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Theorem nmlnoubi 21390
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 5541 . . . . . . . 8  |-  ( x  =  Z  ->  ( T `  x )  =  ( T `  Z ) )
21fveq2d 5545 . . . . . . 7  |-  ( x  =  Z  ->  ( M `  ( T `  x ) )  =  ( M `  ( T `  Z )
) )
3 fveq2 5541 . . . . . . . 8  |-  ( x  =  Z  ->  ( K `  x )  =  ( K `  Z ) )
43oveq2d 5890 . . . . . . 7  |-  ( x  =  Z  ->  ( A  x.  ( K `  x ) )  =  ( A  x.  ( K `  Z )
) )
52, 4breq12d 4052 . . . . . 6  |-  ( x  =  Z  ->  (
( M `  ( T `  x )
)  <_  ( A  x.  ( K `  x
) )  <->  ( M `  ( T `  Z
) )  <_  ( A  x.  ( K `  Z ) ) ) )
6 id 19 . . . . . . . 8  |-  ( ( x  =/=  Z  -> 
( M `  ( T `  x )
)  <_  ( A  x.  ( K `  x
) ) )  -> 
( x  =/=  Z  ->  ( M `  ( T `  x )
)  <_  ( A  x.  ( K `  x
) ) ) )
76imp 418 . . . . . . 7  |-  ( ( ( x  =/=  Z  ->  ( M `  ( T `  x )
)  <_  ( A  x.  ( K `  x
) ) )  /\  x  =/=  Z )  -> 
( M `  ( T `  x )
)  <_  ( A  x.  ( K `  x
) ) )
87adantll 694 . . . . . 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 9843 . . . . . . . 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 2296 . . . . . . . . . . . . . 14  |-  ( BaseSet `  W )  =  (
BaseSet `  W )
14 nmlnoubi.z . . . . . . . . . . . . . 14  |-  Z  =  ( 0vec `  U
)
15 eqid 2296 . . . . . . . . . . . . . 14  |-  ( 0vec `  W )  =  (
0vec `  W )
16 nmlnoubi.7 . . . . . . . . . . . . . 14  |-  L  =  ( U  LnOp  W
)
1712, 13, 14, 15, 16lno0 21350 . . . . . . . . . . . . 13  |-  ( ( U  e.  NrmCVec  /\  W  e.  NrmCVec  /\  T  e.  L )  ->  ( T `  Z )  =  ( 0vec `  W
) )
1810, 11, 17mp3an12 1267 . . . . . . . . . . . 12  |-  ( T  e.  L  ->  ( T `  Z )  =  ( 0vec `  W
) )
1918fveq2d 5545 . . . . . . . . . . 11  |-  ( T  e.  L  ->  ( M `  ( T `  Z ) )  =  ( M `  ( 0vec `  W ) ) )
20 nmlnoubi.m . . . . . . . . . . . . 13  |-  M  =  ( normCV `  W )
2115, 20nvz0 21250 . . . . . . . . . . . 12  |-  ( W  e.  NrmCVec  ->  ( M `  ( 0vec `  W )
)  =  0 )
2211, 21ax-mp 8 . . . . . . . . . . 11  |-  ( M `
 ( 0vec `  W
) )  =  0
2319, 22syl6eq 2344 . . . . . . . . . 10  |-  ( T  e.  L  ->  ( M `  ( T `  Z ) )  =  0 )
2423adantr 451 . . . . . . . . 9  |-  ( ( T  e.  L  /\  ( A  e.  RR  /\  0  <_  A )
)  ->  ( M `  ( T `  Z
) )  =  0 )
25 nmlnoubi.k . . . . . . . . . . . . . 14  |-  K  =  ( normCV `  U )
2614, 25nvz0 21250 . . . . . . . . . . . . 13  |-  ( U  e.  NrmCVec  ->  ( K `  Z )  =  0 )
2710, 26ax-mp 8 . . . . . . . . . . . 12  |-  ( K `
 Z )  =  0
2827oveq2i 5885 . . . . . . . . . . 11  |-  ( A  x.  ( K `  Z ) )  =  ( A  x.  0 )
29 recn 8843 . . . . . . . . . . . 12  |-  ( A  e.  RR  ->  A  e.  CC )
3029mul01d 9027 . . . . . . . . . . 11  |-  ( A  e.  RR  ->  ( A  x.  0 )  =  0 )
3128, 30syl5eq 2340 . . . . . . . . . 10  |-  ( A  e.  RR  ->  ( A  x.  ( K `  Z ) )  =  0 )
3231ad2antrl 708 . . . . . . . . 9  |-  ( ( T  e.  L  /\  ( A  e.  RR  /\  0  <_  A )
)  ->  ( A  x.  ( K `  Z
) )  =  0 )
3324, 32breq12d 4052 . . . . . . . 8  |-  ( ( T  e.  L  /\  ( A  e.  RR  /\  0  <_  A )
)  ->  ( ( M `  ( T `  Z ) )  <_ 
( A  x.  ( K `  Z )
)  <->  0  <_  0
) )
349, 33mpbiri 224 . . . . . . 7  |-  ( ( T  e.  L  /\  ( A  e.  RR  /\  0  <_  A )
)  ->  ( M `  ( T `  Z
) )  <_  ( A  x.  ( K `  Z ) ) )
3534adantr 451 . . . . . 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 2534 . . . . 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 423 . . . 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 2635 . . 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 1148 . 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 21349 . . . 4  |-  ( ( U  e.  NrmCVec  /\  W  e.  NrmCVec  /\  T  e.  L )  ->  T : X --> ( BaseSet `  W
) )
4110, 11, 40mp3an12 1267 . . 3  |-  ( T  e.  L  ->  T : X --> ( BaseSet `  W
) )
42 nmlnoubi.3 . . . 4  |-  N  =  ( U normOp OLD W
)
4312, 13, 25, 20, 42, 10, 11nmoub2i 21368 . . 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 1215 . 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 1227 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 358    /\ w3a 934    = wceq 1632    e. wcel 1696    =/= wne 2459   A.wral 2556   class class class wbr 4039   -->wf 5267   ` cfv 5271  (class class class)co 5874   RRcr 8752   0cc0 8753    x. cmul 8758    <_ cle 8884   NrmCVeccnv 21156   BaseSetcba 21158   0veccn0v 21160   normCVcnmcv 21162    LnOp clno 21334   normOp OLDcnmoo 21335
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-cnex 8809  ax-resscn 8810  ax-1cn 8811  ax-icn 8812  ax-addcl 8813  ax-addrcl 8814  ax-mulcl 8815  ax-mulrcl 8816  ax-mulcom 8817  ax-addass 8818  ax-mulass 8819  ax-distr 8820  ax-i2m1 8821  ax-1ne0 8822  ax-1rid 8823  ax-rnegex 8824  ax-rrecex 8825  ax-cnre 8826  ax-pre-lttri 8827  ax-pre-lttrn 8828  ax-pre-ltadd 8829  ax-pre-mulgt0 8830  ax-pre-sup 8831
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 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rmo 2564  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-1st 6138  df-2nd 6139  df-riota 6320  df-recs 6404  df-rdg 6439  df-er 6676  df-map 6790  df-en 6880  df-dom 6881  df-sdom 6882  df-sup 7210  df-pnf 8885  df-mnf 8886  df-xr 8887  df-ltxr 8888  df-le 8889  df-sub 9055  df-neg 9056  df-div 9440  df-nn 9763  df-2 9820  df-3 9821  df-n0 9982  df-z 10041  df-uz 10247  df-rp 10371  df-seq 11063  df-exp 11121  df-cj 11600  df-re 11601  df-im 11602  df-sqr 11736  df-abs 11737  df-grpo 20874  df-gid 20875  df-ginv 20876  df-ablo 20965  df-vc 21118  df-nv 21164  df-va 21167  df-ba 21168  df-sm 21169  df-0v 21170  df-nmcv 21172  df-lno 21338  df-nmoo 21339
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