Theorem ringlz 9487

Description: The zero of a unital ring is a left absorbing element. (Contributed by FL, 31-Aug-2009.)

Hypotheses

Ref	Expression
ringlz.1	|- Z = (Id` G)
ringlz.2	|- X = ran G
ringlz.3	|- G = (1st` R)
ringlz.4	|- H = (2nd` R)

Assertion

Ref	Expression
ringlz	|- ((R e. Ring /\ A e. X) -> (ZHA) = Z)

Proof of Theorem ringlz

Step	Hyp	Ref	Expression
1		ringlz.3	. . . . . . 7 |- G = (1st` R)
2	1	ringgrp 9476	. . . . . 6 |- (R e. Ring -> G e. Grp)
3		ringlz.2	. . . . . . . 8 |- X = ran G
4		ringlz.1	. . . . . . . 8 |- Z = (Id` G)
5	3, 4	grpidcl 9343	. . . . . . 7 |- (G e. Grp -> Z e. X)
6	3, 4	grplid 9345	. . . . . . 7 |- ((G e. Grp /\ Z e. X) -> (ZGZ) = Z)
7	5, 6	mpdan 768	. . . . . 6 |- (G e. Grp -> (ZGZ) = Z)
8	2, 7	syl 12	. . . . 5 |- (R e. Ring -> (ZGZ) = Z)
9	8	adantr 425	. . . 4 |- ((R e. Ring /\ A e. X) -> (ZGZ) = Z)
10	9	opreq1d 4897	. . 3 |- ((R e. Ring /\ A e. X) -> ((ZGZ)HA) = (ZHA))
11	1, 3, 4	ring0cl 9484	. . . . . 6 |- (R e. Ring -> Z e. X)
12	11	adantr 425	. . . . 5 |- ((R e. Ring /\ A e. X) -> Z e. X)
13		simpr 350	. . . . 5 |- ((R e. Ring /\ A e. X) -> A e. X)
14	12, 12, 13	3jca 1050	. . . 4 |- ((R e. Ring /\ A e. X) -> (Z e. X /\ Z e. X /\ A e. X))
15		ringlz.4	. . . . 5 |- H = (2nd` R)
16	1, 15, 3	ringdir 9472	. . . 4 |- ((R e. Ring /\ (Z e. X /\ Z e. X /\ A e. X)) -> ((ZGZ)HA) = ((ZHA)G(ZHA)))
17	14, 16	syldan 516	. . 3 |- ((R e. Ring /\ A e. X) -> ((ZGZ)HA) = ((ZHA)G(ZHA)))
18	2	adantr 425	. . . 4 |- ((R e. Ring /\ A e. X) -> G e. Grp)
19		simpl 346	. . . . 5 |- ((R e. Ring /\ A e. X) -> R e. Ring)
20	1, 15, 3	ringcl 9468	. . . . 5 |- ((R e. Ring /\ Z e. X /\ A e. X) -> (ZHA) e. X)
21	19, 12, 13, 20	syl111anc 1100	. . . 4 |- ((R e. Ring /\ A e. X) -> (ZHA) e. X)
22	3, 4	grprid 9346	. . . . 5 |- ((G e. Grp /\ (ZHA) e. X) -> ((ZHA)GZ) = (ZHA))
23	22	eqcomd 1889	. . . 4 |- ((G e. Grp /\ (ZHA) e. X) -> (ZHA) = ((ZHA)GZ))
24	18, 21, 23	syl11anc 524	. . 3 |- ((R e. Ring /\ A e. X) -> (ZHA) = ((ZHA)GZ))
25	10, 17, 24	3eqtr3d 1934	. 2 |- ((R e. Ring /\ A e. X) -> ((ZHA)G(ZHA)) = ((ZHA)GZ))
26	3	grplcan 9359	. . 3 |- ((G e. Grp /\ ((ZHA) e. X /\ Z e. X /\ (ZHA) e. X)) -> (((ZHA)G(ZHA)) = ((ZHA)GZ) <-> (ZHA) = Z))
27	18, 21, 12, 21, 26	syl13anc 1102	. 2 |- ((R e. Ring /\ A e. X) -> (((ZHA)G(ZHA)) = ((ZHA)GZ) <-> (ZHA) = Z))
28	25, 27	mpbid 212	1 |- ((R e. Ring /\ A e. X) -> (ZHA) = Z)

Syntax hints:   -> wi 3   <-> wb 163   /\ wa 240   /\ w3a 858   = wceq 1298   e. wcel 1300  ran crn 3987  ` cfv 3998  (class class class)co 4884  1stc1st 5018  2ndc2nd 5019  Grpcgr 9311  Idcgi 9312  Ringcring 9463

This theorem is referenced by:  multinv 14771  ringnegmn1l 16102  isdivrng3 16112  0idl 16173  keridl 16180  prnc 16215

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1304  ax-gen 1305  ax-8 1306  ax-9 1307  ax-10 1308  ax-11 1309  ax-12 1310  ax-13 1311  ax-14 1312  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-10o 1500  ax-16 1580  ax-11o 1588  ax-ext 1865  ax-rep 3428  ax-sep 3438  ax-nul 3445  ax-pow 3481  ax-pr 3524  ax-un 3790

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-3an 860  df-ex 1327  df-sb 1536  df-eu 1775  df-mo 1776  df-clab 1872  df-cleq 1877  df-clel 1880  df-ne 2019  df-ral 2109  df-rex 2110  df-reu 2111  df-rab 2112  df-v 2294  df-sbc 2454  df-csb 2541  df-dif 2597  df-un 2600  df-in 2603  df-ss 2605  df-nul 2876  df-if 2983  df-pw 3035  df-sn 3049  df-pr 3050  df-op 3053  df-uni 3178  df-br 3339  df-opab 3396  df-id 3586  df-xp 4000  df-rel 4001  df-cnv 4002  df-co 4003  df-dm 4004  df-rn 4005  df-res 4006  df-ima 4007  df-fun 4008  df-fn 4009  df-f 4010  df-f1 4011  df-fo 4012  df-f1o 4013  df-fv 4014  df-opr 4886  df-1st 5020  df-2nd 5021  df-grp 9316  df-gid 9317  df-ginv 9318  df-abl 9408  df-ring 9464

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