Theorem ringlz 14055

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

Hypotheses

Ref	Expression
rngz.b	|- B = ( Base ` R )
rngz.t	|- .x. = ( .r ` R )
rngz.z	|- .0. = ( 0g ` R )

Assertion

Ref	Expression
ringlz	|- ( ( R e. Ring /\ X e. B ) -> ( .0. .x. X ) = .0. )

Proof of Theorem ringlz

Step	Hyp	Ref	Expression
1		ringgrp 14013	. . . . . 6 |- ( R e. Ring -> R e. Grp )
2		rngz.b	. . . . . . 7 |- B = ( Base ` R )
3		rngz.z	. . . . . . 7 |- .0. = ( 0g ` R )
4	2, 3	grpidcl 13611	. . . . . 6 |- ( R e. Grp -> .0. e. B )
5		eqid 2231	. . . . . . 7 |- ( +g ` R ) = ( +g ` R )
6	2, 5, 3	grplid 13613	. . . . . 6 |- ( ( R e. Grp /\ .0. e. B ) -> ( .0. ( +g ` R ) .0. ) = .0. )
7	1, 4, 6	syl2anc2 412	. . . . 5 |- ( R e. Ring -> ( .0. ( +g ` R ) .0. ) = .0. )
8	7	adantr 276	. . . 4 |- ( ( R e. Ring /\ X e. B ) -> ( .0. ( +g ` R ) .0. ) = .0. )
9	8	oveq1d 6032	. . 3 |- ( ( R e. Ring /\ X e. B ) -> ( ( .0. ( +g ` R ) .0. ) .x. X ) = ( .0. .x. X ) )
10	1, 4	syl 14	. . . . . 6 |- ( R e. Ring -> .0. e. B )
11	10	adantr 276	. . . . 5 |- ( ( R e. Ring /\ X e. B ) -> .0. e. B )
12		simpr 110	. . . . 5 |- ( ( R e. Ring /\ X e. B ) -> X e. B )
13	11, 11, 12	3jca 1203	. . . 4 |- ( ( R e. Ring /\ X e. B ) -> ( .0. e. B /\ .0. e. B /\ X e. B ) )
14		rngz.t	. . . . 5 |- .x. = ( .r ` R )
15	2, 5, 14	ringdir 14031	. . . 4 |- ( ( R e. Ring /\ ( .0. e. B /\ .0. e. B /\ X e. B ) ) -> ( ( .0. ( +g ` R ) .0. ) .x. X ) = ( ( .0. .x. X ) ( +g ` R ) ( .0. .x. X ) ) )
16	13, 15	syldan 282	. . 3 |- ( ( R e. Ring /\ X e. B ) -> ( ( .0. ( +g ` R ) .0. ) .x. X ) = ( ( .0. .x. X ) ( +g ` R ) ( .0. .x. X ) ) )
17	1	adantr 276	. . . 4 |- ( ( R e. Ring /\ X e. B ) -> R e. Grp )
18		simpl 109	. . . . 5 |- ( ( R e. Ring /\ X e. B ) -> R e. Ring )
19	2, 14	ringcl 14025	. . . . 5 |- ( ( R e. Ring /\ .0. e. B /\ X e. B ) -> ( .0. .x. X ) e. B )
20	18, 11, 12, 19	syl3anc 1273	. . . 4 |- ( ( R e. Ring /\ X e. B ) -> ( .0. .x. X ) e. B )
21	2, 5, 3	grprid 13614	. . . . 5 |- ( ( R e. Grp /\ ( .0. .x. X ) e. B ) -> ( ( .0. .x. X ) ( +g ` R ) .0. ) = ( .0. .x. X ) )
22	21	eqcomd 2237	. . . 4 |- ( ( R e. Grp /\ ( .0. .x. X ) e. B ) -> ( .0. .x. X ) = ( ( .0. .x. X ) ( +g ` R ) .0. ) )
23	17, 20, 22	syl2anc 411	. . 3 |- ( ( R e. Ring /\ X e. B ) -> ( .0. .x. X ) = ( ( .0. .x. X ) ( +g ` R ) .0. ) )
24	9, 16, 23	3eqtr3d 2272	. 2 |- ( ( R e. Ring /\ X e. B ) -> ( ( .0. .x. X ) ( +g ` R ) ( .0. .x. X ) ) = ( ( .0. .x. X ) ( +g ` R ) .0. ) )
25	2, 5	grplcan 13644	. . 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. ) )
26	17, 20, 11, 20, 25	syl13anc 1275	. 2 |- ( ( R e. Ring /\ X e. B ) -> ( ( ( .0. .x. X ) ( +g ` R ) ( .0. .x. X ) ) = ( ( .0. .x. X ) ( +g ` R ) .0. ) <-> ( .0. .x. X ) = .0. ) )
27	24, 26	mpbid 147	1 |- ( ( R e. Ring /\ X e. B ) -> ( .0. .x. X ) = .0. )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   /\ w3a 1004   = wceq 1397   e. wcel 2202   ` cfv 5326  (class class class)co 6017   Basecbs 13081   +g cplusg 13159   .rcmulr 13160   0gc0g 13338   Grpcgrp 13582   Ringcrg 14008

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 619  ax-in2 620  ax-io 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-13 2204  ax-14 2205  ax-ext 2213  ax-coll 4204  ax-sep 4207  ax-pow 4264  ax-pr 4299  ax-un 4530  ax-setind 4635  ax-cnex 8122  ax-resscn 8123  ax-1cn 8124  ax-1re 8125  ax-icn 8126  ax-addcl 8127  ax-addrcl 8128  ax-mulcl 8129  ax-addcom 8131  ax-addass 8133  ax-i2m1 8136  ax-0lt1 8137  ax-0id 8139  ax-rnegex 8140  ax-pre-ltirr 8143  ax-pre-ltadd 8147

This theorem depends on definitions:  df-bi 117  df-3an 1006  df-tru 1400  df-fal 1403  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ne 2403  df-nel 2498  df-ral 2515  df-rex 2516  df-reu 2517  df-rmo 2518  df-rab 2519  df-v 2804  df-sbc 3032  df-csb 3128  df-dif 3202  df-un 3204  df-in 3206  df-ss 3213  df-nul 3495  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-uni 3894  df-int 3929  df-iun 3972  df-br 4089  df-opab 4151  df-mpt 4152  df-id 4390  df-xp 4731  df-rel 4732  df-cnv 4733  df-co 4734  df-dm 4735  df-rn 4736  df-res 4737  df-ima 4738  df-iota 5286  df-fun 5328  df-fn 5329  df-f 5330  df-f1 5331  df-fo 5332  df-f1o 5333  df-fv 5334  df-riota 5970  df-ov 6020  df-oprab 6021  df-mpo 6022  df-pnf 8215  df-mnf 8216  df-ltxr 8218  df-inn 9143  df-2 9201  df-3 9202  df-ndx 13084  df-slot 13085  df-base 13087  df-sets 13088  df-plusg 13172  df-mulr 13173  df-0g 13340  df-mgm 13438  df-sgrp 13484  df-mnd 13499  df-grp 13585  df-minusg 13586  df-mgp 13933  df-ring 14010

This theorem is referenced by:  ringlzd  14057  ringsrg  14059  ring1eq0  14060  ringnegl  14063  mulgass2  14070  dvdsr01  14117  0unit  14142  domneq0  14285