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Theorem nrgdsdi 20221
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
nrgdsdi  |-  ( ( R  e. NrmRing  /\  ( A  e.  X  /\  B  e.  X  /\  C  e.  X )
)  ->  ( ( N `  A )  x.  ( B D C ) )  =  ( ( A  .x.  B
) D ( A 
.x.  C ) ) )

Proof of Theorem nrgdsdi
StepHypRef Expression
1 simpl 457 . . . 4  |-  ( ( R  e. NrmRing  /\  ( A  e.  X  /\  B  e.  X  /\  C  e.  X )
)  ->  R  e. NrmRing )
2 simpr1 994 . . . 4  |-  ( ( R  e. NrmRing  /\  ( A  e.  X  /\  B  e.  X  /\  C  e.  X )
)  ->  A  e.  X )
3 nrgrng 20219 . . . . . . 7  |-  ( R  e. NrmRing  ->  R  e.  Ring )
43adantr 465 . . . . . 6  |-  ( ( R  e. NrmRing  /\  ( A  e.  X  /\  B  e.  X  /\  C  e.  X )
)  ->  R  e.  Ring )
5 rnggrp 16638 . . . . . 6  |-  ( R  e.  Ring  ->  R  e. 
Grp )
64, 5syl 16 . . . . 5  |-  ( ( R  e. NrmRing  /\  ( A  e.  X  /\  B  e.  X  /\  C  e.  X )
)  ->  R  e.  Grp )
7 simpr2 995 . . . . 5  |-  ( ( R  e. NrmRing  /\  ( A  e.  X  /\  B  e.  X  /\  C  e.  X )
)  ->  B  e.  X )
8 simpr3 996 . . . . 5  |-  ( ( R  e. NrmRing  /\  ( A  e.  X  /\  B  e.  X  /\  C  e.  X )
)  ->  C  e.  X )
9 nmmul.x . . . . . 6  |-  X  =  ( Base `  R
)
10 eqid 2438 . . . . . 6  |-  ( -g `  R )  =  (
-g `  R )
119, 10grpsubcl 15597 . . . . 5  |-  ( ( R  e.  Grp  /\  B  e.  X  /\  C  e.  X )  ->  ( B ( -g `  R ) C )  e.  X )
126, 7, 8, 11syl3anc 1218 . . . 4  |-  ( ( R  e. NrmRing  /\  ( A  e.  X  /\  B  e.  X  /\  C  e.  X )
)  ->  ( B
( -g `  R ) C )  e.  X
)
13 nmmul.n . . . . 5  |-  N  =  ( norm `  R
)
14 nmmul.t . . . . 5  |-  .x.  =  ( .r `  R )
159, 13, 14nmmul 20220 . . . 4  |-  ( ( R  e. NrmRing  /\  A  e.  X  /\  ( B ( -g `  R
) C )  e.  X )  ->  ( N `  ( A  .x.  ( B ( -g `  R ) C ) ) )  =  ( ( N `  A
)  x.  ( N `
 ( B (
-g `  R ) C ) ) ) )
161, 2, 12, 15syl3anc 1218 . . 3  |-  ( ( R  e. NrmRing  /\  ( A  e.  X  /\  B  e.  X  /\  C  e.  X )
)  ->  ( N `  ( A  .x.  ( B ( -g `  R
) C ) ) )  =  ( ( N `  A )  x.  ( N `  ( B ( -g `  R
) C ) ) ) )
179, 14, 10, 4, 2, 7, 8rngsubdi 16678 . . . 4  |-  ( ( R  e. NrmRing  /\  ( A  e.  X  /\  B  e.  X  /\  C  e.  X )
)  ->  ( A  .x.  ( B ( -g `  R ) C ) )  =  ( ( A  .x.  B ) ( -g `  R
) ( A  .x.  C ) ) )
1817fveq2d 5690 . . 3  |-  ( ( R  e. NrmRing  /\  ( A  e.  X  /\  B  e.  X  /\  C  e.  X )
)  ->  ( N `  ( A  .x.  ( B ( -g `  R
) C ) ) )  =  ( N `
 ( ( A 
.x.  B ) (
-g `  R )
( A  .x.  C
) ) ) )
1916, 18eqtr3d 2472 . 2  |-  ( ( R  e. NrmRing  /\  ( A  e.  X  /\  B  e.  X  /\  C  e.  X )
)  ->  ( ( N `  A )  x.  ( N `  ( B ( -g `  R
) C ) ) )  =  ( N `
 ( ( A 
.x.  B ) (
-g `  R )
( A  .x.  C
) ) ) )
20 nrgngp 20218 . . . . 5  |-  ( R  e. NrmRing  ->  R  e. NrmGrp )
2120adantr 465 . . . 4  |-  ( ( R  e. NrmRing  /\  ( A  e.  X  /\  B  e.  X  /\  C  e.  X )
)  ->  R  e. NrmGrp )
22 nrgdsdi.d . . . . 5  |-  D  =  ( dist `  R
)
2313, 9, 10, 22ngpds 20170 . . . 4  |-  ( ( R  e. NrmGrp  /\  B  e.  X  /\  C  e.  X )  ->  ( B D C )  =  ( N `  ( B ( -g `  R
) C ) ) )
2421, 7, 8, 23syl3anc 1218 . . 3  |-  ( ( R  e. NrmRing  /\  ( A  e.  X  /\  B  e.  X  /\  C  e.  X )
)  ->  ( B D C )  =  ( N `  ( B ( -g `  R
) C ) ) )
2524oveq2d 6102 . 2  |-  ( ( R  e. NrmRing  /\  ( A  e.  X  /\  B  e.  X  /\  C  e.  X )
)  ->  ( ( N `  A )  x.  ( B D C ) )  =  ( ( N `  A
)  x.  ( N `
 ( B (
-g `  R ) C ) ) ) )
269, 14rngcl 16646 . . . 4  |-  ( ( R  e.  Ring  /\  A  e.  X  /\  B  e.  X )  ->  ( A  .x.  B )  e.  X )
274, 2, 7, 26syl3anc 1218 . . 3  |-  ( ( R  e. NrmRing  /\  ( A  e.  X  /\  B  e.  X  /\  C  e.  X )
)  ->  ( A  .x.  B )  e.  X
)
289, 14rngcl 16646 . . . 4  |-  ( ( R  e.  Ring  /\  A  e.  X  /\  C  e.  X )  ->  ( A  .x.  C )  e.  X )
294, 2, 8, 28syl3anc 1218 . . 3  |-  ( ( R  e. NrmRing  /\  ( A  e.  X  /\  B  e.  X  /\  C  e.  X )
)  ->  ( A  .x.  C )  e.  X
)
3013, 9, 10, 22ngpds 20170 . . 3  |-  ( ( R  e. NrmGrp  /\  ( A  .x.  B )  e.  X  /\  ( A 
.x.  C )  e.  X )  ->  (
( A  .x.  B
) D ( A 
.x.  C ) )  =  ( N `  ( ( A  .x.  B ) ( -g `  R ) ( A 
.x.  C ) ) ) )
3121, 27, 29, 30syl3anc 1218 . 2  |-  ( ( R  e. NrmRing  /\  ( A  e.  X  /\  B  e.  X  /\  C  e.  X )
)  ->  ( ( A  .x.  B ) D ( A  .x.  C
) )  =  ( N `  ( ( A  .x.  B ) ( -g `  R
) ( A  .x.  C ) ) ) )
3219, 25, 313eqtr4d 2480 1  |-  ( ( R  e. NrmRing  /\  ( A  e.  X  /\  B  e.  X  /\  C  e.  X )
)  ->  ( ( N `  A )  x.  ( B D C ) )  =  ( ( A  .x.  B
) D ( A 
.x.  C ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    /\ w3a 965    = wceq 1369    e. wcel 1756   ` cfv 5413  (class class class)co 6086    x. cmul 9279   Basecbs 14166   .rcmulr 14231   distcds 14239   Grpcgrp 15402   -gcsg 15405   Ringcrg 16633   normcnm 20144  NrmGrpcngp 20145  NrmRingcnrg 20147
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 2419  ax-rep 4398  ax-sep 4408  ax-nul 4416  ax-pow 4465  ax-pr 4526  ax-un 6367  ax-cnex 9330  ax-resscn 9331  ax-1cn 9332  ax-icn 9333  ax-addcl 9334  ax-addrcl 9335  ax-mulcl 9336  ax-mulrcl 9337  ax-mulcom 9338  ax-addass 9339  ax-mulass 9340  ax-distr 9341  ax-i2m1 9342  ax-1ne0 9343  ax-1rid 9344  ax-rnegex 9345  ax-rrecex 9346  ax-cnre 9347  ax-pre-lttri 9348  ax-pre-lttrn 9349  ax-pre-ltadd 9350  ax-pre-mulgt0 9351  ax-pre-sup 9352
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 2256  df-mo 2257  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-nel 2604  df-ral 2715  df-rex 2716  df-reu 2717  df-rmo 2718  df-rab 2719  df-v 2969  df-sbc 3182  df-csb 3284  df-dif 3326  df-un 3328  df-in 3330  df-ss 3337  df-pss 3339  df-nul 3633  df-if 3787  df-pw 3857  df-sn 3873  df-pr 3875  df-tp 3877  df-op 3879  df-uni 4087  df-iun 4168  df-br 4288  df-opab 4346  df-mpt 4347  df-tr 4381  df-eprel 4627  df-id 4631  df-po 4636  df-so 4637  df-fr 4674  df-we 4676  df-ord 4717  df-on 4718  df-lim 4719  df-suc 4720  df-xp 4841  df-rel 4842  df-cnv 4843  df-co 4844  df-dm 4845  df-rn 4846  df-res 4847  df-ima 4848  df-iota 5376  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-riota 6047  df-ov 6089  df-oprab 6090  df-mpt2 6091  df-om 6472  df-1st 6572  df-2nd 6573  df-recs 6824  df-rdg 6858  df-er 7093  df-map 7208  df-en 7303  df-dom 7304  df-sdom 7305  df-sup 7683  df-pnf 9412  df-mnf 9413  df-xr 9414  df-ltxr 9415  df-le 9416  df-sub 9589  df-neg 9590  df-div 9986  df-nn 10315  df-2 10372  df-n0 10572  df-z 10639  df-uz 10854  df-q 10946  df-rp 10984  df-xneg 11081  df-xadd 11082  df-xmul 11083  df-ndx 14169  df-slot 14170  df-base 14171  df-sets 14172  df-plusg 14243  df-0g 14372  df-topgen 14374  df-mnd 15407  df-grp 15536  df-minusg 15537  df-sbg 15538  df-mgp 16580  df-ur 16592  df-rng 16635  df-abv 16880  df-psmet 17784  df-xmet 17785  df-met 17786  df-bl 17787  df-mopn 17788  df-top 18478  df-bases 18480  df-topon 18481  df-topsp 18482  df-xms 19870  df-ms 19871  df-nm 20150  df-ngp 20151  df-nrg 20153
This theorem is referenced by: (None)
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