MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  nmoeq0 Structured version   Unicode version

Theorem nmoeq0 18770
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 9091 . . . . . . . . . . 11  |-  0  e.  RR
31, 2syl6eqel 2524 . . . . . . . . . 10  |-  ( ( N `  F )  =  0  ->  ( N `  F )  e.  RR )
4 nmo0.1 . . . . . . . . . . . 12  |-  N  =  ( S normOp T )
54isnghm2 18758 . . . . . . . . . . 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 2436 . . . . . . . . . 10  |-  ( norm `  S )  =  (
norm `  S )
10 eqid 2436 . . . . . . . . . 10  |-  ( norm `  T )  =  (
norm `  T )
114, 8, 9, 10nmoi 18762 . . . . . . . . 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 6096 . . . . . . . . 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 18662 . . . . . . . . . . . 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 9114 . . . . . . . . . 10  |-  ( ( ( ( S  e. NrmGrp  /\  T  e. NrmGrp  /\  F  e.  ( S  GrpHom  T ) )  /\  ( N `
 F )  =  0 )  /\  x  e.  V )  ->  (
( norm `  S ) `  x )  e.  CC )
1918mul02d 9264 . . . . . . . . 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 2468 . . . . . . . 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 4236 . . . . . . 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 2436 . . . . . . . . . . 11  |-  ( Base `  T )  =  (
Base `  T )
258, 24ghmf 15010 . . . . . . . . . 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 5870 . . . . . . . 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 18663 . . . . . . . 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 18662 . . . . . . . . 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 9160 . . . . . . . 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 18664 . . . . . . 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 4295 . . . 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 5779 . . . 4  |-  ( ( ( S  e. NrmGrp  /\  T  e. NrmGrp  /\  F  e.  ( S  GrpHom  T ) )  /\  ( N `  F )  =  0 )  ->  F  =  ( x  e.  V  |->  ( F `  x
) ) )
41 fconstmpt 4921 . . . . 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 2478 . . 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 18769 . . . 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 5728 . . . 4  |-  ( F  =  ( V  X.  {  .0.  } )  -> 
( N `  F
)  =  ( N `
 ( V  X.  {  .0.  } ) ) )
4847eqeq1d 2444 . . 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 1652    e. wcel 1725   {csn 3814   class class class wbr 4212    e. cmpt 4266    X. cxp 4876   -->wf 5450   ` cfv 5454  (class class class)co 6081   RRcr 8989   0cc0 8990    x. cmul 8995    <_ cle 9121   Basecbs 13469   0gc0g 13723    GrpHom cghm 15003   normcnm 18624  NrmGrpcngp 18625   normOpcnmo 18739   NGHom cnghm 18740
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-rep 4320  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701  ax-cnex 9046  ax-resscn 9047  ax-1cn 9048  ax-icn 9049  ax-addcl 9050  ax-addrcl 9051  ax-mulcl 9052  ax-mulrcl 9053  ax-mulcom 9054  ax-addass 9055  ax-mulass 9056  ax-distr 9057  ax-i2m1 9058  ax-1ne0 9059  ax-1rid 9060  ax-rnegex 9061  ax-rrecex 9062  ax-cnre 9063  ax-pre-lttri 9064  ax-pre-lttrn 9065  ax-pre-ltadd 9066  ax-pre-mulgt0 9067  ax-pre-sup 9068
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-nel 2602  df-ral 2710  df-rex 2711  df-reu 2712  df-rmo 2713  df-rab 2714  df-v 2958  df-sbc 3162  df-csb 3252  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-pss 3336  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-tp 3822  df-op 3823  df-uni 4016  df-iun 4095  df-br 4213  df-opab 4267  df-mpt 4268  df-tr 4303  df-eprel 4494  df-id 4498  df-po 4503  df-so 4504  df-fr 4541  df-we 4543  df-ord 4584  df-on 4585  df-lim 4586  df-suc 4587  df-om 4846  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-f1 5459  df-fo 5460  df-f1o 5461  df-fv 5462  df-ov 6084  df-oprab 6085  df-mpt2 6086  df-1st 6349  df-2nd 6350  df-riota 6549  df-recs 6633  df-rdg 6668  df-er 6905  df-map 7020  df-en 7110  df-dom 7111  df-sdom 7112  df-sup 7446  df-pnf 9122  df-mnf 9123  df-xr 9124  df-ltxr 9125  df-le 9126  df-sub 9293  df-neg 9294  df-div 9678  df-nn 10001  df-2 10058  df-n0 10222  df-z 10283  df-uz 10489  df-q 10575  df-rp 10613  df-xneg 10710  df-xadd 10711  df-xmul 10712  df-ico 10922  df-topgen 13667  df-0g 13727  df-mnd 14690  df-mhm 14738  df-grp 14812  df-ghm 15004  df-psmet 16694  df-xmet 16695  df-met 16696  df-bl 16697  df-mopn 16698  df-top 16963  df-bases 16965  df-topon 16966  df-topsp 16967  df-xms 18350  df-ms 18351  df-nm 18630  df-ngp 18631  df-nmo 18742  df-nghm 18743
  Copyright terms: Public domain W3C validator