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Theorem rngnegr 13021
Description: Negation in a ring is the same as right multiplication by -1. (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 12977 . . . . . . 7  |-  ( R  e.  Ring  ->  R  e. 
Grp )
41, 3syl 14 . . . . . 6  |-  ( ph  ->  R  e.  Grp )
5 ringnegl.b . . . . . . . 8  |-  B  =  ( Base `  R
)
6 ringnegl.u . . . . . . . 8  |-  .1.  =  ( 1r `  R )
75, 6ringidcl 12996 . . . . . . 7  |-  ( R  e.  Ring  ->  .1.  e.  B )
81, 7syl 14 . . . . . 6  |-  ( ph  ->  .1.  e.  B )
9 ringnegl.n . . . . . . 7  |-  N  =  ( invg `  R )
105, 9grpinvcl 12781 . . . . . 6  |-  ( ( R  e.  Grp  /\  .1.  e.  B )  -> 
( N `  .1.  )  e.  B )
114, 8, 10syl2anc 411 . . . . 5  |-  ( ph  ->  ( N `  .1.  )  e.  B )
12 eqid 2175 . . . . . 6  |-  ( +g  `  R )  =  ( +g  `  R )
13 ringnegl.t . . . . . 6  |-  .x.  =  ( .r `  R )
145, 12, 13ringdi 12994 . . . . 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 1240 . . . 4  |-  ( ph  ->  ( X  .x.  (
( N `  .1.  ) ( +g  `  R
)  .1.  ) )  =  ( ( X 
.x.  ( N `  .1.  ) ) ( +g  `  R ) ( X 
.x.  .1.  ) )
)
16 eqid 2175 . . . . . . . 8  |-  ( 0g
`  R )  =  ( 0g `  R
)
175, 12, 16, 9grplinv 12782 . . . . . . 7  |-  ( ( R  e.  Grp  /\  .1.  e.  B )  -> 
( ( N `  .1.  ) ( +g  `  R
)  .1.  )  =  ( 0g `  R
) )
184, 8, 17syl2anc 411 . . . . . 6  |-  ( ph  ->  ( ( N `  .1.  ) ( +g  `  R
)  .1.  )  =  ( 0g `  R
) )
1918oveq2d 5881 . . . . 5  |-  ( ph  ->  ( X  .x.  (
( N `  .1.  ) ( +g  `  R
)  .1.  ) )  =  ( X  .x.  ( 0g `  R ) ) )
205, 13, 16ringrz 13015 . . . . . 6  |-  ( ( R  e.  Ring  /\  X  e.  B )  ->  ( X  .x.  ( 0g `  R ) )  =  ( 0g `  R
) )
211, 2, 20syl2anc 411 . . . . 5  |-  ( ph  ->  ( X  .x.  ( 0g `  R ) )  =  ( 0g `  R ) )
2219, 21eqtrd 2208 . . . 4  |-  ( ph  ->  ( X  .x.  (
( N `  .1.  ) ( +g  `  R
)  .1.  ) )  =  ( 0g `  R ) )
235, 13, 6ringridm 13000 . . . . . 6  |-  ( ( R  e.  Ring  /\  X  e.  B )  ->  ( X  .x.  .1.  )  =  X )
241, 2, 23syl2anc 411 . . . . 5  |-  ( ph  ->  ( X  .x.  .1.  )  =  X )
2524oveq2d 5881 . . . 4  |-  ( ph  ->  ( ( X  .x.  ( N `  .1.  )
) ( +g  `  R
) ( X  .x.  .1.  ) )  =  ( ( X  .x.  ( N `  .1.  ) ) ( +g  `  R
) X ) )
2615, 22, 253eqtr3rd 2217 . . 3  |-  ( ph  ->  ( ( X  .x.  ( N `  .1.  )
) ( +g  `  R
) X )  =  ( 0g `  R
) )
275, 13ringcl 12989 . . . . 5  |-  ( ( R  e.  Ring  /\  X  e.  B  /\  ( N `  .1.  )  e.  B )  ->  ( X  .x.  ( N `  .1.  ) )  e.  B
)
281, 2, 11, 27syl3anc 1238 . . . 4  |-  ( ph  ->  ( X  .x.  ( N `  .1.  ) )  e.  B )
295, 12, 16, 9grpinvid2 12785 . . . 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 1238 . . 3  |-  ( ph  ->  ( ( N `  X )  =  ( X  .x.  ( N `
 .1.  ) )  <-> 
( ( X  .x.  ( N `  .1.  )
) ( +g  `  R
) X )  =  ( 0g `  R
) ) )
3126, 30mpbird 167 . 2  |-  ( ph  ->  ( N `  X
)  =  ( X 
.x.  ( N `  .1.  ) ) )
3231eqcomd 2181 1  |-  ( ph  ->  ( X  .x.  ( N `  .1.  ) )  =  ( N `  X ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 105    = wceq 1353    e. wcel 2146   ` cfv 5208  (class class class)co 5865   Basecbs 12427   +g cplusg 12491   .rcmulr 12492   0gc0g 12625   Grpcgrp 12737   invgcminusg 12738   1rcur 12935   Ringcrg 12972
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 614  ax-in2 615  ax-io 709  ax-5 1445  ax-7 1446  ax-gen 1447  ax-ie1 1491  ax-ie2 1492  ax-8 1502  ax-10 1503  ax-11 1504  ax-i12 1505  ax-bndl 1507  ax-4 1508  ax-17 1524  ax-i9 1528  ax-ial 1532  ax-i5r 1533  ax-13 2148  ax-14 2149  ax-ext 2157  ax-coll 4113  ax-sep 4116  ax-pow 4169  ax-pr 4203  ax-un 4427  ax-setind 4530  ax-cnex 7877  ax-resscn 7878  ax-1cn 7879  ax-1re 7880  ax-icn 7881  ax-addcl 7882  ax-addrcl 7883  ax-mulcl 7884  ax-addcom 7886  ax-addass 7888  ax-i2m1 7891  ax-0lt1 7892  ax-0id 7894  ax-rnegex 7895  ax-pre-ltirr 7898  ax-pre-ltadd 7902
This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1459  df-sb 1761  df-eu 2027  df-mo 2028  df-clab 2162  df-cleq 2168  df-clel 2171  df-nfc 2306  df-ne 2346  df-nel 2441  df-ral 2458  df-rex 2459  df-reu 2460  df-rmo 2461  df-rab 2462  df-v 2737  df-sbc 2961  df-csb 3056  df-dif 3129  df-un 3131  df-in 3133  df-ss 3140  df-nul 3421  df-pw 3574  df-sn 3595  df-pr 3596  df-op 3598  df-uni 3806  df-int 3841  df-iun 3884  df-br 3999  df-opab 4060  df-mpt 4061  df-id 4287  df-xp 4626  df-rel 4627  df-cnv 4628  df-co 4629  df-dm 4630  df-rn 4631  df-res 4632  df-ima 4633  df-iota 5170  df-fun 5210  df-fn 5211  df-f 5212  df-f1 5213  df-fo 5214  df-f1o 5215  df-fv 5216  df-riota 5821  df-ov 5868  df-oprab 5869  df-mpo 5870  df-pnf 7968  df-mnf 7969  df-ltxr 7971  df-inn 8891  df-2 8949  df-3 8950  df-ndx 12430  df-slot 12431  df-base 12433  df-sets 12434  df-plusg 12504  df-mulr 12505  df-0g 12627  df-mgm 12639  df-sgrp 12672  df-mnd 12682  df-grp 12740  df-minusg 12741  df-mgp 12926  df-ur 12936  df-ring 12974
This theorem is referenced by:  ringmneg2  13023
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