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Theorem rngorz 23808
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 23796 . . . . . 6  |-  ( R  e.  RingOps  ->  G  e.  GrpOp )
3 ringlz.2 . . . . . . . 8  |-  X  =  ran  G
4 ringlz.1 . . . . . . . 8  |-  Z  =  (GId `  G )
53, 4grpoidcl 23623 . . . . . . 7  |-  ( G  e.  GrpOp  ->  Z  e.  X )
63, 4grpolid 23625 . . . . . . 7  |-  ( ( G  e.  GrpOp  /\  Z  e.  X )  ->  ( Z G Z )  =  Z )
75, 6mpdan 663 . . . . . 6  |-  ( G  e.  GrpOp  ->  ( Z G Z )  =  Z )
82, 7syl 16 . . . . 5  |-  ( R  e.  RingOps  ->  ( Z G Z )  =  Z )
98adantr 462 . . . 4  |-  ( ( R  e.  RingOps  /\  A  e.  X )  ->  ( Z G Z )  =  Z )
109oveq2d 6106 . . 3  |-  ( ( R  e.  RingOps  /\  A  e.  X )  ->  ( A H ( Z G Z ) )  =  ( A H Z ) )
11 simpr 458 . . . . 5  |-  ( ( R  e.  RingOps  /\  A  e.  X )  ->  A  e.  X )
121, 3, 4rngo0cl 23804 . . . . . 6  |-  ( R  e.  RingOps  ->  Z  e.  X
)
1312adantr 462 . . . . 5  |-  ( ( R  e.  RingOps  /\  A  e.  X )  ->  Z  e.  X )
1411, 13, 133jca 1163 . . . 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 23791 . . . 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 467 . . 3  |-  ( ( R  e.  RingOps  /\  A  e.  X )  ->  ( A H ( Z G Z ) )  =  ( ( A H Z ) G ( A H Z ) ) )
182adantr 462 . . . 4  |-  ( ( R  e.  RingOps  /\  A  e.  X )  ->  G  e.  GrpOp )
191, 15, 3rngocl 23788 . . . . 5  |-  ( ( R  e.  RingOps  /\  A  e.  X  /\  Z  e.  X )  ->  ( A H Z )  e.  X )
2013, 19mpd3an3 1310 . . . 4  |-  ( ( R  e.  RingOps  /\  A  e.  X )  ->  ( A H Z )  e.  X )
213, 4grpolid 23625 . . . . 5  |-  ( ( G  e.  GrpOp  /\  ( A H Z )  e.  X )  ->  ( Z G ( A H Z ) )  =  ( A H Z ) )
2221eqcomd 2446 . . . 4  |-  ( ( G  e.  GrpOp  /\  ( A H Z )  e.  X )  ->  ( A H Z )  =  ( Z G ( A H Z ) ) )
2318, 20, 22syl2anc 656 . . 3  |-  ( ( R  e.  RingOps  /\  A  e.  X )  ->  ( A H Z )  =  ( Z G ( A H Z ) ) )
2410, 17, 233eqtr3d 2481 . 2  |-  ( ( R  e.  RingOps  /\  A  e.  X )  ->  (
( A H Z ) G ( A H Z ) )  =  ( Z G ( A H Z ) ) )
253grporcan 23627 . . 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 1215 . 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 960    = wceq 1364    e. wcel 1761   ran crn 4837   ` cfv 5415  (class class class)co 6090   1stc1st 6574   2ndc2nd 6575   GrpOpcgr 23592  GIdcgi 23593   RingOpscrngo 23781
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1713  ax-7 1733  ax-8 1763  ax-9 1765  ax-10 1780  ax-11 1785  ax-12 1797  ax-13 1948  ax-ext 2422  ax-sep 4410  ax-nul 4418  ax-pow 4467  ax-pr 4528  ax-un 6371
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 962  df-tru 1367  df-ex 1592  df-nf 1595  df-sb 1706  df-eu 2263  df-mo 2264  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ne 2606  df-ral 2718  df-rex 2719  df-reu 2720  df-rab 2722  df-v 2972  df-sbc 3184  df-csb 3286  df-dif 3328  df-un 3330  df-in 3332  df-ss 3339  df-nul 3635  df-if 3789  df-sn 3875  df-pr 3877  df-op 3881  df-uni 4089  df-iun 4170  df-br 4290  df-opab 4348  df-mpt 4349  df-id 4632  df-xp 4842  df-rel 4843  df-cnv 4844  df-co 4845  df-dm 4846  df-rn 4847  df-iota 5378  df-fun 5417  df-fn 5418  df-f 5419  df-fo 5421  df-fv 5423  df-riota 6049  df-ov 6093  df-1st 6576  df-2nd 6577  df-grpo 23597  df-gid 23598  df-ablo 23688  df-rngo 23782
This theorem is referenced by:  rngoueqz  23836  zerdivemp1  23840  rngoridfz  23841  rngonegmn1r  28665  zerdivemp1x  28670  0idl  28734  keridl  28741
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