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Theorem nmoi2 18239
Description: The operator norm is a bound on the growth of a vector under the action of the operator. (Contributed by Mario Carneiro, 19-Oct-2015.)
Hypotheses
Ref Expression
nmofval.1  |-  N  =  ( S normOp T )
nmoi.2  |-  V  =  ( Base `  S
)
nmoi.3  |-  L  =  ( norm `  S
)
nmoi.4  |-  M  =  ( norm `  T
)
nmoleub.z  |-  .0.  =  ( 0g `  S )
Assertion
Ref Expression
nmoi2  |-  ( ( ( S  e. NrmGrp  /\  T  e. NrmGrp  /\  F  e.  ( S  GrpHom  T ) )  /\  ( X  e.  V  /\  X  =/= 
.0.  ) )  -> 
( ( M `  ( F `  X ) )  /  ( L `
 X ) )  <_  ( N `  F ) )

Proof of Theorem nmoi2
StepHypRef Expression
1 simpl2 959 . . . . 5  |-  ( ( ( S  e. NrmGrp  /\  T  e. NrmGrp  /\  F  e.  ( S  GrpHom  T ) )  /\  ( X  e.  V  /\  X  =/= 
.0.  ) )  ->  T  e. NrmGrp )
2 simpl3 960 . . . . . . 7  |-  ( ( ( S  e. NrmGrp  /\  T  e. NrmGrp  /\  F  e.  ( S  GrpHom  T ) )  /\  ( X  e.  V  /\  X  =/= 
.0.  ) )  ->  F  e.  ( S  GrpHom  T ) )
3 nmoi.2 . . . . . . . 8  |-  V  =  ( Base `  S
)
4 eqid 2283 . . . . . . . 8  |-  ( Base `  T )  =  (
Base `  T )
53, 4ghmf 14687 . . . . . . 7  |-  ( F  e.  ( S  GrpHom  T )  ->  F : V
--> ( Base `  T
) )
62, 5syl 15 . . . . . 6  |-  ( ( ( S  e. NrmGrp  /\  T  e. NrmGrp  /\  F  e.  ( S  GrpHom  T ) )  /\  ( X  e.  V  /\  X  =/= 
.0.  ) )  ->  F : V --> ( Base `  T ) )
7 simprl 732 . . . . . 6  |-  ( ( ( S  e. NrmGrp  /\  T  e. NrmGrp  /\  F  e.  ( S  GrpHom  T ) )  /\  ( X  e.  V  /\  X  =/= 
.0.  ) )  ->  X  e.  V )
8 ffvelrn 5663 . . . . . 6  |-  ( ( F : V --> ( Base `  T )  /\  X  e.  V )  ->  ( F `  X )  e.  ( Base `  T
) )
96, 7, 8syl2anc 642 . . . . 5  |-  ( ( ( S  e. NrmGrp  /\  T  e. NrmGrp  /\  F  e.  ( S  GrpHom  T ) )  /\  ( X  e.  V  /\  X  =/= 
.0.  ) )  -> 
( F `  X
)  e.  ( Base `  T ) )
10 nmoi.4 . . . . . 6  |-  M  =  ( norm `  T
)
114, 10nmcl 18137 . . . . 5  |-  ( ( T  e. NrmGrp  /\  ( F `  X )  e.  ( Base `  T
) )  ->  ( M `  ( F `  X ) )  e.  RR )
121, 9, 11syl2anc 642 . . . 4  |-  ( ( ( S  e. NrmGrp  /\  T  e. NrmGrp  /\  F  e.  ( S  GrpHom  T ) )  /\  ( X  e.  V  /\  X  =/= 
.0.  ) )  -> 
( M `  ( F `  X )
)  e.  RR )
1312rexrd 8881 . . 3  |-  ( ( ( S  e. NrmGrp  /\  T  e. NrmGrp  /\  F  e.  ( S  GrpHom  T ) )  /\  ( X  e.  V  /\  X  =/= 
.0.  ) )  -> 
( M `  ( F `  X )
)  e.  RR* )
14 nmofval.1 . . . . . 6  |-  N  =  ( S normOp T )
1514nmocl 18229 . . . . 5  |-  ( ( S  e. NrmGrp  /\  T  e. NrmGrp  /\  F  e.  ( S  GrpHom  T ) )  ->  ( N `  F )  e.  RR* )
1615adantr 451 . . . 4  |-  ( ( ( S  e. NrmGrp  /\  T  e. NrmGrp  /\  F  e.  ( S  GrpHom  T ) )  /\  ( X  e.  V  /\  X  =/= 
.0.  ) )  -> 
( N `  F
)  e.  RR* )
17 nmoi.3 . . . . . . . 8  |-  L  =  ( norm `  S
)
18 nmoleub.z . . . . . . . 8  |-  .0.  =  ( 0g `  S )
193, 17, 18nmrpcl 18141 . . . . . . 7  |-  ( ( S  e. NrmGrp  /\  X  e.  V  /\  X  =/= 
.0.  )  ->  ( L `  X )  e.  RR+ )
20193expb 1152 . . . . . 6  |-  ( ( S  e. NrmGrp  /\  ( X  e.  V  /\  X  =/=  .0.  ) )  ->  ( L `  X )  e.  RR+ )
21203ad2antl1 1117 . . . . 5  |-  ( ( ( S  e. NrmGrp  /\  T  e. NrmGrp  /\  F  e.  ( S  GrpHom  T ) )  /\  ( X  e.  V  /\  X  =/= 
.0.  ) )  -> 
( L `  X
)  e.  RR+ )
2221rpxrd 10391 . . . 4  |-  ( ( ( S  e. NrmGrp  /\  T  e. NrmGrp  /\  F  e.  ( S  GrpHom  T ) )  /\  ( X  e.  V  /\  X  =/= 
.0.  ) )  -> 
( L `  X
)  e.  RR* )
2316, 22xmulcld 10622 . . 3  |-  ( ( ( S  e. NrmGrp  /\  T  e. NrmGrp  /\  F  e.  ( S  GrpHom  T ) )  /\  ( X  e.  V  /\  X  =/= 
.0.  ) )  -> 
( ( N `  F ) x e ( L `  X
) )  e.  RR* )
2421rpreccld 10400 . . . . 5  |-  ( ( ( S  e. NrmGrp  /\  T  e. NrmGrp  /\  F  e.  ( S  GrpHom  T ) )  /\  ( X  e.  V  /\  X  =/= 
.0.  ) )  -> 
( 1  /  ( L `  X )
)  e.  RR+ )
2524rpxrd 10391 . . . 4  |-  ( ( ( S  e. NrmGrp  /\  T  e. NrmGrp  /\  F  e.  ( S  GrpHom  T ) )  /\  ( X  e.  V  /\  X  =/= 
.0.  ) )  -> 
( 1  /  ( L `  X )
)  e.  RR* )
2624rpge0d 10394 . . . 4  |-  ( ( ( S  e. NrmGrp  /\  T  e. NrmGrp  /\  F  e.  ( S  GrpHom  T ) )  /\  ( X  e.  V  /\  X  =/= 
.0.  ) )  -> 
0  <_  ( 1  /  ( L `  X ) ) )
2725, 26jca 518 . . 3  |-  ( ( ( S  e. NrmGrp  /\  T  e. NrmGrp  /\  F  e.  ( S  GrpHom  T ) )  /\  ( X  e.  V  /\  X  =/= 
.0.  ) )  -> 
( ( 1  / 
( L `  X
) )  e.  RR*  /\  0  <_  ( 1  /  ( L `  X ) ) ) )
2814, 3, 17, 10nmoix 18238 . . . 4  |-  ( ( ( S  e. NrmGrp  /\  T  e. NrmGrp  /\  F  e.  ( S  GrpHom  T ) )  /\  X  e.  V
)  ->  ( M `  ( F `  X
) )  <_  (
( N `  F
) x e ( L `  X ) ) )
2928adantrr 697 . . 3  |-  ( ( ( S  e. NrmGrp  /\  T  e. NrmGrp  /\  F  e.  ( S  GrpHom  T ) )  /\  ( X  e.  V  /\  X  =/= 
.0.  ) )  -> 
( M `  ( F `  X )
)  <_  ( ( N `  F ) x e ( L `  X ) ) )
30 xlemul1a 10608 . . 3  |-  ( ( ( ( M `  ( F `  X ) )  e.  RR*  /\  (
( N `  F
) x e ( L `  X ) )  e.  RR*  /\  (
( 1  /  ( L `  X )
)  e.  RR*  /\  0  <_  ( 1  /  ( L `  X )
) ) )  /\  ( M `  ( F `
 X ) )  <_  ( ( N `
 F ) x e ( L `  X ) ) )  ->  ( ( M `
 ( F `  X ) ) x e ( 1  / 
( L `  X
) ) )  <_ 
( ( ( N `
 F ) x e ( L `  X ) ) x e ( 1  / 
( L `  X
) ) ) )
3113, 23, 27, 29, 30syl31anc 1185 . 2  |-  ( ( ( S  e. NrmGrp  /\  T  e. NrmGrp  /\  F  e.  ( S  GrpHom  T ) )  /\  ( X  e.  V  /\  X  =/= 
.0.  ) )  -> 
( ( M `  ( F `  X ) ) x e ( 1  /  ( L `
 X ) ) )  <_  ( (
( N `  F
) x e ( L `  X ) ) x e ( 1  /  ( L `
 X ) ) ) )
3224rpred 10390 . . . 4  |-  ( ( ( S  e. NrmGrp  /\  T  e. NrmGrp  /\  F  e.  ( S  GrpHom  T ) )  /\  ( X  e.  V  /\  X  =/= 
.0.  ) )  -> 
( 1  /  ( L `  X )
)  e.  RR )
33 rexmul 10591 . . . 4  |-  ( ( ( M `  ( F `  X )
)  e.  RR  /\  ( 1  /  ( L `  X )
)  e.  RR )  ->  ( ( M `
 ( F `  X ) ) x e ( 1  / 
( L `  X
) ) )  =  ( ( M `  ( F `  X ) )  x.  ( 1  /  ( L `  X ) ) ) )
3412, 32, 33syl2anc 642 . . 3  |-  ( ( ( S  e. NrmGrp  /\  T  e. NrmGrp  /\  F  e.  ( S  GrpHom  T ) )  /\  ( X  e.  V  /\  X  =/= 
.0.  ) )  -> 
( ( M `  ( F `  X ) ) x e ( 1  /  ( L `
 X ) ) )  =  ( ( M `  ( F `
 X ) )  x.  ( 1  / 
( L `  X
) ) ) )
3512recnd 8861 . . . 4  |-  ( ( ( S  e. NrmGrp  /\  T  e. NrmGrp  /\  F  e.  ( S  GrpHom  T ) )  /\  ( X  e.  V  /\  X  =/= 
.0.  ) )  -> 
( M `  ( F `  X )
)  e.  CC )
3621rpcnd 10392 . . . 4  |-  ( ( ( S  e. NrmGrp  /\  T  e. NrmGrp  /\  F  e.  ( S  GrpHom  T ) )  /\  ( X  e.  V  /\  X  =/= 
.0.  ) )  -> 
( L `  X
)  e.  CC )
3721rpne0d 10395 . . . 4  |-  ( ( ( S  e. NrmGrp  /\  T  e. NrmGrp  /\  F  e.  ( S  GrpHom  T ) )  /\  ( X  e.  V  /\  X  =/= 
.0.  ) )  -> 
( L `  X
)  =/=  0 )
3835, 36, 37divrecd 9539 . . 3  |-  ( ( ( S  e. NrmGrp  /\  T  e. NrmGrp  /\  F  e.  ( S  GrpHom  T ) )  /\  ( X  e.  V  /\  X  =/= 
.0.  ) )  -> 
( ( M `  ( F `  X ) )  /  ( L `
 X ) )  =  ( ( M `
 ( F `  X ) )  x.  ( 1  /  ( L `  X )
) ) )
3934, 38eqtr4d 2318 . 2  |-  ( ( ( S  e. NrmGrp  /\  T  e. NrmGrp  /\  F  e.  ( S  GrpHom  T ) )  /\  ( X  e.  V  /\  X  =/= 
.0.  ) )  -> 
( ( M `  ( F `  X ) ) x e ( 1  /  ( L `
 X ) ) )  =  ( ( M `  ( F `
 X ) )  /  ( L `  X ) ) )
40 xmulass 10607 . . . 4  |-  ( ( ( N `  F
)  e.  RR*  /\  ( L `  X )  e.  RR*  /\  ( 1  /  ( L `  X ) )  e. 
RR* )  ->  (
( ( N `  F ) x e ( L `  X
) ) x e ( 1  /  ( L `  X )
) )  =  ( ( N `  F
) x e ( ( L `  X
) x e ( 1  /  ( L `
 X ) ) ) ) )
4116, 22, 25, 40syl3anc 1182 . . 3  |-  ( ( ( S  e. NrmGrp  /\  T  e. NrmGrp  /\  F  e.  ( S  GrpHom  T ) )  /\  ( X  e.  V  /\  X  =/= 
.0.  ) )  -> 
( ( ( N `
 F ) x e ( L `  X ) ) x e ( 1  / 
( L `  X
) ) )  =  ( ( N `  F ) x e ( ( L `  X ) x e ( 1  /  ( L `  X )
) ) ) )
4221rpred 10390 . . . . . 6  |-  ( ( ( S  e. NrmGrp  /\  T  e. NrmGrp  /\  F  e.  ( S  GrpHom  T ) )  /\  ( X  e.  V  /\  X  =/= 
.0.  ) )  -> 
( L `  X
)  e.  RR )
43 rexmul 10591 . . . . . 6  |-  ( ( ( L `  X
)  e.  RR  /\  ( 1  /  ( L `  X )
)  e.  RR )  ->  ( ( L `
 X ) x e ( 1  / 
( L `  X
) ) )  =  ( ( L `  X )  x.  (
1  /  ( L `
 X ) ) ) )
4442, 32, 43syl2anc 642 . . . . 5  |-  ( ( ( S  e. NrmGrp  /\  T  e. NrmGrp  /\  F  e.  ( S  GrpHom  T ) )  /\  ( X  e.  V  /\  X  =/= 
.0.  ) )  -> 
( ( L `  X ) x e ( 1  /  ( L `  X )
) )  =  ( ( L `  X
)  x.  ( 1  /  ( L `  X ) ) ) )
4536, 37recidd 9531 . . . . 5  |-  ( ( ( S  e. NrmGrp  /\  T  e. NrmGrp  /\  F  e.  ( S  GrpHom  T ) )  /\  ( X  e.  V  /\  X  =/= 
.0.  ) )  -> 
( ( L `  X )  x.  (
1  /  ( L `
 X ) ) )  =  1 )
4644, 45eqtrd 2315 . . . 4  |-  ( ( ( S  e. NrmGrp  /\  T  e. NrmGrp  /\  F  e.  ( S  GrpHom  T ) )  /\  ( X  e.  V  /\  X  =/= 
.0.  ) )  -> 
( ( L `  X ) x e ( 1  /  ( L `  X )
) )  =  1 )
4746oveq2d 5874 . . 3  |-  ( ( ( S  e. NrmGrp  /\  T  e. NrmGrp  /\  F  e.  ( S  GrpHom  T ) )  /\  ( X  e.  V  /\  X  =/= 
.0.  ) )  -> 
( ( N `  F ) x e ( ( L `  X ) x e ( 1  /  ( L `  X )
) ) )  =  ( ( N `  F ) x e 1 ) )
48 xmulid1 10599 . . . 4  |-  ( ( N `  F )  e.  RR*  ->  ( ( N `  F ) x e 1 )  =  ( N `  F ) )
4916, 48syl 15 . . 3  |-  ( ( ( S  e. NrmGrp  /\  T  e. NrmGrp  /\  F  e.  ( S  GrpHom  T ) )  /\  ( X  e.  V  /\  X  =/= 
.0.  ) )  -> 
( ( N `  F ) x e 1 )  =  ( N `  F ) )
5041, 47, 493eqtrd 2319 . 2  |-  ( ( ( S  e. NrmGrp  /\  T  e. NrmGrp  /\  F  e.  ( S  GrpHom  T ) )  /\  ( X  e.  V  /\  X  =/= 
.0.  ) )  -> 
( ( ( N `
 F ) x e ( L `  X ) ) x e ( 1  / 
( L `  X
) ) )  =  ( N `  F
) )
5131, 39, 503brtr3d 4052 1  |-  ( ( ( S  e. NrmGrp  /\  T  e. NrmGrp  /\  F  e.  ( S  GrpHom  T ) )  /\  ( X  e.  V  /\  X  =/= 
.0.  ) )  -> 
( ( M `  ( F `  X ) )  /  ( L `
 X ) )  <_  ( N `  F ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    /\ w3a 934    = wceq 1623    e. wcel 1684    =/= wne 2446   class class class wbr 4023   -->wf 5251   ` cfv 5255  (class class class)co 5858   RRcr 8736   0cc0 8737   1c1 8738    x. cmul 8742   RR*cxr 8866    <_ cle 8868    / cdiv 9423   RR+crp 10354   x ecxmu 10451   Basecbs 13148   0gc0g 13400    GrpHom cghm 14680   normcnm 18099  NrmGrpcngp 18100   normOpcnmo 18214
This theorem is referenced by:  nmoleub  18240
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-cnex 8793  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813  ax-pre-mulgt0 8814  ax-pre-sup 8815
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 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-riota 6304  df-recs 6388  df-rdg 6423  df-er 6660  df-map 6774  df-en 6864  df-dom 6865  df-sdom 6866  df-sup 7194  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-sub 9039  df-neg 9040  df-div 9424  df-nn 9747  df-2 9804  df-n0 9966  df-z 10025  df-uz 10231  df-q 10317  df-rp 10355  df-xneg 10452  df-xadd 10453  df-xmul 10454  df-ico 10662  df-topgen 13344  df-0g 13404  df-mnd 14367  df-grp 14489  df-ghm 14681  df-xmet 16373  df-met 16374  df-bl 16375  df-mopn 16376  df-top 16636  df-bases 16638  df-topon 16639  df-topsp 16640  df-xms 17885  df-ms 17886  df-nm 18105  df-ngp 18106  df-nmo 18217  df-nghm 18218
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