Theorem opprnegg 13639

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 2196	. . . 4 |- ( Base ` R ) = ( Base ` R )
3	1, 2	opprbasg 13631	. . 3 |- ( R e. V -> ( Base ` R ) = ( Base ` O ) )
4		eqid 2196	. . . . . . 7 |- ( +g ` R ) = ( +g ` R )
5	1, 4	oppraddg 13632	. . . . . 6 |- ( R e. V -> ( +g ` R ) = ( +g ` O ) )
6	5	oveqd 5939	. . . . 5 |- ( R e. V -> ( y ( +g ` R ) x ) = ( y ( +g ` O ) x ) )
7		eqid 2196	. . . . . 6 |- ( 0g ` R ) = ( 0g ` R )
8	1, 7	oppr0g 13637	. . . . 5 |- ( R e. V -> ( 0g ` R ) = ( 0g ` O ) )
9	6, 8	eqeq12d 2211	. . . 4 |- ( R e. V -> ( ( y ( +g ` R ) x ) = ( 0g ` R ) <-> ( y ( +g ` O ) x ) = ( 0g ` O ) ) )
10	3, 9	riotaeqbidv 5880	. . 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 4115	. 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 13174	. 2 |- ( R e. V -> N = ( x e. ( Base ` R ) |-> ( iota_ y e. ( Base ` R ) ( y ( +g ` R ) x ) = ( 0g ` R ) ) ) )
14	1	opprex 13629	. . 3 |- ( R e. V -> O e. _V )
15		eqid 2196	. . . 4 |- ( Base ` O ) = ( Base ` O )
16		eqid 2196	. . . 4 |- ( +g ` O ) = ( +g ` O )
17		eqid 2196	. . . 4 |- ( 0g ` O ) = ( 0g ` O )
18		eqid 2196	. . . 4 |- ( invg ` O ) = ( invg ` O )
19	15, 16, 17, 18	grpinvfvalg 13174	. . 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 2239	1 |- ( R e. V -> N = ( invg ` O ) )

Syntax hints:   -> wi 4   = wceq 1364   e. wcel 2167   _Vcvv 2763   |-> cmpt 4094   ` cfv 5258   iota_crio 5876  (class class class)co 5922   Basecbs 12678   +g cplusg 12755   0gc0g 12927   invgcminusg 13133  opp_rcoppr 13623

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 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-13 2169  ax-14 2170  ax-ext 2178  ax-coll 4148  ax-sep 4151  ax-nul 4159  ax-pow 4207  ax-pr 4242  ax-un 4468  ax-setind 4573  ax-cnex 7970  ax-resscn 7971  ax-1cn 7972  ax-1re 7973  ax-icn 7974  ax-addcl 7975  ax-addrcl 7976  ax-mulcl 7977  ax-addcom 7979  ax-addass 7981  ax-i2m1 7984  ax-0lt1 7985  ax-0id 7987  ax-rnegex 7988  ax-pre-ltirr 7991  ax-pre-lttrn 7993  ax-pre-ltadd 7995

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ne 2368  df-nel 2463  df-ral 2480  df-rex 2481  df-reu 2482  df-rab 2484  df-v 2765  df-sbc 2990  df-csb 3085  df-dif 3159  df-un 3161  df-in 3163  df-ss 3170  df-nul 3451  df-pw 3607  df-sn 3628  df-pr 3629  df-op 3631  df-uni 3840  df-int 3875  df-iun 3918  df-br 4034  df-opab 4095  df-mpt 4096  df-id 4328  df-xp 4669  df-rel 4670  df-cnv 4671  df-co 4672  df-dm 4673  df-rn 4674  df-res 4675  df-ima 4676  df-iota 5219  df-fun 5260  df-fn 5261  df-f 5262  df-f1 5263  df-fo 5264  df-f1o 5265  df-fv 5266  df-riota 5877  df-ov 5925  df-oprab 5926  df-mpo 5927  df-tpos 6303  df-pnf 8063  df-mnf 8064  df-ltxr 8066  df-inn 8991  df-2 9049  df-3 9050  df-ndx 12681  df-slot 12682  df-base 12684  df-sets 12685  df-plusg 12768  df-mulr 12769  df-0g 12929  df-minusg 13136  df-oppr 13624

This theorem is referenced by:  unitnegcl  13686