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Theorem nmoid 18251
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 2283 . . 3  |-  ( norm `  S )  =  (
norm `  S )
4 nmoid.3 . . 3  |-  .0.  =  ( 0g `  S )
5 simpl 443 . . 3  |-  ( ( S  e. NrmGrp  /\  {  .0.  } 
C.  V )  ->  S  e. NrmGrp )
6 ngpgrp 18121 . . . . 5  |-  ( S  e. NrmGrp  ->  S  e.  Grp )
76adantr 451 . . . 4  |-  ( ( S  e. NrmGrp  /\  {  .0.  } 
C.  V )  ->  S  e.  Grp )
82idghm 14698 . . . 4  |-  ( S  e.  Grp  ->  (  _I  |`  V )  e.  ( S  GrpHom  S ) )
97, 8syl 15 . . 3  |-  ( ( S  e. NrmGrp  /\  {  .0.  } 
C.  V )  -> 
(  _I  |`  V )  e.  ( S  GrpHom  S ) )
10 1re 8837 . . . 4  |-  1  e.  RR
1110a1i 10 . . 3  |-  ( ( S  e. NrmGrp  /\  {  .0.  } 
C.  V )  -> 
1  e.  RR )
12 0le1 9297 . . . 4  |-  0  <_  1
1312a1i 10 . . 3  |-  ( ( S  e. NrmGrp  /\  {  .0.  } 
C.  V )  -> 
0  <_  1 )
142, 3nmcl 18137 . . . . . 6  |-  ( ( S  e. NrmGrp  /\  x  e.  V )  ->  (
( norm `  S ) `  x )  e.  RR )
1514ad2ant2r 727 . . . . 5  |-  ( ( ( S  e. NrmGrp  /\  {  .0.  }  C.  V )  /\  ( x  e.  V  /\  x  =/= 
.0.  ) )  -> 
( ( norm `  S
) `  x )  e.  RR )
1615leidd 9339 . . . 4  |-  ( ( ( S  e. NrmGrp  /\  {  .0.  }  C.  V )  /\  ( x  e.  V  /\  x  =/= 
.0.  ) )  -> 
( ( norm `  S
) `  x )  <_  ( ( norm `  S
) `  x )
)
17 fvresi 5711 . . . . . 6  |-  ( x  e.  V  ->  (
(  _I  |`  V ) `
 x )  =  x )
1817ad2antrl 708 . . . . 5  |-  ( ( ( S  e. NrmGrp  /\  {  .0.  }  C.  V )  /\  ( x  e.  V  /\  x  =/= 
.0.  ) )  -> 
( (  _I  |`  V ) `
 x )  =  x )
1918fveq2d 5529 . . . 4  |-  ( ( ( S  e. NrmGrp  /\  {  .0.  }  C.  V )  /\  ( x  e.  V  /\  x  =/= 
.0.  ) )  -> 
( ( norm `  S
) `  ( (  _I  |`  V ) `  x ) )  =  ( ( norm `  S
) `  x )
)
2015recnd 8861 . . . . 5  |-  ( ( ( S  e. NrmGrp  /\  {  .0.  }  C.  V )  /\  ( x  e.  V  /\  x  =/= 
.0.  ) )  -> 
( ( norm `  S
) `  x )  e.  CC )
2120mulid2d 8853 . . . 4  |-  ( ( ( S  e. NrmGrp  /\  {  .0.  }  C.  V )  /\  ( x  e.  V  /\  x  =/= 
.0.  ) )  -> 
( 1  x.  (
( norm `  S ) `  x ) )  =  ( ( norm `  S
) `  x )
)
2216, 19, 213brtr4d 4053 . . 3  |-  ( ( ( S  e. NrmGrp  /\  {  .0.  }  C.  V )  /\  ( x  e.  V  /\  x  =/= 
.0.  ) )  -> 
( ( norm `  S
) `  ( (  _I  |`  V ) `  x ) )  <_ 
( 1  x.  (
( norm `  S ) `  x ) ) )
231, 2, 3, 3, 4, 5, 5, 9, 11, 13, 22nmolb2d 18227 . 2  |-  ( ( S  e. NrmGrp  /\  {  .0.  } 
C.  V )  -> 
( N `  (  _I  |`  V ) )  <_  1 )
24 pssnel 3519 . . . 4  |-  ( {  .0.  }  C.  V  ->  E. x ( x  e.  V  /\  -.  x  e.  {  .0.  } ) )
2524adantl 452 . . 3  |-  ( ( S  e. NrmGrp  /\  {  .0.  } 
C.  V )  ->  E. x ( x  e.  V  /\  -.  x  e.  {  .0.  } ) )
26 elsn 3655 . . . . . . . 8  |-  ( x  e.  {  .0.  }  <->  x  =  .0.  )
2726biimpri 197 . . . . . . 7  |-  ( x  =  .0.  ->  x  e.  {  .0.  } )
2827necon3bi 2487 . . . . . 6  |-  ( -.  x  e.  {  .0.  }  ->  x  =/=  .0.  )
2921, 19eqtr4d 2318 . . . . . . . 8  |-  ( ( ( S  e. NrmGrp  /\  {  .0.  }  C.  V )  /\  ( x  e.  V  /\  x  =/= 
.0.  ) )  -> 
( 1  x.  (
( norm `  S ) `  x ) )  =  ( ( norm `  S
) `  ( (  _I  |`  V ) `  x ) ) )
301nmocl 18229 . . . . . . . . . . . . 13  |-  ( ( S  e. NrmGrp  /\  S  e. NrmGrp  /\  (  _I  |`  V )  e.  ( S  GrpHom  S ) )  ->  ( N `  (  _I  |`  V ) )  e. 
RR* )
315, 5, 9, 30syl3anc 1182 . . . . . . . . . . . 12  |-  ( ( S  e. NrmGrp  /\  {  .0.  } 
C.  V )  -> 
( N `  (  _I  |`  V ) )  e.  RR* )
321nmoge0 18230 . . . . . . . . . . . . 13  |-  ( ( S  e. NrmGrp  /\  S  e. NrmGrp  /\  (  _I  |`  V )  e.  ( S  GrpHom  S ) )  ->  0  <_  ( N `  (  _I  |`  V ) ) )
335, 5, 9, 32syl3anc 1182 . . . . . . . . . . . 12  |-  ( ( S  e. NrmGrp  /\  {  .0.  } 
C.  V )  -> 
0  <_  ( N `  (  _I  |`  V ) ) )
34 xrrege0 10503 . . . . . . . . . . . 12  |-  ( ( ( ( N `  (  _I  |`  V ) )  e.  RR*  /\  1  e.  RR )  /\  (
0  <_  ( N `  (  _I  |`  V ) )  /\  ( N `
 (  _I  |`  V ) )  <_  1 ) )  ->  ( N `  (  _I  |`  V ) )  e.  RR )
3531, 11, 33, 23, 34syl22anc 1183 . . . . . . . . . . 11  |-  ( ( S  e. NrmGrp  /\  {  .0.  } 
C.  V )  -> 
( N `  (  _I  |`  V ) )  e.  RR )
361isnghm2 18233 . . . . . . . . . . . 12  |-  ( ( S  e. NrmGrp  /\  S  e. NrmGrp  /\  (  _I  |`  V )  e.  ( S  GrpHom  S ) )  ->  (
(  _I  |`  V )  e.  ( S NGHom  S
)  <->  ( N `  (  _I  |`  V ) )  e.  RR ) )
375, 5, 9, 36syl3anc 1182 . . . . . . . . . . 11  |-  ( ( S  e. NrmGrp  /\  {  .0.  } 
C.  V )  -> 
( (  _I  |`  V )  e.  ( S NGHom  S
)  <->  ( N `  (  _I  |`  V ) )  e.  RR ) )
3835, 37mpbird 223 . . . . . . . . . 10  |-  ( ( S  e. NrmGrp  /\  {  .0.  } 
C.  V )  -> 
(  _I  |`  V )  e.  ( S NGHom  S
) )
3938adantr 451 . . . . . . . . 9  |-  ( ( ( S  e. NrmGrp  /\  {  .0.  }  C.  V )  /\  ( x  e.  V  /\  x  =/= 
.0.  ) )  -> 
(  _I  |`  V )  e.  ( S NGHom  S
) )
40 simprl 732 . . . . . . . . 9  |-  ( ( ( S  e. NrmGrp  /\  {  .0.  }  C.  V )  /\  ( x  e.  V  /\  x  =/= 
.0.  ) )  ->  x  e.  V )
411, 2, 3, 3nmoi 18237 . . . . . . . . 9  |-  ( ( (  _I  |`  V )  e.  ( S NGHom  S
)  /\  x  e.  V )  ->  (
( norm `  S ) `  ( (  _I  |`  V ) `
 x ) )  <_  ( ( N `
 (  _I  |`  V ) )  x.  ( (
norm `  S ) `  x ) ) )
4239, 40, 41syl2anc 642 . . . . . . . 8  |-  ( ( ( S  e. NrmGrp  /\  {  .0.  }  C.  V )  /\  ( x  e.  V  /\  x  =/= 
.0.  ) )  -> 
( ( norm `  S
) `  ( (  _I  |`  V ) `  x ) )  <_ 
( ( N `  (  _I  |`  V ) )  x.  ( (
norm `  S ) `  x ) ) )
4329, 42eqbrtrd 4043 . . . . . . 7  |-  ( ( ( S  e. NrmGrp  /\  {  .0.  }  C.  V )  /\  ( x  e.  V  /\  x  =/= 
.0.  ) )  -> 
( 1  x.  (
( norm `  S ) `  x ) )  <_ 
( ( N `  (  _I  |`  V ) )  x.  ( (
norm `  S ) `  x ) ) )
4410a1i 10 . . . . . . . 8  |-  ( ( ( S  e. NrmGrp  /\  {  .0.  }  C.  V )  /\  ( x  e.  V  /\  x  =/= 
.0.  ) )  -> 
1  e.  RR )
4535adantr 451 . . . . . . . 8  |-  ( ( ( S  e. NrmGrp  /\  {  .0.  }  C.  V )  /\  ( x  e.  V  /\  x  =/= 
.0.  ) )  -> 
( N `  (  _I  |`  V ) )  e.  RR )
462, 3, 4nmrpcl 18141 . . . . . . . . . 10  |-  ( ( S  e. NrmGrp  /\  x  e.  V  /\  x  =/=  .0.  )  ->  (
( norm `  S ) `  x )  e.  RR+ )
47463expb 1152 . . . . . . . . 9  |-  ( ( S  e. NrmGrp  /\  (
x  e.  V  /\  x  =/=  .0.  ) )  ->  ( ( norm `  S ) `  x
)  e.  RR+ )
4847adantlr 695 . . . . . . . 8  |-  ( ( ( S  e. NrmGrp  /\  {  .0.  }  C.  V )  /\  ( x  e.  V  /\  x  =/= 
.0.  ) )  -> 
( ( norm `  S
) `  x )  e.  RR+ )
4944, 45, 48lemul1d 10429 . . . . . . 7  |-  ( ( ( 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
) ) ) )
5043, 49mpbird 223 . . . . . 6  |-  ( ( ( S  e. NrmGrp  /\  {  .0.  }  C.  V )  /\  ( x  e.  V  /\  x  =/= 
.0.  ) )  -> 
1  <_  ( N `  (  _I  |`  V ) ) )
5128, 50sylanr2 634 . . . . 5  |-  ( ( ( S  e. NrmGrp  /\  {  .0.  }  C.  V )  /\  ( x  e.  V  /\  -.  x  e.  {  .0.  } ) )  ->  1  <_  ( N `  (  _I  |`  V ) ) )
5251ex 423 . . . 4  |-  ( ( S  e. NrmGrp  /\  {  .0.  } 
C.  V )  -> 
( ( x  e.  V  /\  -.  x  e.  {  .0.  } )  ->  1  <_  ( N `  (  _I  |`  V ) ) ) )
5352exlimdv 1664 . . 3  |-  ( ( S  e. NrmGrp  /\  {  .0.  } 
C.  V )  -> 
( E. x ( x  e.  V  /\  -.  x  e.  {  .0.  } )  ->  1  <_  ( N `  (  _I  |`  V ) ) ) )
5425, 53mpd 14 . 2  |-  ( ( S  e. NrmGrp  /\  {  .0.  } 
C.  V )  -> 
1  <_  ( N `  (  _I  |`  V ) ) )
55 rexr 8877 . . . 4  |-  ( 1  e.  RR  ->  1  e.  RR* )
5610, 55ax-mp 8 . . 3  |-  1  e.  RR*
57 xrletri3 10486 . . 3  |-  ( ( ( N `  (  _I  |`  V ) )  e.  RR*  /\  1  e.  RR* )  ->  (
( N `  (  _I  |`  V ) )  =  1  <->  ( ( N `  (  _I  |`  V ) )  <_ 
1  /\  1  <_  ( N `  (  _I  |`  V ) ) ) ) )
5831, 56, 57sylancl 643 . 2  |-  ( ( S  e. NrmGrp  /\  {  .0.  } 
C.  V )  -> 
( ( N `  (  _I  |`  V ) )  =  1  <->  (
( N `  (  _I  |`  V ) )  <_  1  /\  1  <_  ( N `  (  _I  |`  V ) ) ) ) )
5923, 54, 58mpbir2and 888 1  |-  ( ( S  e. NrmGrp  /\  {  .0.  } 
C.  V )  -> 
( N `  (  _I  |`  V ) )  =  1 )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 176    /\ wa 358   E.wex 1528    = wceq 1623    e. wcel 1684    =/= wne 2446    C. wpss 3153   {csn 3640   class class class wbr 4023    _I cid 4304    |` cres 4691   ` cfv 5255  (class class class)co 5858   RRcr 8736   0cc0 8737   1c1 8738    x. cmul 8742   RR*cxr 8866    <_ cle 8868   RR+crp 10354   Basecbs 13148   0gc0g 13400   Grpcgrp 14362    GrpHom cghm 14680   normcnm 18099  NrmGrpcngp 18100   normOpcnmo 18214   NGHom cnghm 18215
This theorem is referenced by:  idnghm  18252
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-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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