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Theorem isngp3 18136
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 2296 . . 3  |-  ( D  |`  ( X  X.  X
) )  =  ( D  |`  ( X  X.  X ) )
61, 2, 3, 4, 5isngp2 18135 . 2  |-  ( G  e. NrmGrp 
<->  ( G  e.  Grp  /\  G  e.  MetSp  /\  ( N  o.  .-  )  =  ( D  |`  ( X  X.  X ) ) ) )
74, 3msmet2 18022 . . . . . . . . 9  |-  ( G  e.  MetSp  ->  ( D  |`  ( X  X.  X
) )  e.  ( Met `  X ) )
81, 4, 3, 5nmf2 18131 . . . . . . . . 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 14561 . . . . . . . . 9  |-  ( G  e.  Grp  ->  .-  :
( X  X.  X
) --> X )
1110adantr 451 . . . . . . . 8  |-  ( ( G  e.  Grp  /\  G  e.  MetSp )  ->  .-  : ( X  X.  X ) --> X )
12 fco 5414 . . . . . . . 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 5405 . . . . . . 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 17911 . . . . . . 7  |-  ( ( D  |`  ( X  X.  X ) )  e.  ( Met `  X
)  ->  ( D  |`  ( X  X.  X
) ) : ( X  X.  X ) --> RR )
18 ffn 5405 . . . . . . 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 5967 . . . . . 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 4737 . . . . . . . . . 10  |-  ( ( x  e.  X  /\  y  e.  X )  -> 
<. x ,  y >.  e.  ( X  X.  X
) )
23 fvco3 5612 . . . . . . . . . 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 5877 . . . . . . . . 9  |-  ( x ( N  o.  .-  ) y )  =  ( ( N  o.  .-  ) `  <. x ,  y >. )
26 df-ov 5877 . . . . . . . . . 10  |-  ( x 
.-  y )  =  (  .-  `  <. x ,  y >. )
2726fveq2i 5544 . . . . . . . . 9  |-  ( N `
 ( x  .-  y ) )  =  ( N `  (  .-  `  <. x ,  y
>. ) )
2824, 25, 273eqtr4g 2353 . . . . . . . 8  |-  ( ( ( G  e.  Grp  /\  G  e.  MetSp )  /\  ( x  e.  X  /\  y  e.  X
) )  ->  (
x ( N  o.  .-  ) y )  =  ( N `  (
x  .-  y )
) )
29 ovres 6003 . . . . . . . . 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 2310 . . . . . . 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 2298 . . . . . . 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 2596 . . . . 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 1632    e. wcel 1696   A.wral 2556   <.cop 3656    X. cxp 4703    |` cres 4707    o. ccom 4709    Fn wfn 5266   -->wf 5267   ` cfv 5271  (class class class)co 5874   RRcr 8752   Basecbs 13164   distcds 13233   Grpcgrp 14378   -gcsg 14381   Metcme 16386   MetSpcmt 17899   normcnm 18115  NrmGrpcngp 18116
This theorem is referenced by:  isngp4  18149  subgngp  18167
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-cnex 8809  ax-resscn 8810  ax-1cn 8811  ax-icn 8812  ax-addcl 8813  ax-addrcl 8814  ax-mulcl 8815  ax-mulrcl 8816  ax-mulcom 8817  ax-addass 8818  ax-mulass 8819  ax-distr 8820  ax-i2m1 8821  ax-1ne0 8822  ax-1rid 8823  ax-rnegex 8824  ax-rrecex 8825  ax-cnre 8826  ax-pre-lttri 8827  ax-pre-lttrn 8828  ax-pre-ltadd 8829  ax-pre-mulgt0 8830  ax-pre-sup 8831
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 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rmo 2564  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-1st 6138  df-2nd 6139  df-riota 6320  df-recs 6404  df-rdg 6439  df-er 6676  df-map 6790  df-en 6880  df-dom 6881  df-sdom 6882  df-sup 7210  df-pnf 8885  df-mnf 8886  df-xr 8887  df-ltxr 8888  df-le 8889  df-sub 9055  df-neg 9056  df-div 9440  df-nn 9763  df-2 9820  df-n0 9982  df-z 10041  df-uz 10247  df-q 10333  df-rp 10371  df-xneg 10468  df-xadd 10469  df-xmul 10470  df-topgen 13360  df-0g 13420  df-mnd 14383  df-grp 14505  df-minusg 14506  df-sbg 14507  df-xmet 16389  df-met 16390  df-bl 16391  df-mopn 16392  df-top 16652  df-bases 16654  df-topon 16655  df-topsp 16656  df-xms 17901  df-ms 17902  df-nm 18121  df-ngp 18122
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