Mathbox for Jeff Madsen < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  rngonegmn1r Structured version   Unicode version

Theorem rngonegmn1r 26580
 Description: Negation in a ring is the same as right multiplication by . (Contributed by Jeff Madsen, 19-Jun-2010.)
Hypotheses
Ref Expression
ringneg.1
ringneg.2
ringneg.3
ringneg.4
ringneg.5 GId
Assertion
Ref Expression
rngonegmn1r

Proof of Theorem rngonegmn1r
StepHypRef Expression
1 ringneg.3 . . . . . . . . 9
2 ringneg.1 . . . . . . . . . 10
32rneqi 5099 . . . . . . . . 9
41, 3eqtri 2458 . . . . . . . 8
5 ringneg.2 . . . . . . . 8
6 ringneg.5 . . . . . . . 8 GId
74, 5, 6rngo1cl 22022 . . . . . . 7
8 ringneg.4 . . . . . . . 8
92, 1, 8rngonegcl 26575 . . . . . . 7
107, 9mpdan 651 . . . . . 6
1110adantr 453 . . . . 5
127adantr 453 . . . . 5
1311, 12jca 520 . . . 4
142, 5, 1rngodi 21978 . . . . . 6
15143exp2 1172 . . . . 5
1615imp43 580 . . . 4
1713, 16mpdan 651 . . 3
18 eqid 2438 . . . . . . . 8 GId GId
192, 1, 8, 18rngoaddneg2 26577 . . . . . . 7 GId
207, 19mpdan 651 . . . . . 6 GId
2120adantr 453 . . . . 5 GId
2221oveq2d 6100 . . . 4 GId
2318, 1, 2, 5rngorz 21995 . . . 4 GId GId
2422, 23eqtrd 2470 . . 3 GId
255, 4, 6rngoridm 22018 . . . 4
2625oveq2d 6100 . . 3
2717, 24, 263eqtr3rd 2479 . 2 GId
282, 5, 1rngocl 21975 . . . 4
2911, 28mpd3an3 1281 . . 3
302rngogrpo 21983 . . . 4
311, 18, 8grpoinvid2 21824 . . . 4 GId
3230, 31syl3an1 1218 . . 3 GId
3329, 32mpd3an3 1281 . 2 GId
3427, 33mpbird 225 1
 Colors of variables: wff set class Syntax hints:   wi 4   wb 178   wa 360   wceq 1653   wcel 1726   crn 4882  cfv 5457  (class class class)co 6084  c1st 6350  c2nd 6351  cgr 21779  GIdcgi 21780  cgn 21781  crngo 21968 This theorem is referenced by:  rngonegrmul  26582 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-rep 4323  ax-sep 4333  ax-nul 4341  ax-pow 4380  ax-pr 4406  ax-un 4704 This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-reu 2714  df-rmo 2715  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-iun 4097  df-br 4216  df-opab 4270  df-mpt 4271  df-id 4501  df-xp 4887  df-rel 4888  df-cnv 4889  df-co 4890  df-dm 4891  df-rn 4892  df-res 4893  df-ima 4894  df-iota 5421  df-fun 5459  df-fn 5460  df-f 5461  df-f1 5462  df-fo 5463  df-f1o 5464  df-fv 5465  df-ov 6087  df-1st 6352  df-2nd 6353  df-riota 6552  df-grpo 21784  df-gid 21785  df-ginv 21786  df-ablo 21875  df-ass 21906  df-exid 21908  df-mgm 21912  df-sgr 21924  df-mndo 21931  df-rngo 21969
 Copyright terms: Public domain W3C validator