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Theorem nmoeq0 18723
Description: The operator norm is zero only for 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
nmoeq0  |-  ( ( S  e. NrmGrp  /\  T  e. NrmGrp  /\  F  e.  ( S  GrpHom  T ) )  ->  ( ( N `
 F )  =  0  <->  F  =  ( V  X.  {  .0.  }
) ) )

Proof of Theorem nmoeq0
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 id 20 . . . . . . . . . . 11  |-  ( ( N `  F )  =  0  ->  ( N `  F )  =  0 )
2 0re 9047 . . . . . . . . . . 11  |-  0  e.  RR
31, 2syl6eqel 2492 . . . . . . . . . 10  |-  ( ( N `  F )  =  0  ->  ( N `  F )  e.  RR )
4 nmo0.1 . . . . . . . . . . . 12  |-  N  =  ( S normOp T )
54isnghm2 18711 . . . . . . . . . . 11  |-  ( ( S  e. NrmGrp  /\  T  e. NrmGrp  /\  F  e.  ( S  GrpHom  T ) )  ->  ( F  e.  ( S NGHom  T )  <-> 
( N `  F
)  e.  RR ) )
65biimpar 472 . . . . . . . . . 10  |-  ( ( ( S  e. NrmGrp  /\  T  e. NrmGrp  /\  F  e.  ( S  GrpHom  T ) )  /\  ( N `  F )  e.  RR )  ->  F  e.  ( S NGHom  T ) )
73, 6sylan2 461 . . . . . . . . 9  |-  ( ( ( S  e. NrmGrp  /\  T  e. NrmGrp  /\  F  e.  ( S  GrpHom  T ) )  /\  ( N `  F )  =  0 )  ->  F  e.  ( S NGHom  T ) )
8 nmo0.2 . . . . . . . . . 10  |-  V  =  ( Base `  S
)
9 eqid 2404 . . . . . . . . . 10  |-  ( norm `  S )  =  (
norm `  S )
10 eqid 2404 . . . . . . . . . 10  |-  ( norm `  T )  =  (
norm `  T )
114, 8, 9, 10nmoi 18715 . . . . . . . . 9  |-  ( ( F  e.  ( S NGHom 
T )  /\  x  e.  V )  ->  (
( norm `  T ) `  ( F `  x
) )  <_  (
( N `  F
)  x.  ( (
norm `  S ) `  x ) ) )
127, 11sylan 458 . . . . . . . 8  |-  ( ( ( ( S  e. NrmGrp  /\  T  e. NrmGrp  /\  F  e.  ( S  GrpHom  T ) )  /\  ( N `
 F )  =  0 )  /\  x  e.  V )  ->  (
( norm `  T ) `  ( F `  x
) )  <_  (
( N `  F
)  x.  ( (
norm `  S ) `  x ) ) )
13 simplr 732 . . . . . . . . . 10  |-  ( ( ( ( S  e. NrmGrp  /\  T  e. NrmGrp  /\  F  e.  ( S  GrpHom  T ) )  /\  ( N `
 F )  =  0 )  /\  x  e.  V )  ->  ( N `  F )  =  0 )
1413oveq1d 6055 . . . . . . . . 9  |-  ( ( ( ( S  e. NrmGrp  /\  T  e. NrmGrp  /\  F  e.  ( S  GrpHom  T ) )  /\  ( N `
 F )  =  0 )  /\  x  e.  V )  ->  (
( N `  F
)  x.  ( (
norm `  S ) `  x ) )  =  ( 0  x.  (
( norm `  S ) `  x ) ) )
15 simpl1 960 . . . . . . . . . . . 12  |-  ( ( ( S  e. NrmGrp  /\  T  e. NrmGrp  /\  F  e.  ( S  GrpHom  T ) )  /\  ( N `  F )  =  0 )  ->  S  e. NrmGrp )
168, 9nmcl 18615 . . . . . . . . . . . 12  |-  ( ( S  e. NrmGrp  /\  x  e.  V )  ->  (
( norm `  S ) `  x )  e.  RR )
1715, 16sylan 458 . . . . . . . . . . 11  |-  ( ( ( ( S  e. NrmGrp  /\  T  e. NrmGrp  /\  F  e.  ( S  GrpHom  T ) )  /\  ( N `
 F )  =  0 )  /\  x  e.  V )  ->  (
( norm `  S ) `  x )  e.  RR )
1817recnd 9070 . . . . . . . . . 10  |-  ( ( ( ( S  e. NrmGrp  /\  T  e. NrmGrp  /\  F  e.  ( S  GrpHom  T ) )  /\  ( N `
 F )  =  0 )  /\  x  e.  V )  ->  (
( norm `  S ) `  x )  e.  CC )
1918mul02d 9220 . . . . . . . . 9  |-  ( ( ( ( S  e. NrmGrp  /\  T  e. NrmGrp  /\  F  e.  ( S  GrpHom  T ) )  /\  ( N `
 F )  =  0 )  /\  x  e.  V )  ->  (
0  x.  ( (
norm `  S ) `  x ) )  =  0 )
2014, 19eqtrd 2436 . . . . . . . 8  |-  ( ( ( ( S  e. NrmGrp  /\  T  e. NrmGrp  /\  F  e.  ( S  GrpHom  T ) )  /\  ( N `
 F )  =  0 )  /\  x  e.  V )  ->  (
( N `  F
)  x.  ( (
norm `  S ) `  x ) )  =  0 )
2112, 20breqtrd 4196 . . . . . . 7  |-  ( ( ( ( S  e. NrmGrp  /\  T  e. NrmGrp  /\  F  e.  ( S  GrpHom  T ) )  /\  ( N `
 F )  =  0 )  /\  x  e.  V )  ->  (
( norm `  T ) `  ( F `  x
) )  <_  0
)
22 simpll2 997 . . . . . . . 8  |-  ( ( ( ( S  e. NrmGrp  /\  T  e. NrmGrp  /\  F  e.  ( S  GrpHom  T ) )  /\  ( N `
 F )  =  0 )  /\  x  e.  V )  ->  T  e. NrmGrp )
23 simpl3 962 . . . . . . . . . 10  |-  ( ( ( S  e. NrmGrp  /\  T  e. NrmGrp  /\  F  e.  ( S  GrpHom  T ) )  /\  ( N `  F )  =  0 )  ->  F  e.  ( S  GrpHom  T ) )
24 eqid 2404 . . . . . . . . . . 11  |-  ( Base `  T )  =  (
Base `  T )
258, 24ghmf 14965 . . . . . . . . . 10  |-  ( F  e.  ( S  GrpHom  T )  ->  F : V
--> ( Base `  T
) )
2623, 25syl 16 . . . . . . . . 9  |-  ( ( ( S  e. NrmGrp  /\  T  e. NrmGrp  /\  F  e.  ( S  GrpHom  T ) )  /\  ( N `  F )  =  0 )  ->  F : V
--> ( Base `  T
) )
2726ffvelrnda 5829 . . . . . . . 8  |-  ( ( ( ( S  e. NrmGrp  /\  T  e. NrmGrp  /\  F  e.  ( S  GrpHom  T ) )  /\  ( N `
 F )  =  0 )  /\  x  e.  V )  ->  ( F `  x )  e.  ( Base `  T
) )
2824, 10nmge0 18616 . . . . . . . 8  |-  ( ( T  e. NrmGrp  /\  ( F `  x )  e.  ( Base `  T
) )  ->  0  <_  ( ( norm `  T
) `  ( F `  x ) ) )
2922, 27, 28syl2anc 643 . . . . . . 7  |-  ( ( ( ( S  e. NrmGrp  /\  T  e. NrmGrp  /\  F  e.  ( S  GrpHom  T ) )  /\  ( N `
 F )  =  0 )  /\  x  e.  V )  ->  0  <_  ( ( norm `  T
) `  ( F `  x ) ) )
3024, 10nmcl 18615 . . . . . . . . 9  |-  ( ( T  e. NrmGrp  /\  ( F `  x )  e.  ( Base `  T
) )  ->  (
( norm `  T ) `  ( F `  x
) )  e.  RR )
3122, 27, 30syl2anc 643 . . . . . . . 8  |-  ( ( ( ( S  e. NrmGrp  /\  T  e. NrmGrp  /\  F  e.  ( S  GrpHom  T ) )  /\  ( N `
 F )  =  0 )  /\  x  e.  V )  ->  (
( norm `  T ) `  ( F `  x
) )  e.  RR )
32 letri3 9116 . . . . . . . 8  |-  ( ( ( ( norm `  T
) `  ( F `  x ) )  e.  RR  /\  0  e.  RR )  ->  (
( ( norm `  T
) `  ( F `  x ) )  =  0  <->  ( ( (
norm `  T ) `  ( F `  x
) )  <_  0  /\  0  <_  ( (
norm `  T ) `  ( F `  x
) ) ) ) )
3331, 2, 32sylancl 644 . . . . . . 7  |-  ( ( ( ( S  e. NrmGrp  /\  T  e. NrmGrp  /\  F  e.  ( S  GrpHom  T ) )  /\  ( N `
 F )  =  0 )  /\  x  e.  V )  ->  (
( ( norm `  T
) `  ( F `  x ) )  =  0  <->  ( ( (
norm `  T ) `  ( F `  x
) )  <_  0  /\  0  <_  ( (
norm `  T ) `  ( F `  x
) ) ) ) )
3421, 29, 33mpbir2and 889 . . . . . 6  |-  ( ( ( ( S  e. NrmGrp  /\  T  e. NrmGrp  /\  F  e.  ( S  GrpHom  T ) )  /\  ( N `
 F )  =  0 )  /\  x  e.  V )  ->  (
( norm `  T ) `  ( F `  x
) )  =  0 )
35 nmo0.3 . . . . . . . 8  |-  .0.  =  ( 0g `  T )
3624, 10, 35nmeq0 18617 . . . . . . 7  |-  ( ( T  e. NrmGrp  /\  ( F `  x )  e.  ( Base `  T
) )  ->  (
( ( norm `  T
) `  ( F `  x ) )  =  0  <->  ( F `  x )  =  .0.  ) )
3722, 27, 36syl2anc 643 . . . . . 6  |-  ( ( ( ( S  e. NrmGrp  /\  T  e. NrmGrp  /\  F  e.  ( S  GrpHom  T ) )  /\  ( N `
 F )  =  0 )  /\  x  e.  V )  ->  (
( ( norm `  T
) `  ( F `  x ) )  =  0  <->  ( F `  x )  =  .0.  ) )
3834, 37mpbid 202 . . . . 5  |-  ( ( ( ( S  e. NrmGrp  /\  T  e. NrmGrp  /\  F  e.  ( S  GrpHom  T ) )  /\  ( N `
 F )  =  0 )  /\  x  e.  V )  ->  ( F `  x )  =  .0.  )
3938mpteq2dva 4255 . . . 4  |-  ( ( ( S  e. NrmGrp  /\  T  e. NrmGrp  /\  F  e.  ( S  GrpHom  T ) )  /\  ( N `  F )  =  0 )  ->  ( x  e.  V  |->  ( F `
 x ) )  =  ( x  e.  V  |->  .0.  ) )
4026feqmptd 5738 . . . 4  |-  ( ( ( S  e. NrmGrp  /\  T  e. NrmGrp  /\  F  e.  ( S  GrpHom  T ) )  /\  ( N `  F )  =  0 )  ->  F  =  ( x  e.  V  |->  ( F `  x
) ) )
41 fconstmpt 4880 . . . . 5  |-  ( V  X.  {  .0.  }
)  =  ( x  e.  V  |->  .0.  )
4241a1i 11 . . . 4  |-  ( ( ( S  e. NrmGrp  /\  T  e. NrmGrp  /\  F  e.  ( S  GrpHom  T ) )  /\  ( N `  F )  =  0 )  ->  ( V  X.  {  .0.  } )  =  ( x  e.  V  |->  .0.  ) )
4339, 40, 423eqtr4d 2446 . . 3  |-  ( ( ( S  e. NrmGrp  /\  T  e. NrmGrp  /\  F  e.  ( S  GrpHom  T ) )  /\  ( N `  F )  =  0 )  ->  F  =  ( V  X.  {  .0.  } ) )
4443ex 424 . 2  |-  ( ( S  e. NrmGrp  /\  T  e. NrmGrp  /\  F  e.  ( S  GrpHom  T ) )  ->  ( ( N `
 F )  =  0  ->  F  =  ( V  X.  {  .0.  } ) ) )
454, 8, 35nmo0 18722 . . . 4  |-  ( ( S  e. NrmGrp  /\  T  e. NrmGrp
)  ->  ( N `  ( V  X.  {  .0.  } ) )  =  0 )
46453adant3 977 . . 3  |-  ( ( S  e. NrmGrp  /\  T  e. NrmGrp  /\  F  e.  ( S  GrpHom  T ) )  ->  ( N `  ( V  X.  {  .0.  } ) )  =  0 )
47 fveq2 5687 . . . 4  |-  ( F  =  ( V  X.  {  .0.  } )  -> 
( N `  F
)  =  ( N `
 ( V  X.  {  .0.  } ) ) )
4847eqeq1d 2412 . . 3  |-  ( F  =  ( V  X.  {  .0.  } )  -> 
( ( N `  F )  =  0  <-> 
( N `  ( V  X.  {  .0.  }
) )  =  0 ) )
4946, 48syl5ibrcom 214 . 2  |-  ( ( S  e. NrmGrp  /\  T  e. NrmGrp  /\  F  e.  ( S  GrpHom  T ) )  ->  ( F  =  ( V  X.  {  .0.  } )  ->  ( N `  F )  =  0 ) )
5044, 49impbid 184 1  |-  ( ( S  e. NrmGrp  /\  T  e. NrmGrp  /\  F  e.  ( S  GrpHom  T ) )  ->  ( ( N `
 F )  =  0  <->  F  =  ( V  X.  {  .0.  }
) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    /\ w3a 936    = wceq 1649    e. wcel 1721   {csn 3774   class class class wbr 4172    e. cmpt 4226    X. cxp 4835   -->wf 5409   ` cfv 5413  (class class class)co 6040   RRcr 8945   0cc0 8946    x. cmul 8951    <_ cle 9077   Basecbs 13424   0gc0g 13678    GrpHom cghm 14958   normcnm 18577  NrmGrpcngp 18578   normOpcnmo 18692   NGHom cnghm 18693
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-rep 4280  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660  ax-cnex 9002  ax-resscn 9003  ax-1cn 9004  ax-icn 9005  ax-addcl 9006  ax-addrcl 9007  ax-mulcl 9008  ax-mulrcl 9009  ax-mulcom 9010  ax-addass 9011  ax-mulass 9012  ax-distr 9013  ax-i2m1 9014  ax-1ne0 9015  ax-1rid 9016  ax-rnegex 9017  ax-rrecex 9018  ax-cnre 9019  ax-pre-lttri 9020  ax-pre-lttrn 9021  ax-pre-ltadd 9022  ax-pre-mulgt0 9023  ax-pre-sup 9024
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-nel 2570  df-ral 2671  df-rex 2672  df-reu 2673  df-rmo 2674  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-pss 3296  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-tp 3782  df-op 3783  df-uni 3976  df-iun 4055  df-br 4173  df-opab 4227  df-mpt 4228  df-tr 4263  df-eprel 4454  df-id 4458  df-po 4463  df-so 4464  df-fr 4501  df-we 4503  df-ord 4544  df-on 4545  df-lim 4546  df-suc 4547  df-om 4805  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-ov 6043  df-oprab 6044  df-mpt2 6045  df-1st 6308  df-2nd 6309  df-riota 6508  df-recs 6592  df-rdg 6627  df-er 6864  df-map 6979  df-en 7069  df-dom 7070  df-sdom 7071  df-sup 7404  df-pnf 9078  df-mnf 9079  df-xr 9080  df-ltxr 9081  df-le 9082  df-sub 9249  df-neg 9250  df-div 9634  df-nn 9957  df-2 10014  df-n0 10178  df-z 10239  df-uz 10445  df-q 10531  df-rp 10569  df-xneg 10666  df-xadd 10667  df-xmul 10668  df-ico 10878  df-topgen 13622  df-0g 13682  df-mnd 14645  df-mhm 14693  df-grp 14767  df-ghm 14959  df-psmet 16649  df-xmet 16650  df-met 16651  df-bl 16652  df-mopn 16653  df-top 16918  df-bases 16920  df-topon 16921  df-topsp 16922  df-xms 18303  df-ms 18304  df-nm 18583  df-ngp 18584  df-nmo 18695  df-nghm 18696
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