Theorem ringnegl 13233

Description: Negation in a ring is the same as left multiplication by -1. (Contributed by Jeff Madsen, 19-Jun-2010.) (Revised by Mario Carneiro, 2-Jul-2014.)

Hypotheses

Ref	Expression
ringnegl.b	|- B = ( Base ` R )
ringnegl.t	|- .x. = ( .r ` R )
ringnegl.u	|- .1. = ( 1r ` R )
ringnegl.n	|- N = ( invg ` R )
ringnegl.r	|- ( ph -> R e. Ring )
ringnegl.x	|- ( ph -> X e. B )

Assertion

Ref	Expression
ringnegl	|- ( ph -> ( ( N ` .1. ) .x. X ) = ( N ` X ) )

Proof of Theorem ringnegl

Step	Hyp	Ref	Expression
1		ringnegl.r	. . . . 5 |- ( ph -> R e. Ring )
2		ringnegl.b	. . . . . . 7 |- B = ( Base ` R )
3		ringnegl.u	. . . . . . 7 |- .1. = ( 1r ` R )
4	2, 3	ringidcl 13208	. . . . . 6 |- ( R e. Ring -> .1. e. B )
5	1, 4	syl 14	. . . . 5 |- ( ph -> .1. e. B )
6		ringgrp 13189	. . . . . . 7 |- ( R e. Ring -> R e. Grp )
7	1, 6	syl 14	. . . . . 6 |- ( ph -> R e. Grp )
8		ringnegl.n	. . . . . . 7 |- N = ( invg ` R )
9	2, 8	grpinvcl 12926	. . . . . 6 |- ( ( R e. Grp /\ .1. e. B ) -> ( N ` .1. ) e. B )
10	7, 5, 9	syl2anc 411	. . . . 5 |- ( ph -> ( N ` .1. ) e. B )
11		ringnegl.x	. . . . 5 |- ( ph -> X e. B )
12		eqid 2177	. . . . . 6 |- ( +g ` R ) = ( +g ` R )
13		ringnegl.t	. . . . . 6 |- .x. = ( .r ` R )
14	2, 12, 13	ringdir 13207	. . . . 5 |- ( ( R e. Ring /\ ( .1. e. B /\ ( N ` .1. ) e. B /\ X e. B ) ) -> ( ( .1. ( +g ` R ) ( N ` .1. ) ) .x. X ) = ( ( .1. .x. X ) ( +g ` R ) ( ( N ` .1. ) .x. X ) ) )
15	1, 5, 10, 11, 14	syl13anc 1240	. . . 4 |- ( ph -> ( ( .1. ( +g ` R ) ( N ` .1. ) ) .x. X ) = ( ( .1. .x. X ) ( +g ` R ) ( ( N ` .1. ) .x. X ) ) )
16		eqid 2177	. . . . . . . 8 |- ( 0g ` R ) = ( 0g ` R )
17	2, 12, 16, 8	grprinv 12928	. . . . . . 7 |- ( ( R e. Grp /\ .1. e. B ) -> ( .1. ( +g ` R ) ( N ` .1. ) ) = ( 0g ` R ) )
18	7, 5, 17	syl2anc 411	. . . . . 6 |- ( ph -> ( .1. ( +g ` R ) ( N ` .1. ) ) = ( 0g ` R ) )
19	18	oveq1d 5892	. . . . 5 |- ( ph -> ( ( .1. ( +g ` R ) ( N ` .1. ) ) .x. X ) = ( ( 0g ` R ) .x. X ) )
20	2, 13, 16	ringlz 13227	. . . . . 6 |- ( ( R e. Ring /\ X e. B ) -> ( ( 0g ` R ) .x. X ) = ( 0g ` R ) )
21	1, 11, 20	syl2anc 411	. . . . 5 |- ( ph -> ( ( 0g ` R ) .x. X ) = ( 0g ` R ) )
22	19, 21	eqtrd 2210	. . . 4 |- ( ph -> ( ( .1. ( +g ` R ) ( N ` .1. ) ) .x. X ) = ( 0g ` R ) )
23	2, 13, 3	ringlidm 13211	. . . . . 6 |- ( ( R e. Ring /\ X e. B ) -> ( .1. .x. X ) = X )
24	1, 11, 23	syl2anc 411	. . . . 5 |- ( ph -> ( .1. .x. X ) = X )
25	24	oveq1d 5892	. . . 4 |- ( ph -> ( ( .1. .x. X ) ( +g ` R ) ( ( N ` .1. ) .x. X ) ) = ( X ( +g ` R ) ( ( N ` .1. ) .x. X ) ) )
26	15, 22, 25	3eqtr3rd 2219	. . 3 |- ( ph -> ( X ( +g ` R ) ( ( N ` .1. ) .x. X ) ) = ( 0g ` R ) )
27	2, 13	ringcl 13201	. . . . 5 |- ( ( R e. Ring /\ ( N ` .1. ) e. B /\ X e. B ) -> ( ( N ` .1. ) .x. X ) e. B )
28	1, 10, 11, 27	syl3anc 1238	. . . 4 |- ( ph -> ( ( N ` .1. ) .x. X ) e. B )
29	2, 12, 16, 8	grpinvid1 12929	. . . 4 |- ( ( R e. Grp /\ X e. B /\ ( ( N ` .1. ) .x. X ) e. B ) -> ( ( N ` X ) = ( ( N ` .1. ) .x. X ) <-> ( X ( +g ` R ) ( ( N ` .1. ) .x. X ) ) = ( 0g ` R ) ) )
30	7, 11, 28, 29	syl3anc 1238	. . 3 |- ( ph -> ( ( N ` X ) = ( ( N ` .1. ) .x. X ) <-> ( X ( +g ` R ) ( ( N ` .1. ) .x. X ) ) = ( 0g ` R ) ) )
31	26, 30	mpbird 167	. 2 |- ( ph -> ( N ` X ) = ( ( N ` .1. ) .x. X ) )
32	31	eqcomd 2183	1 |- ( ph -> ( ( N ` .1. ) .x. X ) = ( N ` X ) )

Syntax hints:   -> wi 4   <-> wb 105   = wceq 1353   e. wcel 2148   ` cfv 5218  (class class class)co 5877   Basecbs 12464   +g cplusg 12538   .rcmulr 12539   0gc0g 12710   Grpcgrp 12882   invgcminusg 12883   1rcur 13147   Ringcrg 13184

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 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-coll 4120  ax-sep 4123  ax-pow 4176  ax-pr 4211  ax-un 4435  ax-setind 4538  ax-cnex 7904  ax-resscn 7905  ax-1cn 7906  ax-1re 7907  ax-icn 7908  ax-addcl 7909  ax-addrcl 7910  ax-mulcl 7911  ax-addcom 7913  ax-addass 7915  ax-i2m1 7918  ax-0lt1 7919  ax-0id 7921  ax-rnegex 7922  ax-pre-ltirr 7925  ax-pre-ltadd 7929

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-nel 2443  df-ral 2460  df-rex 2461  df-reu 2462  df-rmo 2463  df-rab 2464  df-v 2741  df-sbc 2965  df-csb 3060  df-dif 3133  df-un 3135  df-in 3137  df-ss 3144  df-nul 3425  df-pw 3579  df-sn 3600  df-pr 3601  df-op 3603  df-uni 3812  df-int 3847  df-iun 3890  df-br 4006  df-opab 4067  df-mpt 4068  df-id 4295  df-xp 4634  df-rel 4635  df-cnv 4636  df-co 4637  df-dm 4638  df-rn 4639  df-res 4640  df-ima 4641  df-iota 5180  df-fun 5220  df-fn 5221  df-f 5222  df-f1 5223  df-fo 5224  df-f1o 5225  df-fv 5226  df-riota 5833  df-ov 5880  df-oprab 5881  df-mpo 5882  df-pnf 7996  df-mnf 7997  df-ltxr 7999  df-inn 8922  df-2 8980  df-3 8981  df-ndx 12467  df-slot 12468  df-base 12470  df-sets 12471  df-plusg 12551  df-mulr 12552  df-0g 12712  df-mgm 12780  df-sgrp 12813  df-mnd 12823  df-grp 12885  df-minusg 12886  df-mgp 13136  df-ur 13148  df-ring 13186

This theorem is referenced by:  ringmneg1  13235  dvdsrneg  13277  lmodvsneg  13426  lmodsubvs  13438  lmodsubdi  13439  lmodsubdir  13440