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

Theorem nmo0 16932
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 2064 . . 3  |-  ( norm `  S )  =  (
norm `  S )
4 eqid 2064 . . 3  |-  ( norm `  T )  =  (
norm `  T )
5 eqid 2064 . . 3  |-  ( 0g
`  S )  =  ( 0g `  S
)
6 simpl 437 . . 3  |-  ( ( S  e. NrmGrp  /\  T  e. NrmGrp
)  ->  S  e. NrmGrp )
7 simpr 441 . . 3  |-  ( ( S  e. NrmGrp  /\  T  e. NrmGrp
)  ->  T  e. NrmGrp )
8 ngpgrp 16809 . . . 4  |-  ( S  e. NrmGrp  ->  S  e.  Grp )
9 ngpgrp 16809 . . . 4  |-  ( T  e. NrmGrp  ->  T  e.  Grp )
10 nmo0.3 . . . . 5  |-  .0.  =  ( 0g `  T )
1110, 20ghm 13406 . . . 4  |-  ( ( S  e.  Grp  /\  T  e.  Grp )  ->  ( V  X.  {  .0.  } )  e.  ( S  GrpHom  T ) )
128, 9, 11syl2an 457 . . 3  |-  ( ( S  e. NrmGrp  /\  T  e. NrmGrp
)  ->  ( V  X.  {  .0.  } )  e.  ( S  GrpHom  T ) )
13 0re 8256 . . . 4  |-  0  e.  RR
1413a1i 10 . . 3  |-  ( ( S  e. NrmGrp  /\  T  e. NrmGrp
)  ->  0  e.  RR )
1513leidi 8706 . . . 4  |-  0  <_  0
1615a1i 10 . . 3  |-  ( ( S  e. NrmGrp  /\  T  e. NrmGrp
)  ->  0  <_  0 )
17 fvex 5058 . . . . . . . . 9  |-  ( 0g
`  T )  e. 
_V
1810, 17eqeltri 2134 . . . . . . . 8  |-  .0.  e.  _V
1918fvconst2 5257 . . . . . . 7  |-  ( x  e.  V  ->  (
( V  X.  {  .0.  } ) `  x
)  =  .0.  )
2019ad2antrl 700 . . . . . 6  |-  ( ( ( S  e. NrmGrp  /\  T  e. NrmGrp )  /\  ( x  e.  V  /\  x  =/=  ( 0g `  S
) ) )  -> 
( ( V  X.  {  .0.  } ) `  x )  =  .0.  )
2120fveq2d 5048 . . . . 5  |-  ( ( ( S  e. NrmGrp  /\  T  e. NrmGrp )  /\  ( x  e.  V  /\  x  =/=  ( 0g `  S
) ) )  -> 
( ( norm `  T
) `  ( ( V  X.  {  .0.  }
) `  x )
)  =  ( (
norm `  T ) `  .0.  ) )
224, 10nm0 16836 . . . . . 6  |-  ( T  e. NrmGrp  ->  ( ( norm `  T ) `  .0.  )  =  0 )
2322ad2antlr 699 . . . . 5  |-  ( ( ( S  e. NrmGrp  /\  T  e. NrmGrp )  /\  ( x  e.  V  /\  x  =/=  ( 0g `  S
) ) )  -> 
( ( norm `  T
) `  .0.  )  =  0 )
2421, 23eqtrd 2096 . . . 4  |-  ( ( ( S  e. NrmGrp  /\  T  e. NrmGrp )  /\  ( x  e.  V  /\  x  =/=  ( 0g `  S
) ) )  -> 
( ( norm `  T
) `  ( ( V  X.  {  .0.  }
) `  x )
)  =  0 )
252, 3nmcl 16825 . . . . . . . 8  |-  ( ( S  e. NrmGrp  /\  x  e.  V )  ->  (
( norm `  S ) `  x )  e.  RR )
2625ad2ant2r 703 . . . . . . 7  |-  ( ( ( S  e. NrmGrp  /\  T  e. NrmGrp )  /\  ( x  e.  V  /\  x  =/=  ( 0g `  S
) ) )  -> 
( ( norm `  S
) `  x )  e.  RR )
2726recnd 8279 . . . . . 6  |-  ( ( ( S  e. NrmGrp  /\  T  e. NrmGrp )  /\  ( x  e.  V  /\  x  =/=  ( 0g `  S
) ) )  -> 
( ( norm `  S
) `  x )  e.  CC )
2827mul02d 8427 . . . . 5  |-  ( ( ( S  e. NrmGrp  /\  T  e. NrmGrp )  /\  ( x  e.  V  /\  x  =/=  ( 0g `  S
) ) )  -> 
( 0  x.  (
( norm `  S ) `  x ) )  =  0 )
2915, 28syl5breqr 3622 . . . 4  |-  ( ( ( S  e. NrmGrp  /\  T  e. NrmGrp )  /\  ( x  e.  V  /\  x  =/=  ( 0g `  S
) ) )  -> 
0  <_  ( 0  x.  ( ( norm `  S ) `  x
) ) )
3024, 29eqbrtrd 3606 . . 3  |-  ( ( ( S  e. NrmGrp  /\  T  e. NrmGrp )  /\  ( x  e.  V  /\  x  =/=  ( 0g `  S
) ) )  -> 
( ( norm `  T
) `  ( ( V  X.  {  .0.  }
) `  x )
)  <_  ( 0  x.  ( ( norm `  S ) `  x
) ) )
311, 2, 3, 4, 5, 6, 7, 12, 14, 16, 30nmolb2d 16915 . 2  |-  ( ( S  e. NrmGrp  /\  T  e. NrmGrp
)  ->  ( N `  ( V  X.  {  .0.  } ) )  <_ 
0 )
321nmoge0 16918 . . 3  |-  ( ( S  e. NrmGrp  /\  T  e. NrmGrp  /\  ( V  X.  {  .0.  } )  e.  ( S  GrpHom  T ) )  ->  0  <_  ( N `  ( V  X.  {  .0.  } ) ) )
3312, 32mpd3an3 1236 . 2  |-  ( ( S  e. NrmGrp  /\  T  e. NrmGrp
)  ->  0  <_  ( N `  ( V  X.  {  .0.  }
) ) )
341nmocl 16917 . . . 4  |-  ( ( S  e. NrmGrp  /\  T  e. NrmGrp  /\  ( V  X.  {  .0.  } )  e.  ( S  GrpHom  T ) )  ->  ( N `  ( V  X.  {  .0.  } ) )  e.  RR* )
3512, 34mpd3an3 1236 . . 3  |-  ( ( S  e. NrmGrp  /\  T  e. NrmGrp
)  ->  ( N `  ( V  X.  {  .0.  } ) )  e. 
RR* )
36 0xr 8296 . . 3  |-  0  e.  RR*
37 xrletri3 9855 . . 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.  } ) ) ) ) )
3835, 36, 37sylancl 636 . 2  |-  ( ( S  e. NrmGrp  /\  T  e. NrmGrp
)  ->  ( ( N `  ( V  X.  {  .0.  } ) )  =  0  <->  (
( N `  ( V  X.  {  .0.  }
) )  <_  0  /\  0  <_  ( N `
 ( V  X.  {  .0.  } ) ) ) ) )
3931, 33, 38mpbir2and 850 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 1520    e. wcel 1522    =/= wne 2182   _Vcvv 2475   {csn 3255   class class class wbr 3586    X. cxp 4254   ` cfv 4268  (class class class)co 5360   RRcr 8157   0cc0 8158    x. cmul 8163    <_ cle 8282   RR*cxr 8285   Basecbs 12252   0gc0g 12484   Grpcgrp 12913    GrpHom cghm 13389   normcnm 16787  NrmGrpcngp 16788   normOpcnmo 16902
This theorem is referenced by:  nmoeq0  16933  0nghm  16938  idnghm  16940
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-5 1442  ax-6 1443  ax-7 1444  ax-gen 1445  ax-8 1524  ax-11 1525  ax-13 1526  ax-14 1527  ax-17 1529  ax-12o 1562  ax-10 1576  ax-9 1582  ax-4 1589  ax-16 1775  ax-ext 2046  ax-rep 3691  ax-sep 3701  ax-nul 3709  ax-pow 3745  ax-pr 3769  ax-un 4061  ax-cnex 8213  ax-resscn 8214  ax-1cn 8215  ax-icn 8216  ax-addcl 8217  ax-addrcl 8218  ax-mulcl 8219  ax-mulrcl 8220  ax-mulcom 8221  ax-addass 8222  ax-mulass 8223  ax-distr 8224  ax-i2m1 8225  ax-1ne0 8226  ax-1rid 8227  ax-rnegex 8228  ax-rrecex 8229  ax-cnre 8230  ax-pre-lttri 8231  ax-pre-lttrn 8232  ax-pre-ltadd 8233  ax-pre-mulgt0 8234  ax-pre-sup 8235
This theorem depends on definitions:  df-bi 175  df-or 357  df-an 358  df-3or 897  df-3an 898  df-tru 1259  df-ex 1447  df-sb 1736  df-eu 1958  df-mo 1959  df-clab 2052  df-cleq 2057  df-clel 2060  df-ne 2184  df-nel 2185  df-ral 2278  df-rex 2279  df-reu 2280  df-rab 2281  df-v 2477  df-sbc 2651  df-csb 2733  df-dif 2796  df-un 2798  df-in 2800  df-ss 2804  df-pss 2806  df-nul 3073  df-if 3182  df-pw 3243  df-sn 3261  df-pr 3262  df-tp 3263  df-op 3264  df-uni 3425  df-iun 3502  df-br 3587  df-opab 3641  df-mpt 3642  df-tr 3674  df-eprel 3856  df-id 3860  df-po 3865  df-so 3866  df-fr 3903  df-we 3905  df-ord 3946  df-on 3947  df-lim 3948  df-suc 3949  df-om 4224  df-xp 4270  df-rel 4271  df-cnv 4272  df-co 4273  df-dm 4274  df-rn 4275  df-res 4276  df-ima 4277  df-fun 4278  df-fn 4279  df-f 4280  df-f1 4281  df-fo 4282  df-f1o 4283  df-fv 4284  df-ov 5363  df-oprab 5364  df-mpt2 5365  df-1st 5614  df-2nd 5615  df-iota 5770  df-recs 5843  df-rdg 5878  df-er 6115  df-map 6219  df-en 6302  df-dom 6303  df-sdom 6304  df-riota 6468  df-sup 6676  df-pnf 8287  df-mnf 8288  df-xr 8289  df-ltxr 8290  df-le 8291  df-sub 8452  df-neg 8453  df-div 8816  df-n 9125  df-2 9182  df-n0 9335  df-z 9394  df-uz 9600  df-q 9686  df-rp 9724  df-xneg 9821  df-xadd 9822  df-xmul 9823  df-ico 10030  df-topgen 12432  df-0g 12488  df-mnd 12918  df-mhm 12969  df-grp 13198  df-ghm 13390  df-xmet 15080  df-met 15081  df-bl 15082  df-mopn 15083  df-top 15342  df-bases 15344  df-topon 15345  df-topsp 15346  df-xms 16572  df-ms 16573  df-nm 16793  df-ngp 16794  df-nmo 16905
Copyright terms: Public domain