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Theorem nmoid 18267
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 2296 . . 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 18137 . . . . 5  |-  ( S  e. NrmGrp  ->  S  e.  Grp )
76adantr 451 . . . 4  |-  ( ( S  e. NrmGrp  /\  {  .0.  } 
C.  V )  ->  S  e.  Grp )
82idghm 14714 . . . 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 8853 . . . 4  |-  1  e.  RR
1110a1i 10 . . 3  |-  ( ( S  e. NrmGrp  /\  {  .0.  } 
C.  V )  -> 
1  e.  RR )
12 0le1 9313 . . . 4  |-  0  <_  1
1312a1i 10 . . 3  |-  ( ( S  e. NrmGrp  /\  {  .0.  } 
C.  V )  -> 
0  <_  1 )
142, 3nmcl 18153 . . . . . 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 9355 . . . 4  |-  ( ( ( S  e. NrmGrp  /\  {  .0.  }  C.  V )  /\  ( x  e.  V  /\  x  =/= 
.0.  ) )  -> 
( ( norm `  S
) `  x )  <_  ( ( norm `  S
) `  x )
)
17 fvresi 5727 . . . . . 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 5545 . . . 4  |-  ( ( ( S  e. NrmGrp  /\  {  .0.  }  C.  V )  /\  ( x  e.  V  /\  x  =/= 
.0.  ) )  -> 
( ( norm `  S
) `  ( (  _I  |`  V ) `  x ) )  =  ( ( norm `  S
) `  x )
)
2015recnd 8877 . . . . 5  |-  ( ( ( S  e. NrmGrp  /\  {  .0.  }  C.  V )  /\  ( x  e.  V  /\  x  =/= 
.0.  ) )  -> 
( ( norm `  S
) `  x )  e.  CC )
2120mulid2d 8869 . . . 4  |-  ( ( ( S  e. NrmGrp  /\  {  .0.  }  C.  V )  /\  ( x  e.  V  /\  x  =/= 
.0.  ) )  -> 
( 1  x.  (
( norm `  S ) `  x ) )  =  ( ( norm `  S
) `  x )
)
2216, 19, 213brtr4d 4069 . . 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 18243 . 2  |-  ( ( S  e. NrmGrp  /\  {  .0.  } 
C.  V )  -> 
( N `  (  _I  |`  V ) )  <_  1 )
24 pssnel 3532 . . . 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 3668 . . . . . . . 8  |-  ( x  e.  {  .0.  }  <->  x  =  .0.  )
2726biimpri 197 . . . . . . 7  |-  ( x  =  .0.  ->  x  e.  {  .0.  } )
2827necon3bi 2500 . . . . . 6  |-  ( -.  x  e.  {  .0.  }  ->  x  =/=  .0.  )
2921, 19eqtr4d 2331 . . . . . . . 8  |-  ( ( ( S  e. NrmGrp  /\  {  .0.  }  C.  V )  /\  ( x  e.  V  /\  x  =/= 
.0.  ) )  -> 
( 1  x.  (
( norm `  S ) `  x ) )  =  ( ( norm `  S
) `  ( (  _I  |`  V ) `  x ) ) )
301nmocl 18245 . . . . . . . . . . . . 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 18246 . . . . . . . . . . . . 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 10519 . . . . . . . . . . . 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 18249 . . . . . . . . . . . 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 18253 . . . . . . . . 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 4059 . . . . . . 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 18157 . . . . . . . . . 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 10445 . . . . . . 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 1626 . . 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 8893 . . . 4  |-  ( 1  e.  RR  ->  1  e.  RR* )
5610, 55ax-mp 8 . . 3  |-  1  e.  RR*
57 xrletri3 10502 . . 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 1531    = wceq 1632    e. wcel 1696    =/= wne 2459    C. wpss 3166   {csn 3653   class class class wbr 4039    _I cid 4320    |` cres 4707   ` cfv 5271  (class class class)co 5874   RRcr 8752   0cc0 8753   1c1 8754    x. cmul 8758   RR*cxr 8882    <_ cle 8884   RR+crp 10370   Basecbs 13164   0gc0g 13416   Grpcgrp 14378    GrpHom cghm 14696   normcnm 18115  NrmGrpcngp 18116   normOpcnmo 18230   NGHom cnghm 18231
This theorem is referenced by:  idnghm  18268
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-cnex 8809  ax-resscn 8810  ax-1cn 8811  ax-icn 8812  ax-addcl 8813  ax-addrcl 8814  ax-mulcl 8815  ax-mulrcl 8816  ax-mulcom 8817  ax-addass 8818  ax-mulass 8819  ax-distr 8820  ax-i2m1 8821  ax-1ne0 8822  ax-1rid 8823  ax-rnegex 8824  ax-rrecex 8825  ax-cnre 8826  ax-pre-lttri 8827  ax-pre-lttrn 8828  ax-pre-ltadd 8829  ax-pre-mulgt0 8830  ax-pre-sup 8831
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 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rmo 2564  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-1st 6138  df-2nd 6139  df-riota 6320  df-recs 6404  df-rdg 6439  df-er 6676  df-map 6790  df-en 6880  df-dom 6881  df-sdom 6882  df-sup 7210  df-pnf 8885  df-mnf 8886  df-xr 8887  df-ltxr 8888  df-le 8889  df-sub 9055  df-neg 9056  df-div 9440  df-nn 9763  df-2 9820  df-n0 9982  df-z 10041  df-uz 10247  df-q 10333  df-rp 10371  df-xneg 10468  df-xadd 10469  df-xmul 10470  df-ico 10678  df-topgen 13360  df-0g 13420  df-mnd 14383  df-grp 14505  df-ghm 14697  df-xmet 16389  df-met 16390  df-bl 16391  df-mopn 16392  df-top 16652  df-bases 16654  df-topon 16655  df-topsp 16656  df-xms 17901  df-ms 17902  df-nm 18121  df-ngp 18122  df-nmo 18233  df-nghm 18234
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