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Theorem nmoid 20340
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 2443 . . 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 20210 . . . . 5  |-  ( S  e. NrmGrp  ->  S  e.  Grp )
76adantr 465 . . . 4  |-  ( ( S  e. NrmGrp  /\  {  .0.  } 
C.  V )  ->  S  e.  Grp )
82idghm 15781 . . . 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 9420 . . 3  |-  ( ( S  e. NrmGrp  /\  {  .0.  } 
C.  V )  -> 
1  e.  RR )
11 0le1 9882 . . . 4  |-  0  <_  1
1211a1i 11 . . 3  |-  ( ( S  e. NrmGrp  /\  {  .0.  } 
C.  V )  -> 
0  <_  1 )
132, 3nmcl 20226 . . . . . 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 9925 . . . 4  |-  ( ( ( S  e. NrmGrp  /\  {  .0.  }  C.  V )  /\  ( x  e.  V  /\  x  =/=  .0.  ) )  ->  (
( norm `  S ) `  x )  <_  (
( norm `  S ) `  x ) )
16 fvresi 5923 . . . . . 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 5714 . . . 4  |-  ( ( ( S  e. NrmGrp  /\  {  .0.  }  C.  V )  /\  ( x  e.  V  /\  x  =/=  .0.  ) )  ->  (
( norm `  S ) `  ( (  _I  |`  V ) `
 x ) )  =  ( ( norm `  S ) `  x
) )
1914recnd 9431 . . . . 5  |-  ( ( ( S  e. NrmGrp  /\  {  .0.  }  C.  V )  /\  ( x  e.  V  /\  x  =/=  .0.  ) )  ->  (
( norm `  S ) `  x )  e.  CC )
2019mulid2d 9423 . . . 4  |-  ( ( ( S  e. NrmGrp  /\  {  .0.  }  C.  V )  /\  ( x  e.  V  /\  x  =/=  .0.  ) )  ->  (
1  x.  ( (
norm `  S ) `  x ) )  =  ( ( norm `  S
) `  x )
)
2115, 18, 203brtr4d 4341 . . 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 20316 . 2  |-  ( ( S  e. NrmGrp  /\  {  .0.  } 
C.  V )  -> 
( N `  (  _I  |`  V ) )  <_  1 )
23 pssnel 3763 . . . 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 3910 . . . . . 6  |-  ( x  e.  {  .0.  }  <->  x  =  .0.  )
2625biimpri 206 . . . . 5  |-  ( x  =  .0.  ->  x  e.  {  .0.  } )
2726necon3bi 2674 . . . 4  |-  ( -.  x  e.  {  .0.  }  ->  x  =/=  .0.  )
2820, 18eqtr4d 2478 . . . . . 6  |-  ( ( ( S  e. NrmGrp  /\  {  .0.  }  C.  V )  /\  ( x  e.  V  /\  x  =/=  .0.  ) )  ->  (
1  x.  ( (
norm `  S ) `  x ) )  =  ( ( norm `  S
) `  ( (  _I  |`  V ) `  x ) ) )
291nmocl 20318 . . . . . . . . . . 11  |-  ( ( S  e. NrmGrp  /\  S  e. NrmGrp  /\  (  _I  |`  V )  e.  ( S  GrpHom  S ) )  ->  ( N `  (  _I  |`  V ) )  e. 
RR* )
305, 5, 9, 29syl3anc 1218 . . . . . . . . . 10  |-  ( ( S  e. NrmGrp  /\  {  .0.  } 
C.  V )  -> 
( N `  (  _I  |`  V ) )  e.  RR* )
311nmoge0 20319 . . . . . . . . . . 11  |-  ( ( S  e. NrmGrp  /\  S  e. NrmGrp  /\  (  _I  |`  V )  e.  ( S  GrpHom  S ) )  ->  0  <_  ( N `  (  _I  |`  V ) ) )
325, 5, 9, 31syl3anc 1218 . . . . . . . . . 10  |-  ( ( S  e. NrmGrp  /\  {  .0.  } 
C.  V )  -> 
0  <_  ( N `  (  _I  |`  V ) ) )
33 xrrege0 11165 . . . . . . . . . 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 1219 . . . . . . . . 9  |-  ( ( S  e. NrmGrp  /\  {  .0.  } 
C.  V )  -> 
( N `  (  _I  |`  V ) )  e.  RR )
351isnghm2 20322 . . . . . . . . . 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 1218 . . . . . . . . 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 20326 . . . . . . 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 4331 . . . . 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 9420 . . . . . 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 20230 . . . . . . . 8  |-  ( ( S  e. NrmGrp  /\  x  e.  V  /\  x  =/=  .0.  )  ->  (
( norm `  S ) `  x )  e.  RR+ )
46453expb 1188 . . . . . . 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 11085 . . . . 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 1692 . 2  |-  ( ( S  e. NrmGrp  /\  {  .0.  } 
C.  V )  -> 
1  <_  ( N `  (  _I  |`  V ) ) )
52 1re 9404 . . . 4  |-  1  e.  RR
5352rexri 9455 . . 3  |-  1  e.  RR*
54 xrletri3 11148 . . 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 913 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 1369   E.wex 1586    e. wcel 1756    =/= wne 2620    C. wpss 3348   {csn 3896   class class class wbr 4311    _I cid 4650    |` cres 4861   ` cfv 5437  (class class class)co 6110   RRcr 9300   0cc0 9301   1c1 9302    x. cmul 9306   RR*cxr 9436    <_ cle 9438   RR+crp 11010   Basecbs 14193   0gc0g 14397   Grpcgrp 15429    GrpHom cghm 15763   normcnm 20188  NrmGrpcngp 20189   normOpcnmo 20303   NGHom cnghm 20304
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 4422  ax-sep 4432  ax-nul 4440  ax-pow 4489  ax-pr 4550  ax-un 6391  ax-cnex 9357  ax-resscn 9358  ax-1cn 9359  ax-icn 9360  ax-addcl 9361  ax-addrcl 9362  ax-mulcl 9363  ax-mulrcl 9364  ax-mulcom 9365  ax-addass 9366  ax-mulass 9367  ax-distr 9368  ax-i2m1 9369  ax-1ne0 9370  ax-1rid 9371  ax-rnegex 9372  ax-rrecex 9373  ax-cnre 9374  ax-pre-lttri 9375  ax-pre-lttrn 9376  ax-pre-ltadd 9377  ax-pre-mulgt0 9378  ax-pre-sup 9379
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 2577  df-ne 2622  df-nel 2623  df-ral 2739  df-rex 2740  df-reu 2741  df-rmo 2742  df-rab 2743  df-v 2993  df-sbc 3206  df-csb 3308  df-dif 3350  df-un 3352  df-in 3354  df-ss 3361  df-pss 3363  df-nul 3657  df-if 3811  df-pw 3881  df-sn 3897  df-pr 3899  df-tp 3901  df-op 3903  df-uni 4111  df-iun 4192  df-br 4312  df-opab 4370  df-mpt 4371  df-tr 4405  df-eprel 4651  df-id 4655  df-po 4660  df-so 4661  df-fr 4698  df-we 4700  df-ord 4741  df-on 4742  df-lim 4743  df-suc 4744  df-xp 4865  df-rel 4866  df-cnv 4867  df-co 4868  df-dm 4869  df-rn 4870  df-res 4871  df-ima 4872  df-iota 5400  df-fun 5439  df-fn 5440  df-f 5441  df-f1 5442  df-fo 5443  df-f1o 5444  df-fv 5445  df-riota 6071  df-ov 6113  df-oprab 6114  df-mpt2 6115  df-om 6496  df-1st 6596  df-2nd 6597  df-recs 6851  df-rdg 6885  df-er 7120  df-map 7235  df-en 7330  df-dom 7331  df-sdom 7332  df-sup 7710  df-pnf 9439  df-mnf 9440  df-xr 9441  df-ltxr 9442  df-le 9443  df-sub 9616  df-neg 9617  df-div 10013  df-nn 10342  df-2 10399  df-n0 10599  df-z 10666  df-uz 10881  df-q 10973  df-rp 11011  df-xneg 11108  df-xadd 11109  df-xmul 11110  df-ico 11325  df-0g 14399  df-topgen 14401  df-mnd 15434  df-grp 15564  df-ghm 15764  df-psmet 17828  df-xmet 17829  df-met 17830  df-bl 17831  df-mopn 17832  df-top 18522  df-bases 18524  df-topon 18525  df-topsp 18526  df-xms 19914  df-ms 19915  df-nm 20194  df-ngp 20195  df-nmo 20306  df-nghm 20307
This theorem is referenced by:  idnghm  20341
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