pr 4629  ax-un 6472  ax-cnex 9439  ax-resscn 9440  ax-1cn 9441  ax-icn 9442  ax-addcl 9443  ax-addrcl 9444  ax-mulcl 9445  ax-mulrcl 9446  ax-mulcom 9447  ax-addass 9448  ax-mulass 9449  ax-distr 9450  ax-i2m1 9451  ax-1ne0 9452  ax-1rid 9453  ax-rnegex 9454  ax-rrecex 9455  ax-cnre 9456  ax-pre-lttri 9457  ax-pre-lttrn 9458  ax-pre-ltadd 9459  ax-pre-mulgt0 9460  ax-pre-sup 9461
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 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ne 2646  df-nel 2647  df-ral 2800  df-rex 2801  df-reu 2802  df-rmo 2803  df-rab 2804  df-v 3070  df-sbc 3285  df-csb 3387  df-dif 3429  df-un 3431  df-in 3433  df-ss 3440  df-pss 3442  df-nul 3736  df-if 3890  df-pw 3960  df-sn 3976  df-pr 3978  df-tp 3980  df-op 3982  df-uni 4190  df-iun 4271  df-br 4391  df-opab 4449  df-mpt 4450  df-tr 4484  df-eprel 4730  df-id 4734  df-po 4739  df-so 4740  df-fr 4777  df-we 4779  df-ord 4820  df-on 4821  df-lim 4822  df-suc 4823  df-xp 4944  df-rel 4945  df-cnv 4946  df-co 4947  df-dm 4948  df-rn 4949  df-res 4950  df-ima 4951  df-iota 5479  df-fun 5518  df-fn 5519  df-f 5520  df-f1 5521  df-fo 5522  df-f1o 5523  df-fv 5524  df-riota 6151  df-ov 6193  df-oprab 6194  df-mpt2 6195  df-om 6577  df-1st 6677  df-2nd 6678  df-recs 6932  df-rdg 6966  df-er 7201  df-map 7316  df-en 7411  df-dom 7412  df-sdom 7413  df-sup 7792  df-pnf 9521  df-mnf 9522  df-xr 9523  df-ltxr 9524  df-le 9525  df-sub 9698  df-neg 9699  df-div 10095  df-nn 10424  df-2 10481  df-3 10482  df-n0 10681  df-z 10748  df-uz 10963  df-rp 11093  df-seq 11908  df-exp 11967  df-cj 12690  df-re 12691  df-im 12692  df-sqr 12826  df-abs 12827  df-grpo 23813  df-gid 23814  df-ginv 23815  df-ablo 23904  df-vc 24059  df-nv 24105  df-va 24108  df-ba 24109  df-sm 24110  df-0v 24111  df-nmcv 24113  df-nmoo 24280
This theorem is referenced by:  nmlnogt0  24332  htthlem  24454
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