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Theorem isngp2 18119
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 18118 . 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 4979 . . . . . . 7  |-  ( D  |`  ( X  X.  X
) )  C_  D
75, 6eqsstri 3208 . . . . . 6  |-  E  C_  D
8 sseq1 3199 . . . . . 6  |-  ( ( N  o.  .-  )  =  E  ->  ( ( N  o.  .-  )  C_  D  <->  E  C_  D ) )
97, 8mpbiri 224 . . . . 5  |-  ( ( N  o.  .-  )  =  E  ->  ( N  o.  .-  )  C_  D )
10 isngp2.x . . . . . . . . . . . . 13  |-  X  =  ( Base `  G
)
113reseq1i 4951 . . . . . . . . . . . . . 14  |-  ( D  |`  ( X  X.  X
) )  =  ( ( dist `  G
)  |`  ( X  X.  X ) )
125, 11eqtri 2303 . . . . . . . . . . . . 13  |-  E  =  ( ( dist `  G
)  |`  ( X  X.  X ) )
1310, 12msmet 18003 . . . . . . . . . . . 12  |-  ( G  e.  MetSp  ->  E  e.  ( Met `  X ) )
141, 10, 3, 5nmf2 18115 . . . . . . . . . . . 12  |-  ( ( G  e.  Grp  /\  E  e.  ( Met `  X ) )  ->  N : X --> RR )
1513, 14sylan2 460 . . . . . . . . . . 11  |-  ( ( G  e.  Grp  /\  G  e.  MetSp )  ->  N : X --> RR )
1615adantr 451 . . . . . . . . . 10  |-  ( ( ( G  e.  Grp  /\  G  e.  MetSp )  /\  ( N  o.  .-  )  C_  D )  ->  N : X --> RR )
1710, 2grpsubf 14545 . . . . . . . . . . 11  |-  ( G  e.  Grp  ->  .-  :
( X  X.  X
) --> X )
1817ad2antrr 706 . . . . . . . . . 10  |-  ( ( ( G  e.  Grp  /\  G  e.  MetSp )  /\  ( N  o.  .-  )  C_  D )  ->  .-  :
( X  X.  X
) --> X )
19 fco 5398 . . . . . . . . . 10  |-  ( ( N : X --> RR  /\  .-  : ( X  X.  X ) --> X )  ->  ( N  o.  .-  ) : ( X  X.  X ) --> RR )
2016, 18, 19syl2anc 642 . . . . . . . . 9  |-  ( ( ( G  e.  Grp  /\  G  e.  MetSp )  /\  ( N  o.  .-  )  C_  D )  ->  ( N  o.  .-  ) : ( X  X.  X
) --> RR )
21 fdm 5393 . . . . . . . . 9  |-  ( ( N  o.  .-  ) : ( X  X.  X ) --> RR  ->  dom  ( N  o.  .-  )  =  ( X  X.  X ) )
2220, 21syl 15 . . . . . . . 8  |-  ( ( ( G  e.  Grp  /\  G  e.  MetSp )  /\  ( N  o.  .-  )  C_  D )  ->  dom  ( N  o.  .-  )  =  ( X  X.  X ) )
2322reseq2d 4955 . . . . . . 7  |-  ( ( ( G  e.  Grp  /\  G  e.  MetSp )  /\  ( N  o.  .-  )  C_  D )  ->  ( E  |`  dom  ( N  o.  .-  ) )  =  ( E  |`  ( X  X.  X
) ) )
2410, 12msf 18004 . . . . . . . . . 10  |-  ( G  e.  MetSp  ->  E :
( X  X.  X
) --> RR )
2524ad2antlr 707 . . . . . . . . 9  |-  ( ( ( G  e.  Grp  /\  G  e.  MetSp )  /\  ( N  o.  .-  )  C_  D )  ->  E : ( X  X.  X ) --> RR )
26 ffun 5391 . . . . . . . . 9  |-  ( E : ( X  X.  X ) --> RR  ->  Fun 
E )
2725, 26syl 15 . . . . . . . 8  |-  ( ( ( G  e.  Grp  /\  G  e.  MetSp )  /\  ( N  o.  .-  )  C_  D )  ->  Fun  E )
28 simpr 447 . . . . . . . . . 10  |-  ( ( ( G  e.  Grp  /\  G  e.  MetSp )  /\  ( N  o.  .-  )  C_  D )  ->  ( N  o.  .-  )  C_  D )
29 ssv 3198 . . . . . . . . . . . 12  |-  RR  C_  _V
30 fss 5397 . . . . . . . . . . . 12  |-  ( ( ( N  o.  .-  ) : ( X  X.  X ) --> RR  /\  RR  C_  _V )  -> 
( N  o.  .-  ) : ( X  X.  X ) --> _V )
3120, 29, 30sylancl 643 . . . . . . . . . . 11  |-  ( ( ( G  e.  Grp  /\  G  e.  MetSp )  /\  ( N  o.  .-  )  C_  D )  ->  ( N  o.  .-  ) : ( X  X.  X
) --> _V )
32 fssxp 5400 . . . . . . . . . . 11  |-  ( ( N  o.  .-  ) : ( X  X.  X ) --> _V  ->  ( N  o.  .-  )  C_  ( ( X  X.  X )  X.  _V ) )
3331, 32syl 15 . . . . . . . . . 10  |-  ( ( ( G  e.  Grp  /\  G  e.  MetSp )  /\  ( N  o.  .-  )  C_  D )  ->  ( N  o.  .-  )  C_  ( ( X  X.  X )  X.  _V ) )
3428, 33ssind 3393 . . . . . . . . 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 4701 . . . . . . . . . 10  |-  ( D  |`  ( X  X.  X
) )  =  ( D  i^i  ( ( X  X.  X )  X.  _V ) )
365, 35eqtri 2303 . . . . . . . . 9  |-  E  =  ( D  i^i  (
( X  X.  X
)  X.  _V )
)
3734, 36syl6sseqr 3225 . . . . . . . 8  |-  ( ( ( G  e.  Grp  /\  G  e.  MetSp )  /\  ( N  o.  .-  )  C_  D )  ->  ( N  o.  .-  )  C_  E )
38 funssres 5294 . . . . . . . 8  |-  ( ( Fun  E  /\  ( N  o.  .-  )  C_  E )  ->  ( E  |`  dom  ( N  o.  .-  ) )  =  ( N  o.  .-  ) )
3927, 37, 38syl2anc 642 . . . . . . 7  |-  ( ( ( G  e.  Grp  /\  G  e.  MetSp )  /\  ( N  o.  .-  )  C_  D )  ->  ( E  |`  dom  ( N  o.  .-  ) )  =  ( N  o.  .-  ) )
40 ffn 5389 . . . . . . . 8  |-  ( E : ( X  X.  X ) --> RR  ->  E  Fn  ( X  X.  X ) )
41 fnresdm 5353 . . . . . . . 8  |-  ( E  Fn  ( X  X.  X )  ->  ( E  |`  ( X  X.  X ) )  =  E )
4225, 40, 413syl 18 . . . . . . 7  |-  ( ( ( G  e.  Grp  /\  G  e.  MetSp )  /\  ( N  o.  .-  )  C_  D )  ->  ( E  |`  ( X  X.  X ) )  =  E )
4323, 39, 423eqtr3d 2323 . . . . . 6  |-  ( ( ( G  e.  Grp  /\  G  e.  MetSp )  /\  ( N  o.  .-  )  C_  D )  ->  ( N  o.  .-  )  =  E )
4443ex 423 . . . . 5  |-  ( ( G  e.  Grp  /\  G  e.  MetSp )  -> 
( ( N  o.  .-  )  C_  D  ->  ( N  o.  .-  )  =  E ) )
459, 44impbid2 195 . . . 4  |-  ( ( G  e.  Grp  /\  G  e.  MetSp )  -> 
( ( N  o.  .-  )  =  E  <->  ( N  o.  .-  )  C_  D
) )
4645pm5.32i 618 . . 3  |-  ( ( ( G  e.  Grp  /\  G  e.  MetSp )  /\  ( N  o.  .-  )  =  E )  <->  ( ( G  e.  Grp  /\  G  e.  MetSp )  /\  ( N  o.  .-  )  C_  D ) )
47 df-3an 936 . . 3  |-  ( ( G  e.  Grp  /\  G  e.  MetSp  /\  ( N  o.  .-  )  =  E )  <->  ( ( G  e.  Grp  /\  G  e.  MetSp )  /\  ( N  o.  .-  )  =  E ) )
48 df-3an 936 . . 3  |-  ( ( G  e.  Grp  /\  G  e.  MetSp  /\  ( N  o.  .-  )  C_  D )  <->  ( ( G  e.  Grp  /\  G  e.  MetSp )  /\  ( N  o.  .-  )  C_  D ) )
4946, 47, 483bitr4i 268 . 2  |-  ( ( G  e.  Grp  /\  G  e.  MetSp  /\  ( N  o.  .-  )  =  E )  <->  ( G  e.  Grp  /\  G  e. 
MetSp  /\  ( N  o.  .-  )  C_  D )
)
504, 49bitr4i 243 1  |-  ( G  e. NrmGrp 
<->  ( G  e.  Grp  /\  G  e.  MetSp  /\  ( N  o.  .-  )  =  E ) )
Colors of variables: wff set class
Syntax hints:    <-> wb 176    /\ wa 358    /\ w3a 934    = wceq 1623    e. wcel 1684   _Vcvv 2788    i^i cin 3151    C_ wss 3152    X. cxp 4687   dom cdm 4689    |` cres 4691    o. ccom 4693   Fun wfun 5249    Fn wfn 5250   -->wf 5251   ` cfv 5255   RRcr 8736   Basecbs 13148   distcds 13217   Grpcgrp 14362   -gcsg 14365   Metcme 16370   MetSpcmt 17883   normcnm 18099  NrmGrpcngp 18100
This theorem is referenced by:  isngp3  18120  ngpds  18125  ngppropd  18153
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-cnex 8793  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813  ax-pre-mulgt0 8814  ax-pre-sup 8815
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-riota 6304  df-recs 6388  df-rdg 6423  df-er 6660  df-map 6774  df-en 6864  df-dom 6865  df-sdom 6866  df-sup 7194  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-sub 9039  df-neg 9040  df-div 9424  df-nn 9747  df-2 9804  df-n0 9966  df-z 10025  df-uz 10231  df-q 10317  df-rp 10355  df-xneg 10452  df-xadd 10453  df-xmul 10454  df-topgen 13344  df-0g 13404  df-mnd 14367  df-grp 14489  df-minusg 14490  df-sbg 14491  df-xmet 16373  df-met 16374  df-bl 16375  df-mopn 16376  df-top 16636  df-bases 16638  df-topon 16639  df-topsp 16640  df-xms 17885  df-ms 17886  df-nm 18105  df-ngp 18106
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