Theorem nrgdsdir 22470

Description: Distribute a distance calculation. (Contributed by Mario Carneiro, 5-Oct-2015.)

Hypotheses

Ref	Expression
nmmul.x	|- X = ( Base ` R )
nmmul.n	|- N = ( norm ` R )
nmmul.t	|- .x. = ( .r ` R )
nrgdsdi.d	|- D = ( dist ` R )

Assertion

Ref	Expression
nrgdsdir	|- ( ( R e. NrmRing /\ ( A e. X /\ B e. X /\ C e. X ) ) -> ( ( A D B ) x. ( N ` C ) ) = ( ( A .x. C ) D ( B .x. C ) ) )

Proof of Theorem nrgdsdir

Step	Hyp	Ref	Expression
1		simpl 473	. . . 4 |- ( ( R e. NrmRing /\ ( A e. X /\ B e. X /\ C e. X ) ) -> R e. NrmRing )
2		nrgring 22467	. . . . . . 7 |- ( R e. NrmRing -> R e. Ring )
3	2	adantr 481	. . . . . 6 |- ( ( R e. NrmRing /\ ( A e. X /\ B e. X /\ C e. X ) ) -> R e. Ring )
4		ringgrp 18552	. . . . . 6 |- ( R e. Ring -> R e. Grp )
5	3, 4	syl 17	. . . . 5 |- ( ( R e. NrmRing /\ ( A e. X /\ B e. X /\ C e. X ) ) -> R e. Grp )
6		simpr1 1067	. . . . 5 |- ( ( R e. NrmRing /\ ( A e. X /\ B e. X /\ C e. X ) ) -> A e. X )
7		simpr2 1068	. . . . 5 |- ( ( R e. NrmRing /\ ( A e. X /\ B e. X /\ C e. X ) ) -> B e. X )
8		nmmul.x	. . . . . 6 |- X = ( Base ` R )
9		eqid 2622	. . . . . 6 |- ( -g ` R ) = ( -g ` R )
10	8, 9	grpsubcl 17495	. . . . 5 |- ( ( R e. Grp /\ A e. X /\ B e. X ) -> ( A ( -g ` R ) B ) e. X )
11	5, 6, 7, 10	syl3anc 1326	. . . 4 |- ( ( R e. NrmRing /\ ( A e. X /\ B e. X /\ C e. X ) ) -> ( A ( -g ` R ) B ) e. X )
12		simpr3 1069	. . . 4 |- ( ( R e. NrmRing /\ ( A e. X /\ B e. X /\ C e. X ) ) -> C e. X )
13		nmmul.n	. . . . 5 |- N = ( norm ` R )
14		nmmul.t	. . . . 5 |- .x. = ( .r ` R )
15	8, 13, 14	nmmul 22468	. . . 4 |- ( ( R e. NrmRing /\ ( A ( -g ` R ) B ) e. X /\ C e. X ) -> ( N ` ( ( A ( -g ` R ) B ) .x. C ) ) = ( ( N ` ( A ( -g ` R ) B ) ) x. ( N ` C ) ) )
16	1, 11, 12, 15	syl3anc 1326	. . 3 |- ( ( R e. NrmRing /\ ( A e. X /\ B e. X /\ C e. X ) ) -> ( N ` ( ( A ( -g ` R ) B ) .x. C ) ) = ( ( N ` ( A ( -g ` R ) B ) ) x. ( N ` C ) ) )
17	8, 14, 9, 3, 6, 7, 12	rngsubdir 18600	. . . 4 |- ( ( R e. NrmRing /\ ( A e. X /\ B e. X /\ C e. X ) ) -> ( ( A ( -g ` R ) B ) .x. C ) = ( ( A .x. C ) ( -g ` R ) ( B .x. C ) ) )
18	17	fveq2d 6195	. . 3 |- ( ( R e. NrmRing /\ ( A e. X /\ B e. X /\ C e. X ) ) -> ( N ` ( ( A ( -g ` R ) B ) .x. C ) ) = ( N ` ( ( A .x. C ) ( -g ` R ) ( B .x. C ) ) ) )
19	16, 18	eqtr3d 2658	. 2 |- ( ( R e. NrmRing /\ ( A e. X /\ B e. X /\ C e. X ) ) -> ( ( N ` ( A ( -g ` R ) B ) ) x. ( N ` C ) ) = ( N ` ( ( A .x. C ) ( -g ` R ) ( B .x. C ) ) ) )
20		nrgngp 22466	. . . . 5 |- ( R e. NrmRing -> R e. NrmGrp )
21	20	adantr 481	. . . 4 |- ( ( R e. NrmRing /\ ( A e. X /\ B e. X /\ C e. X ) ) -> R e. NrmGrp )
22		nrgdsdi.d	. . . . 5 |- D = ( dist ` R )
23	13, 8, 9, 22	ngpds 22408	. . . 4 |- ( ( R e. NrmGrp /\ A e. X /\ B e. X ) -> ( A D B ) = ( N ` ( A ( -g ` R ) B ) ) )
24	21, 6, 7, 23	syl3anc 1326	. . 3 |- ( ( R e. NrmRing /\ ( A e. X /\ B e. X /\ C e. X ) ) -> ( A D B ) = ( N ` ( A ( -g ` R ) B ) ) )
25	24	oveq1d 6665	. 2 |- ( ( R e. NrmRing /\ ( A e. X /\ B e. X /\ C e. X ) ) -> ( ( A D B ) x. ( N ` C ) ) = ( ( N ` ( A ( -g ` R ) B ) ) x. ( N ` C ) ) )
26	8, 14	ringcl 18561	. . . 4 |- ( ( R e. Ring /\ A e. X /\ C e. X ) -> ( A .x. C ) e. X )
27	3, 6, 12, 26	syl3anc 1326	. . 3 |- ( ( R e. NrmRing /\ ( A e. X /\ B e. X /\ C e. X ) ) -> ( A .x. C ) e. X )
28	8, 14	ringcl 18561	. . . 4 |- ( ( R e. Ring /\ B e. X /\ C e. X ) -> ( B .x. C ) e. X )
29	3, 7, 12, 28	syl3anc 1326	. . 3 |- ( ( R e. NrmRing /\ ( A e. X /\ B e. X /\ C e. X ) ) -> ( B .x. C ) e. X )
30	13, 8, 9, 22	ngpds 22408	. . 3 |- ( ( R e. NrmGrp /\ ( A .x. C ) e. X /\ ( B .x. C ) e. X ) -> ( ( A .x. C ) D ( B .x. C ) ) = ( N ` ( ( A .x. C ) ( -g ` R ) ( B .x. C ) ) ) )
31	21, 27, 29, 30	syl3anc 1326	. 2 |- ( ( R e. NrmRing /\ ( A e. X /\ B e. X /\ C e. X ) ) -> ( ( A .x. C ) D ( B .x. C ) ) = ( N ` ( ( A .x. C ) ( -g ` R ) ( B .x. C ) ) ) )
32	19, 25, 31	3eqtr4d 2666	1 |- ( ( R e. NrmRing /\ ( A e. X /\ B e. X /\ C e. X ) ) -> ( ( A D B ) x. ( N ` C ) ) = ( ( A .x. C ) D ( B .x. C ) ) )

Syntax hints:   -> wi 4   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   ` cfv 5888  (class class class)co 6650   x. cmul 9941   Basecbs 15857   .rcmulr 15942   distcds 15950   Grpcgrp 17422   -gcsg 17424   Ringcrg 18547   normcnm 22381  NrmGrpcngp 22382  NrmRingcnrg 22384

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-ndx 15860  df-slot 15861  df-base 15863  df-sets 15864  df-plusg 15954  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-mgp 18490  df-ur 18502  df-ring 18549  df-abv 18817  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  df-nrg 22390

This theorem is referenced by: (None)