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Theorem rnglz 15393
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 15362 . . . . . 6  |-  ( R  e.  Ring  ->  R  e. 
Grp )
2 rngz.b . . . . . . . 8  |-  B  =  ( Base `  R
)
3 rngz.z . . . . . . . 8  |-  .0.  =  ( 0g `  R )
42, 3grpidcl 14526 . . . . . . 7  |-  ( R  e.  Grp  ->  .0.  e.  B )
5 eqid 2296 . . . . . . . 8  |-  ( +g  `  R )  =  ( +g  `  R )
62, 5, 3grplid 14528 . . . . . . 7  |-  ( ( R  e.  Grp  /\  .0.  e.  B )  -> 
(  .0.  ( +g  `  R )  .0.  )  =  .0.  )
74, 6mpdan 649 . . . . . 6  |-  ( R  e.  Grp  ->  (  .0.  ( +g  `  R
)  .0.  )  =  .0.  )
81, 7syl 15 . . . . 5  |-  ( R  e.  Ring  ->  (  .0.  ( +g  `  R
)  .0.  )  =  .0.  )
98adantr 451 . . . 4  |-  ( ( R  e.  Ring  /\  X  e.  B )  ->  (  .0.  ( +g  `  R
)  .0.  )  =  .0.  )
109oveq1d 5889 . . 3  |-  ( ( R  e.  Ring  /\  X  e.  B )  ->  (
(  .0.  ( +g  `  R )  .0.  )  .x.  X )  =  (  .0.  .x.  X )
)
111, 4syl 15 . . . . . 6  |-  ( R  e.  Ring  ->  .0.  e.  B )
1211adantr 451 . . . . 5  |-  ( ( R  e.  Ring  /\  X  e.  B )  ->  .0.  e.  B )
13 simpr 447 . . . . 5  |-  ( ( R  e.  Ring  /\  X  e.  B )  ->  X  e.  B )
1412, 12, 133jca 1132 . . . 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 15376 . . . 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 456 . . 3  |-  ( ( R  e.  Ring  /\  X  e.  B )  ->  (
(  .0.  ( +g  `  R )  .0.  )  .x.  X )  =  ( (  .0.  .x.  X
) ( +g  `  R
) (  .0.  .x.  X ) ) )
181adantr 451 . . . 4  |-  ( ( R  e.  Ring  /\  X  e.  B )  ->  R  e.  Grp )
19 simpl 443 . . . . 5  |-  ( ( R  e.  Ring  /\  X  e.  B )  ->  R  e.  Ring )
202, 15rngcl 15370 . . . . 5  |-  ( ( R  e.  Ring  /\  .0.  e.  B  /\  X  e.  B )  ->  (  .0.  .x.  X )  e.  B )
2119, 12, 13, 20syl3anc 1182 . . . 4  |-  ( ( R  e.  Ring  /\  X  e.  B )  ->  (  .0.  .x.  X )  e.  B )
222, 5, 3grprid 14529 . . . . 5  |-  ( ( R  e.  Grp  /\  (  .0.  .x.  X )  e.  B )  ->  (
(  .0.  .x.  X
) ( +g  `  R
)  .0.  )  =  (  .0.  .x.  X
) )
2322eqcomd 2301 . . . 4  |-  ( ( R  e.  Grp  /\  (  .0.  .x.  X )  e.  B )  ->  (  .0.  .x.  X )  =  ( (  .0.  .x.  X ) ( +g  `  R )  .0.  )
)
2418, 21, 23syl2anc 642 . . 3  |-  ( ( R  e.  Ring  /\  X  e.  B )  ->  (  .0.  .x.  X )  =  ( (  .0.  .x.  X ) ( +g  `  R )  .0.  )
)
2510, 17, 243eqtr3d 2336 . 2  |-  ( ( R  e.  Ring  /\  X  e.  B )  ->  (
(  .0.  .x.  X
) ( +g  `  R
) (  .0.  .x.  X ) )  =  ( (  .0.  .x.  X ) ( +g  `  R )  .0.  )
)
262, 5grplcan 14550 . . 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 1184 . 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 201 1  |-  ( ( R  e.  Ring  /\  X  e.  B )  ->  (  .0.  .x.  X )  =  .0.  )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    /\ w3a 934    = wceq 1632    e. wcel 1696   ` cfv 5271  (class class class)co 5874   Basecbs 13164   +g cplusg 13224   .rcmulr 13225   0gc0g 13416   Grpcgrp 14378   Ringcrg 15353
This theorem is referenced by:  rng1eq0  15395  rngnegl  15396  mulgass2  15403  gsumdixp  15408  dvdsr01  15453  0unit  15478  irredn0  15501  drngmul0or  15549  cntzsubr  15593  isabvd  15601  domneq0  16054  psrlidm  16164  mplsubrglem  16199  mplmonmul  16224  evlslem4  16261  coe1tmmul  16369  evlslem6  19413  evlslem3  19414  mdegmullem  19480  coe1mul3  19501  fta1glem1  19567  mamulid  27561  cntzsdrg  27613  lflsc0N  29895  hdmapinvlem3  32735  hdmapinvlem4  32736
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  ax-cnex 8809  ax-resscn 8810  ax-1cn 8811  ax-icn 8812  ax-addcl 8813  ax-addrcl 8814  ax-mulcl 8815  ax-mulrcl 8816  ax-mulcom 8817  ax-addass 8818  ax-mulass 8819  ax-distr 8820  ax-i2m1 8821  ax-1ne0 8822  ax-1rid 8823  ax-rnegex 8824  ax-rrecex 8825  ax-cnre 8826  ax-pre-lttri 8827  ax-pre-lttrn 8828  ax-pre-ltadd 8829  ax-pre-mulgt0 8830
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  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-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rmo 2564  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-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  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-oprab 5878  df-mpt2 5879  df-riota 6320  df-recs 6404  df-rdg 6439  df-er 6676  df-en 6880  df-dom 6881  df-sdom 6882  df-pnf 8885  df-mnf 8886  df-xr 8887  df-ltxr 8888  df-le 8889  df-sub 9055  df-neg 9056  df-nn 9763  df-2 9820  df-ndx 13167  df-slot 13168  df-base 13169  df-sets 13170  df-plusg 13237  df-0g 13420  df-mnd 14383  df-grp 14505  df-minusg 14506  df-mgp 15342  df-rng 15356
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