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Theorem rngnegl 16675
Description: Negation in a ring is the same as left multiplication by -1. (rngonegmn1l 28726 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 16655 . . . . . 6  |-  ( R  e.  Ring  ->  .1.  e.  B )
51, 4syl 16 . . . . 5  |-  ( ph  ->  .1.  e.  B )
6 rnggrp 16640 . . . . . . 7  |-  ( R  e.  Ring  ->  R  e. 
Grp )
71, 6syl 16 . . . . . 6  |-  ( ph  ->  R  e.  Grp )
8 rngnegl.n . . . . . . 7  |-  N  =  ( invg `  R )
92, 8grpinvcl 15574 . . . . . 6  |-  ( ( R  e.  Grp  /\  .1.  e.  B )  -> 
( N `  .1.  )  e.  B )
107, 5, 9syl2anc 661 . . . . 5  |-  ( ph  ->  ( N `  .1.  )  e.  B )
11 rngnegl.x . . . . 5  |-  ( ph  ->  X  e.  B )
12 eqid 2438 . . . . . 6  |-  ( +g  `  R )  =  ( +g  `  R )
13 rngnegl.t . . . . . 6  |-  .x.  =  ( .r `  R )
142, 12, 13rngdir 16654 . . . . 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 1220 . . . 4  |-  ( ph  ->  ( (  .1.  ( +g  `  R ) ( N `  .1.  )
)  .x.  X )  =  ( (  .1. 
.x.  X ) ( +g  `  R ) ( ( N `  .1.  )  .x.  X ) ) )
16 eqid 2438 . . . . . . . 8  |-  ( 0g
`  R )  =  ( 0g `  R
)
172, 12, 16, 8grprinv 15576 . . . . . . 7  |-  ( ( R  e.  Grp  /\  .1.  e.  B )  -> 
(  .1.  ( +g  `  R ) ( N `
 .1.  ) )  =  ( 0g `  R ) )
187, 5, 17syl2anc 661 . . . . . 6  |-  ( ph  ->  (  .1.  ( +g  `  R ) ( N `
 .1.  ) )  =  ( 0g `  R ) )
1918oveq1d 6101 . . . . 5  |-  ( ph  ->  ( (  .1.  ( +g  `  R ) ( N `  .1.  )
)  .x.  X )  =  ( ( 0g
`  R )  .x.  X ) )
202, 13, 16rnglz 16671 . . . . . 6  |-  ( ( R  e.  Ring  /\  X  e.  B )  ->  (
( 0g `  R
)  .x.  X )  =  ( 0g `  R ) )
211, 11, 20syl2anc 661 . . . . 5  |-  ( ph  ->  ( ( 0g `  R )  .x.  X
)  =  ( 0g
`  R ) )
2219, 21eqtrd 2470 . . . 4  |-  ( ph  ->  ( (  .1.  ( +g  `  R ) ( N `  .1.  )
)  .x.  X )  =  ( 0g `  R ) )
232, 13, 3rnglidm 16658 . . . . . 6  |-  ( ( R  e.  Ring  /\  X  e.  B )  ->  (  .1.  .x.  X )  =  X )
241, 11, 23syl2anc 661 . . . . 5  |-  ( ph  ->  (  .1.  .x.  X
)  =  X )
2524oveq1d 6101 . . . 4  |-  ( ph  ->  ( (  .1.  .x.  X ) ( +g  `  R ) ( ( N `  .1.  )  .x.  X ) )  =  ( X ( +g  `  R ) ( ( N `  .1.  )  .x.  X ) ) )
2615, 22, 253eqtr3rd 2479 . . 3  |-  ( ph  ->  ( X ( +g  `  R ) ( ( N `  .1.  )  .x.  X ) )  =  ( 0g `  R
) )
272, 13rngcl 16648 . . . . 5  |-  ( ( R  e.  Ring  /\  ( N `  .1.  )  e.  B  /\  X  e.  B )  ->  (
( N `  .1.  )  .x.  X )  e.  B )
281, 10, 11, 27syl3anc 1218 . . . 4  |-  ( ph  ->  ( ( N `  .1.  )  .x.  X )  e.  B )
292, 12, 16, 8grpinvid1 15577 . . . 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 1218 . . 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 2443 1  |-  ( ph  ->  ( ( N `  .1.  )  .x.  X )  =  ( N `  X ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    = wceq 1369    e. wcel 1756   ` cfv 5413  (class class class)co 6086   Basecbs 14166   +g cplusg 14230   .rcmulr 14231   0gc0g 14370   Grpcgrp 15402   invgcminusg 15403   1rcur 16593   Ringcrg 16635
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2419  ax-rep 4398  ax-sep 4408  ax-nul 4416  ax-pow 4465  ax-pr 4526  ax-un 6367  ax-cnex 9330  ax-resscn 9331  ax-1cn 9332  ax-icn 9333  ax-addcl 9334  ax-addrcl 9335  ax-mulcl 9336  ax-mulrcl 9337  ax-mulcom 9338  ax-addass 9339  ax-mulass 9340  ax-distr 9341  ax-i2m1 9342  ax-1ne0 9343  ax-1rid 9344  ax-rnegex 9345  ax-rrecex 9346  ax-cnre 9347  ax-pre-lttri 9348  ax-pre-lttrn 9349  ax-pre-ltadd 9350  ax-pre-mulgt0 9351
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2256  df-mo 2257  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-nel 2604  df-ral 2715  df-rex 2716  df-reu 2717  df-rmo 2718  df-rab 2719  df-v 2969  df-sbc 3182  df-csb 3284  df-dif 3326  df-un 3328  df-in 3330  df-ss 3337  df-pss 3339  df-nul 3633  df-if 3787  df-pw 3857  df-sn 3873  df-pr 3875  df-tp 3877  df-op 3879  df-uni 4087  df-iun 4168  df-br 4288  df-opab 4346  df-mpt 4347  df-tr 4381  df-eprel 4627  df-id 4631  df-po 4636  df-so 4637  df-fr 4674  df-we 4676  df-ord 4717  df-on 4718  df-lim 4719  df-suc 4720  df-xp 4841  df-rel 4842  df-cnv 4843  df-co 4844  df-dm 4845  df-rn 4846  df-res 4847  df-ima 4848  df-iota 5376  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-riota 6047  df-ov 6089  df-oprab 6090  df-mpt2 6091  df-om 6472  df-recs 6824  df-rdg 6858  df-er 7093  df-en 7303  df-dom 7304  df-sdom 7305  df-pnf 9412  df-mnf 9413  df-xr 9414  df-ltxr 9415  df-le 9416  df-sub 9589  df-neg 9590  df-nn 10315  df-2 10372  df-ndx 14169  df-slot 14170  df-base 14171  df-sets 14172  df-plusg 14243  df-0g 14372  df-mnd 15407  df-grp 15536  df-minusg 15537  df-mgp 16582  df-ur 16594  df-rng 16637
This theorem is referenced by:  rngmneg1  16677  dvdsrneg  16736  abvneg  16899  lmodvsneg  16969  lmodsubvs  16981  lmodsubdi  16982  lmodsubdir  16983  lmodvsinv  17097  mplind  17564  mdetralt  18394  m2detleiblem7  18413  lflsub  32605  baerlem3lem1  35245
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