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Theorem nmlnoubi 22146
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 5669 . . . . . . . 8  |-  ( x  =  Z  ->  ( T `  x )  =  ( T `  Z ) )
21fveq2d 5673 . . . . . . 7  |-  ( x  =  Z  ->  ( M `  ( T `  x ) )  =  ( M `  ( T `  Z )
) )
3 fveq2 5669 . . . . . . . 8  |-  ( x  =  Z  ->  ( K `  x )  =  ( K `  Z ) )
43oveq2d 6037 . . . . . . 7  |-  ( x  =  Z  ->  ( A  x.  ( K `  x ) )  =  ( A  x.  ( K `  Z )
) )
52, 4breq12d 4167 . . . . . 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 10014 . . . . . . . 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 2388 . . . . . . . . . . . . . 14  |-  ( BaseSet `  W )  =  (
BaseSet `  W )
14 nmlnoubi.z . . . . . . . . . . . . . 14  |-  Z  =  ( 0vec `  U
)
15 eqid 2388 . . . . . . . . . . . . . 14  |-  ( 0vec `  W )  =  (
0vec `  W )
16 nmlnoubi.7 . . . . . . . . . . . . . 14  |-  L  =  ( U  LnOp  W
)
1712, 13, 14, 15, 16lno0 22106 . . . . . . . . . . . . 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 5673 . . . . . . . . . . 11  |-  ( T  e.  L  ->  ( M `  ( T `  Z ) )  =  ( M `  ( 0vec `  W ) ) )
20 nmlnoubi.m . . . . . . . . . . . . 13  |-  M  =  ( normCV `  W )
2115, 20nvz0 22006 . . . . . . . . . . . 12  |-  ( W  e.  NrmCVec  ->  ( M `  ( 0vec `  W )
)  =  0 )
2211, 21ax-mp 8 . . . . . . . . . . 11  |-  ( M `
 ( 0vec `  W
) )  =  0
2319, 22syl6eq 2436 . . . . . . . . . 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 22006 . . . . . . . . . . . . 13  |-  ( U  e.  NrmCVec  ->  ( K `  Z )  =  0 )
2710, 26ax-mp 8 . . . . . . . . . . . 12  |-  ( K `
 Z )  =  0
2827oveq2i 6032 . . . . . . . . . . 11  |-  ( A  x.  ( K `  Z ) )  =  ( A  x.  0 )
29 recn 9014 . . . . . . . . . . . 12  |-  ( A  e.  RR  ->  A  e.  CC )
3029mul01d 9198 . . . . . . . . . . 11  |-  ( A  e.  RR  ->  ( A  x.  0 )  =  0 )
3128, 30syl5eq 2432 . . . . . . . . . 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 4167 . . . . . . . 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 2626 . . . . 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 2729 . . 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 22105 . . . 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 22124 . . 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 1649    e. wcel 1717    =/= wne 2551   A.wral 2650   class class class wbr 4154   -->wf 5391   ` cfv 5395  (class class class)co 6021   RRcr 8923   0cc0 8924    x. cmul 8929    <_ cle 9055   NrmCVeccnv 21912   BaseSetcba 21914   0veccn0v 21916   normCVcnmcv 21918    LnOp clno 22090   normOp OLDcnmoo 22091
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2369  ax-rep 4262  ax-sep 4272  ax-nul 4280  ax-pow 4319  ax-pr 4345  ax-un 4642  ax-cnex 8980  ax-resscn 8981  ax-1cn 8982  ax-icn 8983  ax-addcl 8984  ax-addrcl 8985  ax-mulcl 8986  ax-mulrcl 8987  ax-mulcom 8988  ax-addass 8989  ax-mulass 8990  ax-distr 8991  ax-i2m1 8992  ax-1ne0 8993  ax-1rid 8994  ax-rnegex 8995  ax-rrecex 8996  ax-cnre 8997  ax-pre-lttri 8998  ax-pre-lttrn 8999  ax-pre-ltadd 9000  ax-pre-mulgt0 9001  ax-pre-sup 9002
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2243  df-mo 2244  df-clab 2375  df-cleq 2381  df-clel 2384  df-nfc 2513  df-ne 2553  df-nel 2554  df-ral 2655  df-rex 2656  df-reu 2657  df-rmo 2658  df-rab 2659  df-v 2902  df-sbc 3106  df-csb 3196  df-dif 3267  df-un 3269  df-in 3271  df-ss 3278  df-pss 3280  df-nul 3573  df-if 3684  df-pw 3745  df-sn 3764  df-pr 3765  df-tp 3766  df-op 3767  df-uni 3959  df-iun 4038  df-br 4155  df-opab 4209  df-mpt 4210  df-tr 4245  df-eprel 4436  df-id 4440  df-po 4445  df-so 4446  df-fr 4483  df-we 4485  df-ord 4526  df-on 4527  df-lim 4528  df-suc 4529  df-om 4787  df-xp 4825  df-rel 4826  df-cnv 4827  df-co 4828  df-dm 4829  df-rn 4830  df-res 4831  df-ima 4832  df-iota 5359  df-fun 5397  df-fn 5398  df-f 5399  df-f1 5400  df-fo 5401  df-f1o 5402  df-fv 5403  df-ov 6024  df-oprab 6025  df-mpt2 6026  df-1st 6289  df-2nd 6290  df-riota 6486  df-recs 6570  df-rdg 6605  df-er 6842  df-map 6957  df-en 7047  df-dom 7048  df-sdom 7049  df-sup 7382  df-pnf 9056  df-mnf 9057  df-xr 9058  df-ltxr 9059  df-le 9060  df-sub 9226  df-neg 9227  df-div 9611  df-nn 9934  df-2 9991  df-3 9992  df-n0 10155  df-z 10216  df-uz 10422  df-rp 10546  df-seq 11252  df-exp 11311  df-cj 11832  df-re 11833  df-im 11834  df-sqr 11968  df-abs 11969  df-grpo 21628  df-gid 21629  df-ginv 21630  df-ablo 21719  df-vc 21874  df-nv 21920  df-va 21923  df-ba 21924  df-sm 21925  df-0v 21926  df-nmcv 21928  df-lno 22094  df-nmoo 22095
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