MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  tngngp2 Unicode version

Theorem tngngp2 18184
Description: A norm turns a group into a normed group iff the generated metric is in fact a metric. (Contributed by Mario Carneiro, 4-Oct-2015.)
Hypotheses
Ref Expression
tngngp2.t  |-  T  =  ( G toNrmGrp  N )
tngngp2.x  |-  X  =  ( Base `  G
)
tngngp2.d  |-  D  =  ( dist `  T
)
Assertion
Ref Expression
tngngp2  |-  ( N : X --> RR  ->  ( T  e. NrmGrp  <->  ( G  e. 
Grp  /\  D  e.  ( Met `  X ) ) ) )

Proof of Theorem tngngp2
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ngpgrp 18137 . . . . 5  |-  ( T  e. NrmGrp  ->  T  e.  Grp )
2 tngngp2.x . . . . . . . 8  |-  X  =  ( Base `  G
)
3 fvex 5555 . . . . . . . 8  |-  ( Base `  G )  e.  _V
42, 3eqeltri 2366 . . . . . . 7  |-  X  e. 
_V
5 reex 8844 . . . . . . 7  |-  RR  e.  _V
6 fex2 5417 . . . . . . 7  |-  ( ( N : X --> RR  /\  X  e.  _V  /\  RR  e.  _V )  ->  N  e.  _V )
74, 5, 6mp3an23 1269 . . . . . 6  |-  ( N : X --> RR  ->  N  e.  _V )
82a1i 10 . . . . . . 7  |-  ( N  e.  _V  ->  X  =  ( Base `  G
) )
9 tngngp2.t . . . . . . . 8  |-  T  =  ( G toNrmGrp  N )
109, 2tngbas 18173 . . . . . . 7  |-  ( N  e.  _V  ->  X  =  ( Base `  T
) )
11 eqid 2296 . . . . . . . . 9  |-  ( +g  `  G )  =  ( +g  `  G )
129, 11tngplusg 18174 . . . . . . . 8  |-  ( N  e.  _V  ->  ( +g  `  G )  =  ( +g  `  T
) )
1312proplem3 13609 . . . . . . 7  |-  ( ( N  e.  _V  /\  ( x  e.  X  /\  y  e.  X
) )  ->  (
x ( +g  `  G
) y )  =  ( x ( +g  `  T ) y ) )
148, 10, 13grppropd 14516 . . . . . 6  |-  ( N  e.  _V  ->  ( G  e.  Grp  <->  T  e.  Grp ) )
157, 14syl 15 . . . . 5  |-  ( N : X --> RR  ->  ( G  e.  Grp  <->  T  e.  Grp ) )
161, 15syl5ibr 212 . . . 4  |-  ( N : X --> RR  ->  ( T  e. NrmGrp  ->  G  e. 
Grp ) )
1716imp 418 . . 3  |-  ( ( N : X --> RR  /\  T  e. NrmGrp )  ->  G  e.  Grp )
18 ngpms 18138 . . . . . 6  |-  ( T  e. NrmGrp  ->  T  e.  MetSp )
1918adantl 452 . . . . 5  |-  ( ( N : X --> RR  /\  T  e. NrmGrp )  ->  T  e.  MetSp )
20 eqid 2296 . . . . . 6  |-  ( Base `  T )  =  (
Base `  T )
21 tngngp2.d . . . . . . 7  |-  D  =  ( dist `  T
)
2221reseq1i 4967 . . . . . 6  |-  ( D  |`  ( ( Base `  T
)  X.  ( Base `  T ) ) )  =  ( ( dist `  T )  |`  (
( Base `  T )  X.  ( Base `  T
) ) )
2320, 22msmet 18019 . . . . 5  |-  ( T  e.  MetSp  ->  ( D  |`  ( ( Base `  T
)  X.  ( Base `  T ) ) )  e.  ( Met `  ( Base `  T ) ) )
2419, 23syl 15 . . . 4  |-  ( ( N : X --> RR  /\  T  e. NrmGrp )  ->  ( D  |`  ( ( Base `  T )  X.  ( Base `  T
) ) )  e.  ( Met `  ( Base `  T ) ) )
25 eqid 2296 . . . . . . . . . . 11  |-  ( -g `  G )  =  (
-g `  G )
262, 25grpsubf 14561 . . . . . . . . . 10  |-  ( G  e.  Grp  ->  ( -g `  G ) : ( X  X.  X
) --> X )
2717, 26syl 15 . . . . . . . . 9  |-  ( ( N : X --> RR  /\  T  e. NrmGrp )  ->  (
-g `  G ) : ( X  X.  X ) --> X )
28 fco 5414 . . . . . . . . 9  |-  ( ( N : X --> RR  /\  ( -g `  G ) : ( X  X.  X ) --> X )  ->  ( N  o.  ( -g `  G ) ) : ( X  X.  X ) --> RR )
2927, 28syldan 456 . . . . . . . 8  |-  ( ( N : X --> RR  /\  T  e. NrmGrp )  ->  ( N  o.  ( -g `  G ) ) : ( X  X.  X
) --> RR )
307adantr 451 . . . . . . . . . . 11  |-  ( ( N : X --> RR  /\  T  e. NrmGrp )  ->  N  e.  _V )
319, 25tngds 18180 . . . . . . . . . . 11  |-  ( N  e.  _V  ->  ( N  o.  ( -g `  G ) )  =  ( dist `  T
) )
3230, 31syl 15 . . . . . . . . . 10  |-  ( ( N : X --> RR  /\  T  e. NrmGrp )  ->  ( N  o.  ( -g `  G ) )  =  ( dist `  T
) )
3332, 21syl6reqr 2347 . . . . . . . . 9  |-  ( ( N : X --> RR  /\  T  e. NrmGrp )  ->  D  =  ( N  o.  ( -g `  G ) ) )
3433feq1d 5395 . . . . . . . 8  |-  ( ( N : X --> RR  /\  T  e. NrmGrp )  ->  ( D : ( X  X.  X ) --> RR  <->  ( N  o.  ( -g `  G ) ) : ( X  X.  X
) --> RR ) )
3529, 34mpbird 223 . . . . . . 7  |-  ( ( N : X --> RR  /\  T  e. NrmGrp )  ->  D : ( X  X.  X ) --> RR )
36 ffn 5405 . . . . . . 7  |-  ( D : ( X  X.  X ) --> RR  ->  D  Fn  ( X  X.  X ) )
37 fnresdm 5369 . . . . . . 7  |-  ( D  Fn  ( X  X.  X )  ->  ( D  |`  ( X  X.  X ) )  =  D )
3835, 36, 373syl 18 . . . . . 6  |-  ( ( N : X --> RR  /\  T  e. NrmGrp )  ->  ( D  |`  ( X  X.  X ) )  =  D )
3930, 10syl 15 . . . . . . . 8  |-  ( ( N : X --> RR  /\  T  e. NrmGrp )  ->  X  =  ( Base `  T
) )
4039, 39xpeq12d 4730 . . . . . . 7  |-  ( ( N : X --> RR  /\  T  e. NrmGrp )  ->  ( X  X.  X )  =  ( ( Base `  T )  X.  ( Base `  T ) ) )
4140reseq2d 4971 . . . . . 6  |-  ( ( N : X --> RR  /\  T  e. NrmGrp )  ->  ( D  |`  ( X  X.  X ) )  =  ( D  |`  (
( Base `  T )  X.  ( Base `  T
) ) ) )
4238, 41eqtr3d 2330 . . . . 5  |-  ( ( N : X --> RR  /\  T  e. NrmGrp )  ->  D  =  ( D  |`  ( ( Base `  T
)  X.  ( Base `  T ) ) ) )
4339fveq2d 5545 . . . . 5  |-  ( ( N : X --> RR  /\  T  e. NrmGrp )  ->  ( Met `  X )  =  ( Met `  ( Base `  T ) ) )
4442, 43eleq12d 2364 . . . 4  |-  ( ( N : X --> RR  /\  T  e. NrmGrp )  ->  ( D  e.  ( Met `  X )  <->  ( D  |`  ( ( Base `  T
)  X.  ( Base `  T ) ) )  e.  ( Met `  ( Base `  T ) ) ) )
4524, 44mpbird 223 . . 3  |-  ( ( N : X --> RR  /\  T  e. NrmGrp )  ->  D  e.  ( Met `  X
) )
4617, 45jca 518 . 2  |-  ( ( N : X --> RR  /\  T  e. NrmGrp )  ->  ( G  e.  Grp  /\  D  e.  ( Met `  X ) ) )
4715biimpa 470 . . . 4  |-  ( ( N : X --> RR  /\  G  e.  Grp )  ->  T  e.  Grp )
4847adantrr 697 . . 3  |-  ( ( N : X --> RR  /\  ( G  e.  Grp  /\  D  e.  ( Met `  X ) ) )  ->  T  e.  Grp )
49 simprr 733 . . . . . . . 8  |-  ( ( N : X --> RR  /\  ( G  e.  Grp  /\  D  e.  ( Met `  X ) ) )  ->  D  e.  ( Met `  X ) )
507adantr 451 . . . . . . . . . 10  |-  ( ( N : X --> RR  /\  ( G  e.  Grp  /\  D  e.  ( Met `  X ) ) )  ->  N  e.  _V )
5150, 10syl 15 . . . . . . . . 9  |-  ( ( N : X --> RR  /\  ( G  e.  Grp  /\  D  e.  ( Met `  X ) ) )  ->  X  =  (
Base `  T )
)
5251fveq2d 5545 . . . . . . . 8  |-  ( ( N : X --> RR  /\  ( G  e.  Grp  /\  D  e.  ( Met `  X ) ) )  ->  ( Met `  X
)  =  ( Met `  ( Base `  T
) ) )
5349, 52eleqtrd 2372 . . . . . . 7  |-  ( ( N : X --> RR  /\  ( G  e.  Grp  /\  D  e.  ( Met `  X ) ) )  ->  D  e.  ( Met `  ( Base `  T ) ) )
54 metf 17911 . . . . . . 7  |-  ( D  e.  ( Met `  ( Base `  T ) )  ->  D : ( ( Base `  T
)  X.  ( Base `  T ) ) --> RR )
5553, 54syl 15 . . . . . 6  |-  ( ( N : X --> RR  /\  ( G  e.  Grp  /\  D  e.  ( Met `  X ) ) )  ->  D : ( ( Base `  T
)  X.  ( Base `  T ) ) --> RR )
56 ffn 5405 . . . . . 6  |-  ( D : ( ( Base `  T )  X.  ( Base `  T ) ) --> RR  ->  D  Fn  ( ( Base `  T
)  X.  ( Base `  T ) ) )
57 fnresdm 5369 . . . . . 6  |-  ( D  Fn  ( ( Base `  T )  X.  ( Base `  T ) )  ->  ( D  |`  ( ( Base `  T
)  X.  ( Base `  T ) ) )  =  D )
5855, 56, 573syl 18 . . . . 5  |-  ( ( N : X --> RR  /\  ( G  e.  Grp  /\  D  e.  ( Met `  X ) ) )  ->  ( D  |`  ( ( Base `  T
)  X.  ( Base `  T ) ) )  =  D )
5958, 53eqeltrd 2370 . . . 4  |-  ( ( N : X --> RR  /\  ( G  e.  Grp  /\  D  e.  ( Met `  X ) ) )  ->  ( D  |`  ( ( Base `  T
)  X.  ( Base `  T ) ) )  e.  ( Met `  ( Base `  T ) ) )
6058fveq2d 5545 . . . . 5  |-  ( ( N : X --> RR  /\  ( G  e.  Grp  /\  D  e.  ( Met `  X ) ) )  ->  ( MetOpen `  ( D  |`  ( ( Base `  T )  X.  ( Base `  T ) ) ) )  =  (
MetOpen `  D ) )
61 simprl 732 . . . . . 6  |-  ( ( N : X --> RR  /\  ( G  e.  Grp  /\  D  e.  ( Met `  X ) ) )  ->  G  e.  Grp )
62 eqid 2296 . . . . . . 7  |-  ( MetOpen `  D )  =  (
MetOpen `  D )
639, 21, 62tngtopn 18182 . . . . . 6  |-  ( ( G  e.  Grp  /\  N  e.  _V )  ->  ( MetOpen `  D )  =  ( TopOpen `  T
) )
6461, 50, 63syl2anc 642 . . . . 5  |-  ( ( N : X --> RR  /\  ( G  e.  Grp  /\  D  e.  ( Met `  X ) ) )  ->  ( MetOpen `  D
)  =  ( TopOpen `  T ) )
6560, 64eqtr2d 2329 . . . 4  |-  ( ( N : X --> RR  /\  ( G  e.  Grp  /\  D  e.  ( Met `  X ) ) )  ->  ( TopOpen `  T
)  =  ( MetOpen `  ( D  |`  ( (
Base `  T )  X.  ( Base `  T
) ) ) ) )
66 eqid 2296 . . . . 5  |-  ( TopOpen `  T )  =  (
TopOpen `  T )
6766, 20, 22isms2 18012 . . . 4  |-  ( T  e.  MetSp 
<->  ( ( D  |`  ( ( Base `  T
)  X.  ( Base `  T ) ) )  e.  ( Met `  ( Base `  T ) )  /\  ( TopOpen `  T
)  =  ( MetOpen `  ( D  |`  ( (
Base `  T )  X.  ( Base `  T
) ) ) ) ) )
6859, 65, 67sylanbrc 645 . . 3  |-  ( ( N : X --> RR  /\  ( G  e.  Grp  /\  D  e.  ( Met `  X ) ) )  ->  T  e.  MetSp )
69 simpl 443 . . . . . . 7  |-  ( ( N : X --> RR  /\  ( G  e.  Grp  /\  D  e.  ( Met `  X ) ) )  ->  N : X --> RR )
709, 2, 5tngnm 18183 . . . . . . 7  |-  ( ( G  e.  Grp  /\  N : X --> RR )  ->  N  =  (
norm `  T )
)
7161, 69, 70syl2anc 642 . . . . . 6  |-  ( ( N : X --> RR  /\  ( G  e.  Grp  /\  D  e.  ( Met `  X ) ) )  ->  N  =  (
norm `  T )
)
728, 10eqtr3d 2330 . . . . . . . 8  |-  ( N  e.  _V  ->  ( Base `  G )  =  ( Base `  T
) )
7372, 12grpsubpropd 14582 . . . . . . 7  |-  ( N  e.  _V  ->  ( -g `  G )  =  ( -g `  T
) )
7450, 73syl 15 . . . . . 6  |-  ( ( N : X --> RR  /\  ( G  e.  Grp  /\  D  e.  ( Met `  X ) ) )  ->  ( -g `  G
)  =  ( -g `  T ) )
7571, 74coeq12d 4864 . . . . 5  |-  ( ( N : X --> RR  /\  ( G  e.  Grp  /\  D  e.  ( Met `  X ) ) )  ->  ( N  o.  ( -g `  G ) )  =  ( (
norm `  T )  o.  ( -g `  T
) ) )
7650, 31syl 15 . . . . 5  |-  ( ( N : X --> RR  /\  ( G  e.  Grp  /\  D  e.  ( Met `  X ) ) )  ->  ( N  o.  ( -g `  G ) )  =  ( dist `  T ) )
7775, 76eqtr3d 2330 . . . 4  |-  ( ( N : X --> RR  /\  ( G  e.  Grp  /\  D  e.  ( Met `  X ) ) )  ->  ( ( norm `  T )  o.  ( -g `  T ) )  =  ( dist `  T
) )
78 eqimss 3243 . . . 4  |-  ( ( ( norm `  T
)  o.  ( -g `  T ) )  =  ( dist `  T
)  ->  ( ( norm `  T )  o.  ( -g `  T
) )  C_  ( dist `  T ) )
7977, 78syl 15 . . 3  |-  ( ( N : X --> RR  /\  ( G  e.  Grp  /\  D  e.  ( Met `  X ) ) )  ->  ( ( norm `  T )  o.  ( -g `  T ) ) 
C_  ( dist `  T
) )
80 eqid 2296 . . . 4  |-  ( norm `  T )  =  (
norm `  T )
81 eqid 2296 . . . 4  |-  ( -g `  T )  =  (
-g `  T )
82 eqid 2296 . . . 4  |-  ( dist `  T )  =  (
dist `  T )
8380, 81, 82isngp 18134 . . 3  |-  ( T  e. NrmGrp 
<->  ( T  e.  Grp  /\  T  e.  MetSp  /\  (
( norm `  T )  o.  ( -g `  T
) )  C_  ( dist `  T ) ) )
8448, 68, 79, 83syl3anbrc 1136 . 2  |-  ( ( N : X --> RR  /\  ( G  e.  Grp  /\  D  e.  ( Met `  X ) ) )  ->  T  e. NrmGrp )
8546, 84impbida 805 1  |-  ( N : X --> RR  ->  ( T  e. NrmGrp  <->  ( G  e. 
Grp  /\  D  e.  ( Met `  X ) ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    = wceq 1632    e. wcel 1696   _Vcvv 2801    C_ wss 3165    X. cxp 4703    |` cres 4707    o. ccom 4709    Fn wfn 5266   -->wf 5267   ` cfv 5271  (class class class)co 5874   RRcr 8752   Basecbs 13164   +g cplusg 13224   distcds 13233   TopOpenctopn 13342   Grpcgrp 14378   -gcsg 14381   Metcme 16386   MetOpencmopn 16388   MetSpcmt 17899   normcnm 18115  NrmGrpcngp 18116   toNrmGrp ctng 18117
This theorem is referenced by:  tngngpd  18185  tngngp  18186  tngnrg  18201
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-3 9821  df-4 9822  df-5 9823  df-6 9824  df-7 9825  df-8 9826  df-9 9827  df-10 9828  df-n0 9982  df-z 10041  df-dec 10141  df-uz 10247  df-q 10333  df-rp 10371  df-xneg 10468  df-xadd 10469  df-xmul 10470  df-ndx 13167  df-slot 13168  df-base 13169  df-sets 13170  df-plusg 13237  df-tset 13243  df-ds 13246  df-rest 13343  df-topn 13344  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  df-tng 18123
  Copyright terms: Public domain W3C validator