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

Theorem rngolz 21068
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 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 )
109oveq1d 5873 . . 3  |-  ( ( R  e.  RingOps  /\  A  e.  X )  ->  (
( Z G Z ) H A )  =  ( Z H A ) )
111, 3, 4rngo0cl 21065 . . . . . 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 21053 . . . 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 21049 . . . . 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 20887 . . . . 5  |-  ( ( G  e.  GrpOp  /\  ( Z H A )  e.  X )  ->  (
( Z H A ) G Z )  =  ( Z H A ) )
2322eqcomd 2288 . . . 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 2323 . 2  |-  ( ( R  e.  RingOps  /\  A  e.  X )  ->  (
( Z H A ) G ( Z H A ) )  =  ( ( Z H A ) G Z ) )
263grpolcan 20900 . . 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 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:  multinv  25422  rngonegmn1l  26580  isdrngo3  26590  0idl  26650  keridl  26657  prnc  26692
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-rep 4131  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-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-1st 6122  df-2nd 6123  df-riota 6304  df-grpo 20858  df-gid 20859  df-ginv 20860  df-ablo 20949  df-rngo 21043
  Copyright terms: Public domain W3C validator