Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  rngorz Structured version   Unicode version

Theorem rngorz 21992
 Description: The zero of a unital ring is a right absorbing element. (Contributed by FL, 31-Aug-2009.) (New usage is discouraged.)
Hypotheses
Ref Expression
ringlz.1 GId
ringlz.2
ringlz.3
ringlz.4
Assertion
Ref Expression
rngorz

Proof of Theorem rngorz
StepHypRef Expression
1 ringlz.3 . . . . . . 7
21rngogrpo 21980 . . . . . 6
3 ringlz.2 . . . . . . . 8
4 ringlz.1 . . . . . . . 8 GId
53, 4grpoidcl 21807 . . . . . . 7
63, 4grpolid 21809 . . . . . . 7
75, 6mpdan 651 . . . . . 6
82, 7syl 16 . . . . 5
98adantr 453 . . . 4
109oveq2d 6099 . . 3
11 simpr 449 . . . . 5
121, 3, 4rngo0cl 21988 . . . . . 6
1312adantr 453 . . . . 5
1411, 13, 133jca 1135 . . . 4
15 ringlz.4 . . . . 5
161, 15, 3rngodi 21975 . . . 4
1714, 16syldan 458 . . 3
182adantr 453 . . . 4
191, 15, 3rngocl 21972 . . . . 5
2013, 19mpd3an3 1281 . . . 4
213, 4grpolid 21809 . . . . 5
2221eqcomd 2443 . . . 4
2318, 20, 22syl2anc 644 . . 3
2410, 17, 233eqtr3d 2478 . 2
253grporcan 21811 . . 3
2618, 20, 13, 20, 25syl13anc 1187 . 2
2724, 26mpbid 203 1
 Colors of variables: wff set class Syntax hints:   wi 4   wb 178   wa 360   w3a 937   wceq 1653   wcel 1726   crn 4881  cfv 5456  (class class class)co 6083  c1st 6349  c2nd 6350  cgr 21776  GIdcgi 21777  crngo 21965 This theorem is referenced by:  rngoueqz  22020  zerdivemp1  22024  rngoridfz  22025  rngonegmn1r  26568  zerdivemp1x  26573  0idl  26637  keridl  26644 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 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-sep 4332  ax-nul 4340  ax-pow 4379  ax-pr 4405  ax-un 4703 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-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 4215  df-opab 4269  df-mpt 4270  df-id 4500  df-xp 4886  df-rel 4887  df-cnv 4888  df-co 4889  df-dm 4890  df-rn 4891  df-iota 5420  df-fun 5458  df-fn 5459  df-f 5460  df-fo 5462  df-fv 5464  df-ov 6086  df-1st 6351  df-2nd 6352  df-riota 6551  df-grpo 21781  df-gid 21782  df-ablo 21872  df-rngo 21966
 Copyright terms: Public domain W3C validator