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Theorem rngnegr 15397
Description: Negation in a ring is the same as right multiplication by -1. (rngonegmn1r 26684 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  =  ( inv g `  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 15362 . . . . . . 7  |-  ( R  e.  Ring  ->  R  e. 
Grp )
41, 3syl 15 . . . . . 6  |-  ( ph  ->  R  e.  Grp )
5 rngnegl.b . . . . . . . 8  |-  B  =  ( Base `  R
)
6 rngnegl.u . . . . . . . 8  |-  .1.  =  ( 1r `  R )
75, 6rngidcl 15377 . . . . . . 7  |-  ( R  e.  Ring  ->  .1.  e.  B )
81, 7syl 15 . . . . . 6  |-  ( ph  ->  .1.  e.  B )
9 rngnegl.n . . . . . . 7  |-  N  =  ( inv g `  R )
105, 9grpinvcl 14543 . . . . . 6  |-  ( ( R  e.  Grp  /\  .1.  e.  B )  -> 
( N `  .1.  )  e.  B )
114, 8, 10syl2anc 642 . . . . 5  |-  ( ph  ->  ( N `  .1.  )  e.  B )
12 eqid 2296 . . . . . 6  |-  ( +g  `  R )  =  ( +g  `  R )
13 rngnegl.t . . . . . 6  |-  .x.  =  ( .r `  R )
145, 12, 13rngdi 15375 . . . . 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 1184 . . . 4  |-  ( ph  ->  ( X  .x.  (
( N `  .1.  ) ( +g  `  R
)  .1.  ) )  =  ( ( X 
.x.  ( N `  .1.  ) ) ( +g  `  R ) ( X 
.x.  .1.  ) )
)
16 eqid 2296 . . . . . . . 8  |-  ( 0g
`  R )  =  ( 0g `  R
)
175, 12, 16, 9grplinv 14544 . . . . . . 7  |-  ( ( R  e.  Grp  /\  .1.  e.  B )  -> 
( ( N `  .1.  ) ( +g  `  R
)  .1.  )  =  ( 0g `  R
) )
184, 8, 17syl2anc 642 . . . . . 6  |-  ( ph  ->  ( ( N `  .1.  ) ( +g  `  R
)  .1.  )  =  ( 0g `  R
) )
1918oveq2d 5890 . . . . 5  |-  ( ph  ->  ( X  .x.  (
( N `  .1.  ) ( +g  `  R
)  .1.  ) )  =  ( X  .x.  ( 0g `  R ) ) )
205, 13, 16rngrz 15394 . . . . . 6  |-  ( ( R  e.  Ring  /\  X  e.  B )  ->  ( X  .x.  ( 0g `  R ) )  =  ( 0g `  R
) )
211, 2, 20syl2anc 642 . . . . 5  |-  ( ph  ->  ( X  .x.  ( 0g `  R ) )  =  ( 0g `  R ) )
2219, 21eqtrd 2328 . . . 4  |-  ( ph  ->  ( X  .x.  (
( N `  .1.  ) ( +g  `  R
)  .1.  ) )  =  ( 0g `  R ) )
235, 13, 6rngridm 15381 . . . . . 6  |-  ( ( R  e.  Ring  /\  X  e.  B )  ->  ( X  .x.  .1.  )  =  X )
241, 2, 23syl2anc 642 . . . . 5  |-  ( ph  ->  ( X  .x.  .1.  )  =  X )
2524oveq2d 5890 . . . 4  |-  ( ph  ->  ( ( X  .x.  ( N `  .1.  )
) ( +g  `  R
) ( X  .x.  .1.  ) )  =  ( ( X  .x.  ( N `  .1.  ) ) ( +g  `  R
) X ) )
2615, 22, 253eqtr3rd 2337 . . 3  |-  ( ph  ->  ( ( X  .x.  ( N `  .1.  )
) ( +g  `  R
) X )  =  ( 0g `  R
) )
275, 13rngcl 15370 . . . . 5  |-  ( ( R  e.  Ring  /\  X  e.  B  /\  ( N `  .1.  )  e.  B )  ->  ( X  .x.  ( N `  .1.  ) )  e.  B
)
281, 2, 11, 27syl3anc 1182 . . . 4  |-  ( ph  ->  ( X  .x.  ( N `  .1.  ) )  e.  B )
295, 12, 16, 9grpinvid2 14547 . . . 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 1182 . . 3  |-  ( ph  ->  ( ( N `  X )  =  ( X  .x.  ( N `
 .1.  ) )  <-> 
( ( X  .x.  ( N `  .1.  )
) ( +g  `  R
) X )  =  ( 0g `  R
) ) )
3126, 30mpbird 223 . 2  |-  ( ph  ->  ( N `  X
)  =  ( X 
.x.  ( N `  .1.  ) ) )
3231eqcomd 2301 1  |-  ( ph  ->  ( X  .x.  ( N `  .1.  ) )  =  ( N `  X ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    = wceq 1632    e. wcel 1696   ` cfv 5271  (class class class)co 5874   Basecbs 13164   +g cplusg 13224   .rcmulr 13225   0gc0g 13416   Grpcgrp 14378   inv gcminusg 14379   Ringcrg 15353   1rcur 15355
This theorem is referenced by:  rngmneg2  15399  irredneg  15508  lmodsubdi  15698  ldualvsubval  29969  lcdvsubval  32430  mapdpglem30  32514
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
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-riota 6320  df-recs 6404  df-rdg 6439  df-er 6676  df-en 6880  df-dom 6881  df-sdom 6882  df-pnf 8885  df-mnf 8886  df-xr 8887  df-ltxr 8888  df-le 8889  df-sub 9055  df-neg 9056  df-nn 9763  df-2 9820  df-ndx 13167  df-slot 13168  df-base 13169  df-sets 13170  df-plusg 13237  df-0g 13420  df-mnd 14383  df-grp 14505  df-minusg 14506  df-mgp 15342  df-rng 15356  df-ur 15358
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