Theorem ringrz 14002

Description: The zero of a unital ring is a right-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
ringrz	|- ( ( R e. Ring /\ X e. B ) -> ( X .x. .0. ) = .0. )

Proof of Theorem ringrz

Step	Hyp	Ref	Expression
1		ringgrp 13959	. . . . . 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 13557	. . . . . 6 |- ( R e. Grp -> .0. e. B )
5		eqid 2229	. . . . . . 7 |- ( +g ` R ) = ( +g ` R )
6	2, 5, 3	grplid 13559	. . . . . 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	oveq2d 6016	. . 3 |- ( ( R e. Ring /\ X e. B ) -> ( X .x. ( .0. ( +g ` R ) .0. ) ) = ( X .x. .0. ) )
10		simpr 110	. . . . 5 |- ( ( R e. Ring /\ X e. B ) -> X e. B )
11	1, 4	syl 14	. . . . . 6 |- ( R e. Ring -> .0. e. B )
12	11	adantr 276	. . . . 5 |- ( ( R e. Ring /\ X e. B ) -> .0. e. B )
13	10, 12, 12	3jca 1201	. . . 4 |- ( ( R e. Ring /\ X e. B ) -> ( X e. B /\ .0. e. B /\ .0. e. B ) )
14		rngz.t	. . . . 5 |- .x. = ( .r ` R )
15	2, 5, 14	ringdi 13976	. . . 4 |- ( ( R e. Ring /\ ( X e. B /\ .0. e. B /\ .0. e. B ) ) -> ( X .x. ( .0. ( +g ` R ) .0. ) ) = ( ( X .x. .0. ) ( +g ` R ) ( X .x. .0. ) ) )
16	13, 15	syldan 282	. . 3 |- ( ( R e. Ring /\ X e. B ) -> ( X .x. ( .0. ( +g ` R ) .0. ) ) = ( ( X .x. .0. ) ( +g ` R ) ( X .x. .0. ) ) )
17	1	adantr 276	. . . 4 |- ( ( R e. Ring /\ X e. B ) -> R e. Grp )
18	2, 14	ringcl 13971	. . . . 5 |- ( ( R e. Ring /\ X e. B /\ .0. e. B ) -> ( X .x. .0. ) e. B )
19	12, 18	mpd3an3 1372	. . . 4 |- ( ( R e. Ring /\ X e. B ) -> ( X .x. .0. ) e. B )
20	2, 5, 3	grplid 13559	. . . . 5 |- ( ( R e. Grp /\ ( X .x. .0. ) e. B ) -> ( .0. ( +g ` R ) ( X .x. .0. ) ) = ( X .x. .0. ) )
21	20	eqcomd 2235	. . . 4 |- ( ( R e. Grp /\ ( X .x. .0. ) e. B ) -> ( X .x. .0. ) = ( .0. ( +g ` R ) ( X .x. .0. ) ) )
22	17, 19, 21	syl2anc 411	. . 3 |- ( ( R e. Ring /\ X e. B ) -> ( X .x. .0. ) = ( .0. ( +g ` R ) ( X .x. .0. ) ) )
23	9, 16, 22	3eqtr3d 2270	. 2 |- ( ( R e. Ring /\ X e. B ) -> ( ( X .x. .0. ) ( +g ` R ) ( X .x. .0. ) ) = ( .0. ( +g ` R ) ( X .x. .0. ) ) )
24	2, 5	grprcan 13565	. . 3 |- ( ( R e. Grp /\ ( ( X .x. .0. ) e. B /\ .0. e. B /\ ( X .x. .0. ) e. B ) ) -> ( ( ( X .x. .0. ) ( +g ` R ) ( X .x. .0. ) ) = ( .0. ( +g ` R ) ( X .x. .0. ) ) <-> ( X .x. .0. ) = .0. ) )
25	17, 19, 12, 19, 24	syl13anc 1273	. 2 |- ( ( R e. Ring /\ X e. B ) -> ( ( ( X .x. .0. ) ( +g ` R ) ( X .x. .0. ) ) = ( .0. ( +g ` R ) ( X .x. .0. ) ) <-> ( X .x. .0. ) = .0. ) )
26	23, 25	mpbid 147	1 |- ( ( R e. Ring /\ X e. B ) -> ( X .x. .0. ) = .0. )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   /\ w3a 1002   = wceq 1395   e. wcel 2200   ` cfv 5317  (class class class)co 6000   Basecbs 13027   +g cplusg 13105   .rcmulr 13106   0gc0g 13284   Grpcgrp 13528   Ringcrg 13954

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-sep 4201  ax-pow 4257  ax-pr 4292  ax-un 4523  ax-setind 4628  ax-cnex 8086  ax-resscn 8087  ax-1cn 8088  ax-1re 8089  ax-icn 8090  ax-addcl 8091  ax-addrcl 8092  ax-mulcl 8093  ax-addcom 8095  ax-addass 8097  ax-i2m1 8100  ax-0lt1 8101  ax-0id 8103  ax-rnegex 8104  ax-pre-ltirr 8107  ax-pre-ltadd 8111

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 3888  df-int 3923  df-br 4083  df-opab 4145  df-mpt 4146  df-id 4383  df-xp 4724  df-rel 4725  df-cnv 4726  df-co 4727  df-dm 4728  df-rn 4729  df-res 4730  df-iota 5277  df-fun 5319  df-fn 5320  df-fv 5325  df-riota 5953  df-ov 6003  df-oprab 6004  df-mpo 6005  df-pnf 8179  df-mnf 8180  df-ltxr 8182  df-inn 9107  df-2 9165  df-3 9166  df-ndx 13030  df-slot 13031  df-base 13033  df-sets 13034  df-plusg 13118  df-mulr 13119  df-0g 13286  df-mgm 13384  df-sgrp 13430  df-mnd 13445  df-grp 13531  df-mgp 13879  df-ring 13956

This theorem is referenced by:  ringrzd  14004  ringsrg  14005  ringinvnz1ne0  14007  ringinvnzdiv  14008  ringnegr  14010  dvdsr02  14063  rrgeq0  14223  unitrrg  14225  domneq0  14230  lmodvs0  14280