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Theorem nmoo0 21483
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 2358 . . . . 5  |-  ( BaseSet `  U )  =  (
BaseSet `  U )
2 eqid 2358 . . . . 5  |-  ( BaseSet `  W )  =  (
BaseSet `  W )
3 nmoo0.0 . . . . 5  |-  Z  =  ( U  0op  W
)
41, 2, 30oo 21481 . . . 4  |-  ( ( U  e.  NrmCVec  /\  W  e.  NrmCVec )  ->  Z : ( BaseSet `  U
) --> ( BaseSet `  W
) )
5 eqid 2358 . . . . 5  |-  ( normCV `  U )  =  (
normCV
`  U )
6 eqid 2358 . . . . 5  |-  ( normCV `  W )  =  (
normCV
`  W )
7 nmoo0.3 . . . . 5  |-  N  =  ( U normOp OLD W
)
81, 2, 5, 6, 7nmooval 21455 . . . 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 1278 . . 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 3722 . . . . 5  |-  { 0 }  =  { x  |  x  =  0 }
11 eqid 2358 . . . . . . . . . . 11  |-  ( 0vec `  U )  =  (
0vec `  U )
121, 11nvzcl 21306 . . . . . . . . . 10  |-  ( U  e.  NrmCVec  ->  ( 0vec `  U
)  e.  ( BaseSet `  U ) )
1311, 5nvz0 21348 . . . . . . . . . . 11  |-  ( U  e.  NrmCVec  ->  ( ( normCV `  U ) `  ( 0vec `  U ) )  =  0 )
14 0le1 9387 . . . . . . . . . . 11  |-  0  <_  1
1513, 14syl6eqbr 4141 . . . . . . . . . 10  |-  ( U  e.  NrmCVec  ->  ( ( normCV `  U ) `  ( 0vec `  U ) )  <_  1 )
16 fveq2 5608 . . . . . . . . . . . 12  |-  ( z  =  ( 0vec `  U
)  ->  ( ( normCV `  U ) `  z
)  =  ( (
normCV
`  U ) `  ( 0vec `  U )
) )
1716breq1d 4114 . . . . . . . . . . 11  |-  ( z  =  ( 0vec `  U
)  ->  ( (
( normCV `  U ) `  z )  <_  1  <->  ( ( normCV `  U ) `  ( 0vec `  U )
)  <_  1 ) )
1817rspcev 2960 . . . . . . . . . 10  |-  ( ( ( 0vec `  U
)  e.  ( BaseSet `  U )  /\  (
( normCV `  U ) `  ( 0vec `  U )
)  <_  1 )  ->  E. z  e.  (
BaseSet `  U ) ( ( normCV `  U ) `  z )  <_  1
)
1912, 15, 18syl2anc 642 . . . . . . . . 9  |-  ( U  e.  NrmCVec  ->  E. z  e.  (
BaseSet `  U ) ( ( normCV `  U ) `  z )  <_  1
)
2019biantrurd 494 . . . . . . . 8  |-  ( U  e.  NrmCVec  ->  ( x  =  0  <->  ( E. z  e.  ( BaseSet `  U )
( ( normCV `  U
) `  z )  <_  1  /\  x  =  0 ) ) )
2120adantr 451 . . . . . . 7  |-  ( ( U  e.  NrmCVec  /\  W  e.  NrmCVec )  ->  (
x  =  0  <->  ( E. z  e.  ( BaseSet
`  U ) ( ( normCV `  U ) `  z )  <_  1  /\  x  =  0
) ) )
22 eqid 2358 . . . . . . . . . . . . . . 15  |-  ( 0vec `  W )  =  (
0vec `  W )
231, 22, 30oval 21480 . . . . . . . . . . . . . 14  |-  ( ( U  e.  NrmCVec  /\  W  e.  NrmCVec  /\  z  e.  ( BaseSet `  U )
)  ->  ( Z `  z )  =  (
0vec `  W )
)
24233expa 1151 . . . . . . . . . . . . 13  |-  ( ( ( U  e.  NrmCVec  /\  W  e.  NrmCVec )  /\  z  e.  ( BaseSet `  U ) )  -> 
( Z `  z
)  =  ( 0vec `  W ) )
2524fveq2d 5612 . . . . . . . . . . . 12  |-  ( ( ( U  e.  NrmCVec  /\  W  e.  NrmCVec )  /\  z  e.  ( BaseSet `  U ) )  -> 
( ( normCV `  W
) `  ( Z `  z ) )  =  ( ( normCV `  W
) `  ( 0vec `  W ) ) )
2622, 6nvz0 21348 . . . . . . . . . . . . 13  |-  ( W  e.  NrmCVec  ->  ( ( normCV `  W ) `  ( 0vec `  W ) )  =  0 )
2726ad2antlr 707 . . . . . . . . . . . 12  |-  ( ( ( U  e.  NrmCVec  /\  W  e.  NrmCVec )  /\  z  e.  ( BaseSet `  U ) )  -> 
( ( normCV `  W
) `  ( 0vec `  W ) )  =  0 )
2825, 27eqtrd 2390 . . . . . . . . . . 11  |-  ( ( ( U  e.  NrmCVec  /\  W  e.  NrmCVec )  /\  z  e.  ( BaseSet `  U ) )  -> 
( ( normCV `  W
) `  ( Z `  z ) )  =  0 )
2928eqeq2d 2369 . . . . . . . . . 10  |-  ( ( ( U  e.  NrmCVec  /\  W  e.  NrmCVec )  /\  z  e.  ( BaseSet `  U ) )  -> 
( x  =  ( ( normCV `  W ) `  ( Z `  z ) )  <->  x  =  0
) )
3029anbi2d 684 . . . . . . . . 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 2636 . . . . . . . 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 2769 . . . . . . . 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 253 . . . . . . 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 244 . . . . . 6  |-  ( ( U  e.  NrmCVec  /\  W  e.  NrmCVec )  ->  (
x  =  0  <->  E. z  e.  ( BaseSet `  U ) ( ( ( normCV `  U ) `  z )  <_  1  /\  x  =  (
( normCV `  W ) `  ( Z `  z ) ) ) ) )
3534abbidv 2472 . . . . 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 2403 . . . 4  |-  ( ( U  e.  NrmCVec  /\  W  e.  NrmCVec )  ->  { x  |  E. z  e.  (
BaseSet `  U ) ( ( ( normCV `  U
) `  z )  <_  1  /\  x  =  ( ( normCV `  W
) `  ( Z `  z ) ) ) }  =  { 0 } )
3736supeq1d 7289 . . 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 2390 . 2  |-  ( ( U  e.  NrmCVec  /\  W  e.  NrmCVec )  ->  ( N `  Z )  =  sup ( { 0 } ,  RR* ,  <  ) )
39 xrltso 10567 . . 3  |-  <  Or  RR*
40 0xr 8968 . . 3  |-  0  e.  RR*
41 supsn 7310 . . 3  |-  ( (  <  Or  RR*  /\  0  e.  RR* )  ->  sup ( { 0 } ,  RR* ,  <  )  =  0 )
4239, 40, 41mp2an 653 . 2  |-  sup ( { 0 } ,  RR* ,  <  )  =  0
4338, 42syl6eq 2406 1  |-  ( ( U  e.  NrmCVec  /\  W  e.  NrmCVec )  ->  ( N `  Z )  =  0 )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    = wceq 1642    e. wcel 1710   {cab 2344   E.wrex 2620   {csn 3716   class class class wbr 4104    Or wor 4395   -->wf 5333   ` cfv 5337  (class class class)co 5945   supcsup 7283   0cc0 8827   1c1 8828   RR*cxr 8956    < clt 8957    <_ cle 8958   NrmCVeccnv 21254   BaseSetcba 21256   0veccn0v 21258   normCVcnmcv 21260   normOp OLDcnmoo 21433    0op c0o 21435
This theorem is referenced by:  0blo  21484  nmlno0lem  21485
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-13 1712  ax-14 1714  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1930  ax-ext 2339  ax-rep 4212  ax-sep 4222  ax-nul 4230  ax-pow 4269  ax-pr 4295  ax-un 4594  ax-cnex 8883  ax-resscn 8884  ax-1cn 8885  ax-icn 8886  ax-addcl 8887  ax-addrcl 8888  ax-mulcl 8889  ax-mulrcl 8890  ax-mulcom 8891  ax-addass 8892  ax-mulass 8893  ax-distr 8894  ax-i2m1 8895  ax-1ne0 8896  ax-1rid 8897  ax-rnegex 8898  ax-rrecex 8899  ax-cnre 8900  ax-pre-lttri 8901  ax-pre-lttrn 8902  ax-pre-ltadd 8903  ax-pre-mulgt0 8904  ax-pre-sup 8905
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2213  df-mo 2214  df-clab 2345  df-cleq 2351  df-clel 2354  df-nfc 2483  df-ne 2523  df-nel 2524  df-ral 2624  df-rex 2625  df-reu 2626  df-rmo 2627  df-rab 2628  df-v 2866  df-sbc 3068  df-csb 3158  df-dif 3231  df-un 3233  df-in 3235  df-ss 3242  df-pss 3244  df-nul 3532  df-if 3642  df-pw 3703  df-sn 3722  df-pr 3723  df-tp 3724  df-op 3725  df-uni 3909  df-iun 3988  df-br 4105  df-opab 4159  df-mpt 4160  df-tr 4195  df-eprel 4387  df-id 4391  df-po 4396  df-so 4397  df-fr 4434  df-we 4436  df-ord 4477  df-on 4478  df-lim 4479  df-suc 4480  df-om 4739  df-xp 4777  df-rel 4778  df-cnv 4779  df-co 4780  df-dm 4781  df-rn 4782  df-res 4783  df-ima 4784  df-iota 5301  df-fun 5339  df-fn 5340  df-f 5341  df-f1 5342  df-fo 5343  df-f1o 5344  df-fv 5345  df-ov 5948  df-oprab 5949  df-mpt2 5950  df-1st 6209  df-2nd 6210  df-riota 6391  df-recs 6475  df-rdg 6510  df-er 6747  df-map 6862  df-en 6952  df-dom 6953  df-sdom 6954  df-sup 7284  df-pnf 8959  df-mnf 8960  df-xr 8961  df-ltxr 8962  df-le 8963  df-sub 9129  df-neg 9130  df-div 9514  df-nn 9837  df-2 9894  df-3 9895  df-n0 10058  df-z 10117  df-uz 10323  df-rp 10447  df-seq 11139  df-exp 11198  df-cj 11680  df-re 11681  df-im 11682  df-sqr 11816  df-abs 11817  df-grpo 20970  df-gid 20971  df-ginv 20972  df-ablo 21061  df-vc 21216  df-nv 21262  df-va 21265  df-ba 21266  df-sm 21267  df-0v 21268  df-nmcv 21270  df-nmoo 21437  df-0o 21439
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