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Theorem rnglz 16684
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 16653 . . . . . 6  |-  ( R  e.  Ring  ->  R  e. 
Grp )
2 rngz.b . . . . . . . 8  |-  B  =  ( Base `  R
)
3 rngz.z . . . . . . . 8  |-  .0.  =  ( 0g `  R )
42, 3grpidcl 15569 . . . . . . 7  |-  ( R  e.  Grp  ->  .0.  e.  B )
5 eqid 2443 . . . . . . . 8  |-  ( +g  `  R )  =  ( +g  `  R )
62, 5, 3grplid 15571 . . . . . . 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 6109 . . 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 16667 . . . 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 16661 . . . . 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 15572 . . . . 5  |-  ( ( R  e.  Grp  /\  (  .0.  .x.  X )  e.  B )  ->  (
(  .0.  .x.  X
) ( +g  `  R
)  .0.  )  =  (  .0.  .x.  X
) )
2322eqcomd 2448 . . . 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 2483 . 2  |-  ( ( R  e.  Ring  /\  X  e.  B )  ->  (
(  .0.  .x.  X
) ( +g  `  R
) (  .0.  .x.  X ) )  =  ( (  .0.  .x.  X ) ( +g  `  R )  .0.  )
)
262, 5grplcan 15593 . . 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 5421  (class class class)co 6094   Basecbs 14177   +g cplusg 14241   .rcmulr 14242   0gc0g 14381   Grpcgrp 15413   Ringcrg 16648
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 4406  ax-sep 4416  ax-nul 4424  ax-pow 4473  ax-pr 4534  ax-un 6375  ax-cnex 9341  ax-resscn 9342  ax-1cn 9343  ax-icn 9344  ax-addcl 9345  ax-addrcl 9346  ax-mulcl 9347  ax-mulrcl 9348  ax-mulcom 9349  ax-addass 9350  ax-mulass 9351  ax-distr 9352  ax-i2m1 9353  ax-1ne0 9354  ax-1rid 9355  ax-rnegex 9356  ax-rrecex 9357  ax-cnre 9358  ax-pre-lttri 9359  ax-pre-lttrn 9360  ax-pre-ltadd 9361  ax-pre-mulgt0 9362
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 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2571  df-ne 2611  df-nel 2612  df-ral 2723  df-rex 2724  df-reu 2725  df-rmo 2726  df-rab 2727  df-v 2977  df-sbc 3190  df-csb 3292  df-dif 3334  df-un 3336  df-in 3338  df-ss 3345  df-pss 3347  df-nul 3641  df-if 3795  df-pw 3865  df-sn 3881  df-pr 3883  df-tp 3885  df-op 3887  df-uni 4095  df-iun 4176  df-br 4296  df-opab 4354  df-mpt 4355  df-tr 4389  df-eprel 4635  df-id 4639  df-po 4644  df-so 4645  df-fr 4682  df-we 4684  df-ord 4725  df-on 4726  df-lim 4727  df-suc 4728  df-xp 4849  df-rel 4850  df-cnv 4851  df-co 4852  df-dm 4853  df-rn 4854  df-res 4855  df-ima 4856  df-iota 5384  df-fun 5423  df-fn 5424  df-f 5425  df-f1 5426  df-fo 5427  df-f1o 5428  df-fv 5429  df-riota 6055  df-ov 6097  df-oprab 6098  df-mpt2 6099  df-om 6480  df-recs 6835  df-rdg 6869  df-er 7104  df-en 7314  df-dom 7315  df-sdom 7316  df-pnf 9423  df-mnf 9424  df-xr 9425  df-ltxr 9426  df-le 9427  df-sub 9600  df-neg 9601  df-nn 10326  df-2 10383  df-ndx 14180  df-slot 14181  df-base 14182  df-sets 14183  df-plusg 14254  df-0g 14383  df-mnd 15418  df-grp 15548  df-minusg 15549  df-mgp 16595  df-rng 16650
This theorem is referenced by:  rngsrg  16686  rng1eq0  16687  rngnegl  16688  mulgass2  16695  gsumdixpOLD  16703  gsumdixp  16704  dvdsr01  16750  0unit  16775  irredn0  16798  drngmul0or  16856  cntzsubr  16900  isabvd  16908  domneq0  17372  psrlidm  17477  psrlidmOLD  17478  mplsubrglem  17520  mplsubrglemOLD  17521  mplmonmul  17546  evlslem4OLD  17593  evlslem4  17594  evlslem6  17601  evlslem6OLD  17602  evlslem3  17603  coe1tmmul  17733  frlmphllem  18208  mamulid  18307  1mavmul  18362  mdet1  18411  mdetr0  18415  mdegmullem  21552  coe1mul3  21574  fta1glem1  21640  cntzsdrg  29562  rmsupp0  30784  ply1mulgsumlem2  30848  dmatmul  30879  mdetdiaglem  30938  lflsc0N  32731  hdmapinvlem3  35571  hdmapinvlem4  35572
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