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Theorem nmoi2 18255
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 2296 . . . . . . . 8  |-  ( Base `  T )  =  (
Base `  T )
53, 4ghmf 14703 . . . . . . 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 5679 . . . . . 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 18153 . . . . 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 8897 . . 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 18245 . . . . 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 18157 . . . . . . 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 10407 . . . 4  |-  ( ( ( S  e. NrmGrp  /\  T  e. NrmGrp  /\  F  e.  ( S  GrpHom  T ) )  /\  ( X  e.  V  /\  X  =/= 
.0.  ) )  -> 
( L `  X
)  e.  RR* )
2316, 22xmulcld 10638 . . 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 10416 . . . . 5  |-  ( ( ( S  e. NrmGrp  /\  T  e. NrmGrp  /\  F  e.  ( S  GrpHom  T ) )  /\  ( X  e.  V  /\  X  =/= 
.0.  ) )  -> 
( 1  /  ( L `  X )
)  e.  RR+ )
2524rpxrd 10407 . . . 4  |-  ( ( ( S  e. NrmGrp  /\  T  e. NrmGrp  /\  F  e.  ( S  GrpHom  T ) )  /\  ( X  e.  V  /\  X  =/= 
.0.  ) )  -> 
( 1  /  ( L `  X )
)  e.  RR* )
2624rpge0d 10410 . . . 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 18254 . . . 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 10624 . . 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 10406 . . . 4  |-  ( ( ( S  e. NrmGrp  /\  T  e. NrmGrp  /\  F  e.  ( S  GrpHom  T ) )  /\  ( X  e.  V  /\  X  =/= 
.0.  ) )  -> 
( 1  /  ( L `  X )
)  e.  RR )
33 rexmul 10607 . . . 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 8877 . . . 4  |-  ( ( ( S  e. NrmGrp  /\  T  e. NrmGrp  /\  F  e.  ( S  GrpHom  T ) )  /\  ( X  e.  V  /\  X  =/= 
.0.  ) )  -> 
( M `  ( F `  X )
)  e.  CC )
3621rpcnd 10408 . . . 4  |-  ( ( ( S  e. NrmGrp  /\  T  e. NrmGrp  /\  F  e.  ( S  GrpHom  T ) )  /\  ( X  e.  V  /\  X  =/= 
.0.  ) )  -> 
( L `  X
)  e.  CC )
3721rpne0d 10411 . . . 4  |-  ( ( ( S  e. NrmGrp  /\  T  e. NrmGrp  /\  F  e.  ( S  GrpHom  T ) )  /\  ( X  e.  V  /\  X  =/= 
.0.  ) )  -> 
( L `  X
)  =/=  0 )
3835, 36, 37divrecd 9555 . . 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 2331 . 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 10623 . . . 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 10406 . . . . . 6  |-  ( ( ( S  e. NrmGrp  /\  T  e. NrmGrp  /\  F  e.  ( S  GrpHom  T ) )  /\  ( X  e.  V  /\  X  =/= 
.0.  ) )  -> 
( L `  X
)  e.  RR )
43 rexmul 10607 . . . . . 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 9547 . . . . 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 2328 . . . 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 5890 . . 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 10615 . . . 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 2332 . 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 4068 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 1632    e. wcel 1696    =/= wne 2459   class class class wbr 4039   -->wf 5267   ` cfv 5271  (class class class)co 5874   RRcr 8752   0cc0 8753   1c1 8754    x. cmul 8758   RR*cxr 8882    <_ cle 8884    / cdiv 9439   RR+crp 10370   x ecxmu 10467   Basecbs 13164   0gc0g 13416    GrpHom cghm 14696   normcnm 18115  NrmGrpcngp 18116   normOpcnmo 18230
This theorem is referenced by:  nmoleub  18256
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-met 16390  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  df-nghm 18234
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