Theorem rngrz 13502

Description: The zero of a non-unital ring is a right-absorbing element. (Contributed by FL, 31-Aug-2009.) Generalization of ringrz 13600. (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 13494	. . . . . 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 13161	. . . . . 6 |- ( R e. Grp -> .0. e. B )
5		eqid 2196	. . . . . . 7 |- ( +g ` R ) = ( +g ` R )
6	2, 5, 3	grplid 13163	. . . . . 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 5938	. . 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 13499	. . . . . 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 1179	. . . 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 13496	. . . 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 13500	. . . . 5 |- ( ( R e. Rng /\ X e. B /\ .0. e. B ) -> ( X .x. .0. ) e. B )
19	12, 18	mpd3an3 1349	. . . 4 |- ( ( R e. Rng /\ X e. B ) -> ( X .x. .0. ) e. B )
20	2, 5, 3	grplid 13163	. . . . 5 |- ( ( R e. Grp /\ ( X .x. .0. ) e. B ) -> ( .0. ( +g ` R ) ( X .x. .0. ) ) = ( X .x. .0. ) )
21	20	eqcomd 2202	. . . 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 2237	. 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 13169	. . 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 1251	. 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 980   = wceq 1364   e. wcel 2167   ` cfv 5258  (class class class)co 5922   Basecbs 12678   +g cplusg 12755   .rcmulr 12756   0gc0g 12927   Grpcgrp 13132  Rngcrng 13488

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 615  ax-in2 616  ax-io 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-13 2169  ax-14 2170  ax-ext 2178  ax-sep 4151  ax-pow 4207  ax-pr 4242  ax-un 4468  ax-setind 4573  ax-cnex 7970  ax-resscn 7971  ax-1cn 7972  ax-1re 7973  ax-icn 7974  ax-addcl 7975  ax-addrcl 7976  ax-mulcl 7977  ax-addcom 7979  ax-addass 7981  ax-i2m1 7984  ax-0lt1 7985  ax-0id 7987  ax-rnegex 7988  ax-pre-ltirr 7991  ax-pre-ltadd 7995

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ne 2368  df-nel 2463  df-ral 2480  df-rex 2481  df-reu 2482  df-rmo 2483  df-rab 2484  df-v 2765  df-sbc 2990  df-csb 3085  df-dif 3159  df-un 3161  df-in 3163  df-ss 3170  df-nul 3451  df-pw 3607  df-sn 3628  df-pr 3629  df-op 3631  df-uni 3840  df-int 3875  df-br 4034  df-opab 4095  df-mpt 4096  df-id 4328  df-xp 4669  df-rel 4670  df-cnv 4671  df-co 4672  df-dm 4673  df-rn 4674  df-res 4675  df-iota 5219  df-fun 5260  df-fn 5261  df-fv 5266  df-riota 5877  df-ov 5925  df-oprab 5926  df-mpo 5927  df-pnf 8063  df-mnf 8064  df-ltxr 8066  df-inn 8991  df-2 9049  df-3 9050  df-ndx 12681  df-slot 12682  df-base 12684  df-sets 12685  df-plusg 12768  df-mulr 12769  df-0g 12929  df-mgm 12999  df-sgrp 13045  df-mnd 13058  df-grp 13135  df-abl 13417  df-mgp 13477  df-rng 13489

This theorem is referenced by:  rngmneg2  13504