Theorem ringnegr 13785

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 13734	. . . . . . 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 13753	. . . . . . 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 13351	. . . . . 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 2204	. . . . . 6 |- ( +g ` R ) = ( +g ` R )
13		ringnegl.t	. . . . . 6 |- .x. = ( .r ` R )
14	5, 12, 13	ringdi 13751	. . . . 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 2204	. . . . . . . 8 |- ( 0g ` R ) = ( 0g ` R )
17	5, 12, 16, 9	grplinv 13353	. . . . . . 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 5959	. . . . 5 |- ( ph -> ( X .x. ( ( N ` .1. ) ( +g ` R ) .1. ) ) = ( X .x. ( 0g ` R ) ) )
20	5, 13, 16	ringrz 13777	. . . . . 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 2237	. . . 4 |- ( ph -> ( X .x. ( ( N ` .1. ) ( +g ` R ) .1. ) ) = ( 0g ` R ) )
23	5, 13, 6	ringridm 13757	. . . . . 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 5959	. . . 4 |- ( ph -> ( ( X .x. ( N ` .1. ) ) ( +g ` R ) ( X .x. .1. ) ) = ( ( X .x. ( N ` .1. ) ) ( +g ` R ) X ) )
26	15, 22, 25	3eqtr3rd 2246	. . 3 |- ( ph -> ( ( X .x. ( N ` .1. ) ) ( +g ` R ) X ) = ( 0g ` R ) )
27	5, 13	ringcl 13746	. . . . 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 13356	. . . 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 2210	1 |- ( ph -> ( X .x. ( N ` .1. ) ) = ( N ` X ) )

Syntax hints:   -> wi 4   <-> wb 105   = wceq 1372   e. wcel 2175   ` cfv 5270  (class class class)co 5943   Basecbs 12803   +g cplusg 12880   .rcmulr 12881   0gc0g 13059   Grpcgrp 13303   invgcminusg 13304   1rcur 13692   Ringcrg 13729

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 1469  ax-7 1470  ax-gen 1471  ax-ie1 1515  ax-ie2 1516  ax-8 1526  ax-10 1527  ax-11 1528  ax-i12 1529  ax-bndl 1531  ax-4 1532  ax-17 1548  ax-i9 1552  ax-ial 1556  ax-i5r 1557  ax-13 2177  ax-14 2178  ax-ext 2186  ax-coll 4158  ax-sep 4161  ax-pow 4217  ax-pr 4252  ax-un 4479  ax-setind 4584  ax-cnex 8015  ax-resscn 8016  ax-1cn 8017  ax-1re 8018  ax-icn 8019  ax-addcl 8020  ax-addrcl 8021  ax-mulcl 8022  ax-addcom 8024  ax-addass 8026  ax-i2m1 8029  ax-0lt1 8030  ax-0id 8032  ax-rnegex 8033  ax-pre-ltirr 8036  ax-pre-ltadd 8040

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1375  df-fal 1378  df-nf 1483  df-sb 1785  df-eu 2056  df-mo 2057  df-clab 2191  df-cleq 2197  df-clel 2200  df-nfc 2336  df-ne 2376  df-nel 2471  df-ral 2488  df-rex 2489  df-reu 2490  df-rmo 2491  df-rab 2492  df-v 2773  df-sbc 2998  df-csb 3093  df-dif 3167  df-un 3169  df-in 3171  df-ss 3178  df-nul 3460  df-pw 3617  df-sn 3638  df-pr 3639  df-op 3641  df-uni 3850  df-int 3885  df-iun 3928  df-br 4044  df-opab 4105  df-mpt 4106  df-id 4339  df-xp 4680  df-rel 4681  df-cnv 4682  df-co 4683  df-dm 4684  df-rn 4685  df-res 4686  df-ima 4687  df-iota 5231  df-fun 5272  df-fn 5273  df-f 5274  df-f1 5275  df-fo 5276  df-f1o 5277  df-fv 5278  df-riota 5898  df-ov 5946  df-oprab 5947  df-mpo 5948  df-pnf 8108  df-mnf 8109  df-ltxr 8111  df-inn 9036  df-2 9094  df-3 9095  df-ndx 12806  df-slot 12807  df-base 12809  df-sets 12810  df-plusg 12893  df-mulr 12894  df-0g 13061  df-mgm 13159  df-sgrp 13205  df-mnd 13220  df-grp 13306  df-minusg 13307  df-mgp 13654  df-ur 13693  df-ring 13731

This theorem is referenced by:  ringmneg2  13787  lmodsubdi  14077