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Theorem nmoid 21117
Description: The operator norm of the identity function on a nontrivial group. (Contributed by Mario Carneiro, 20-Oct-2015.)
Hypotheses
Ref Expression
nmoid.1  |-  N  =  ( S normOp S )
nmoid.2  |-  V  =  ( Base `  S
)
nmoid.3  |-  .0.  =  ( 0g `  S )
Assertion
Ref Expression
nmoid  |-  ( ( S  e. NrmGrp  /\  {  .0.  } 
C.  V )  -> 
( N `  (  _I  |`  V ) )  =  1 )

Proof of Theorem nmoid
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 nmoid.1 . . 3  |-  N  =  ( S normOp S )
2 nmoid.2 . . 3  |-  V  =  ( Base `  S
)
3 eqid 2467 . . 3  |-  ( norm `  S )  =  (
norm `  S )
4 nmoid.3 . . 3  |-  .0.  =  ( 0g `  S )
5 simpl 457 . . 3  |-  ( ( S  e. NrmGrp  /\  {  .0.  } 
C.  V )  ->  S  e. NrmGrp )
6 ngpgrp 20987 . . . . 5  |-  ( S  e. NrmGrp  ->  S  e.  Grp )
76adantr 465 . . . 4  |-  ( ( S  e. NrmGrp  /\  {  .0.  } 
C.  V )  ->  S  e.  Grp )
82idghm 16154 . . . 4  |-  ( S  e.  Grp  ->  (  _I  |`  V )  e.  ( S  GrpHom  S ) )
97, 8syl 16 . . 3  |-  ( ( S  e. NrmGrp  /\  {  .0.  } 
C.  V )  -> 
(  _I  |`  V )  e.  ( S  GrpHom  S ) )
10 1red 9623 . . 3  |-  ( ( S  e. NrmGrp  /\  {  .0.  } 
C.  V )  -> 
1  e.  RR )
11 0le1 10088 . . . 4  |-  0  <_  1
1211a1i 11 . . 3  |-  ( ( S  e. NrmGrp  /\  {  .0.  } 
C.  V )  -> 
0  <_  1 )
132, 3nmcl 21003 . . . . . 6  |-  ( ( S  e. NrmGrp  /\  x  e.  V )  ->  (
( norm `  S ) `  x )  e.  RR )
1413ad2ant2r 746 . . . . 5  |-  ( ( ( S  e. NrmGrp  /\  {  .0.  }  C.  V )  /\  ( x  e.  V  /\  x  =/=  .0.  ) )  ->  (
( norm `  S ) `  x )  e.  RR )
1514leidd 10131 . . . 4  |-  ( ( ( S  e. NrmGrp  /\  {  .0.  }  C.  V )  /\  ( x  e.  V  /\  x  =/=  .0.  ) )  ->  (
( norm `  S ) `  x )  <_  (
( norm `  S ) `  x ) )
16 fvresi 6098 . . . . . 6  |-  ( x  e.  V  ->  (
(  _I  |`  V ) `
 x )  =  x )
1716ad2antrl 727 . . . . 5  |-  ( ( ( S  e. NrmGrp  /\  {  .0.  }  C.  V )  /\  ( x  e.  V  /\  x  =/=  .0.  ) )  ->  (
(  _I  |`  V ) `
 x )  =  x )
1817fveq2d 5876 . . . 4  |-  ( ( ( S  e. NrmGrp  /\  {  .0.  }  C.  V )  /\  ( x  e.  V  /\  x  =/=  .0.  ) )  ->  (
( norm `  S ) `  ( (  _I  |`  V ) `
 x ) )  =  ( ( norm `  S ) `  x
) )
1914recnd 9634 . . . . 5  |-  ( ( ( S  e. NrmGrp  /\  {  .0.  }  C.  V )  /\  ( x  e.  V  /\  x  =/=  .0.  ) )  ->  (
( norm `  S ) `  x )  e.  CC )
2019mulid2d 9626 . . . 4  |-  ( ( ( S  e. NrmGrp  /\  {  .0.  }  C.  V )  /\  ( x  e.  V  /\  x  =/=  .0.  ) )  ->  (
1  x.  ( (
norm `  S ) `  x ) )  =  ( ( norm `  S
) `  x )
)
2115, 18, 203brtr4d 4483 . . 3  |-  ( ( ( S  e. NrmGrp  /\  {  .0.  }  C.  V )  /\  ( x  e.  V  /\  x  =/=  .0.  ) )  ->  (
( norm `  S ) `  ( (  _I  |`  V ) `
 x ) )  <_  ( 1  x.  ( ( norm `  S
) `  x )
) )
221, 2, 3, 3, 4, 5, 5, 9, 10, 12, 21nmolb2d 21093 . 2  |-  ( ( S  e. NrmGrp  /\  {  .0.  } 
C.  V )  -> 
( N `  (  _I  |`  V ) )  <_  1 )
23 pssnel 3898 . . . 4  |-  ( {  .0.  }  C.  V  ->  E. x ( x  e.  V  /\  -.  x  e.  {  .0.  } ) )
2423adantl 466 . . 3  |-  ( ( S  e. NrmGrp  /\  {  .0.  } 
C.  V )  ->  E. x ( x  e.  V  /\  -.  x  e.  {  .0.  } ) )
25 elsn 4047 . . . . . 6  |-  ( x  e.  {  .0.  }  <->  x  =  .0.  )
2625biimpri 206 . . . . 5  |-  ( x  =  .0.  ->  x  e.  {  .0.  } )
2726necon3bi 2696 . . . 4  |-  ( -.  x  e.  {  .0.  }  ->  x  =/=  .0.  )
2820, 18eqtr4d 2511 . . . . . 6  |-  ( ( ( S  e. NrmGrp  /\  {  .0.  }  C.  V )  /\  ( x  e.  V  /\  x  =/=  .0.  ) )  ->  (
1  x.  ( (
norm `  S ) `  x ) )  =  ( ( norm `  S
) `  ( (  _I  |`  V ) `  x ) ) )
291nmocl 21095 . . . . . . . . . . 11  |-  ( ( S  e. NrmGrp  /\  S  e. NrmGrp  /\  (  _I  |`  V )  e.  ( S  GrpHom  S ) )  ->  ( N `  (  _I  |`  V ) )  e. 
RR* )
305, 5, 9, 29syl3anc 1228 . . . . . . . . . 10  |-  ( ( S  e. NrmGrp  /\  {  .0.  } 
C.  V )  -> 
( N `  (  _I  |`  V ) )  e.  RR* )
311nmoge0 21096 . . . . . . . . . . 11  |-  ( ( S  e. NrmGrp  /\  S  e. NrmGrp  /\  (  _I  |`  V )  e.  ( S  GrpHom  S ) )  ->  0  <_  ( N `  (  _I  |`  V ) ) )
325, 5, 9, 31syl3anc 1228 . . . . . . . . . 10  |-  ( ( S  e. NrmGrp  /\  {  .0.  } 
C.  V )  -> 
0  <_  ( N `  (  _I  |`  V ) ) )
33 xrrege0 11387 . . . . . . . . . 10  |-  ( ( ( ( N `  (  _I  |`  V ) )  e.  RR*  /\  1  e.  RR )  /\  (
0  <_  ( N `  (  _I  |`  V ) )  /\  ( N `
 (  _I  |`  V ) )  <_  1 ) )  ->  ( N `  (  _I  |`  V ) )  e.  RR )
3430, 10, 32, 22, 33syl22anc 1229 . . . . . . . . 9  |-  ( ( S  e. NrmGrp  /\  {  .0.  } 
C.  V )  -> 
( N `  (  _I  |`  V ) )  e.  RR )
351isnghm2 21099 . . . . . . . . . 10  |-  ( ( S  e. NrmGrp  /\  S  e. NrmGrp  /\  (  _I  |`  V )  e.  ( S  GrpHom  S ) )  ->  (
(  _I  |`  V )  e.  ( S NGHom  S
)  <->  ( N `  (  _I  |`  V ) )  e.  RR ) )
365, 5, 9, 35syl3anc 1228 . . . . . . . . 9  |-  ( ( S  e. NrmGrp  /\  {  .0.  } 
C.  V )  -> 
( (  _I  |`  V )  e.  ( S NGHom  S
)  <->  ( N `  (  _I  |`  V ) )  e.  RR ) )
3734, 36mpbird 232 . . . . . . . 8  |-  ( ( S  e. NrmGrp  /\  {  .0.  } 
C.  V )  -> 
(  _I  |`  V )  e.  ( S NGHom  S
) )
3837adantr 465 . . . . . . 7  |-  ( ( ( S  e. NrmGrp  /\  {  .0.  }  C.  V )  /\  ( x  e.  V  /\  x  =/=  .0.  ) )  ->  (  _I  |`  V )  e.  ( S NGHom  S ) )
39 simprl 755 . . . . . . 7  |-  ( ( ( S  e. NrmGrp  /\  {  .0.  }  C.  V )  /\  ( x  e.  V  /\  x  =/=  .0.  ) )  ->  x  e.  V )
401, 2, 3, 3nmoi 21103 . . . . . . 7  |-  ( ( (  _I  |`  V )  e.  ( S NGHom  S
)  /\  x  e.  V )  ->  (
( norm `  S ) `  ( (  _I  |`  V ) `
 x ) )  <_  ( ( N `
 (  _I  |`  V ) )  x.  ( (
norm `  S ) `  x ) ) )
4138, 39, 40syl2anc 661 . . . . . 6  |-  ( ( ( S  e. NrmGrp  /\  {  .0.  }  C.  V )  /\  ( x  e.  V  /\  x  =/=  .0.  ) )  ->  (
( norm `  S ) `  ( (  _I  |`  V ) `
 x ) )  <_  ( ( N `
 (  _I  |`  V ) )  x.  ( (
norm `  S ) `  x ) ) )
4228, 41eqbrtrd 4473 . . . . 5  |-  ( ( ( S  e. NrmGrp  /\  {  .0.  }  C.  V )  /\  ( x  e.  V  /\  x  =/=  .0.  ) )  ->  (
1  x.  ( (
norm `  S ) `  x ) )  <_ 
( ( N `  (  _I  |`  V ) )  x.  ( (
norm `  S ) `  x ) ) )
43 1red 9623 . . . . . 6  |-  ( ( ( S  e. NrmGrp  /\  {  .0.  }  C.  V )  /\  ( x  e.  V  /\  x  =/=  .0.  ) )  ->  1  e.  RR )
4434adantr 465 . . . . . 6  |-  ( ( ( S  e. NrmGrp  /\  {  .0.  }  C.  V )  /\  ( x  e.  V  /\  x  =/=  .0.  ) )  ->  ( N `  (  _I  |`  V ) )  e.  RR )
452, 3, 4nmrpcl 21007 . . . . . . . 8  |-  ( ( S  e. NrmGrp  /\  x  e.  V  /\  x  =/=  .0.  )  ->  (
( norm `  S ) `  x )  e.  RR+ )
46453expb 1197 . . . . . . 7  |-  ( ( S  e. NrmGrp  /\  (
x  e.  V  /\  x  =/=  .0.  ) )  ->  ( ( norm `  S ) `  x
)  e.  RR+ )
4746adantlr 714 . . . . . 6  |-  ( ( ( S  e. NrmGrp  /\  {  .0.  }  C.  V )  /\  ( x  e.  V  /\  x  =/=  .0.  ) )  ->  (
( norm `  S ) `  x )  e.  RR+ )
4843, 44, 47lemul1d 11307 . . . . 5  |-  ( ( ( S  e. NrmGrp  /\  {  .0.  }  C.  V )  /\  ( x  e.  V  /\  x  =/=  .0.  ) )  ->  (
1  <_  ( N `  (  _I  |`  V ) )  <->  ( 1  x.  ( ( norm `  S
) `  x )
)  <_  ( ( N `  (  _I  |`  V ) )  x.  ( ( norm `  S
) `  x )
) ) )
4942, 48mpbird 232 . . . 4  |-  ( ( ( S  e. NrmGrp  /\  {  .0.  }  C.  V )  /\  ( x  e.  V  /\  x  =/=  .0.  ) )  ->  1  <_  ( N `  (  _I  |`  V ) ) )
5027, 49sylanr2 653 . . 3  |-  ( ( ( S  e. NrmGrp  /\  {  .0.  }  C.  V )  /\  ( x  e.  V  /\  -.  x  e.  {  .0.  } ) )  -> 
1  <_  ( N `  (  _I  |`  V ) ) )
5124, 50exlimddv 1702 . 2  |-  ( ( S  e. NrmGrp  /\  {  .0.  } 
C.  V )  -> 
1  <_  ( N `  (  _I  |`  V ) ) )
52 1re 9607 . . . 4  |-  1  e.  RR
5352rexri 9658 . . 3  |-  1  e.  RR*
54 xrletri3 11370 . . 3  |-  ( ( ( N `  (  _I  |`  V ) )  e.  RR*  /\  1  e.  RR* )  ->  (
( N `  (  _I  |`  V ) )  =  1  <->  ( ( N `  (  _I  |`  V ) )  <_ 
1  /\  1  <_  ( N `  (  _I  |`  V ) ) ) ) )
5530, 53, 54sylancl 662 . 2  |-  ( ( S  e. NrmGrp  /\  {  .0.  } 
C.  V )  -> 
( ( N `  (  _I  |`  V ) )  =  1  <->  (
( N `  (  _I  |`  V ) )  <_  1  /\  1  <_  ( N `  (  _I  |`  V ) ) ) ) )
5622, 51, 55mpbir2and 920 1  |-  ( ( S  e. NrmGrp  /\  {  .0.  } 
C.  V )  -> 
( N `  (  _I  |`  V ) )  =  1 )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1379   E.wex 1596    e. wcel 1767    =/= wne 2662    C. wpss 3482   {csn 4033   class class class wbr 4453    _I cid 4796    |` cres 5007   ` cfv 5594  (class class class)co 6295   RRcr 9503   0cc0 9504   1c1 9505    x. cmul 9509   RR*cxr 9639    <_ cle 9641   RR+crp 11232   Basecbs 14507   0gc0g 14712   Grpcgrp 15925    GrpHom cghm 16136   normcnm 20965  NrmGrpcngp 20966   normOpcnmo 21080   NGHom cnghm 21081
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4564  ax-sep 4574  ax-nul 4582  ax-pow 4631  ax-pr 4692  ax-un 6587  ax-cnex 9560  ax-resscn 9561  ax-1cn 9562  ax-icn 9563  ax-addcl 9564  ax-addrcl 9565  ax-mulcl 9566  ax-mulrcl 9567  ax-mulcom 9568  ax-addass 9569  ax-mulass 9570  ax-distr 9571  ax-i2m1 9572  ax-1ne0 9573  ax-1rid 9574  ax-rnegex 9575  ax-rrecex 9576  ax-cnre 9577  ax-pre-lttri 9578  ax-pre-lttrn 9579  ax-pre-ltadd 9580  ax-pre-mulgt0 9581  ax-pre-sup 9582
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2822  df-rex 2823  df-reu 2824  df-rmo 2825  df-rab 2826  df-v 3120  df-sbc 3337  df-csb 3441  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-pss 3497  df-nul 3791  df-if 3946  df-pw 4018  df-sn 4034  df-pr 4036  df-tp 4038  df-op 4040  df-uni 4252  df-iun 4333  df-br 4454  df-opab 4512  df-mpt 4513  df-tr 4547  df-eprel 4797  df-id 4801  df-po 4806  df-so 4807  df-fr 4844  df-we 4846  df-ord 4887  df-on 4888  df-lim 4889  df-suc 4890  df-xp 5011  df-rel 5012  df-cnv 5013  df-co 5014  df-dm 5015  df-rn 5016  df-res 5017  df-ima 5018  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-riota 6256  df-ov 6298  df-oprab 6299  df-mpt2 6300  df-om 6696  df-1st 6795  df-2nd 6796  df-recs 7054  df-rdg 7088  df-er 7323  df-map 7434  df-en 7529  df-dom 7530  df-sdom 7531  df-sup 7913  df-pnf 9642  df-mnf 9643  df-xr 9644  df-ltxr 9645  df-le 9646  df-sub 9819  df-neg 9820  df-div 10219  df-nn 10549  df-2 10606  df-n0 10808  df-z 10877  df-uz 11095  df-q 11195  df-rp 11233  df-xneg 11330  df-xadd 11331  df-xmul 11332  df-ico 11547  df-0g 14714  df-topgen 14716  df-mgm 15746  df-sgrp 15785  df-mnd 15795  df-grp 15929  df-ghm 16137  df-psmet 18281  df-xmet 18282  df-met 18283  df-bl 18284  df-mopn 18285  df-top 19268  df-bases 19270  df-topon 19271  df-topsp 19272  df-xms 20691  df-ms 20692  df-nm 20971  df-ngp 20972  df-nmo 21083  df-nghm 21084
This theorem is referenced by:  idnghm  21118
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