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Theorem nmo0 18240
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
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 nmo0.1 . . 3  |-  N  =  ( S normOp T )
2 nmo0.2 . . 3  |-  V  =  ( Base `  S
)
3 eqid 2284 . . 3  |-  ( norm `  S )  =  (
norm `  S )
4 eqid 2284 . . 3  |-  ( norm `  T )  =  (
norm `  T )
5 eqid 2284 . . 3  |-  ( 0g
`  S )  =  ( 0g `  S
)
6 simpl 443 . . 3  |-  ( ( S  e. NrmGrp  /\  T  e. NrmGrp
)  ->  S  e. NrmGrp )
7 simpr 447 . . 3  |-  ( ( S  e. NrmGrp  /\  T  e. NrmGrp
)  ->  T  e. NrmGrp )
8 ngpgrp 18117 . . . 4  |-  ( S  e. NrmGrp  ->  S  e.  Grp )
9 ngpgrp 18117 . . . 4  |-  ( T  e. NrmGrp  ->  T  e.  Grp )
10 nmo0.3 . . . . 5  |-  .0.  =  ( 0g `  T )
1110, 20ghm 14693 . . . 4  |-  ( ( S  e.  Grp  /\  T  e.  Grp )  ->  ( V  X.  {  .0.  } )  e.  ( S  GrpHom  T ) )
128, 9, 11syl2an 463 . . 3  |-  ( ( S  e. NrmGrp  /\  T  e. NrmGrp
)  ->  ( V  X.  {  .0.  } )  e.  ( S  GrpHom  T ) )
13 0re 8834 . . . 4  |-  0  e.  RR
1413a1i 10 . . 3  |-  ( ( S  e. NrmGrp  /\  T  e. NrmGrp
)  ->  0  e.  RR )
15 0le0 9823 . . . 4  |-  0  <_  0
1615a1i 10 . . 3  |-  ( ( S  e. NrmGrp  /\  T  e. NrmGrp
)  ->  0  <_  0 )
17 fvex 5500 . . . . . . . . 9  |-  ( 0g
`  T )  e. 
_V
1810, 17eqeltri 2354 . . . . . . . 8  |-  .0.  e.  _V
1918fvconst2 5691 . . . . . . 7  |-  ( x  e.  V  ->  (
( V  X.  {  .0.  } ) `  x
)  =  .0.  )
2019ad2antrl 708 . . . . . 6  |-  ( ( ( S  e. NrmGrp  /\  T  e. NrmGrp )  /\  ( x  e.  V  /\  x  =/=  ( 0g `  S
) ) )  -> 
( ( V  X.  {  .0.  } ) `  x )  =  .0.  )
2120fveq2d 5490 . . . . 5  |-  ( ( ( S  e. NrmGrp  /\  T  e. NrmGrp )  /\  ( x  e.  V  /\  x  =/=  ( 0g `  S
) ) )  -> 
( ( norm `  T
) `  ( ( V  X.  {  .0.  }
) `  x )
)  =  ( (
norm `  T ) `  .0.  ) )
224, 10nm0 18144 . . . . . 6  |-  ( T  e. NrmGrp  ->  ( ( norm `  T ) `  .0.  )  =  0 )
2322ad2antlr 707 . . . . 5  |-  ( ( ( S  e. NrmGrp  /\  T  e. NrmGrp )  /\  ( x  e.  V  /\  x  =/=  ( 0g `  S
) ) )  -> 
( ( norm `  T
) `  .0.  )  =  0 )
2421, 23eqtrd 2316 . . . 4  |-  ( ( ( S  e. NrmGrp  /\  T  e. NrmGrp )  /\  ( x  e.  V  /\  x  =/=  ( 0g `  S
) ) )  -> 
( ( norm `  T
) `  ( ( V  X.  {  .0.  }
) `  x )
)  =  0 )
252, 3nmcl 18133 . . . . . . . 8  |-  ( ( S  e. NrmGrp  /\  x  e.  V )  ->  (
( norm `  S ) `  x )  e.  RR )
2625ad2ant2r 727 . . . . . . 7  |-  ( ( ( S  e. NrmGrp  /\  T  e. NrmGrp )  /\  ( x  e.  V  /\  x  =/=  ( 0g `  S
) ) )  -> 
( ( norm `  S
) `  x )  e.  RR )
2726recnd 8857 . . . . . 6  |-  ( ( ( S  e. NrmGrp  /\  T  e. NrmGrp )  /\  ( x  e.  V  /\  x  =/=  ( 0g `  S
) ) )  -> 
( ( norm `  S
) `  x )  e.  CC )
2827mul02d 9006 . . . . 5  |-  ( ( ( S  e. NrmGrp  /\  T  e. NrmGrp )  /\  ( x  e.  V  /\  x  =/=  ( 0g `  S
) ) )  -> 
( 0  x.  (
( norm `  S ) `  x ) )  =  0 )
2915, 28syl5breqr 4060 . . . 4  |-  ( ( ( S  e. NrmGrp  /\  T  e. NrmGrp )  /\  ( x  e.  V  /\  x  =/=  ( 0g `  S
) ) )  -> 
0  <_  ( 0  x.  ( ( norm `  S ) `  x
) ) )
3024, 29eqbrtrd 4044 . . 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 18223 . 2  |-  ( ( S  e. NrmGrp  /\  T  e. NrmGrp
)  ->  ( N `  ( V  X.  {  .0.  } ) )  <_ 
0 )
321nmoge0 18226 . . 3  |-  ( ( S  e. NrmGrp  /\  T  e. NrmGrp  /\  ( V  X.  {  .0.  } )  e.  ( S  GrpHom  T ) )  ->  0  <_  ( N `  ( V  X.  {  .0.  } ) ) )
3312, 32mpd3an3 1278 . 2  |-  ( ( S  e. NrmGrp  /\  T  e. NrmGrp
)  ->  0  <_  ( N `  ( V  X.  {  .0.  }
) ) )
341nmocl 18225 . . . 4  |-  ( ( S  e. NrmGrp  /\  T  e. NrmGrp  /\  ( V  X.  {  .0.  } )  e.  ( S  GrpHom  T ) )  ->  ( N `  ( V  X.  {  .0.  } ) )  e.  RR* )
3512, 34mpd3an3 1278 . . 3  |-  ( ( S  e. NrmGrp  /\  T  e. NrmGrp
)  ->  ( N `  ( V  X.  {  .0.  } ) )  e. 
RR* )
36 0xr 8874 . . 3  |-  0  e.  RR*
37 xrletri3 10482 . . 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 643 . 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 888 1  |-  ( ( S  e. NrmGrp  /\  T  e. NrmGrp
)  ->  ( N `  ( V  X.  {  .0.  } ) )  =  0 )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    = wceq 1623    e. wcel 1685    =/= wne 2447   _Vcvv 2789   {csn 3641   class class class wbr 4024    X. cxp 4686   ` cfv 5221  (class class class)co 5820   RRcr 8732   0cc0 8733    x. cmul 8738   RR*cxr 8862    <_ cle 8864   Basecbs 13144   0gc0g 13396   Grpcgrp 14358    GrpHom cghm 14676   normcnm 18095  NrmGrpcngp 18096   normOpcnmo 18210
This theorem is referenced by:  nmoeq0  18241  0nghm  18246  idnghm  18248
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 1636  ax-8 1644  ax-13 1687  ax-14 1689  ax-6 1704  ax-7 1709  ax-11 1716  ax-12 1868  ax-ext 2265  ax-rep 4132  ax-sep 4142  ax-nul 4150  ax-pow 4187  ax-pr 4213  ax-un 4511  ax-cnex 8789  ax-resscn 8790  ax-1cn 8791  ax-icn 8792  ax-addcl 8793  ax-addrcl 8794  ax-mulcl 8795  ax-mulrcl 8796  ax-mulcom 8797  ax-addass 8798  ax-mulass 8799  ax-distr 8800  ax-i2m1 8801  ax-1ne0 8802  ax-1rid 8803  ax-rnegex 8804  ax-rrecex 8805  ax-cnre 8806  ax-pre-lttri 8807  ax-pre-lttrn 8808  ax-pre-ltadd 8809  ax-pre-mulgt0 8810  ax-pre-sup 8811
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 1631  df-eu 2148  df-mo 2149  df-clab 2271  df-cleq 2277  df-clel 2280  df-nfc 2409  df-ne 2449  df-nel 2450  df-ral 2549  df-rex 2550  df-reu 2551  df-rmo 2552  df-rab 2553  df-v 2791  df-sbc 2993  df-csb 3083  df-dif 3156  df-un 3158  df-in 3160  df-ss 3167  df-pss 3169  df-nul 3457  df-if 3567  df-pw 3628  df-sn 3647  df-pr 3648  df-tp 3649  df-op 3650  df-uni 3829  df-iun 3908  df-br 4025  df-opab 4079  df-mpt 4080  df-tr 4115  df-eprel 4304  df-id 4308  df-po 4313  df-so 4314  df-fr 4351  df-we 4353  df-ord 4394  df-on 4395  df-lim 4396  df-suc 4397  df-om 4656  df-xp 4694  df-rel 4695  df-cnv 4696  df-co 4697  df-dm 4698  df-rn 4699  df-res 4700  df-ima 4701  df-fun 5223  df-fn 5224  df-f 5225  df-f1 5226  df-fo 5227  df-f1o 5228  df-fv 5229  df-ov 5823  df-oprab 5824  df-mpt2 5825  df-1st 6084  df-2nd 6085  df-iota 6253  df-riota 6300  df-recs 6384  df-rdg 6419  df-er 6656  df-map 6770  df-en 6860  df-dom 6861  df-sdom 6862  df-sup 7190  df-pnf 8865  df-mnf 8866  df-xr 8867  df-ltxr 8868  df-le 8869  df-sub 9035  df-neg 9036  df-div 9420  df-nn 9743  df-2 9800  df-n0 9962  df-z 10021  df-uz 10227  df-q 10313  df-rp 10351  df-xneg 10448  df-xadd 10449  df-xmul 10450  df-ico 10658  df-topgen 13340  df-0g 13400  df-mnd 14363  df-mhm 14411  df-grp 14485  df-ghm 14677  df-xmet 16369  df-met 16370  df-bl 16371  df-mopn 16372  df-top 16632  df-bases 16634  df-topon 16635  df-topsp 16636  df-xms 17881  df-ms 17882  df-nm 18101  df-ngp 18102  df-nmo 18213
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