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Theorem rnglz 39156
Description: The zero of a nonunital ring is a left-absorbing element. (Contributed by AV, 17-Apr-2020.)
Hypotheses
Ref Expression
rngcl.b  |-  B  =  ( Base `  R
)
rngcl.t  |-  .x.  =  ( .r `  R )
rnglz.z  |-  .0.  =  ( 0g `  R )
Assertion
Ref Expression
rnglz  |-  ( ( R  e. Rng  /\  X  e.  B )  ->  (  .0.  .x.  X )  =  .0.  )

Proof of Theorem rnglz
StepHypRef Expression
1 rngabl 39149 . . . . . . 7  |-  ( R  e. Rng  ->  R  e.  Abel )
2 ablgrp 17423 . . . . . . 7  |-  ( R  e.  Abel  ->  R  e. 
Grp )
31, 2syl 17 . . . . . 6  |-  ( R  e. Rng  ->  R  e.  Grp )
4 rngcl.b . . . . . . . 8  |-  B  =  ( Base `  R
)
5 rnglz.z . . . . . . . 8  |-  .0.  =  ( 0g `  R )
64, 5grpidcl 16682 . . . . . . 7  |-  ( R  e.  Grp  ->  .0.  e.  B )
7 eqid 2422 . . . . . . . 8  |-  ( +g  `  R )  =  ( +g  `  R )
84, 7, 5grplid 16684 . . . . . . 7  |-  ( ( R  e.  Grp  /\  .0.  e.  B )  -> 
(  .0.  ( +g  `  R )  .0.  )  =  .0.  )
96, 8mpdan 672 . . . . . 6  |-  ( R  e.  Grp  ->  (  .0.  ( +g  `  R
)  .0.  )  =  .0.  )
103, 9syl 17 . . . . 5  |-  ( R  e. Rng  ->  (  .0.  ( +g  `  R )  .0.  )  =  .0.  )
1110adantr 466 . . . 4  |-  ( ( R  e. Rng  /\  X  e.  B )  ->  (  .0.  ( +g  `  R
)  .0.  )  =  .0.  )
1211oveq1d 6317 . . 3  |-  ( ( R  e. Rng  /\  X  e.  B )  ->  (
(  .0.  ( +g  `  R )  .0.  )  .x.  X )  =  (  .0.  .x.  X )
)
13 simpl 458 . . . 4  |-  ( ( R  e. Rng  /\  X  e.  B )  ->  R  e. Rng )
143, 6syl 17 . . . . . . 7  |-  ( R  e. Rng  ->  .0.  e.  B
)
1514, 14jca 534 . . . . . 6  |-  ( R  e. Rng  ->  (  .0.  e.  B  /\  .0.  e.  B
) )
1615anim1i 570 . . . . 5  |-  ( ( R  e. Rng  /\  X  e.  B )  ->  (
(  .0.  e.  B  /\  .0.  e.  B )  /\  X  e.  B
) )
17 df-3an 984 . . . . 5  |-  ( (  .0.  e.  B  /\  .0.  e.  B  /\  X  e.  B )  <->  ( (  .0.  e.  B  /\  .0.  e.  B )  /\  X  e.  B ) )
1816, 17sylibr 215 . . . 4  |-  ( ( R  e. Rng  /\  X  e.  B )  ->  (  .0.  e.  B  /\  .0.  e.  B  /\  X  e.  B ) )
19 rngcl.t . . . . 5  |-  .x.  =  ( .r `  R )
204, 7, 19rngdir 39154 . . . 4  |-  ( ( R  e. Rng  /\  (  .0.  e.  B  /\  .0.  e.  B  /\  X  e.  B ) )  -> 
( (  .0.  ( +g  `  R )  .0.  )  .x.  X )  =  ( (  .0. 
.x.  X ) ( +g  `  R ) (  .0.  .x.  X
) ) )
2113, 18, 20syl2anc 665 . . 3  |-  ( ( R  e. Rng  /\  X  e.  B )  ->  (
(  .0.  ( +g  `  R )  .0.  )  .x.  X )  =  ( (  .0.  .x.  X
) ( +g  `  R
) (  .0.  .x.  X ) ) )
223adantr 466 . . . 4  |-  ( ( R  e. Rng  /\  X  e.  B )  ->  R  e.  Grp )
2314adantr 466 . . . . 5  |-  ( ( R  e. Rng  /\  X  e.  B )  ->  .0.  e.  B )
24 simpr 462 . . . . 5  |-  ( ( R  e. Rng  /\  X  e.  B )  ->  X  e.  B )
254, 19rngcl 39155 . . . . 5  |-  ( ( R  e. Rng  /\  .0.  e.  B  /\  X  e.  B )  ->  (  .0.  .x.  X )  e.  B )
2613, 23, 24, 25syl3anc 1264 . . . 4  |-  ( ( R  e. Rng  /\  X  e.  B )  ->  (  .0.  .x.  X )  e.  B )
274, 7, 5grprid 16685 . . . . 5  |-  ( ( R  e.  Grp  /\  (  .0.  .x.  X )  e.  B )  ->  (
(  .0.  .x.  X
) ( +g  `  R
)  .0.  )  =  (  .0.  .x.  X
) )
2827eqcomd 2430 . . . 4  |-  ( ( R  e.  Grp  /\  (  .0.  .x.  X )  e.  B )  ->  (  .0.  .x.  X )  =  ( (  .0.  .x.  X ) ( +g  `  R )  .0.  )
)
2922, 26, 28syl2anc 665 . . 3  |-  ( ( R  e. Rng  /\  X  e.  B )  ->  (  .0.  .x.  X )  =  ( (  .0.  .x.  X ) ( +g  `  R )  .0.  )
)
3012, 21, 293eqtr3d 2471 . 2  |-  ( ( R  e. Rng  /\  X  e.  B )  ->  (
(  .0.  .x.  X
) ( +g  `  R
) (  .0.  .x.  X ) )  =  ( (  .0.  .x.  X ) ( +g  `  R )  .0.  )
)
314, 7grplcan 16706 . . 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.  ) )
3222, 26, 23, 26, 31syl13anc 1266 . 2  |-  ( ( R  e. Rng  /\  X  e.  B )  ->  (
( (  .0.  .x.  X ) ( +g  `  R ) (  .0. 
.x.  X ) )  =  ( (  .0. 
.x.  X ) ( +g  `  R )  .0.  )  <->  (  .0.  .x. 
X )  =  .0.  ) )
3330, 32mpbid 213 1  |-  ( ( R  e. Rng  /\  X  e.  B )  ->  (  .0.  .x.  X )  =  .0.  )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 187    /\ wa 370    /\ w3a 982    = wceq 1437    e. wcel 1868   ` cfv 5598  (class class class)co 6302   Basecbs 15109   +g cplusg 15178   .rcmulr 15179   0gc0g 15326   Grpcgrp 16657   Abelcabl 17419  Rngcrng 39146
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1748  ax-6 1794  ax-7 1839  ax-8 1870  ax-9 1872  ax-10 1887  ax-11 1892  ax-12 1905  ax-13 2053  ax-ext 2400  ax-rep 4533  ax-sep 4543  ax-nul 4552  ax-pow 4599  ax-pr 4657  ax-un 6594  ax-cnex 9596  ax-resscn 9597  ax-1cn 9598  ax-icn 9599  ax-addcl 9600  ax-addrcl 9601  ax-mulcl 9602  ax-mulrcl 9603  ax-mulcom 9604  ax-addass 9605  ax-mulass 9606  ax-distr 9607  ax-i2m1 9608  ax-1ne0 9609  ax-1rid 9610  ax-rnegex 9611  ax-rrecex 9612  ax-cnre 9613  ax-pre-lttri 9614  ax-pre-lttrn 9615  ax-pre-ltadd 9616  ax-pre-mulgt0 9617
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1787  df-eu 2269  df-mo 2270  df-clab 2408  df-cleq 2414  df-clel 2417  df-nfc 2572  df-ne 2620  df-nel 2621  df-ral 2780  df-rex 2781  df-reu 2782  df-rmo 2783  df-rab 2784  df-v 3083  df-sbc 3300  df-csb 3396  df-dif 3439  df-un 3441  df-in 3443  df-ss 3450  df-pss 3452  df-nul 3762  df-if 3910  df-pw 3981  df-sn 3997  df-pr 3999  df-tp 4001  df-op 4003  df-uni 4217  df-iun 4298  df-br 4421  df-opab 4480  df-mpt 4481  df-tr 4516  df-eprel 4761  df-id 4765  df-po 4771  df-so 4772  df-fr 4809  df-we 4811  df-xp 4856  df-rel 4857  df-cnv 4858  df-co 4859  df-dm 4860  df-rn 4861  df-res 4862  df-ima 4863  df-pred 5396  df-ord 5442  df-on 5443  df-lim 5444  df-suc 5445  df-iota 5562  df-fun 5600  df-fn 5601  df-f 5602  df-f1 5603  df-fo 5604  df-f1o 5605  df-fv 5606  df-riota 6264  df-ov 6305  df-oprab 6306  df-mpt2 6307  df-om 6704  df-wrecs 7033  df-recs 7095  df-rdg 7133  df-er 7368  df-en 7575  df-dom 7576  df-sdom 7577  df-pnf 9678  df-mnf 9679  df-xr 9680  df-ltxr 9681  df-le 9682  df-sub 9863  df-neg 9864  df-nn 10611  df-2 10669  df-ndx 15112  df-slot 15113  df-base 15114  df-sets 15115  df-plusg 15191  df-0g 15328  df-mgm 16476  df-sgrp 16515  df-mnd 16525  df-grp 16661  df-minusg 16662  df-abl 17421  df-mgp 17712  df-rng0 39147
This theorem is referenced by:  zrrnghm  39189
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