Theorem ringrzNEW 17157

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

Hypotheses

Ref	Expression
ringz.1NEW	|- B = (base` R)
ringz.3NEW	|- T = (.r` R)
ringz.4NEW	|- Z = (0g` R)

Assertion

Ref	Expression
ringrzNEW	|- ((R e. RingNEW /\ X e. B) -> (XTZ) = Z)

Proof of Theorem ringrzNEW

Step	Hyp	Ref	Expression
1		ringgrpNEW 17152	. . . . . 6 |- (R e. RingNEW -> R e. GrpNEW)
2		ringz.1NEW	. . . . . . . 8 |- B = (base` R)
3		ringz.4NEW	. . . . . . . 8 |- Z = (0g` R)
4	2, 3	grpidclNEW 17118	. . . . . . 7 |- (R e. GrpNEW -> Z e. B)
5		eqid 1884	. . . . . . . 8 |- (+g` R) = (+g` R)
6	2, 5, 3	grplidNEW 17120	. . . . . . 7 |- ((R e. GrpNEW /\ Z e. B) -> (Z(+g` R)Z) = Z)
7	4, 6	mpdan 768	. . . . . 6 |- (R e. GrpNEW -> (Z(+g` R)Z) = Z)
8	1, 7	syl 12	. . . . 5 |- (R e. RingNEW -> (Z(+g` R)Z) = Z)
9	8	adantr 425	. . . 4 |- ((R e. RingNEW /\ X e. B) -> (Z(+g` R)Z) = Z)
10	9	opreq2d 4898	. . 3 |- ((R e. RingNEW /\ X e. B) -> (XT(Z(+g` R)Z)) = (XTZ))
11		simpr 350	. . . . 5 |- ((R e. RingNEW /\ X e. B) -> X e. B)
12	1, 4	syl 12	. . . . . 6 |- (R e. RingNEW -> Z e. B)
13	12	adantr 425	. . . . 5 |- ((R e. RingNEW /\ X e. B) -> Z e. B)
14	11, 13, 13	3jca 1050	. . . 4 |- ((R e. RingNEW /\ X e. B) -> (X e. B /\ Z e. B /\ Z e. B))
15		ringz.3NEW	. . . . 5 |- T = (.r` R)
16	2, 5, 15	ringdiNEW 17147	. . . 4 |- ((R e. RingNEW /\ (X e. B /\ Z e. B /\ Z e. B)) -> (XT(Z(+g` R)Z)) = ((XTZ)(+g` R)(XTZ)))
17	14, 16	syldan 516	. . 3 |- ((R e. RingNEW /\ X e. B) -> (XT(Z(+g` R)Z)) = ((XTZ)(+g` R)(XTZ)))
18	1	adantr 425	. . . 4 |- ((R e. RingNEW /\ X e. B) -> R e. GrpNEW)
19	2, 15	ringclNEW 17144	. . . . 5 |- ((R e. RingNEW /\ X e. B /\ Z e. B) -> (XTZ) e. B)
20	13, 19	mpd3an3 1192	. . . 4 |- ((R e. RingNEW /\ X e. B) -> (XTZ) e. B)
21	2, 5, 3	grplidNEW 17120	. . . . 5 |- ((R e. GrpNEW /\ (XTZ) e. B) -> (Z(+g` R)(XTZ)) = (XTZ))
22	21	eqcomd 1889	. . . 4 |- ((R e. GrpNEW /\ (XTZ) e. B) -> (XTZ) = (Z(+g` R)(XTZ)))
23	18, 20, 22	syl11anc 524	. . 3 |- ((R e. RingNEW /\ X e. B) -> (XTZ) = (Z(+g` R)(XTZ)))
24	10, 17, 23	3eqtr3d 1934	. 2 |- ((R e. RingNEW /\ X e. B) -> ((XTZ)(+g` R)(XTZ)) = (Z(+g` R)(XTZ)))
25	2, 5	grprcanNEW 17122	. . 3 |- ((R e. GrpNEW /\ ((XTZ) e. B /\ Z e. B /\ (XTZ) e. B)) -> (((XTZ)(+g` R)(XTZ)) = (Z(+g` R)(XTZ)) <-> (XTZ) = Z))
26	18, 20, 13, 20, 25	syl13anc 1102	. 2 |- ((R e. RingNEW /\ X e. B) -> (((XTZ)(+g` R)(XTZ)) = (Z(+g` R)(XTZ)) <-> (XTZ) = Z))
27	24, 26	mpbid 212	1 |- ((R e. RingNEW /\ X e. B) -> (XTZ) = Z)

Syntax hints:   -> wi 3   <-> wb 163   /\ wa 240   /\ w3a 858   = wceq 1298   e. wcel 1300  ` cfv 3998  (class class class)co 4884  basecbs 16758  +_gcplusg 17080  GrpNEWcgrp 17081  0_gc0g 17082  ._rcmulr 17085  RingNEWcrg 17086

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-tru 1262  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-dif 2597  df-un 2600  df-in 2603  df-ss 2605  df-nul 2876  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-fv 4014  df-opr 4886  df-mpt 5006  df-iota 5089  df-struct 16708  df-grpNEW 17089  df-0g 17090  df-ablNEW 17092  df-ringNEW 17094

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