Theorem ringnegr 13229

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 13184	. . . . . . 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 13203	. . . . . . 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 12921	. . . . . 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 2177	. . . . . 6 |- ( +g ` R ) = ( +g ` R )
13		ringnegl.t	. . . . . 6 |- .x. = ( .r ` R )
14	5, 12, 13	ringdi 13201	. . . . 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 1240	. . . 4 |- ( ph -> ( X .x. ( ( N ` .1. ) ( +g ` R ) .1. ) ) = ( ( X .x. ( N ` .1. ) ) ( +g ` R ) ( X .x. .1. ) ) )
16		eqid 2177	. . . . . . . 8 |- ( 0g ` R ) = ( 0g ` R )
17	5, 12, 16, 9	grplinv 12922	. . . . . . 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 5891	. . . . 5 |- ( ph -> ( X .x. ( ( N ` .1. ) ( +g ` R ) .1. ) ) = ( X .x. ( 0g ` R ) ) )
20	5, 13, 16	ringrz 13223	. . . . . 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 2210	. . . 4 |- ( ph -> ( X .x. ( ( N ` .1. ) ( +g ` R ) .1. ) ) = ( 0g ` R ) )
23	5, 13, 6	ringridm 13207	. . . . . 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 5891	. . . 4 |- ( ph -> ( ( X .x. ( N ` .1. ) ) ( +g ` R ) ( X .x. .1. ) ) = ( ( X .x. ( N ` .1. ) ) ( +g ` R ) X ) )
26	15, 22, 25	3eqtr3rd 2219	. . 3 |- ( ph -> ( ( X .x. ( N ` .1. ) ) ( +g ` R ) X ) = ( 0g ` R ) )
27	5, 13	ringcl 13196	. . . . 5 |- ( ( R e. Ring /\ X e. B /\ ( N ` .1. ) e. B ) -> ( X .x. ( N ` .1. ) ) e. B )
28	1, 2, 11, 27	syl3anc 1238	. . . 4 |- ( ph -> ( X .x. ( N ` .1. ) ) e. B )
29	5, 12, 16, 9	grpinvid2 12925	. . . 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 1238	. . 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 2183	1 |- ( ph -> ( X .x. ( N ` .1. ) ) = ( N ` X ) )

Syntax hints:   -> wi 4   <-> wb 105   = wceq 1353   e. wcel 2148   ` cfv 5217  (class class class)co 5875   Basecbs 12462   +g cplusg 12536   .rcmulr 12537   0gc0g 12705   Grpcgrp 12877   invgcminusg 12878   1rcur 13142   Ringcrg 13179

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 4119  ax-sep 4122  ax-pow 4175  ax-pr 4210  ax-un 4434  ax-setind 4537  ax-cnex 7902  ax-resscn 7903  ax-1cn 7904  ax-1re 7905  ax-icn 7906  ax-addcl 7907  ax-addrcl 7908  ax-mulcl 7909  ax-addcom 7911  ax-addass 7913  ax-i2m1 7916  ax-0lt1 7917  ax-0id 7919  ax-rnegex 7920  ax-pre-ltirr 7923  ax-pre-ltadd 7927

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 2740  df-sbc 2964  df-csb 3059  df-dif 3132  df-un 3134  df-in 3136  df-ss 3143  df-nul 3424  df-pw 3578  df-sn 3599  df-pr 3600  df-op 3602  df-uni 3811  df-int 3846  df-iun 3889  df-br 4005  df-opab 4066  df-mpt 4067  df-id 4294  df-xp 4633  df-rel 4634  df-cnv 4635  df-co 4636  df-dm 4637  df-rn 4638  df-res 4639  df-ima 4640  df-iota 5179  df-fun 5219  df-fn 5220  df-f 5221  df-f1 5222  df-fo 5223  df-f1o 5224  df-fv 5225  df-riota 5831  df-ov 5878  df-oprab 5879  df-mpo 5880  df-pnf 7994  df-mnf 7995  df-ltxr 7997  df-inn 8920  df-2 8978  df-3 8979  df-ndx 12465  df-slot 12466  df-base 12468  df-sets 12469  df-plusg 12549  df-mulr 12550  df-0g 12707  df-mgm 12775  df-sgrp 12808  df-mnd 12818  df-grp 12880  df-minusg 12881  df-mgp 13131  df-ur 13143  df-ring 13181

This theorem is referenced by:  ringmneg2  13231  lmodsubdi  13434