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Theorem nmoo0 22297
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 normOp OLD 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 2438 . . . . 5  |-  ( BaseSet `  U )  =  (
BaseSet `  U )
2 eqid 2438 . . . . 5  |-  ( BaseSet `  W )  =  (
BaseSet `  W )
3 nmoo0.0 . . . . 5  |-  Z  =  ( U  0op  W
)
41, 2, 30oo 22295 . . . 4  |-  ( ( U  e.  NrmCVec  /\  W  e.  NrmCVec )  ->  Z : ( BaseSet `  U
) --> ( BaseSet `  W
) )
5 eqid 2438 . . . . 5  |-  ( normCV `  U )  =  (
normCV
`  U )
6 eqid 2438 . . . . 5  |-  ( normCV `  W )  =  (
normCV
`  W )
7 nmoo0.3 . . . . 5  |-  N  =  ( U normOp OLD W
)
81, 2, 5, 6, 7nmooval 22269 . . . 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 1281 . . 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 3822 . . . . 5  |-  { 0 }  =  { x  |  x  =  0 }
11 eqid 2438 . . . . . . . . . . 11  |-  ( 0vec `  U )  =  (
0vec `  U )
121, 11nvzcl 22120 . . . . . . . . . 10  |-  ( U  e.  NrmCVec  ->  ( 0vec `  U
)  e.  ( BaseSet `  U ) )
1311, 5nvz0 22162 . . . . . . . . . . 11  |-  ( U  e.  NrmCVec  ->  ( ( normCV `  U ) `  ( 0vec `  U ) )  =  0 )
14 0le1 9556 . . . . . . . . . . 11  |-  0  <_  1
1513, 14syl6eqbr 4252 . . . . . . . . . 10  |-  ( U  e.  NrmCVec  ->  ( ( normCV `  U ) `  ( 0vec `  U ) )  <_  1 )
16 fveq2 5731 . . . . . . . . . . . 12  |-  ( z  =  ( 0vec `  U
)  ->  ( ( normCV `  U ) `  z
)  =  ( (
normCV
`  U ) `  ( 0vec `  U )
) )
1716breq1d 4225 . . . . . . . . . . 11  |-  ( z  =  ( 0vec `  U
)  ->  ( (
( normCV `  U ) `  z )  <_  1  <->  ( ( normCV `  U ) `  ( 0vec `  U )
)  <_  1 ) )
1817rspcev 3054 . . . . . . . . . 10  |-  ( ( ( 0vec `  U
)  e.  ( BaseSet `  U )  /\  (
( normCV `  U ) `  ( 0vec `  U )
)  <_  1 )  ->  E. z  e.  (
BaseSet `  U ) ( ( normCV `  U ) `  z )  <_  1
)
1912, 15, 18syl2anc 644 . . . . . . . . 9  |-  ( U  e.  NrmCVec  ->  E. z  e.  (
BaseSet `  U ) ( ( normCV `  U ) `  z )  <_  1
)
2019biantrurd 496 . . . . . . . 8  |-  ( U  e.  NrmCVec  ->  ( x  =  0  <->  ( E. z  e.  ( BaseSet `  U )
( ( normCV `  U
) `  z )  <_  1  /\  x  =  0 ) ) )
2120adantr 453 . . . . . . 7  |-  ( ( U  e.  NrmCVec  /\  W  e.  NrmCVec )  ->  (
x  =  0  <->  ( E. z  e.  ( BaseSet
`  U ) ( ( normCV `  U ) `  z )  <_  1  /\  x  =  0
) ) )
22 eqid 2438 . . . . . . . . . . . . . . 15  |-  ( 0vec `  W )  =  (
0vec `  W )
231, 22, 30oval 22294 . . . . . . . . . . . . . 14  |-  ( ( U  e.  NrmCVec  /\  W  e.  NrmCVec  /\  z  e.  ( BaseSet `  U )
)  ->  ( Z `  z )  =  (
0vec `  W )
)
24233expa 1154 . . . . . . . . . . . . 13  |-  ( ( ( U  e.  NrmCVec  /\  W  e.  NrmCVec )  /\  z  e.  ( BaseSet `  U ) )  -> 
( Z `  z
)  =  ( 0vec `  W ) )
2524fveq2d 5735 . . . . . . . . . . . 12  |-  ( ( ( U  e.  NrmCVec  /\  W  e.  NrmCVec )  /\  z  e.  ( BaseSet `  U ) )  -> 
( ( normCV `  W
) `  ( Z `  z ) )  =  ( ( normCV `  W
) `  ( 0vec `  W ) ) )
2622, 6nvz0 22162 . . . . . . . . . . . . 13  |-  ( W  e.  NrmCVec  ->  ( ( normCV `  W ) `  ( 0vec `  W ) )  =  0 )
2726ad2antlr 709 . . . . . . . . . . . 12  |-  ( ( ( U  e.  NrmCVec  /\  W  e.  NrmCVec )  /\  z  e.  ( BaseSet `  U ) )  -> 
( ( normCV `  W
) `  ( 0vec `  W ) )  =  0 )
2825, 27eqtrd 2470 . . . . . . . . . . 11  |-  ( ( ( U  e.  NrmCVec  /\  W  e.  NrmCVec )  /\  z  e.  ( BaseSet `  U ) )  -> 
( ( normCV `  W
) `  ( Z `  z ) )  =  0 )
2928eqeq2d 2449 . . . . . . . . . 10  |-  ( ( ( U  e.  NrmCVec  /\  W  e.  NrmCVec )  /\  z  e.  ( BaseSet `  U ) )  -> 
( x  =  ( ( normCV `  W ) `  ( Z `  z ) )  <->  x  =  0
) )
3029anbi2d 686 . . . . . . . . 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 2724 . . . . . . . 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 2863 . . . . . . . 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 255 . . . . . . 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 246 . . . . . 6  |-  ( ( U  e.  NrmCVec  /\  W  e.  NrmCVec )  ->  (
x  =  0  <->  E. z  e.  ( BaseSet `  U ) ( ( ( normCV `  U ) `  z )  <_  1  /\  x  =  (
( normCV `  W ) `  ( Z `  z ) ) ) ) )
3534abbidv 2552 . . . . 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 2483 . . . 4  |-  ( ( U  e.  NrmCVec  /\  W  e.  NrmCVec )  ->  { x  |  E. z  e.  (
BaseSet `  U ) ( ( ( normCV `  U
) `  z )  <_  1  /\  x  =  ( ( normCV `  W
) `  ( Z `  z ) ) ) }  =  { 0 } )
3736supeq1d 7454 . . 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 2470 . 2  |-  ( ( U  e.  NrmCVec  /\  W  e.  NrmCVec )  ->  ( N `  Z )  =  sup ( { 0 } ,  RR* ,  <  ) )
39 xrltso 10739 . . 3  |-  <  Or  RR*
40 0xr 9136 . . 3  |-  0  e.  RR*
41 supsn 7477 . . 3  |-  ( (  <  Or  RR*  /\  0  e.  RR* )  ->  sup ( { 0 } ,  RR* ,  <  )  =  0 )
4239, 40, 41mp2an 655 . 2  |-  sup ( { 0 } ,  RR* ,  <  )  =  0
4338, 42syl6eq 2486 1  |-  ( ( U  e.  NrmCVec  /\  W  e.  NrmCVec )  ->  ( N `  Z )  =  0 )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 178    /\ wa 360    = wceq 1653    e. wcel 1726   {cab 2424   E.wrex 2708   {csn 3816   class class class wbr 4215    Or wor 4505   -->wf 5453   ` cfv 5457  (class class class)co 6084   supcsup 7448   0cc0 8995   1c1 8996   RR*cxr 9124    < clt 9125    <_ cle 9126   NrmCVeccnv 22068   BaseSetcba 22070   0veccn0v 22072   normCVcnmcv 22074   normOp OLDcnmoo 22247    0op c0o 22249
This theorem is referenced by:  0blo  22298  nmlno0lem  22299
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-rep 4323  ax-sep 4333  ax-nul 4341  ax-pow 4380  ax-pr 4406  ax-un 4704  ax-cnex 9051  ax-resscn 9052  ax-1cn 9053  ax-icn 9054  ax-addcl 9055  ax-addrcl 9056  ax-mulcl 9057  ax-mulrcl 9058  ax-mulcom 9059  ax-addass 9060  ax-mulass 9061  ax-distr 9062  ax-i2m1 9063  ax-1ne0 9064  ax-1rid 9065  ax-rnegex 9066  ax-rrecex 9067  ax-cnre 9068  ax-pre-lttri 9069  ax-pre-lttrn 9070  ax-pre-ltadd 9071  ax-pre-mulgt0 9072  ax-pre-sup 9073
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-nel 2604  df-ral 2712  df-rex 2713  df-reu 2714  df-rmo 2715  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-pss 3338  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-tp 3824  df-op 3825  df-uni 4018  df-iun 4097  df-br 4216  df-opab 4270  df-mpt 4271  df-tr 4306  df-eprel 4497  df-id 4501  df-po 4506  df-so 4507  df-fr 4544  df-we 4546  df-ord 4587  df-on 4588  df-lim 4589  df-suc 4590  df-om 4849  df-xp 4887  df-rel 4888  df-cnv 4889  df-co 4890  df-dm 4891  df-rn 4892  df-res 4893  df-ima 4894  df-iota 5421  df-fun 5459  df-fn 5460  df-f 5461  df-f1 5462  df-fo 5463  df-f1o 5464  df-fv 5465  df-ov 6087  df-oprab 6088  df-mpt2 6089  df-1st 6352  df-2nd 6353  df-riota 6552  df-recs 6636  df-rdg 6671  df-er 6908  df-map 7023  df-en 7113  df-dom 7114  df-sdom 7115  df-sup 7449  df-pnf 9127  df-mnf 9128  df-xr 9129  df-ltxr 9130  df-le 9131  df-sub 9298  df-neg 9299  df-div 9683  df-nn 10006  df-2 10063  df-3 10064  df-n0 10227  df-z 10288  df-uz 10494  df-rp 10618  df-seq 11329  df-exp 11388  df-cj 11909  df-re 11910  df-im 11911  df-sqr 12045  df-abs 12046  df-grpo 21784  df-gid 21785  df-ginv 21786  df-ablo 21875  df-vc 22030  df-nv 22076  df-va 22079  df-ba 22080  df-sm 22081  df-0v 22082  df-nmcv 22084  df-nmoo 22251  df-0o 22253
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