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Theorem nmlnoubi 24343
Description: An upper bound for the operator norm of a linear operator, using only the properties of nonzero arguments. (Contributed by NM, 1-Jan-2008.) (New usage is discouraged.)
Hypotheses
Ref Expression
nmlnoubi.1  |-  X  =  ( BaseSet `  U )
nmlnoubi.z  |-  Z  =  ( 0vec `  U
)
nmlnoubi.k  |-  K  =  ( normCV `  U )
nmlnoubi.m  |-  M  =  ( normCV `  W )
nmlnoubi.3  |-  N  =  ( U normOpOLD W
)
nmlnoubi.7  |-  L  =  ( U  LnOp  W
)
nmlnoubi.u  |-  U  e.  NrmCVec
nmlnoubi.w  |-  W  e.  NrmCVec
Assertion
Ref Expression
nmlnoubi  |-  ( ( T  e.  L  /\  ( A  e.  RR  /\  0  <_  A )  /\  A. x  e.  X  ( x  =/=  Z  ->  ( M `  ( T `  x )
)  <_  ( A  x.  ( K `  x
) ) ) )  ->  ( N `  T )  <_  A
)
Distinct variable groups:    x, A    x, K    x, L    x, M    x, T    x, U    x, W    x, X
Allowed substitution hints:    N( x)    Z( x)

Proof of Theorem nmlnoubi
StepHypRef Expression
1 fveq2 5794 . . . . . . . 8  |-  ( x  =  Z  ->  ( T `  x )  =  ( T `  Z ) )
21fveq2d 5798 . . . . . . 7  |-  ( x  =  Z  ->  ( M `  ( T `  x ) )  =  ( M `  ( T `  Z )
) )
3 fveq2 5794 . . . . . . . 8  |-  ( x  =  Z  ->  ( K `  x )  =  ( K `  Z ) )
43oveq2d 6211 . . . . . . 7  |-  ( x  =  Z  ->  ( A  x.  ( K `  x ) )  =  ( A  x.  ( K `  Z )
) )
52, 4breq12d 4408 . . . . . 6  |-  ( x  =  Z  ->  (
( M `  ( T `  x )
)  <_  ( A  x.  ( K `  x
) )  <->  ( M `  ( T `  Z
) )  <_  ( A  x.  ( K `  Z ) ) ) )
6 id 22 . . . . . . . 8  |-  ( ( x  =/=  Z  -> 
( M `  ( T `  x )
)  <_  ( A  x.  ( K `  x
) ) )  -> 
( x  =/=  Z  ->  ( M `  ( T `  x )
)  <_  ( A  x.  ( K `  x
) ) ) )
76imp 429 . . . . . . 7  |-  ( ( ( x  =/=  Z  ->  ( M `  ( T `  x )
)  <_  ( A  x.  ( K `  x
) ) )  /\  x  =/=  Z )  -> 
( M `  ( T `  x )
)  <_  ( A  x.  ( K `  x
) ) )
87adantll 713 . . . . . 6  |-  ( ( ( ( T  e.  L  /\  ( A  e.  RR  /\  0  <_  A ) )  /\  ( x  =/=  Z  ->  ( M `  ( T `  x )
)  <_  ( A  x.  ( K `  x
) ) ) )  /\  x  =/=  Z
)  ->  ( M `  ( T `  x
) )  <_  ( A  x.  ( K `  x ) ) )
9 0le0 10517 . . . . . . . 8  |-  0  <_  0
10 nmlnoubi.u . . . . . . . . . . . . 13  |-  U  e.  NrmCVec
11 nmlnoubi.w . . . . . . . . . . . . 13  |-  W  e.  NrmCVec
12 nmlnoubi.1 . . . . . . . . . . . . . 14  |-  X  =  ( BaseSet `  U )
13 eqid 2452 . . . . . . . . . . . . . 14  |-  ( BaseSet `  W )  =  (
BaseSet `  W )
14 nmlnoubi.z . . . . . . . . . . . . . 14  |-  Z  =  ( 0vec `  U
)
15 eqid 2452 . . . . . . . . . . . . . 14  |-  ( 0vec `  W )  =  (
0vec `  W )
16 nmlnoubi.7 . . . . . . . . . . . . . 14  |-  L  =  ( U  LnOp  W
)
1712, 13, 14, 15, 16lno0 24303 . . . . . . . . . . . . 13  |-  ( ( U  e.  NrmCVec  /\  W  e.  NrmCVec  /\  T  e.  L )  ->  ( T `  Z )  =  ( 0vec `  W
) )
1810, 11, 17mp3an12 1305 . . . . . . . . . . . 12  |-  ( T  e.  L  ->  ( T `  Z )  =  ( 0vec `  W
) )
1918fveq2d 5798 . . . . . . . . . . 11  |-  ( T  e.  L  ->  ( M `  ( T `  Z ) )  =  ( M `  ( 0vec `  W ) ) )
20 nmlnoubi.m . . . . . . . . . . . . 13  |-  M  =  ( normCV `  W )
2115, 20nvz0 24203 . . . . . . . . . . . 12  |-  ( W  e.  NrmCVec  ->  ( M `  ( 0vec `  W )
)  =  0 )
2211, 21ax-mp 5 . . . . . . . . . . 11  |-  ( M `
 ( 0vec `  W
) )  =  0
2319, 22syl6eq 2509 . . . . . . . . . 10  |-  ( T  e.  L  ->  ( M `  ( T `  Z ) )  =  0 )
2423adantr 465 . . . . . . . . 9  |-  ( ( T  e.  L  /\  ( A  e.  RR  /\  0  <_  A )
)  ->  ( M `  ( T `  Z
) )  =  0 )
25 nmlnoubi.k . . . . . . . . . . . . . 14  |-  K  =  ( normCV `  U )
2614, 25nvz0 24203 . . . . . . . . . . . . 13  |-  ( U  e.  NrmCVec  ->  ( K `  Z )  =  0 )
2710, 26ax-mp 5 . . . . . . . . . . . 12  |-  ( K `
 Z )  =  0
2827oveq2i 6206 . . . . . . . . . . 11  |-  ( A  x.  ( K `  Z ) )  =  ( A  x.  0 )
29 recn 9478 . . . . . . . . . . . 12  |-  ( A  e.  RR  ->  A  e.  CC )
3029mul01d 9674 . . . . . . . . . . 11  |-  ( A  e.  RR  ->  ( A  x.  0 )  =  0 )
3128, 30syl5eq 2505 . . . . . . . . . 10  |-  ( A  e.  RR  ->  ( A  x.  ( K `  Z ) )  =  0 )
3231ad2antrl 727 . . . . . . . . 9  |-  ( ( T  e.  L  /\  ( A  e.  RR  /\  0  <_  A )
)  ->  ( A  x.  ( K `  Z
) )  =  0 )
3324, 32breq12d 4408 . . . . . . . 8  |-  ( ( T  e.  L  /\  ( A  e.  RR  /\  0  <_  A )
)  ->  ( ( M `  ( T `  Z ) )  <_ 
( A  x.  ( K `  Z )
)  <->  0  <_  0
) )
349, 33mpbiri 233 . . . . . . 7  |-  ( ( T  e.  L  /\  ( A  e.  RR  /\  0  <_  A )
)  ->  ( M `  ( T `  Z
) )  <_  ( A  x.  ( K `  Z ) ) )
3534adantr 465 . . . . . 6  |-  ( ( ( T  e.  L  /\  ( A  e.  RR  /\  0  <_  A )
)  /\  ( x  =/=  Z  ->  ( M `  ( T `  x
) )  <_  ( A  x.  ( K `  x ) ) ) )  ->  ( M `  ( T `  Z
) )  <_  ( A  x.  ( K `  Z ) ) )
365, 8, 35pm2.61ne 2764 . . . . 5  |-  ( ( ( T  e.  L  /\  ( A  e.  RR  /\  0  <_  A )
)  /\  ( x  =/=  Z  ->  ( M `  ( T `  x
) )  <_  ( A  x.  ( K `  x ) ) ) )  ->  ( M `  ( T `  x
) )  <_  ( A  x.  ( K `  x ) ) )
3736ex 434 . . . 4  |-  ( ( T  e.  L  /\  ( A  e.  RR  /\  0  <_  A )
)  ->  ( (
x  =/=  Z  -> 
( M `  ( T `  x )
)  <_  ( A  x.  ( K `  x
) ) )  -> 
( M `  ( T `  x )
)  <_  ( A  x.  ( K `  x
) ) ) )
3837ralimdv 2831 . . 3  |-  ( ( T  e.  L  /\  ( A  e.  RR  /\  0  <_  A )
)  ->  ( A. x  e.  X  (
x  =/=  Z  -> 
( M `  ( T `  x )
)  <_  ( A  x.  ( K `  x
) ) )  ->  A. x  e.  X  ( M `  ( T `
 x ) )  <_  ( A  x.  ( K `  x ) ) ) )
39383impia 1185 . 2  |-  ( ( T  e.  L  /\  ( A  e.  RR  /\  0  <_  A )  /\  A. x  e.  X  ( x  =/=  Z  ->  ( M `  ( T `  x )
)  <_  ( A  x.  ( K `  x
) ) ) )  ->  A. x  e.  X  ( M `  ( T `
 x ) )  <_  ( A  x.  ( K `  x ) ) )
4012, 13, 16lnof 24302 . . . 4  |-  ( ( U  e.  NrmCVec  /\  W  e.  NrmCVec  /\  T  e.  L )  ->  T : X --> ( BaseSet `  W
) )
4110, 11, 40mp3an12 1305 . . 3  |-  ( T  e.  L  ->  T : X --> ( BaseSet `  W
) )
42 nmlnoubi.3 . . . 4  |-  N  =  ( U normOpOLD W
)
4312, 13, 25, 20, 42, 10, 11nmoub2i 24321 . . 3  |-  ( ( T : X --> ( BaseSet `  W )  /\  ( A  e.  RR  /\  0  <_  A )  /\  A. x  e.  X  ( M `  ( T `  x ) )  <_ 
( A  x.  ( K `  x )
) )  ->  ( N `  T )  <_  A )
4441, 43syl3an1 1252 . 2  |-  ( ( T  e.  L  /\  ( A  e.  RR  /\  0  <_  A )  /\  A. x  e.  X  ( M `  ( T `
 x ) )  <_  ( A  x.  ( K `  x ) ) )  ->  ( N `  T )  <_  A )
4539, 44syld3an3 1264 1  |-  ( ( T  e.  L  /\  ( A  e.  RR  /\  0  <_  A )  /\  A. x  e.  X  ( x  =/=  Z  ->  ( M `  ( T `  x )
)  <_  ( A  x.  ( K `  x
) ) ) )  ->  ( N `  T )  <_  A
)
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    /\ w3a 965    = wceq 1370    e. wcel 1758    =/= wne 2645   A.wral 2796   class class class wbr 4395   -->wf 5517   ` cfv 5521  (class class class)co 6195   RRcr 9387   0cc0 9388    x. cmul 9393    <_ cle 9525   NrmCVeccnv 24109   BaseSetcba 24111   0veccn0v 24113   normCVcnmcv 24115    LnOp clno 24287   normOpOLDcnmoo 24288
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1954  ax-ext 2431  ax-rep 4506  ax-sep 4516  ax-nul 4524  ax-pow 4573  ax-pr 4634  ax-un 6477  ax-cnex 9444  ax-resscn 9445  ax-1cn 9446  ax-icn 9447  ax-addcl 9448  ax-addrcl 9449  ax-mulcl 9450  ax-mulrcl 9451  ax-mulcom 9452  ax-addass 9453  ax-mulass 9454  ax-distr 9455  ax-i2m1 9456  ax-1ne0 9457  ax-1rid 9458  ax-rnegex 9459  ax-rrecex 9460  ax-cnre 9461  ax-pre-lttri 9462  ax-pre-lttrn 9463  ax-pre-ltadd 9464  ax-pre-mulgt0 9465  ax-pre-sup 9466
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2265  df-mo 2266  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2602  df-ne 2647  df-nel 2648  df-ral 2801  df-rex 2802  df-reu 2803  df-rmo 2804  df-rab 2805  df-v 3074  df-sbc 3289  df-csb 3391  df-dif 3434  df-un 3436  df-in 3438  df-ss 3445  df-pss 3447  df-nul 3741  df-if 3895  df-pw 3965  df-sn 3981  df-pr 3983  df-tp 3985  df-op 3987  df-uni 4195  df-iun 4276  df-br 4396  df-opab 4454  df-mpt 4455  df-tr 4489  df-eprel 4735  df-id 4739  df-po 4744  df-so 4745  df-fr 4782  df-we 4784  df-ord 4825  df-on 4826  df-lim 4827  df-suc 4828  df-xp 4949  df-rel 4950  df-cnv 4951  df-co 4952  df-dm 4953  df-rn 4954  df-res 4955  df-ima 4956  df-iota 5484  df-fun 5523  df-fn 5524  df-f 5525  df-f1 5526  df-fo 5527  df-f1o 5528  df-fv 5529  df-riota 6156  df-ov 6198  df-oprab 6199  df-mpt2 6200  df-om 6582  df-1st 6682  df-2nd 6683  df-recs 6937  df-rdg 6971  df-er 7206  df-map 7321  df-en 7416  df-dom 7417  df-sdom 7418  df-sup 7797  df-pnf 9526  df-mnf 9527  df-xr 9528  df-ltxr 9529  df-le 9530  df-sub 9703  df-neg 9704  df-div 10100  df-nn 10429  df-2 10486  df-3 10487  df-n0 10686  df-z 10753  df-uz 10968  df-rp 11098  df-seq 11919  df-exp 11978  df-cj 12701  df-re 12702  df-im 12703  df-sqr 12837  df-abs 12838  df-grpo 23825  df-gid 23826  df-ginv 23827  df-ablo 23916  df-vc 24071  df-nv 24117  df-va 24120  df-ba 24121  df-sm 24122  df-0v 24123  df-nmcv 24125  df-lno 24291  df-nmoo 24292
This theorem is referenced by: (None)
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