MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  rngnegr Structured version   Unicode version

Theorem rngnegr 17453
Description: Negation in a ring is the same as right multiplication by -1. (rngonegmn1r 31616 analog.) (Contributed by Jeff Madsen, 19-Jun-2010.) (Revised by Mario Carneiro, 2-Jul-2014.)
Hypotheses
Ref Expression
ringnegl.b  |-  B  =  ( Base `  R
)
ringnegl.t  |-  .x.  =  ( .r `  R )
ringnegl.u  |-  .1.  =  ( 1r `  R )
ringnegl.n  |-  N  =  ( invg `  R )
ringnegl.r  |-  ( ph  ->  R  e.  Ring )
ringnegl.x  |-  ( ph  ->  X  e.  B )
Assertion
Ref Expression
rngnegr  |-  ( ph  ->  ( X  .x.  ( N `  .1.  ) )  =  ( N `  X ) )

Proof of Theorem rngnegr
StepHypRef Expression
1 ringnegl.r . . . . 5  |-  ( ph  ->  R  e.  Ring )
2 ringnegl.x . . . . 5  |-  ( ph  ->  X  e.  B )
3 ringgrp 17415 . . . . . . 7  |-  ( R  e.  Ring  ->  R  e. 
Grp )
41, 3syl 17 . . . . . 6  |-  ( ph  ->  R  e.  Grp )
5 ringnegl.b . . . . . . . 8  |-  B  =  ( Base `  R
)
6 ringnegl.u . . . . . . . 8  |-  .1.  =  ( 1r `  R )
75, 6ringidcl 17431 . . . . . . 7  |-  ( R  e.  Ring  ->  .1.  e.  B )
81, 7syl 17 . . . . . 6  |-  ( ph  ->  .1.  e.  B )
9 ringnegl.n . . . . . . 7  |-  N  =  ( invg `  R )
105, 9grpinvcl 16311 . . . . . 6  |-  ( ( R  e.  Grp  /\  .1.  e.  B )  -> 
( N `  .1.  )  e.  B )
114, 8, 10syl2anc 659 . . . . 5  |-  ( ph  ->  ( N `  .1.  )  e.  B )
12 eqid 2402 . . . . . 6  |-  ( +g  `  R )  =  ( +g  `  R )
13 ringnegl.t . . . . . 6  |-  .x.  =  ( .r `  R )
145, 12, 13ringdi 17429 . . . . 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 1232 . . . 4  |-  ( ph  ->  ( X  .x.  (
( N `  .1.  ) ( +g  `  R
)  .1.  ) )  =  ( ( X 
.x.  ( N `  .1.  ) ) ( +g  `  R ) ( X 
.x.  .1.  ) )
)
16 eqid 2402 . . . . . . . 8  |-  ( 0g
`  R )  =  ( 0g `  R
)
175, 12, 16, 9grplinv 16312 . . . . . . 7  |-  ( ( R  e.  Grp  /\  .1.  e.  B )  -> 
( ( N `  .1.  ) ( +g  `  R
)  .1.  )  =  ( 0g `  R
) )
184, 8, 17syl2anc 659 . . . . . 6  |-  ( ph  ->  ( ( N `  .1.  ) ( +g  `  R
)  .1.  )  =  ( 0g `  R
) )
1918oveq2d 6250 . . . . 5  |-  ( ph  ->  ( X  .x.  (
( N `  .1.  ) ( +g  `  R
)  .1.  ) )  =  ( X  .x.  ( 0g `  R ) ) )
205, 13, 16ringrz 17448 . . . . . 6  |-  ( ( R  e.  Ring  /\  X  e.  B )  ->  ( X  .x.  ( 0g `  R ) )  =  ( 0g `  R
) )
211, 2, 20syl2anc 659 . . . . 5  |-  ( ph  ->  ( X  .x.  ( 0g `  R ) )  =  ( 0g `  R ) )
2219, 21eqtrd 2443 . . . 4  |-  ( ph  ->  ( X  .x.  (
( N `  .1.  ) ( +g  `  R
)  .1.  ) )  =  ( 0g `  R ) )
235, 13, 6ringridm 17435 . . . . . 6  |-  ( ( R  e.  Ring  /\  X  e.  B )  ->  ( X  .x.  .1.  )  =  X )
241, 2, 23syl2anc 659 . . . . 5  |-  ( ph  ->  ( X  .x.  .1.  )  =  X )
2524oveq2d 6250 . . . 4  |-  ( ph  ->  ( ( X  .x.  ( N `  .1.  )
) ( +g  `  R
) ( X  .x.  .1.  ) )  =  ( ( X  .x.  ( N `  .1.  ) ) ( +g  `  R
) X ) )
2615, 22, 253eqtr3rd 2452 . . 3  |-  ( ph  ->  ( ( X  .x.  ( N `  .1.  )
) ( +g  `  R
) X )  =  ( 0g `  R
) )
275, 13ringcl 17424 . . . . 5  |-  ( ( R  e.  Ring  /\  X  e.  B  /\  ( N `  .1.  )  e.  B )  ->  ( X  .x.  ( N `  .1.  ) )  e.  B
)
281, 2, 11, 27syl3anc 1230 . . . 4  |-  ( ph  ->  ( X  .x.  ( N `  .1.  ) )  e.  B )
295, 12, 16, 9grpinvid2 16315 . . . 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 1230 . . 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 2410 1  |-  ( ph  ->  ( X  .x.  ( N `  .1.  ) )  =  ( N `  X ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    = wceq 1405    e. wcel 1842   ` cfv 5525  (class class class)co 6234   Basecbs 14733   +g cplusg 14801   .rcmulr 14802   0gc0g 14946   Grpcgrp 16269   invgcminusg 16270   1rcur 17365   Ringcrg 17410
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-8 1844  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-rep 4506  ax-sep 4516  ax-nul 4524  ax-pow 4571  ax-pr 4629  ax-un 6530  ax-cnex 9498  ax-resscn 9499  ax-1cn 9500  ax-icn 9501  ax-addcl 9502  ax-addrcl 9503  ax-mulcl 9504  ax-mulrcl 9505  ax-mulcom 9506  ax-addass 9507  ax-mulass 9508  ax-distr 9509  ax-i2m1 9510  ax-1ne0 9511  ax-1rid 9512  ax-rnegex 9513  ax-rrecex 9514  ax-cnre 9515  ax-pre-lttri 9516  ax-pre-lttrn 9517  ax-pre-ltadd 9518  ax-pre-mulgt0 9519
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 975  df-3an 976  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-nel 2601  df-ral 2758  df-rex 2759  df-reu 2760  df-rmo 2761  df-rab 2762  df-v 3060  df-sbc 3277  df-csb 3373  df-dif 3416  df-un 3418  df-in 3420  df-ss 3427  df-pss 3429  df-nul 3738  df-if 3885  df-pw 3956  df-sn 3972  df-pr 3974  df-tp 3976  df-op 3978  df-uni 4191  df-iun 4272  df-br 4395  df-opab 4453  df-mpt 4454  df-tr 4489  df-eprel 4733  df-id 4737  df-po 4743  df-so 4744  df-fr 4781  df-we 4783  df-ord 4824  df-on 4825  df-lim 4826  df-suc 4827  df-xp 4948  df-rel 4949  df-cnv 4950  df-co 4951  df-dm 4952  df-rn 4953  df-res 4954  df-ima 4955  df-iota 5489  df-fun 5527  df-fn 5528  df-f 5529  df-f1 5530  df-fo 5531  df-f1o 5532  df-fv 5533  df-riota 6196  df-ov 6237  df-oprab 6238  df-mpt2 6239  df-om 6639  df-recs 6999  df-rdg 7033  df-er 7268  df-en 7475  df-dom 7476  df-sdom 7477  df-pnf 9580  df-mnf 9581  df-xr 9582  df-ltxr 9583  df-le 9584  df-sub 9763  df-neg 9764  df-nn 10497  df-2 10555  df-ndx 14736  df-slot 14737  df-base 14738  df-sets 14739  df-plusg 14814  df-0g 14948  df-mgm 16088  df-sgrp 16127  df-mnd 16137  df-grp 16273  df-minusg 16274  df-mgp 17354  df-ur 17366  df-ring 17412
This theorem is referenced by:  ringmneg2  17455  irredneg  17571  lmodsubdi  17779  mdetunilem7  19304  ldualvsubval  32156  lcdvsubval  34619  mapdpglem30  34703
  Copyright terms: Public domain W3C validator