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Theorem nmlnoubi 22289
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 normOp OLD 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 5720 . . . . . . . 8  |-  ( x  =  Z  ->  ( T `  x )  =  ( T `  Z ) )
21fveq2d 5724 . . . . . . 7  |-  ( x  =  Z  ->  ( M `  ( T `  x ) )  =  ( M `  ( T `  Z )
) )
3 fveq2 5720 . . . . . . . 8  |-  ( x  =  Z  ->  ( K `  x )  =  ( K `  Z ) )
43oveq2d 6089 . . . . . . 7  |-  ( x  =  Z  ->  ( A  x.  ( K `  x ) )  =  ( A  x.  ( K `  Z )
) )
52, 4breq12d 4217 . . . . . 6  |-  ( x  =  Z  ->  (
( M `  ( T `  x )
)  <_  ( A  x.  ( K `  x
) )  <->  ( M `  ( T `  Z
) )  <_  ( A  x.  ( K `  Z ) ) ) )
6 id 20 . . . . . . . 8  |-  ( ( x  =/=  Z  -> 
( M `  ( T `  x )
)  <_  ( A  x.  ( K `  x
) ) )  -> 
( x  =/=  Z  ->  ( M `  ( T `  x )
)  <_  ( A  x.  ( K `  x
) ) ) )
76imp 419 . . . . . . 7  |-  ( ( ( x  =/=  Z  ->  ( M `  ( T `  x )
)  <_  ( A  x.  ( K `  x
) ) )  /\  x  =/=  Z )  -> 
( M `  ( T `  x )
)  <_  ( A  x.  ( K `  x
) ) )
87adantll 695 . . . . . 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 10073 . . . . . . . 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 2435 . . . . . . . . . . . . . 14  |-  ( BaseSet `  W )  =  (
BaseSet `  W )
14 nmlnoubi.z . . . . . . . . . . . . . 14  |-  Z  =  ( 0vec `  U
)
15 eqid 2435 . . . . . . . . . . . . . 14  |-  ( 0vec `  W )  =  (
0vec `  W )
16 nmlnoubi.7 . . . . . . . . . . . . . 14  |-  L  =  ( U  LnOp  W
)
1712, 13, 14, 15, 16lno0 22249 . . . . . . . . . . . . 13  |-  ( ( U  e.  NrmCVec  /\  W  e.  NrmCVec  /\  T  e.  L )  ->  ( T `  Z )  =  ( 0vec `  W
) )
1810, 11, 17mp3an12 1269 . . . . . . . . . . . 12  |-  ( T  e.  L  ->  ( T `  Z )  =  ( 0vec `  W
) )
1918fveq2d 5724 . . . . . . . . . . 11  |-  ( T  e.  L  ->  ( M `  ( T `  Z ) )  =  ( M `  ( 0vec `  W ) ) )
20 nmlnoubi.m . . . . . . . . . . . . 13  |-  M  =  ( normCV `  W )
2115, 20nvz0 22149 . . . . . . . . . . . 12  |-  ( W  e.  NrmCVec  ->  ( M `  ( 0vec `  W )
)  =  0 )
2211, 21ax-mp 8 . . . . . . . . . . 11  |-  ( M `
 ( 0vec `  W
) )  =  0
2319, 22syl6eq 2483 . . . . . . . . . 10  |-  ( T  e.  L  ->  ( M `  ( T `  Z ) )  =  0 )
2423adantr 452 . . . . . . . . 9  |-  ( ( T  e.  L  /\  ( A  e.  RR  /\  0  <_  A )
)  ->  ( M `  ( T `  Z
) )  =  0 )
25 nmlnoubi.k . . . . . . . . . . . . . 14  |-  K  =  ( normCV `  U )
2614, 25nvz0 22149 . . . . . . . . . . . . 13  |-  ( U  e.  NrmCVec  ->  ( K `  Z )  =  0 )
2710, 26ax-mp 8 . . . . . . . . . . . 12  |-  ( K `
 Z )  =  0
2827oveq2i 6084 . . . . . . . . . . 11  |-  ( A  x.  ( K `  Z ) )  =  ( A  x.  0 )
29 recn 9072 . . . . . . . . . . . 12  |-  ( A  e.  RR  ->  A  e.  CC )
3029mul01d 9257 . . . . . . . . . . 11  |-  ( A  e.  RR  ->  ( A  x.  0 )  =  0 )
3128, 30syl5eq 2479 . . . . . . . . . 10  |-  ( A  e.  RR  ->  ( A  x.  ( K `  Z ) )  =  0 )
3231ad2antrl 709 . . . . . . . . 9  |-  ( ( T  e.  L  /\  ( A  e.  RR  /\  0  <_  A )
)  ->  ( A  x.  ( K `  Z
) )  =  0 )
3324, 32breq12d 4217 . . . . . . . 8  |-  ( ( T  e.  L  /\  ( A  e.  RR  /\  0  <_  A )
)  ->  ( ( M `  ( T `  Z ) )  <_ 
( A  x.  ( K `  Z )
)  <->  0  <_  0
) )
349, 33mpbiri 225 . . . . . . 7  |-  ( ( T  e.  L  /\  ( A  e.  RR  /\  0  <_  A )
)  ->  ( M `  ( T `  Z
) )  <_  ( A  x.  ( K `  Z ) ) )
3534adantr 452 . . . . . 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 2673 . . . . 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 424 . . . 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 2777 . . 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 1150 . 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 22248 . . . 4  |-  ( ( U  e.  NrmCVec  /\  W  e.  NrmCVec  /\  T  e.  L )  ->  T : X --> ( BaseSet `  W
) )
4110, 11, 40mp3an12 1269 . . 3  |-  ( T  e.  L  ->  T : X --> ( BaseSet `  W
) )
42 nmlnoubi.3 . . . 4  |-  N  =  ( U normOp OLD W
)
4312, 13, 25, 20, 42, 10, 11nmoub2i 22267 . . 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 1217 . 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 1229 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 set class
Syntax hints:    -> wi 4    /\ wa 359    /\ w3a 936    = wceq 1652    e. wcel 1725    =/= wne 2598   A.wral 2697   class class class wbr 4204   -->wf 5442   ` cfv 5446  (class class class)co 6073   RRcr 8981   0cc0 8982    x. cmul 8987    <_ cle 9113   NrmCVeccnv 22055   BaseSetcba 22057   0veccn0v 22059   normCVcnmcv 22061    LnOp clno 22233   normOp OLDcnmoo 22234
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-rep 4312  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693  ax-cnex 9038  ax-resscn 9039  ax-1cn 9040  ax-icn 9041  ax-addcl 9042  ax-addrcl 9043  ax-mulcl 9044  ax-mulrcl 9045  ax-mulcom 9046  ax-addass 9047  ax-mulass 9048  ax-distr 9049  ax-i2m1 9050  ax-1ne0 9051  ax-1rid 9052  ax-rnegex 9053  ax-rrecex 9054  ax-cnre 9055  ax-pre-lttri 9056  ax-pre-lttrn 9057  ax-pre-ltadd 9058  ax-pre-mulgt0 9059  ax-pre-sup 9060
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-nel 2601  df-ral 2702  df-rex 2703  df-reu 2704  df-rmo 2705  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-pss 3328  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-tp 3814  df-op 3815  df-uni 4008  df-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  df-tr 4295  df-eprel 4486  df-id 4490  df-po 4495  df-so 4496  df-fr 4533  df-we 4535  df-ord 4576  df-on 4577  df-lim 4578  df-suc 4579  df-om 4838  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-ov 6076  df-oprab 6077  df-mpt2 6078  df-1st 6341  df-2nd 6342  df-riota 6541  df-recs 6625  df-rdg 6660  df-er 6897  df-map 7012  df-en 7102  df-dom 7103  df-sdom 7104  df-sup 7438  df-pnf 9114  df-mnf 9115  df-xr 9116  df-ltxr 9117  df-le 9118  df-sub 9285  df-neg 9286  df-div 9670  df-nn 9993  df-2 10050  df-3 10051  df-n0 10214  df-z 10275  df-uz 10481  df-rp 10605  df-seq 11316  df-exp 11375  df-cj 11896  df-re 11897  df-im 11898  df-sqr 12032  df-abs 12033  df-grpo 21771  df-gid 21772  df-ginv 21773  df-ablo 21862  df-vc 22017  df-nv 22063  df-va 22066  df-ba 22067  df-sm 22068  df-0v 22069  df-nmcv 22071  df-lno 22237  df-nmoo 22238
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