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Theorem nmoid 21375
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 2457 . . 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 21245 . . . . 5  |-  ( S  e. NrmGrp  ->  S  e.  Grp )
76adantr 465 . . . 4  |-  ( ( S  e. NrmGrp  /\  {  .0.  } 
C.  V )  ->  S  e.  Grp )
82idghm 16409 . . . 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 9628 . . 3  |-  ( ( S  e. NrmGrp  /\  {  .0.  } 
C.  V )  -> 
1  e.  RR )
11 0le1 10097 . . . 4  |-  0  <_  1
1211a1i 11 . . 3  |-  ( ( S  e. NrmGrp  /\  {  .0.  } 
C.  V )  -> 
0  <_  1 )
132, 3nmcl 21261 . . . . . 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 10140 . . . 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 9639 . . . . 5  |-  ( ( ( S  e. NrmGrp  /\  {  .0.  }  C.  V )  /\  ( x  e.  V  /\  x  =/=  .0.  ) )  ->  (
( norm `  S ) `  x )  e.  CC )
2019mulid2d 9631 . . . 4  |-  ( ( ( S  e. NrmGrp  /\  {  .0.  }  C.  V )  /\  ( x  e.  V  /\  x  =/=  .0.  ) )  ->  (
1  x.  ( (
norm `  S ) `  x ) )  =  ( ( norm `  S
) `  x )
)
2115, 18, 203brtr4d 4486 . . 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 21351 . 2  |-  ( ( S  e. NrmGrp  /\  {  .0.  } 
C.  V )  -> 
( N `  (  _I  |`  V ) )  <_  1 )
23 pssnel 3896 . . . 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 4046 . . . . . 6  |-  ( x  e.  {  .0.  }  <->  x  =  .0.  )
2625biimpri 206 . . . . 5  |-  ( x  =  .0.  ->  x  e.  {  .0.  } )
2726necon3bi 2686 . . . 4  |-  ( -.  x  e.  {  .0.  }  ->  x  =/=  .0.  )
2820, 18eqtr4d 2501 . . . . . 6  |-  ( ( ( S  e. NrmGrp  /\  {  .0.  }  C.  V )  /\  ( x  e.  V  /\  x  =/=  .0.  ) )  ->  (
1  x.  ( (
norm `  S ) `  x ) )  =  ( ( norm `  S
) `  ( (  _I  |`  V ) `  x ) ) )
291nmocl 21353 . . . . . . . . . . 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 21354 . . . . . . . . . . 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 11400 . . . . . . . . . 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 21357 . . . . . . . . . 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 756 . . . . . . 7  |-  ( ( ( S  e. NrmGrp  /\  {  .0.  }  C.  V )  /\  ( x  e.  V  /\  x  =/=  .0.  ) )  ->  x  e.  V )
401, 2, 3, 3nmoi 21361 . . . . . . 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 4476 . . . . 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 9628 . . . . . 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 21265 . . . . . . . 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 11320 . . . . 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 1727 . 2  |-  ( ( S  e. NrmGrp  /\  {  .0.  } 
C.  V )  -> 
1  <_  ( N `  (  _I  |`  V ) ) )
52 1re 9612 . . . 4  |-  1  e.  RR
5352rexri 9663 . . 3  |-  1  e.  RR*
54 xrletri3 11383 . . 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 922 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 1395   E.wex 1613    e. wcel 1819    =/= wne 2652    C. wpss 3472   {csn 4032   class class class wbr 4456    _I cid 4799    |` cres 5010   ` cfv 5594  (class class class)co 6296   RRcr 9508   0cc0 9509   1c1 9510    x. cmul 9514   RR*cxr 9644    <_ cle 9646   RR+crp 11245   Basecbs 14644   0gc0g 14857   Grpcgrp 16180    GrpHom cghm 16391   normcnm 21223  NrmGrpcngp 21224   normOpcnmo 21338   NGHom cnghm 21339
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-rep 4568  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695  ax-un 6591  ax-cnex 9565  ax-resscn 9566  ax-1cn 9567  ax-icn 9568  ax-addcl 9569  ax-addrcl 9570  ax-mulcl 9571  ax-mulrcl 9572  ax-mulcom 9573  ax-addass 9574  ax-mulass 9575  ax-distr 9576  ax-i2m1 9577  ax-1ne0 9578  ax-1rid 9579  ax-rnegex 9580  ax-rrecex 9581  ax-cnre 9582  ax-pre-lttri 9583  ax-pre-lttrn 9584  ax-pre-ltadd 9585  ax-pre-mulgt0 9586  ax-pre-sup 9587
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-nel 2655  df-ral 2812  df-rex 2813  df-reu 2814  df-rmo 2815  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-pss 3487  df-nul 3794  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-tp 4037  df-op 4039  df-uni 4252  df-iun 4334  df-br 4457  df-opab 4516  df-mpt 4517  df-tr 4551  df-eprel 4800  df-id 4804  df-po 4809  df-so 4810  df-fr 4847  df-we 4849  df-ord 4890  df-on 4891  df-lim 4892  df-suc 4893  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  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 6258  df-ov 6299  df-oprab 6300  df-mpt2 6301  df-om 6700  df-1st 6799  df-2nd 6800  df-recs 7060  df-rdg 7094  df-er 7329  df-map 7440  df-en 7536  df-dom 7537  df-sdom 7538  df-sup 7919  df-pnf 9647  df-mnf 9648  df-xr 9649  df-ltxr 9650  df-le 9651  df-sub 9826  df-neg 9827  df-div 10228  df-nn 10557  df-2 10615  df-n0 10817  df-z 10886  df-uz 11107  df-q 11208  df-rp 11246  df-xneg 11343  df-xadd 11344  df-xmul 11345  df-ico 11560  df-0g 14859  df-topgen 14861  df-mgm 15999  df-sgrp 16038  df-mnd 16048  df-grp 16184  df-ghm 16392  df-psmet 18538  df-xmet 18539  df-met 18540  df-bl 18541  df-mopn 18542  df-top 19526  df-bases 19528  df-topon 19529  df-topsp 19530  df-xms 20949  df-ms 20950  df-nm 21229  df-ngp 21230  df-nmo 21341  df-nghm 21342
This theorem is referenced by:  idnghm  21376
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