Theorem rngrz 14220

Description: The zero of a non-unital ring is a right-absorbing element. (Contributed by FL, 31-Aug-2009.) Generalization of ringrz 14322. (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 14212	. . . . . 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 13811	. . . . . 6 |- ( R e. Grp -> .0. e. B )
5		eqid 2238	. . . . . . 7 |- ( +g ` R ) = ( +g ` R )
6	2, 5, 3	grplid 13813	. . . . . 6 |- ( ( R e. Grp /\ .0. e. B ) -> ( .0. ( +g ` R ) .0. ) = .0. )
7	1, 4, 6	syl2anc2 416	. . . . 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 6091	. . 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 14217	. . . . . 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 1208	. . . 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 14214	. . . 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 14218	. . . . 5 |- ( ( R e. Rng /\ X e. B /\ .0. e. B ) -> ( X .x. .0. ) e. B )
19	12, 18	mpd3an3 1379	. . . 4 |- ( ( R e. Rng /\ X e. B ) -> ( X .x. .0. ) e. B )
20	2, 5, 3	grplid 13813	. . . . 5 |- ( ( R e. Grp /\ ( X .x. .0. ) e. B ) -> ( .0. ( +g ` R ) ( X .x. .0. ) ) = ( X .x. .0. ) )
21	20	eqcomd 2244	. . . 4 |- ( ( R e. Grp /\ ( X .x. .0. ) e. B ) -> ( X .x. .0. ) = ( .0. ( +g ` R ) ( X .x. .0. ) ) )
22	17, 19, 21	syl2anc 415	. . 3 |- ( ( R e. Rng /\ X e. B ) -> ( X .x. .0. ) = ( .0. ( +g ` R ) ( X .x. .0. ) ) )
23	9, 16, 22	3eqtr3d 2279	. 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 13819	. . 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 1280	. 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 1009   = wceq 1402   e. wcel 2209   ` cfv 5372  (class class class)co 6075   Basecbs 13330   +g cplusg 13408   .rcmulr 13409   0gc0g 13587   Grpcgrp 13782  Rngcrng 14206

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 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4244  ax-pow 4306  ax-pr 4341  ax-un 4573  ax-setind 4679  ax-cnex 8260  ax-resscn 8261  ax-1cn 8262  ax-1re 8263  ax-icn 8264  ax-addcl 8265  ax-addrcl 8266  ax-mulcl 8267  ax-addcom 8269  ax-addass 8271  ax-i2m1 8274  ax-0lt1 8275  ax-0id 8277  ax-rnegex 8278  ax-pre-ltirr 8281  ax-pre-ltadd 8285

This theorem depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-nel 2516  df-ral 2533  df-rex 2534  df-reu 2535  df-rmo 2536  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-pw 3687  df-sn 3711  df-pr 3712  df-op 3714  df-uni 3931  df-int 3966  df-br 4126  df-opab 4188  df-mpt 4189  df-id 4433  df-xp 4775  df-rel 4776  df-cnv 4777  df-co 4778  df-dm 4779  df-rn 4780  df-res 4781  df-iota 5332  df-fun 5374  df-fn 5375  df-fv 5380  df-riota 6028  df-ov 6078  df-oprab 6079  df-mpo 6080  df-pnf 8352  df-mnf 8353  df-ltxr 8355  df-inn 9284  df-2 9342  df-3 9343  df-ndx 13333  df-slot 13334  df-base 13336  df-sets 13337  df-plusg 13421  df-mulr 13422  df-0g 13589  df-mgm 13653  df-sgrp 13694  df-mnd 13707  df-grp 13785  df-abl 14067  df-mgp 14195  df-rng 14207

This theorem is referenced by:  rngmneg2  14222