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Theorem nmoo0 24191
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 2443 . . . . 5  |-  ( BaseSet `  U )  =  (
BaseSet `  U )
2 eqid 2443 . . . . 5  |-  ( BaseSet `  W )  =  (
BaseSet `  W )
3 nmoo0.0 . . . . 5  |-  Z  =  ( U  0op  W
)
41, 2, 30oo 24189 . . . 4  |-  ( ( U  e.  NrmCVec  /\  W  e.  NrmCVec )  ->  Z : ( BaseSet `  U
) --> ( BaseSet `  W
) )
5 eqid 2443 . . . . 5  |-  ( normCV `  U )  =  (
normCV
`  U )
6 eqid 2443 . . . . 5  |-  ( normCV `  W )  =  (
normCV
`  W )
7 nmoo0.3 . . . . 5  |-  N  =  ( U normOpOLD W
)
81, 2, 5, 6, 7nmooval 24163 . . . 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 1315 . . 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 3878 . . . . 5  |-  { 0 }  =  { x  |  x  =  0 }
11 eqid 2443 . . . . . . . . . . 11  |-  ( 0vec `  U )  =  (
0vec `  U )
121, 11nvzcl 24014 . . . . . . . . . 10  |-  ( U  e.  NrmCVec  ->  ( 0vec `  U
)  e.  ( BaseSet `  U ) )
1311, 5nvz0 24056 . . . . . . . . . . 11  |-  ( U  e.  NrmCVec  ->  ( ( normCV `  U ) `  ( 0vec `  U ) )  =  0 )
14 0le1 9863 . . . . . . . . . . 11  |-  0  <_  1
1513, 14syl6eqbr 4329 . . . . . . . . . 10  |-  ( U  e.  NrmCVec  ->  ( ( normCV `  U ) `  ( 0vec `  U ) )  <_  1 )
16 fveq2 5691 . . . . . . . . . . . 12  |-  ( z  =  ( 0vec `  U
)  ->  ( ( normCV `  U ) `  z
)  =  ( (
normCV
`  U ) `  ( 0vec `  U )
) )
1716breq1d 4302 . . . . . . . . . . 11  |-  ( z  =  ( 0vec `  U
)  ->  ( (
( normCV `  U ) `  z )  <_  1  <->  ( ( normCV `  U ) `  ( 0vec `  U )
)  <_  1 ) )
1817rspcev 3073 . . . . . . . . . 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 2443 . . . . . . . . . . . . . . 15  |-  ( 0vec `  W )  =  (
0vec `  W )
231, 22, 30oval 24188 . . . . . . . . . . . . . 14  |-  ( ( U  e.  NrmCVec  /\  W  e.  NrmCVec  /\  z  e.  ( BaseSet `  U )
)  ->  ( Z `  z )  =  (
0vec `  W )
)
24233expa 1187 . . . . . . . . . . . . 13  |-  ( ( ( U  e.  NrmCVec  /\  W  e.  NrmCVec )  /\  z  e.  ( BaseSet `  U ) )  -> 
( Z `  z
)  =  ( 0vec `  W ) )
2524fveq2d 5695 . . . . . . . . . . . 12  |-  ( ( ( U  e.  NrmCVec  /\  W  e.  NrmCVec )  /\  z  e.  ( BaseSet `  U ) )  -> 
( ( normCV `  W
) `  ( Z `  z ) )  =  ( ( normCV `  W
) `  ( 0vec `  W ) ) )
2622, 6nvz0 24056 . . . . . . . . . . . . 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 2475 . . . . . . . . . . 11  |-  ( ( ( U  e.  NrmCVec  /\  W  e.  NrmCVec )  /\  z  e.  ( BaseSet `  U ) )  -> 
( ( normCV `  W
) `  ( Z `  z ) )  =  0 )
2928eqeq2d 2454 . . . . . . . . . 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 2732 . . . . . . . 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 2873 . . . . . . . 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 2557 . . . . 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 2488 . . . 4  |-  ( ( U  e.  NrmCVec  /\  W  e.  NrmCVec )  ->  { x  |  E. z  e.  (
BaseSet `  U ) ( ( ( normCV `  U
) `  z )  <_  1  /\  x  =  ( ( normCV `  W
) `  ( Z `  z ) ) ) }  =  { 0 } )
3736supeq1d 7696 . . 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 2475 . 2  |-  ( ( U  e.  NrmCVec  /\  W  e.  NrmCVec )  ->  ( N `  Z )  =  sup ( { 0 } ,  RR* ,  <  ) )
39 xrltso 11118 . . 3  |-  <  Or  RR*
40 0xr 9430 . . 3  |-  0  e.  RR*
41 supsn 7719 . . 3  |-  ( (  <  Or  RR*  /\  0  e.  RR* )  ->  sup ( { 0 } ,  RR* ,  <  )  =  0 )
4239, 40, 41mp2an 672 . 2  |-  sup ( { 0 } ,  RR* ,  <  )  =  0
4338, 42syl6eq 2491 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 1369    e. wcel 1756   {cab 2429   E.wrex 2716   {csn 3877   class class class wbr 4292    Or wor 4640   -->wf 5414   ` cfv 5418  (class class class)co 6091   supcsup 7690   0cc0 9282   1c1 9283   RR*cxr 9417    < clt 9418    <_ cle 9419   NrmCVeccnv 23962   BaseSetcba 23964   0veccn0v 23966   normCVcnmcv 23968   normOpOLDcnmoo 24141    0op c0o 24143
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4403  ax-sep 4413  ax-nul 4421  ax-pow 4470  ax-pr 4531  ax-un 6372  ax-cnex 9338  ax-resscn 9339  ax-1cn 9340  ax-icn 9341  ax-addcl 9342  ax-addrcl 9343  ax-mulcl 9344  ax-mulrcl 9345  ax-mulcom 9346  ax-addass 9347  ax-mulass 9348  ax-distr 9349  ax-i2m1 9350  ax-1ne0 9351  ax-1rid 9352  ax-rnegex 9353  ax-rrecex 9354  ax-cnre 9355  ax-pre-lttri 9356  ax-pre-lttrn 9357  ax-pre-ltadd 9358  ax-pre-mulgt0 9359  ax-pre-sup 9360
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2568  df-ne 2608  df-nel 2609  df-ral 2720  df-rex 2721  df-reu 2722  df-rmo 2723  df-rab 2724  df-v 2974  df-sbc 3187  df-csb 3289  df-dif 3331  df-un 3333  df-in 3335  df-ss 3342  df-pss 3344  df-nul 3638  df-if 3792  df-pw 3862  df-sn 3878  df-pr 3880  df-tp 3882  df-op 3884  df-uni 4092  df-iun 4173  df-br 4293  df-opab 4351  df-mpt 4352  df-tr 4386  df-eprel 4632  df-id 4636  df-po 4641  df-so 4642  df-fr 4679  df-we 4681  df-ord 4722  df-on 4723  df-lim 4724  df-suc 4725  df-xp 4846  df-rel 4847  df-cnv 4848  df-co 4849  df-dm 4850  df-rn 4851  df-res 4852  df-ima 4853  df-iota 5381  df-fun 5420  df-fn 5421  df-f 5422  df-f1 5423  df-fo 5424  df-f1o 5425  df-fv 5426  df-riota 6052  df-ov 6094  df-oprab 6095  df-mpt2 6096  df-om 6477  df-1st 6577  df-2nd 6578  df-recs 6832  df-rdg 6866  df-er 7101  df-map 7216  df-en 7311  df-dom 7312  df-sdom 7313  df-sup 7691  df-pnf 9420  df-mnf 9421  df-xr 9422  df-ltxr 9423  df-le 9424  df-sub 9597  df-neg 9598  df-div 9994  df-nn 10323  df-2 10380  df-3 10381  df-n0 10580  df-z 10647  df-uz 10862  df-rp 10992  df-seq 11807  df-exp 11866  df-cj 12588  df-re 12589  df-im 12590  df-sqr 12724  df-abs 12725  df-grpo 23678  df-gid 23679  df-ginv 23680  df-ablo 23769  df-vc 23924  df-nv 23970  df-va 23973  df-ba 23974  df-sm 23975  df-0v 23976  df-nmcv 23978  df-nmoo 24145  df-0o 24147
This theorem is referenced by:  0blo  24192  nmlno0lem  24193
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