MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  isngp2 Structured version   Visualization version   Unicode version
Theorem isngp2 22401
Description: The property of being a normed group. (Contributed by Mario Carneiro, 2-Oct-2015.)
Hypotheses
Ref Expression
isngp.n |- N = ( norm ` G )
isngp.z |- .- = ( -g ` G )
isngp.d |- D = ( dist ` G )
isngp2.x |- X = ( Base ` G )
isngp2.e |- E = ( D |` ( X X. X ) )
Assertion
Ref Expression
isngp2 |- ( G e. NrmGrp <-> ( G e. Grp /\ G e. MetSp /\ ( N o. .- ) = E ) )
Proof of Theorem isngp2
StepHypRef Expression
1 isngp.n . . 3 |- N = ( norm ` G )
2 isngp.z . . 3 |- .- = ( -g ` G )
3 isngp.d . . 3 |- D = ( dist ` G )
41, 2, 3isngp 22400 . 2 |- ( G e. NrmGrp <-> ( G e. Grp /\ G e. MetSp /\ ( N o. .- ) C_ D ) )
5 isngp2.e . . . . . . 7 |- E = ( D |` ( X X. X ) )
6 resss 5422 . . . . . . 7 |- ( D |` ( X X. X ) ) C_ D
75, 6eqsstri 3635 . . . . . 6 |- E C_ D
8 sseq1 3626 . . . . . 6 |- ( ( N o. .- ) = E -> ( ( N o. .- ) C_ D <-> E C_ D ) )
97, 8mpbiri 248 . . . . 5 |- ( ( N o. .- ) = E -> ( N o. .- ) C_ D )
10 isngp2.x . . . . . . . . . . . . 13 |- X = ( Base ` G )
113reseq1i 5392 . . . . . . . . . . . . . 14 |- ( D |` ( X X. X ) ) = ( ( dist ` G ) |` ( X X. X ) )
125, 11eqtri 2644 . . . . . . . . . . . . 13 |- E = ( ( dist ` G ) |` ( X X. X ) )
1310, 12msmet 22262 . . . . . . . . . . . 12 |- ( G e. MetSp -> E e. ( Met ` X ) )
141, 10, 3, 5nmf2 22397 . . . . . . . . . . . 12 |- ( ( G e. Grp /\ E e. ( Met ` X ) ) -> N : X --> RR )
1513, 14sylan2 491 . . . . . . . . . . 11 |- ( ( G e. Grp /\ G e. MetSp ) -> N : X --> RR )
1615adantr 481 . . . . . . . . . 10 |- ( ( ( G e. Grp /\ G e. MetSp ) /\ ( N o. .- ) C_ D ) -> N : X --> RR )
1710, 2grpsubf 17494 . . . . . . . . . . 11 |- ( G e. Grp -> .- : ( X X. X ) --> X )
1817ad2antrr 762 . . . . . . . . . 10 |- ( ( ( G e. Grp /\ G e. MetSp ) /\ ( N o. .- ) C_ D ) -> .- : ( X X. X ) --> X )
19 fco 6058 . . . . . . . . . 10 |- ( ( N : X --> RR /\ .- : ( X X. X ) --> X ) -> ( N o. .- ) : ( X X. X ) --> RR )
2016, 18, 19syl2anc 693 . . . . . . . . 9 |- ( ( ( G e. Grp /\ G e. MetSp ) /\ ( N o. .- ) C_ D ) -> ( N o. .- ) : ( X X. X ) --> RR )
21 fdm 6051 . . . . . . . . 9 |- ( ( N o. .- ) : ( X X. X ) --> RR -> dom ( N o. .- ) = ( X X. X ) )
2220, 21syl 17 . . . . . . . 8 |- ( ( ( G e. Grp /\ G e. MetSp ) /\ ( N o. .- ) C_ D ) -> dom ( N o. .- ) = ( X X. X ) )
2322reseq2d 5396 . . . . . . 7 |- ( ( ( G e. Grp /\ G e. MetSp ) /\ ( N o. .- ) C_ D ) -> ( E |` dom ( N o. .- ) ) = ( E |` ( X X. X ) ) )
2410, 12msf 22263 . . . . . . . . . 10 |- ( G e. MetSp -> E : ( X X. X ) --> RR )
2524ad2antlr 763 . . . . . . . . 9 |- ( ( ( G e. Grp /\ G e. MetSp ) /\ ( N o. .- ) C_ D ) -> E : ( X X. X ) --> RR )
26 ffun 6048 . . . . . . . . 9 |- ( E : ( X X. X ) --> RR -> Fun E )
2725, 26syl 17 . . . . . . . 8 |- ( ( ( G e. Grp /\ G e. MetSp ) /\ ( N o. .- ) C_ D ) -> Fun E )
28 simpr 477 . . . . . . . . . 10 |- ( ( ( G e. Grp /\ G e. MetSp ) /\ ( N o. .- ) C_ D ) -> ( N o. .- ) C_ D )
29 ssv 3625 . . . . . . . . . . . 12 |- RR C_ _V
30 fss 6056 . . . . . . . . . . . 12 |- ( ( ( N o. .- ) : ( X X. X ) --> RR /\ RR C_ _V ) -> ( N o. .- ) : ( X X. X ) --> _V )
3120, 29, 30sylancl 694 . . . . . . . . . . 11 |- ( ( ( G e. Grp /\ G e. MetSp ) /\ ( N o. .- ) C_ D ) -> ( N o. .- ) : ( X X. X ) --> _V )
32 fssxp 6060 . . . . . . . . . . 11 |- ( ( N o. .- ) : ( X X. X ) --> _V -> ( N o. .- ) C_ ( ( X X. X ) X. _V ) )
3331, 32syl 17 . . . . . . . . . 10 |- ( ( ( G e. Grp /\ G e. MetSp ) /\ ( N o. .- ) C_ D ) -> ( N o. .- ) C_ ( ( X X. X ) X. _V ) )
3428, 33ssind 3837 . . . . . . . . 9 |- ( ( ( G e. Grp /\ G e. MetSp ) /\ ( N o. .- ) C_ D ) -> ( N o. .- ) C_ ( D i^i ( ( X X. X ) X. _V ) ) )
35 df-res 5126 . . . . . . . . . 10 |- ( D |` ( X X. X ) ) = ( D i^i ( ( X X. X ) X. _V ) )
365, 35eqtri 2644 . . . . . . . . 9 |- E = ( D i^i ( ( X X. X ) X. _V ) )
3734, 36syl6sseqr 3652 . . . . . . . 8 |- ( ( ( G e. Grp /\ G e. MetSp ) /\ ( N o. .- ) C_ D ) -> ( N o. .- ) C_ E )
38 funssres 5930 . . . . . . . 8 |- ( ( Fun E /\ ( N o. .- ) C_ E ) -> ( E |` dom ( N o. .- ) ) = ( N o. .- ) )
3927, 37, 38syl2anc 693 . . . . . . 7 |- ( ( ( G e. Grp /\ G e. MetSp ) /\ ( N o. .- ) C_ D ) -> ( E |` dom ( N o. .- ) ) = ( N o. .- ) )
40 ffn 6045 . . . . . . . 8 |- ( E : ( X X. X ) --> RR -> E Fn ( X X. X ) )
41 fnresdm 6000 . . . . . . . 8 |- ( E Fn ( X X. X ) -> ( E |` ( X X. X ) ) = E )
4225, 40, 413syl 18 . . . . . . 7 |- ( ( ( G e. Grp /\ G e. MetSp ) /\ ( N o. .- ) C_ D ) -> ( E |` ( X X. X ) ) = E )
4323, 39, 423eqtr3d 2664 . . . . . 6 |- ( ( ( G e. Grp /\ G e. MetSp ) /\ ( N o. .- ) C_ D ) -> ( N o. .- ) = E )
4443ex 450 . . . . 5 |- ( ( G e. Grp /\ G e. MetSp ) -> ( ( N o. .- ) C_ D -> ( N o. .- ) = E ) )
459, 44impbid2 216 . . . 4 |- ( ( G e. Grp /\ G e. MetSp ) -> ( ( N o. .- ) = E <-> ( N o. .- ) C_ D ) )
4645pm5.32i 669 . . 3 |- ( ( ( G e. Grp /\ G e. MetSp ) /\ ( N o. .- ) = E ) <-> ( ( G e. Grp /\ G e. MetSp ) /\ ( N o. .- ) C_ D ) )
47 df-3an 1039 . . 3 |- ( ( G e. Grp /\ G e. MetSp /\ ( N o. .- ) = E ) <-> ( ( G e. Grp /\ G e. MetSp ) /\ ( N o. .- ) = E ) )
48 df-3an 1039 . . 3 |- ( ( G e. Grp /\ G e. MetSp /\ ( N o. .- ) C_ D ) <-> ( ( G e. Grp /\ G e. MetSp ) /\ ( N o. .- ) C_ D ) )
4946, 47, 483bitr4i 292 . 2 |- ( ( G e. Grp /\ G e. MetSp /\ ( N o. .- ) = E ) <-> ( G e. Grp /\ G e. MetSp /\ ( N o. .- ) C_ D ) )
504, 49bitr4i 267 1 |- ( G e. NrmGrp <-> ( G e. Grp /\ G e. MetSp /\ ( N o. .- ) = E ) )
Colors of variables: wff setvar class
Syntax hints:   <-> wb 196   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   _Vcvv 3200   i^i cin 3573   C_ wss 3574   X. cxp 5112   dom cdm 5114   |` cres 5116   o. ccom 5118   Fun wfun 5882   Fn wfn 5883   -->wf 5884   ` cfv 5888   RRcr 9935   Basecbs 15857   distcds 15950   Grpcgrp 17422   -gcsg 17424   Metcme 19732   MetSpcmt 22123   normcnm 22381  NrmGrpcngp 22382
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013  ax-pre-sup 10014
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-er 7742  df-map 7859  df-en 7956  df-dom 7957  df-sdom 7958  df-sup 8348  df-inf 8349  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-div 10685  df-nn 11021  df-2 11079  df-n0 11293  df-z 11378  df-uz 11688  df-q 11789  df-rp 11833  df-xneg 11946  df-xadd 11947  df-xmul 11948  df-0g 16102  df-topgen 16104  df-mgm 17242  df-sgrp 17284  df-mnd 17295  df-grp 17425  df-minusg 17426  df-sbg 17427  df-psmet 19738  df-xmet 19739  df-met 19740  df-bl 19741  df-mopn 19742  df-top 20699  df-topon 20716  df-topsp 20737  df-bases 20750  df-xms 22125  df-ms 22126  df-nm 22387  df-ngp 22388
This theorem is referenced by:  isngp3  22402  ngpds  22408  ngppropd  22441  nrmtngdist  22461  nrmtngnrm  22462
  Copyright terms: Public domain W3C validator