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Theorem rngorz 25276
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  |-  Z  =  (GId `  G )
ringlz.2  |-  X  =  ran  G
ringlz.3  |-  G  =  ( 1st `  R
)
ringlz.4  |-  H  =  ( 2nd `  R
)
Assertion
Ref Expression
rngorz  |-  ( ( R  e.  RingOps  /\  A  e.  X )  ->  ( A H Z )  =  Z )

Proof of Theorem rngorz
StepHypRef Expression
1 ringlz.3 . . . . . . 7  |-  G  =  ( 1st `  R
)
21rngogrpo 25264 . . . . . 6  |-  ( R  e.  RingOps  ->  G  e.  GrpOp )
3 ringlz.2 . . . . . . . 8  |-  X  =  ran  G
4 ringlz.1 . . . . . . . 8  |-  Z  =  (GId `  G )
53, 4grpoidcl 25091 . . . . . . 7  |-  ( G  e.  GrpOp  ->  Z  e.  X )
63, 4grpolid 25093 . . . . . . 7  |-  ( ( G  e.  GrpOp  /\  Z  e.  X )  ->  ( Z G Z )  =  Z )
75, 6mpdan 668 . . . . . 6  |-  ( G  e.  GrpOp  ->  ( Z G Z )  =  Z )
82, 7syl 16 . . . . 5  |-  ( R  e.  RingOps  ->  ( Z G Z )  =  Z )
98adantr 465 . . . 4  |-  ( ( R  e.  RingOps  /\  A  e.  X )  ->  ( Z G Z )  =  Z )
109oveq2d 6297 . . 3  |-  ( ( R  e.  RingOps  /\  A  e.  X )  ->  ( A H ( Z G Z ) )  =  ( A H Z ) )
11 simpr 461 . . . . 5  |-  ( ( R  e.  RingOps  /\  A  e.  X )  ->  A  e.  X )
121, 3, 4rngo0cl 25272 . . . . . 6  |-  ( R  e.  RingOps  ->  Z  e.  X
)
1312adantr 465 . . . . 5  |-  ( ( R  e.  RingOps  /\  A  e.  X )  ->  Z  e.  X )
1411, 13, 133jca 1177 . . . 4  |-  ( ( R  e.  RingOps  /\  A  e.  X )  ->  ( A  e.  X  /\  Z  e.  X  /\  Z  e.  X )
)
15 ringlz.4 . . . . 5  |-  H  =  ( 2nd `  R
)
161, 15, 3rngodi 25259 . . . 4  |-  ( ( R  e.  RingOps  /\  ( A  e.  X  /\  Z  e.  X  /\  Z  e.  X )
)  ->  ( A H ( Z G Z ) )  =  ( ( A H Z ) G ( A H Z ) ) )
1714, 16syldan 470 . . 3  |-  ( ( R  e.  RingOps  /\  A  e.  X )  ->  ( A H ( Z G Z ) )  =  ( ( A H Z ) G ( A H Z ) ) )
182adantr 465 . . . 4  |-  ( ( R  e.  RingOps  /\  A  e.  X )  ->  G  e.  GrpOp )
191, 15, 3rngocl 25256 . . . . 5  |-  ( ( R  e.  RingOps  /\  A  e.  X  /\  Z  e.  X )  ->  ( A H Z )  e.  X )
2013, 19mpd3an3 1326 . . . 4  |-  ( ( R  e.  RingOps  /\  A  e.  X )  ->  ( A H Z )  e.  X )
213, 4grpolid 25093 . . . . 5  |-  ( ( G  e.  GrpOp  /\  ( A H Z )  e.  X )  ->  ( Z G ( A H Z ) )  =  ( A H Z ) )
2221eqcomd 2451 . . . 4  |-  ( ( G  e.  GrpOp  /\  ( A H Z )  e.  X )  ->  ( A H Z )  =  ( Z G ( A H Z ) ) )
2318, 20, 22syl2anc 661 . . 3  |-  ( ( R  e.  RingOps  /\  A  e.  X )  ->  ( A H Z )  =  ( Z G ( A H Z ) ) )
2410, 17, 233eqtr3d 2492 . 2  |-  ( ( R  e.  RingOps  /\  A  e.  X )  ->  (
( A H Z ) G ( A H Z ) )  =  ( Z G ( A H Z ) ) )
253grporcan 25095 . . 3  |-  ( ( G  e.  GrpOp  /\  (
( A H Z )  e.  X  /\  Z  e.  X  /\  ( A H Z )  e.  X ) )  ->  ( ( ( A H Z ) G ( A H Z ) )  =  ( Z G ( A H Z ) )  <->  ( A H Z )  =  Z ) )
2618, 20, 13, 20, 25syl13anc 1231 . 2  |-  ( ( R  e.  RingOps  /\  A  e.  X )  ->  (
( ( A H Z ) G ( A H Z ) )  =  ( Z G ( A H Z ) )  <->  ( A H Z )  =  Z ) )
2724, 26mpbid 210 1  |-  ( ( R  e.  RingOps  /\  A  e.  X )  ->  ( A H Z )  =  Z )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 974    = wceq 1383    e. wcel 1804   ran crn 4990   ` cfv 5578  (class class class)co 6281   1stc1st 6783   2ndc2nd 6784   GrpOpcgr 25060  GIdcgi 25061   RingOpscrngo 25249
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1605  ax-4 1618  ax-5 1691  ax-6 1734  ax-7 1776  ax-8 1806  ax-9 1808  ax-10 1823  ax-11 1828  ax-12 1840  ax-13 1985  ax-ext 2421  ax-sep 4558  ax-nul 4566  ax-pow 4615  ax-pr 4676  ax-un 6577
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 976  df-tru 1386  df-ex 1600  df-nf 1604  df-sb 1727  df-eu 2272  df-mo 2273  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2593  df-ne 2640  df-ral 2798  df-rex 2799  df-reu 2800  df-rab 2802  df-v 3097  df-sbc 3314  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3771  df-if 3927  df-sn 4015  df-pr 4017  df-op 4021  df-uni 4235  df-iun 4317  df-br 4438  df-opab 4496  df-mpt 4497  df-id 4785  df-xp 4995  df-rel 4996  df-cnv 4997  df-co 4998  df-dm 4999  df-rn 5000  df-iota 5541  df-fun 5580  df-fn 5581  df-f 5582  df-fo 5584  df-fv 5586  df-riota 6242  df-ov 6284  df-1st 6785  df-2nd 6786  df-grpo 25065  df-gid 25066  df-ablo 25156  df-rngo 25250
This theorem is referenced by:  rngoueqz  25304  zerdivemp1  25308  rngoridfz  25309  rngonegmn1r  30328  zerdivemp1x  30333  0idl  30397  keridl  30404
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