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Theorem isngp2 20169
Description: The property of being a normed group. (Contributed by Mario Carneiro, 2-Oct-2015.)
Hypotheses
Ref Expression
isngp.n  |-  N  =  ( norm `  G
)
isngp.z  |-  .-  =  ( -g `  G )
isngp.d  |-  D  =  ( dist `  G
)
isngp2.x  |-  X  =  ( Base `  G
)
isngp2.e  |-  E  =  ( D  |`  ( X  X.  X ) )
Assertion
Ref Expression
isngp2  |-  ( G  e. NrmGrp 
<->  ( G  e.  Grp  /\  G  e.  MetSp  /\  ( N  o.  .-  )  =  E ) )

Proof of Theorem isngp2
StepHypRef Expression
1 isngp.n . . 3  |-  N  =  ( norm `  G
)
2 isngp.z . . 3  |-  .-  =  ( -g `  G )
3 isngp.d . . 3  |-  D  =  ( dist `  G
)
41, 2, 3isngp 20168 . 2  |-  ( G  e. NrmGrp 
<->  ( G  e.  Grp  /\  G  e.  MetSp  /\  ( N  o.  .-  )  C_  D ) )
5 isngp2.e . . . . . . 7  |-  E  =  ( D  |`  ( X  X.  X ) )
6 resss 5129 . . . . . . 7  |-  ( D  |`  ( X  X.  X
) )  C_  D
75, 6eqsstri 3381 . . . . . 6  |-  E  C_  D
8 sseq1 3372 . . . . . 6  |-  ( ( N  o.  .-  )  =  E  ->  ( ( N  o.  .-  )  C_  D  <->  E  C_  D ) )
97, 8mpbiri 233 . . . . 5  |-  ( ( N  o.  .-  )  =  E  ->  ( N  o.  .-  )  C_  D )
10 isngp2.x . . . . . . . . . . . . 13  |-  X  =  ( Base `  G
)
113reseq1i 5101 . . . . . . . . . . . . . 14  |-  ( D  |`  ( X  X.  X
) )  =  ( ( dist `  G
)  |`  ( X  X.  X ) )
125, 11eqtri 2458 . . . . . . . . . . . . 13  |-  E  =  ( ( dist `  G
)  |`  ( X  X.  X ) )
1310, 12msmet 20012 . . . . . . . . . . . 12  |-  ( G  e.  MetSp  ->  E  e.  ( Met `  X ) )
141, 10, 3, 5nmf2 20165 . . . . . . . . . . . 12  |-  ( ( G  e.  Grp  /\  E  e.  ( Met `  X ) )  ->  N : X --> RR )
1513, 14sylan2 474 . . . . . . . . . . 11  |-  ( ( G  e.  Grp  /\  G  e.  MetSp )  ->  N : X --> RR )
1615adantr 465 . . . . . . . . . 10  |-  ( ( ( G  e.  Grp  /\  G  e.  MetSp )  /\  ( N  o.  .-  )  C_  D )  ->  N : X --> RR )
1710, 2grpsubf 15596 . . . . . . . . . . 11  |-  ( G  e.  Grp  ->  .-  :
( X  X.  X
) --> X )
1817ad2antrr 725 . . . . . . . . . 10  |-  ( ( ( G  e.  Grp  /\  G  e.  MetSp )  /\  ( N  o.  .-  )  C_  D )  ->  .-  :
( X  X.  X
) --> X )
19 fco 5563 . . . . . . . . . 10  |-  ( ( N : X --> RR  /\  .-  : ( X  X.  X ) --> X )  ->  ( N  o.  .-  ) : ( X  X.  X ) --> RR )
2016, 18, 19syl2anc 661 . . . . . . . . 9  |-  ( ( ( G  e.  Grp  /\  G  e.  MetSp )  /\  ( N  o.  .-  )  C_  D )  ->  ( N  o.  .-  ) : ( X  X.  X
) --> RR )
21 fdm 5558 . . . . . . . . 9  |-  ( ( N  o.  .-  ) : ( X  X.  X ) --> RR  ->  dom  ( N  o.  .-  )  =  ( X  X.  X ) )
2220, 21syl 16 . . . . . . . 8  |-  ( ( ( G  e.  Grp  /\  G  e.  MetSp )  /\  ( N  o.  .-  )  C_  D )  ->  dom  ( N  o.  .-  )  =  ( X  X.  X ) )
2322reseq2d 5105 . . . . . . 7  |-  ( ( ( G  e.  Grp  /\  G  e.  MetSp )  /\  ( N  o.  .-  )  C_  D )  ->  ( E  |`  dom  ( N  o.  .-  ) )  =  ( E  |`  ( X  X.  X
) ) )
2410, 12msf 20013 . . . . . . . . . 10  |-  ( G  e.  MetSp  ->  E :
( X  X.  X
) --> RR )
2524ad2antlr 726 . . . . . . . . 9  |-  ( ( ( G  e.  Grp  /\  G  e.  MetSp )  /\  ( N  o.  .-  )  C_  D )  ->  E : ( X  X.  X ) --> RR )
26 ffun 5556 . . . . . . . . 9  |-  ( E : ( X  X.  X ) --> RR  ->  Fun 
E )
2725, 26syl 16 . . . . . . . 8  |-  ( ( ( G  e.  Grp  /\  G  e.  MetSp )  /\  ( N  o.  .-  )  C_  D )  ->  Fun  E )
28 simpr 461 . . . . . . . . . 10  |-  ( ( ( G  e.  Grp  /\  G  e.  MetSp )  /\  ( N  o.  .-  )  C_  D )  ->  ( N  o.  .-  )  C_  D )
29 ssv 3371 . . . . . . . . . . . 12  |-  RR  C_  _V
30 fss 5562 . . . . . . . . . . . 12  |-  ( ( ( N  o.  .-  ) : ( X  X.  X ) --> RR  /\  RR  C_  _V )  -> 
( N  o.  .-  ) : ( X  X.  X ) --> _V )
3120, 29, 30sylancl 662 . . . . . . . . . . 11  |-  ( ( ( G  e.  Grp  /\  G  e.  MetSp )  /\  ( N  o.  .-  )  C_  D )  ->  ( N  o.  .-  ) : ( X  X.  X
) --> _V )
32 fssxp 5565 . . . . . . . . . . 11  |-  ( ( N  o.  .-  ) : ( X  X.  X ) --> _V  ->  ( N  o.  .-  )  C_  ( ( X  X.  X )  X.  _V ) )
3331, 32syl 16 . . . . . . . . . 10  |-  ( ( ( G  e.  Grp  /\  G  e.  MetSp )  /\  ( N  o.  .-  )  C_  D )  ->  ( N  o.  .-  )  C_  ( ( X  X.  X )  X.  _V ) )
3428, 33ssind 3569 . . . . . . . . 9  |-  ( ( ( G  e.  Grp  /\  G  e.  MetSp )  /\  ( N  o.  .-  )  C_  D )  ->  ( N  o.  .-  )  C_  ( D  i^i  (
( X  X.  X
)  X.  _V )
) )
35 df-res 4847 . . . . . . . . . 10  |-  ( D  |`  ( X  X.  X
) )  =  ( D  i^i  ( ( X  X.  X )  X.  _V ) )
365, 35eqtri 2458 . . . . . . . . 9  |-  E  =  ( D  i^i  (
( X  X.  X
)  X.  _V )
)
3734, 36syl6sseqr 3398 . . . . . . . 8  |-  ( ( ( G  e.  Grp  /\  G  e.  MetSp )  /\  ( N  o.  .-  )  C_  D )  ->  ( N  o.  .-  )  C_  E )
38 funssres 5453 . . . . . . . 8  |-  ( ( Fun  E  /\  ( N  o.  .-  )  C_  E )  ->  ( E  |`  dom  ( N  o.  .-  ) )  =  ( N  o.  .-  ) )
3927, 37, 38syl2anc 661 . . . . . . 7  |-  ( ( ( G  e.  Grp  /\  G  e.  MetSp )  /\  ( N  o.  .-  )  C_  D )  ->  ( E  |`  dom  ( N  o.  .-  ) )  =  ( N  o.  .-  ) )
40 ffn 5554 . . . . . . . 8  |-  ( E : ( X  X.  X ) --> RR  ->  E  Fn  ( X  X.  X ) )
41 fnresdm 5515 . . . . . . . 8  |-  ( E  Fn  ( X  X.  X )  ->  ( E  |`  ( X  X.  X ) )  =  E )
4225, 40, 413syl 20 . . . . . . 7  |-  ( ( ( G  e.  Grp  /\  G  e.  MetSp )  /\  ( N  o.  .-  )  C_  D )  ->  ( E  |`  ( X  X.  X ) )  =  E )
4323, 39, 423eqtr3d 2478 . . . . . 6  |-  ( ( ( G  e.  Grp  /\  G  e.  MetSp )  /\  ( N  o.  .-  )  C_  D )  ->  ( N  o.  .-  )  =  E )
4443ex 434 . . . . 5  |-  ( ( G  e.  Grp  /\  G  e.  MetSp )  -> 
( ( N  o.  .-  )  C_  D  ->  ( N  o.  .-  )  =  E ) )
459, 44impbid2 204 . . . 4  |-  ( ( G  e.  Grp  /\  G  e.  MetSp )  -> 
( ( N  o.  .-  )  =  E  <->  ( N  o.  .-  )  C_  D
) )
4645pm5.32i 637 . . 3  |-  ( ( ( G  e.  Grp  /\  G  e.  MetSp )  /\  ( N  o.  .-  )  =  E )  <->  ( ( G  e.  Grp  /\  G  e.  MetSp )  /\  ( N  o.  .-  )  C_  D ) )
47 df-3an 967 . . 3  |-  ( ( G  e.  Grp  /\  G  e.  MetSp  /\  ( N  o.  .-  )  =  E )  <->  ( ( G  e.  Grp  /\  G  e.  MetSp )  /\  ( N  o.  .-  )  =  E ) )
48 df-3an 967 . . 3  |-  ( ( G  e.  Grp  /\  G  e.  MetSp  /\  ( N  o.  .-  )  C_  D )  <->  ( ( G  e.  Grp  /\  G  e.  MetSp )  /\  ( N  o.  .-  )  C_  D ) )
4946, 47, 483bitr4i 277 . 2  |-  ( ( G  e.  Grp  /\  G  e.  MetSp  /\  ( N  o.  .-  )  =  E )  <->  ( G  e.  Grp  /\  G  e. 
MetSp  /\  ( N  o.  .-  )  C_  D )
)
504, 49bitr4i 252 1  |-  ( G  e. NrmGrp 
<->  ( G  e.  Grp  /\  G  e.  MetSp  /\  ( N  o.  .-  )  =  E ) )
Colors of variables: wff setvar class
Syntax hints:    <-> wb 184    /\ wa 369    /\ w3a 965    = wceq 1369    e. wcel 1756   _Vcvv 2967    i^i cin 3322    C_ wss 3323    X. cxp 4833   dom cdm 4835    |` cres 4837    o. ccom 4839   Fun wfun 5407    Fn wfn 5408   -->wf 5409   ` cfv 5413   RRcr 9273   Basecbs 14166   distcds 14239   Grpcgrp 15402   -gcsg 15405   Metcme 17782   MetSpcmt 19873   normcnm 20149  NrmGrpcngp 20150
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 2419  ax-rep 4398  ax-sep 4408  ax-nul 4416  ax-pow 4465  ax-pr 4526  ax-un 6367  ax-cnex 9330  ax-resscn 9331  ax-1cn 9332  ax-icn 9333  ax-addcl 9334  ax-addrcl 9335  ax-mulcl 9336  ax-mulrcl 9337  ax-mulcom 9338  ax-addass 9339  ax-mulass 9340  ax-distr 9341  ax-i2m1 9342  ax-1ne0 9343  ax-1rid 9344  ax-rnegex 9345  ax-rrecex 9346  ax-cnre 9347  ax-pre-lttri 9348  ax-pre-lttrn 9349  ax-pre-ltadd 9350  ax-pre-mulgt0 9351  ax-pre-sup 9352
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 2256  df-mo 2257  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-nel 2604  df-ral 2715  df-rex 2716  df-reu 2717  df-rmo 2718  df-rab 2719  df-v 2969  df-sbc 3182  df-csb 3284  df-dif 3326  df-un 3328  df-in 3330  df-ss 3337  df-pss 3339  df-nul 3633  df-if 3787  df-pw 3857  df-sn 3873  df-pr 3875  df-tp 3877  df-op 3879  df-uni 4087  df-iun 4168  df-br 4288  df-opab 4346  df-mpt 4347  df-tr 4381  df-eprel 4627  df-id 4631  df-po 4636  df-so 4637  df-fr 4674  df-we 4676  df-ord 4717  df-on 4718  df-lim 4719  df-suc 4720  df-xp 4841  df-rel 4842  df-cnv 4843  df-co 4844  df-dm 4845  df-rn 4846  df-res 4847  df-ima 4848  df-iota 5376  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-riota 6047  df-ov 6089  df-oprab 6090  df-mpt2 6091  df-om 6472  df-1st 6572  df-2nd 6573  df-recs 6824  df-rdg 6858  df-er 7093  df-map 7208  df-en 7303  df-dom 7304  df-sdom 7305  df-sup 7683  df-pnf 9412  df-mnf 9413  df-xr 9414  df-ltxr 9415  df-le 9416  df-sub 9589  df-neg 9590  df-div 9986  df-nn 10315  df-2 10372  df-n0 10572  df-z 10639  df-uz 10854  df-q 10946  df-rp 10984  df-xneg 11081  df-xadd 11082  df-xmul 11083  df-0g 14372  df-topgen 14374  df-mnd 15407  df-grp 15536  df-minusg 15537  df-sbg 15538  df-psmet 17789  df-xmet 17790  df-met 17791  df-bl 17792  df-mopn 17793  df-top 18483  df-bases 18485  df-topon 18486  df-topsp 18487  df-xms 19875  df-ms 19876  df-nm 20155  df-ngp 20156
This theorem is referenced by:  isngp3  20170  ngpds  20175  ngppropd  20203
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