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Theorem rngolz 25226
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 25215 . . . . . 6  |-  ( R  e.  RingOps  ->  G  e.  GrpOp )
3 ringlz.2 . . . . . . . 8  |-  X  =  ran  G
4 ringlz.1 . . . . . . . 8  |-  Z  =  (GId `  G )
53, 4grpoidcl 25042 . . . . . . 7  |-  ( G  e.  GrpOp  ->  Z  e.  X )
63, 4grpolid 25044 . . . . . . 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 6310 . . 3  |-  ( ( R  e.  RingOps  /\  A  e.  X )  ->  (
( Z G Z ) H A )  =  ( Z H A ) )
111, 3, 4rngo0cl 25223 . . . . . 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 1176 . . . 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 25211 . . . 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 25207 . . . . 5  |-  ( ( R  e.  RingOps  /\  Z  e.  X  /\  A  e.  X )  ->  ( Z H A )  e.  X )
2119, 12, 13, 20syl3anc 1228 . . . 4  |-  ( ( R  e.  RingOps  /\  A  e.  X )  ->  ( Z H A )  e.  X )
223, 4grporid 25045 . . . . 5  |-  ( ( G  e.  GrpOp  /\  ( Z H A )  e.  X )  ->  (
( Z H A ) G Z )  =  ( Z H A ) )
2322eqcomd 2475 . . . 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 2516 . 2  |-  ( ( R  e.  RingOps  /\  A  e.  X )  ->  (
( Z H A ) G ( Z H A ) )  =  ( ( Z H A ) G Z ) )
263grpolcan 25058 . . 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 1230 . 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 973    = wceq 1379    e. wcel 1767   ran crn 5006   ` cfv 5594  (class class class)co 6295   1stc1st 6793   2ndc2nd 6794   GrpOpcgr 25011  GIdcgi 25012   RingOpscrngo 25200
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4564  ax-sep 4574  ax-nul 4582  ax-pow 4631  ax-pr 4692  ax-un 6587
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2822  df-rex 2823  df-reu 2824  df-rab 2826  df-v 3120  df-sbc 3337  df-csb 3441  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-nul 3791  df-if 3946  df-sn 4034  df-pr 4036  df-op 4040  df-uni 4252  df-iun 4333  df-br 4454  df-opab 4512  df-mpt 4513  df-id 4801  df-xp 5011  df-rel 5012  df-cnv 5013  df-co 5014  df-dm 5015  df-rn 5016  df-res 5017  df-ima 5018  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-riota 6256  df-ov 6298  df-1st 6795  df-2nd 6796  df-grpo 25016  df-gid 25017  df-ginv 25018  df-ablo 25107  df-rngo 25201
This theorem is referenced by:  rngonegmn1l  30279  isdrngo3  30289  0idl  30349  keridl  30356  prnc  30391
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