Theorem ringlz 14006

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 13964	. . . . . 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 13562	. . . . . 6 |- ( R e. Grp -> .0. e. B )
5		eqid 2229	. . . . . . 7 |- ( +g ` R ) = ( +g ` R )
6	2, 5, 3	grplid 13564	. . . . . 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 6016	. . 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 1201	. . . 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 13982	. . . 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 13976	. . . . 5 |- ( ( R e. Ring /\ .0. e. B /\ X e. B ) -> ( .0. .x. X ) e. B )
20	18, 11, 12, 19	syl3anc 1271	. . . 4 |- ( ( R e. Ring /\ X e. B ) -> ( .0. .x. X ) e. B )
21	2, 5, 3	grprid 13565	. . . . 5 |- ( ( R e. Grp /\ ( .0. .x. X ) e. B ) -> ( ( .0. .x. X ) ( +g ` R ) .0. ) = ( .0. .x. X ) )
22	21	eqcomd 2235	. . . 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 2270	. 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 13595	. . 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 1273	. 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 1002   = wceq 1395   e. wcel 2200   ` cfv 5318  (class class class)co 6001   Basecbs 13032   +g cplusg 13110   .rcmulr 13111   0gc0g 13289   Grpcgrp 13533   Ringcrg 13959

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 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-coll 4199  ax-sep 4202  ax-pow 4258  ax-pr 4293  ax-un 4524  ax-setind 4629  ax-cnex 8090  ax-resscn 8091  ax-1cn 8092  ax-1re 8093  ax-icn 8094  ax-addcl 8095  ax-addrcl 8096  ax-mulcl 8097  ax-addcom 8099  ax-addass 8101  ax-i2m1 8104  ax-0lt1 8105  ax-0id 8107  ax-rnegex 8108  ax-pre-ltirr 8111  ax-pre-ltadd 8115

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-nel 2496  df-ral 2513  df-rex 2514  df-reu 2515  df-rmo 2516  df-rab 2517  df-v 2801  df-sbc 3029  df-csb 3125  df-dif 3199  df-un 3201  df-in 3203  df-ss 3210  df-nul 3492  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3889  df-int 3924  df-iun 3967  df-br 4084  df-opab 4146  df-mpt 4147  df-id 4384  df-xp 4725  df-rel 4726  df-cnv 4727  df-co 4728  df-dm 4729  df-rn 4730  df-res 4731  df-ima 4732  df-iota 5278  df-fun 5320  df-fn 5321  df-f 5322  df-f1 5323  df-fo 5324  df-f1o 5325  df-fv 5326  df-riota 5954  df-ov 6004  df-oprab 6005  df-mpo 6006  df-pnf 8183  df-mnf 8184  df-ltxr 8186  df-inn 9111  df-2 9169  df-3 9170  df-ndx 13035  df-slot 13036  df-base 13038  df-sets 13039  df-plusg 13123  df-mulr 13124  df-0g 13291  df-mgm 13389  df-sgrp 13435  df-mnd 13450  df-grp 13536  df-minusg 13537  df-mgp 13884  df-ring 13961

This theorem is referenced by:  ringlzd  14008  ringsrg  14010  ring1eq0  14011  ringnegl  14014  mulgass2  14021  dvdsr01  14068  0unit  14093  domneq0  14236