Theorem opprringbg 14387

Description: Bidirectional form of opprring 14386. (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 14386	. 2 |- ( R e. Ring -> O e. Ring )
3		eqid 2238	. . . . . 6 |- (oppr ` O ) = (oppr ` O )
4	3	opprring 14386	. . . . 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 2239	. . . . 5 |- ( ( R e. V /\ O e. Ring ) -> ( Base ` R ) = ( Base ` R ) )
7		eqid 2238	. . . . . . 7 |- ( Base ` R ) = ( Base ` R )
8	1, 7	opprbasg 14382	. . . . . 6 |- ( R e. V -> ( Base ` R ) = ( Base ` O ) )
9		eqid 2238	. . . . . . 7 |- ( Base ` O ) = ( Base ` O )
10	3, 9	opprbasg 14382	. . . . . 6 |- ( O e. Ring -> ( Base ` O ) = ( Base ` (oppr ` O ) ) )
11	8, 10	sylan9eq 2291	. . . . 5 |- ( ( R e. V /\ O e. Ring ) -> ( Base ` R ) = ( Base ` (oppr ` O ) ) )
12		eqid 2238	. . . . . . . 8 |- ( +g ` R ) = ( +g ` R )
13	1, 12	oppraddg 14383	. . . . . . 7 |- ( R e. V -> ( +g ` R ) = ( +g ` O ) )
14		eqid 2238	. . . . . . . 8 |- ( +g ` O ) = ( +g ` O )
15	3, 14	oppraddg 14383	. . . . . . 7 |- ( O e. Ring -> ( +g ` O ) = ( +g ` (oppr ` O ) ) )
16	13, 15	sylan9eq 2291	. . . . . 6 |- ( ( R e. V /\ O e. Ring ) -> ( +g ` R ) = ( +g ` (oppr ` O ) ) )
17	16	oveqdr 6113	. . . . 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 2238	. . . . . . . . 9 |- ( .r ` O ) = ( .r ` O )
19		eqid 2238	. . . . . . . . 9 |- ( .r ` (oppr ` O ) ) = ( .r ` (oppr ` O ) )
20	9, 18, 3, 19	opprmulg 14378	. . . . . . . 8 |- ( ( O e. Ring /\ x e. ( Base ` R ) /\ y e. ( Base ` R ) ) -> ( x ( .r ` (oppr ` O ) ) y ) = ( y ( .r ` O ) x ) )
21	20	3adant1l 1261	. . . . . . 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 1052	. . . . . . . 8 |- ( ( ( R e. V /\ O e. Ring ) /\ x e. ( Base ` R ) /\ y e. ( Base ` R ) ) -> R e. V )
23		simp3 1030	. . . . . . . 8 |- ( ( ( R e. V /\ O e. Ring ) /\ x e. ( Base ` R ) /\ y e. ( Base ` R ) ) -> y e. ( Base ` R ) )
24		simp2 1029	. . . . . . . 8 |- ( ( ( R e. V /\ O e. Ring ) /\ x e. ( Base ` R ) /\ y e. ( Base ` R ) ) -> x e. ( Base ` R ) )
25		eqid 2238	. . . . . . . . 9 |- ( .r ` R ) = ( .r ` R )
26	7, 25, 1, 18	opprmulg 14378	. . . . . . . 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 1278	. . . . . . 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 2272	. . . . . 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 1235	. . . . 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 14345	. . . 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 ) )

This proof depends on syntax axioms:   -> wi 4   /\ wa 104   <-> wb 105   /\ w3a 1009   = wceq 1402   e. wcel 2209   ` cfv 5377  (class class class)co 6085   Basecbs 13354   +g cplusg 13433   .rcmulr 13434   Ringcrg 14302  opp_rcoppr 14374

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-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-rmo 2536  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-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-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 9306  df-2 9364  df-3 9365  df-ndx 13357  df-slot 13358  df-base 13360  df-sets 13361  df-plusg 13446  df-mulr 13447  df-0g 13614  df-mgm 13678  df-sgrp 13719  df-mnd 13732  df-grp 13810  df-mgp 14220  df-ur 14265  df-ring 14304  df-oppr 14375

This theorem is used by:  opprringb  14388  rhmopp  14485  opprnzrbg  14494