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Theorem rngnegl 15380
Description: Negation in a ring is the same as left multiplication by -1. (rngonegmn1l 26580 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
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 15361 . . . . . 6  |-  ( R  e.  Ring  ->  .1.  e.  B )
51, 4syl 15 . . . . 5  |-  ( ph  ->  .1.  e.  B )
6 rnggrp 15346 . . . . . . 7  |-  ( R  e.  Ring  ->  R  e. 
Grp )
71, 6syl 15 . . . . . 6  |-  ( ph  ->  R  e.  Grp )
8 rngnegl.n . . . . . . 7  |-  N  =  ( inv g `  R )
92, 8grpinvcl 14527 . . . . . 6  |-  ( ( R  e.  Grp  /\  .1.  e.  B )  -> 
( N `  .1.  )  e.  B )
107, 5, 9syl2anc 642 . . . . 5  |-  ( ph  ->  ( N `  .1.  )  e.  B )
11 rngnegl.x . . . . 5  |-  ( ph  ->  X  e.  B )
12 eqid 2283 . . . . . 6  |-  ( +g  `  R )  =  ( +g  `  R )
13 rngnegl.t . . . . . 6  |-  .x.  =  ( .r `  R )
142, 12, 13rngdir 15360 . . . . 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 1184 . . . 4  |-  ( ph  ->  ( (  .1.  ( +g  `  R ) ( N `  .1.  )
)  .x.  X )  =  ( (  .1. 
.x.  X ) ( +g  `  R ) ( ( N `  .1.  )  .x.  X ) ) )
16 eqid 2283 . . . . . . . 8  |-  ( 0g
`  R )  =  ( 0g `  R
)
172, 12, 16, 8grprinv 14529 . . . . . . 7  |-  ( ( R  e.  Grp  /\  .1.  e.  B )  -> 
(  .1.  ( +g  `  R ) ( N `
 .1.  ) )  =  ( 0g `  R ) )
187, 5, 17syl2anc 642 . . . . . 6  |-  ( ph  ->  (  .1.  ( +g  `  R ) ( N `
 .1.  ) )  =  ( 0g `  R ) )
1918oveq1d 5873 . . . . 5  |-  ( ph  ->  ( (  .1.  ( +g  `  R ) ( N `  .1.  )
)  .x.  X )  =  ( ( 0g
`  R )  .x.  X ) )
202, 13, 16rnglz 15377 . . . . . 6  |-  ( ( R  e.  Ring  /\  X  e.  B )  ->  (
( 0g `  R
)  .x.  X )  =  ( 0g `  R ) )
211, 11, 20syl2anc 642 . . . . 5  |-  ( ph  ->  ( ( 0g `  R )  .x.  X
)  =  ( 0g
`  R ) )
2219, 21eqtrd 2315 . . . 4  |-  ( ph  ->  ( (  .1.  ( +g  `  R ) ( N `  .1.  )
)  .x.  X )  =  ( 0g `  R ) )
232, 13, 3rnglidm 15364 . . . . . 6  |-  ( ( R  e.  Ring  /\  X  e.  B )  ->  (  .1.  .x.  X )  =  X )
241, 11, 23syl2anc 642 . . . . 5  |-  ( ph  ->  (  .1.  .x.  X
)  =  X )
2524oveq1d 5873 . . . 4  |-  ( ph  ->  ( (  .1.  .x.  X ) ( +g  `  R ) ( ( N `  .1.  )  .x.  X ) )  =  ( X ( +g  `  R ) ( ( N `  .1.  )  .x.  X ) ) )
2615, 22, 253eqtr3rd 2324 . . 3  |-  ( ph  ->  ( X ( +g  `  R ) ( ( N `  .1.  )  .x.  X ) )  =  ( 0g `  R
) )
272, 13rngcl 15354 . . . . 5  |-  ( ( R  e.  Ring  /\  ( N `  .1.  )  e.  B  /\  X  e.  B )  ->  (
( N `  .1.  )  .x.  X )  e.  B )
281, 10, 11, 27syl3anc 1182 . . . 4  |-  ( ph  ->  ( ( N `  .1.  )  .x.  X )  e.  B )
292, 12, 16, 8grpinvid1 14530 . . . 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 1182 . . 3  |-  ( ph  ->  ( ( N `  X )  =  ( ( N `  .1.  )  .x.  X )  <->  ( X
( +g  `  R ) ( ( N `  .1.  )  .x.  X ) )  =  ( 0g
`  R ) ) )
3126, 30mpbird 223 . 2  |-  ( ph  ->  ( N `  X
)  =  ( ( N `  .1.  )  .x.  X ) )
3231eqcomd 2288 1  |-  ( ph  ->  ( ( N `  .1.  )  .x.  X )  =  ( N `  X ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    = wceq 1623    e. wcel 1684   ` cfv 5255  (class class class)co 5858   Basecbs 13148   +g cplusg 13208   .rcmulr 13209   0gc0g 13400   Grpcgrp 14362   inv gcminusg 14363   Ringcrg 15337   1rcur 15339
This theorem is referenced by:  rngmneg1  15382  dvdsrneg  15436  abvneg  15599  lmodvsnegOLD  15668  lmodvsneg  15669  lmodsubvs  15681  lmodsubdi  15682  lmodsubdir  15683  lmodvsinv  15793  mplind  16243  lflsub  29257  baerlem3lem1  31897
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-cnex 8793  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813  ax-pre-mulgt0 8814
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 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-riota 6304  df-recs 6388  df-rdg 6423  df-er 6660  df-en 6864  df-dom 6865  df-sdom 6866  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-sub 9039  df-neg 9040  df-nn 9747  df-2 9804  df-ndx 13151  df-slot 13152  df-base 13153  df-sets 13154  df-plusg 13221  df-0g 13404  df-mnd 14367  df-grp 14489  df-minusg 14490  df-mgp 15326  df-rng 15340  df-ur 15342
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