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Theorem nmlnoubi 25828
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 5774 . . . . . . . 8  |-  ( x  =  Z  ->  ( T `  x )  =  ( T `  Z ) )
21fveq2d 5778 . . . . . . 7  |-  ( x  =  Z  ->  ( M `  ( T `  x ) )  =  ( M `  ( T `  Z )
) )
3 fveq2 5774 . . . . . . . 8  |-  ( x  =  Z  ->  ( K `  x )  =  ( K `  Z ) )
43oveq2d 6212 . . . . . . 7  |-  ( x  =  Z  ->  ( A  x.  ( K `  x ) )  =  ( A  x.  ( K `  Z )
) )
52, 4breq12d 4380 . . . . . 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 427 . . . . . . 7  |-  ( ( ( x  =/=  Z  ->  ( M `  ( T `  x )
)  <_  ( A  x.  ( K `  x
) ) )  /\  x  =/=  Z )  -> 
( M `  ( T `  x )
)  <_  ( A  x.  ( K `  x
) ) )
87adantll 711 . . . . . 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 10542 . . . . . . . 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 2382 . . . . . . . . . . . . . 14  |-  ( BaseSet `  W )  =  (
BaseSet `  W )
14 nmlnoubi.z . . . . . . . . . . . . . 14  |-  Z  =  ( 0vec `  U
)
15 eqid 2382 . . . . . . . . . . . . . 14  |-  ( 0vec `  W )  =  (
0vec `  W )
16 nmlnoubi.7 . . . . . . . . . . . . . 14  |-  L  =  ( U  LnOp  W
)
1712, 13, 14, 15, 16lno0 25788 . . . . . . . . . . . . 13  |-  ( ( U  e.  NrmCVec  /\  W  e.  NrmCVec  /\  T  e.  L )  ->  ( T `  Z )  =  ( 0vec `  W
) )
1810, 11, 17mp3an12 1312 . . . . . . . . . . . 12  |-  ( T  e.  L  ->  ( T `  Z )  =  ( 0vec `  W
) )
1918fveq2d 5778 . . . . . . . . . . 11  |-  ( T  e.  L  ->  ( M `  ( T `  Z ) )  =  ( M `  ( 0vec `  W ) ) )
20 nmlnoubi.m . . . . . . . . . . . . 13  |-  M  =  ( normCV `  W )
2115, 20nvz0 25688 . . . . . . . . . . . 12  |-  ( W  e.  NrmCVec  ->  ( M `  ( 0vec `  W )
)  =  0 )
2211, 21ax-mp 5 . . . . . . . . . . 11  |-  ( M `
 ( 0vec `  W
) )  =  0
2319, 22syl6eq 2439 . . . . . . . . . 10  |-  ( T  e.  L  ->  ( M `  ( T `  Z ) )  =  0 )
2423adantr 463 . . . . . . . . 9  |-  ( ( T  e.  L  /\  ( A  e.  RR  /\  0  <_  A )
)  ->  ( M `  ( T `  Z
) )  =  0 )
25 nmlnoubi.k . . . . . . . . . . . . . 14  |-  K  =  ( normCV `  U )
2614, 25nvz0 25688 . . . . . . . . . . . . 13  |-  ( U  e.  NrmCVec  ->  ( K `  Z )  =  0 )
2710, 26ax-mp 5 . . . . . . . . . . . 12  |-  ( K `
 Z )  =  0
2827oveq2i 6207 . . . . . . . . . . 11  |-  ( A  x.  ( K `  Z ) )  =  ( A  x.  0 )
29 recn 9493 . . . . . . . . . . . 12  |-  ( A  e.  RR  ->  A  e.  CC )
3029mul01d 9690 . . . . . . . . . . 11  |-  ( A  e.  RR  ->  ( A  x.  0 )  =  0 )
3128, 30syl5eq 2435 . . . . . . . . . 10  |-  ( A  e.  RR  ->  ( A  x.  ( K `  Z ) )  =  0 )
3231ad2antrl 725 . . . . . . . . 9  |-  ( ( T  e.  L  /\  ( A  e.  RR  /\  0  <_  A )
)  ->  ( A  x.  ( K `  Z
) )  =  0 )
3324, 32breq12d 4380 . . . . . . . 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 463 . . . . . 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 2697 . . . . 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 432 . . . 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 2792 . . 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 1191 . 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 25787 . . . 4  |-  ( ( U  e.  NrmCVec  /\  W  e.  NrmCVec  /\  T  e.  L )  ->  T : X --> ( BaseSet `  W
) )
4110, 11, 40mp3an12 1312 . . 3  |-  ( T  e.  L  ->  T : X --> ( BaseSet `  W
) )
42 nmlnoubi.3 . . . 4  |-  N  =  ( U normOpOLD W
)
4312, 13, 25, 20, 42, 10, 11nmoub2i 25806 . . 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 1259 . 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 1271 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 367    /\ w3a 971    = wceq 1399    e. wcel 1826    =/= wne 2577   A.wral 2732   class class class wbr 4367   -->wf 5492   ` cfv 5496  (class class class)co 6196   RRcr 9402   0cc0 9403    x. cmul 9408    <_ cle 9540   NrmCVeccnv 25594   BaseSetcba 25596   0veccn0v 25598   normCVcnmcv 25600    LnOp clno 25772   normOpOLDcnmoo 25773
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1626  ax-4 1639  ax-5 1712  ax-6 1755  ax-7 1798  ax-8 1828  ax-9 1830  ax-10 1845  ax-11 1850  ax-12 1862  ax-13 2006  ax-ext 2360  ax-rep 4478  ax-sep 4488  ax-nul 4496  ax-pow 4543  ax-pr 4601  ax-un 6491  ax-cnex 9459  ax-resscn 9460  ax-1cn 9461  ax-icn 9462  ax-addcl 9463  ax-addrcl 9464  ax-mulcl 9465  ax-mulrcl 9466  ax-mulcom 9467  ax-addass 9468  ax-mulass 9469  ax-distr 9470  ax-i2m1 9471  ax-1ne0 9472  ax-1rid 9473  ax-rnegex 9474  ax-rrecex 9475  ax-cnre 9476  ax-pre-lttri 9477  ax-pre-lttrn 9478  ax-pre-ltadd 9479  ax-pre-mulgt0 9480  ax-pre-sup 9481
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 972  df-3an 973  df-tru 1402  df-ex 1621  df-nf 1625  df-sb 1748  df-eu 2222  df-mo 2223  df-clab 2368  df-cleq 2374  df-clel 2377  df-nfc 2532  df-ne 2579  df-nel 2580  df-ral 2737  df-rex 2738  df-reu 2739  df-rmo 2740  df-rab 2741  df-v 3036  df-sbc 3253  df-csb 3349  df-dif 3392  df-un 3394  df-in 3396  df-ss 3403  df-pss 3405  df-nul 3712  df-if 3858  df-pw 3929  df-sn 3945  df-pr 3947  df-tp 3949  df-op 3951  df-uni 4164  df-iun 4245  df-br 4368  df-opab 4426  df-mpt 4427  df-tr 4461  df-eprel 4705  df-id 4709  df-po 4714  df-so 4715  df-fr 4752  df-we 4754  df-ord 4795  df-on 4796  df-lim 4797  df-suc 4798  df-xp 4919  df-rel 4920  df-cnv 4921  df-co 4922  df-dm 4923  df-rn 4924  df-res 4925  df-ima 4926  df-iota 5460  df-fun 5498  df-fn 5499  df-f 5500  df-f1 5501  df-fo 5502  df-f1o 5503  df-fv 5504  df-riota 6158  df-ov 6199  df-oprab 6200  df-mpt2 6201  df-om 6600  df-1st 6699  df-2nd 6700  df-recs 6960  df-rdg 6994  df-er 7229  df-map 7340  df-en 7436  df-dom 7437  df-sdom 7438  df-sup 7816  df-pnf 9541  df-mnf 9542  df-xr 9543  df-ltxr 9544  df-le 9545  df-sub 9720  df-neg 9721  df-div 10124  df-nn 10453  df-2 10511  df-3 10512  df-n0 10713  df-z 10782  df-uz 11002  df-rp 11140  df-seq 12011  df-exp 12070  df-cj 12934  df-re 12935  df-im 12936  df-sqrt 13070  df-abs 13071  df-grpo 25310  df-gid 25311  df-ginv 25312  df-ablo 25401  df-vc 25556  df-nv 25602  df-va 25605  df-ba 25606  df-sm 25607  df-0v 25608  df-nmcv 25610  df-lno 25776  df-nmoo 25777
This theorem is referenced by: (None)
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