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Theorem rngorz 21069
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 21057 . . . . . 6  |-  ( R  e.  RingOps  ->  G  e.  GrpOp )
3 ringlz.2 . . . . . . . 8  |-  X  =  ran  G
4 ringlz.1 . . . . . . . 8  |-  Z  =  (GId `  G )
53, 4grpoidcl 20884 . . . . . . 7  |-  ( G  e.  GrpOp  ->  Z  e.  X )
63, 4grpolid 20886 . . . . . . 7  |-  ( ( G  e.  GrpOp  /\  Z  e.  X )  ->  ( Z G Z )  =  Z )
75, 6mpdan 649 . . . . . 6  |-  ( G  e.  GrpOp  ->  ( Z G Z )  =  Z )
82, 7syl 15 . . . . 5  |-  ( R  e.  RingOps  ->  ( Z G Z )  =  Z )
98adantr 451 . . . 4  |-  ( ( R  e.  RingOps  /\  A  e.  X )  ->  ( Z G Z )  =  Z )
109oveq2d 5874 . . 3  |-  ( ( R  e.  RingOps  /\  A  e.  X )  ->  ( A H ( Z G Z ) )  =  ( A H Z ) )
11 simpr 447 . . . . 5  |-  ( ( R  e.  RingOps  /\  A  e.  X )  ->  A  e.  X )
121, 3, 4rngo0cl 21065 . . . . . 6  |-  ( R  e.  RingOps  ->  Z  e.  X
)
1312adantr 451 . . . . 5  |-  ( ( R  e.  RingOps  /\  A  e.  X )  ->  Z  e.  X )
1411, 13, 133jca 1132 . . . 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 21052 . . . 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 456 . . 3  |-  ( ( R  e.  RingOps  /\  A  e.  X )  ->  ( A H ( Z G Z ) )  =  ( ( A H Z ) G ( A H Z ) ) )
182adantr 451 . . . 4  |-  ( ( R  e.  RingOps  /\  A  e.  X )  ->  G  e.  GrpOp )
191, 15, 3rngocl 21049 . . . . 5  |-  ( ( R  e.  RingOps  /\  A  e.  X  /\  Z  e.  X )  ->  ( A H Z )  e.  X )
2013, 19mpd3an3 1278 . . . 4  |-  ( ( R  e.  RingOps  /\  A  e.  X )  ->  ( A H Z )  e.  X )
213, 4grpolid 20886 . . . . 5  |-  ( ( G  e.  GrpOp  /\  ( A H Z )  e.  X )  ->  ( Z G ( A H Z ) )  =  ( A H Z ) )
2221eqcomd 2288 . . . 4  |-  ( ( G  e.  GrpOp  /\  ( A H Z )  e.  X )  ->  ( A H Z )  =  ( Z G ( A H Z ) ) )
2318, 20, 22syl2anc 642 . . 3  |-  ( ( R  e.  RingOps  /\  A  e.  X )  ->  ( A H Z )  =  ( Z G ( A H Z ) ) )
2410, 17, 233eqtr3d 2323 . 2  |-  ( ( R  e.  RingOps  /\  A  e.  X )  ->  (
( A H Z ) G ( A H Z ) )  =  ( Z G ( A H Z ) ) )
253grporcan 20888 . . 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 1184 . 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 201 1  |-  ( ( R  e.  RingOps  /\  A  e.  X )  ->  ( A H Z )  =  Z )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    /\ w3a 934    = wceq 1623    e. wcel 1684   ran crn 4690   ` cfv 5255  (class class class)co 5858   1stc1st 6120   2ndc2nd 6121   GrpOpcgr 20853  GIdcgi 20854   RingOpscrngo 21042
This theorem is referenced by:  rngoueqz  21097  multinvb  25423  zerdivemp1  25436  rngoridfz  25437  rngonegmn1r  26581  zerdivemp1x  26586  0idl  26650  keridl  26657
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-reu 2550  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-fo 5261  df-fv 5263  df-ov 5861  df-1st 6122  df-2nd 6123  df-riota 6304  df-grpo 20858  df-gid 20859  df-ablo 20949  df-rngo 21043
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