Theorem ringrz 13015

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 12977	. . . . . 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 12764	. . . . . 6 |- ( R e. Grp -> .0. e. B )
5		eqid 2175	. . . . . . 7 |- ( +g ` R ) = ( +g ` R )
6	2, 5, 3	grplid 12766	. . . . . 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 5881	. . 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 1177	. . . 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 12994	. . . 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 12989	. . . . 5 |- ( ( R e. Ring /\ X e. B /\ .0. e. B ) -> ( X .x. .0. ) e. B )
19	12, 18	mpd3an3 1338	. . . 4 |- ( ( R e. Ring /\ X e. B ) -> ( X .x. .0. ) e. B )
20	2, 5, 3	grplid 12766	. . . . 5 |- ( ( R e. Grp /\ ( X .x. .0. ) e. B ) -> ( .0. ( +g ` R ) ( X .x. .0. ) ) = ( X .x. .0. ) )
21	20	eqcomd 2181	. . . 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 2216	. 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 12770	. . 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 1240	. 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 978   = wceq 1353   e. wcel 2146   ` cfv 5208  (class class class)co 5865   Basecbs 12427   +g cplusg 12491   .rcmulr 12492   0gc0g 12625   Grpcgrp 12737   Ringcrg 12972

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 614  ax-in2 615  ax-io 709  ax-5 1445  ax-7 1446  ax-gen 1447  ax-ie1 1491  ax-ie2 1492  ax-8 1502  ax-10 1503  ax-11 1504  ax-i12 1505  ax-bndl 1507  ax-4 1508  ax-17 1524  ax-i9 1528  ax-ial 1532  ax-i5r 1533  ax-13 2148  ax-14 2149  ax-ext 2157  ax-sep 4116  ax-pow 4169  ax-pr 4203  ax-un 4427  ax-setind 4530  ax-cnex 7877  ax-resscn 7878  ax-1cn 7879  ax-1re 7880  ax-icn 7881  ax-addcl 7882  ax-addrcl 7883  ax-mulcl 7884  ax-addcom 7886  ax-addass 7888  ax-i2m1 7891  ax-0lt1 7892  ax-0id 7894  ax-rnegex 7895  ax-pre-ltirr 7898  ax-pre-ltadd 7902

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1459  df-sb 1761  df-eu 2027  df-mo 2028  df-clab 2162  df-cleq 2168  df-clel 2171  df-nfc 2306  df-ne 2346  df-nel 2441  df-ral 2458  df-rex 2459  df-reu 2460  df-rmo 2461  df-rab 2462  df-v 2737  df-sbc 2961  df-csb 3056  df-dif 3129  df-un 3131  df-in 3133  df-ss 3140  df-nul 3421  df-pw 3574  df-sn 3595  df-pr 3596  df-op 3598  df-uni 3806  df-int 3841  df-br 3999  df-opab 4060  df-mpt 4061  df-id 4287  df-xp 4626  df-rel 4627  df-cnv 4628  df-co 4629  df-dm 4630  df-rn 4631  df-res 4632  df-iota 5170  df-fun 5210  df-fn 5211  df-fv 5216  df-riota 5821  df-ov 5868  df-oprab 5869  df-mpo 5870  df-pnf 7968  df-mnf 7969  df-ltxr 7971  df-inn 8891  df-2 8949  df-3 8950  df-ndx 12430  df-slot 12431  df-base 12433  df-sets 12434  df-plusg 12504  df-mulr 12505  df-0g 12627  df-mgm 12639  df-sgrp 12672  df-mnd 12682  df-grp 12740  df-mgp 12926  df-ring 12974

This theorem is referenced by:  ringsrg  13016  ringinvnz1ne0  13018  ringinvnzdiv  13019  rngnegr  13021