Theorem rnglz 41884

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

Step	Hyp	Ref	Expression
1		rngabl 41877	. . . . . . 7 |- ( R e. Rng -> R e. Abel )
2		ablgrp 18198	. . . . . . 7 |- ( R e. Abel -> R e. Grp )
3	1, 2	syl 17	. . . . . 6 |- ( R e. Rng -> R e. Grp )
4		rngcl.b	. . . . . . . 8 |- B = ( Base ` R )
5		rnglz.z	. . . . . . . 8 |- .0. = ( 0g ` R )
6	4, 5	grpidcl 17450	. . . . . . 7 |- ( R e. Grp -> .0. e. B )
7		eqid 2622	. . . . . . . 8 |- ( +g ` R ) = ( +g ` R )
8	4, 7, 5	grplid 17452	. . . . . . 7 |- ( ( R e. Grp /\ .0. e. B ) -> ( .0. ( +g ` R ) .0. ) = .0. )
9	6, 8	mpdan 702	. . . . . 6 |- ( R e. Grp -> ( .0. ( +g ` R ) .0. ) = .0. )
10	3, 9	syl 17	. . . . 5 |- ( R e. Rng -> ( .0. ( +g ` R ) .0. ) = .0. )
11	10	adantr 481	. . . 4 |- ( ( R e. Rng /\ X e. B ) -> ( .0. ( +g ` R ) .0. ) = .0. )
12	11	oveq1d 6665	. . 3 |- ( ( R e. Rng /\ X e. B ) -> ( ( .0. ( +g ` R ) .0. ) .x. X ) = ( .0. .x. X ) )
13		simpl 473	. . . 4 |- ( ( R e. Rng /\ X e. B ) -> R e. Rng )
14	3, 6	syl 17	. . . . . . 7 |- ( R e. Rng -> .0. e. B )
15	14, 14	jca 554	. . . . . 6 |- ( R e. Rng -> ( .0. e. B /\ .0. e. B ) )
16	15	anim1i 592	. . . . 5 |- ( ( R e. Rng /\ X e. B ) -> ( ( .0. e. B /\ .0. e. B ) /\ X e. B ) )
17		df-3an 1039	. . . . 5 |- ( ( .0. e. B /\ .0. e. B /\ X e. B ) <-> ( ( .0. e. B /\ .0. e. B ) /\ X e. B ) )
18	16, 17	sylibr 224	. . . 4 |- ( ( R e. Rng /\ X e. B ) -> ( .0. e. B /\ .0. e. B /\ X e. B ) )
19		rngcl.t	. . . . 5 |- .x. = ( .r ` R )
20	4, 7, 19	rngdir 41882	. . . 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 ) ) )
21	13, 18, 20	syl2anc 693	. . 3 |- ( ( R e. Rng /\ X e. B ) -> ( ( .0. ( +g ` R ) .0. ) .x. X ) = ( ( .0. .x. X ) ( +g ` R ) ( .0. .x. X ) ) )
22	3	adantr 481	. . . 4 |- ( ( R e. Rng /\ X e. B ) -> R e. Grp )
23	14	adantr 481	. . . . 5 |- ( ( R e. Rng /\ X e. B ) -> .0. e. B )
24		simpr 477	. . . . 5 |- ( ( R e. Rng /\ X e. B ) -> X e. B )
25	4, 19	rngcl 41883	. . . . 5 |- ( ( R e. Rng /\ .0. e. B /\ X e. B ) -> ( .0. .x. X ) e. B )
26	13, 23, 24, 25	syl3anc 1326	. . . 4 |- ( ( R e. Rng /\ X e. B ) -> ( .0. .x. X ) e. B )
27	4, 7, 5	grprid 17453	. . . . 5 |- ( ( R e. Grp /\ ( .0. .x. X ) e. B ) -> ( ( .0. .x. X ) ( +g ` R ) .0. ) = ( .0. .x. X ) )
28	27	eqcomd 2628	. . . 4 |- ( ( R e. Grp /\ ( .0. .x. X ) e. B ) -> ( .0. .x. X ) = ( ( .0. .x. X ) ( +g ` R ) .0. ) )
29	22, 26, 28	syl2anc 693	. . 3 |- ( ( R e. Rng /\ X e. B ) -> ( .0. .x. X ) = ( ( .0. .x. X ) ( +g ` R ) .0. ) )
30	12, 21, 29	3eqtr3d 2664	. 2 |- ( ( R e. Rng /\ X e. B ) -> ( ( .0. .x. X ) ( +g ` R ) ( .0. .x. X ) ) = ( ( .0. .x. X ) ( +g ` R ) .0. ) )
31	4, 7	grplcan 17477	. . 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. ) )
32	22, 26, 23, 26, 31	syl13anc 1328	. 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. ) )
33	30, 32	mpbid 222	1 |- ( ( R e. Rng /\ X e. B ) -> ( .0. .x. X ) = .0. )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   ` cfv 5888  (class class class)co 6650   Basecbs 15857   +g cplusg 15941   .rcmulr 15942   0gc0g 16100   Grpcgrp 17422   Abelcabl 18194  Rngcrng 41874

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-er 7742  df-en 7956  df-dom 7957  df-sdom 7958  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-nn 11021  df-2 11079  df-ndx 15860  df-slot 15861  df-base 15863  df-sets 15864  df-plusg 15954  df-0g 16102  df-mgm 17242  df-sgrp 17284  df-mnd 17295  df-grp 17425  df-minusg 17426  df-abl 18196  df-mgp 18490  df-rng0 41875

This theorem is referenced by:  zrrnghm  41917