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Theorem ringnegl 13041
Description: Negation in a ring is the same as left 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
ringnegl  |-  ( ph  ->  ( ( N `  .1.  )  .x.  X )  =  ( N `  X ) )

Proof of Theorem ringnegl
StepHypRef Expression
1 ringnegl.r . . . . 5  |-  ( ph  ->  R  e.  Ring )
2 ringnegl.b . . . . . . 7  |-  B  =  ( Base `  R
)
3 ringnegl.u . . . . . . 7  |-  .1.  =  ( 1r `  R )
42, 3ringidcl 13016 . . . . . 6  |-  ( R  e.  Ring  ->  .1.  e.  B )
51, 4syl 14 . . . . 5  |-  ( ph  ->  .1.  e.  B )
6 ringgrp 12997 . . . . . . 7  |-  ( R  e.  Ring  ->  R  e. 
Grp )
71, 6syl 14 . . . . . 6  |-  ( ph  ->  R  e.  Grp )
8 ringnegl.n . . . . . . 7  |-  N  =  ( invg `  R )
92, 8grpinvcl 12798 . . . . . 6  |-  ( ( R  e.  Grp  /\  .1.  e.  B )  -> 
( N `  .1.  )  e.  B )
107, 5, 9syl2anc 411 . . . . 5  |-  ( ph  ->  ( N `  .1.  )  e.  B )
11 ringnegl.x . . . . 5  |-  ( ph  ->  X  e.  B )
12 eqid 2177 . . . . . 6  |-  ( +g  `  R )  =  ( +g  `  R )
13 ringnegl.t . . . . . 6  |-  .x.  =  ( .r `  R )
142, 12, 13ringdir 13015 . . . . 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 1240 . . . 4  |-  ( ph  ->  ( (  .1.  ( +g  `  R ) ( N `  .1.  )
)  .x.  X )  =  ( (  .1. 
.x.  X ) ( +g  `  R ) ( ( N `  .1.  )  .x.  X ) ) )
16 eqid 2177 . . . . . . . 8  |-  ( 0g
`  R )  =  ( 0g `  R
)
172, 12, 16, 8grprinv 12800 . . . . . . 7  |-  ( ( R  e.  Grp  /\  .1.  e.  B )  -> 
(  .1.  ( +g  `  R ) ( N `
 .1.  ) )  =  ( 0g `  R ) )
187, 5, 17syl2anc 411 . . . . . 6  |-  ( ph  ->  (  .1.  ( +g  `  R ) ( N `
 .1.  ) )  =  ( 0g `  R ) )
1918oveq1d 5883 . . . . 5  |-  ( ph  ->  ( (  .1.  ( +g  `  R ) ( N `  .1.  )
)  .x.  X )  =  ( ( 0g
`  R )  .x.  X ) )
202, 13, 16ringlz 13035 . . . . . 6  |-  ( ( R  e.  Ring  /\  X  e.  B )  ->  (
( 0g `  R
)  .x.  X )  =  ( 0g `  R ) )
211, 11, 20syl2anc 411 . . . . 5  |-  ( ph  ->  ( ( 0g `  R )  .x.  X
)  =  ( 0g
`  R ) )
2219, 21eqtrd 2210 . . . 4  |-  ( ph  ->  ( (  .1.  ( +g  `  R ) ( N `  .1.  )
)  .x.  X )  =  ( 0g `  R ) )
232, 13, 3ringlidm 13019 . . . . . 6  |-  ( ( R  e.  Ring  /\  X  e.  B )  ->  (  .1.  .x.  X )  =  X )
241, 11, 23syl2anc 411 . . . . 5  |-  ( ph  ->  (  .1.  .x.  X
)  =  X )
2524oveq1d 5883 . . . 4  |-  ( ph  ->  ( (  .1.  .x.  X ) ( +g  `  R ) ( ( N `  .1.  )  .x.  X ) )  =  ( X ( +g  `  R ) ( ( N `  .1.  )  .x.  X ) ) )
2615, 22, 253eqtr3rd 2219 . . 3  |-  ( ph  ->  ( X ( +g  `  R ) ( ( N `  .1.  )  .x.  X ) )  =  ( 0g `  R
) )
272, 13ringcl 13009 . . . . 5  |-  ( ( R  e.  Ring  /\  ( N `  .1.  )  e.  B  /\  X  e.  B )  ->  (
( N `  .1.  )  .x.  X )  e.  B )
281, 10, 11, 27syl3anc 1238 . . . 4  |-  ( ph  ->  ( ( N `  .1.  )  .x.  X )  e.  B )
292, 12, 16, 8grpinvid1 12801 . . . 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 1238 . . 3  |-  ( ph  ->  ( ( N `  X )  =  ( ( N `  .1.  )  .x.  X )  <->  ( X
( +g  `  R ) ( ( N `  .1.  )  .x.  X ) )  =  ( 0g
`  R ) ) )
3126, 30mpbird 167 . 2  |-  ( ph  ->  ( N `  X
)  =  ( ( N `  .1.  )  .x.  X ) )
3231eqcomd 2183 1  |-  ( ph  ->  ( ( N `  .1.  )  .x.  X )  =  ( N `  X ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 105    = wceq 1353    e. wcel 2148   ` cfv 5211  (class class class)co 5868   Basecbs 12432   +g cplusg 12505   .rcmulr 12506   0gc0g 12640   Grpcgrp 12754   invgcminusg 12755   1rcur 12955   Ringcrg 12992
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 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-coll 4115  ax-sep 4118  ax-pow 4171  ax-pr 4205  ax-un 4429  ax-setind 4532  ax-cnex 7880  ax-resscn 7881  ax-1cn 7882  ax-1re 7883  ax-icn 7884  ax-addcl 7885  ax-addrcl 7886  ax-mulcl 7887  ax-addcom 7889  ax-addass 7891  ax-i2m1 7894  ax-0lt1 7895  ax-0id 7897  ax-rnegex 7898  ax-pre-ltirr 7901  ax-pre-ltadd 7905
This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-nel 2443  df-ral 2460  df-rex 2461  df-reu 2462  df-rmo 2463  df-rab 2464  df-v 2739  df-sbc 2963  df-csb 3058  df-dif 3131  df-un 3133  df-in 3135  df-ss 3142  df-nul 3423  df-pw 3576  df-sn 3597  df-pr 3598  df-op 3600  df-uni 3808  df-int 3843  df-iun 3886  df-br 4001  df-opab 4062  df-mpt 4063  df-id 4289  df-xp 4628  df-rel 4629  df-cnv 4630  df-co 4631  df-dm 4632  df-rn 4633  df-res 4634  df-ima 4635  df-iota 5173  df-fun 5213  df-fn 5214  df-f 5215  df-f1 5216  df-fo 5217  df-f1o 5218  df-fv 5219  df-riota 5824  df-ov 5871  df-oprab 5872  df-mpo 5873  df-pnf 7971  df-mnf 7972  df-ltxr 7974  df-inn 8896  df-2 8954  df-3 8955  df-ndx 12435  df-slot 12436  df-base 12438  df-sets 12439  df-plusg 12518  df-mulr 12519  df-0g 12642  df-mgm 12654  df-sgrp 12687  df-mnd 12697  df-grp 12757  df-minusg 12758  df-mgp 12945  df-ur 12956  df-ring 12994
This theorem is referenced by:  ringmneg1  13043  dvdsrneg  13084
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