Theorem rngrz 13895

Description: The zero of a non-unital ring is a right-absorbing element. (Contributed by FL, 31-Aug-2009.) Generalization of ringrz 13993. (Revised by AV, 16-Feb-2025.)

Hypotheses

Ref	Expression
rngcl.b	|- B = ( Base ` R )
rngcl.t	|- .x. = ( .r ` R )
rnglz.z	|- .0. = ( 0g ` R )

Assertion

Ref	Expression
rngrz	|- ( ( R e. Rng /\ X e. B ) -> ( X .x. .0. ) = .0. )

Proof of Theorem rngrz

Step	Hyp	Ref	Expression
1		rnggrp 13887	. . . . . 6 |- ( R e. Rng -> R e. Grp )
2		rngcl.b	. . . . . . 7 |- B = ( Base ` R )
3		rnglz.z	. . . . . . 7 |- .0. = ( 0g ` R )
4	2, 3	grpidcl 13548	. . . . . 6 |- ( R e. Grp -> .0. e. B )
5		eqid 2229	. . . . . . 7 |- ( +g ` R ) = ( +g ` R )
6	2, 5, 3	grplid 13550	. . . . . 6 |- ( ( R e. Grp /\ .0. e. B ) -> ( .0. ( +g ` R ) .0. ) = .0. )
7	1, 4, 6	syl2anc2 412	. . . . 5 |- ( R e. Rng -> ( .0. ( +g ` R ) .0. ) = .0. )
8	7	adantr 276	. . . 4 |- ( ( R e. Rng /\ X e. B ) -> ( .0. ( +g ` R ) .0. ) = .0. )
9	8	oveq2d 6010	. . 3 |- ( ( R e. Rng /\ X e. B ) -> ( X .x. ( .0. ( +g ` R ) .0. ) ) = ( X .x. .0. ) )
10		simpr 110	. . . . 5 |- ( ( R e. Rng /\ X e. B ) -> X e. B )
11	2, 3	rng0cl 13892	. . . . . 6 |- ( R e. Rng -> .0. e. B )
12	11	adantr 276	. . . . 5 |- ( ( R e. Rng /\ X e. B ) -> .0. e. B )
13	10, 12, 12	3jca 1201	. . . 4 |- ( ( R e. Rng /\ X e. B ) -> ( X e. B /\ .0. e. B /\ .0. e. B ) )
14		rngcl.t	. . . . 5 |- .x. = ( .r ` R )
15	2, 5, 14	rngdi 13889	. . . 4 |- ( ( R e. Rng /\ ( X e. B /\ .0. e. B /\ .0. e. B ) ) -> ( X .x. ( .0. ( +g ` R ) .0. ) ) = ( ( X .x. .0. ) ( +g ` R ) ( X .x. .0. ) ) )
16	13, 15	syldan 282	. . 3 |- ( ( R e. Rng /\ X e. B ) -> ( X .x. ( .0. ( +g ` R ) .0. ) ) = ( ( X .x. .0. ) ( +g ` R ) ( X .x. .0. ) ) )
17	1	adantr 276	. . . 4 |- ( ( R e. Rng /\ X e. B ) -> R e. Grp )
18	2, 14	rngcl 13893	. . . . 5 |- ( ( R e. Rng /\ X e. B /\ .0. e. B ) -> ( X .x. .0. ) e. B )
19	12, 18	mpd3an3 1372	. . . 4 |- ( ( R e. Rng /\ X e. B ) -> ( X .x. .0. ) e. B )
20	2, 5, 3	grplid 13550	. . . . 5 |- ( ( R e. Grp /\ ( X .x. .0. ) e. B ) -> ( .0. ( +g ` R ) ( X .x. .0. ) ) = ( X .x. .0. ) )
21	20	eqcomd 2235	. . . 4 |- ( ( R e. Grp /\ ( X .x. .0. ) e. B ) -> ( X .x. .0. ) = ( .0. ( +g ` R ) ( X .x. .0. ) ) )
22	17, 19, 21	syl2anc 411	. . 3 |- ( ( R e. Rng /\ X e. B ) -> ( X .x. .0. ) = ( .0. ( +g ` R ) ( X .x. .0. ) ) )
23	9, 16, 22	3eqtr3d 2270	. 2 |- ( ( R e. Rng /\ X e. B ) -> ( ( X .x. .0. ) ( +g ` R ) ( X .x. .0. ) ) = ( .0. ( +g ` R ) ( X .x. .0. ) ) )
24	2, 5	grprcan 13556	. . 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. ) )
25	17, 19, 12, 19, 24	syl13anc 1273	. 2 |- ( ( R e. Rng /\ X e. B ) -> ( ( ( X .x. .0. ) ( +g ` R ) ( X .x. .0. ) ) = ( .0. ( +g ` R ) ( X .x. .0. ) ) <-> ( X .x. .0. ) = .0. ) )
26	23, 25	mpbid 147	1 |- ( ( R e. Rng /\ X e. B ) -> ( X .x. .0. ) = .0. )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   /\ w3a 1002   = wceq 1395   e. wcel 2200   ` cfv 5314  (class class class)co 5994   Basecbs 13018   +g cplusg 13096   .rcmulr 13097   0gc0g 13275   Grpcgrp 13519  Rngcrng 13881

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 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-sep 4201  ax-pow 4257  ax-pr 4292  ax-un 4521  ax-setind 4626  ax-cnex 8078  ax-resscn 8079  ax-1cn 8080  ax-1re 8081  ax-icn 8082  ax-addcl 8083  ax-addrcl 8084  ax-mulcl 8085  ax-addcom 8087  ax-addass 8089  ax-i2m1 8092  ax-0lt1 8093  ax-0id 8095  ax-rnegex 8096  ax-pre-ltirr 8099  ax-pre-ltadd 8103

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-nel 2496  df-ral 2513  df-rex 2514  df-reu 2515  df-rmo 2516  df-rab 2517  df-v 2801  df-sbc 3029  df-csb 3125  df-dif 3199  df-un 3201  df-in 3203  df-ss 3210  df-nul 3492  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3888  df-int 3923  df-br 4083  df-opab 4145  df-mpt 4146  df-id 4381  df-xp 4722  df-rel 4723  df-cnv 4724  df-co 4725  df-dm 4726  df-rn 4727  df-res 4728  df-iota 5274  df-fun 5316  df-fn 5317  df-fv 5322  df-riota 5947  df-ov 5997  df-oprab 5998  df-mpo 5999  df-pnf 8171  df-mnf 8172  df-ltxr 8174  df-inn 9099  df-2 9157  df-3 9158  df-ndx 13021  df-slot 13022  df-base 13024  df-sets 13025  df-plusg 13109  df-mulr 13110  df-0g 13277  df-mgm 13375  df-sgrp 13421  df-mnd 13436  df-grp 13522  df-abl 13810  df-mgp 13870  df-rng 13882

This theorem is referenced by:  rngmneg2  13897