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Theorem nrgdsdi 18192
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 443 . . . 4  |-  ( ( R  e. NrmRing  /\  ( A  e.  X  /\  B  e.  X  /\  C  e.  X )
)  ->  R  e. NrmRing )
2 simpr1 961 . . . 4  |-  ( ( R  e. NrmRing  /\  ( A  e.  X  /\  B  e.  X  /\  C  e.  X )
)  ->  A  e.  X )
3 nrgrng 18190 . . . . . . 7  |-  ( R  e. NrmRing  ->  R  e.  Ring )
43adantr 451 . . . . . 6  |-  ( ( R  e. NrmRing  /\  ( A  e.  X  /\  B  e.  X  /\  C  e.  X )
)  ->  R  e.  Ring )
5 rnggrp 15362 . . . . . 6  |-  ( R  e.  Ring  ->  R  e. 
Grp )
64, 5syl 15 . . . . 5  |-  ( ( R  e. NrmRing  /\  ( A  e.  X  /\  B  e.  X  /\  C  e.  X )
)  ->  R  e.  Grp )
7 simpr2 962 . . . . 5  |-  ( ( R  e. NrmRing  /\  ( A  e.  X  /\  B  e.  X  /\  C  e.  X )
)  ->  B  e.  X )
8 simpr3 963 . . . . 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 2296 . . . . . 6  |-  ( -g `  R )  =  (
-g `  R )
119, 10grpsubcl 14562 . . . . 5  |-  ( ( R  e.  Grp  /\  B  e.  X  /\  C  e.  X )  ->  ( B ( -g `  R ) C )  e.  X )
126, 7, 8, 11syl3anc 1182 . . . 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 18191 . . . 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 1182 . . 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 15401 . . . 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 5545 . . 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 2330 . 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 18189 . . . . 5  |-  ( R  e. NrmRing  ->  R  e. NrmGrp )
2120adantr 451 . . . 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 18141 . . . 4  |-  ( ( R  e. NrmGrp  /\  B  e.  X  /\  C  e.  X )  ->  ( B D C )  =  ( N `  ( B ( -g `  R
) C ) ) )
2421, 7, 8, 23syl3anc 1182 . . 3  |-  ( ( R  e. NrmRing  /\  ( A  e.  X  /\  B  e.  X  /\  C  e.  X )
)  ->  ( B D C )  =  ( N `  ( B ( -g `  R
) C ) ) )
2524oveq2d 5890 . 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 15370 . . . 4  |-  ( ( R  e.  Ring  /\  A  e.  X  /\  B  e.  X )  ->  ( A  .x.  B )  e.  X )
274, 2, 7, 26syl3anc 1182 . . 3  |-  ( ( R  e. NrmRing  /\  ( A  e.  X  /\  B  e.  X  /\  C  e.  X )
)  ->  ( A  .x.  B )  e.  X
)
289, 14rngcl 15370 . . . 4  |-  ( ( R  e.  Ring  /\  A  e.  X  /\  C  e.  X )  ->  ( A  .x.  C )  e.  X )
294, 2, 8, 28syl3anc 1182 . . 3  |-  ( ( R  e. NrmRing  /\  ( A  e.  X  /\  B  e.  X  /\  C  e.  X )
)  ->  ( A  .x.  C )  e.  X
)
3013, 9, 10, 22ngpds 18141 . . 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 1182 . 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 2338 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 set class
Syntax hints:    -> wi 4    /\ wa 358    /\ w3a 934    = wceq 1632    e. wcel 1696   ` cfv 5271  (class class class)co 5874    x. cmul 8758   Basecbs 13164   .rcmulr 13225   distcds 13233   Grpcgrp 14378   -gcsg 14381   Ringcrg 15353   normcnm 18115  NrmGrpcngp 18116  NrmRingcnrg 18118
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-cnex 8809  ax-resscn 8810  ax-1cn 8811  ax-icn 8812  ax-addcl 8813  ax-addrcl 8814  ax-mulcl 8815  ax-mulrcl 8816  ax-mulcom 8817  ax-addass 8818  ax-mulass 8819  ax-distr 8820  ax-i2m1 8821  ax-1ne0 8822  ax-1rid 8823  ax-rnegex 8824  ax-rrecex 8825  ax-cnre 8826  ax-pre-lttri 8827  ax-pre-lttrn 8828  ax-pre-ltadd 8829  ax-pre-mulgt0 8830  ax-pre-sup 8831
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rmo 2564  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-1st 6138  df-2nd 6139  df-riota 6320  df-recs 6404  df-rdg 6439  df-er 6676  df-map 6790  df-en 6880  df-dom 6881  df-sdom 6882  df-sup 7210  df-pnf 8885  df-mnf 8886  df-xr 8887  df-ltxr 8888  df-le 8889  df-sub 9055  df-neg 9056  df-div 9440  df-nn 9763  df-2 9820  df-n0 9982  df-z 10041  df-uz 10247  df-q 10333  df-rp 10371  df-xneg 10468  df-xadd 10469  df-xmul 10470  df-ndx 13167  df-slot 13168  df-base 13169  df-sets 13170  df-plusg 13237  df-topgen 13360  df-0g 13420  df-mnd 14383  df-grp 14505  df-minusg 14506  df-sbg 14507  df-mgp 15342  df-rng 15356  df-ur 15358  df-abv 15598  df-xmet 16389  df-met 16390  df-bl 16391  df-mopn 16392  df-top 16652  df-bases 16654  df-topon 16655  df-topsp 16656  df-xms 17901  df-ms 17902  df-nm 18121  df-ngp 18122  df-nrg 18124
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