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Theorem nmoo0 21369
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 2283 . . . . 5  |-  ( BaseSet `  U )  =  (
BaseSet `  U )
2 eqid 2283 . . . . 5  |-  ( BaseSet `  W )  =  (
BaseSet `  W )
3 nmoo0.0 . . . . 5  |-  Z  =  ( U  0op  W
)
41, 2, 30oo 21367 . . . 4  |-  ( ( U  e.  NrmCVec  /\  W  e.  NrmCVec )  ->  Z : ( BaseSet `  U
) --> ( BaseSet `  W
) )
5 eqid 2283 . . . . 5  |-  ( normCV `  U )  =  (
normCV
`  U )
6 eqid 2283 . . . . 5  |-  ( normCV `  W )  =  (
normCV
`  W )
7 nmoo0.3 . . . . 5  |-  N  =  ( U normOp OLD W
)
81, 2, 5, 6, 7nmooval 21341 . . . 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 3646 . . . . 5  |-  { 0 }  =  { x  |  x  =  0 }
11 eqid 2283 . . . . . . . . . . 11  |-  ( 0vec `  U )  =  (
0vec `  U )
121, 11nvzcl 21192 . . . . . . . . . 10  |-  ( U  e.  NrmCVec  ->  ( 0vec `  U
)  e.  ( BaseSet `  U ) )
1311, 5nvz0 21234 . . . . . . . . . . 11  |-  ( U  e.  NrmCVec  ->  ( ( normCV `  U ) `  ( 0vec `  U ) )  =  0 )
14 0le1 9297 . . . . . . . . . . 11  |-  0  <_  1
1513, 14syl6eqbr 4060 . . . . . . . . . 10  |-  ( U  e.  NrmCVec  ->  ( ( normCV `  U ) `  ( 0vec `  U ) )  <_  1 )
16 fveq2 5525 . . . . . . . . . . . 12  |-  ( z  =  ( 0vec `  U
)  ->  ( ( normCV `  U ) `  z
)  =  ( (
normCV
`  U ) `  ( 0vec `  U )
) )
1716breq1d 4033 . . . . . . . . . . 11  |-  ( z  =  ( 0vec `  U
)  ->  ( (
( normCV `  U ) `  z )  <_  1  <->  ( ( normCV `  U ) `  ( 0vec `  U )
)  <_  1 ) )
1817rspcev 2884 . . . . . . . . . 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 2283 . . . . . . . . . . . . . . 15  |-  ( 0vec `  W )  =  (
0vec `  W )
231, 22, 30oval 21366 . . . . . . . . . . . . . 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 5529 . . . . . . . . . . . 12  |-  ( ( ( U  e.  NrmCVec  /\  W  e.  NrmCVec )  /\  z  e.  ( BaseSet `  U ) )  -> 
( ( normCV `  W
) `  ( Z `  z ) )  =  ( ( normCV `  W
) `  ( 0vec `  W ) ) )
2622, 6nvz0 21234 . . . . . . . . . . . . 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 2315 . . . . . . . . . . 11  |-  ( ( ( U  e.  NrmCVec  /\  W  e.  NrmCVec )  /\  z  e.  ( BaseSet `  U ) )  -> 
( ( normCV `  W
) `  ( Z `  z ) )  =  0 )
2928eqeq2d 2294 . . . . . . . . . 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 2560 . . . . . . . 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 2693 . . . . . . . 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 2397 . . . . 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 2328 . . . 4  |-  ( ( U  e.  NrmCVec  /\  W  e.  NrmCVec )  ->  { x  |  E. z  e.  (
BaseSet `  U ) ( ( ( normCV `  U
) `  z )  <_  1  /\  x  =  ( ( normCV `  W
) `  ( Z `  z ) ) ) }  =  { 0 } )
3736supeq1d 7199 . . 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 2315 . 2  |-  ( ( U  e.  NrmCVec  /\  W  e.  NrmCVec )  ->  ( N `  Z )  =  sup ( { 0 } ,  RR* ,  <  ) )
39 xrltso 10475 . . 3  |-  <  Or  RR*
40 0xr 8878 . . 3  |-  0  e.  RR*
41 supsn 7220 . . 3  |-  ( (  <  Or  RR*  /\  0  e.  RR* )  ->  sup ( { 0 } ,  RR* ,  <  )  =  0 )
4239, 40, 41mp2an 653 . 2  |-  sup ( { 0 } ,  RR* ,  <  )  =  0
4338, 42syl6eq 2331 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 1623    e. wcel 1684   {cab 2269   E.wrex 2544   {csn 3640   class class class wbr 4023    Or wor 4313   -->wf 5251   ` cfv 5255  (class class class)co 5858   supcsup 7193   0cc0 8737   1c1 8738   RR*cxr 8866    < clt 8867    <_ cle 8868   NrmCVeccnv 21140   BaseSetcba 21142   0veccn0v 21144   normCVcnmcv 21146   normOp OLDcnmoo 21319    0op c0o 21321
This theorem is referenced by:  0blo  21370  nmlno0lem  21371
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-cnex 8793  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813  ax-pre-mulgt0 8814  ax-pre-sup 8815
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-riota 6304  df-recs 6388  df-rdg 6423  df-er 6660  df-map 6774  df-en 6864  df-dom 6865  df-sdom 6866  df-sup 7194  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-sub 9039  df-neg 9040  df-div 9424  df-nn 9747  df-2 9804  df-3 9805  df-n0 9966  df-z 10025  df-uz 10231  df-rp 10355  df-seq 11047  df-exp 11105  df-cj 11584  df-re 11585  df-im 11586  df-sqr 11720  df-abs 11721  df-grpo 20858  df-gid 20859  df-ginv 20860  df-ablo 20949  df-vc 21102  df-nv 21148  df-va 21151  df-ba 21152  df-sm 21153  df-0v 21154  df-nmcv 21156  df-nmoo 21323  df-0o 21325
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