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Theorem nmo0 20317
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 2443 . . 3  |-  ( norm `  S )  =  (
norm `  S )
4 eqid 2443 . . 3  |-  ( norm `  T )  =  (
norm `  T )
5 eqid 2443 . . 3  |-  ( 0g
`  S )  =  ( 0g `  S
)
6 simpl 457 . . 3  |-  ( ( S  e. NrmGrp  /\  T  e. NrmGrp
)  ->  S  e. NrmGrp )
7 simpr 461 . . 3  |-  ( ( S  e. NrmGrp  /\  T  e. NrmGrp
)  ->  T  e. NrmGrp )
8 ngpgrp 20194 . . . 4  |-  ( S  e. NrmGrp  ->  S  e.  Grp )
9 ngpgrp 20194 . . . 4  |-  ( T  e. NrmGrp  ->  T  e.  Grp )
10 nmo0.3 . . . . 5  |-  .0.  =  ( 0g `  T )
1110, 20ghm 15764 . . . 4  |-  ( ( S  e.  Grp  /\  T  e.  Grp )  ->  ( V  X.  {  .0.  } )  e.  ( S  GrpHom  T ) )
128, 9, 11syl2an 477 . . 3  |-  ( ( S  e. NrmGrp  /\  T  e. NrmGrp
)  ->  ( V  X.  {  .0.  } )  e.  ( S  GrpHom  T ) )
13 0red 9390 . . 3  |-  ( ( S  e. NrmGrp  /\  T  e. NrmGrp
)  ->  0  e.  RR )
14 0le0 10414 . . . 4  |-  0  <_  0
1514a1i 11 . . 3  |-  ( ( S  e. NrmGrp  /\  T  e. NrmGrp
)  ->  0  <_  0 )
16 fvex 5704 . . . . . . . . 9  |-  ( 0g
`  T )  e. 
_V
1710, 16eqeltri 2513 . . . . . . . 8  |-  .0.  e.  _V
1817fvconst2 5936 . . . . . . 7  |-  ( x  e.  V  ->  (
( V  X.  {  .0.  } ) `  x
)  =  .0.  )
1918ad2antrl 727 . . . . . 6  |-  ( ( ( S  e. NrmGrp  /\  T  e. NrmGrp )  /\  ( x  e.  V  /\  x  =/=  ( 0g `  S
) ) )  -> 
( ( V  X.  {  .0.  } ) `  x )  =  .0.  )
2019fveq2d 5698 . . . . 5  |-  ( ( ( S  e. NrmGrp  /\  T  e. NrmGrp )  /\  ( x  e.  V  /\  x  =/=  ( 0g `  S
) ) )  -> 
( ( norm `  T
) `  ( ( V  X.  {  .0.  }
) `  x )
)  =  ( (
norm `  T ) `  .0.  ) )
214, 10nm0 20221 . . . . . 6  |-  ( T  e. NrmGrp  ->  ( ( norm `  T ) `  .0.  )  =  0 )
2221ad2antlr 726 . . . . 5  |-  ( ( ( S  e. NrmGrp  /\  T  e. NrmGrp )  /\  ( x  e.  V  /\  x  =/=  ( 0g `  S
) ) )  -> 
( ( norm `  T
) `  .0.  )  =  0 )
2320, 22eqtrd 2475 . . . 4  |-  ( ( ( S  e. NrmGrp  /\  T  e. NrmGrp )  /\  ( x  e.  V  /\  x  =/=  ( 0g `  S
) ) )  -> 
( ( norm `  T
) `  ( ( V  X.  {  .0.  }
) `  x )
)  =  0 )
242, 3nmcl 20210 . . . . . . . 8  |-  ( ( S  e. NrmGrp  /\  x  e.  V )  ->  (
( norm `  S ) `  x )  e.  RR )
2524ad2ant2r 746 . . . . . . 7  |-  ( ( ( S  e. NrmGrp  /\  T  e. NrmGrp )  /\  ( x  e.  V  /\  x  =/=  ( 0g `  S
) ) )  -> 
( ( norm `  S
) `  x )  e.  RR )
2625recnd 9415 . . . . . 6  |-  ( ( ( S  e. NrmGrp  /\  T  e. NrmGrp )  /\  ( x  e.  V  /\  x  =/=  ( 0g `  S
) ) )  -> 
( ( norm `  S
) `  x )  e.  CC )
2726mul02d 9570 . . . . 5  |-  ( ( ( S  e. NrmGrp  /\  T  e. NrmGrp )  /\  ( x  e.  V  /\  x  =/=  ( 0g `  S
) ) )  -> 
( 0  x.  (
( norm `  S ) `  x ) )  =  0 )
2814, 27syl5breqr 4331 . . . 4  |-  ( ( ( S  e. NrmGrp  /\  T  e. NrmGrp )  /\  ( x  e.  V  /\  x  =/=  ( 0g `  S
) ) )  -> 
0  <_  ( 0  x.  ( ( norm `  S ) `  x
) ) )
2923, 28eqbrtrd 4315 . . 3  |-  ( ( ( S  e. NrmGrp  /\  T  e. NrmGrp )  /\  ( x  e.  V  /\  x  =/=  ( 0g `  S
) ) )  -> 
( ( norm `  T
) `  ( ( V  X.  {  .0.  }
) `  x )
)  <_  ( 0  x.  ( ( norm `  S ) `  x
) ) )
301, 2, 3, 4, 5, 6, 7, 12, 13, 15, 29nmolb2d 20300 . 2  |-  ( ( S  e. NrmGrp  /\  T  e. NrmGrp
)  ->  ( N `  ( V  X.  {  .0.  } ) )  <_ 
0 )
311nmoge0 20303 . . 3  |-  ( ( S  e. NrmGrp  /\  T  e. NrmGrp  /\  ( V  X.  {  .0.  } )  e.  ( S  GrpHom  T ) )  ->  0  <_  ( N `  ( V  X.  {  .0.  } ) ) )
3212, 31mpd3an3 1315 . 2  |-  ( ( S  e. NrmGrp  /\  T  e. NrmGrp
)  ->  0  <_  ( N `  ( V  X.  {  .0.  }
) ) )
331nmocl 20302 . . . 4  |-  ( ( S  e. NrmGrp  /\  T  e. NrmGrp  /\  ( V  X.  {  .0.  } )  e.  ( S  GrpHom  T ) )  ->  ( N `  ( V  X.  {  .0.  } ) )  e.  RR* )
3412, 33mpd3an3 1315 . . 3  |-  ( ( S  e. NrmGrp  /\  T  e. NrmGrp
)  ->  ( N `  ( V  X.  {  .0.  } ) )  e. 
RR* )
35 0xr 9433 . . 3  |-  0  e.  RR*
36 xrletri3 11132 . . 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.  } ) ) ) ) )
3734, 35, 36sylancl 662 . 2  |-  ( ( S  e. NrmGrp  /\  T  e. NrmGrp
)  ->  ( ( N `  ( V  X.  {  .0.  } ) )  =  0  <->  (
( N `  ( V  X.  {  .0.  }
) )  <_  0  /\  0  <_  ( N `
 ( V  X.  {  .0.  } ) ) ) ) )
3830, 32, 37mpbir2and 913 1  |-  ( ( S  e. NrmGrp  /\  T  e. NrmGrp
)  ->  ( N `  ( V  X.  {  .0.  } ) )  =  0 )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1369    e. wcel 1756    =/= wne 2609   _Vcvv 2975   {csn 3880   class class class wbr 4295    X. cxp 4841   ` cfv 5421  (class class class)co 6094   RRcr 9284   0cc0 9285    x. cmul 9290   RR*cxr 9420    <_ cle 9422   Basecbs 14177   0gc0g 14381   Grpcgrp 15413    GrpHom cghm 15747   normcnm 20172  NrmGrpcngp 20173   normOpcnmo 20287
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4406  ax-sep 4416  ax-nul 4424  ax-pow 4473  ax-pr 4534  ax-un 6375  ax-cnex 9341  ax-resscn 9342  ax-1cn 9343  ax-icn 9344  ax-addcl 9345  ax-addrcl 9346  ax-mulcl 9347  ax-mulrcl 9348  ax-mulcom 9349  ax-addass 9350  ax-mulass 9351  ax-distr 9352  ax-i2m1 9353  ax-1ne0 9354  ax-1rid 9355  ax-rnegex 9356  ax-rrecex 9357  ax-cnre 9358  ax-pre-lttri 9359  ax-pre-lttrn 9360  ax-pre-ltadd 9361  ax-pre-mulgt0 9362  ax-pre-sup 9363
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2571  df-ne 2611  df-nel 2612  df-ral 2723  df-rex 2724  df-reu 2725  df-rmo 2726  df-rab 2727  df-v 2977  df-sbc 3190  df-csb 3292  df-dif 3334  df-un 3336  df-in 3338  df-ss 3345  df-pss 3347  df-nul 3641  df-if 3795  df-pw 3865  df-sn 3881  df-pr 3883  df-tp 3885  df-op 3887  df-uni 4095  df-iun 4176  df-br 4296  df-opab 4354  df-mpt 4355  df-tr 4389  df-eprel 4635  df-id 4639  df-po 4644  df-so 4645  df-fr 4682  df-we 4684  df-ord 4725  df-on 4726  df-lim 4727  df-suc 4728  df-xp 4849  df-rel 4850  df-cnv 4851  df-co 4852  df-dm 4853  df-rn 4854  df-res 4855  df-ima 4856  df-iota 5384  df-fun 5423  df-fn 5424  df-f 5425  df-f1 5426  df-fo 5427  df-f1o 5428  df-fv 5429  df-riota 6055  df-ov 6097  df-oprab 6098  df-mpt2 6099  df-om 6480  df-1st 6580  df-2nd 6581  df-recs 6835  df-rdg 6869  df-er 7104  df-map 7219  df-en 7314  df-dom 7315  df-sdom 7316  df-sup 7694  df-pnf 9423  df-mnf 9424  df-xr 9425  df-ltxr 9426  df-le 9427  df-sub 9600  df-neg 9601  df-div 9997  df-nn 10326  df-2 10383  df-n0 10583  df-z 10650  df-uz 10865  df-q 10957  df-rp 10995  df-xneg 11092  df-xadd 11093  df-xmul 11094  df-ico 11309  df-0g 14383  df-topgen 14385  df-mnd 15418  df-mhm 15467  df-grp 15548  df-ghm 15748  df-psmet 17812  df-xmet 17813  df-met 17814  df-bl 17815  df-mopn 17816  df-top 18506  df-bases 18508  df-topon 18509  df-topsp 18510  df-xms 19898  df-ms 19899  df-nm 20178  df-ngp 20179  df-nmo 20290
This theorem is referenced by:  nmoeq0  20318  0nghm  20323  idnghm  20325
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