Theorem rngonegmn1r 33741

Description: Negation in a ring is the same as right multiplication by -u 1. (Contributed by Jeff Madsen, 19-Jun-2010.)

Hypotheses

Ref	Expression
ringneg.1	|- G = ( 1st ` R )
ringneg.2	|- H = ( 2nd ` R )
ringneg.3	|- X = ran G
ringneg.4	|- N = ( inv ` G )
ringneg.5	|- U = (GId ` H )

Assertion

Ref	Expression
rngonegmn1r	|- ( ( R e. RingOps /\ A e. X ) -> ( N ` A ) = ( A H ( N ` U ) ) )

Proof of Theorem rngonegmn1r

Step	Hyp	Ref	Expression
1		ringneg.3	. . . . . . . . 9 |- X = ran G
2		ringneg.1	. . . . . . . . . 10 |- G = ( 1st ` R )
3	2	rneqi 5352	. . . . . . . . 9 |- ran G = ran ( 1st ` R )
4	1, 3	eqtri 2644	. . . . . . . 8 |- X = ran ( 1st ` R )
5		ringneg.2	. . . . . . . 8 |- H = ( 2nd ` R )
6		ringneg.5	. . . . . . . 8 |- U = (GId ` H )
7	4, 5, 6	rngo1cl 33738	. . . . . . 7 |- ( R e. RingOps -> U e. X )
8		ringneg.4	. . . . . . . 8 |- N = ( inv ` G )
9	2, 1, 8	rngonegcl 33726	. . . . . . 7 |- ( ( R e. RingOps /\ U e. X ) -> ( N ` U ) e. X )
10	7, 9	mpdan 702	. . . . . 6 |- ( R e. RingOps -> ( N ` U ) e. X )
11	10	adantr 481	. . . . 5 |- ( ( R e. RingOps /\ A e. X ) -> ( N ` U ) e. X )
12	7	adantr 481	. . . . 5 |- ( ( R e. RingOps /\ A e. X ) -> U e. X )
13	11, 12	jca 554	. . . 4 |- ( ( R e. RingOps /\ A e. X ) -> ( ( N ` U ) e. X /\ U e. X ) )
14	2, 5, 1	rngodi 33703	. . . . . 6 |- ( ( R e. RingOps /\ ( A e. X /\ ( N ` U ) e. X /\ U e. X ) ) -> ( A H ( ( N ` U ) G U ) ) = ( ( A H ( N ` U ) ) G ( A H U ) ) )
15	14	3exp2 1285	. . . . 5 |- ( R e. RingOps -> ( A e. X -> ( ( N ` U ) e. X -> ( U e. X -> ( A H ( ( N ` U ) G U ) ) = ( ( A H ( N ` U ) ) G ( A H U ) ) ) ) ) )
16	15	imp43 621	. . . 4 |- ( ( ( R e. RingOps /\ A e. X ) /\ ( ( N ` U ) e. X /\ U e. X ) ) -> ( A H ( ( N ` U ) G U ) ) = ( ( A H ( N ` U ) ) G ( A H U ) ) )
17	13, 16	mpdan 702	. . 3 |- ( ( R e. RingOps /\ A e. X ) -> ( A H ( ( N ` U ) G U ) ) = ( ( A H ( N ` U ) ) G ( A H U ) ) )
18		eqid 2622	. . . . . . . 8 |- (GId ` G ) = (GId ` G )
19	2, 1, 8, 18	rngoaddneg2 33728	. . . . . . 7 |- ( ( R e. RingOps /\ U e. X ) -> ( ( N ` U ) G U ) = (GId ` G ) )
20	7, 19	mpdan 702	. . . . . 6 |- ( R e. RingOps -> ( ( N ` U ) G U ) = (GId ` G ) )
21	20	adantr 481	. . . . 5 |- ( ( R e. RingOps /\ A e. X ) -> ( ( N ` U ) G U ) = (GId ` G ) )
22	21	oveq2d 6666	. . . 4 |- ( ( R e. RingOps /\ A e. X ) -> ( A H ( ( N ` U ) G U ) ) = ( A H (GId ` G ) ) )
23	18, 1, 2, 5	rngorz 33722	. . . 4 |- ( ( R e. RingOps /\ A e. X ) -> ( A H (GId ` G ) ) = (GId ` G ) )
24	22, 23	eqtrd 2656	. . 3 |- ( ( R e. RingOps /\ A e. X ) -> ( A H ( ( N ` U ) G U ) ) = (GId ` G ) )
25	5, 4, 6	rngoridm 33737	. . . 4 |- ( ( R e. RingOps /\ A e. X ) -> ( A H U ) = A )
26	25	oveq2d 6666	. . 3 |- ( ( R e. RingOps /\ A e. X ) -> ( ( A H ( N ` U ) ) G ( A H U ) ) = ( ( A H ( N ` U ) ) G A ) )
27	17, 24, 26	3eqtr3rd 2665	. 2 |- ( ( R e. RingOps /\ A e. X ) -> ( ( A H ( N ` U ) ) G A ) = (GId ` G ) )
28	2, 5, 1	rngocl 33700	. . . 4 |- ( ( R e. RingOps /\ A e. X /\ ( N ` U ) e. X ) -> ( A H ( N ` U ) ) e. X )
29	11, 28	mpd3an3 1425	. . 3 |- ( ( R e. RingOps /\ A e. X ) -> ( A H ( N ` U ) ) e. X )
30	2	rngogrpo 33709	. . . 4 |- ( R e. RingOps -> G e. GrpOp )
31	1, 18, 8	grpoinvid2 27383	. . . 4 |- ( ( G e. GrpOp /\ A e. X /\ ( A H ( N ` U ) ) e. X ) -> ( ( N ` A ) = ( A H ( N ` U ) ) <-> ( ( A H ( N ` U ) ) G A ) = (GId ` G ) ) )
32	30, 31	syl3an1 1359	. . 3 |- ( ( R e. RingOps /\ A e. X /\ ( A H ( N ` U ) ) e. X ) -> ( ( N ` A ) = ( A H ( N ` U ) ) <-> ( ( A H ( N ` U ) ) G A ) = (GId ` G ) ) )
33	29, 32	mpd3an3 1425	. 2 |- ( ( R e. RingOps /\ A e. X ) -> ( ( N ` A ) = ( A H ( N ` U ) ) <-> ( ( A H ( N ` U ) ) G A ) = (GId ` G ) ) )
34	27, 33	mpbird 247	1 |- ( ( R e. RingOps /\ A e. X ) -> ( N ` A ) = ( A H ( N ` U ) ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   ran crn 5115   ` cfv 5888  (class class class)co 6650   1stc1st 7166   2ndc2nd 7167   GrpOpcgr 27343  GIdcgi 27344   invcgn 27345   RingOpscrngo 33693

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-riota 6611  df-ov 6653  df-1st 7168  df-2nd 7169  df-grpo 27347  df-gid 27348  df-ginv 27349  df-ablo 27399  df-ass 33642  df-exid 33644  df-mgmOLD 33648  df-sgrOLD 33660  df-mndo 33666  df-rngo 33694

This theorem is referenced by:  rngonegrmul  33743