Theorem isngp3 22402

Description: The property of being a normed group. (Contributed by Mario Carneiro, 4-Oct-2015.)

Hypotheses

Ref	Expression
isngp.n	|- N = ( norm ` G )
isngp.z	|- .- = ( -g ` G )
isngp.d	|- D = ( dist ` G )
isngp2.x	|- X = ( Base ` G )

Assertion

Ref	Expression
isngp3	|- ( G e. NrmGrp <-> ( G e. Grp /\ G e. MetSp /\ A. x e. X A. y e. X ( x D y ) = ( N ` ( x .- y ) ) ) )

Distinct variable groups:   x, y, D   x, G, y   x, .- , y   x, N, y   x, X, y

Proof of Theorem isngp3

Step	Hyp	Ref	Expression
1		isngp.n	. . 3 |- N = ( norm ` G )
2		isngp.z	. . 3 |- .- = ( -g ` G )
3		isngp.d	. . 3 |- D = ( dist ` G )
4		isngp2.x	. . 3 |- X = ( Base ` G )
5		eqid 2622	. . 3 |- ( D |` ( X X. X ) ) = ( D |` ( X X. X ) )
6	1, 2, 3, 4, 5	isngp2 22401	. 2 |- ( G e. NrmGrp <-> ( G e. Grp /\ G e. MetSp /\ ( N o. .- ) = ( D |` ( X X. X ) ) ) )
7	4, 3	msmet2 22265	. . . . . . . . 9 |- ( G e. MetSp -> ( D |` ( X X. X ) ) e. ( Met ` X ) )
8	1, 4, 3, 5	nmf2 22397	. . . . . . . . 9 |- ( ( G e. Grp /\ ( D |` ( X X. X ) ) e. ( Met ` X ) ) -> N : X --> RR )
9	7, 8	sylan2 491	. . . . . . . 8 |- ( ( G e. Grp /\ G e. MetSp ) -> N : X --> RR )
10	4, 2	grpsubf 17494	. . . . . . . . 9 |- ( G e. Grp -> .- : ( X X. X ) --> X )
11	10	adantr 481	. . . . . . . 8 |- ( ( G e. Grp /\ G e. MetSp ) -> .- : ( X X. X ) --> X )
12		fco 6058	. . . . . . . 8 |- ( ( N : X --> RR /\ .- : ( X X. X ) --> X ) -> ( N o. .- ) : ( X X. X ) --> RR )
13	9, 11, 12	syl2anc 693	. . . . . . 7 |- ( ( G e. Grp /\ G e. MetSp ) -> ( N o. .- ) : ( X X. X ) --> RR )
14		ffn 6045	. . . . . . 7 |- ( ( N o. .- ) : ( X X. X ) --> RR -> ( N o. .- ) Fn ( X X. X ) )
15	13, 14	syl 17	. . . . . 6 |- ( ( G e. Grp /\ G e. MetSp ) -> ( N o. .- ) Fn ( X X. X ) )
16	7	adantl 482	. . . . . . 7 |- ( ( G e. Grp /\ G e. MetSp ) -> ( D |` ( X X. X ) ) e. ( Met ` X ) )
17		metf 22135	. . . . . . 7 |- ( ( D |` ( X X. X ) ) e. ( Met ` X ) -> ( D |` ( X X. X ) ) : ( X X. X ) --> RR )
18		ffn 6045	. . . . . . 7 |- ( ( D |` ( X X. X ) ) : ( X X. X ) --> RR -> ( D |` ( X X. X ) ) Fn ( X X. X ) )
19	16, 17, 18	3syl 18	. . . . . 6 |- ( ( G e. Grp /\ G e. MetSp ) -> ( D |` ( X X. X ) ) Fn ( X X. X ) )
20		eqfnov2 6767	. . . . . 6 |- ( ( ( N o. .- ) Fn ( X X. X ) /\ ( D |` ( X X. X ) ) Fn ( X X. X ) ) -> ( ( N o. .- ) = ( D |` ( X X. X ) ) <-> A. x e. X A. y e. X ( x ( N o. .- ) y ) = ( x ( D |` ( X X. X ) ) y ) ) )
21	15, 19, 20	syl2anc 693	. . . . 5 |- ( ( G e. Grp /\ G e. MetSp ) -> ( ( N o. .- ) = ( D |` ( X X. X ) ) <-> A. x e. X A. y e. X ( x ( N o. .- ) y ) = ( x ( D |` ( X X. X ) ) y ) ) )
22		opelxpi 5148	. . . . . . . . . 10 |- ( ( x e. X /\ y e. X ) -> <. x , y >. e. ( X X. X ) )
23		fvco3 6275	. . . . . . . . . 10 |- ( ( .- : ( X X. X ) --> X /\ <. x , y >. e. ( X X. X ) ) -> ( ( N o. .- ) ` <. x , y >. ) = ( N ` ( .- ` <. x , y >. ) ) )
24	11, 22, 23	syl2an 494	. . . . . . . . 9 |- ( ( ( G e. Grp /\ G e. MetSp ) /\ ( x e. X /\ y e. X ) ) -> ( ( N o. .- ) ` <. x , y >. ) = ( N ` ( .- ` <. x , y >. ) ) )
25		df-ov 6653	. . . . . . . . 9 |- ( x ( N o. .- ) y ) = ( ( N o. .- ) ` <. x , y >. )
26		df-ov 6653	. . . . . . . . . 10 |- ( x .- y ) = ( .- ` <. x , y >. )
27	26	fveq2i 6194	. . . . . . . . 9 |- ( N ` ( x .- y ) ) = ( N ` ( .- ` <. x , y >. ) )
28	24, 25, 27	3eqtr4g 2681	. . . . . . . 8 |- ( ( ( G e. Grp /\ G e. MetSp ) /\ ( x e. X /\ y e. X ) ) -> ( x ( N o. .- ) y ) = ( N ` ( x .- y ) ) )
29		ovres 6800	. . . . . . . . 9 |- ( ( x e. X /\ y e. X ) -> ( x ( D |` ( X X. X ) ) y ) = ( x D y ) )
30	29	adantl 482	. . . . . . . 8 |- ( ( ( G e. Grp /\ G e. MetSp ) /\ ( x e. X /\ y e. X ) ) -> ( x ( D |` ( X X. X ) ) y ) = ( x D y ) )
31	28, 30	eqeq12d 2637	. . . . . . 7 |- ( ( ( G e. Grp /\ G e. MetSp ) /\ ( x e. X /\ y e. X ) ) -> ( ( x ( N o. .- ) y ) = ( x ( D |` ( X X. X ) ) y ) <-> ( N ` ( x .- y ) ) = ( x D y ) ) )
32		eqcom 2629	. . . . . . 7 |- ( ( N ` ( x .- y ) ) = ( x D y ) <-> ( x D y ) = ( N ` ( x .- y ) ) )
33	31, 32	syl6bb 276	. . . . . 6 |- ( ( ( G e. Grp /\ G e. MetSp ) /\ ( x e. X /\ y e. X ) ) -> ( ( x ( N o. .- ) y ) = ( x ( D |` ( X X. X ) ) y ) <-> ( x D y ) = ( N ` ( x .- y ) ) ) )
34	33	2ralbidva 2988	. . . . 5 |- ( ( G e. Grp /\ G e. MetSp ) -> ( A. x e. X A. y e. X ( x ( N o. .- ) y ) = ( x ( D |` ( X X. X ) ) y ) <-> A. x e. X A. y e. X ( x D y ) = ( N ` ( x .- y ) ) ) )
35	21, 34	bitrd 268	. . . 4 |- ( ( G e. Grp /\ G e. MetSp ) -> ( ( N o. .- ) = ( D |` ( X X. X ) ) <-> A. x e. X A. y e. X ( x D y ) = ( N ` ( x .- y ) ) ) )
36	35	pm5.32i 669	. . 3 |- ( ( ( G e. Grp /\ G e. MetSp ) /\ ( N o. .- ) = ( D |` ( X X. X ) ) ) <-> ( ( G e. Grp /\ G e. MetSp ) /\ A. x e. X A. y e. X ( x D y ) = ( N ` ( x .- y ) ) ) )
37		df-3an 1039	. . 3 |- ( ( G e. Grp /\ G e. MetSp /\ ( N o. .- ) = ( D |` ( X X. X ) ) ) <-> ( ( G e. Grp /\ G e. MetSp ) /\ ( N o. .- ) = ( D |` ( X X. X ) ) ) )
38		df-3an 1039	. . 3 |- ( ( G e. Grp /\ G e. MetSp /\ A. x e. X A. y e. X ( x D y ) = ( N ` ( x .- y ) ) ) <-> ( ( G e. Grp /\ G e. MetSp ) /\ A. x e. X A. y e. X ( x D y ) = ( N ` ( x .- y ) ) ) )
39	36, 37, 38	3bitr4i 292	. 2 |- ( ( G e. Grp /\ G e. MetSp /\ ( N o. .- ) = ( D |` ( X X. X ) ) ) <-> ( G e. Grp /\ G e. MetSp /\ A. x e. X A. y e. X ( x D y ) = ( N ` ( x .- y ) ) ) )
40	6, 39	bitri 264	1 |- ( G e. NrmGrp <-> ( G e. Grp /\ G e. MetSp /\ A. x e. X A. y e. X ( x D y ) = ( N ` ( x .- y ) ) ) )

Syntax hints:   <-> wb 196   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   A.wral 2912   <.cop 4183   X. cxp 5112   |` cres 5116   o. ccom 5118   Fn wfn 5883   -->wf 5884   ` cfv 5888  (class class class)co 6650   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:  isngp4  22416  subgngp  22439