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

Theorem rngorz 25195
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 25183 . . . . . 6  |-  ( R  e.  RingOps  ->  G  e.  GrpOp )
3 ringlz.2 . . . . . . . 8  |-  X  =  ran  G
4 ringlz.1 . . . . . . . 8  |-  Z  =  (GId `  G )
53, 4grpoidcl 25010 . . . . . . 7  |-  ( G  e.  GrpOp  ->  Z  e.  X )
63, 4grpolid 25012 . . . . . . 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 6310 . . 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 25191 . . . . . 6  |-  ( R  e.  RingOps  ->  Z  e.  X
)
1312adantr 465 . . . . 5  |-  ( ( R  e.  RingOps  /\  A  e.  X )  ->  Z  e.  X )
1411, 13, 133jca 1176 . . . 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 25178 . . . 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 25175 . . . . 5  |-  ( ( R  e.  RingOps  /\  A  e.  X  /\  Z  e.  X )  ->  ( A H Z )  e.  X )
2013, 19mpd3an3 1325 . . . 4  |-  ( ( R  e.  RingOps  /\  A  e.  X )  ->  ( A H Z )  e.  X )
213, 4grpolid 25012 . . . . 5  |-  ( ( G  e.  GrpOp  /\  ( A H Z )  e.  X )  ->  ( Z G ( A H Z ) )  =  ( A H Z ) )
2221eqcomd 2475 . . . 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 2516 . 2  |-  ( ( R  e.  RingOps  /\  A  e.  X )  ->  (
( A H Z ) G ( A H Z ) )  =  ( Z G ( A H Z ) ) )
253grporcan 25014 . . 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 1230 . 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 973    = wceq 1379    e. wcel 1767   ran crn 5005   ` cfv 5593  (class class class)co 6294   1stc1st 6792   2ndc2nd 6793   GrpOpcgr 24979  GIdcgi 24980   RingOpscrngo 25168
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4573  ax-nul 4581  ax-pow 4630  ax-pr 4691  ax-un 6586
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2822  df-rex 2823  df-reu 2824  df-rab 2826  df-v 3120  df-sbc 3337  df-csb 3441  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-nul 3791  df-if 3945  df-sn 4033  df-pr 4035  df-op 4039  df-uni 4251  df-iun 4332  df-br 4453  df-opab 4511  df-mpt 4512  df-id 4800  df-xp 5010  df-rel 5011  df-cnv 5012  df-co 5013  df-dm 5014  df-rn 5015  df-iota 5556  df-fun 5595  df-fn 5596  df-f 5597  df-fo 5599  df-fv 5601  df-riota 6255  df-ov 6297  df-1st 6794  df-2nd 6795  df-grpo 24984  df-gid 24985  df-ablo 25075  df-rngo 25169
This theorem is referenced by:  rngoueqz  25223  zerdivemp1  25227  rngoridfz  25228  rngonegmn1r  30248  zerdivemp1x  30253  0idl  30317  keridl  30324
  Copyright terms: Public domain W3C validator