Theorem ringlz 13880

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 13838	. . . . . 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 13436	. . . . . 6 |- ( R e. Grp -> .0. e. B )
5		eqid 2206	. . . . . . 7 |- ( +g ` R ) = ( +g ` R )
6	2, 5, 3	grplid 13438	. . . . . 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 5972	. . 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 1180	. . . 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 13856	. . . 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 13850	. . . . 5 |- ( ( R e. Ring /\ .0. e. B /\ X e. B ) -> ( .0. .x. X ) e. B )
20	18, 11, 12, 19	syl3anc 1250	. . . 4 |- ( ( R e. Ring /\ X e. B ) -> ( .0. .x. X ) e. B )
21	2, 5, 3	grprid 13439	. . . . 5 |- ( ( R e. Grp /\ ( .0. .x. X ) e. B ) -> ( ( .0. .x. X ) ( +g ` R ) .0. ) = ( .0. .x. X ) )
22	21	eqcomd 2212	. . . 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 2247	. 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 13469	. . 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 1252	. 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 981   = wceq 1373   e. wcel 2177   ` cfv 5280  (class class class)co 5957   Basecbs 12907   +g cplusg 12984   .rcmulr 12985   0gc0g 13163   Grpcgrp 13407   Ringcrg 13833

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 615  ax-in2 616  ax-io 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-13 2179  ax-14 2180  ax-ext 2188  ax-coll 4167  ax-sep 4170  ax-pow 4226  ax-pr 4261  ax-un 4488  ax-setind 4593  ax-cnex 8036  ax-resscn 8037  ax-1cn 8038  ax-1re 8039  ax-icn 8040  ax-addcl 8041  ax-addrcl 8042  ax-mulcl 8043  ax-addcom 8045  ax-addass 8047  ax-i2m1 8050  ax-0lt1 8051  ax-0id 8053  ax-rnegex 8054  ax-pre-ltirr 8057  ax-pre-ltadd 8061

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-fal 1379  df-nf 1485  df-sb 1787  df-eu 2058  df-mo 2059  df-clab 2193  df-cleq 2199  df-clel 2202  df-nfc 2338  df-ne 2378  df-nel 2473  df-ral 2490  df-rex 2491  df-reu 2492  df-rmo 2493  df-rab 2494  df-v 2775  df-sbc 3003  df-csb 3098  df-dif 3172  df-un 3174  df-in 3176  df-ss 3183  df-nul 3465  df-pw 3623  df-sn 3644  df-pr 3645  df-op 3647  df-uni 3857  df-int 3892  df-iun 3935  df-br 4052  df-opab 4114  df-mpt 4115  df-id 4348  df-xp 4689  df-rel 4690  df-cnv 4691  df-co 4692  df-dm 4693  df-rn 4694  df-res 4695  df-ima 4696  df-iota 5241  df-fun 5282  df-fn 5283  df-f 5284  df-f1 5285  df-fo 5286  df-f1o 5287  df-fv 5288  df-riota 5912  df-ov 5960  df-oprab 5961  df-mpo 5962  df-pnf 8129  df-mnf 8130  df-ltxr 8132  df-inn 9057  df-2 9115  df-3 9116  df-ndx 12910  df-slot 12911  df-base 12913  df-sets 12914  df-plusg 12997  df-mulr 12998  df-0g 13165  df-mgm 13263  df-sgrp 13309  df-mnd 13324  df-grp 13410  df-minusg 13411  df-mgp 13758  df-ring 13835

This theorem is referenced by:  ringlzd  13882  ringsrg  13884  ring1eq0  13885  ringnegl  13888  mulgass2  13895  dvdsr01  13941  0unit  13966  domneq0  14109