Theorem ringnegl 13685

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 13654	. . . . . 6 |- ( R e. Ring -> .1. e. B )
5	1, 4	syl 14	. . . . 5 |- ( ph -> .1. e. B )
6		ringgrp 13635	. . . . . . 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 13252	. . . . . 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 2196	. . . . . 6 |- ( +g ` R ) = ( +g ` R )
13		ringnegl.t	. . . . . 6 |- .x. = ( .r ` R )
14	2, 12, 13	ringdir 13653	. . . . 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 1251	. . . 4 |- ( ph -> ( ( .1. ( +g ` R ) ( N ` .1. ) ) .x. X ) = ( ( .1. .x. X ) ( +g ` R ) ( ( N ` .1. ) .x. X ) ) )
16		eqid 2196	. . . . . . . 8 |- ( 0g ` R ) = ( 0g ` R )
17	2, 12, 16, 8	grprinv 13255	. . . . . . 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 5940	. . . . 5 |- ( ph -> ( ( .1. ( +g ` R ) ( N ` .1. ) ) .x. X ) = ( ( 0g ` R ) .x. X ) )
20	2, 13, 16	ringlz 13677	. . . . . 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 2229	. . . 4 |- ( ph -> ( ( .1. ( +g ` R ) ( N ` .1. ) ) .x. X ) = ( 0g ` R ) )
23	2, 13, 3	ringlidm 13657	. . . . . 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 5940	. . . 4 |- ( ph -> ( ( .1. .x. X ) ( +g ` R ) ( ( N ` .1. ) .x. X ) ) = ( X ( +g ` R ) ( ( N ` .1. ) .x. X ) ) )
26	15, 22, 25	3eqtr3rd 2238	. . 3 |- ( ph -> ( X ( +g ` R ) ( ( N ` .1. ) .x. X ) ) = ( 0g ` R ) )
27	2, 13	ringcl 13647	. . . . 5 |- ( ( R e. Ring /\ ( N ` .1. ) e. B /\ X e. B ) -> ( ( N ` .1. ) .x. X ) e. B )
28	1, 10, 11, 27	syl3anc 1249	. . . 4 |- ( ph -> ( ( N ` .1. ) .x. X ) e. B )
29	2, 12, 16, 8	grpinvid1 13256	. . . 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 1249	. . 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 2202	1 |- ( ph -> ( ( N ` .1. ) .x. X ) = ( N ` X ) )

Syntax hints:   -> wi 4   <-> wb 105   = wceq 1364   e. wcel 2167   ` cfv 5259  (class class class)co 5925   Basecbs 12705   +g cplusg 12782   .rcmulr 12783   0gc0g 12960   Grpcgrp 13204   invgcminusg 13205   1rcur 13593   Ringcrg 13630

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 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-13 2169  ax-14 2170  ax-ext 2178  ax-coll 4149  ax-sep 4152  ax-pow 4208  ax-pr 4243  ax-un 4469  ax-setind 4574  ax-cnex 7989  ax-resscn 7990  ax-1cn 7991  ax-1re 7992  ax-icn 7993  ax-addcl 7994  ax-addrcl 7995  ax-mulcl 7996  ax-addcom 7998  ax-addass 8000  ax-i2m1 8003  ax-0lt1 8004  ax-0id 8006  ax-rnegex 8007  ax-pre-ltirr 8010  ax-pre-ltadd 8014

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ne 2368  df-nel 2463  df-ral 2480  df-rex 2481  df-reu 2482  df-rmo 2483  df-rab 2484  df-v 2765  df-sbc 2990  df-csb 3085  df-dif 3159  df-un 3161  df-in 3163  df-ss 3170  df-nul 3452  df-pw 3608  df-sn 3629  df-pr 3630  df-op 3632  df-uni 3841  df-int 3876  df-iun 3919  df-br 4035  df-opab 4096  df-mpt 4097  df-id 4329  df-xp 4670  df-rel 4671  df-cnv 4672  df-co 4673  df-dm 4674  df-rn 4675  df-res 4676  df-ima 4677  df-iota 5220  df-fun 5261  df-fn 5262  df-f 5263  df-f1 5264  df-fo 5265  df-f1o 5266  df-fv 5267  df-riota 5880  df-ov 5928  df-oprab 5929  df-mpo 5930  df-pnf 8082  df-mnf 8083  df-ltxr 8085  df-inn 9010  df-2 9068  df-3 9069  df-ndx 12708  df-slot 12709  df-base 12711  df-sets 12712  df-plusg 12795  df-mulr 12796  df-0g 12962  df-mgm 13060  df-sgrp 13106  df-mnd 13121  df-grp 13207  df-minusg 13208  df-mgp 13555  df-ur 13594  df-ring 13632

This theorem is referenced by:  ringmneg1  13687  dvdsrneg  13737  lmodvsneg  13965  lmodsubvs  13977  lmodsubdi  13978  lmodsubdir  13979