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Theorem ringrz 13223
Description: The zero of a unital ring is a right-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
ringrz  |-  ( ( R  e.  Ring  /\  X  e.  B )  ->  ( X  .x.  .0.  )  =  .0.  )

Proof of Theorem ringrz
StepHypRef Expression
1 ringgrp 13184 . . . . . 6  |-  ( R  e.  Ring  ->  R  e. 
Grp )
2 rngz.b . . . . . . 7  |-  B  =  ( Base `  R
)
3 rngz.z . . . . . . 7  |-  .0.  =  ( 0g `  R )
42, 3grpidcl 12904 . . . . . 6  |-  ( R  e.  Grp  ->  .0.  e.  B )
5 eqid 2177 . . . . . . 7  |-  ( +g  `  R )  =  ( +g  `  R )
62, 5, 3grplid 12906 . . . . . 6  |-  ( ( R  e.  Grp  /\  .0.  e.  B )  -> 
(  .0.  ( +g  `  R )  .0.  )  =  .0.  )
71, 4, 6syl2anc2 412 . . . . 5  |-  ( R  e.  Ring  ->  (  .0.  ( +g  `  R
)  .0.  )  =  .0.  )
87adantr 276 . . . 4  |-  ( ( R  e.  Ring  /\  X  e.  B )  ->  (  .0.  ( +g  `  R
)  .0.  )  =  .0.  )
98oveq2d 5891 . . 3  |-  ( ( R  e.  Ring  /\  X  e.  B )  ->  ( X  .x.  (  .0.  ( +g  `  R )  .0.  ) )  =  ( X  .x.  .0.  )
)
10 simpr 110 . . . . 5  |-  ( ( R  e.  Ring  /\  X  e.  B )  ->  X  e.  B )
111, 4syl 14 . . . . . 6  |-  ( R  e.  Ring  ->  .0.  e.  B )
1211adantr 276 . . . . 5  |-  ( ( R  e.  Ring  /\  X  e.  B )  ->  .0.  e.  B )
1310, 12, 123jca 1177 . . . 4  |-  ( ( R  e.  Ring  /\  X  e.  B )  ->  ( X  e.  B  /\  .0.  e.  B  /\  .0.  e.  B ) )
14 rngz.t . . . . 5  |-  .x.  =  ( .r `  R )
152, 5, 14ringdi 13201 . . . 4  |-  ( ( R  e.  Ring  /\  ( X  e.  B  /\  .0.  e.  B  /\  .0.  e.  B ) )  -> 
( X  .x.  (  .0.  ( +g  `  R
)  .0.  ) )  =  ( ( X 
.x.  .0.  ) ( +g  `  R ) ( X  .x.  .0.  )
) )
1613, 15syldan 282 . . 3  |-  ( ( R  e.  Ring  /\  X  e.  B )  ->  ( X  .x.  (  .0.  ( +g  `  R )  .0.  ) )  =  ( ( X  .x.  .0.  ) ( +g  `  R
) ( X  .x.  .0.  ) ) )
171adantr 276 . . . 4  |-  ( ( R  e.  Ring  /\  X  e.  B )  ->  R  e.  Grp )
182, 14ringcl 13196 . . . . 5  |-  ( ( R  e.  Ring  /\  X  e.  B  /\  .0.  e.  B )  ->  ( X  .x.  .0.  )  e.  B )
1912, 18mpd3an3 1338 . . . 4  |-  ( ( R  e.  Ring  /\  X  e.  B )  ->  ( X  .x.  .0.  )  e.  B )
202, 5, 3grplid 12906 . . . . 5  |-  ( ( R  e.  Grp  /\  ( X  .x.  .0.  )  e.  B )  ->  (  .0.  ( +g  `  R
) ( X  .x.  .0.  ) )  =  ( X  .x.  .0.  )
)
2120eqcomd 2183 . . . 4  |-  ( ( R  e.  Grp  /\  ( X  .x.  .0.  )  e.  B )  ->  ( X  .x.  .0.  )  =  (  .0.  ( +g  `  R ) ( X 
.x.  .0.  ) )
)
2217, 19, 21syl2anc 411 . . 3  |-  ( ( R  e.  Ring  /\  X  e.  B )  ->  ( X  .x.  .0.  )  =  (  .0.  ( +g  `  R ) ( X 
.x.  .0.  ) )
)
239, 16, 223eqtr3d 2218 . 2  |-  ( ( R  e.  Ring  /\  X  e.  B )  ->  (
( X  .x.  .0.  ) ( +g  `  R
) ( X  .x.  .0.  ) )  =  (  .0.  ( +g  `  R
) ( X  .x.  .0.  ) ) )
242, 5grprcan 12910 . . 3  |-  ( ( R  e.  Grp  /\  ( ( X  .x.  .0.  )  e.  B  /\  .0.  e.  B  /\  ( X  .x.  .0.  )  e.  B ) )  -> 
( ( ( X 
.x.  .0.  ) ( +g  `  R ) ( X  .x.  .0.  )
)  =  (  .0.  ( +g  `  R
) ( X  .x.  .0.  ) )  <->  ( X  .x.  .0.  )  =  .0.  ) )
2517, 19, 12, 19, 24syl13anc 1240 . 2  |-  ( ( R  e.  Ring  /\  X  e.  B )  ->  (
( ( X  .x.  .0.  ) ( +g  `  R
) ( X  .x.  .0.  ) )  =  (  .0.  ( +g  `  R
) ( X  .x.  .0.  ) )  <->  ( X  .x.  .0.  )  =  .0.  ) )
2623, 25mpbid 147 1  |-  ( ( R  e.  Ring  /\  X  e.  B )  ->  ( X  .x.  .0.  )  =  .0.  )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 104    <-> wb 105    /\ w3a 978    = wceq 1353    e. wcel 2148   ` cfv 5217  (class class class)co 5875   Basecbs 12462   +g cplusg 12536   .rcmulr 12537   0gc0g 12705   Grpcgrp 12877   Ringcrg 13179
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-sep 4122  ax-pow 4175  ax-pr 4210  ax-un 4434  ax-setind 4537  ax-cnex 7902  ax-resscn 7903  ax-1cn 7904  ax-1re 7905  ax-icn 7906  ax-addcl 7907  ax-addrcl 7908  ax-mulcl 7909  ax-addcom 7911  ax-addass 7913  ax-i2m1 7916  ax-0lt1 7917  ax-0id 7919  ax-rnegex 7920  ax-pre-ltirr 7923  ax-pre-ltadd 7927
This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-nel 2443  df-ral 2460  df-rex 2461  df-reu 2462  df-rmo 2463  df-rab 2464  df-v 2740  df-sbc 2964  df-csb 3059  df-dif 3132  df-un 3134  df-in 3136  df-ss 3143  df-nul 3424  df-pw 3578  df-sn 3599  df-pr 3600  df-op 3602  df-uni 3811  df-int 3846  df-br 4005  df-opab 4066  df-mpt 4067  df-id 4294  df-xp 4633  df-rel 4634  df-cnv 4635  df-co 4636  df-dm 4637  df-rn 4638  df-res 4639  df-iota 5179  df-fun 5219  df-fn 5220  df-fv 5225  df-riota 5831  df-ov 5878  df-oprab 5879  df-mpo 5880  df-pnf 7994  df-mnf 7995  df-ltxr 7997  df-inn 8920  df-2 8978  df-3 8979  df-ndx 12465  df-slot 12466  df-base 12468  df-sets 12469  df-plusg 12549  df-mulr 12550  df-0g 12707  df-mgm 12775  df-sgrp 12808  df-mnd 12818  df-grp 12880  df-mgp 13131  df-ring 13181
This theorem is referenced by:  ringsrg  13224  ringinvnz1ne0  13226  ringinvnzdiv  13227  ringnegr  13229  dvdsr02  13274  lmodvs0  13412
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