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Theorem rngolz 21084
Description: The zero of a unital ring is a left 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
rngolz  |-  ( ( R  e.  RingOps  /\  A  e.  X )  ->  ( Z H A )  =  Z )

Proof of Theorem rngolz
StepHypRef Expression
1 ringlz.3 . . . . . . 7  |-  G  =  ( 1st `  R
)
21rngogrpo 21073 . . . . . 6  |-  ( R  e.  RingOps  ->  G  e.  GrpOp )
3 ringlz.2 . . . . . . . 8  |-  X  =  ran  G
4 ringlz.1 . . . . . . . 8  |-  Z  =  (GId `  G )
53, 4grpoidcl 20900 . . . . . . 7  |-  ( G  e.  GrpOp  ->  Z  e.  X )
63, 4grpolid 20902 . . . . . . 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 )
109oveq1d 5889 . . 3  |-  ( ( R  e.  RingOps  /\  A  e.  X )  ->  (
( Z G Z ) H A )  =  ( Z H A ) )
111, 3, 4rngo0cl 21081 . . . . . 6  |-  ( R  e.  RingOps  ->  Z  e.  X
)
1211adantr 451 . . . . 5  |-  ( ( R  e.  RingOps  /\  A  e.  X )  ->  Z  e.  X )
13 simpr 447 . . . . 5  |-  ( ( R  e.  RingOps  /\  A  e.  X )  ->  A  e.  X )
1412, 12, 133jca 1132 . . . 4  |-  ( ( R  e.  RingOps  /\  A  e.  X )  ->  ( Z  e.  X  /\  Z  e.  X  /\  A  e.  X )
)
15 ringlz.4 . . . . 5  |-  H  =  ( 2nd `  R
)
161, 15, 3rngodir 21069 . . . 4  |-  ( ( R  e.  RingOps  /\  ( Z  e.  X  /\  Z  e.  X  /\  A  e.  X )
)  ->  ( ( Z G Z ) H A )  =  ( ( Z H A ) G ( Z H A ) ) )
1714, 16syldan 456 . . 3  |-  ( ( R  e.  RingOps  /\  A  e.  X )  ->  (
( Z G Z ) H A )  =  ( ( Z H A ) G ( Z H A ) ) )
182adantr 451 . . . 4  |-  ( ( R  e.  RingOps  /\  A  e.  X )  ->  G  e.  GrpOp )
19 simpl 443 . . . . 5  |-  ( ( R  e.  RingOps  /\  A  e.  X )  ->  R  e.  RingOps )
201, 15, 3rngocl 21065 . . . . 5  |-  ( ( R  e.  RingOps  /\  Z  e.  X  /\  A  e.  X )  ->  ( Z H A )  e.  X )
2119, 12, 13, 20syl3anc 1182 . . . 4  |-  ( ( R  e.  RingOps  /\  A  e.  X )  ->  ( Z H A )  e.  X )
223, 4grporid 20903 . . . . 5  |-  ( ( G  e.  GrpOp  /\  ( Z H A )  e.  X )  ->  (
( Z H A ) G Z )  =  ( Z H A ) )
2322eqcomd 2301 . . . 4  |-  ( ( G  e.  GrpOp  /\  ( Z H A )  e.  X )  ->  ( Z H A )  =  ( ( Z H A ) G Z ) )
2418, 21, 23syl2anc 642 . . 3  |-  ( ( R  e.  RingOps  /\  A  e.  X )  ->  ( Z H A )  =  ( ( Z H A ) G Z ) )
2510, 17, 243eqtr3d 2336 . 2  |-  ( ( R  e.  RingOps  /\  A  e.  X )  ->  (
( Z H A ) G ( Z H A ) )  =  ( ( Z H A ) G Z ) )
263grpolcan 20916 . . 3  |-  ( ( G  e.  GrpOp  /\  (
( Z H A )  e.  X  /\  Z  e.  X  /\  ( Z H A )  e.  X ) )  ->  ( ( ( Z H A ) G ( Z H A ) )  =  ( ( Z H A ) G Z )  <->  ( Z H A )  =  Z ) )
2718, 21, 12, 21, 26syl13anc 1184 . 2  |-  ( ( R  e.  RingOps  /\  A  e.  X )  ->  (
( ( Z H A ) G ( Z H A ) )  =  ( ( Z H A ) G Z )  <->  ( Z H A )  =  Z ) )
2825, 27mpbid 201 1  |-  ( ( R  e.  RingOps  /\  A  e.  X )  ->  ( Z H A )  =  Z )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    /\ w3a 934    = wceq 1632    e. wcel 1696   ran crn 4706   ` cfv 5271  (class class class)co 5874   1stc1st 6136   2ndc2nd 6137   GrpOpcgr 20869  GIdcgi 20870   RingOpscrngo 21058
This theorem is referenced by:  multinv  25525  rngonegmn1l  26683  isdrngo3  26693  0idl  26753  keridl  26760  prnc  26795
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-reu 2563  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-1st 6138  df-2nd 6139  df-riota 6320  df-grpo 20874  df-gid 20875  df-ginv 20876  df-ablo 20965  df-rngo 21059
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