Theorem ringnegr 13548

Description: Negation in a ring is the same as right 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
ringnegr	|- ( ph -> ( X .x. ( N ` .1. ) ) = ( N ` X ) )

Proof of Theorem ringnegr

Step	Hyp	Ref	Expression
1		ringnegl.r	. . . . 5 |- ( ph -> R e. Ring )
2		ringnegl.x	. . . . 5 |- ( ph -> X e. B )
3		ringgrp 13497	. . . . . . 7 |- ( R e. Ring -> R e. Grp )
4	1, 3	syl 14	. . . . . 6 |- ( ph -> R e. Grp )
5		ringnegl.b	. . . . . . . 8 |- B = ( Base ` R )
6		ringnegl.u	. . . . . . . 8 |- .1. = ( 1r ` R )
7	5, 6	ringidcl 13516	. . . . . . 7 |- ( R e. Ring -> .1. e. B )
8	1, 7	syl 14	. . . . . 6 |- ( ph -> .1. e. B )
9		ringnegl.n	. . . . . . 7 |- N = ( invg ` R )
10	5, 9	grpinvcl 13120	. . . . . 6 |- ( ( R e. Grp /\ .1. e. B ) -> ( N ` .1. ) e. B )
11	4, 8, 10	syl2anc 411	. . . . 5 |- ( ph -> ( N ` .1. ) e. B )
12		eqid 2193	. . . . . 6 |- ( +g ` R ) = ( +g ` R )
13		ringnegl.t	. . . . . 6 |- .x. = ( .r ` R )
14	5, 12, 13	ringdi 13514	. . . . 5 |- ( ( R e. Ring /\ ( X e. B /\ ( N ` .1. ) e. B /\ .1. e. B ) ) -> ( X .x. ( ( N ` .1. ) ( +g ` R ) .1. ) ) = ( ( X .x. ( N ` .1. ) ) ( +g ` R ) ( X .x. .1. ) ) )
15	1, 2, 11, 8, 14	syl13anc 1251	. . . 4 |- ( ph -> ( X .x. ( ( N ` .1. ) ( +g ` R ) .1. ) ) = ( ( X .x. ( N ` .1. ) ) ( +g ` R ) ( X .x. .1. ) ) )
16		eqid 2193	. . . . . . . 8 |- ( 0g ` R ) = ( 0g ` R )
17	5, 12, 16, 9	grplinv 13122	. . . . . . 7 |- ( ( R e. Grp /\ .1. e. B ) -> ( ( N ` .1. ) ( +g ` R ) .1. ) = ( 0g ` R ) )
18	4, 8, 17	syl2anc 411	. . . . . 6 |- ( ph -> ( ( N ` .1. ) ( +g ` R ) .1. ) = ( 0g ` R ) )
19	18	oveq2d 5934	. . . . 5 |- ( ph -> ( X .x. ( ( N ` .1. ) ( +g ` R ) .1. ) ) = ( X .x. ( 0g ` R ) ) )
20	5, 13, 16	ringrz 13540	. . . . . 6 |- ( ( R e. Ring /\ X e. B ) -> ( X .x. ( 0g ` R ) ) = ( 0g ` R ) )
21	1, 2, 20	syl2anc 411	. . . . 5 |- ( ph -> ( X .x. ( 0g ` R ) ) = ( 0g ` R ) )
22	19, 21	eqtrd 2226	. . . 4 |- ( ph -> ( X .x. ( ( N ` .1. ) ( +g ` R ) .1. ) ) = ( 0g ` R ) )
23	5, 13, 6	ringridm 13520	. . . . . 6 |- ( ( R e. Ring /\ X e. B ) -> ( X .x. .1. ) = X )
24	1, 2, 23	syl2anc 411	. . . . 5 |- ( ph -> ( X .x. .1. ) = X )
25	24	oveq2d 5934	. . . 4 |- ( ph -> ( ( X .x. ( N ` .1. ) ) ( +g ` R ) ( X .x. .1. ) ) = ( ( X .x. ( N ` .1. ) ) ( +g ` R ) X ) )
26	15, 22, 25	3eqtr3rd 2235	. . 3 |- ( ph -> ( ( X .x. ( N ` .1. ) ) ( +g ` R ) X ) = ( 0g ` R ) )
27	5, 13	ringcl 13509	. . . . 5 |- ( ( R e. Ring /\ X e. B /\ ( N ` .1. ) e. B ) -> ( X .x. ( N ` .1. ) ) e. B )
28	1, 2, 11, 27	syl3anc 1249	. . . 4 |- ( ph -> ( X .x. ( N ` .1. ) ) e. B )
29	5, 12, 16, 9	grpinvid2 13125	. . . 4 |- ( ( R e. Grp /\ X e. B /\ ( X .x. ( N ` .1. ) ) e. B ) -> ( ( N ` X ) = ( X .x. ( N ` .1. ) ) <-> ( ( X .x. ( N ` .1. ) ) ( +g ` R ) X ) = ( 0g ` R ) ) )
30	4, 2, 28, 29	syl3anc 1249	. . 3 |- ( ph -> ( ( N ` X ) = ( X .x. ( N ` .1. ) ) <-> ( ( X .x. ( N ` .1. ) ) ( +g ` R ) X ) = ( 0g ` R ) ) )
31	26, 30	mpbird 167	. 2 |- ( ph -> ( N ` X ) = ( X .x. ( N ` .1. ) ) )
32	31	eqcomd 2199	1 |- ( ph -> ( X .x. ( N ` .1. ) ) = ( N ` X ) )

Syntax hints:   -> wi 4   <-> wb 105   = wceq 1364   e. wcel 2164   ` cfv 5254  (class class class)co 5918   Basecbs 12618   +g cplusg 12695   .rcmulr 12696   0gc0g 12867   Grpcgrp 13072   invgcminusg 13073   1rcur 13455   Ringcrg 13492

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 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2166  ax-14 2167  ax-ext 2175  ax-coll 4144  ax-sep 4147  ax-pow 4203  ax-pr 4238  ax-un 4464  ax-setind 4569  ax-cnex 7963  ax-resscn 7964  ax-1cn 7965  ax-1re 7966  ax-icn 7967  ax-addcl 7968  ax-addrcl 7969  ax-mulcl 7970  ax-addcom 7972  ax-addass 7974  ax-i2m1 7977  ax-0lt1 7978  ax-0id 7980  ax-rnegex 7981  ax-pre-ltirr 7984  ax-pre-ltadd 7988

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ne 2365  df-nel 2460  df-ral 2477  df-rex 2478  df-reu 2479  df-rmo 2480  df-rab 2481  df-v 2762  df-sbc 2986  df-csb 3081  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3447  df-pw 3603  df-sn 3624  df-pr 3625  df-op 3627  df-uni 3836  df-int 3871  df-iun 3914  df-br 4030  df-opab 4091  df-mpt 4092  df-id 4324  df-xp 4665  df-rel 4666  df-cnv 4667  df-co 4668  df-dm 4669  df-rn 4670  df-res 4671  df-ima 4672  df-iota 5215  df-fun 5256  df-fn 5257  df-f 5258  df-f1 5259  df-fo 5260  df-f1o 5261  df-fv 5262  df-riota 5873  df-ov 5921  df-oprab 5922  df-mpo 5923  df-pnf 8056  df-mnf 8057  df-ltxr 8059  df-inn 8983  df-2 9041  df-3 9042  df-ndx 12621  df-slot 12622  df-base 12624  df-sets 12625  df-plusg 12708  df-mulr 12709  df-0g 12869  df-mgm 12939  df-sgrp 12985  df-mnd 12998  df-grp 13075  df-minusg 13076  df-mgp 13417  df-ur 13456  df-ring 13494

This theorem is referenced by:  ringmneg2  13550  lmodsubdi  13840