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Theorem tngngp2 20260
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 20213 . . . . 5  |-  ( T  e. NrmGrp  ->  T  e.  Grp )
2 tngngp2.x . . . . . . . 8  |-  X  =  ( Base `  G
)
3 fvex 5722 . . . . . . . 8  |-  ( Base `  G )  e.  _V
42, 3eqeltri 2513 . . . . . . 7  |-  X  e. 
_V
5 reex 9394 . . . . . . 7  |-  RR  e.  _V
6 fex2 6553 . . . . . . 7  |-  ( ( N : X --> RR  /\  X  e.  _V  /\  RR  e.  _V )  ->  N  e.  _V )
74, 5, 6mp3an23 1306 . . . . . 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 20249 . . . . . . 7  |-  ( N  e.  _V  ->  X  =  ( Base `  T
) )
11 eqid 2443 . . . . . . . . 9  |-  ( +g  `  G )  =  ( +g  `  G )
129, 11tngplusg 20250 . . . . . . . 8  |-  ( N  e.  _V  ->  ( +g  `  G )  =  ( +g  `  T
) )
1312proplem3 14650 . . . . . . 7  |-  ( ( N  e.  _V  /\  ( x  e.  X  /\  y  e.  X
) )  ->  (
x ( +g  `  G
) y )  =  ( x ( +g  `  T ) y ) )
148, 10, 13grppropd 15577 . . . . . 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 20214 . . . . . 6  |-  ( T  e. NrmGrp  ->  T  e.  MetSp )
1918adantl 466 . . . . 5  |-  ( ( N : X --> RR  /\  T  e. NrmGrp )  ->  T  e.  MetSp )
20 eqid 2443 . . . . . 6  |-  ( Base `  T )  =  (
Base `  T )
21 tngngp2.d . . . . . 6  |-  D  =  ( dist `  T
)
2220, 21msmet2 20057 . . . . 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 2443 . . . . . . . . . 10  |-  ( -g `  G )  =  (
-g `  G )
252, 24grpsubf 15626 . . . . . . . . 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 5589 . . . . . . . 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 20256 . . . . . . . . . 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 2494 . . . . . . . 8  |-  ( ( N : X --> RR  /\  T  e. NrmGrp )  ->  D  =  ( N  o.  ( -g `  G ) ) )
3332feq1d 5567 . . . . . . 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 5580 . . . . . 6  |-  ( D : ( X  X.  X ) --> RR  ->  D  Fn  ( X  X.  X ) )
36 fnresdm 5541 . . . . . 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 4886 . . . . . 6  |-  ( ( N : X --> RR  /\  T  e. NrmGrp )  ->  ( X  X.  X )  =  ( ( Base `  T )  X.  ( Base `  T ) ) )
4039reseq2d 5131 . . . . 5  |-  ( ( N : X --> RR  /\  T  e. NrmGrp )  ->  ( D  |`  ( X  X.  X ) )  =  ( D  |`  (
( Base `  T )  X.  ( Base `  T
) ) ) )
4137, 40eqtr3d 2477 . . . 4  |-  ( ( N : X --> RR  /\  T  e. NrmGrp )  ->  D  =  ( D  |`  ( ( Base `  T
)  X.  ( Base `  T ) ) ) )
4238fveq2d 5716 . . . 4  |-  ( ( N : X --> RR  /\  T  e. NrmGrp )  ->  ( Met `  X )  =  ( Met `  ( Base `  T ) ) )
4323, 41, 423eltr4d 2524 . . 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 5716 . . . . . . . 8  |-  ( ( N : X --> RR  /\  ( G  e.  Grp  /\  D  e.  ( Met `  X ) ) )  ->  ( Met `  X
)  =  ( Met `  ( Base `  T
) ) )
5147, 50eleqtrd 2519 . . . . . . 7  |-  ( ( N : X --> RR  /\  ( G  e.  Grp  /\  D  e.  ( Met `  X ) ) )  ->  D  e.  ( Met `  ( Base `  T ) ) )
52 metf 19927 . . . . . . 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 5580 . . . . . 6  |-  ( D : ( ( Base `  T )  X.  ( Base `  T ) ) --> RR  ->  D  Fn  ( ( Base `  T
)  X.  ( Base `  T ) ) )
55 fnresdm 5541 . . . . . 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 2517 . . . 4  |-  ( ( N : X --> RR  /\  ( G  e.  Grp  /\  D  e.  ( Met `  X ) ) )  ->  ( D  |`  ( ( Base `  T
)  X.  ( Base `  T ) ) )  e.  ( Met `  ( Base `  T ) ) )
5856fveq2d 5716 . . . . 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 2443 . . . . . . 7  |-  ( MetOpen `  D )  =  (
MetOpen `  D )
619, 21, 60tngtopn 20258 . . . . . 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 2476 . . . 4  |-  ( ( N : X --> RR  /\  ( G  e.  Grp  /\  D  e.  ( Met `  X ) ) )  ->  ( TopOpen `  T
)  =  ( MetOpen `  ( D  |`  ( (
Base `  T )  X.  ( Base `  T
) ) ) ) )
64 eqid 2443 . . . . 5  |-  ( TopOpen `  T )  =  (
TopOpen `  T )
6521reseq1i 5127 . . . . 5  |-  ( D  |`  ( ( Base `  T
)  X.  ( Base `  T ) ) )  =  ( ( dist `  T )  |`  (
( Base `  T )  X.  ( Base `  T
) ) )
6664, 20, 65isms2 20047 . . . 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 20259 . . . . . . 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 2477 . . . . . . . 8  |-  ( N  e.  _V  ->  ( Base `  G )  =  ( Base `  T
) )
7271, 12grpsubpropd 15647 . . . . . . 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 5025 . . . . 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 2477 . . . 4  |-  ( ( N : X --> RR  /\  ( G  e.  Grp  /\  D  e.  ( Met `  X ) ) )  ->  ( ( norm `  T )  o.  ( -g `  T ) )  =  ( dist `  T
) )
77 eqimss 3429 . . . 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 2443 . . . 4  |-  ( norm `  T )  =  (
norm `  T )
80 eqid 2443 . . . 4  |-  ( -g `  T )  =  (
-g `  T )
81 eqid 2443 . . . 4  |-  ( dist `  T )  =  (
dist `  T )
8279, 80, 81isngp 20210 . . 3  |-  ( T  e. NrmGrp 
<->  ( T  e.  Grp  /\  T  e.  MetSp  /\  (
( norm `  T )  o.  ( -g `  T
) )  C_  ( dist `  T ) ) )
8346, 67, 78, 82syl3anbrc 1172 . 2  |-  ( ( N : X --> RR  /\  ( G  e.  Grp  /\  D  e.  ( Met `  X ) ) )  ->  T  e. NrmGrp )
8444, 83impbida 828 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 1369    e. wcel 1756   _Vcvv 2993    C_ wss 3349    X. cxp 4859    |` cres 4863    o. ccom 4865    Fn wfn 5434   -->wf 5435   ` cfv 5439  (class class class)co 6112   RRcr 9302   Basecbs 14195   +g cplusg 14259   distcds 14268   TopOpenctopn 14381   Grpcgrp 15431   -gcsg 15434   Metcme 17824   MetOpencmopn 17828   MetSpcmt 19915   normcnm 20191  NrmGrpcngp 20192   toNrmGrp ctng 20193
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4424  ax-sep 4434  ax-nul 4442  ax-pow 4491  ax-pr 4552  ax-un 6393  ax-cnex 9359  ax-resscn 9360  ax-1cn 9361  ax-icn 9362  ax-addcl 9363  ax-addrcl 9364  ax-mulcl 9365  ax-mulrcl 9366  ax-mulcom 9367  ax-addass 9368  ax-mulass 9369  ax-distr 9370  ax-i2m1 9371  ax-1ne0 9372  ax-1rid 9373  ax-rnegex 9374  ax-rrecex 9375  ax-cnre 9376  ax-pre-lttri 9377  ax-pre-lttrn 9378  ax-pre-ltadd 9379  ax-pre-mulgt0 9380  ax-pre-sup 9381
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2577  df-ne 2622  df-nel 2623  df-ral 2741  df-rex 2742  df-reu 2743  df-rmo 2744  df-rab 2745  df-v 2995  df-sbc 3208  df-csb 3310  df-dif 3352  df-un 3354  df-in 3356  df-ss 3363  df-pss 3365  df-nul 3659  df-if 3813  df-pw 3883  df-sn 3899  df-pr 3901  df-tp 3903  df-op 3905  df-uni 4113  df-iun 4194  df-br 4314  df-opab 4372  df-mpt 4373  df-tr 4407  df-eprel 4653  df-id 4657  df-po 4662  df-so 4663  df-fr 4700  df-we 4702  df-ord 4743  df-on 4744  df-lim 4745  df-suc 4746  df-xp 4867  df-rel 4868  df-cnv 4869  df-co 4870  df-dm 4871  df-rn 4872  df-res 4873  df-ima 4874  df-iota 5402  df-fun 5441  df-fn 5442  df-f 5443  df-f1 5444  df-fo 5445  df-f1o 5446  df-fv 5447  df-riota 6073  df-ov 6115  df-oprab 6116  df-mpt2 6117  df-om 6498  df-1st 6598  df-2nd 6599  df-recs 6853  df-rdg 6887  df-er 7122  df-map 7237  df-en 7332  df-dom 7333  df-sdom 7334  df-sup 7712  df-pnf 9441  df-mnf 9442  df-xr 9443  df-ltxr 9444  df-le 9445  df-sub 9618  df-neg 9619  df-div 10015  df-nn 10344  df-2 10401  df-3 10402  df-4 10403  df-5 10404  df-6 10405  df-7 10406  df-8 10407  df-9 10408  df-10 10409  df-n0 10601  df-z 10668  df-dec 10777  df-uz 10883  df-q 10975  df-rp 11013  df-xneg 11110  df-xadd 11111  df-xmul 11112  df-ndx 14198  df-slot 14199  df-base 14200  df-sets 14201  df-plusg 14272  df-tset 14278  df-ds 14281  df-rest 14382  df-topn 14383  df-0g 14401  df-topgen 14403  df-mnd 15436  df-grp 15566  df-minusg 15567  df-sbg 15568  df-psmet 17831  df-xmet 17832  df-met 17833  df-bl 17834  df-mopn 17835  df-top 18525  df-bases 18527  df-topon 18528  df-topsp 18529  df-xms 19917  df-ms 19918  df-nm 20197  df-ngp 20198  df-tng 20199
This theorem is referenced by:  tngngpd  20261  tngngp  20262  tngnrg  20277
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