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Theorem nmolb2d 18243
Description: Any upper bound on the values of a linear operator at nonzero vectors translates to an upper bound on the operator norm. (Contributed by Mario Carneiro, 18-Oct-2015.)
Hypotheses
Ref Expression
nmofval.1  |-  N  =  ( S normOp T )
nmofval.2  |-  V  =  ( Base `  S
)
nmofval.3  |-  L  =  ( norm `  S
)
nmofval.4  |-  M  =  ( norm `  T
)
nmolb2d.z  |-  .0.  =  ( 0g `  S )
nmolb2d.1  |-  ( ph  ->  S  e. NrmGrp )
nmolb2d.2  |-  ( ph  ->  T  e. NrmGrp )
nmolb2d.3  |-  ( ph  ->  F  e.  ( S 
GrpHom  T ) )
nmolb2d.4  |-  ( ph  ->  A  e.  RR )
nmolb2d.5  |-  ( ph  ->  0  <_  A )
nmolb2d.6  |-  ( (
ph  /\  ( x  e.  V  /\  x  =/=  .0.  ) )  -> 
( M `  ( F `  x )
)  <_  ( A  x.  ( L `  x
) ) )
Assertion
Ref Expression
nmolb2d  |-  ( ph  ->  ( N `  F
)  <_  A )
Distinct variable groups:    x, L    x, M    x, S    x, T    x, A    x, F    ph, x    x, V    x, N
Allowed substitution hint:    .0. ( x)

Proof of Theorem nmolb2d
StepHypRef Expression
1 fveq2 5541 . . . . . 6  |-  ( x  =  .0.  ->  ( F `  x )  =  ( F `  .0.  ) )
21fveq2d 5545 . . . . 5  |-  ( x  =  .0.  ->  ( M `  ( F `  x ) )  =  ( M `  ( F `  .0.  ) ) )
3 fveq2 5541 . . . . . 6  |-  ( x  =  .0.  ->  ( L `  x )  =  ( L `  .0.  ) )
43oveq2d 5890 . . . . 5  |-  ( x  =  .0.  ->  ( A  x.  ( L `  x ) )  =  ( A  x.  ( L `  .0.  ) ) )
52, 4breq12d 4052 . . . 4  |-  ( x  =  .0.  ->  (
( M `  ( F `  x )
)  <_  ( A  x.  ( L `  x
) )  <->  ( M `  ( F `  .0.  ) )  <_  ( A  x.  ( L `  .0.  ) ) ) )
6 nmolb2d.6 . . . . 5  |-  ( (
ph  /\  ( x  e.  V  /\  x  =/=  .0.  ) )  -> 
( M `  ( F `  x )
)  <_  ( A  x.  ( L `  x
) ) )
76anassrs 629 . . . 4  |-  ( ( ( ph  /\  x  e.  V )  /\  x  =/=  .0.  )  ->  ( M `  ( F `  x ) )  <_ 
( A  x.  ( L `  x )
) )
8 0le0 9843 . . . . . . 7  |-  0  <_  0
9 nmolb2d.4 . . . . . . . . 9  |-  ( ph  ->  A  e.  RR )
109recnd 8877 . . . . . . . 8  |-  ( ph  ->  A  e.  CC )
1110mul01d 9027 . . . . . . 7  |-  ( ph  ->  ( A  x.  0 )  =  0 )
128, 11syl5breqr 4075 . . . . . 6  |-  ( ph  ->  0  <_  ( A  x.  0 ) )
13 nmolb2d.3 . . . . . . . . 9  |-  ( ph  ->  F  e.  ( S 
GrpHom  T ) )
14 nmolb2d.z . . . . . . . . . 10  |-  .0.  =  ( 0g `  S )
15 eqid 2296 . . . . . . . . . 10  |-  ( 0g
`  T )  =  ( 0g `  T
)
1614, 15ghmid 14705 . . . . . . . . 9  |-  ( F  e.  ( S  GrpHom  T )  ->  ( F `  .0.  )  =  ( 0g `  T ) )
1713, 16syl 15 . . . . . . . 8  |-  ( ph  ->  ( F `  .0.  )  =  ( 0g `  T ) )
1817fveq2d 5545 . . . . . . 7  |-  ( ph  ->  ( M `  ( F `  .0.  ) )  =  ( M `  ( 0g `  T ) ) )
19 nmolb2d.2 . . . . . . . 8  |-  ( ph  ->  T  e. NrmGrp )
20 nmofval.4 . . . . . . . . 9  |-  M  =  ( norm `  T
)
2120, 15nm0 18164 . . . . . . . 8  |-  ( T  e. NrmGrp  ->  ( M `  ( 0g `  T ) )  =  0 )
2219, 21syl 15 . . . . . . 7  |-  ( ph  ->  ( M `  ( 0g `  T ) )  =  0 )
2318, 22eqtrd 2328 . . . . . 6  |-  ( ph  ->  ( M `  ( F `  .0.  ) )  =  0 )
24 nmolb2d.1 . . . . . . . 8  |-  ( ph  ->  S  e. NrmGrp )
25 nmofval.3 . . . . . . . . 9  |-  L  =  ( norm `  S
)
2625, 14nm0 18164 . . . . . . . 8  |-  ( S  e. NrmGrp  ->  ( L `  .0.  )  =  0
)
2724, 26syl 15 . . . . . . 7  |-  ( ph  ->  ( L `  .0.  )  =  0 )
2827oveq2d 5890 . . . . . 6  |-  ( ph  ->  ( A  x.  ( L `  .0.  ) )  =  ( A  x.  0 ) )
2912, 23, 283brtr4d 4069 . . . . 5  |-  ( ph  ->  ( M `  ( F `  .0.  ) )  <_  ( A  x.  ( L `  .0.  )
) )
3029adantr 451 . . . 4  |-  ( (
ph  /\  x  e.  V )  ->  ( M `  ( F `  .0.  ) )  <_ 
( A  x.  ( L `  .0.  ) ) )
315, 7, 30pm2.61ne 2534 . . 3  |-  ( (
ph  /\  x  e.  V )  ->  ( M `  ( F `  x ) )  <_ 
( A  x.  ( L `  x )
) )
3231ralrimiva 2639 . 2  |-  ( ph  ->  A. x  e.  V  ( M `  ( F `
 x ) )  <_  ( A  x.  ( L `  x ) ) )
33 nmolb2d.5 . . 3  |-  ( ph  ->  0  <_  A )
34 nmofval.1 . . . 4  |-  N  =  ( S normOp T )
35 nmofval.2 . . . 4  |-  V  =  ( Base `  S
)
3634, 35, 25, 20nmolb 18242 . . 3  |-  ( ( ( S  e. NrmGrp  /\  T  e. NrmGrp  /\  F  e.  ( S  GrpHom  T ) )  /\  A  e.  RR  /\  0  <_  A )  ->  ( A. x  e.  V  ( M `  ( F `  x ) )  <_  ( A  x.  ( L `  x
) )  ->  ( N `  F )  <_  A ) )
3724, 19, 13, 9, 33, 36syl311anc 1196 . 2  |-  ( ph  ->  ( A. x  e.  V  ( M `  ( F `  x ) )  <_  ( A  x.  ( L `  x
) )  ->  ( N `  F )  <_  A ) )
3832, 37mpd 14 1  |-  ( ph  ->  ( N `  F
)  <_  A )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1632    e. wcel 1696    =/= wne 2459   A.wral 2556   class class class wbr 4039   ` cfv 5271  (class class class)co 5874   RRcr 8752   0cc0 8753    x. cmul 8758    <_ cle 8884   Basecbs 13164   0gc0g 13416    GrpHom cghm 14696   normcnm 18115  NrmGrpcngp 18116   normOpcnmo 18230
This theorem is referenced by:  nmo0  18260  nmoco  18262  nmotri  18264  nmoid  18267  nmoleub2lem  18611
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-n0 9982  df-z 10041  df-uz 10247  df-q 10333  df-rp 10371  df-xneg 10468  df-xadd 10469  df-xmul 10470  df-ico 10678  df-topgen 13360  df-0g 13420  df-mnd 14383  df-grp 14505  df-ghm 14697  df-xmet 16389  df-bl 16391  df-mopn 16392  df-top 16652  df-bases 16654  df-topon 16655  df-topsp 16656  df-xms 17901  df-ms 17902  df-nm 18121  df-ngp 18122  df-nmo 18233
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