HomeHome Metamath Proof Explorer < Previous   Next >
Related theorems
Unicode version

Theorem nmo0 16060
Description: The operator norm of the zero operator. (Contributed by Mario Carneiro, 20-Oct-2015.)
Hypotheses
Ref Expression
nmo0.1  |-  N  =  ( S normOp T )
nmo0.2  |-  V  =  ( Base `  S
)
nmo0.3  |-  .0.  =  ( 0g `  T )
Assertion
Ref Expression
nmo0  |-  ( ( S  e. NrmGrp  /\  T  e. NrmGrp
)  ->  ( N `  ( V  X.  {  .0.  } ) )  =  0 )

Proof of Theorem nmo0
StepHypRef Expression
1 nmo0.1 . . 3  |-  N  =  ( S normOp T )
2 nmo0.2 . . 3  |-  V  =  ( Base `  S
)
3 eqid 2061 . . 3  |-  ( norm `  S )  =  (
norm `  S )
4 eqid 2061 . . 3  |-  ( norm `  T )  =  (
norm `  T )
5 eqid 2061 . . 3  |-  ( 0g
`  S )  =  ( 0g `  S
)
6 simpl 436 . . 3  |-  ( ( S  e. NrmGrp  /\  T  e. NrmGrp
)  ->  S  e. NrmGrp )
7 simpr 440 . . 3  |-  ( ( S  e. NrmGrp  /\  T  e. NrmGrp
)  ->  T  e. NrmGrp )
8 ngpgrp 15937 . . . 4  |-  ( S  e. NrmGrp  ->  S  e.  Grp )
9 ngpgrp 15937 . . . 4  |-  ( T  e. NrmGrp  ->  T  e.  Grp )
10 nmo0.3 . . . . 5  |-  .0.  =  ( 0g `  T )
1110, 20ghm 12710 . . . 4  |-  ( ( S  e.  Grp  /\  T  e.  Grp )  ->  ( V  X.  {  .0.  } )  e.  ( S  GrpHom  T ) )
128, 9, 11syl2an 456 . . 3  |-  ( ( S  e. NrmGrp  /\  T  e. NrmGrp
)  ->  ( V  X.  {  .0.  } )  e.  ( S  GrpHom  T ) )
13 0re 8259 . . . 4  |-  0  e.  RR
1413a1i 10 . . 3  |-  ( ( S  e. NrmGrp  /\  T  e. NrmGrp
)  ->  0  e.  RR )
1513leidi 8535 . . . 4  |-  0  <_  0
1615a1i 10 . . 3  |-  ( ( S  e. NrmGrp  /\  T  e. NrmGrp
)  ->  0  <_  0 )
17 fvex 5052 . . . . . . . . 9  |-  ( 0g
`  T )  e. 
_V
1810, 17eqeltri 2131 . . . . . . . 8  |-  .0.  e.  _V
1918fvconst2 5250 . . . . . . 7  |-  ( x  e.  V  ->  (
( V  X.  {  .0.  } ) `  x
)  =  .0.  )
2019ad2antrl 698 . . . . . 6  |-  ( ( ( S  e. NrmGrp  /\  T  e. NrmGrp )  /\  ( x  e.  V  /\  x  =/=  ( 0g `  S
) ) )  -> 
( ( V  X.  {  .0.  } ) `  x )  =  .0.  )
2120fveq2d 5042 . . . . 5  |-  ( ( ( S  e. NrmGrp  /\  T  e. NrmGrp )  /\  ( x  e.  V  /\  x  =/=  ( 0g `  S
) ) )  -> 
( ( norm `  T
) `  ( ( V  X.  {  .0.  }
) `  x )
)  =  ( (
norm `  T ) `  .0.  ) )
224, 10nm0 15964 . . . . . 6  |-  ( T  e. NrmGrp  ->  ( ( norm `  T ) `  .0.  )  =  0 )
2322ad2antlr 697 . . . . 5  |-  ( ( ( S  e. NrmGrp  /\  T  e. NrmGrp )  /\  ( x  e.  V  /\  x  =/=  ( 0g `  S
) ) )  -> 
( ( norm `  T
) `  .0.  )  =  0 )
2421, 23eqtrd 2093 . . . 4  |-  ( ( ( S  e. NrmGrp  /\  T  e. NrmGrp )  /\  ( x  e.  V  /\  x  =/=  ( 0g `  S
) ) )  -> 
( ( norm `  T
) `  ( ( V  X.  {  .0.  }
) `  x )
)  =  0 )
252, 3nmcl 15953 . . . . . . . 8  |-  ( ( S  e. NrmGrp  /\  x  e.  V )  ->  (
( norm `  S ) `  x )  e.  RR )
2625ad2ant2r 701 . . . . . . 7  |-  ( ( ( S  e. NrmGrp  /\  T  e. NrmGrp )  /\  ( x  e.  V  /\  x  =/=  ( 0g `  S
) ) )  -> 
( ( norm `  S
) `  x )  e.  RR )
2726recnd 8239 . . . . . 6  |-  ( ( ( S  e. NrmGrp  /\  T  e. NrmGrp )  /\  ( x  e.  V  /\  x  =/=  ( 0g `  S
) ) )  -> 
( ( norm `  S
) `  x )  e.  CC )
28 mul02 8366 . . . . . 6  |-  ( ( ( norm `  S
) `  x )  e.  CC  ->  ( 0  x.  ( ( norm `  S ) `  x
) )  =  0 )
2927, 28syl 15 . . . . 5  |-  ( ( ( S  e. NrmGrp  /\  T  e. NrmGrp )  /\  ( x  e.  V  /\  x  =/=  ( 0g `  S
) ) )  -> 
( 0  x.  (
( norm `  S ) `  x ) )  =  0 )
3015, 29syl5breqr 3618 . . . 4  |-  ( ( ( S  e. NrmGrp  /\  T  e. NrmGrp )  /\  ( x  e.  V  /\  x  =/=  ( 0g `  S
) ) )  -> 
0  <_  ( 0  x.  ( ( norm `  S ) `  x
) ) )
3124, 30eqbrtrd 3602 . . 3  |-  ( ( ( S  e. NrmGrp  /\  T  e. NrmGrp )  /\  ( x  e.  V  /\  x  =/=  ( 0g `  S
) ) )  -> 
( ( norm `  T
) `  ( ( V  X.  {  .0.  }
) `  x )
)  <_  ( 0  x.  ( ( norm `  S ) `  x
) ) )
321, 2, 3, 4, 5, 6, 7, 12, 14, 16, 31nmolb2d 16043 . 2  |-  ( ( S  e. NrmGrp  /\  T  e. NrmGrp
)  ->  ( N `  ( V  X.  {  .0.  } ) )  <_ 
0 )
331nmoge0 16046 . . 3  |-  ( ( S  e. NrmGrp  /\  T  e. NrmGrp  /\  ( V  X.  {  .0.  } )  e.  ( S  GrpHom  T ) )  ->  0  <_  ( N `  ( V  X.  {  .0.  } ) ) )
3412, 33mpd3an3 1233 . 2  |-  ( ( S  e. NrmGrp  /\  T  e. NrmGrp
)  ->  0  <_  ( N `  ( V  X.  {  .0.  }
) ) )
351nmocl 16045 . . . 4  |-  ( ( S  e. NrmGrp  /\  T  e. NrmGrp  /\  ( V  X.  {  .0.  } )  e.  ( S  GrpHom  T ) )  ->  ( N `  ( V  X.  {  .0.  } ) )  e.  RR* )
3612, 35mpd3an3 1233 . . 3  |-  ( ( S  e. NrmGrp  /\  T  e. NrmGrp
)  ->  ( N `  ( V  X.  {  .0.  } ) )  e. 
RR* )
37 0xr 8274 . . 3  |-  0  e.  RR*
38 xrletri3 9505 . . 3  |-  ( ( ( N `  ( V  X.  {  .0.  }
) )  e.  RR*  /\  0  e.  RR* )  ->  ( ( N `  ( V  X.  {  .0.  } ) )  =  0  <-> 
( ( N `  ( V  X.  {  .0.  } ) )  <_  0  /\  0  <_  ( N `
 ( V  X.  {  .0.  } ) ) ) ) )
3936, 37, 38sylancl 634 . 2  |-  ( ( S  e. NrmGrp  /\  T  e. NrmGrp
)  ->  ( ( N `  ( V  X.  {  .0.  } ) )  =  0  <->  (
( N `  ( V  X.  {  .0.  }
) )  <_  0  /\  0  <_  ( N `
 ( V  X.  {  .0.  } ) ) ) ) )
4032, 34, 39mpbir2and 847 1  |-  ( ( S  e. NrmGrp  /\  T  e. NrmGrp
)  ->  ( N `  ( V  X.  {  .0.  } ) )  =  0 )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 174    /\ wa 356    = wceq 1517    e. wcel 1519    =/= wne 2179   _Vcvv 2472   {csn 3251   class class class wbr 3582    X. cxp 4250   ` cfv 4264  (class class class)co 5353   CCcc 8145   RRcr 8146   0cc0 8147    x. cmul 8152    <_ cle 8260   RR*cxr 8263   Basecbs 11630   0gc0g 11862   Grpcgrp 12226    GrpHom cghm 12693   normcnm 15915  NrmGrpcngp 15916   normOpcnmo 16030
This theorem is referenced by:  nmoeq0  16061  0nghm  16066  idnghm  16068
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-5 1439  ax-6 1440  ax-7 1441  ax-gen 1442  ax-8 1521  ax-11 1522  ax-13 1523  ax-14 1524  ax-17 1526  ax-12o 1559  ax-10 1573  ax-9 1579  ax-4 1586  ax-16 1772  ax-ext 2043  ax-rep 3687  ax-sep 3697  ax-nul 3705  ax-pow 3741  ax-pr 3765  ax-un 4057  ax-cnex 8202  ax-resscn 8203  ax-1cn 8204  ax-icn 8205  ax-addcl 8206  ax-addrcl 8207  ax-mulcl 8208  ax-mulrcl 8209  ax-mulcom 8210  ax-addass 8211  ax-mulass 8212  ax-distr 8213  ax-i2m1 8214  ax-1ne0 8215  ax-1rid 8216  ax-rnegex 8217  ax-rrecex 8218  ax-cnre 8219  ax-pre-lttri 8220  ax-pre-lttrn 8221  ax-pre-ltadd 8222  ax-pre-mulgt0 8223  ax-pre-sup 8224
This theorem depends on definitions:  df-bi 175  df-or 357  df-an 358  df-3or 894  df-3an 895  df-tru 1256  df-ex 1444  df-sb 1733  df-eu 1955  df-mo 1956  df-clab 2049  df-cleq 2054  df-clel 2057  df-ne 2181  df-nel 2182  df-ral 2275  df-rex 2276  df-reu 2277  df-rab 2278  df-v 2474  df-sbc 2648  df-csb 2730  df-dif 2793  df-un 2795  df-in 2797  df-ss 2801  df-pss 2803  df-nul 3070  df-if 3178  df-pw 3239  df-sn 3257  df-pr 3258  df-tp 3259  df-op 3260  df-uni 3421  df-iun 3498  df-br 3583  df-opab 3637  df-mpt 3638  df-tr 3670  df-eprel 3852  df-id 3856  df-po 3861  df-so 3862  df-fr 3899  df-we 3901  df-ord 3942  df-on 3943  df-lim 3944  df-suc 3945  df-om 4220  df-xp 4266  df-rel 4267  df-cnv 4268  df-co 4269  df-dm 4270  df-rn 4271  df-res 4272  df-ima 4273  df-fun 4274  df-fn 4275  df-f 4276  df-f1 4277  df-fo 4278  df-f1o 4279  df-fv 4280  df-ov 5356  df-oprab 5357  df-mpt2 5358  df-1st 5607  df-2nd 5608  df-iota 5762  df-recs 5835  df-rdg 5870  df-er 6107  df-map 6211  df-en 6294  df-dom 6295  df-sdom 6296  df-riota 6460  df-sup 6666  df-pnf 8265  df-mnf 8266  df-xr 8267  df-ltxr 8268  df-le 8269  df-sub 8402  df-neg 8403  df-div 8631  df-n 8873  df-2 8920  df-n0 9066  df-z 9118  df-uz 9315  df-q 9400  df-rp 9437  df-xneg 9471  df-xadd 9472  df-xmul 9473  df-ico 9677  df-topgen 11810  df-0g 11866  df-mnd 12231  df-mhm 12279  df-grp 12506  df-ghm 12694  df-xmet 14139  df-met 14140  df-bl 14141  df-mopn 14142  df-top 14470  df-bases 14472  df-topon 14473  df-topsp 14474  df-xms 15700  df-ms 15701  df-nm 15921  df-ngp 15922  df-nmo 16033
Copyright terms: Public domain