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Theorem nmolb2d 18227
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 5525 . . . . . 6  |-  ( x  =  .0.  ->  ( F `  x )  =  ( F `  .0.  ) )
21fveq2d 5529 . . . . 5  |-  ( x  =  .0.  ->  ( M `  ( F `  x ) )  =  ( M `  ( F `  .0.  ) ) )
3 fveq2 5525 . . . . . 6  |-  ( x  =  .0.  ->  ( L `  x )  =  ( L `  .0.  ) )
43oveq2d 5874 . . . . 5  |-  ( x  =  .0.  ->  ( A  x.  ( L `  x ) )  =  ( A  x.  ( L `  .0.  ) ) )
52, 4breq12d 4036 . . . 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 9827 . . . . . . 7  |-  0  <_  0
9 nmolb2d.4 . . . . . . . . 9  |-  ( ph  ->  A  e.  RR )
109recnd 8861 . . . . . . . 8  |-  ( ph  ->  A  e.  CC )
1110mul01d 9011 . . . . . . 7  |-  ( ph  ->  ( A  x.  0 )  =  0 )
128, 11syl5breqr 4059 . . . . . 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 2283 . . . . . . . . . 10  |-  ( 0g
`  T )  =  ( 0g `  T
)
1614, 15ghmid 14689 . . . . . . . . 9  |-  ( F  e.  ( S  GrpHom  T )  ->  ( F `  .0.  )  =  ( 0g `  T ) )
1713, 16syl 15 . . . . . . . 8  |-  ( ph  ->  ( F `  .0.  )  =  ( 0g `  T ) )
1817fveq2d 5529 . . . . . . 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 18148 . . . . . . . 8  |-  ( T  e. NrmGrp  ->  ( M `  ( 0g `  T ) )  =  0 )
2219, 21syl 15 . . . . . . 7  |-  ( ph  ->  ( M `  ( 0g `  T ) )  =  0 )
2318, 22eqtrd 2315 . . . . . 6  |-  ( ph  ->  ( M `  ( F `  .0.  ) )  =  0 )
24 nmolb2d.1 . . . . . . . 8  |-  ( ph  ->  S  e. NrmGrp )
25 nmofval.3 . . . . . . . . 9  |-  L  =  ( norm `  S
)
2625, 14nm0 18148 . . . . . . . 8  |-  ( S  e. NrmGrp  ->  ( L `  .0.  )  =  0
)
2724, 26syl 15 . . . . . . 7  |-  ( ph  ->  ( L `  .0.  )  =  0 )
2827oveq2d 5874 . . . . . 6  |-  ( ph  ->  ( A  x.  ( L `  .0.  ) )  =  ( A  x.  0 ) )
2912, 23, 283brtr4d 4053 . . . . 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 2521 . . 3  |-  ( (
ph  /\  x  e.  V )  ->  ( M `  ( F `  x ) )  <_ 
( A  x.  ( L `  x )
) )
3231ralrimiva 2626 . 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 18226 . . 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 1623    e. wcel 1684    =/= wne 2446   A.wral 2543   class class class wbr 4023   ` cfv 5255  (class class class)co 5858   RRcr 8736   0cc0 8737    x. cmul 8742    <_ cle 8868   Basecbs 13148   0gc0g 13400    GrpHom cghm 14680   normcnm 18099  NrmGrpcngp 18100   normOpcnmo 18214
This theorem is referenced by:  nmo0  18244  nmoco  18246  nmotri  18248  nmoid  18251  nmoleub2lem  18595
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-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
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