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Theorem nmoi2 18756
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 961 . . . . 5  |-  ( ( ( S  e. NrmGrp  /\  T  e. NrmGrp  /\  F  e.  ( S  GrpHom  T ) )  /\  ( X  e.  V  /\  X  =/= 
.0.  ) )  ->  T  e. NrmGrp )
2 simpl3 962 . . . . . . 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 2435 . . . . . . . 8  |-  ( Base `  T )  =  (
Base `  T )
53, 4ghmf 15002 . . . . . . 7  |-  ( F  e.  ( S  GrpHom  T )  ->  F : V
--> ( Base `  T
) )
62, 5syl 16 . . . . . 6  |-  ( ( ( S  e. NrmGrp  /\  T  e. NrmGrp  /\  F  e.  ( S  GrpHom  T ) )  /\  ( X  e.  V  /\  X  =/= 
.0.  ) )  ->  F : V --> ( Base `  T ) )
7 simprl 733 . . . . . 6  |-  ( ( ( S  e. NrmGrp  /\  T  e. NrmGrp  /\  F  e.  ( S  GrpHom  T ) )  /\  ( X  e.  V  /\  X  =/= 
.0.  ) )  ->  X  e.  V )
86, 7ffvelrnd 5863 . . . . 5  |-  ( ( ( S  e. NrmGrp  /\  T  e. NrmGrp  /\  F  e.  ( S  GrpHom  T ) )  /\  ( X  e.  V  /\  X  =/= 
.0.  ) )  -> 
( F `  X
)  e.  ( Base `  T ) )
9 nmoi.4 . . . . . 6  |-  M  =  ( norm `  T
)
104, 9nmcl 18654 . . . . 5  |-  ( ( T  e. NrmGrp  /\  ( F `  X )  e.  ( Base `  T
) )  ->  ( M `  ( F `  X ) )  e.  RR )
111, 8, 10syl2anc 643 . . . 4  |-  ( ( ( S  e. NrmGrp  /\  T  e. NrmGrp  /\  F  e.  ( S  GrpHom  T ) )  /\  ( X  e.  V  /\  X  =/= 
.0.  ) )  -> 
( M `  ( F `  X )
)  e.  RR )
1211rexrd 9126 . . 3  |-  ( ( ( S  e. NrmGrp  /\  T  e. NrmGrp  /\  F  e.  ( S  GrpHom  T ) )  /\  ( X  e.  V  /\  X  =/= 
.0.  ) )  -> 
( M `  ( F `  X )
)  e.  RR* )
13 nmofval.1 . . . . . 6  |-  N  =  ( S normOp T )
1413nmocl 18746 . . . . 5  |-  ( ( S  e. NrmGrp  /\  T  e. NrmGrp  /\  F  e.  ( S  GrpHom  T ) )  ->  ( N `  F )  e.  RR* )
1514adantr 452 . . . 4  |-  ( ( ( S  e. NrmGrp  /\  T  e. NrmGrp  /\  F  e.  ( S  GrpHom  T ) )  /\  ( X  e.  V  /\  X  =/= 
.0.  ) )  -> 
( N `  F
)  e.  RR* )
16 nmoi.3 . . . . . . . 8  |-  L  =  ( norm `  S
)
17 nmoleub.z . . . . . . . 8  |-  .0.  =  ( 0g `  S )
183, 16, 17nmrpcl 18658 . . . . . . 7  |-  ( ( S  e. NrmGrp  /\  X  e.  V  /\  X  =/= 
.0.  )  ->  ( L `  X )  e.  RR+ )
19183expb 1154 . . . . . 6  |-  ( ( S  e. NrmGrp  /\  ( X  e.  V  /\  X  =/=  .0.  ) )  ->  ( L `  X )  e.  RR+ )
20193ad2antl1 1119 . . . . 5  |-  ( ( ( S  e. NrmGrp  /\  T  e. NrmGrp  /\  F  e.  ( S  GrpHom  T ) )  /\  ( X  e.  V  /\  X  =/= 
.0.  ) )  -> 
( L `  X
)  e.  RR+ )
2120rpxrd 10641 . . . 4  |-  ( ( ( S  e. NrmGrp  /\  T  e. NrmGrp  /\  F  e.  ( S  GrpHom  T ) )  /\  ( X  e.  V  /\  X  =/= 
.0.  ) )  -> 
( L `  X
)  e.  RR* )
2215, 21xmulcld 10873 . . 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* )
2320rpreccld 10650 . . . . 5  |-  ( ( ( S  e. NrmGrp  /\  T  e. NrmGrp  /\  F  e.  ( S  GrpHom  T ) )  /\  ( X  e.  V  /\  X  =/= 
.0.  ) )  -> 
( 1  /  ( L `  X )
)  e.  RR+ )
2423rpxrd 10641 . . . 4  |-  ( ( ( S  e. NrmGrp  /\  T  e. NrmGrp  /\  F  e.  ( S  GrpHom  T ) )  /\  ( X  e.  V  /\  X  =/= 
.0.  ) )  -> 
( 1  /  ( L `  X )
)  e.  RR* )
2523rpge0d 10644 . . . 4  |-  ( ( ( S  e. NrmGrp  /\  T  e. NrmGrp  /\  F  e.  ( S  GrpHom  T ) )  /\  ( X  e.  V  /\  X  =/= 
.0.  ) )  -> 
0  <_  ( 1  /  ( L `  X ) ) )
2624, 25jca 519 . . 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 ) ) ) )
2713, 3, 16, 9nmoix 18755 . . . 4  |-  ( ( ( S  e. NrmGrp  /\  T  e. NrmGrp  /\  F  e.  ( S  GrpHom  T ) )  /\  X  e.  V
)  ->  ( M `  ( F `  X
) )  <_  (
( N `  F
) x e ( L `  X ) ) )
2827adantrr 698 . . 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 ) ) )
29 xlemul1a 10859 . . 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
) ) ) )
3012, 22, 26, 28, 29syl31anc 1187 . 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 ) ) ) )
3123rpred 10640 . . . 4  |-  ( ( ( S  e. NrmGrp  /\  T  e. NrmGrp  /\  F  e.  ( S  GrpHom  T ) )  /\  ( X  e.  V  /\  X  =/= 
.0.  ) )  -> 
( 1  /  ( L `  X )
)  e.  RR )
32 rexmul 10842 . . . 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 ) ) ) )
3311, 31, 32syl2anc 643 . . 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
) ) ) )
3411recnd 9106 . . . 4  |-  ( ( ( S  e. NrmGrp  /\  T  e. NrmGrp  /\  F  e.  ( S  GrpHom  T ) )  /\  ( X  e.  V  /\  X  =/= 
.0.  ) )  -> 
( M `  ( F `  X )
)  e.  CC )
3520rpcnd 10642 . . . 4  |-  ( ( ( S  e. NrmGrp  /\  T  e. NrmGrp  /\  F  e.  ( S  GrpHom  T ) )  /\  ( X  e.  V  /\  X  =/= 
.0.  ) )  -> 
( L `  X
)  e.  CC )
3620rpne0d 10645 . . . 4  |-  ( ( ( S  e. NrmGrp  /\  T  e. NrmGrp  /\  F  e.  ( S  GrpHom  T ) )  /\  ( X  e.  V  /\  X  =/= 
.0.  ) )  -> 
( L `  X
)  =/=  0 )
3734, 35, 36divrecd 9785 . . 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 )
) ) )
3833, 37eqtr4d 2470 . 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 ) ) )
39 xmulass 10858 . . . 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 ) ) ) ) )
4015, 21, 24, 39syl3anc 1184 . . 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 )
) ) ) )
4120rpred 10640 . . . . . 6  |-  ( ( ( S  e. NrmGrp  /\  T  e. NrmGrp  /\  F  e.  ( S  GrpHom  T ) )  /\  ( X  e.  V  /\  X  =/= 
.0.  ) )  -> 
( L `  X
)  e.  RR )
42 rexmul 10842 . . . . . 6  |-  ( ( ( L `  X
)  e.  RR  /\  ( 1  /  ( L `  X )
)  e.  RR )  ->  ( ( L `
 X ) x e ( 1  / 
( L `  X
) ) )  =  ( ( L `  X )  x.  (
1  /  ( L `
 X ) ) ) )
4341, 31, 42syl2anc 643 . . . . 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 ) ) ) )
4435, 36recidd 9777 . . . . 5  |-  ( ( ( S  e. NrmGrp  /\  T  e. NrmGrp  /\  F  e.  ( S  GrpHom  T ) )  /\  ( X  e.  V  /\  X  =/= 
.0.  ) )  -> 
( ( L `  X )  x.  (
1  /  ( L `
 X ) ) )  =  1 )
4543, 44eqtrd 2467 . . . 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 )
4645oveq2d 6089 . . 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 ) )
47 xmulid1 10850 . . . 4  |-  ( ( N `  F )  e.  RR*  ->  ( ( N `  F ) x e 1 )  =  ( N `  F ) )
4815, 47syl 16 . . 3  |-  ( ( ( S  e. NrmGrp  /\  T  e. NrmGrp  /\  F  e.  ( S  GrpHom  T ) )  /\  ( X  e.  V  /\  X  =/= 
.0.  ) )  -> 
( ( N `  F ) x e 1 )  =  ( N `  F ) )
4940, 46, 483eqtrd 2471 . 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
) )
5030, 38, 493brtr3d 4233 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 359    /\ w3a 936    = wceq 1652    e. wcel 1725    =/= wne 2598   class class class wbr 4204   -->wf 5442   ` cfv 5446  (class class class)co 6073   RRcr 8981   0cc0 8982   1c1 8983    x. cmul 8987   RR*cxr 9111    <_ cle 9113    / cdiv 9669   RR+crp 10604   x ecxmu 10701   Basecbs 13461   0gc0g 13715    GrpHom cghm 14995   normcnm 18616  NrmGrpcngp 18617   normOpcnmo 18731
This theorem is referenced by:  nmoleub  18757
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-rep 4312  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693  ax-cnex 9038  ax-resscn 9039  ax-1cn 9040  ax-icn 9041  ax-addcl 9042  ax-addrcl 9043  ax-mulcl 9044  ax-mulrcl 9045  ax-mulcom 9046  ax-addass 9047  ax-mulass 9048  ax-distr 9049  ax-i2m1 9050  ax-1ne0 9051  ax-1rid 9052  ax-rnegex 9053  ax-rrecex 9054  ax-cnre 9055  ax-pre-lttri 9056  ax-pre-lttrn 9057  ax-pre-ltadd 9058  ax-pre-mulgt0 9059  ax-pre-sup 9060
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-nel 2601  df-ral 2702  df-rex 2703  df-reu 2704  df-rmo 2705  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-pss 3328  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-tp 3814  df-op 3815  df-uni 4008  df-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  df-tr 4295  df-eprel 4486  df-id 4490  df-po 4495  df-so 4496  df-fr 4533  df-we 4535  df-ord 4576  df-on 4577  df-lim 4578  df-suc 4579  df-om 4838  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-ov 6076  df-oprab 6077  df-mpt2 6078  df-1st 6341  df-2nd 6342  df-riota 6541  df-recs 6625  df-rdg 6660  df-er 6897  df-map 7012  df-en 7102  df-dom 7103  df-sdom 7104  df-sup 7438  df-pnf 9114  df-mnf 9115  df-xr 9116  df-ltxr 9117  df-le 9118  df-sub 9285  df-neg 9286  df-div 9670  df-nn 9993  df-2 10050  df-n0 10214  df-z 10275  df-uz 10481  df-q 10567  df-rp 10605  df-xneg 10702  df-xadd 10703  df-xmul 10704  df-ico 10914  df-topgen 13659  df-0g 13719  df-mnd 14682  df-grp 14804  df-ghm 14996  df-psmet 16686  df-xmet 16687  df-met 16688  df-bl 16689  df-mopn 16690  df-top 16955  df-bases 16957  df-topon 16958  df-topsp 16959  df-xms 18342  df-ms 18343  df-nm 18622  df-ngp 18623  df-nmo 18734  df-nghm 18735
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