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Theorem tngngp2 20079
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 20032 . . . . 5  |-  ( T  e. NrmGrp  ->  T  e.  Grp )
2 tngngp2.x . . . . . . . 8  |-  X  =  ( Base `  G
)
3 fvex 5689 . . . . . . . 8  |-  ( Base `  G )  e.  _V
42, 3eqeltri 2503 . . . . . . 7  |-  X  e. 
_V
5 reex 9360 . . . . . . 7  |-  RR  e.  _V
6 fex2 6521 . . . . . . 7  |-  ( ( N : X --> RR  /\  X  e.  _V  /\  RR  e.  _V )  ->  N  e.  _V )
74, 5, 6mp3an23 1299 . . . . . 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 20068 . . . . . . 7  |-  ( N  e.  _V  ->  X  =  ( Base `  T
) )
11 eqid 2433 . . . . . . . . 9  |-  ( +g  `  G )  =  ( +g  `  G )
129, 11tngplusg 20069 . . . . . . . 8  |-  ( N  e.  _V  ->  ( +g  `  G )  =  ( +g  `  T
) )
1312proplem3 14611 . . . . . . 7  |-  ( ( N  e.  _V  /\  ( x  e.  X  /\  y  e.  X
) )  ->  (
x ( +g  `  G
) y )  =  ( x ( +g  `  T ) y ) )
148, 10, 13grppropd 15535 . . . . . 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 20033 . . . . . 6  |-  ( T  e. NrmGrp  ->  T  e.  MetSp )
1918adantl 463 . . . . 5  |-  ( ( N : X --> RR  /\  T  e. NrmGrp )  ->  T  e.  MetSp )
20 eqid 2433 . . . . . 6  |-  ( Base `  T )  =  (
Base `  T )
21 tngngp2.d . . . . . 6  |-  D  =  ( dist `  T
)
2220, 21msmet2 19876 . . . . 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 2433 . . . . . . . . . 10  |-  ( -g `  G )  =  (
-g `  G )
252, 24grpsubf 15584 . . . . . . . . 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 5556 . . . . . . . 8  |-  ( ( N : X --> RR  /\  ( -g `  G ) : ( X  X.  X ) --> X )  ->  ( N  o.  ( -g `  G ) ) : ( X  X.  X ) --> RR )
2826, 27syldan 467 . . . . . . 7  |-  ( ( N : X --> RR  /\  T  e. NrmGrp )  ->  ( N  o.  ( -g `  G ) ) : ( X  X.  X
) --> RR )
297adantr 462 . . . . . . . . . 10  |-  ( ( N : X --> RR  /\  T  e. NrmGrp )  ->  N  e.  _V )
309, 24tngds 20075 . . . . . . . . . 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 2484 . . . . . . . 8  |-  ( ( N : X --> RR  /\  T  e. NrmGrp )  ->  D  =  ( N  o.  ( -g `  G ) ) )
3332feq1d 5534 . . . . . . 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 5547 . . . . . 6  |-  ( D : ( X  X.  X ) --> RR  ->  D  Fn  ( X  X.  X ) )
36 fnresdm 5508 . . . . . 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 4852 . . . . . 6  |-  ( ( N : X --> RR  /\  T  e. NrmGrp )  ->  ( X  X.  X )  =  ( ( Base `  T )  X.  ( Base `  T ) ) )
4039reseq2d 5097 . . . . 5  |-  ( ( N : X --> RR  /\  T  e. NrmGrp )  ->  ( D  |`  ( X  X.  X ) )  =  ( D  |`  (
( Base `  T )  X.  ( Base `  T
) ) ) )
4137, 40eqtr3d 2467 . . . 4  |-  ( ( N : X --> RR  /\  T  e. NrmGrp )  ->  D  =  ( D  |`  ( ( Base `  T
)  X.  ( Base `  T ) ) ) )
4238fveq2d 5683 . . . 4  |-  ( ( N : X --> RR  /\  T  e. NrmGrp )  ->  ( Met `  X )  =  ( Met `  ( Base `  T ) ) )
4323, 41, 423eltr4d 2514 . . 3  |-  ( ( N : X --> RR  /\  T  e. NrmGrp )  ->  D  e.  ( Met `  X
) )
4417, 43jca 529 . 2  |-  ( ( N : X --> RR  /\  T  e. NrmGrp )  ->  ( G  e.  Grp  /\  D  e.  ( Met `  X ) ) )
4515biimpa 481 . . . 4  |-  ( ( N : X --> RR  /\  G  e.  Grp )  ->  T  e.  Grp )
4645adantrr 709 . . 3  |-  ( ( N : X --> RR  /\  ( G  e.  Grp  /\  D  e.  ( Met `  X ) ) )  ->  T  e.  Grp )
47 simprr 749 . . . . . . . 8  |-  ( ( N : X --> RR  /\  ( G  e.  Grp  /\  D  e.  ( Met `  X ) ) )  ->  D  e.  ( Met `  X ) )
487adantr 462 . . . . . . . . . 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 5683 . . . . . . . 8  |-  ( ( N : X --> RR  /\  ( G  e.  Grp  /\  D  e.  ( Met `  X ) ) )  ->  ( Met `  X
)  =  ( Met `  ( Base `  T
) ) )
5147, 50eleqtrd 2509 . . . . . . 7  |-  ( ( N : X --> RR  /\  ( G  e.  Grp  /\  D  e.  ( Met `  X ) ) )  ->  D  e.  ( Met `  ( Base `  T ) ) )
52 metf 19746 . . . . . . 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 5547 . . . . . 6  |-  ( D : ( ( Base `  T )  X.  ( Base `  T ) ) --> RR  ->  D  Fn  ( ( Base `  T
)  X.  ( Base `  T ) ) )
55 fnresdm 5508 . . . . . 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 2507 . . . 4  |-  ( ( N : X --> RR  /\  ( G  e.  Grp  /\  D  e.  ( Met `  X ) ) )  ->  ( D  |`  ( ( Base `  T
)  X.  ( Base `  T ) ) )  e.  ( Met `  ( Base `  T ) ) )
5856fveq2d 5683 . . . . 5  |-  ( ( N : X --> RR  /\  ( G  e.  Grp  /\  D  e.  ( Met `  X ) ) )  ->  ( MetOpen `  ( D  |`  ( ( Base `  T )  X.  ( Base `  T ) ) ) )  =  (
MetOpen `  D ) )
59 simprl 748 . . . . . 6  |-  ( ( N : X --> RR  /\  ( G  e.  Grp  /\  D  e.  ( Met `  X ) ) )  ->  G  e.  Grp )
60 eqid 2433 . . . . . . 7  |-  ( MetOpen `  D )  =  (
MetOpen `  D )
619, 21, 60tngtopn 20077 . . . . . 6  |-  ( ( G  e.  Grp  /\  N  e.  _V )  ->  ( MetOpen `  D )  =  ( TopOpen `  T
) )
6259, 48, 61syl2anc 654 . . . . 5  |-  ( ( N : X --> RR  /\  ( G  e.  Grp  /\  D  e.  ( Met `  X ) ) )  ->  ( MetOpen `  D
)  =  ( TopOpen `  T ) )
6358, 62eqtr2d 2466 . . . 4  |-  ( ( N : X --> RR  /\  ( G  e.  Grp  /\  D  e.  ( Met `  X ) ) )  ->  ( TopOpen `  T
)  =  ( MetOpen `  ( D  |`  ( (
Base `  T )  X.  ( Base `  T
) ) ) ) )
64 eqid 2433 . . . . 5  |-  ( TopOpen `  T )  =  (
TopOpen `  T )
6521reseq1i 5093 . . . . 5  |-  ( D  |`  ( ( Base `  T
)  X.  ( Base `  T ) ) )  =  ( ( dist `  T )  |`  (
( Base `  T )  X.  ( Base `  T
) ) )
6664, 20, 65isms2 19866 . . . 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 657 . . 3  |-  ( ( N : X --> RR  /\  ( G  e.  Grp  /\  D  e.  ( Met `  X ) ) )  ->  T  e.  MetSp )
68 simpl 454 . . . . . . 7  |-  ( ( N : X --> RR  /\  ( G  e.  Grp  /\  D  e.  ( Met `  X ) ) )  ->  N : X --> RR )
699, 2, 5tngnm 20078 . . . . . . 7  |-  ( ( G  e.  Grp  /\  N : X --> RR )  ->  N  =  (
norm `  T )
)
7059, 68, 69syl2anc 654 . . . . . 6  |-  ( ( N : X --> RR  /\  ( G  e.  Grp  /\  D  e.  ( Met `  X ) ) )  ->  N  =  (
norm `  T )
)
718, 10eqtr3d 2467 . . . . . . . 8  |-  ( N  e.  _V  ->  ( Base `  G )  =  ( Base `  T
) )
7271, 12grpsubpropd 15605 . . . . . . 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 4991 . . . . 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 2467 . . . 4  |-  ( ( N : X --> RR  /\  ( G  e.  Grp  /\  D  e.  ( Met `  X ) ) )  ->  ( ( norm `  T )  o.  ( -g `  T ) )  =  ( dist `  T
) )
77 eqimss 3396 . . . 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 2433 . . . 4  |-  ( norm `  T )  =  (
norm `  T )
80 eqid 2433 . . . 4  |-  ( -g `  T )  =  (
-g `  T )
81 eqid 2433 . . . 4  |-  ( dist `  T )  =  (
dist `  T )
8279, 80, 81isngp 20029 . . 3  |-  ( T  e. NrmGrp 
<->  ( T  e.  Grp  /\  T  e.  MetSp  /\  (
( norm `  T )  o.  ( -g `  T
) )  C_  ( dist `  T ) ) )
8346, 67, 78, 82syl3anbrc 1165 . 2  |-  ( ( N : X --> RR  /\  ( G  e.  Grp  /\  D  e.  ( Met `  X ) ) )  ->  T  e. NrmGrp )
8444, 83impbida 821 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 1362    e. wcel 1755   _Vcvv 2962    C_ wss 3316    X. cxp 4825    |` cres 4829    o. ccom 4831    Fn wfn 5401   -->wf 5402   ` cfv 5406  (class class class)co 6080   RRcr 9268   Basecbs 14156   +g cplusg 14220   distcds 14229   TopOpenctopn 14342   Grpcgrp 15392   -gcsg 15395   Metcme 17645   MetOpencmopn 17649   MetSpcmt 19734   normcnm 20010  NrmGrpcngp 20011   toNrmGrp ctng 20012
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1594  ax-4 1605  ax-5 1669  ax-6 1707  ax-7 1727  ax-8 1757  ax-9 1759  ax-10 1774  ax-11 1779  ax-12 1791  ax-13 1942  ax-ext 2414  ax-rep 4391  ax-sep 4401  ax-nul 4409  ax-pow 4458  ax-pr 4519  ax-un 6361  ax-cnex 9325  ax-resscn 9326  ax-1cn 9327  ax-icn 9328  ax-addcl 9329  ax-addrcl 9330  ax-mulcl 9331  ax-mulrcl 9332  ax-mulcom 9333  ax-addass 9334  ax-mulass 9335  ax-distr 9336  ax-i2m1 9337  ax-1ne0 9338  ax-1rid 9339  ax-rnegex 9340  ax-rrecex 9341  ax-cnre 9342  ax-pre-lttri 9343  ax-pre-lttrn 9344  ax-pre-ltadd 9345  ax-pre-mulgt0 9346  ax-pre-sup 9347
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 959  df-3an 960  df-tru 1365  df-ex 1590  df-nf 1593  df-sb 1700  df-eu 2258  df-mo 2259  df-clab 2420  df-cleq 2426  df-clel 2429  df-nfc 2558  df-ne 2598  df-nel 2599  df-ral 2710  df-rex 2711  df-reu 2712  df-rmo 2713  df-rab 2714  df-v 2964  df-sbc 3176  df-csb 3277  df-dif 3319  df-un 3321  df-in 3323  df-ss 3330  df-pss 3332  df-nul 3626  df-if 3780  df-pw 3850  df-sn 3866  df-pr 3868  df-tp 3870  df-op 3872  df-uni 4080  df-iun 4161  df-br 4281  df-opab 4339  df-mpt 4340  df-tr 4374  df-eprel 4619  df-id 4623  df-po 4628  df-so 4629  df-fr 4666  df-we 4668  df-ord 4709  df-on 4710  df-lim 4711  df-suc 4712  df-xp 4833  df-rel 4834  df-cnv 4835  df-co 4836  df-dm 4837  df-rn 4838  df-res 4839  df-ima 4840  df-iota 5369  df-fun 5408  df-fn 5409  df-f 5410  df-f1 5411  df-fo 5412  df-f1o 5413  df-fv 5414  df-riota 6039  df-ov 6083  df-oprab 6084  df-mpt2 6085  df-om 6466  df-1st 6566  df-2nd 6567  df-recs 6818  df-rdg 6852  df-er 7089  df-map 7204  df-en 7299  df-dom 7300  df-sdom 7301  df-sup 7679  df-pnf 9407  df-mnf 9408  df-xr 9409  df-ltxr 9410  df-le 9411  df-sub 9584  df-neg 9585  df-div 9981  df-nn 10310  df-2 10367  df-3 10368  df-4 10369  df-5 10370  df-6 10371  df-7 10372  df-8 10373  df-9 10374  df-10 10375  df-n0 10567  df-z 10634  df-dec 10743  df-uz 10849  df-q 10941  df-rp 10979  df-xneg 11076  df-xadd 11077  df-xmul 11078  df-ndx 14159  df-slot 14160  df-base 14161  df-sets 14162  df-plusg 14233  df-tset 14239  df-ds 14242  df-rest 14343  df-topn 14344  df-0g 14362  df-topgen 14364  df-mnd 15397  df-grp 15524  df-minusg 15525  df-sbg 15526  df-psmet 17652  df-xmet 17653  df-met 17654  df-bl 17655  df-mopn 17656  df-top 18344  df-bases 18346  df-topon 18347  df-topsp 18348  df-xms 19736  df-ms 19737  df-nm 20016  df-ngp 20017  df-tng 20018
This theorem is referenced by:  tngngpd  20080  tngngp  20081  tngnrg  20096
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