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Theorem rnglz 38201
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 38194 . . . . . . 7  |-  ( R  e. Rng  ->  R  e.  Abel )
2 ablgrp 17127 . . . . . . 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 16402 . . . . . . 7  |-  ( R  e.  Grp  ->  .0.  e.  B )
7 eqid 2402 . . . . . . . 8  |-  ( +g  `  R )  =  ( +g  `  R )
84, 7, 5grplid 16404 . . . . . . 7  |-  ( ( R  e.  Grp  /\  .0.  e.  B )  -> 
(  .0.  ( +g  `  R )  .0.  )  =  .0.  )
96, 8mpdan 666 . . . . . 6  |-  ( R  e.  Grp  ->  (  .0.  ( +g  `  R
)  .0.  )  =  .0.  )
103, 9syl 17 . . . . 5  |-  ( R  e. Rng  ->  (  .0.  ( +g  `  R )  .0.  )  =  .0.  )
1110adantr 463 . . . 4  |-  ( ( R  e. Rng  /\  X  e.  B )  ->  (  .0.  ( +g  `  R
)  .0.  )  =  .0.  )
1211oveq1d 6293 . . 3  |-  ( ( R  e. Rng  /\  X  e.  B )  ->  (
(  .0.  ( +g  `  R )  .0.  )  .x.  X )  =  (  .0.  .x.  X )
)
13 simpl 455 . . . 4  |-  ( ( R  e. Rng  /\  X  e.  B )  ->  R  e. Rng )
143, 6syl 17 . . . . . . 7  |-  ( R  e. Rng  ->  .0.  e.  B
)
1514, 14jca 530 . . . . . 6  |-  ( R  e. Rng  ->  (  .0.  e.  B  /\  .0.  e.  B
) )
1615anim1i 566 . . . . 5  |-  ( ( R  e. Rng  /\  X  e.  B )  ->  (
(  .0.  e.  B  /\  .0.  e.  B )  /\  X  e.  B
) )
17 df-3an 976 . . . . 5  |-  ( (  .0.  e.  B  /\  .0.  e.  B  /\  X  e.  B )  <->  ( (  .0.  e.  B  /\  .0.  e.  B )  /\  X  e.  B ) )
1816, 17sylibr 212 . . . 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 38199 . . . 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 659 . . 3  |-  ( ( R  e. Rng  /\  X  e.  B )  ->  (
(  .0.  ( +g  `  R )  .0.  )  .x.  X )  =  ( (  .0.  .x.  X
) ( +g  `  R
) (  .0.  .x.  X ) ) )
223adantr 463 . . . 4  |-  ( ( R  e. Rng  /\  X  e.  B )  ->  R  e.  Grp )
2314adantr 463 . . . . 5  |-  ( ( R  e. Rng  /\  X  e.  B )  ->  .0.  e.  B )
24 simpr 459 . . . . 5  |-  ( ( R  e. Rng  /\  X  e.  B )  ->  X  e.  B )
254, 19rngcl 38200 . . . . 5  |-  ( ( R  e. Rng  /\  .0.  e.  B  /\  X  e.  B )  ->  (  .0.  .x.  X )  e.  B )
2613, 23, 24, 25syl3anc 1230 . . . 4  |-  ( ( R  e. Rng  /\  X  e.  B )  ->  (  .0.  .x.  X )  e.  B )
274, 7, 5grprid 16405 . . . . 5  |-  ( ( R  e.  Grp  /\  (  .0.  .x.  X )  e.  B )  ->  (
(  .0.  .x.  X
) ( +g  `  R
)  .0.  )  =  (  .0.  .x.  X
) )
2827eqcomd 2410 . . . 4  |-  ( ( R  e.  Grp  /\  (  .0.  .x.  X )  e.  B )  ->  (  .0.  .x.  X )  =  ( (  .0.  .x.  X ) ( +g  `  R )  .0.  )
)
2922, 26, 28syl2anc 659 . . 3  |-  ( ( R  e. Rng  /\  X  e.  B )  ->  (  .0.  .x.  X )  =  ( (  .0.  .x.  X ) ( +g  `  R )  .0.  )
)
3012, 21, 293eqtr3d 2451 . 2  |-  ( ( R  e. Rng  /\  X  e.  B )  ->  (
(  .0.  .x.  X
) ( +g  `  R
) (  .0.  .x.  X ) )  =  ( (  .0.  .x.  X ) ( +g  `  R )  .0.  )
)
314, 7grplcan 16426 . . 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 1232 . 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 210 1  |-  ( ( R  e. Rng  /\  X  e.  B )  ->  (  .0.  .x.  X )  =  .0.  )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 367    /\ w3a 974    = wceq 1405    e. wcel 1842   ` cfv 5569  (class class class)co 6278   Basecbs 14841   +g cplusg 14909   .rcmulr 14910   0gc0g 15054   Grpcgrp 16377   Abelcabl 17123  Rngcrng 38191
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-8 1844  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-rep 4507  ax-sep 4517  ax-nul 4525  ax-pow 4572  ax-pr 4630  ax-un 6574  ax-cnex 9578  ax-resscn 9579  ax-1cn 9580  ax-icn 9581  ax-addcl 9582  ax-addrcl 9583  ax-mulcl 9584  ax-mulrcl 9585  ax-mulcom 9586  ax-addass 9587  ax-mulass 9588  ax-distr 9589  ax-i2m1 9590  ax-1ne0 9591  ax-1rid 9592  ax-rnegex 9593  ax-rrecex 9594  ax-cnre 9595  ax-pre-lttri 9596  ax-pre-lttrn 9597  ax-pre-ltadd 9598  ax-pre-mulgt0 9599
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 975  df-3an 976  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-nel 2601  df-ral 2759  df-rex 2760  df-reu 2761  df-rmo 2762  df-rab 2763  df-v 3061  df-sbc 3278  df-csb 3374  df-dif 3417  df-un 3419  df-in 3421  df-ss 3428  df-pss 3430  df-nul 3739  df-if 3886  df-pw 3957  df-sn 3973  df-pr 3975  df-tp 3977  df-op 3979  df-uni 4192  df-iun 4273  df-br 4396  df-opab 4454  df-mpt 4455  df-tr 4490  df-eprel 4734  df-id 4738  df-po 4744  df-so 4745  df-fr 4782  df-we 4784  df-xp 4829  df-rel 4830  df-cnv 4831  df-co 4832  df-dm 4833  df-rn 4834  df-res 4835  df-ima 4836  df-pred 5367  df-ord 5413  df-on 5414  df-lim 5415  df-suc 5416  df-iota 5533  df-fun 5571  df-fn 5572  df-f 5573  df-f1 5574  df-fo 5575  df-f1o 5576  df-fv 5577  df-riota 6240  df-ov 6281  df-oprab 6282  df-mpt2 6283  df-om 6684  df-wrecs 7013  df-recs 7075  df-rdg 7113  df-er 7348  df-en 7555  df-dom 7556  df-sdom 7557  df-pnf 9660  df-mnf 9661  df-xr 9662  df-ltxr 9663  df-le 9664  df-sub 9843  df-neg 9844  df-nn 10577  df-2 10635  df-ndx 14844  df-slot 14845  df-base 14846  df-sets 14847  df-plusg 14922  df-0g 15056  df-mgm 16196  df-sgrp 16235  df-mnd 16245  df-grp 16381  df-minusg 16382  df-abl 17125  df-mgp 17462  df-rng0 38192
This theorem is referenced by:  zrrnghm  38234
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