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Theorem nmoo0 25537
Description: The operator norm of the zero operator. (Contributed by NM, 27-Nov-2007.) (New usage is discouraged.)
Hypotheses
Ref Expression
nmoo0.3  |-  N  =  ( U normOpOLD W
)
nmoo0.0  |-  Z  =  ( U  0op  W
)
Assertion
Ref Expression
nmoo0  |-  ( ( U  e.  NrmCVec  /\  W  e.  NrmCVec )  ->  ( N `  Z )  =  0 )

Proof of Theorem nmoo0
Dummy variables  x  z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqid 2467 . . . . 5  |-  ( BaseSet `  U )  =  (
BaseSet `  U )
2 eqid 2467 . . . . 5  |-  ( BaseSet `  W )  =  (
BaseSet `  W )
3 nmoo0.0 . . . . 5  |-  Z  =  ( U  0op  W
)
41, 2, 30oo 25535 . . . 4  |-  ( ( U  e.  NrmCVec  /\  W  e.  NrmCVec )  ->  Z : ( BaseSet `  U
) --> ( BaseSet `  W
) )
5 eqid 2467 . . . . 5  |-  ( normCV `  U )  =  (
normCV
`  U )
6 eqid 2467 . . . . 5  |-  ( normCV `  W )  =  (
normCV
`  W )
7 nmoo0.3 . . . . 5  |-  N  =  ( U normOpOLD W
)
81, 2, 5, 6, 7nmooval 25509 . . . 4  |-  ( ( U  e.  NrmCVec  /\  W  e.  NrmCVec  /\  Z :
( BaseSet `  U ) --> ( BaseSet `  W )
)  ->  ( N `  Z )  =  sup ( { x  |  E. z  e.  ( BaseSet `  U ) ( ( ( normCV `  U ) `  z )  <_  1  /\  x  =  (
( normCV `  W ) `  ( Z `  z ) ) ) } ,  RR* ,  <  ) )
94, 8mpd3an3 1325 . . 3  |-  ( ( U  e.  NrmCVec  /\  W  e.  NrmCVec )  ->  ( N `  Z )  =  sup ( { x  |  E. z  e.  (
BaseSet `  U ) ( ( ( normCV `  U
) `  z )  <_  1  /\  x  =  ( ( normCV `  W
) `  ( Z `  z ) ) ) } ,  RR* ,  <  ) )
10 df-sn 4034 . . . . 5  |-  { 0 }  =  { x  |  x  =  0 }
11 eqid 2467 . . . . . . . . . . 11  |-  ( 0vec `  U )  =  (
0vec `  U )
121, 11nvzcl 25360 . . . . . . . . . 10  |-  ( U  e.  NrmCVec  ->  ( 0vec `  U
)  e.  ( BaseSet `  U ) )
1311, 5nvz0 25402 . . . . . . . . . . 11  |-  ( U  e.  NrmCVec  ->  ( ( normCV `  U ) `  ( 0vec `  U ) )  =  0 )
14 0le1 10088 . . . . . . . . . . 11  |-  0  <_  1
1513, 14syl6eqbr 4490 . . . . . . . . . 10  |-  ( U  e.  NrmCVec  ->  ( ( normCV `  U ) `  ( 0vec `  U ) )  <_  1 )
16 fveq2 5872 . . . . . . . . . . . 12  |-  ( z  =  ( 0vec `  U
)  ->  ( ( normCV `  U ) `  z
)  =  ( (
normCV
`  U ) `  ( 0vec `  U )
) )
1716breq1d 4463 . . . . . . . . . . 11  |-  ( z  =  ( 0vec `  U
)  ->  ( (
( normCV `  U ) `  z )  <_  1  <->  ( ( normCV `  U ) `  ( 0vec `  U )
)  <_  1 ) )
1817rspcev 3219 . . . . . . . . . 10  |-  ( ( ( 0vec `  U
)  e.  ( BaseSet `  U )  /\  (
( normCV `  U ) `  ( 0vec `  U )
)  <_  1 )  ->  E. z  e.  (
BaseSet `  U ) ( ( normCV `  U ) `  z )  <_  1
)
1912, 15, 18syl2anc 661 . . . . . . . . 9  |-  ( U  e.  NrmCVec  ->  E. z  e.  (
BaseSet `  U ) ( ( normCV `  U ) `  z )  <_  1
)
2019biantrurd 508 . . . . . . . 8  |-  ( U  e.  NrmCVec  ->  ( x  =  0  <->  ( E. z  e.  ( BaseSet `  U )
( ( normCV `  U
) `  z )  <_  1  /\  x  =  0 ) ) )
2120adantr 465 . . . . . . 7  |-  ( ( U  e.  NrmCVec  /\  W  e.  NrmCVec )  ->  (
x  =  0  <->  ( E. z  e.  ( BaseSet
`  U ) ( ( normCV `  U ) `  z )  <_  1  /\  x  =  0
) ) )
22 eqid 2467 . . . . . . . . . . . . . . 15  |-  ( 0vec `  W )  =  (
0vec `  W )
231, 22, 30oval 25534 . . . . . . . . . . . . . 14  |-  ( ( U  e.  NrmCVec  /\  W  e.  NrmCVec  /\  z  e.  ( BaseSet `  U )
)  ->  ( Z `  z )  =  (
0vec `  W )
)
24233expa 1196 . . . . . . . . . . . . 13  |-  ( ( ( U  e.  NrmCVec  /\  W  e.  NrmCVec )  /\  z  e.  ( BaseSet `  U ) )  -> 
( Z `  z
)  =  ( 0vec `  W ) )
2524fveq2d 5876 . . . . . . . . . . . 12  |-  ( ( ( U  e.  NrmCVec  /\  W  e.  NrmCVec )  /\  z  e.  ( BaseSet `  U ) )  -> 
( ( normCV `  W
) `  ( Z `  z ) )  =  ( ( normCV `  W
) `  ( 0vec `  W ) ) )
2622, 6nvz0 25402 . . . . . . . . . . . . 13  |-  ( W  e.  NrmCVec  ->  ( ( normCV `  W ) `  ( 0vec `  W ) )  =  0 )
2726ad2antlr 726 . . . . . . . . . . . 12  |-  ( ( ( U  e.  NrmCVec  /\  W  e.  NrmCVec )  /\  z  e.  ( BaseSet `  U ) )  -> 
( ( normCV `  W
) `  ( 0vec `  W ) )  =  0 )
2825, 27eqtrd 2508 . . . . . . . . . . 11  |-  ( ( ( U  e.  NrmCVec  /\  W  e.  NrmCVec )  /\  z  e.  ( BaseSet `  U ) )  -> 
( ( normCV `  W
) `  ( Z `  z ) )  =  0 )
2928eqeq2d 2481 . . . . . . . . . 10  |-  ( ( ( U  e.  NrmCVec  /\  W  e.  NrmCVec )  /\  z  e.  ( BaseSet `  U ) )  -> 
( x  =  ( ( normCV `  W ) `  ( Z `  z ) )  <->  x  =  0
) )
3029anbi2d 703 . . . . . . . . 9  |-  ( ( ( U  e.  NrmCVec  /\  W  e.  NrmCVec )  /\  z  e.  ( BaseSet `  U ) )  -> 
( ( ( (
normCV
`  U ) `  z )  <_  1  /\  x  =  (
( normCV `  W ) `  ( Z `  z ) ) )  <->  ( (
( normCV `  U ) `  z )  <_  1  /\  x  =  0
) ) )
3130rexbidva 2975 . . . . . . . 8  |-  ( ( U  e.  NrmCVec  /\  W  e.  NrmCVec )  ->  ( E. z  e.  ( BaseSet
`  U ) ( ( ( normCV `  U
) `  z )  <_  1  /\  x  =  ( ( normCV `  W
) `  ( Z `  z ) ) )  <->  E. z  e.  ( BaseSet
`  U ) ( ( ( normCV `  U
) `  z )  <_  1  /\  x  =  0 ) ) )
32 r19.41v 3019 . . . . . . . 8  |-  ( E. z  e.  ( BaseSet `  U ) ( ( ( normCV `  U ) `  z )  <_  1  /\  x  =  0
)  <->  ( E. z  e.  ( BaseSet `  U )
( ( normCV `  U
) `  z )  <_  1  /\  x  =  0 ) )
3331, 32syl6rbb 262 . . . . . . 7  |-  ( ( U  e.  NrmCVec  /\  W  e.  NrmCVec )  ->  (
( E. z  e.  ( BaseSet `  U )
( ( normCV `  U
) `  z )  <_  1  /\  x  =  0 )  <->  E. z  e.  ( BaseSet `  U )
( ( ( normCV `  U ) `  z
)  <_  1  /\  x  =  ( ( normCV `  W ) `  ( Z `  z )
) ) ) )
3421, 33bitrd 253 . . . . . 6  |-  ( ( U  e.  NrmCVec  /\  W  e.  NrmCVec )  ->  (
x  =  0  <->  E. z  e.  ( BaseSet `  U ) ( ( ( normCV `  U ) `  z )  <_  1  /\  x  =  (
( normCV `  W ) `  ( Z `  z ) ) ) ) )
3534abbidv 2603 . . . . 5  |-  ( ( U  e.  NrmCVec  /\  W  e.  NrmCVec )  ->  { x  |  x  =  0 }  =  { x  |  E. z  e.  (
BaseSet `  U ) ( ( ( normCV `  U
) `  z )  <_  1  /\  x  =  ( ( normCV `  W
) `  ( Z `  z ) ) ) } )
3610, 35syl5req 2521 . . . 4  |-  ( ( U  e.  NrmCVec  /\  W  e.  NrmCVec )  ->  { x  |  E. z  e.  (
BaseSet `  U ) ( ( ( normCV `  U
) `  z )  <_  1  /\  x  =  ( ( normCV `  W
) `  ( Z `  z ) ) ) }  =  { 0 } )
3736supeq1d 7918 . . 3  |-  ( ( U  e.  NrmCVec  /\  W  e.  NrmCVec )  ->  sup ( { x  |  E. z  e.  ( BaseSet `  U ) ( ( ( normCV `  U ) `  z )  <_  1  /\  x  =  (
( normCV `  W ) `  ( Z `  z ) ) ) } ,  RR* ,  <  )  =  sup ( { 0 } ,  RR* ,  <  ) )
389, 37eqtrd 2508 . 2  |-  ( ( U  e.  NrmCVec  /\  W  e.  NrmCVec )  ->  ( N `  Z )  =  sup ( { 0 } ,  RR* ,  <  ) )
39 xrltso 11359 . . 3  |-  <  Or  RR*
40 0xr 9652 . . 3  |-  0  e.  RR*
41 supsn 7942 . . 3  |-  ( (  <  Or  RR*  /\  0  e.  RR* )  ->  sup ( { 0 } ,  RR* ,  <  )  =  0 )
4239, 40, 41mp2an 672 . 2  |-  sup ( { 0 } ,  RR* ,  <  )  =  0
4338, 42syl6eq 2524 1  |-  ( ( U  e.  NrmCVec  /\  W  e.  NrmCVec )  ->  ( N `  Z )  =  0 )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1379    e. wcel 1767   {cab 2452   E.wrex 2818   {csn 4033   class class class wbr 4453    Or wor 4805   -->wf 5590   ` cfv 5594  (class class class)co 6295   supcsup 7912   0cc0 9504   1c1 9505   RR*cxr 9639    < clt 9640    <_ cle 9641   NrmCVeccnv 25308   BaseSetcba 25310   0veccn0v 25312   normCVcnmcv 25314   normOpOLDcnmoo 25487    0op c0o 25489
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4564  ax-sep 4574  ax-nul 4582  ax-pow 4631  ax-pr 4692  ax-un 6587  ax-cnex 9560  ax-resscn 9561  ax-1cn 9562  ax-icn 9563  ax-addcl 9564  ax-addrcl 9565  ax-mulcl 9566  ax-mulrcl 9567  ax-mulcom 9568  ax-addass 9569  ax-mulass 9570  ax-distr 9571  ax-i2m1 9572  ax-1ne0 9573  ax-1rid 9574  ax-rnegex 9575  ax-rrecex 9576  ax-cnre 9577  ax-pre-lttri 9578  ax-pre-lttrn 9579  ax-pre-ltadd 9580  ax-pre-mulgt0 9581  ax-pre-sup 9582
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2822  df-rex 2823  df-reu 2824  df-rmo 2825  df-rab 2826  df-v 3120  df-sbc 3337  df-csb 3441  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-pss 3497  df-nul 3791  df-if 3946  df-pw 4018  df-sn 4034  df-pr 4036  df-tp 4038  df-op 4040  df-uni 4252  df-iun 4333  df-br 4454  df-opab 4512  df-mpt 4513  df-tr 4547  df-eprel 4797  df-id 4801  df-po 4806  df-so 4807  df-fr 4844  df-we 4846  df-ord 4887  df-on 4888  df-lim 4889  df-suc 4890  df-xp 5011  df-rel 5012  df-cnv 5013  df-co 5014  df-dm 5015  df-rn 5016  df-res 5017  df-ima 5018  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-riota 6256  df-ov 6298  df-oprab 6299  df-mpt2 6300  df-om 6696  df-1st 6795  df-2nd 6796  df-recs 7054  df-rdg 7088  df-er 7323  df-map 7434  df-en 7529  df-dom 7530  df-sdom 7531  df-sup 7913  df-pnf 9642  df-mnf 9643  df-xr 9644  df-ltxr 9645  df-le 9646  df-sub 9819  df-neg 9820  df-div 10219  df-nn 10549  df-2 10606  df-3 10607  df-n0 10808  df-z 10877  df-uz 11095  df-rp 11233  df-seq 12088  df-exp 12147  df-cj 12911  df-re 12912  df-im 12913  df-sqrt 13047  df-abs 13048  df-grpo 25024  df-gid 25025  df-ginv 25026  df-ablo 25115  df-vc 25270  df-nv 25316  df-va 25319  df-ba 25320  df-sm 25321  df-0v 25322  df-nmcv 25324  df-nmoo 25491  df-0o 25493
This theorem is referenced by:  0blo  25538  nmlno0lem  25539
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