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Theorem rngnegr 17020
Description: Negation in a ring is the same as right multiplication by -1. (rngonegmn1r 29943 analog.) (Contributed by Jeff Madsen, 19-Jun-2010.) (Revised by Mario Carneiro, 2-Jul-2014.)
Hypotheses
Ref Expression
rngnegl.b  |-  B  =  ( Base `  R
)
rngnegl.t  |-  .x.  =  ( .r `  R )
rngnegl.u  |-  .1.  =  ( 1r `  R )
rngnegl.n  |-  N  =  ( invg `  R )
rngnegl.r  |-  ( ph  ->  R  e.  Ring )
rngnegl.x  |-  ( ph  ->  X  e.  B )
Assertion
Ref Expression
rngnegr  |-  ( ph  ->  ( X  .x.  ( N `  .1.  ) )  =  ( N `  X ) )

Proof of Theorem rngnegr
StepHypRef Expression
1 rngnegl.r . . . . 5  |-  ( ph  ->  R  e.  Ring )
2 rngnegl.x . . . . 5  |-  ( ph  ->  X  e.  B )
3 rnggrp 16984 . . . . . . 7  |-  ( R  e.  Ring  ->  R  e. 
Grp )
41, 3syl 16 . . . . . 6  |-  ( ph  ->  R  e.  Grp )
5 rngnegl.b . . . . . . . 8  |-  B  =  ( Base `  R
)
6 rngnegl.u . . . . . . . 8  |-  .1.  =  ( 1r `  R )
75, 6rngidcl 16999 . . . . . . 7  |-  ( R  e.  Ring  ->  .1.  e.  B )
81, 7syl 16 . . . . . 6  |-  ( ph  ->  .1.  e.  B )
9 rngnegl.n . . . . . . 7  |-  N  =  ( invg `  R )
105, 9grpinvcl 15889 . . . . . 6  |-  ( ( R  e.  Grp  /\  .1.  e.  B )  -> 
( N `  .1.  )  e.  B )
114, 8, 10syl2anc 661 . . . . 5  |-  ( ph  ->  ( N `  .1.  )  e.  B )
12 eqid 2460 . . . . . 6  |-  ( +g  `  R )  =  ( +g  `  R )
13 rngnegl.t . . . . . 6  |-  .x.  =  ( .r `  R )
145, 12, 13rngdi 16997 . . . . 5  |-  ( ( R  e.  Ring  /\  ( X  e.  B  /\  ( N `  .1.  )  e.  B  /\  .1.  e.  B ) )  -> 
( X  .x.  (
( N `  .1.  ) ( +g  `  R
)  .1.  ) )  =  ( ( X 
.x.  ( N `  .1.  ) ) ( +g  `  R ) ( X 
.x.  .1.  ) )
)
151, 2, 11, 8, 14syl13anc 1225 . . . 4  |-  ( ph  ->  ( X  .x.  (
( N `  .1.  ) ( +g  `  R
)  .1.  ) )  =  ( ( X 
.x.  ( N `  .1.  ) ) ( +g  `  R ) ( X 
.x.  .1.  ) )
)
16 eqid 2460 . . . . . . . 8  |-  ( 0g
`  R )  =  ( 0g `  R
)
175, 12, 16, 9grplinv 15890 . . . . . . 7  |-  ( ( R  e.  Grp  /\  .1.  e.  B )  -> 
( ( N `  .1.  ) ( +g  `  R
)  .1.  )  =  ( 0g `  R
) )
184, 8, 17syl2anc 661 . . . . . 6  |-  ( ph  ->  ( ( N `  .1.  ) ( +g  `  R
)  .1.  )  =  ( 0g `  R
) )
1918oveq2d 6291 . . . . 5  |-  ( ph  ->  ( X  .x.  (
( N `  .1.  ) ( +g  `  R
)  .1.  ) )  =  ( X  .x.  ( 0g `  R ) ) )
205, 13, 16rngrz 17016 . . . . . 6  |-  ( ( R  e.  Ring  /\  X  e.  B )  ->  ( X  .x.  ( 0g `  R ) )  =  ( 0g `  R
) )
211, 2, 20syl2anc 661 . . . . 5  |-  ( ph  ->  ( X  .x.  ( 0g `  R ) )  =  ( 0g `  R ) )
2219, 21eqtrd 2501 . . . 4  |-  ( ph  ->  ( X  .x.  (
( N `  .1.  ) ( +g  `  R
)  .1.  ) )  =  ( 0g `  R ) )
235, 13, 6rngridm 17003 . . . . . 6  |-  ( ( R  e.  Ring  /\  X  e.  B )  ->  ( X  .x.  .1.  )  =  X )
241, 2, 23syl2anc 661 . . . . 5  |-  ( ph  ->  ( X  .x.  .1.  )  =  X )
2524oveq2d 6291 . . . 4  |-  ( ph  ->  ( ( X  .x.  ( N `  .1.  )
) ( +g  `  R
) ( X  .x.  .1.  ) )  =  ( ( X  .x.  ( N `  .1.  ) ) ( +g  `  R
) X ) )
2615, 22, 253eqtr3rd 2510 . . 3  |-  ( ph  ->  ( ( X  .x.  ( N `  .1.  )
) ( +g  `  R
) X )  =  ( 0g `  R
) )
275, 13rngcl 16992 . . . . 5  |-  ( ( R  e.  Ring  /\  X  e.  B  /\  ( N `  .1.  )  e.  B )  ->  ( X  .x.  ( N `  .1.  ) )  e.  B
)
281, 2, 11, 27syl3anc 1223 . . . 4  |-  ( ph  ->  ( X  .x.  ( N `  .1.  ) )  e.  B )
295, 12, 16, 9grpinvid2 15893 . . . 4  |-  ( ( R  e.  Grp  /\  X  e.  B  /\  ( X  .x.  ( N `
 .1.  ) )  e.  B )  -> 
( ( N `  X )  =  ( X  .x.  ( N `
 .1.  ) )  <-> 
( ( X  .x.  ( N `  .1.  )
) ( +g  `  R
) X )  =  ( 0g `  R
) ) )
304, 2, 28, 29syl3anc 1223 . . 3  |-  ( ph  ->  ( ( N `  X )  =  ( X  .x.  ( N `
 .1.  ) )  <-> 
( ( X  .x.  ( N `  .1.  )
) ( +g  `  R
) X )  =  ( 0g `  R
) ) )
3126, 30mpbird 232 . 2  |-  ( ph  ->  ( N `  X
)  =  ( X 
.x.  ( N `  .1.  ) ) )
3231eqcomd 2468 1  |-  ( ph  ->  ( X  .x.  ( N `  .1.  ) )  =  ( N `  X ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    = wceq 1374    e. wcel 1762   ` cfv 5579  (class class class)co 6275   Basecbs 14479   +g cplusg 14544   .rcmulr 14545   0gc0g 14684   Grpcgrp 15716   invgcminusg 15717   1rcur 16936   Ringcrg 16979
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-8 1764  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1961  ax-ext 2438  ax-rep 4551  ax-sep 4561  ax-nul 4569  ax-pow 4618  ax-pr 4679  ax-un 6567  ax-cnex 9537  ax-resscn 9538  ax-1cn 9539  ax-icn 9540  ax-addcl 9541  ax-addrcl 9542  ax-mulcl 9543  ax-mulrcl 9544  ax-mulcom 9545  ax-addass 9546  ax-mulass 9547  ax-distr 9548  ax-i2m1 9549  ax-1ne0 9550  ax-1rid 9551  ax-rnegex 9552  ax-rrecex 9553  ax-cnre 9554  ax-pre-lttri 9555  ax-pre-lttrn 9556  ax-pre-ltadd 9557  ax-pre-mulgt0 9558
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 969  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2272  df-mo 2273  df-clab 2446  df-cleq 2452  df-clel 2455  df-nfc 2610  df-ne 2657  df-nel 2658  df-ral 2812  df-rex 2813  df-reu 2814  df-rmo 2815  df-rab 2816  df-v 3108  df-sbc 3325  df-csb 3429  df-dif 3472  df-un 3474  df-in 3476  df-ss 3483  df-pss 3485  df-nul 3779  df-if 3933  df-pw 4005  df-sn 4021  df-pr 4023  df-tp 4025  df-op 4027  df-uni 4239  df-iun 4320  df-br 4441  df-opab 4499  df-mpt 4500  df-tr 4534  df-eprel 4784  df-id 4788  df-po 4793  df-so 4794  df-fr 4831  df-we 4833  df-ord 4874  df-on 4875  df-lim 4876  df-suc 4877  df-xp 4998  df-rel 4999  df-cnv 5000  df-co 5001  df-dm 5002  df-rn 5003  df-res 5004  df-ima 5005  df-iota 5542  df-fun 5581  df-fn 5582  df-f 5583  df-f1 5584  df-fo 5585  df-f1o 5586  df-fv 5587  df-riota 6236  df-ov 6278  df-oprab 6279  df-mpt2 6280  df-om 6672  df-recs 7032  df-rdg 7066  df-er 7301  df-en 7507  df-dom 7508  df-sdom 7509  df-pnf 9619  df-mnf 9620  df-xr 9621  df-ltxr 9622  df-le 9623  df-sub 9796  df-neg 9797  df-nn 10526  df-2 10583  df-ndx 14482  df-slot 14483  df-base 14484  df-sets 14485  df-plusg 14557  df-0g 14686  df-mnd 15721  df-grp 15851  df-minusg 15852  df-mgp 16925  df-ur 16937  df-rng 16981
This theorem is referenced by:  rngmneg2  17022  irredneg  17136  lmodsubdi  17343  mdetunilem7  18880  ldualvsubval  33829  lcdvsubval  36290  mapdpglem30  36374
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