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Theorem tngngp2 21034
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 20987 . . . . 5  |-  ( T  e. NrmGrp  ->  T  e.  Grp )
2 tngngp2.x . . . . . . . 8  |-  X  =  ( Base `  G
)
3 fvex 5882 . . . . . . . 8  |-  ( Base `  G )  e.  _V
42, 3eqeltri 2551 . . . . . . 7  |-  X  e. 
_V
5 reex 9595 . . . . . . 7  |-  RR  e.  _V
6 fex2 6750 . . . . . . 7  |-  ( ( N : X --> RR  /\  X  e.  _V  /\  RR  e.  _V )  ->  N  e.  _V )
74, 5, 6mp3an23 1316 . . . . . 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 21023 . . . . . . 7  |-  ( N  e.  _V  ->  X  =  ( Base `  T
) )
11 eqid 2467 . . . . . . . . 9  |-  ( +g  `  G )  =  ( +g  `  G )
129, 11tngplusg 21024 . . . . . . . 8  |-  ( N  e.  _V  ->  ( +g  `  G )  =  ( +g  `  T
) )
1312proplem3 14963 . . . . . . 7  |-  ( ( N  e.  _V  /\  ( x  e.  X  /\  y  e.  X
) )  ->  (
x ( +g  `  G
) y )  =  ( x ( +g  `  T ) y ) )
148, 10, 13grppropd 15940 . . . . . 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 221 . . . 4  |-  ( N : X --> RR  ->  ( T  e. NrmGrp  ->  G  e. 
Grp ) )
1716imp 429 . . 3  |-  ( ( N : X --> RR  /\  T  e. NrmGrp )  ->  G  e.  Grp )
18 ngpms 20988 . . . . . 6  |-  ( T  e. NrmGrp  ->  T  e.  MetSp )
1918adantl 466 . . . . 5  |-  ( ( N : X --> RR  /\  T  e. NrmGrp )  ->  T  e.  MetSp )
20 eqid 2467 . . . . . 6  |-  ( Base `  T )  =  (
Base `  T )
21 tngngp2.d . . . . . 6  |-  D  =  ( dist `  T
)
2220, 21msmet2 20831 . . . . 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 2467 . . . . . . . . . 10  |-  ( -g `  G )  =  (
-g `  G )
252, 24grpsubf 15989 . . . . . . . . 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 5747 . . . . . . . 8  |-  ( ( N : X --> RR  /\  ( -g `  G ) : ( X  X.  X ) --> X )  ->  ( N  o.  ( -g `  G ) ) : ( X  X.  X ) --> RR )
2826, 27syldan 470 . . . . . . 7  |-  ( ( N : X --> RR  /\  T  e. NrmGrp )  ->  ( N  o.  ( -g `  G ) ) : ( X  X.  X
) --> RR )
297adantr 465 . . . . . . . . . 10  |-  ( ( N : X --> RR  /\  T  e. NrmGrp )  ->  N  e.  _V )
309, 24tngds 21030 . . . . . . . . . 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 2527 . . . . . . . 8  |-  ( ( N : X --> RR  /\  T  e. NrmGrp )  ->  D  =  ( N  o.  ( -g `  G ) ) )
3332feq1d 5723 . . . . . . 7  |-  ( ( N : X --> RR  /\  T  e. NrmGrp )  ->  ( D : ( X  X.  X ) --> RR  <->  ( N  o.  ( -g `  G ) ) : ( X  X.  X
) --> RR ) )
3428, 33mpbird 232 . . . . . 6  |-  ( ( N : X --> RR  /\  T  e. NrmGrp )  ->  D : ( X  X.  X ) --> RR )
35 ffn 5737 . . . . . 6  |-  ( D : ( X  X.  X ) --> RR  ->  D  Fn  ( X  X.  X ) )
36 fnresdm 5696 . . . . . 6  |-  ( D  Fn  ( X  X.  X )  ->  ( D  |`  ( X  X.  X ) )  =  D )
3734, 35, 363syl 20 . . . . 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 5030 . . . . . 6  |-  ( ( N : X --> RR  /\  T  e. NrmGrp )  ->  ( X  X.  X )  =  ( ( Base `  T )  X.  ( Base `  T ) ) )
4039reseq2d 5279 . . . . 5  |-  ( ( N : X --> RR  /\  T  e. NrmGrp )  ->  ( D  |`  ( X  X.  X ) )  =  ( D  |`  (
( Base `  T )  X.  ( Base `  T
) ) ) )
4137, 40eqtr3d 2510 . . . 4  |-  ( ( N : X --> RR  /\  T  e. NrmGrp )  ->  D  =  ( D  |`  ( ( Base `  T
)  X.  ( Base `  T ) ) ) )
4238fveq2d 5876 . . . 4  |-  ( ( N : X --> RR  /\  T  e. NrmGrp )  ->  ( Met `  X )  =  ( Met `  ( Base `  T ) ) )
4323, 41, 423eltr4d 2570 . . 3  |-  ( ( N : X --> RR  /\  T  e. NrmGrp )  ->  D  e.  ( Met `  X
) )
4417, 43jca 532 . 2  |-  ( ( N : X --> RR  /\  T  e. NrmGrp )  ->  ( G  e.  Grp  /\  D  e.  ( Met `  X ) ) )
4515biimpa 484 . . . 4  |-  ( ( N : X --> RR  /\  G  e.  Grp )  ->  T  e.  Grp )
4645adantrr 716 . . 3  |-  ( ( N : X --> RR  /\  ( G  e.  Grp  /\  D  e.  ( Met `  X ) ) )  ->  T  e.  Grp )
47 simprr 756 . . . . . . . 8  |-  ( ( N : X --> RR  /\  ( G  e.  Grp  /\  D  e.  ( Met `  X ) ) )  ->  D  e.  ( Met `  X ) )
487adantr 465 . . . . . . . . . 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 5876 . . . . . . . 8  |-  ( ( N : X --> RR  /\  ( G  e.  Grp  /\  D  e.  ( Met `  X ) ) )  ->  ( Met `  X
)  =  ( Met `  ( Base `  T
) ) )
5147, 50eleqtrd 2557 . . . . . . 7  |-  ( ( N : X --> RR  /\  ( G  e.  Grp  /\  D  e.  ( Met `  X ) ) )  ->  D  e.  ( Met `  ( Base `  T ) ) )
52 metf 20701 . . . . . . 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 5737 . . . . . 6  |-  ( D : ( ( Base `  T )  X.  ( Base `  T ) ) --> RR  ->  D  Fn  ( ( Base `  T
)  X.  ( Base `  T ) ) )
55 fnresdm 5696 . . . . . 6  |-  ( D  Fn  ( ( Base `  T )  X.  ( Base `  T ) )  ->  ( D  |`  ( ( Base `  T
)  X.  ( Base `  T ) ) )  =  D )
5653, 54, 553syl 20 . . . . 5  |-  ( ( N : X --> RR  /\  ( G  e.  Grp  /\  D  e.  ( Met `  X ) ) )  ->  ( D  |`  ( ( Base `  T
)  X.  ( Base `  T ) ) )  =  D )
5756, 51eqeltrd 2555 . . . 4  |-  ( ( N : X --> RR  /\  ( G  e.  Grp  /\  D  e.  ( Met `  X ) ) )  ->  ( D  |`  ( ( Base `  T
)  X.  ( Base `  T ) ) )  e.  ( Met `  ( Base `  T ) ) )
5856fveq2d 5876 . . . . 5  |-  ( ( N : X --> RR  /\  ( G  e.  Grp  /\  D  e.  ( Met `  X ) ) )  ->  ( MetOpen `  ( D  |`  ( ( Base `  T )  X.  ( Base `  T ) ) ) )  =  (
MetOpen `  D ) )
59 simprl 755 . . . . . 6  |-  ( ( N : X --> RR  /\  ( G  e.  Grp  /\  D  e.  ( Met `  X ) ) )  ->  G  e.  Grp )
60 eqid 2467 . . . . . . 7  |-  ( MetOpen `  D )  =  (
MetOpen `  D )
619, 21, 60tngtopn 21032 . . . . . 6  |-  ( ( G  e.  Grp  /\  N  e.  _V )  ->  ( MetOpen `  D )  =  ( TopOpen `  T
) )
6259, 48, 61syl2anc 661 . . . . 5  |-  ( ( N : X --> RR  /\  ( G  e.  Grp  /\  D  e.  ( Met `  X ) ) )  ->  ( MetOpen `  D
)  =  ( TopOpen `  T ) )
6358, 62eqtr2d 2509 . . . 4  |-  ( ( N : X --> RR  /\  ( G  e.  Grp  /\  D  e.  ( Met `  X ) ) )  ->  ( TopOpen `  T
)  =  ( MetOpen `  ( D  |`  ( (
Base `  T )  X.  ( Base `  T
) ) ) ) )
64 eqid 2467 . . . . 5  |-  ( TopOpen `  T )  =  (
TopOpen `  T )
6521reseq1i 5275 . . . . 5  |-  ( D  |`  ( ( Base `  T
)  X.  ( Base `  T ) ) )  =  ( ( dist `  T )  |`  (
( Base `  T )  X.  ( Base `  T
) ) )
6664, 20, 65isms2 20821 . . . 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 664 . . 3  |-  ( ( N : X --> RR  /\  ( G  e.  Grp  /\  D  e.  ( Met `  X ) ) )  ->  T  e.  MetSp )
68 simpl 457 . . . . . . 7  |-  ( ( N : X --> RR  /\  ( G  e.  Grp  /\  D  e.  ( Met `  X ) ) )  ->  N : X --> RR )
699, 2, 5tngnm 21033 . . . . . . 7  |-  ( ( G  e.  Grp  /\  N : X --> RR )  ->  N  =  (
norm `  T )
)
7059, 68, 69syl2anc 661 . . . . . 6  |-  ( ( N : X --> RR  /\  ( G  e.  Grp  /\  D  e.  ( Met `  X ) ) )  ->  N  =  (
norm `  T )
)
718, 10eqtr3d 2510 . . . . . . . 8  |-  ( N  e.  _V  ->  ( Base `  G )  =  ( Base `  T
) )
7271, 12grpsubpropd 16012 . . . . . . 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 5173 . . . . 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 2510 . . . 4  |-  ( ( N : X --> RR  /\  ( G  e.  Grp  /\  D  e.  ( Met `  X ) ) )  ->  ( ( norm `  T )  o.  ( -g `  T ) )  =  ( dist `  T
) )
77 eqimss 3561 . . . 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 2467 . . . 4  |-  ( norm `  T )  =  (
norm `  T )
80 eqid 2467 . . . 4  |-  ( -g `  T )  =  (
-g `  T )
81 eqid 2467 . . . 4  |-  ( dist `  T )  =  (
dist `  T )
8279, 80, 81isngp 20984 . . 3  |-  ( T  e. NrmGrp 
<->  ( T  e.  Grp  /\  T  e.  MetSp  /\  (
( norm `  T )  o.  ( -g `  T
) )  C_  ( dist `  T ) ) )
8346, 67, 78, 82syl3anbrc 1180 . 2  |-  ( ( N : X --> RR  /\  ( G  e.  Grp  /\  D  e.  ( Met `  X ) ) )  ->  T  e. NrmGrp )
8444, 83impbida 830 1  |-  ( N : X --> RR  ->  ( T  e. NrmGrp  <->  ( G  e. 
Grp  /\  D  e.  ( Met `  X ) ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1379    e. wcel 1767   _Vcvv 3118    C_ wss 3481    X. cxp 5003    |` cres 5007    o. ccom 5009    Fn wfn 5589   -->wf 5590   ` cfv 5594  (class class class)co 6295   RRcr 9503   Basecbs 14507   +g cplusg 14572   distcds 14581   TopOpenctopn 14694   Grpcgrp 15925   -gcsg 15927   Metcme 18274   MetOpencmopn 18278   MetSpcmt 20689   normcnm 20965  NrmGrpcngp 20966   toNrmGrp ctng 20967
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4564  ax-sep 4574  ax-nul 4582  ax-pow 4631  ax-pr 4692  ax-un 6587  ax-cnex 9560  ax-resscn 9561  ax-1cn 9562  ax-icn 9563  ax-addcl 9564  ax-addrcl 9565  ax-mulcl 9566  ax-mulrcl 9567  ax-mulcom 9568  ax-addass 9569  ax-mulass 9570  ax-distr 9571  ax-i2m1 9572  ax-1ne0 9573  ax-1rid 9574  ax-rnegex 9575  ax-rrecex 9576  ax-cnre 9577  ax-pre-lttri 9578  ax-pre-lttrn 9579  ax-pre-ltadd 9580  ax-pre-mulgt0 9581  ax-pre-sup 9582
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2822  df-rex 2823  df-reu 2824  df-rmo 2825  df-rab 2826  df-v 3120  df-sbc 3337  df-csb 3441  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-pss 3497  df-nul 3791  df-if 3946  df-pw 4018  df-sn 4034  df-pr 4036  df-tp 4038  df-op 4040  df-uni 4252  df-iun 4333  df-br 4454  df-opab 4512  df-mpt 4513  df-tr 4547  df-eprel 4797  df-id 4801  df-po 4806  df-so 4807  df-fr 4844  df-we 4846  df-ord 4887  df-on 4888  df-lim 4889  df-suc 4890  df-xp 5011  df-rel 5012  df-cnv 5013  df-co 5014  df-dm 5015  df-rn 5016  df-res 5017  df-ima 5018  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-riota 6256  df-ov 6298  df-oprab 6299  df-mpt2 6300  df-om 6696  df-1st 6795  df-2nd 6796  df-recs 7054  df-rdg 7088  df-er 7323  df-map 7434  df-en 7529  df-dom 7530  df-sdom 7531  df-sup 7913  df-pnf 9642  df-mnf 9643  df-xr 9644  df-ltxr 9645  df-le 9646  df-sub 9819  df-neg 9820  df-div 10219  df-nn 10549  df-2 10606  df-3 10607  df-4 10608  df-5 10609  df-6 10610  df-7 10611  df-8 10612  df-9 10613  df-10 10614  df-n0 10808  df-z 10877  df-dec 10989  df-uz 11095  df-q 11195  df-rp 11233  df-xneg 11330  df-xadd 11331  df-xmul 11332  df-ndx 14510  df-slot 14511  df-base 14512  df-sets 14513  df-plusg 14585  df-tset 14591  df-ds 14594  df-rest 14695  df-topn 14696  df-0g 14714  df-topgen 14716  df-mgm 15746  df-sgrp 15785  df-mnd 15795  df-grp 15929  df-minusg 15930  df-sbg 15931  df-psmet 18281  df-xmet 18282  df-met 18283  df-bl 18284  df-mopn 18285  df-top 19268  df-bases 19270  df-topon 19271  df-topsp 19272  df-xms 20691  df-ms 20692  df-nm 20971  df-ngp 20972  df-tng 20973
This theorem is referenced by:  tngngpd  21035  tngngp  21036  tngnrg  21051
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