Theorem opprnegg 14389

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 2238	. . . 4 |- ( Base ` R ) = ( Base ` R )
3	1, 2	opprbasg 14380	. . 3 |- ( R e. V -> ( Base ` R ) = ( Base ` O ) )
4		eqid 2238	. . . . . . 7 |- ( +g ` R ) = ( +g ` R )
5	1, 4	oppraddg 14381	. . . . . 6 |- ( R e. V -> ( +g ` R ) = ( +g ` O ) )
6	5	oveqd 6102	. . . . 5 |- ( R e. V -> ( y ( +g ` R ) x ) = ( y ( +g ` O ) x ) )
7		eqid 2238	. . . . . 6 |- ( 0g ` R ) = ( 0g ` R )
8	1, 7	oppr0g 14387	. . . . 5 |- ( R e. V -> ( 0g ` R ) = ( 0g ` O ) )
9	6, 8	eqeq12d 2253	. . . 4 |- ( R e. V -> ( ( y ( +g ` R ) x ) = ( 0g ` R ) <-> ( y ( +g ` O ) x ) = ( 0g ` O ) ) )
10	3, 9	riotaeqbidv 6041	. . 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 4213	. 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 13847	. 2 |- ( R e. V -> N = ( x e. ( Base ` R ) |-> ( iota_ y e. ( Base ` R ) ( y ( +g ` R ) x ) = ( 0g ` R ) ) ) )
14	1	opprex 14378	. . 3 |- ( R e. V -> O e. _V )
15		eqid 2238	. . . 4 |- ( Base ` O ) = ( Base ` O )
16		eqid 2238	. . . 4 |- ( +g ` O ) = ( +g ` O )
17		eqid 2238	. . . 4 |- ( 0g ` O ) = ( 0g ` O )
18		eqid 2238	. . . 4 |- ( invg ` O ) = ( invg ` O )
19	15, 16, 17, 18	grpinvfvalg 13847	. . 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 2281	1 |- ( R e. V -> N = ( invg ` O ) )

This proof depends on syntax axioms:   -> wi 4   = wceq 1402   e. wcel 2209   _Vcvv 2821   |-> cmpt 4192   ` cfv 5377   iota_crio 6037  (class class class)co 6085   Basecbs 13352   +g cplusg 13431   0gc0g 13610   invgcminusg 13806  opp_rcoppr 14372

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-coll 4246  ax-sep 4249  ax-nul 4259  ax-pow 4311  ax-pr 4346  ax-un 4578  ax-setind 4684  ax-cnex 8270  ax-resscn 8271  ax-1cn 8272  ax-1re 8273  ax-icn 8274  ax-addcl 8275  ax-addrcl 8276  ax-mulcl 8277  ax-addcom 8279  ax-addass 8281  ax-i2m1 8284  ax-0lt1 8285  ax-0id 8287  ax-rnegex 8288  ax-pre-ltirr 8291  ax-pre-lttrn 8293  ax-pre-ltadd 8295

This proof depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-nel 2516  df-ral 2533  df-rex 2534  df-reu 2535  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-pw 3690  df-sn 3715  df-pr 3716  df-op 3718  df-uni 3936  df-int 3971  df-iun 4014  df-br 4131  df-opab 4193  df-mpt 4194  df-id 4438  df-xp 4780  df-rel 4781  df-cnv 4782  df-co 4783  df-dm 4784  df-rn 4785  df-res 4786  df-ima 4787  df-iota 5337  df-fun 5379  df-fn 5380  df-f 5381  df-f1 5382  df-fo 5383  df-f1o 5384  df-fv 5385  df-riota 6038  df-ov 6088  df-oprab 6089  df-mpo 6090  df-tpos 6516  df-pnf 8362  df-mnf 8363  df-ltxr 8365  df-inn 9305  df-2 9363  df-3 9364  df-ndx 13355  df-slot 13356  df-base 13358  df-sets 13359  df-plusg 13444  df-mulr 13445  df-0g 13612  df-minusg 13809  df-oppr 14373

This theorem is used by:  unitnegcl  14437  opprdrng  14620