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Theorem isngp2 21202
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 21201 . 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 5209 . . . . . . 7  |-  ( D  |`  ( X  X.  X
) )  C_  D
75, 6eqsstri 3447 . . . . . 6  |-  E  C_  D
8 sseq1 3438 . . . . . 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 5182 . . . . . . . . . . . . . 14  |-  ( D  |`  ( X  X.  X
) )  =  ( ( dist `  G
)  |`  ( X  X.  X ) )
125, 11eqtri 2411 . . . . . . . . . . . . 13  |-  E  =  ( ( dist `  G
)  |`  ( X  X.  X ) )
1310, 12msmet 21045 . . . . . . . . . . . 12  |-  ( G  e.  MetSp  ->  E  e.  ( Met `  X ) )
141, 10, 3, 5nmf2 21198 . . . . . . . . . . . 12  |-  ( ( G  e.  Grp  /\  E  e.  ( Met `  X ) )  ->  N : X --> RR )
1513, 14sylan2 472 . . . . . . . . . . 11  |-  ( ( G  e.  Grp  /\  G  e.  MetSp )  ->  N : X --> RR )
1615adantr 463 . . . . . . . . . 10  |-  ( ( ( G  e.  Grp  /\  G  e.  MetSp )  /\  ( N  o.  .-  )  C_  D )  ->  N : X --> RR )
1710, 2grpsubf 16234 . . . . . . . . . . 11  |-  ( G  e.  Grp  ->  .-  :
( X  X.  X
) --> X )
1817ad2antrr 723 . . . . . . . . . 10  |-  ( ( ( G  e.  Grp  /\  G  e.  MetSp )  /\  ( N  o.  .-  )  C_  D )  ->  .-  :
( X  X.  X
) --> X )
19 fco 5649 . . . . . . . . . 10  |-  ( ( N : X --> RR  /\  .-  : ( X  X.  X ) --> X )  ->  ( N  o.  .-  ) : ( X  X.  X ) --> RR )
2016, 18, 19syl2anc 659 . . . . . . . . 9  |-  ( ( ( G  e.  Grp  /\  G  e.  MetSp )  /\  ( N  o.  .-  )  C_  D )  ->  ( N  o.  .-  ) : ( X  X.  X
) --> RR )
21 fdm 5643 . . . . . . . . 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 5186 . . . . . . 7  |-  ( ( ( G  e.  Grp  /\  G  e.  MetSp )  /\  ( N  o.  .-  )  C_  D )  ->  ( E  |`  dom  ( N  o.  .-  ) )  =  ( E  |`  ( X  X.  X
) ) )
2410, 12msf 21046 . . . . . . . . . 10  |-  ( G  e.  MetSp  ->  E :
( X  X.  X
) --> RR )
2524ad2antlr 724 . . . . . . . . 9  |-  ( ( ( G  e.  Grp  /\  G  e.  MetSp )  /\  ( N  o.  .-  )  C_  D )  ->  E : ( X  X.  X ) --> RR )
26 ffun 5641 . . . . . . . . 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 459 . . . . . . . . . 10  |-  ( ( ( G  e.  Grp  /\  G  e.  MetSp )  /\  ( N  o.  .-  )  C_  D )  ->  ( N  o.  .-  )  C_  D )
29 ssv 3437 . . . . . . . . . . . 12  |-  RR  C_  _V
30 fss 5647 . . . . . . . . . . . 12  |-  ( ( ( N  o.  .-  ) : ( X  X.  X ) --> RR  /\  RR  C_  _V )  -> 
( N  o.  .-  ) : ( X  X.  X ) --> _V )
3120, 29, 30sylancl 660 . . . . . . . . . . 11  |-  ( ( ( G  e.  Grp  /\  G  e.  MetSp )  /\  ( N  o.  .-  )  C_  D )  ->  ( N  o.  .-  ) : ( X  X.  X
) --> _V )
32 fssxp 5651 . . . . . . . . . . 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 3636 . . . . . . . . 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 4925 . . . . . . . . . 10  |-  ( D  |`  ( X  X.  X
) )  =  ( D  i^i  ( ( X  X.  X )  X.  _V ) )
365, 35eqtri 2411 . . . . . . . . 9  |-  E  =  ( D  i^i  (
( X  X.  X
)  X.  _V )
)
3734, 36syl6sseqr 3464 . . . . . . . 8  |-  ( ( ( G  e.  Grp  /\  G  e.  MetSp )  /\  ( N  o.  .-  )  C_  D )  ->  ( N  o.  .-  )  C_  E )
38 funssres 5536 . . . . . . . 8  |-  ( ( Fun  E  /\  ( N  o.  .-  )  C_  E )  ->  ( E  |`  dom  ( N  o.  .-  ) )  =  ( N  o.  .-  ) )
3927, 37, 38syl2anc 659 . . . . . . 7  |-  ( ( ( G  e.  Grp  /\  G  e.  MetSp )  /\  ( N  o.  .-  )  C_  D )  ->  ( E  |`  dom  ( N  o.  .-  ) )  =  ( N  o.  .-  ) )
40 ffn 5639 . . . . . . . 8  |-  ( E : ( X  X.  X ) --> RR  ->  E  Fn  ( X  X.  X ) )
41 fnresdm 5598 . . . . . . . 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 2431 . . . . . 6  |-  ( ( ( G  e.  Grp  /\  G  e.  MetSp )  /\  ( N  o.  .-  )  C_  D )  ->  ( N  o.  .-  )  =  E )
4443ex 432 . . . . 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 635 . . 3  |-  ( ( ( G  e.  Grp  /\  G  e.  MetSp )  /\  ( N  o.  .-  )  =  E )  <->  ( ( G  e.  Grp  /\  G  e.  MetSp )  /\  ( N  o.  .-  )  C_  D ) )
47 df-3an 973 . . 3  |-  ( ( G  e.  Grp  /\  G  e.  MetSp  /\  ( N  o.  .-  )  =  E )  <->  ( ( G  e.  Grp  /\  G  e.  MetSp )  /\  ( N  o.  .-  )  =  E ) )
48 df-3an 973 . . 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 367    /\ w3a 971    = wceq 1399    e. wcel 1826   _Vcvv 3034    i^i cin 3388    C_ wss 3389    X. cxp 4911   dom cdm 4913    |` cres 4915    o. ccom 4917   Fun wfun 5490    Fn wfn 5491   -->wf 5492   ` cfv 5496   RRcr 9402   Basecbs 14634   distcds 14711   Grpcgrp 16170   -gcsg 16172   Metcme 18517   MetSpcmt 20906   normcnm 21182  NrmGrpcngp 21183
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1626  ax-4 1639  ax-5 1712  ax-6 1755  ax-7 1798  ax-8 1828  ax-9 1830  ax-10 1845  ax-11 1850  ax-12 1862  ax-13 2006  ax-ext 2360  ax-rep 4478  ax-sep 4488  ax-nul 4496  ax-pow 4543  ax-pr 4601  ax-un 6491  ax-cnex 9459  ax-resscn 9460  ax-1cn 9461  ax-icn 9462  ax-addcl 9463  ax-addrcl 9464  ax-mulcl 9465  ax-mulrcl 9466  ax-mulcom 9467  ax-addass 9468  ax-mulass 9469  ax-distr 9470  ax-i2m1 9471  ax-1ne0 9472  ax-1rid 9473  ax-rnegex 9474  ax-rrecex 9475  ax-cnre 9476  ax-pre-lttri 9477  ax-pre-lttrn 9478  ax-pre-ltadd 9479  ax-pre-mulgt0 9480  ax-pre-sup 9481
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 972  df-3an 973  df-tru 1402  df-ex 1621  df-nf 1625  df-sb 1748  df-eu 2222  df-mo 2223  df-clab 2368  df-cleq 2374  df-clel 2377  df-nfc 2532  df-ne 2579  df-nel 2580  df-ral 2737  df-rex 2738  df-reu 2739  df-rmo 2740  df-rab 2741  df-v 3036  df-sbc 3253  df-csb 3349  df-dif 3392  df-un 3394  df-in 3396  df-ss 3403  df-pss 3405  df-nul 3712  df-if 3858  df-pw 3929  df-sn 3945  df-pr 3947  df-tp 3949  df-op 3951  df-uni 4164  df-iun 4245  df-br 4368  df-opab 4426  df-mpt 4427  df-tr 4461  df-eprel 4705  df-id 4709  df-po 4714  df-so 4715  df-fr 4752  df-we 4754  df-ord 4795  df-on 4796  df-lim 4797  df-suc 4798  df-xp 4919  df-rel 4920  df-cnv 4921  df-co 4922  df-dm 4923  df-rn 4924  df-res 4925  df-ima 4926  df-iota 5460  df-fun 5498  df-fn 5499  df-f 5500  df-f1 5501  df-fo 5502  df-f1o 5503  df-fv 5504  df-riota 6158  df-ov 6199  df-oprab 6200  df-mpt2 6201  df-om 6600  df-1st 6699  df-2nd 6700  df-recs 6960  df-rdg 6994  df-er 7229  df-map 7340  df-en 7436  df-dom 7437  df-sdom 7438  df-sup 7816  df-pnf 9541  df-mnf 9542  df-xr 9543  df-ltxr 9544  df-le 9545  df-sub 9720  df-neg 9721  df-div 10124  df-nn 10453  df-2 10511  df-n0 10713  df-z 10782  df-uz 11002  df-q 11102  df-rp 11140  df-xneg 11239  df-xadd 11240  df-xmul 11241  df-0g 14849  df-topgen 14851  df-mgm 15989  df-sgrp 16028  df-mnd 16038  df-grp 16174  df-minusg 16175  df-sbg 16176  df-psmet 18524  df-xmet 18525  df-met 18526  df-bl 18527  df-mopn 18528  df-top 19484  df-bases 19486  df-topon 19487  df-topsp 19488  df-xms 20908  df-ms 20909  df-nm 21188  df-ngp 21189
This theorem is referenced by:  isngp3  21203  ngpds  21208  ngppropd  21236
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