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Theorem rnglz 16667
Description: The zero of a unital ring is a left absorbing element. (Contributed by FL, 31-Aug-2009.)
Hypotheses
Ref Expression
rngz.b  |-  B  =  ( Base `  R
)
rngz.t  |-  .x.  =  ( .r `  R )
rngz.z  |-  .0.  =  ( 0g `  R )
Assertion
Ref Expression
rnglz  |-  ( ( R  e.  Ring  /\  X  e.  B )  ->  (  .0.  .x.  X )  =  .0.  )

Proof of Theorem rnglz
StepHypRef Expression
1 rnggrp 16636 . . . . . 6  |-  ( R  e.  Ring  ->  R  e. 
Grp )
2 rngz.b . . . . . . . 8  |-  B  =  ( Base `  R
)
3 rngz.z . . . . . . . 8  |-  .0.  =  ( 0g `  R )
42, 3grpidcl 15555 . . . . . . 7  |-  ( R  e.  Grp  ->  .0.  e.  B )
5 eqid 2437 . . . . . . . 8  |-  ( +g  `  R )  =  ( +g  `  R )
62, 5, 3grplid 15557 . . . . . . 7  |-  ( ( R  e.  Grp  /\  .0.  e.  B )  -> 
(  .0.  ( +g  `  R )  .0.  )  =  .0.  )
74, 6mpdan 668 . . . . . 6  |-  ( R  e.  Grp  ->  (  .0.  ( +g  `  R
)  .0.  )  =  .0.  )
81, 7syl 16 . . . . 5  |-  ( R  e.  Ring  ->  (  .0.  ( +g  `  R
)  .0.  )  =  .0.  )
98adantr 465 . . . 4  |-  ( ( R  e.  Ring  /\  X  e.  B )  ->  (  .0.  ( +g  `  R
)  .0.  )  =  .0.  )
109oveq1d 6101 . . 3  |-  ( ( R  e.  Ring  /\  X  e.  B )  ->  (
(  .0.  ( +g  `  R )  .0.  )  .x.  X )  =  (  .0.  .x.  X )
)
111, 4syl 16 . . . . . 6  |-  ( R  e.  Ring  ->  .0.  e.  B )
1211adantr 465 . . . . 5  |-  ( ( R  e.  Ring  /\  X  e.  B )  ->  .0.  e.  B )
13 simpr 461 . . . . 5  |-  ( ( R  e.  Ring  /\  X  e.  B )  ->  X  e.  B )
1412, 12, 133jca 1168 . . . 4  |-  ( ( R  e.  Ring  /\  X  e.  B )  ->  (  .0.  e.  B  /\  .0.  e.  B  /\  X  e.  B ) )
15 rngz.t . . . . 5  |-  .x.  =  ( .r `  R )
162, 5, 15rngdir 16650 . . . 4  |-  ( ( R  e.  Ring  /\  (  .0.  e.  B  /\  .0.  e.  B  /\  X  e.  B ) )  -> 
( (  .0.  ( +g  `  R )  .0.  )  .x.  X )  =  ( (  .0. 
.x.  X ) ( +g  `  R ) (  .0.  .x.  X
) ) )
1714, 16syldan 470 . . 3  |-  ( ( R  e.  Ring  /\  X  e.  B )  ->  (
(  .0.  ( +g  `  R )  .0.  )  .x.  X )  =  ( (  .0.  .x.  X
) ( +g  `  R
) (  .0.  .x.  X ) ) )
181adantr 465 . . . 4  |-  ( ( R  e.  Ring  /\  X  e.  B )  ->  R  e.  Grp )
19 simpl 457 . . . . 5  |-  ( ( R  e.  Ring  /\  X  e.  B )  ->  R  e.  Ring )
202, 15rngcl 16644 . . . . 5  |-  ( ( R  e.  Ring  /\  .0.  e.  B  /\  X  e.  B )  ->  (  .0.  .x.  X )  e.  B )
2119, 12, 13, 20syl3anc 1218 . . . 4  |-  ( ( R  e.  Ring  /\  X  e.  B )  ->  (  .0.  .x.  X )  e.  B )
222, 5, 3grprid 15558 . . . . 5  |-  ( ( R  e.  Grp  /\  (  .0.  .x.  X )  e.  B )  ->  (
(  .0.  .x.  X
) ( +g  `  R
)  .0.  )  =  (  .0.  .x.  X
) )
2322eqcomd 2442 . . . 4  |-  ( ( R  e.  Grp  /\  (  .0.  .x.  X )  e.  B )  ->  (  .0.  .x.  X )  =  ( (  .0.  .x.  X ) ( +g  `  R )  .0.  )
)
2418, 21, 23syl2anc 661 . . 3  |-  ( ( R  e.  Ring  /\  X  e.  B )  ->  (  .0.  .x.  X )  =  ( (  .0.  .x.  X ) ( +g  `  R )  .0.  )
)
2510, 17, 243eqtr3d 2477 . 2  |-  ( ( R  e.  Ring  /\  X  e.  B )  ->  (
(  .0.  .x.  X
) ( +g  `  R
) (  .0.  .x.  X ) )  =  ( (  .0.  .x.  X ) ( +g  `  R )  .0.  )
)
262, 5grplcan 15579 . . 3  |-  ( ( R  e.  Grp  /\  ( (  .0.  .x.  X )  e.  B  /\  .0.  e.  B  /\  (  .0.  .x.  X )  e.  B ) )  -> 
( ( (  .0. 
.x.  X ) ( +g  `  R ) (  .0.  .x.  X
) )  =  ( (  .0.  .x.  X
) ( +g  `  R
)  .0.  )  <->  (  .0.  .x. 
X )  =  .0.  ) )
2718, 21, 12, 21, 26syl13anc 1220 . 2  |-  ( ( R  e.  Ring  /\  X  e.  B )  ->  (
( (  .0.  .x.  X ) ( +g  `  R ) (  .0. 
.x.  X ) )  =  ( (  .0. 
.x.  X ) ( +g  `  R )  .0.  )  <->  (  .0.  .x. 
X )  =  .0.  ) )
2825, 27mpbid 210 1  |-  ( ( R  e.  Ring  /\  X  e.  B )  ->  (  .0.  .x.  X )  =  .0.  )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 965    = wceq 1369    e. wcel 1756   ` cfv 5411  (class class class)co 6086   Basecbs 14166   +g cplusg 14230   .rcmulr 14231   0gc0g 14370   Grpcgrp 15402   Ringcrg 16631
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 2418  ax-rep 4396  ax-sep 4406  ax-nul 4414  ax-pow 4463  ax-pr 4524  ax-un 6367  ax-cnex 9330  ax-resscn 9331  ax-1cn 9332  ax-icn 9333  ax-addcl 9334  ax-addrcl 9335  ax-mulcl 9336  ax-mulrcl 9337  ax-mulcom 9338  ax-addass 9339  ax-mulass 9340  ax-distr 9341  ax-i2m1 9342  ax-1ne0 9343  ax-1rid 9344  ax-rnegex 9345  ax-rrecex 9346  ax-cnre 9347  ax-pre-lttri 9348  ax-pre-lttrn 9349  ax-pre-ltadd 9350  ax-pre-mulgt0 9351
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2256  df-mo 2257  df-clab 2424  df-cleq 2430  df-clel 2433  df-nfc 2562  df-ne 2602  df-nel 2603  df-ral 2714  df-rex 2715  df-reu 2716  df-rmo 2717  df-rab 2718  df-v 2968  df-sbc 3180  df-csb 3282  df-dif 3324  df-un 3326  df-in 3328  df-ss 3335  df-pss 3337  df-nul 3631  df-if 3785  df-pw 3855  df-sn 3871  df-pr 3873  df-tp 3875  df-op 3877  df-uni 4085  df-iun 4166  df-br 4286  df-opab 4344  df-mpt 4345  df-tr 4379  df-eprel 4624  df-id 4628  df-po 4633  df-so 4634  df-fr 4671  df-we 4673  df-ord 4714  df-on 4715  df-lim 4716  df-suc 4717  df-xp 4838  df-rel 4839  df-cnv 4840  df-co 4841  df-dm 4842  df-rn 4843  df-res 4844  df-ima 4845  df-iota 5374  df-fun 5413  df-fn 5414  df-f 5415  df-f1 5416  df-fo 5417  df-f1o 5418  df-fv 5419  df-riota 6045  df-ov 6089  df-oprab 6090  df-mpt2 6091  df-om 6472  df-recs 6824  df-rdg 6858  df-er 7093  df-en 7303  df-dom 7304  df-sdom 7305  df-pnf 9412  df-mnf 9413  df-xr 9414  df-ltxr 9415  df-le 9416  df-sub 9589  df-neg 9590  df-nn 10315  df-2 10372  df-ndx 14169  df-slot 14170  df-base 14171  df-sets 14172  df-plusg 14243  df-0g 14372  df-mnd 15407  df-grp 15534  df-minusg 15535  df-mgp 16578  df-rng 16633
This theorem is referenced by:  rngsrg  16669  rng1eq0  16670  rngnegl  16671  mulgass2  16678  gsumdixpOLD  16686  gsumdixp  16687  dvdsr01  16733  0unit  16758  irredn0  16781  drngmul0or  16829  cntzsubr  16873  isabvd  16881  domneq0  17343  psrlidm  17448  psrlidmOLD  17449  mplsubrglem  17491  mplsubrglemOLD  17492  mplmonmul  17517  evlslem4OLD  17562  evlslem4  17563  evlslem6  17570  evlslem6OLD  17571  evlslem3  17572  coe1tmmul  17701  frlmphllem  18174  mamulid  18273  1mavmul  18328  mdet1  18377  mdetr0  18381  mdegmullem  21518  coe1mul3  21540  fta1glem1  21606  cntzsdrg  29502  rmsupp0  30716  dmatmul  30777  mdetdiaglem  30794  lflsc0N  32479  hdmapinvlem3  35319  hdmapinvlem4  35320
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