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Theorem nmolb2d 21350
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 5872 . . . . . 6  |-  ( x  =  .0.  ->  ( F `  x )  =  ( F `  .0.  ) )
21fveq2d 5876 . . . . 5  |-  ( x  =  .0.  ->  ( M `  ( F `  x ) )  =  ( M `  ( F `  .0.  ) ) )
3 fveq2 5872 . . . . . 6  |-  ( x  =  .0.  ->  ( L `  x )  =  ( L `  .0.  ) )
43oveq2d 6312 . . . . 5  |-  ( x  =  .0.  ->  ( A  x.  ( L `  x ) )  =  ( A  x.  ( L `  .0.  ) ) )
52, 4breq12d 4469 . . . 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 648 . . . 4  |-  ( ( ( ph  /\  x  e.  V )  /\  x  =/=  .0.  )  ->  ( M `  ( F `  x ) )  <_ 
( A  x.  ( L `  x )
) )
8 0le0 10646 . . . . . . 7  |-  0  <_  0
9 nmolb2d.4 . . . . . . . . 9  |-  ( ph  ->  A  e.  RR )
109recnd 9639 . . . . . . . 8  |-  ( ph  ->  A  e.  CC )
1110mul01d 9796 . . . . . . 7  |-  ( ph  ->  ( A  x.  0 )  =  0 )
128, 11syl5breqr 4492 . . . . . 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 2457 . . . . . . . . . 10  |-  ( 0g
`  T )  =  ( 0g `  T
)
1614, 15ghmid 16399 . . . . . . . . 9  |-  ( F  e.  ( S  GrpHom  T )  ->  ( F `  .0.  )  =  ( 0g `  T ) )
1713, 16syl 16 . . . . . . . 8  |-  ( ph  ->  ( F `  .0.  )  =  ( 0g `  T ) )
1817fveq2d 5876 . . . . . . 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 21271 . . . . . . . 8  |-  ( T  e. NrmGrp  ->  ( M `  ( 0g `  T ) )  =  0 )
2219, 21syl 16 . . . . . . 7  |-  ( ph  ->  ( M `  ( 0g `  T ) )  =  0 )
2318, 22eqtrd 2498 . . . . . 6  |-  ( ph  ->  ( M `  ( F `  .0.  ) )  =  0 )
24 nmolb2d.1 . . . . . . . 8  |-  ( ph  ->  S  e. NrmGrp )
25 nmofval.3 . . . . . . . . 9  |-  L  =  ( norm `  S
)
2625, 14nm0 21271 . . . . . . . 8  |-  ( S  e. NrmGrp  ->  ( L `  .0.  )  =  0
)
2724, 26syl 16 . . . . . . 7  |-  ( ph  ->  ( L `  .0.  )  =  0 )
2827oveq2d 6312 . . . . . 6  |-  ( ph  ->  ( A  x.  ( L `  .0.  ) )  =  ( A  x.  0 ) )
2912, 23, 283brtr4d 4486 . . . . 5  |-  ( ph  ->  ( M `  ( F `  .0.  ) )  <_  ( A  x.  ( L `  .0.  )
) )
3029adantr 465 . . . 4  |-  ( (
ph  /\  x  e.  V )  ->  ( M `  ( F `  .0.  ) )  <_ 
( A  x.  ( L `  .0.  ) ) )
315, 7, 30pm2.61ne 2772 . . 3  |-  ( (
ph  /\  x  e.  V )  ->  ( M `  ( F `  x ) )  <_ 
( A  x.  ( L `  x )
) )
3231ralrimiva 2871 . 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 21349 . . 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 1242 . 2  |-  ( ph  ->  ( A. x  e.  V  ( M `  ( F `  x ) )  <_  ( A  x.  ( L `  x
) )  ->  ( N `  F )  <_  A ) )
3832, 37mpd 15 1  |-  ( ph  ->  ( N `  F
)  <_  A )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1395    e. wcel 1819    =/= wne 2652   A.wral 2807   class class class wbr 4456   ` cfv 5594  (class class class)co 6296   RRcr 9508   0cc0 9509    x. cmul 9514    <_ cle 9646   Basecbs 14643   0gc0g 14856    GrpHom cghm 16390   normcnm 21222  NrmGrpcngp 21223   normOpcnmo 21337
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-rep 4568  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695  ax-un 6591  ax-cnex 9565  ax-resscn 9566  ax-1cn 9567  ax-icn 9568  ax-addcl 9569  ax-addrcl 9570  ax-mulcl 9571  ax-mulrcl 9572  ax-mulcom 9573  ax-addass 9574  ax-mulass 9575  ax-distr 9576  ax-i2m1 9577  ax-1ne0 9578  ax-1rid 9579  ax-rnegex 9580  ax-rrecex 9581  ax-cnre 9582  ax-pre-lttri 9583  ax-pre-lttrn 9584  ax-pre-ltadd 9585  ax-pre-mulgt0 9586  ax-pre-sup 9587
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-nel 2655  df-ral 2812  df-rex 2813  df-reu 2814  df-rmo 2815  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-pss 3487  df-nul 3794  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-tp 4037  df-op 4039  df-uni 4252  df-iun 4334  df-br 4457  df-opab 4516  df-mpt 4517  df-tr 4551  df-eprel 4800  df-id 4804  df-po 4809  df-so 4810  df-fr 4847  df-we 4849  df-ord 4890  df-on 4891  df-lim 4892  df-suc 4893  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-riota 6258  df-ov 6299  df-oprab 6300  df-mpt2 6301  df-om 6700  df-1st 6799  df-2nd 6800  df-recs 7060  df-rdg 7094  df-er 7329  df-map 7440  df-en 7536  df-dom 7537  df-sdom 7538  df-sup 7919  df-pnf 9647  df-mnf 9648  df-xr 9649  df-ltxr 9650  df-le 9651  df-sub 9826  df-neg 9827  df-div 10228  df-nn 10557  df-2 10615  df-n0 10817  df-z 10886  df-uz 11107  df-q 11208  df-rp 11246  df-xneg 11343  df-xadd 11344  df-xmul 11345  df-ico 11560  df-0g 14858  df-topgen 14860  df-mgm 15998  df-sgrp 16037  df-mnd 16047  df-grp 16183  df-ghm 16391  df-psmet 18537  df-xmet 18538  df-bl 18540  df-mopn 18541  df-top 19525  df-bases 19527  df-topon 19528  df-topsp 19529  df-xms 20948  df-ms 20949  df-nm 21228  df-ngp 21229  df-nmo 21340
This theorem is referenced by:  nmo0  21367  nmoco  21369  nmotri  21371  nmoid  21374  nmoleub2lem  21722
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