Theorem opprringbg 13397

Description: Bidirectional form of opprring 13396. (Contributed by Mario Carneiro, 6-Dec-2014.)

Hypothesis

Ref	Expression
opprbas.1	|- O = (oppr ` R )

Assertion

Ref	Expression
opprringbg	|- ( R e. V -> ( R e. Ring <-> O e. Ring ) )

Proof of Theorem opprringbg

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

Step	Hyp	Ref	Expression
1		opprbas.1	. . 3 |- O = (oppr ` R )
2	1	opprring 13396	. 2 |- ( R e. Ring -> O e. Ring )
3		eqid 2189	. . . . . 6 |- (oppr ` O ) = (oppr ` O )
4	3	opprring 13396	. . . . 5 |- ( O e. Ring -> (oppr ` O ) e. Ring )
5	4	adantl 277	. . . 4 |- ( ( R e. V /\ O e. Ring ) -> (oppr ` O ) e. Ring )
6		eqidd 2190	. . . . 5 |- ( ( R e. V /\ O e. Ring ) -> ( Base ` R ) = ( Base ` R ) )
7		eqid 2189	. . . . . . 7 |- ( Base ` R ) = ( Base ` R )
8	1, 7	opprbasg 13392	. . . . . 6 |- ( R e. V -> ( Base ` R ) = ( Base ` O ) )
9		eqid 2189	. . . . . . 7 |- ( Base ` O ) = ( Base ` O )
10	3, 9	opprbasg 13392	. . . . . 6 |- ( O e. Ring -> ( Base ` O ) = ( Base ` (oppr ` O ) ) )
11	8, 10	sylan9eq 2242	. . . . 5 |- ( ( R e. V /\ O e. Ring ) -> ( Base ` R ) = ( Base ` (oppr ` O ) ) )
12		eqid 2189	. . . . . . . 8 |- ( +g ` R ) = ( +g ` R )
13	1, 12	oppraddg 13393	. . . . . . 7 |- ( R e. V -> ( +g ` R ) = ( +g ` O ) )
14		eqid 2189	. . . . . . . 8 |- ( +g ` O ) = ( +g ` O )
15	3, 14	oppraddg 13393	. . . . . . 7 |- ( O e. Ring -> ( +g ` O ) = ( +g ` (oppr ` O ) ) )
16	13, 15	sylan9eq 2242	. . . . . 6 |- ( ( R e. V /\ O e. Ring ) -> ( +g ` R ) = ( +g ` (oppr ` O ) ) )
17	16	oveqdr 5919	. . . . 5 |- ( ( ( R e. V /\ O e. Ring ) /\ ( x e. ( Base ` R ) /\ y e. ( Base ` R ) ) ) -> ( x ( +g ` R ) y ) = ( x ( +g ` (oppr ` O ) ) y ) )
18		eqid 2189	. . . . . . . . 9 |- ( .r ` O ) = ( .r ` O )
19		eqid 2189	. . . . . . . . 9 |- ( .r ` (oppr ` O ) ) = ( .r ` (oppr ` O ) )
20	9, 18, 3, 19	opprmulg 13388	. . . . . . . 8 |- ( ( O e. Ring /\ x e. ( Base ` R ) /\ y e. ( Base ` R ) ) -> ( x ( .r ` (oppr ` O ) ) y ) = ( y ( .r ` O ) x ) )
21	20	3adant1l 1232	. . . . . . 7 |- ( ( ( R e. V /\ O e. Ring ) /\ x e. ( Base ` R ) /\ y e. ( Base ` R ) ) -> ( x ( .r ` (oppr ` O ) ) y ) = ( y ( .r ` O ) x ) )
22		simp1l 1023	. . . . . . . 8 |- ( ( ( R e. V /\ O e. Ring ) /\ x e. ( Base ` R ) /\ y e. ( Base ` R ) ) -> R e. V )
23		simp3 1001	. . . . . . . 8 |- ( ( ( R e. V /\ O e. Ring ) /\ x e. ( Base ` R ) /\ y e. ( Base ` R ) ) -> y e. ( Base ` R ) )
24		simp2 1000	. . . . . . . 8 |- ( ( ( R e. V /\ O e. Ring ) /\ x e. ( Base ` R ) /\ y e. ( Base ` R ) ) -> x e. ( Base ` R ) )
25		eqid 2189	. . . . . . . . 9 |- ( .r ` R ) = ( .r ` R )
26	7, 25, 1, 18	opprmulg 13388	. . . . . . . 8 |- ( ( R e. V /\ y e. ( Base ` R ) /\ x e. ( Base ` R ) ) -> ( y ( .r ` O ) x ) = ( x ( .r ` R ) y ) )
27	22, 23, 24, 26	syl3anc 1249	. . . . . . 7 |- ( ( ( R e. V /\ O e. Ring ) /\ x e. ( Base ` R ) /\ y e. ( Base ` R ) ) -> ( y ( .r ` O ) x ) = ( x ( .r ` R ) y ) )
28	21, 27	eqtr2d 2223	. . . . . 6 |- ( ( ( R e. V /\ O e. Ring ) /\ x e. ( Base ` R ) /\ y e. ( Base ` R ) ) -> ( x ( .r ` R ) y ) = ( x ( .r ` (oppr ` O ) ) y ) )
29	28	3expb 1206	. . . . 5 |- ( ( ( R e. V /\ O e. Ring ) /\ ( x e. ( Base ` R ) /\ y e. ( Base ` R ) ) ) -> ( x ( .r ` R ) y ) = ( x ( .r ` (oppr ` O ) ) y ) )
30	6, 11, 17, 29	ringpropd 13359	. . . 4 |- ( ( R e. V /\ O e. Ring ) -> ( R e. Ring <-> (oppr ` O ) e. Ring ) )
31	5, 30	mpbird 167	. . 3 |- ( ( R e. V /\ O e. Ring ) -> R e. Ring )
32	31	ex 115	. 2 |- ( R e. V -> ( O e. Ring -> R e. Ring ) )
33	2, 32	impbid2 143	1 |- ( R e. V -> ( R e. Ring <-> O e. Ring ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   /\ w3a 980   = wceq 1364   e. wcel 2160   ` cfv 5231  (class class class)co 5891   Basecbs 12486   +g cplusg 12561   .rcmulr 12562   Ringcrg 13317  opp_rcoppr 13384

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 2162  ax-14 2163  ax-ext 2171  ax-sep 4136  ax-nul 4144  ax-pow 4189  ax-pr 4224  ax-un 4448  ax-setind 4551  ax-cnex 7921  ax-resscn 7922  ax-1cn 7923  ax-1re 7924  ax-icn 7925  ax-addcl 7926  ax-addrcl 7927  ax-mulcl 7928  ax-addcom 7930  ax-addass 7932  ax-i2m1 7935  ax-0lt1 7936  ax-0id 7938  ax-rnegex 7939  ax-pre-ltirr 7942  ax-pre-lttrn 7944  ax-pre-ltadd 7946

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 2041  df-mo 2042  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ne 2361  df-nel 2456  df-ral 2473  df-rex 2474  df-reu 2475  df-rmo 2476  df-rab 2477  df-v 2754  df-sbc 2978  df-csb 3073  df-dif 3146  df-un 3148  df-in 3150  df-ss 3157  df-nul 3438  df-pw 3592  df-sn 3613  df-pr 3614  df-op 3616  df-uni 3825  df-int 3860  df-br 4019  df-opab 4080  df-mpt 4081  df-id 4308  df-xp 4647  df-rel 4648  df-cnv 4649  df-co 4650  df-dm 4651  df-rn 4652  df-res 4653  df-ima 4654  df-iota 5193  df-fun 5233  df-fn 5234  df-fv 5239  df-riota 5847  df-ov 5894  df-oprab 5895  df-mpo 5896  df-tpos 6264  df-pnf 8013  df-mnf 8014  df-ltxr 8016  df-inn 8939  df-2 8997  df-3 8998  df-ndx 12489  df-slot 12490  df-base 12492  df-sets 12493  df-plusg 12574  df-mulr 12575  df-0g 12735  df-mgm 12804  df-sgrp 12837  df-mnd 12850  df-grp 12920  df-mgp 13242  df-ur 13281  df-ring 13319  df-oppr 13385

This theorem is referenced by:  rhmopp  13493