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Theorem rngolz 23887
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 23876 . . . . . 6  |-  ( R  e.  RingOps  ->  G  e.  GrpOp )
3 ringlz.2 . . . . . . . 8  |-  X  =  ran  G
4 ringlz.1 . . . . . . . 8  |-  Z  =  (GId `  G )
53, 4grpoidcl 23703 . . . . . . 7  |-  ( G  e.  GrpOp  ->  Z  e.  X )
63, 4grpolid 23705 . . . . . . 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 )
109oveq1d 6105 . . 3  |-  ( ( R  e.  RingOps  /\  A  e.  X )  ->  (
( Z G Z ) H A )  =  ( Z H A ) )
111, 3, 4rngo0cl 23884 . . . . . 6  |-  ( R  e.  RingOps  ->  Z  e.  X
)
1211adantr 465 . . . . 5  |-  ( ( R  e.  RingOps  /\  A  e.  X )  ->  Z  e.  X )
13 simpr 461 . . . . 5  |-  ( ( R  e.  RingOps  /\  A  e.  X )  ->  A  e.  X )
1412, 12, 133jca 1168 . . . 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 23872 . . . 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 470 . . 3  |-  ( ( R  e.  RingOps  /\  A  e.  X )  ->  (
( Z G Z ) H A )  =  ( ( Z H A ) G ( Z H A ) ) )
182adantr 465 . . . 4  |-  ( ( R  e.  RingOps  /\  A  e.  X )  ->  G  e.  GrpOp )
19 simpl 457 . . . . 5  |-  ( ( R  e.  RingOps  /\  A  e.  X )  ->  R  e.  RingOps )
201, 15, 3rngocl 23868 . . . . 5  |-  ( ( R  e.  RingOps  /\  Z  e.  X  /\  A  e.  X )  ->  ( Z H A )  e.  X )
2119, 12, 13, 20syl3anc 1218 . . . 4  |-  ( ( R  e.  RingOps  /\  A  e.  X )  ->  ( Z H A )  e.  X )
223, 4grporid 23706 . . . . 5  |-  ( ( G  e.  GrpOp  /\  ( Z H A )  e.  X )  ->  (
( Z H A ) G Z )  =  ( Z H A ) )
2322eqcomd 2447 . . . 4  |-  ( ( G  e.  GrpOp  /\  ( Z H A )  e.  X )  ->  ( Z H A )  =  ( ( Z H A ) G Z ) )
2418, 21, 23syl2anc 661 . . 3  |-  ( ( R  e.  RingOps  /\  A  e.  X )  ->  ( Z H A )  =  ( ( Z H A ) G Z ) )
2510, 17, 243eqtr3d 2482 . 2  |-  ( ( R  e.  RingOps  /\  A  e.  X )  ->  (
( Z H A ) G ( Z H A ) )  =  ( ( Z H A ) G Z ) )
263grpolcan 23719 . . 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 1220 . 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 210 1  |-  ( ( R  e.  RingOps  /\  A  e.  X )  ->  ( Z H A )  =  Z )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 965    = wceq 1369    e. wcel 1756   ran crn 4840   ` cfv 5417  (class class class)co 6090   1stc1st 6574   2ndc2nd 6575   GrpOpcgr 23672  GIdcgi 23673   RingOpscrngo 23861
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4402  ax-sep 4412  ax-nul 4420  ax-pow 4469  ax-pr 4530  ax-un 6371
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2567  df-ne 2607  df-ral 2719  df-rex 2720  df-reu 2721  df-rab 2723  df-v 2973  df-sbc 3186  df-csb 3288  df-dif 3330  df-un 3332  df-in 3334  df-ss 3341  df-nul 3637  df-if 3791  df-sn 3877  df-pr 3879  df-op 3883  df-uni 4091  df-iun 4172  df-br 4292  df-opab 4350  df-mpt 4351  df-id 4635  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-res 4851  df-ima 4852  df-iota 5380  df-fun 5419  df-fn 5420  df-f 5421  df-f1 5422  df-fo 5423  df-f1o 5424  df-fv 5425  df-riota 6051  df-ov 6093  df-1st 6576  df-2nd 6577  df-grpo 23677  df-gid 23678  df-ginv 23679  df-ablo 23768  df-rngo 23862
This theorem is referenced by:  rngonegmn1l  28753  isdrngo3  28763  0idl  28823  keridl  28830  prnc  28865
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