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Theorem nmoeq0 18245
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 19 . . . . . . . . . . 11  |-  ( ( N `  F )  =  0  ->  ( N `  F )  =  0 )
2 0re 8838 . . . . . . . . . . 11  |-  0  e.  RR
31, 2syl6eqel 2371 . . . . . . . . . 10  |-  ( ( N `  F )  =  0  ->  ( N `  F )  e.  RR )
4 nmo0.1 . . . . . . . . . . . 12  |-  N  =  ( S normOp T )
54isnghm2 18233 . . . . . . . . . . 11  |-  ( ( S  e. NrmGrp  /\  T  e. NrmGrp  /\  F  e.  ( S  GrpHom  T ) )  ->  ( F  e.  ( S NGHom  T )  <-> 
( N `  F
)  e.  RR ) )
65biimpar 471 . . . . . . . . . 10  |-  ( ( ( S  e. NrmGrp  /\  T  e. NrmGrp  /\  F  e.  ( S  GrpHom  T ) )  /\  ( N `  F )  e.  RR )  ->  F  e.  ( S NGHom  T ) )
73, 6sylan2 460 . . . . . . . . 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 2283 . . . . . . . . . 10  |-  ( norm `  S )  =  (
norm `  S )
10 eqid 2283 . . . . . . . . . 10  |-  ( norm `  T )  =  (
norm `  T )
114, 8, 9, 10nmoi 18237 . . . . . . . . 9  |-  ( ( F  e.  ( S NGHom 
T )  /\  x  e.  V )  ->  (
( norm `  T ) `  ( F `  x
) )  <_  (
( N `  F
)  x.  ( (
norm `  S ) `  x ) ) )
127, 11sylan 457 . . . . . . . 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 731 . . . . . . . . . 10  |-  ( ( ( ( S  e. NrmGrp  /\  T  e. NrmGrp  /\  F  e.  ( S  GrpHom  T ) )  /\  ( N `
 F )  =  0 )  /\  x  e.  V )  ->  ( N `  F )  =  0 )
1413oveq1d 5873 . . . . . . . . 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 958 . . . . . . . . . . . 12  |-  ( ( ( S  e. NrmGrp  /\  T  e. NrmGrp  /\  F  e.  ( S  GrpHom  T ) )  /\  ( N `  F )  =  0 )  ->  S  e. NrmGrp )
168, 9nmcl 18137 . . . . . . . . . . . 12  |-  ( ( S  e. NrmGrp  /\  x  e.  V )  ->  (
( norm `  S ) `  x )  e.  RR )
1715, 16sylan 457 . . . . . . . . . . 11  |-  ( ( ( ( S  e. NrmGrp  /\  T  e. NrmGrp  /\  F  e.  ( S  GrpHom  T ) )  /\  ( N `
 F )  =  0 )  /\  x  e.  V )  ->  (
( norm `  S ) `  x )  e.  RR )
1817recnd 8861 . . . . . . . . . 10  |-  ( ( ( ( S  e. NrmGrp  /\  T  e. NrmGrp  /\  F  e.  ( S  GrpHom  T ) )  /\  ( N `
 F )  =  0 )  /\  x  e.  V )  ->  (
( norm `  S ) `  x )  e.  CC )
1918mul02d 9010 . . . . . . . . 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 2315 . . . . . . . 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 4047 . . . . . . 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 995 . . . . . . . 8  |-  ( ( ( ( S  e. NrmGrp  /\  T  e. NrmGrp  /\  F  e.  ( S  GrpHom  T ) )  /\  ( N `
 F )  =  0 )  /\  x  e.  V )  ->  T  e. NrmGrp )
23 simpl3 960 . . . . . . . . . 10  |-  ( ( ( S  e. NrmGrp  /\  T  e. NrmGrp  /\  F  e.  ( S  GrpHom  T ) )  /\  ( N `  F )  =  0 )  ->  F  e.  ( S  GrpHom  T ) )
24 eqid 2283 . . . . . . . . . . 11  |-  ( Base `  T )  =  (
Base `  T )
258, 24ghmf 14687 . . . . . . . . . 10  |-  ( F  e.  ( S  GrpHom  T )  ->  F : V
--> ( Base `  T
) )
2623, 25syl 15 . . . . . . . . 9  |-  ( ( ( S  e. NrmGrp  /\  T  e. NrmGrp  /\  F  e.  ( S  GrpHom  T ) )  /\  ( N `  F )  =  0 )  ->  F : V
--> ( Base `  T
) )
27 ffvelrn 5663 . . . . . . . . 9  |-  ( ( F : V --> ( Base `  T )  /\  x  e.  V )  ->  ( F `  x )  e.  ( Base `  T
) )
2826, 27sylan 457 . . . . . . . 8  |-  ( ( ( ( S  e. NrmGrp  /\  T  e. NrmGrp  /\  F  e.  ( S  GrpHom  T ) )  /\  ( N `
 F )  =  0 )  /\  x  e.  V )  ->  ( F `  x )  e.  ( Base `  T
) )
2924, 10nmge0 18138 . . . . . . . 8  |-  ( ( T  e. NrmGrp  /\  ( F `  x )  e.  ( Base `  T
) )  ->  0  <_  ( ( norm `  T
) `  ( F `  x ) ) )
3022, 28, 29syl2anc 642 . . . . . . 7  |-  ( ( ( ( S  e. NrmGrp  /\  T  e. NrmGrp  /\  F  e.  ( S  GrpHom  T ) )  /\  ( N `
 F )  =  0 )  /\  x  e.  V )  ->  0  <_  ( ( norm `  T
) `  ( F `  x ) ) )
3124, 10nmcl 18137 . . . . . . . . 9  |-  ( ( T  e. NrmGrp  /\  ( F `  x )  e.  ( Base `  T
) )  ->  (
( norm `  T ) `  ( F `  x
) )  e.  RR )
3222, 28, 31syl2anc 642 . . . . . . . 8  |-  ( ( ( ( S  e. NrmGrp  /\  T  e. NrmGrp  /\  F  e.  ( S  GrpHom  T ) )  /\  ( N `
 F )  =  0 )  /\  x  e.  V )  ->  (
( norm `  T ) `  ( F `  x
) )  e.  RR )
33 letri3 8907 . . . . . . . 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
) ) ) ) )
3432, 2, 33sylancl 643 . . . . . . 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
) ) ) ) )
3521, 30, 34mpbir2and 888 . . . . . 6  |-  ( ( ( ( S  e. NrmGrp  /\  T  e. NrmGrp  /\  F  e.  ( S  GrpHom  T ) )  /\  ( N `
 F )  =  0 )  /\  x  e.  V )  ->  (
( norm `  T ) `  ( F `  x
) )  =  0 )
36 nmo0.3 . . . . . . . 8  |-  .0.  =  ( 0g `  T )
3724, 10, 36nmeq0 18139 . . . . . . 7  |-  ( ( T  e. NrmGrp  /\  ( F `  x )  e.  ( Base `  T
) )  ->  (
( ( norm `  T
) `  ( F `  x ) )  =  0  <->  ( F `  x )  =  .0.  ) )
3822, 28, 37syl2anc 642 . . . . . 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.  ) )
3935, 38mpbid 201 . . . . 5  |-  ( ( ( ( S  e. NrmGrp  /\  T  e. NrmGrp  /\  F  e.  ( S  GrpHom  T ) )  /\  ( N `
 F )  =  0 )  /\  x  e.  V )  ->  ( F `  x )  =  .0.  )
4039mpteq2dva 4106 . . . 4  |-  ( ( ( S  e. NrmGrp  /\  T  e. NrmGrp  /\  F  e.  ( S  GrpHom  T ) )  /\  ( N `  F )  =  0 )  ->  ( x  e.  V  |->  ( F `
 x ) )  =  ( x  e.  V  |->  .0.  ) )
4126feqmptd 5575 . . . 4  |-  ( ( ( S  e. NrmGrp  /\  T  e. NrmGrp  /\  F  e.  ( S  GrpHom  T ) )  /\  ( N `  F )  =  0 )  ->  F  =  ( x  e.  V  |->  ( F `  x
) ) )
42 fconstmpt 4732 . . . . 5  |-  ( V  X.  {  .0.  }
)  =  ( x  e.  V  |->  .0.  )
4342a1i 10 . . . 4  |-  ( ( ( S  e. NrmGrp  /\  T  e. NrmGrp  /\  F  e.  ( S  GrpHom  T ) )  /\  ( N `  F )  =  0 )  ->  ( V  X.  {  .0.  } )  =  ( x  e.  V  |->  .0.  ) )
4440, 41, 433eqtr4d 2325 . . 3  |-  ( ( ( S  e. NrmGrp  /\  T  e. NrmGrp  /\  F  e.  ( S  GrpHom  T ) )  /\  ( N `  F )  =  0 )  ->  F  =  ( V  X.  {  .0.  } ) )
4544ex 423 . 2  |-  ( ( S  e. NrmGrp  /\  T  e. NrmGrp  /\  F  e.  ( S  GrpHom  T ) )  ->  ( ( N `
 F )  =  0  ->  F  =  ( V  X.  {  .0.  } ) ) )
464, 8, 36nmo0 18244 . . . 4  |-  ( ( S  e. NrmGrp  /\  T  e. NrmGrp
)  ->  ( N `  ( V  X.  {  .0.  } ) )  =  0 )
47463adant3 975 . . 3  |-  ( ( S  e. NrmGrp  /\  T  e. NrmGrp  /\  F  e.  ( S  GrpHom  T ) )  ->  ( N `  ( V  X.  {  .0.  } ) )  =  0 )
48 fveq2 5525 . . . 4  |-  ( F  =  ( V  X.  {  .0.  } )  -> 
( N `  F
)  =  ( N `
 ( V  X.  {  .0.  } ) ) )
4948eqeq1d 2291 . . 3  |-  ( F  =  ( V  X.  {  .0.  } )  -> 
( ( N `  F )  =  0  <-> 
( N `  ( V  X.  {  .0.  }
) )  =  0 ) )
5047, 49syl5ibrcom 213 . 2  |-  ( ( S  e. NrmGrp  /\  T  e. NrmGrp  /\  F  e.  ( S  GrpHom  T ) )  ->  ( F  =  ( V  X.  {  .0.  } )  ->  ( N `  F )  =  0 ) )
5145, 50impbid 183 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 176    /\ wa 358    /\ w3a 934    = wceq 1623    e. wcel 1684   {csn 3640   class class class wbr 4023    e. cmpt 4077    X. cxp 4687   -->wf 5251   ` cfv 5255  (class class class)co 5858   RRcr 8736   0cc0 8737    x. cmul 8742    <_ cle 8868   Basecbs 13148   0gc0g 13400    GrpHom cghm 14680   normcnm 18099  NrmGrpcngp 18100   normOpcnmo 18214   NGHom cnghm 18215
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 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-cnex 8793  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813  ax-pre-mulgt0 8814  ax-pre-sup 8815
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 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-riota 6304  df-recs 6388  df-rdg 6423  df-er 6660  df-map 6774  df-en 6864  df-dom 6865  df-sdom 6866  df-sup 7194  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-sub 9039  df-neg 9040  df-div 9424  df-nn 9747  df-2 9804  df-n0 9966  df-z 10025  df-uz 10231  df-q 10317  df-rp 10355  df-xneg 10452  df-xadd 10453  df-xmul 10454  df-ico 10662  df-topgen 13344  df-0g 13404  df-mnd 14367  df-mhm 14415  df-grp 14489  df-ghm 14681  df-xmet 16373  df-met 16374  df-bl 16375  df-mopn 16376  df-top 16636  df-bases 16638  df-topon 16639  df-topsp 16640  df-xms 17885  df-ms 17886  df-nm 18105  df-ngp 18106  df-nmo 18217  df-nghm 18218
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