Theorem opprnegg 13579

Description: The negative function in an opposite ring. (Contributed by Mario Carneiro, 5-Dec-2014.) (Revised by Mario Carneiro, 2-Oct-2015.)

Hypotheses

Ref	Expression
opprbas.1	|- O = (oppr ` R )
opprneg.2	|- N = ( invg ` R )

Assertion

Ref	Expression
opprnegg	|- ( R e. V -> N = ( invg ` O ) )

Proof of Theorem opprnegg

Dummy variables x y are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		opprbas.1	. . . 4 |- O = (oppr ` R )
2		eqid 2193	. . . 4 |- ( Base ` R ) = ( Base ` R )
3	1, 2	opprbasg 13571	. . 3 |- ( R e. V -> ( Base ` R ) = ( Base ` O ) )
4		eqid 2193	. . . . . . 7 |- ( +g ` R ) = ( +g ` R )
5	1, 4	oppraddg 13572	. . . . . 6 |- ( R e. V -> ( +g ` R ) = ( +g ` O ) )
6	5	oveqd 5935	. . . . 5 |- ( R e. V -> ( y ( +g ` R ) x ) = ( y ( +g ` O ) x ) )
7		eqid 2193	. . . . . 6 |- ( 0g ` R ) = ( 0g ` R )
8	1, 7	oppr0g 13577	. . . . 5 |- ( R e. V -> ( 0g ` R ) = ( 0g ` O ) )
9	6, 8	eqeq12d 2208	. . . 4 |- ( R e. V -> ( ( y ( +g ` R ) x ) = ( 0g ` R ) <-> ( y ( +g ` O ) x ) = ( 0g ` O ) ) )
10	3, 9	riotaeqbidv 5876	. . 3 |- ( R e. V -> ( iota_ y e. ( Base ` R ) ( y ( +g ` R ) x ) = ( 0g ` R ) ) = ( iota_ y e. ( Base ` O ) ( y ( +g ` O ) x ) = ( 0g ` O ) ) )
11	3, 10	mpteq12dv 4111	. 2 |- ( R e. V -> ( x e. ( Base ` R ) |-> ( iota_ y e. ( Base ` R ) ( y ( +g ` R ) x ) = ( 0g ` R ) ) ) = ( x e. ( Base ` O ) |-> ( iota_ y e. ( Base ` O ) ( y ( +g ` O ) x ) = ( 0g ` O ) ) ) )
12		opprneg.2	. . 3 |- N = ( invg ` R )
13	2, 4, 7, 12	grpinvfvalg 13114	. 2 |- ( R e. V -> N = ( x e. ( Base ` R ) |-> ( iota_ y e. ( Base ` R ) ( y ( +g ` R ) x ) = ( 0g ` R ) ) ) )
14	1	opprex 13569	. . 3 |- ( R e. V -> O e. _V )
15		eqid 2193	. . . 4 |- ( Base ` O ) = ( Base ` O )
16		eqid 2193	. . . 4 |- ( +g ` O ) = ( +g ` O )
17		eqid 2193	. . . 4 |- ( 0g ` O ) = ( 0g ` O )
18		eqid 2193	. . . 4 |- ( invg ` O ) = ( invg ` O )
19	15, 16, 17, 18	grpinvfvalg 13114	. . 3 |- ( O e. _V -> ( invg ` O ) = ( x e. ( Base ` O ) |-> ( iota_ y e. ( Base ` O ) ( y ( +g ` O ) x ) = ( 0g ` O ) ) ) )
20	14, 19	syl 14	. 2 |- ( R e. V -> ( invg ` O ) = ( x e. ( Base ` O ) |-> ( iota_ y e. ( Base ` O ) ( y ( +g ` O ) x ) = ( 0g ` O ) ) ) )
21	11, 13, 20	3eqtr4d 2236	1 |- ( R e. V -> N = ( invg ` O ) )

Syntax hints:   -> wi 4   = wceq 1364   e. wcel 2164   _Vcvv 2760   |-> cmpt 4090   ` cfv 5254   iota_crio 5872  (class class class)co 5918   Basecbs 12618   +g cplusg 12695   0gc0g 12867   invgcminusg 13073  opp_rcoppr 13563

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-nul 4155  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-lttrn 7986  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-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-tpos 6298  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-minusg 13076  df-oppr 13564

This theorem is referenced by:  unitnegcl  13626