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Theorem isngp2 20305
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 20304 . 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 5232 . . . . . . 7  |-  ( D  |`  ( X  X.  X
) )  C_  D
75, 6eqsstri 3484 . . . . . 6  |-  E  C_  D
8 sseq1 3475 . . . . . 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 5204 . . . . . . . . . . . . . 14  |-  ( D  |`  ( X  X.  X
) )  =  ( ( dist `  G
)  |`  ( X  X.  X ) )
125, 11eqtri 2480 . . . . . . . . . . . . 13  |-  E  =  ( ( dist `  G
)  |`  ( X  X.  X ) )
1310, 12msmet 20148 . . . . . . . . . . . 12  |-  ( G  e.  MetSp  ->  E  e.  ( Met `  X ) )
141, 10, 3, 5nmf2 20301 . . . . . . . . . . . 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 15707 . . . . . . . . . . 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 5666 . . . . . . . . . 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 5661 . . . . . . . . 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 5208 . . . . . . 7  |-  ( ( ( G  e.  Grp  /\  G  e.  MetSp )  /\  ( N  o.  .-  )  C_  D )  ->  ( E  |`  dom  ( N  o.  .-  ) )  =  ( E  |`  ( X  X.  X
) ) )
2410, 12msf 20149 . . . . . . . . . 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 5659 . . . . . . . . 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 3474 . . . . . . . . . . . 12  |-  RR  C_  _V
30 fss 5665 . . . . . . . . . . . 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 5668 . . . . . . . . . . 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 3672 . . . . . . . . 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 4950 . . . . . . . . . 10  |-  ( D  |`  ( X  X.  X
) )  =  ( D  i^i  ( ( X  X.  X )  X.  _V ) )
365, 35eqtri 2480 . . . . . . . . 9  |-  E  =  ( D  i^i  (
( X  X.  X
)  X.  _V )
)
3734, 36syl6sseqr 3501 . . . . . . . 8  |-  ( ( ( G  e.  Grp  /\  G  e.  MetSp )  /\  ( N  o.  .-  )  C_  D )  ->  ( N  o.  .-  )  C_  E )
38 funssres 5556 . . . . . . . 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 5657 . . . . . . . 8  |-  ( E : ( X  X.  X ) --> RR  ->  E  Fn  ( X  X.  X ) )
41 fnresdm 5618 . . . . . . . 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 2500 . . . . . 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 1370    e. wcel 1758   _Vcvv 3068    i^i cin 3425    C_ wss 3426    X. cxp 4936   dom cdm 4938    |` cres 4940    o. ccom 4942   Fun wfun 5510    Fn wfn 5511   -->wf 5512   ` cfv 5516   RRcr 9382   Basecbs 14276   distcds 14349   Grpcgrp 15512   -gcsg 15515   Metcme 17911   MetSpcmt 20009   normcnm 20285  NrmGrpcngp 20286
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-rep 4501  ax-sep 4511  ax-nul 4519  ax-pow 4568  ax-pr 4629  ax-un 6472  ax-cnex 9439  ax-resscn 9440  ax-1cn 9441  ax-icn 9442  ax-addcl 9443  ax-addrcl 9444  ax-mulcl 9445  ax-mulrcl 9446  ax-mulcom 9447  ax-addass 9448  ax-mulass 9449  ax-distr 9450  ax-i2m1 9451  ax-1ne0 9452  ax-1rid 9453  ax-rnegex 9454  ax-rrecex 9455  ax-cnre 9456  ax-pre-lttri 9457  ax-pre-lttrn 9458  ax-pre-ltadd 9459  ax-pre-mulgt0 9460  ax-pre-sup 9461
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ne 2646  df-nel 2647  df-ral 2800  df-rex 2801  df-reu 2802  df-rmo 2803  df-rab 2804  df-v 3070  df-sbc 3285  df-csb 3387  df-dif 3429  df-un 3431  df-in 3433  df-ss 3440  df-pss 3442  df-nul 3736  df-if 3890  df-pw 3960  df-sn 3976  df-pr 3978  df-tp 3980  df-op 3982  df-uni 4190  df-iun 4271  df-br 4391  df-opab 4449  df-mpt 4450  df-tr 4484  df-eprel 4730  df-id 4734  df-po 4739  df-so 4740  df-fr 4777  df-we 4779  df-ord 4820  df-on 4821  df-lim 4822  df-suc 4823  df-xp 4944  df-rel 4945  df-cnv 4946  df-co 4947  df-dm 4948  df-rn 4949  df-res 4950  df-ima 4951  df-iota 5479  df-fun 5518  df-fn 5519  df-f 5520  df-f1 5521  df-fo 5522  df-f1o 5523  df-fv 5524  df-riota 6151  df-ov 6193  df-oprab 6194  df-mpt2 6195  df-om 6577  df-1st 6677  df-2nd 6678  df-recs 6932  df-rdg 6966  df-er 7201  df-map 7316  df-en 7411  df-dom 7412  df-sdom 7413  df-sup 7792  df-pnf 9521  df-mnf 9522  df-xr 9523  df-ltxr 9524  df-le 9525  df-sub 9698  df-neg 9699  df-div 10095  df-nn 10424  df-2 10481  df-n0 10681  df-z 10748  df-uz 10963  df-q 11055  df-rp 11093  df-xneg 11190  df-xadd 11191  df-xmul 11192  df-0g 14482  df-topgen 14484  df-mnd 15517  df-grp 15647  df-minusg 15648  df-sbg 15649  df-psmet 17918  df-xmet 17919  df-met 17920  df-bl 17921  df-mopn 17922  df-top 18619  df-bases 18621  df-topon 18622  df-topsp 18623  df-xms 20011  df-ms 20012  df-nm 20291  df-ngp 20292
This theorem is referenced by:  isngp3  20306  ngpds  20311  ngppropd  20339
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