Theorem ringlz 14075

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 14033	. . . . . 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 13630	. . . . . 6 |- ( R e. Grp -> .0. e. B )
5		eqid 2231	. . . . . . 7 |- ( +g ` R ) = ( +g ` R )
6	2, 5, 3	grplid 13632	. . . . . 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 6033	. . 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 14051	. . . 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 14045	. . . . 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 13633	. . . . 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 13663	. . 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 6018   Basecbs 13100   +g cplusg 13178   .rcmulr 13179   0gc0g 13357   Grpcgrp 13601   Ringcrg 14028

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 8123  ax-resscn 8124  ax-1cn 8125  ax-1re 8126  ax-icn 8127  ax-addcl 8128  ax-addrcl 8129  ax-mulcl 8130  ax-addcom 8132  ax-addass 8134  ax-i2m1 8137  ax-0lt1 8138  ax-0id 8140  ax-rnegex 8141  ax-pre-ltirr 8144  ax-pre-ltadd 8148

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 5971  df-ov 6021  df-oprab 6022  df-mpo 6023  df-pnf 8216  df-mnf 8217  df-ltxr 8219  df-inn 9144  df-2 9202  df-3 9203  df-ndx 13103  df-slot 13104  df-base 13106  df-sets 13107  df-plusg 13191  df-mulr 13192  df-0g 13359  df-mgm 13457  df-sgrp 13503  df-mnd 13518  df-grp 13604  df-minusg 13605  df-mgp 13953  df-ring 14030

This theorem is referenced by:  ringlzd  14077  ringsrg  14079  ring1eq0  14080  ringnegl  14083  mulgass2  14090  dvdsr01  14137  0unit  14162  domneq0  14305