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

Theorem rngnegl 16620
Description: Negation in a ring is the same as left multiplication by -1. (rngonegmn1l 28599 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
rngnegl  |-  ( ph  ->  ( ( N `  .1.  )  .x.  X )  =  ( N `  X ) )

Proof of Theorem rngnegl
StepHypRef Expression
1 rngnegl.r . . . . 5  |-  ( ph  ->  R  e.  Ring )
2 rngnegl.b . . . . . . 7  |-  B  =  ( Base `  R
)
3 rngnegl.u . . . . . . 7  |-  .1.  =  ( 1r `  R )
42, 3rngidcl 16601 . . . . . 6  |-  ( R  e.  Ring  ->  .1.  e.  B )
51, 4syl 16 . . . . 5  |-  ( ph  ->  .1.  e.  B )
6 rnggrp 16586 . . . . . . 7  |-  ( R  e.  Ring  ->  R  e. 
Grp )
71, 6syl 16 . . . . . 6  |-  ( ph  ->  R  e.  Grp )
8 rngnegl.n . . . . . . 7  |-  N  =  ( invg `  R )
92, 8grpinvcl 15563 . . . . . 6  |-  ( ( R  e.  Grp  /\  .1.  e.  B )  -> 
( N `  .1.  )  e.  B )
107, 5, 9syl2anc 654 . . . . 5  |-  ( ph  ->  ( N `  .1.  )  e.  B )
11 rngnegl.x . . . . 5  |-  ( ph  ->  X  e.  B )
12 eqid 2433 . . . . . 6  |-  ( +g  `  R )  =  ( +g  `  R )
13 rngnegl.t . . . . . 6  |-  .x.  =  ( .r `  R )
142, 12, 13rngdir 16600 . . . . 5  |-  ( ( R  e.  Ring  /\  (  .1.  e.  B  /\  ( N `  .1.  )  e.  B  /\  X  e.  B ) )  -> 
( (  .1.  ( +g  `  R ) ( N `  .1.  )
)  .x.  X )  =  ( (  .1. 
.x.  X ) ( +g  `  R ) ( ( N `  .1.  )  .x.  X ) ) )
151, 5, 10, 11, 14syl13anc 1213 . . . 4  |-  ( ph  ->  ( (  .1.  ( +g  `  R ) ( N `  .1.  )
)  .x.  X )  =  ( (  .1. 
.x.  X ) ( +g  `  R ) ( ( N `  .1.  )  .x.  X ) ) )
16 eqid 2433 . . . . . . . 8  |-  ( 0g
`  R )  =  ( 0g `  R
)
172, 12, 16, 8grprinv 15565 . . . . . . 7  |-  ( ( R  e.  Grp  /\  .1.  e.  B )  -> 
(  .1.  ( +g  `  R ) ( N `
 .1.  ) )  =  ( 0g `  R ) )
187, 5, 17syl2anc 654 . . . . . 6  |-  ( ph  ->  (  .1.  ( +g  `  R ) ( N `
 .1.  ) )  =  ( 0g `  R ) )
1918oveq1d 6095 . . . . 5  |-  ( ph  ->  ( (  .1.  ( +g  `  R ) ( N `  .1.  )
)  .x.  X )  =  ( ( 0g
`  R )  .x.  X ) )
202, 13, 16rnglz 16617 . . . . . 6  |-  ( ( R  e.  Ring  /\  X  e.  B )  ->  (
( 0g `  R
)  .x.  X )  =  ( 0g `  R ) )
211, 11, 20syl2anc 654 . . . . 5  |-  ( ph  ->  ( ( 0g `  R )  .x.  X
)  =  ( 0g
`  R ) )
2219, 21eqtrd 2465 . . . 4  |-  ( ph  ->  ( (  .1.  ( +g  `  R ) ( N `  .1.  )
)  .x.  X )  =  ( 0g `  R ) )
232, 13, 3rnglidm 16604 . . . . . 6  |-  ( ( R  e.  Ring  /\  X  e.  B )  ->  (  .1.  .x.  X )  =  X )
241, 11, 23syl2anc 654 . . . . 5  |-  ( ph  ->  (  .1.  .x.  X
)  =  X )
2524oveq1d 6095 . . . 4  |-  ( ph  ->  ( (  .1.  .x.  X ) ( +g  `  R ) ( ( N `  .1.  )  .x.  X ) )  =  ( X ( +g  `  R ) ( ( N `  .1.  )  .x.  X ) ) )
2615, 22, 253eqtr3rd 2474 . . 3  |-  ( ph  ->  ( X ( +g  `  R ) ( ( N `  .1.  )  .x.  X ) )  =  ( 0g `  R
) )
272, 13rngcl 16594 . . . . 5  |-  ( ( R  e.  Ring  /\  ( N `  .1.  )  e.  B  /\  X  e.  B )  ->  (
( N `  .1.  )  .x.  X )  e.  B )
281, 10, 11, 27syl3anc 1211 . . . 4  |-  ( ph  ->  ( ( N `  .1.  )  .x.  X )  e.  B )
292, 12, 16, 8grpinvid1 15566 . . . 4  |-  ( ( R  e.  Grp  /\  X  e.  B  /\  ( ( N `  .1.  )  .x.  X )  e.  B )  -> 
( ( N `  X )  =  ( ( N `  .1.  )  .x.  X )  <->  ( X
( +g  `  R ) ( ( N `  .1.  )  .x.  X ) )  =  ( 0g
`  R ) ) )
307, 11, 28, 29syl3anc 1211 . . 3  |-  ( ph  ->  ( ( N `  X )  =  ( ( N `  .1.  )  .x.  X )  <->  ( X
( +g  `  R ) ( ( N `  .1.  )  .x.  X ) )  =  ( 0g
`  R ) ) )
3126, 30mpbird 232 . 2  |-  ( ph  ->  ( N `  X
)  =  ( ( N `  .1.  )  .x.  X ) )
3231eqcomd 2438 1  |-  ( ph  ->  ( ( N `  .1.  )  .x.  X )  =  ( N `  X ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    = wceq 1362    e. wcel 1755   ` cfv 5406  (class class class)co 6080   Basecbs 14157   +g cplusg 14221   .rcmulr 14222   0gc0g 14361   Grpcgrp 15393   invgcminusg 15394   Ringcrg 16577   1rcur 16579
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1594  ax-4 1605  ax-5 1669  ax-6 1707  ax-7 1727  ax-8 1757  ax-9 1759  ax-10 1774  ax-11 1779  ax-12 1791  ax-13 1942  ax-ext 2414  ax-rep 4391  ax-sep 4401  ax-nul 4409  ax-pow 4458  ax-pr 4519  ax-un 6361  ax-cnex 9326  ax-resscn 9327  ax-1cn 9328  ax-icn 9329  ax-addcl 9330  ax-addrcl 9331  ax-mulcl 9332  ax-mulrcl 9333  ax-mulcom 9334  ax-addass 9335  ax-mulass 9336  ax-distr 9337  ax-i2m1 9338  ax-1ne0 9339  ax-1rid 9340  ax-rnegex 9341  ax-rrecex 9342  ax-cnre 9343  ax-pre-lttri 9344  ax-pre-lttrn 9345  ax-pre-ltadd 9346  ax-pre-mulgt0 9347
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 959  df-3an 960  df-tru 1365  df-ex 1590  df-nf 1593  df-sb 1700  df-eu 2258  df-mo 2259  df-clab 2420  df-cleq 2426  df-clel 2429  df-nfc 2558  df-ne 2598  df-nel 2599  df-ral 2710  df-rex 2711  df-reu 2712  df-rmo 2713  df-rab 2714  df-v 2964  df-sbc 3176  df-csb 3277  df-dif 3319  df-un 3321  df-in 3323  df-ss 3330  df-pss 3332  df-nul 3626  df-if 3780  df-pw 3850  df-sn 3866  df-pr 3868  df-tp 3870  df-op 3872  df-uni 4080  df-iun 4161  df-br 4281  df-opab 4339  df-mpt 4340  df-tr 4374  df-eprel 4619  df-id 4623  df-po 4628  df-so 4629  df-fr 4666  df-we 4668  df-ord 4709  df-on 4710  df-lim 4711  df-suc 4712  df-xp 4833  df-rel 4834  df-cnv 4835  df-co 4836  df-dm 4837  df-rn 4838  df-res 4839  df-ima 4840  df-iota 5369  df-fun 5408  df-fn 5409  df-f 5410  df-f1 5411  df-fo 5412  df-f1o 5413  df-fv 5414  df-riota 6039  df-ov 6083  df-oprab 6084  df-mpt2 6085  df-om 6466  df-recs 6818  df-rdg 6852  df-er 7089  df-en 7299  df-dom 7300  df-sdom 7301  df-pnf 9408  df-mnf 9409  df-xr 9410  df-ltxr 9411  df-le 9412  df-sub 9585  df-neg 9586  df-nn 10311  df-2 10368  df-ndx 14160  df-slot 14161  df-base 14162  df-sets 14163  df-plusg 14234  df-0g 14363  df-mnd 15398  df-grp 15525  df-minusg 15526  df-mgp 16566  df-rng 16580  df-ur 16582
This theorem is referenced by:  rngmneg1  16622  dvdsrneg  16680  abvneg  16843  lmodvsneg  16913  lmodsubvs  16925  lmodsubdi  16926  lmodsubdir  16927  lmodvsinv  17039  mplind  17516  mdetralt  18256  m2detleiblem7  18275  lflsub  32285  baerlem3lem1  34925
  Copyright terms: Public domain W3C validator