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Theorem isngp2 18517
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 18516 . 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 5112 . . . . . . 7  |-  ( D  |`  ( X  X.  X
) )  C_  D
75, 6eqsstri 3323 . . . . . 6  |-  E  C_  D
8 sseq1 3314 . . . . . 6  |-  ( ( N  o.  .-  )  =  E  ->  ( ( N  o.  .-  )  C_  D  <->  E  C_  D ) )
97, 8mpbiri 225 . . . . 5  |-  ( ( N  o.  .-  )  =  E  ->  ( N  o.  .-  )  C_  D )
10 isngp2.x . . . . . . . . . . . . 13  |-  X  =  ( Base `  G
)
113reseq1i 5084 . . . . . . . . . . . . . 14  |-  ( D  |`  ( X  X.  X
) )  =  ( ( dist `  G
)  |`  ( X  X.  X ) )
125, 11eqtri 2409 . . . . . . . . . . . . 13  |-  E  =  ( ( dist `  G
)  |`  ( X  X.  X ) )
1310, 12msmet 18379 . . . . . . . . . . . 12  |-  ( G  e.  MetSp  ->  E  e.  ( Met `  X ) )
141, 10, 3, 5nmf2 18513 . . . . . . . . . . . 12  |-  ( ( G  e.  Grp  /\  E  e.  ( Met `  X ) )  ->  N : X --> RR )
1513, 14sylan2 461 . . . . . . . . . . 11  |-  ( ( G  e.  Grp  /\  G  e.  MetSp )  ->  N : X --> RR )
1615adantr 452 . . . . . . . . . 10  |-  ( ( ( G  e.  Grp  /\  G  e.  MetSp )  /\  ( N  o.  .-  )  C_  D )  ->  N : X --> RR )
1710, 2grpsubf 14797 . . . . . . . . . . 11  |-  ( G  e.  Grp  ->  .-  :
( X  X.  X
) --> X )
1817ad2antrr 707 . . . . . . . . . 10  |-  ( ( ( G  e.  Grp  /\  G  e.  MetSp )  /\  ( N  o.  .-  )  C_  D )  ->  .-  :
( X  X.  X
) --> X )
19 fco 5542 . . . . . . . . . 10  |-  ( ( N : X --> RR  /\  .-  : ( X  X.  X ) --> X )  ->  ( N  o.  .-  ) : ( X  X.  X ) --> RR )
2016, 18, 19syl2anc 643 . . . . . . . . 9  |-  ( ( ( G  e.  Grp  /\  G  e.  MetSp )  /\  ( N  o.  .-  )  C_  D )  ->  ( N  o.  .-  ) : ( X  X.  X
) --> RR )
21 fdm 5537 . . . . . . . . 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 5088 . . . . . . 7  |-  ( ( ( G  e.  Grp  /\  G  e.  MetSp )  /\  ( N  o.  .-  )  C_  D )  ->  ( E  |`  dom  ( N  o.  .-  ) )  =  ( E  |`  ( X  X.  X
) ) )
2410, 12msf 18380 . . . . . . . . . 10  |-  ( G  e.  MetSp  ->  E :
( X  X.  X
) --> RR )
2524ad2antlr 708 . . . . . . . . 9  |-  ( ( ( G  e.  Grp  /\  G  e.  MetSp )  /\  ( N  o.  .-  )  C_  D )  ->  E : ( X  X.  X ) --> RR )
26 ffun 5535 . . . . . . . . 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 448 . . . . . . . . . 10  |-  ( ( ( G  e.  Grp  /\  G  e.  MetSp )  /\  ( N  o.  .-  )  C_  D )  ->  ( N  o.  .-  )  C_  D )
29 ssv 3313 . . . . . . . . . . . 12  |-  RR  C_  _V
30 fss 5541 . . . . . . . . . . . 12  |-  ( ( ( N  o.  .-  ) : ( X  X.  X ) --> RR  /\  RR  C_  _V )  -> 
( N  o.  .-  ) : ( X  X.  X ) --> _V )
3120, 29, 30sylancl 644 . . . . . . . . . . 11  |-  ( ( ( G  e.  Grp  /\  G  e.  MetSp )  /\  ( N  o.  .-  )  C_  D )  ->  ( N  o.  .-  ) : ( X  X.  X
) --> _V )
32 fssxp 5544 . . . . . . . . . . 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 3510 . . . . . . . . 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 4832 . . . . . . . . . 10  |-  ( D  |`  ( X  X.  X
) )  =  ( D  i^i  ( ( X  X.  X )  X.  _V ) )
365, 35eqtri 2409 . . . . . . . . 9  |-  E  =  ( D  i^i  (
( X  X.  X
)  X.  _V )
)
3734, 36syl6sseqr 3340 . . . . . . . 8  |-  ( ( ( G  e.  Grp  /\  G  e.  MetSp )  /\  ( N  o.  .-  )  C_  D )  ->  ( N  o.  .-  )  C_  E )
38 funssres 5435 . . . . . . . 8  |-  ( ( Fun  E  /\  ( N  o.  .-  )  C_  E )  ->  ( E  |`  dom  ( N  o.  .-  ) )  =  ( N  o.  .-  ) )
3927, 37, 38syl2anc 643 . . . . . . 7  |-  ( ( ( G  e.  Grp  /\  G  e.  MetSp )  /\  ( N  o.  .-  )  C_  D )  ->  ( E  |`  dom  ( N  o.  .-  ) )  =  ( N  o.  .-  ) )
40 ffn 5533 . . . . . . . 8  |-  ( E : ( X  X.  X ) --> RR  ->  E  Fn  ( X  X.  X ) )
41 fnresdm 5496 . . . . . . . 8  |-  ( E  Fn  ( X  X.  X )  ->  ( E  |`  ( X  X.  X ) )  =  E )
4225, 40, 413syl 19 . . . . . . 7  |-  ( ( ( G  e.  Grp  /\  G  e.  MetSp )  /\  ( N  o.  .-  )  C_  D )  ->  ( E  |`  ( X  X.  X ) )  =  E )
4323, 39, 423eqtr3d 2429 . . . . . 6  |-  ( ( ( G  e.  Grp  /\  G  e.  MetSp )  /\  ( N  o.  .-  )  C_  D )  ->  ( N  o.  .-  )  =  E )
4443ex 424 . . . . 5  |-  ( ( G  e.  Grp  /\  G  e.  MetSp )  -> 
( ( N  o.  .-  )  C_  D  ->  ( N  o.  .-  )  =  E ) )
459, 44impbid2 196 . . . 4  |-  ( ( G  e.  Grp  /\  G  e.  MetSp )  -> 
( ( N  o.  .-  )  =  E  <->  ( N  o.  .-  )  C_  D
) )
4645pm5.32i 619 . . 3  |-  ( ( ( G  e.  Grp  /\  G  e.  MetSp )  /\  ( N  o.  .-  )  =  E )  <->  ( ( G  e.  Grp  /\  G  e.  MetSp )  /\  ( N  o.  .-  )  C_  D ) )
47 df-3an 938 . . 3  |-  ( ( G  e.  Grp  /\  G  e.  MetSp  /\  ( N  o.  .-  )  =  E )  <->  ( ( G  e.  Grp  /\  G  e.  MetSp )  /\  ( N  o.  .-  )  =  E ) )
48 df-3an 938 . . 3  |-  ( ( G  e.  Grp  /\  G  e.  MetSp  /\  ( N  o.  .-  )  C_  D )  <->  ( ( G  e.  Grp  /\  G  e.  MetSp )  /\  ( N  o.  .-  )  C_  D ) )
4946, 47, 483bitr4i 269 . 2  |-  ( ( G  e.  Grp  /\  G  e.  MetSp  /\  ( N  o.  .-  )  =  E )  <->  ( G  e.  Grp  /\  G  e. 
MetSp  /\  ( N  o.  .-  )  C_  D )
)
504, 49bitr4i 244 1  |-  ( G  e. NrmGrp 
<->  ( G  e.  Grp  /\  G  e.  MetSp  /\  ( N  o.  .-  )  =  E ) )
Colors of variables: wff set class
Syntax hints:    <-> wb 177    /\ wa 359    /\ w3a 936    = wceq 1649    e. wcel 1717   _Vcvv 2901    i^i cin 3264    C_ wss 3265    X. cxp 4818   dom cdm 4820    |` cres 4822    o. ccom 4824   Fun wfun 5390    Fn wfn 5391   -->wf 5392   ` cfv 5396   RRcr 8924   Basecbs 13398   distcds 13467   Grpcgrp 14614   -gcsg 14617   Metcme 16615   MetSpcmt 18259   normcnm 18497  NrmGrpcngp 18498
This theorem is referenced by:  isngp3  18518  ngpds  18523  ngppropd  18551
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2370  ax-rep 4263  ax-sep 4273  ax-nul 4281  ax-pow 4320  ax-pr 4346  ax-un 4643  ax-cnex 8981  ax-resscn 8982  ax-1cn 8983  ax-icn 8984  ax-addcl 8985  ax-addrcl 8986  ax-mulcl 8987  ax-mulrcl 8988  ax-mulcom 8989  ax-addass 8990  ax-mulass 8991  ax-distr 8992  ax-i2m1 8993  ax-1ne0 8994  ax-1rid 8995  ax-rnegex 8996  ax-rrecex 8997  ax-cnre 8998  ax-pre-lttri 8999  ax-pre-lttrn 9000  ax-pre-ltadd 9001  ax-pre-mulgt0 9002  ax-pre-sup 9003
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2244  df-mo 2245  df-clab 2376  df-cleq 2382  df-clel 2385  df-nfc 2514  df-ne 2554  df-nel 2555  df-ral 2656  df-rex 2657  df-reu 2658  df-rmo 2659  df-rab 2660  df-v 2903  df-sbc 3107  df-csb 3197  df-dif 3268  df-un 3270  df-in 3272  df-ss 3279  df-pss 3281  df-nul 3574  df-if 3685  df-pw 3746  df-sn 3765  df-pr 3766  df-tp 3767  df-op 3768  df-uni 3960  df-iun 4039  df-br 4156  df-opab 4210  df-mpt 4211  df-tr 4246  df-eprel 4437  df-id 4441  df-po 4446  df-so 4447  df-fr 4484  df-we 4486  df-ord 4527  df-on 4528  df-lim 4529  df-suc 4530  df-om 4788  df-xp 4826  df-rel 4827  df-cnv 4828  df-co 4829  df-dm 4830  df-rn 4831  df-res 4832  df-ima 4833  df-iota 5360  df-fun 5398  df-fn 5399  df-f 5400  df-f1 5401  df-fo 5402  df-f1o 5403  df-fv 5404  df-ov 6025  df-oprab 6026  df-mpt2 6027  df-1st 6290  df-2nd 6291  df-riota 6487  df-recs 6571  df-rdg 6606  df-er 6843  df-map 6958  df-en 7048  df-dom 7049  df-sdom 7050  df-sup 7383  df-pnf 9057  df-mnf 9058  df-xr 9059  df-ltxr 9060  df-le 9061  df-sub 9227  df-neg 9228  df-div 9612  df-nn 9935  df-2 9992  df-n0 10156  df-z 10217  df-uz 10423  df-q 10509  df-rp 10547  df-xneg 10644  df-xadd 10645  df-xmul 10646  df-topgen 13596  df-0g 13656  df-mnd 14619  df-grp 14741  df-minusg 14742  df-sbg 14743  df-xmet 16621  df-met 16622  df-bl 16623  df-mopn 16624  df-top 16888  df-bases 16890  df-topon 16891  df-topsp 16892  df-xms 18261  df-ms 18262  df-nm 18503  df-ngp 18504
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