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Theorem isngp3 18120
Description: The property of being a normed group. (Contributed by Mario Carneiro, 4-Oct-2015.)
Hypotheses
Ref Expression
isngp.n  |-  N  =  ( norm `  G
)
isngp.z  |-  .-  =  ( -g `  G )
isngp.d  |-  D  =  ( dist `  G
)
isngp2.x  |-  X  =  ( Base `  G
)
Assertion
Ref Expression
isngp3  |-  ( G  e. NrmGrp 
<->  ( G  e.  Grp  /\  G  e.  MetSp  /\  A. x  e.  X  A. y  e.  X  (
x D y )  =  ( N `  ( x  .-  y ) ) ) )
Distinct variable groups:    x, y, D    x, G, y    x,  .- , y    x, N, y   
x, X, y

Proof of Theorem isngp3
StepHypRef Expression
1 isngp.n . . 3  |-  N  =  ( norm `  G
)
2 isngp.z . . 3  |-  .-  =  ( -g `  G )
3 isngp.d . . 3  |-  D  =  ( dist `  G
)
4 isngp2.x . . 3  |-  X  =  ( Base `  G
)
5 eqid 2283 . . 3  |-  ( D  |`  ( X  X.  X
) )  =  ( D  |`  ( X  X.  X ) )
61, 2, 3, 4, 5isngp2 18119 . 2  |-  ( G  e. NrmGrp 
<->  ( G  e.  Grp  /\  G  e.  MetSp  /\  ( N  o.  .-  )  =  ( D  |`  ( X  X.  X ) ) ) )
74, 3msmet2 18006 . . . . . . . . 9  |-  ( G  e.  MetSp  ->  ( D  |`  ( X  X.  X
) )  e.  ( Met `  X ) )
81, 4, 3, 5nmf2 18115 . . . . . . . . 9  |-  ( ( G  e.  Grp  /\  ( D  |`  ( X  X.  X ) )  e.  ( Met `  X
) )  ->  N : X --> RR )
97, 8sylan2 460 . . . . . . . 8  |-  ( ( G  e.  Grp  /\  G  e.  MetSp )  ->  N : X --> RR )
104, 2grpsubf 14545 . . . . . . . . 9  |-  ( G  e.  Grp  ->  .-  :
( X  X.  X
) --> X )
1110adantr 451 . . . . . . . 8  |-  ( ( G  e.  Grp  /\  G  e.  MetSp )  ->  .-  : ( X  X.  X ) --> X )
12 fco 5398 . . . . . . . 8  |-  ( ( N : X --> RR  /\  .-  : ( X  X.  X ) --> X )  ->  ( N  o.  .-  ) : ( X  X.  X ) --> RR )
139, 11, 12syl2anc 642 . . . . . . 7  |-  ( ( G  e.  Grp  /\  G  e.  MetSp )  -> 
( N  o.  .-  ) : ( X  X.  X ) --> RR )
14 ffn 5389 . . . . . . 7  |-  ( ( N  o.  .-  ) : ( X  X.  X ) --> RR  ->  ( N  o.  .-  )  Fn  ( X  X.  X
) )
1513, 14syl 15 . . . . . 6  |-  ( ( G  e.  Grp  /\  G  e.  MetSp )  -> 
( N  o.  .-  )  Fn  ( X  X.  X ) )
167adantl 452 . . . . . . 7  |-  ( ( G  e.  Grp  /\  G  e.  MetSp )  -> 
( D  |`  ( X  X.  X ) )  e.  ( Met `  X
) )
17 metf 17895 . . . . . . 7  |-  ( ( D  |`  ( X  X.  X ) )  e.  ( Met `  X
)  ->  ( D  |`  ( X  X.  X
) ) : ( X  X.  X ) --> RR )
18 ffn 5389 . . . . . . 7  |-  ( ( D  |`  ( X  X.  X ) ) : ( X  X.  X
) --> RR  ->  ( D  |`  ( X  X.  X ) )  Fn  ( X  X.  X
) )
1916, 17, 183syl 18 . . . . . 6  |-  ( ( G  e.  Grp  /\  G  e.  MetSp )  -> 
( D  |`  ( X  X.  X ) )  Fn  ( X  X.  X ) )
20 eqfnov2 5951 . . . . . 6  |-  ( ( ( N  o.  .-  )  Fn  ( X  X.  X )  /\  ( D  |`  ( X  X.  X ) )  Fn  ( X  X.  X
) )  ->  (
( N  o.  .-  )  =  ( D  |`  ( X  X.  X
) )  <->  A. x  e.  X  A. y  e.  X  ( x
( N  o.  .-  ) y )  =  ( x ( D  |`  ( X  X.  X
) ) y ) ) )
2115, 19, 20syl2anc 642 . . . . 5  |-  ( ( G  e.  Grp  /\  G  e.  MetSp )  -> 
( ( N  o.  .-  )  =  ( D  |`  ( X  X.  X
) )  <->  A. x  e.  X  A. y  e.  X  ( x
( N  o.  .-  ) y )  =  ( x ( D  |`  ( X  X.  X
) ) y ) ) )
22 opelxpi 4721 . . . . . . . . . 10  |-  ( ( x  e.  X  /\  y  e.  X )  -> 
<. x ,  y >.  e.  ( X  X.  X
) )
23 fvco3 5596 . . . . . . . . . 10  |-  ( ( 
.-  : ( X  X.  X ) --> X  /\  <. x ,  y
>.  e.  ( X  X.  X ) )  -> 
( ( N  o.  .-  ) `  <. x ,  y >. )  =  ( N `  (  .-  `  <. x ,  y >. )
) )
2411, 22, 23syl2an 463 . . . . . . . . 9  |-  ( ( ( G  e.  Grp  /\  G  e.  MetSp )  /\  ( x  e.  X  /\  y  e.  X
) )  ->  (
( N  o.  .-  ) `  <. x ,  y >. )  =  ( N `  (  .-  ` 
<. x ,  y >.
) ) )
25 df-ov 5861 . . . . . . . . 9  |-  ( x ( N  o.  .-  ) y )  =  ( ( N  o.  .-  ) `  <. x ,  y >. )
26 df-ov 5861 . . . . . . . . . 10  |-  ( x 
.-  y )  =  (  .-  `  <. x ,  y >. )
2726fveq2i 5528 . . . . . . . . 9  |-  ( N `
 ( x  .-  y ) )  =  ( N `  (  .-  `  <. x ,  y
>. ) )
2824, 25, 273eqtr4g 2340 . . . . . . . 8  |-  ( ( ( G  e.  Grp  /\  G  e.  MetSp )  /\  ( x  e.  X  /\  y  e.  X
) )  ->  (
x ( N  o.  .-  ) y )  =  ( N `  (
x  .-  y )
) )
29 ovres 5987 . . . . . . . . 9  |-  ( ( x  e.  X  /\  y  e.  X )  ->  ( x ( D  |`  ( X  X.  X
) ) y )  =  ( x D y ) )
3029adantl 452 . . . . . . . 8  |-  ( ( ( G  e.  Grp  /\  G  e.  MetSp )  /\  ( x  e.  X  /\  y  e.  X
) )  ->  (
x ( D  |`  ( X  X.  X
) ) y )  =  ( x D y ) )
3128, 30eqeq12d 2297 . . . . . . 7  |-  ( ( ( G  e.  Grp  /\  G  e.  MetSp )  /\  ( x  e.  X  /\  y  e.  X
) )  ->  (
( x ( N  o.  .-  ) y
)  =  ( x ( D  |`  ( X  X.  X ) ) y )  <->  ( N `  ( x  .-  y
) )  =  ( x D y ) ) )
32 eqcom 2285 . . . . . . 7  |-  ( ( N `  ( x 
.-  y ) )  =  ( x D y )  <->  ( x D y )  =  ( N `  (
x  .-  y )
) )
3331, 32syl6bb 252 . . . . . 6  |-  ( ( ( G  e.  Grp  /\  G  e.  MetSp )  /\  ( x  e.  X  /\  y  e.  X
) )  ->  (
( x ( N  o.  .-  ) y
)  =  ( x ( D  |`  ( X  X.  X ) ) y )  <->  ( x D y )  =  ( N `  (
x  .-  y )
) ) )
34332ralbidva 2583 . . . . 5  |-  ( ( G  e.  Grp  /\  G  e.  MetSp )  -> 
( A. x  e.  X  A. y  e.  X  ( x ( N  o.  .-  )
y )  =  ( x ( D  |`  ( X  X.  X
) ) y )  <->  A. x  e.  X  A. y  e.  X  ( x D y )  =  ( N `
 ( x  .-  y ) ) ) )
3521, 34bitrd 244 . . . 4  |-  ( ( G  e.  Grp  /\  G  e.  MetSp )  -> 
( ( N  o.  .-  )  =  ( D  |`  ( X  X.  X
) )  <->  A. x  e.  X  A. y  e.  X  ( x D y )  =  ( N `  (
x  .-  y )
) ) )
3635pm5.32i 618 . . 3  |-  ( ( ( G  e.  Grp  /\  G  e.  MetSp )  /\  ( N  o.  .-  )  =  ( D  |`  ( X  X.  X
) ) )  <->  ( ( G  e.  Grp  /\  G  e.  MetSp )  /\  A. x  e.  X  A. y  e.  X  (
x D y )  =  ( N `  ( x  .-  y ) ) ) )
37 df-3an 936 . . 3  |-  ( ( G  e.  Grp  /\  G  e.  MetSp  /\  ( N  o.  .-  )  =  ( D  |`  ( X  X.  X ) ) )  <->  ( ( G  e.  Grp  /\  G  e.  MetSp )  /\  ( N  o.  .-  )  =  ( D  |`  ( X  X.  X ) ) ) )
38 df-3an 936 . . 3  |-  ( ( G  e.  Grp  /\  G  e.  MetSp  /\  A. x  e.  X  A. y  e.  X  (
x D y )  =  ( N `  ( x  .-  y ) ) )  <->  ( ( G  e.  Grp  /\  G  e.  MetSp )  /\  A. x  e.  X  A. y  e.  X  (
x D y )  =  ( N `  ( x  .-  y ) ) ) )
3936, 37, 383bitr4i 268 . 2  |-  ( ( G  e.  Grp  /\  G  e.  MetSp  /\  ( N  o.  .-  )  =  ( D  |`  ( X  X.  X ) ) )  <->  ( G  e. 
Grp  /\  G  e.  MetSp  /\  A. x  e.  X  A. y  e.  X  ( x D y )  =  ( N `
 ( x  .-  y ) ) ) )
406, 39bitri 240 1  |-  ( G  e. NrmGrp 
<->  ( G  e.  Grp  /\  G  e.  MetSp  /\  A. x  e.  X  A. y  e.  X  (
x D y )  =  ( N `  ( x  .-  y ) ) ) )
Colors of variables: wff set class
Syntax hints:    <-> wb 176    /\ wa 358    /\ w3a 934    = wceq 1623    e. wcel 1684   A.wral 2543   <.cop 3643    X. cxp 4687    |` cres 4691    o. ccom 4693    Fn wfn 5250   -->wf 5251   ` cfv 5255  (class class class)co 5858   RRcr 8736   Basecbs 13148   distcds 13217   Grpcgrp 14362   -gcsg 14365   Metcme 16370   MetSpcmt 17883   normcnm 18099  NrmGrpcngp 18100
This theorem is referenced by:  isngp4  18133  subgngp  18151
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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