Theorem ringnegr 14071

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 14020	. . . . . . 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 14039	. . . . . . 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 13636	. . . . . 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 2231	. . . . . 6 |- ( +g ` R ) = ( +g ` R )
13		ringnegl.t	. . . . . 6 |- .x. = ( .r ` R )
14	5, 12, 13	ringdi 14037	. . . . 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 1275	. . . 4 |- ( ph -> ( X .x. ( ( N ` .1. ) ( +g ` R ) .1. ) ) = ( ( X .x. ( N ` .1. ) ) ( +g ` R ) ( X .x. .1. ) ) )
16		eqid 2231	. . . . . . . 8 |- ( 0g ` R ) = ( 0g ` R )
17	5, 12, 16, 9	grplinv 13638	. . . . . . 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 6034	. . . . 5 |- ( ph -> ( X .x. ( ( N ` .1. ) ( +g ` R ) .1. ) ) = ( X .x. ( 0g ` R ) ) )
20	5, 13, 16	ringrz 14063	. . . . . 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 2264	. . . 4 |- ( ph -> ( X .x. ( ( N ` .1. ) ( +g ` R ) .1. ) ) = ( 0g ` R ) )
23	5, 13, 6	ringridm 14043	. . . . . 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 6034	. . . 4 |- ( ph -> ( ( X .x. ( N ` .1. ) ) ( +g ` R ) ( X .x. .1. ) ) = ( ( X .x. ( N ` .1. ) ) ( +g ` R ) X ) )
26	15, 22, 25	3eqtr3rd 2273	. . 3 |- ( ph -> ( ( X .x. ( N ` .1. ) ) ( +g ` R ) X ) = ( 0g ` R ) )
27	5, 13	ringcl 14032	. . . . 5 |- ( ( R e. Ring /\ X e. B /\ ( N ` .1. ) e. B ) -> ( X .x. ( N ` .1. ) ) e. B )
28	1, 2, 11, 27	syl3anc 1273	. . . 4 |- ( ph -> ( X .x. ( N ` .1. ) ) e. B )
29	5, 12, 16, 9	grpinvid2 13641	. . . 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 1273	. . 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 2237	1 |- ( ph -> ( X .x. ( N ` .1. ) ) = ( N ` X ) )

Syntax hints:   -> wi 4   <-> wb 105   = wceq 1397   e. wcel 2202   ` cfv 5326  (class class class)co 6018   Basecbs 13087   +g cplusg 13165   .rcmulr 13166   0gc0g 13344   Grpcgrp 13588   invgcminusg 13589   1rcur 13978   Ringcrg 14015

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 13090  df-slot 13091  df-base 13093  df-sets 13094  df-plusg 13178  df-mulr 13179  df-0g 13346  df-mgm 13444  df-sgrp 13490  df-mnd 13505  df-grp 13591  df-minusg 13592  df-mgp 13940  df-ur 13979  df-ring 14017

This theorem is referenced by:  ringmneg2  14073  lmodsubdi  14364