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Theorem tngngp2 18557
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 18510 . . . . 5  |-  ( T  e. NrmGrp  ->  T  e.  Grp )
2 tngngp2.x . . . . . . . 8  |-  X  =  ( Base `  G
)
3 fvex 5675 . . . . . . . 8  |-  ( Base `  G )  e.  _V
42, 3eqeltri 2450 . . . . . . 7  |-  X  e. 
_V
5 reex 9007 . . . . . . 7  |-  RR  e.  _V
6 fex2 5536 . . . . . . 7  |-  ( ( N : X --> RR  /\  X  e.  _V  /\  RR  e.  _V )  ->  N  e.  _V )
74, 5, 6mp3an23 1271 . . . . . 6  |-  ( N : X --> RR  ->  N  e.  _V )
82a1i 11 . . . . . . 7  |-  ( N  e.  _V  ->  X  =  ( Base `  G
) )
9 tngngp2.t . . . . . . . 8  |-  T  =  ( G toNrmGrp  N )
109, 2tngbas 18546 . . . . . . 7  |-  ( N  e.  _V  ->  X  =  ( Base `  T
) )
11 eqid 2380 . . . . . . . . 9  |-  ( +g  `  G )  =  ( +g  `  G )
129, 11tngplusg 18547 . . . . . . . 8  |-  ( N  e.  _V  ->  ( +g  `  G )  =  ( +g  `  T
) )
1312proplem3 13836 . . . . . . 7  |-  ( ( N  e.  _V  /\  ( x  e.  X  /\  y  e.  X
) )  ->  (
x ( +g  `  G
) y )  =  ( x ( +g  `  T ) y ) )
148, 10, 13grppropd 14743 . . . . . 6  |-  ( N  e.  _V  ->  ( G  e.  Grp  <->  T  e.  Grp ) )
157, 14syl 16 . . . . 5  |-  ( N : X --> RR  ->  ( G  e.  Grp  <->  T  e.  Grp ) )
161, 15syl5ibr 213 . . . 4  |-  ( N : X --> RR  ->  ( T  e. NrmGrp  ->  G  e. 
Grp ) )
1716imp 419 . . 3  |-  ( ( N : X --> RR  /\  T  e. NrmGrp )  ->  G  e.  Grp )
18 ngpms 18511 . . . . . 6  |-  ( T  e. NrmGrp  ->  T  e.  MetSp )
1918adantl 453 . . . . 5  |-  ( ( N : X --> RR  /\  T  e. NrmGrp )  ->  T  e.  MetSp )
20 eqid 2380 . . . . . 6  |-  ( Base `  T )  =  (
Base `  T )
21 tngngp2.d . . . . . 6  |-  D  =  ( dist `  T
)
2220, 21msmet2 18373 . . . . 5  |-  ( T  e.  MetSp  ->  ( D  |`  ( ( Base `  T
)  X.  ( Base `  T ) ) )  e.  ( Met `  ( Base `  T ) ) )
2319, 22syl 16 . . . 4  |-  ( ( N : X --> RR  /\  T  e. NrmGrp )  ->  ( D  |`  ( ( Base `  T )  X.  ( Base `  T
) ) )  e.  ( Met `  ( Base `  T ) ) )
24 eqid 2380 . . . . . . . . . 10  |-  ( -g `  G )  =  (
-g `  G )
252, 24grpsubf 14788 . . . . . . . . 9  |-  ( G  e.  Grp  ->  ( -g `  G ) : ( X  X.  X
) --> X )
2617, 25syl 16 . . . . . . . 8  |-  ( ( N : X --> RR  /\  T  e. NrmGrp )  ->  (
-g `  G ) : ( X  X.  X ) --> X )
27 fco 5533 . . . . . . . 8  |-  ( ( N : X --> RR  /\  ( -g `  G ) : ( X  X.  X ) --> X )  ->  ( N  o.  ( -g `  G ) ) : ( X  X.  X ) --> RR )
2826, 27syldan 457 . . . . . . 7  |-  ( ( N : X --> RR  /\  T  e. NrmGrp )  ->  ( N  o.  ( -g `  G ) ) : ( X  X.  X
) --> RR )
297adantr 452 . . . . . . . . . 10  |-  ( ( N : X --> RR  /\  T  e. NrmGrp )  ->  N  e.  _V )
309, 24tngds 18553 . . . . . . . . . 10  |-  ( N  e.  _V  ->  ( N  o.  ( -g `  G ) )  =  ( dist `  T
) )
3129, 30syl 16 . . . . . . . . 9  |-  ( ( N : X --> RR  /\  T  e. NrmGrp )  ->  ( N  o.  ( -g `  G ) )  =  ( dist `  T
) )
3231, 21syl6reqr 2431 . . . . . . . 8  |-  ( ( N : X --> RR  /\  T  e. NrmGrp )  ->  D  =  ( N  o.  ( -g `  G ) ) )
3332feq1d 5513 . . . . . . 7  |-  ( ( N : X --> RR  /\  T  e. NrmGrp )  ->  ( D : ( X  X.  X ) --> RR  <->  ( N  o.  ( -g `  G ) ) : ( X  X.  X
) --> RR ) )
3428, 33mpbird 224 . . . . . 6  |-  ( ( N : X --> RR  /\  T  e. NrmGrp )  ->  D : ( X  X.  X ) --> RR )
35 ffn 5524 . . . . . 6  |-  ( D : ( X  X.  X ) --> RR  ->  D  Fn  ( X  X.  X ) )
36 fnresdm 5487 . . . . . 6  |-  ( D  Fn  ( X  X.  X )  ->  ( D  |`  ( X  X.  X ) )  =  D )
3734, 35, 363syl 19 . . . . 5  |-  ( ( N : X --> RR  /\  T  e. NrmGrp )  ->  ( D  |`  ( X  X.  X ) )  =  D )
3829, 10syl 16 . . . . . . 7  |-  ( ( N : X --> RR  /\  T  e. NrmGrp )  ->  X  =  ( Base `  T
) )
3938, 38xpeq12d 4836 . . . . . 6  |-  ( ( N : X --> RR  /\  T  e. NrmGrp )  ->  ( X  X.  X )  =  ( ( Base `  T )  X.  ( Base `  T ) ) )
4039reseq2d 5079 . . . . 5  |-  ( ( N : X --> RR  /\  T  e. NrmGrp )  ->  ( D  |`  ( X  X.  X ) )  =  ( D  |`  (
( Base `  T )  X.  ( Base `  T
) ) ) )
4137, 40eqtr3d 2414 . . . 4  |-  ( ( N : X --> RR  /\  T  e. NrmGrp )  ->  D  =  ( D  |`  ( ( Base `  T
)  X.  ( Base `  T ) ) ) )
4238fveq2d 5665 . . . 4  |-  ( ( N : X --> RR  /\  T  e. NrmGrp )  ->  ( Met `  X )  =  ( Met `  ( Base `  T ) ) )
4323, 41, 423eltr4d 2461 . . 3  |-  ( ( N : X --> RR  /\  T  e. NrmGrp )  ->  D  e.  ( Met `  X
) )
4417, 43jca 519 . 2  |-  ( ( N : X --> RR  /\  T  e. NrmGrp )  ->  ( G  e.  Grp  /\  D  e.  ( Met `  X ) ) )
4515biimpa 471 . . . 4  |-  ( ( N : X --> RR  /\  G  e.  Grp )  ->  T  e.  Grp )
4645adantrr 698 . . 3  |-  ( ( N : X --> RR  /\  ( G  e.  Grp  /\  D  e.  ( Met `  X ) ) )  ->  T  e.  Grp )
47 simprr 734 . . . . . . . 8  |-  ( ( N : X --> RR  /\  ( G  e.  Grp  /\  D  e.  ( Met `  X ) ) )  ->  D  e.  ( Met `  X ) )
487adantr 452 . . . . . . . . . 10  |-  ( ( N : X --> RR  /\  ( G  e.  Grp  /\  D  e.  ( Met `  X ) ) )  ->  N  e.  _V )
4948, 10syl 16 . . . . . . . . 9  |-  ( ( N : X --> RR  /\  ( G  e.  Grp  /\  D  e.  ( Met `  X ) ) )  ->  X  =  (
Base `  T )
)
5049fveq2d 5665 . . . . . . . 8  |-  ( ( N : X --> RR  /\  ( G  e.  Grp  /\  D  e.  ( Met `  X ) ) )  ->  ( Met `  X
)  =  ( Met `  ( Base `  T
) ) )
5147, 50eleqtrd 2456 . . . . . . 7  |-  ( ( N : X --> RR  /\  ( G  e.  Grp  /\  D  e.  ( Met `  X ) ) )  ->  D  e.  ( Met `  ( Base `  T ) ) )
52 metf 18262 . . . . . . 7  |-  ( D  e.  ( Met `  ( Base `  T ) )  ->  D : ( ( Base `  T
)  X.  ( Base `  T ) ) --> RR )
5351, 52syl 16 . . . . . 6  |-  ( ( N : X --> RR  /\  ( G  e.  Grp  /\  D  e.  ( Met `  X ) ) )  ->  D : ( ( Base `  T
)  X.  ( Base `  T ) ) --> RR )
54 ffn 5524 . . . . . 6  |-  ( D : ( ( Base `  T )  X.  ( Base `  T ) ) --> RR  ->  D  Fn  ( ( Base `  T
)  X.  ( Base `  T ) ) )
55 fnresdm 5487 . . . . . 6  |-  ( D  Fn  ( ( Base `  T )  X.  ( Base `  T ) )  ->  ( D  |`  ( ( Base `  T
)  X.  ( Base `  T ) ) )  =  D )
5653, 54, 553syl 19 . . . . 5  |-  ( ( N : X --> RR  /\  ( G  e.  Grp  /\  D  e.  ( Met `  X ) ) )  ->  ( D  |`  ( ( Base `  T
)  X.  ( Base `  T ) ) )  =  D )
5756, 51eqeltrd 2454 . . . 4  |-  ( ( N : X --> RR  /\  ( G  e.  Grp  /\  D  e.  ( Met `  X ) ) )  ->  ( D  |`  ( ( Base `  T
)  X.  ( Base `  T ) ) )  e.  ( Met `  ( Base `  T ) ) )
5856fveq2d 5665 . . . . 5  |-  ( ( N : X --> RR  /\  ( G  e.  Grp  /\  D  e.  ( Met `  X ) ) )  ->  ( MetOpen `  ( D  |`  ( ( Base `  T )  X.  ( Base `  T ) ) ) )  =  (
MetOpen `  D ) )
59 simprl 733 . . . . . 6  |-  ( ( N : X --> RR  /\  ( G  e.  Grp  /\  D  e.  ( Met `  X ) ) )  ->  G  e.  Grp )
60 eqid 2380 . . . . . . 7  |-  ( MetOpen `  D )  =  (
MetOpen `  D )
619, 21, 60tngtopn 18555 . . . . . 6  |-  ( ( G  e.  Grp  /\  N  e.  _V )  ->  ( MetOpen `  D )  =  ( TopOpen `  T
) )
6259, 48, 61syl2anc 643 . . . . 5  |-  ( ( N : X --> RR  /\  ( G  e.  Grp  /\  D  e.  ( Met `  X ) ) )  ->  ( MetOpen `  D
)  =  ( TopOpen `  T ) )
6358, 62eqtr2d 2413 . . . 4  |-  ( ( N : X --> RR  /\  ( G  e.  Grp  /\  D  e.  ( Met `  X ) ) )  ->  ( TopOpen `  T
)  =  ( MetOpen `  ( D  |`  ( (
Base `  T )  X.  ( Base `  T
) ) ) ) )
64 eqid 2380 . . . . 5  |-  ( TopOpen `  T )  =  (
TopOpen `  T )
6521reseq1i 5075 . . . . 5  |-  ( D  |`  ( ( Base `  T
)  X.  ( Base `  T ) ) )  =  ( ( dist `  T )  |`  (
( Base `  T )  X.  ( Base `  T
) ) )
6664, 20, 65isms2 18363 . . . 4  |-  ( T  e.  MetSp 
<->  ( ( D  |`  ( ( Base `  T
)  X.  ( Base `  T ) ) )  e.  ( Met `  ( Base `  T ) )  /\  ( TopOpen `  T
)  =  ( MetOpen `  ( D  |`  ( (
Base `  T )  X.  ( Base `  T
) ) ) ) ) )
6757, 63, 66sylanbrc 646 . . 3  |-  ( ( N : X --> RR  /\  ( G  e.  Grp  /\  D  e.  ( Met `  X ) ) )  ->  T  e.  MetSp )
68 simpl 444 . . . . . . 7  |-  ( ( N : X --> RR  /\  ( G  e.  Grp  /\  D  e.  ( Met `  X ) ) )  ->  N : X --> RR )
699, 2, 5tngnm 18556 . . . . . . 7  |-  ( ( G  e.  Grp  /\  N : X --> RR )  ->  N  =  (
norm `  T )
)
7059, 68, 69syl2anc 643 . . . . . 6  |-  ( ( N : X --> RR  /\  ( G  e.  Grp  /\  D  e.  ( Met `  X ) ) )  ->  N  =  (
norm `  T )
)
718, 10eqtr3d 2414 . . . . . . . 8  |-  ( N  e.  _V  ->  ( Base `  G )  =  ( Base `  T
) )
7271, 12grpsubpropd 14809 . . . . . . 7  |-  ( N  e.  _V  ->  ( -g `  G )  =  ( -g `  T
) )
7348, 72syl 16 . . . . . 6  |-  ( ( N : X --> RR  /\  ( G  e.  Grp  /\  D  e.  ( Met `  X ) ) )  ->  ( -g `  G
)  =  ( -g `  T ) )
7470, 73coeq12d 4970 . . . . 5  |-  ( ( N : X --> RR  /\  ( G  e.  Grp  /\  D  e.  ( Met `  X ) ) )  ->  ( N  o.  ( -g `  G ) )  =  ( (
norm `  T )  o.  ( -g `  T
) ) )
7548, 30syl 16 . . . . 5  |-  ( ( N : X --> RR  /\  ( G  e.  Grp  /\  D  e.  ( Met `  X ) ) )  ->  ( N  o.  ( -g `  G ) )  =  ( dist `  T ) )
7674, 75eqtr3d 2414 . . . 4  |-  ( ( N : X --> RR  /\  ( G  e.  Grp  /\  D  e.  ( Met `  X ) ) )  ->  ( ( norm `  T )  o.  ( -g `  T ) )  =  ( dist `  T
) )
77 eqimss 3336 . . . 4  |-  ( ( ( norm `  T
)  o.  ( -g `  T ) )  =  ( dist `  T
)  ->  ( ( norm `  T )  o.  ( -g `  T
) )  C_  ( dist `  T ) )
7876, 77syl 16 . . 3  |-  ( ( N : X --> RR  /\  ( G  e.  Grp  /\  D  e.  ( Met `  X ) ) )  ->  ( ( norm `  T )  o.  ( -g `  T ) ) 
C_  ( dist `  T
) )
79 eqid 2380 . . . 4  |-  ( norm `  T )  =  (
norm `  T )
80 eqid 2380 . . . 4  |-  ( -g `  T )  =  (
-g `  T )
81 eqid 2380 . . . 4  |-  ( dist `  T )  =  (
dist `  T )
8279, 80, 81isngp 18507 . . 3  |-  ( T  e. NrmGrp 
<->  ( T  e.  Grp  /\  T  e.  MetSp  /\  (
( norm `  T )  o.  ( -g `  T
) )  C_  ( dist `  T ) ) )
8346, 67, 78, 82syl3anbrc 1138 . 2  |-  ( ( N : X --> RR  /\  ( G  e.  Grp  /\  D  e.  ( Met `  X ) ) )  ->  T  e. NrmGrp )
8444, 83impbida 806 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 177    /\ wa 359    = wceq 1649    e. wcel 1717   _Vcvv 2892    C_ wss 3256    X. cxp 4809    |` cres 4813    o. ccom 4815    Fn wfn 5382   -->wf 5383   ` cfv 5387  (class class class)co 6013   RRcr 8915   Basecbs 13389   +g cplusg 13449   distcds 13458   TopOpenctopn 13569   Grpcgrp 14605   -gcsg 14608   Metcme 16606   MetOpencmopn 16610   MetSpcmt 18250   normcnm 18488  NrmGrpcngp 18489   toNrmGrp ctng 18490
This theorem is referenced by:  tngngpd  18558  tngngp  18559  tngnrg  18574
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2361  ax-rep 4254  ax-sep 4264  ax-nul 4272  ax-pow 4311  ax-pr 4337  ax-un 4634  ax-cnex 8972  ax-resscn 8973  ax-1cn 8974  ax-icn 8975  ax-addcl 8976  ax-addrcl 8977  ax-mulcl 8978  ax-mulrcl 8979  ax-mulcom 8980  ax-addass 8981  ax-mulass 8982  ax-distr 8983  ax-i2m1 8984  ax-1ne0 8985  ax-1rid 8986  ax-rnegex 8987  ax-rrecex 8988  ax-cnre 8989  ax-pre-lttri 8990  ax-pre-lttrn 8991  ax-pre-ltadd 8992  ax-pre-mulgt0 8993  ax-pre-sup 8994
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2235  df-mo 2236  df-clab 2367  df-cleq 2373  df-clel 2376  df-nfc 2505  df-ne 2545  df-nel 2546  df-ral 2647  df-rex 2648  df-reu 2649  df-rmo 2650  df-rab 2651  df-v 2894  df-sbc 3098  df-csb 3188  df-dif 3259  df-un 3261  df-in 3263  df-ss 3270  df-pss 3272  df-nul 3565  df-if 3676  df-pw 3737  df-sn 3756  df-pr 3757  df-tp 3758  df-op 3759  df-uni 3951  df-iun 4030  df-br 4147  df-opab 4201  df-mpt 4202  df-tr 4237  df-eprel 4428  df-id 4432  df-po 4437  df-so 4438  df-fr 4475  df-we 4477  df-ord 4518  df-on 4519  df-lim 4520  df-suc 4521  df-om 4779  df-xp 4817  df-rel 4818  df-cnv 4819  df-co 4820  df-dm 4821  df-rn 4822  df-res 4823  df-ima 4824  df-iota 5351  df-fun 5389  df-fn 5390  df-f 5391  df-f1 5392  df-fo 5393  df-f1o 5394  df-fv 5395  df-ov 6016  df-oprab 6017  df-mpt2 6018  df-1st 6281  df-2nd 6282  df-riota 6478  df-recs 6562  df-rdg 6597  df-er 6834  df-map 6949  df-en 7039  df-dom 7040  df-sdom 7041  df-sup 7374  df-pnf 9048  df-mnf 9049  df-xr 9050  df-ltxr 9051  df-le 9052  df-sub 9218  df-neg 9219  df-div 9603  df-nn 9926  df-2 9983  df-3 9984  df-4 9985  df-5 9986  df-6 9987  df-7 9988  df-8 9989  df-9 9990  df-10 9991  df-n0 10147  df-z 10208  df-dec 10308  df-uz 10414  df-q 10500  df-rp 10538  df-xneg 10635  df-xadd 10636  df-xmul 10637  df-ndx 13392  df-slot 13393  df-base 13394  df-sets 13395  df-plusg 13462  df-tset 13468  df-ds 13471  df-rest 13570  df-topn 13571  df-topgen 13587  df-0g 13647  df-mnd 14610  df-grp 14732  df-minusg 14733  df-sbg 14734  df-xmet 16612  df-met 16613  df-bl 16614  df-mopn 16615  df-top 16879  df-bases 16881  df-topon 16882  df-topsp 16883  df-xms 18252  df-ms 18253  df-nm 18494  df-ngp 18495  df-tng 18496
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