Theorem opprringbg 14326

Description: Bidirectional form of opprring 14325. (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 14325	. 2 |- ( R e. Ring -> O e. Ring )
3		eqid 2234	. . . . . 6 |- (oppr ` O ) = (oppr ` O )
4	3	opprring 14325	. . . . 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 2235	. . . . 5 |- ( ( R e. V /\ O e. Ring ) -> ( Base ` R ) = ( Base ` R ) )
7		eqid 2234	. . . . . . 7 |- ( Base ` R ) = ( Base ` R )
8	1, 7	opprbasg 14321	. . . . . 6 |- ( R e. V -> ( Base ` R ) = ( Base ` O ) )
9		eqid 2234	. . . . . . 7 |- ( Base ` O ) = ( Base ` O )
10	3, 9	opprbasg 14321	. . . . . 6 |- ( O e. Ring -> ( Base ` O ) = ( Base ` (oppr ` O ) ) )
11	8, 10	sylan9eq 2287	. . . . 5 |- ( ( R e. V /\ O e. Ring ) -> ( Base ` R ) = ( Base ` (oppr ` O ) ) )
12		eqid 2234	. . . . . . . 8 |- ( +g ` R ) = ( +g ` R )
13	1, 12	oppraddg 14322	. . . . . . 7 |- ( R e. V -> ( +g ` R ) = ( +g ` O ) )
14		eqid 2234	. . . . . . . 8 |- ( +g ` O ) = ( +g ` O )
15	3, 14	oppraddg 14322	. . . . . . 7 |- ( O e. Ring -> ( +g ` O ) = ( +g ` (oppr ` O ) ) )
16	13, 15	sylan9eq 2287	. . . . . 6 |- ( ( R e. V /\ O e. Ring ) -> ( +g ` R ) = ( +g ` (oppr ` O ) ) )
17	16	oveqdr 6086	. . . . 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 2234	. . . . . . . . 9 |- ( .r ` O ) = ( .r ` O )
19		eqid 2234	. . . . . . . . 9 |- ( .r ` (oppr ` O ) ) = ( .r ` (oppr ` O ) )
20	9, 18, 3, 19	opprmulg 14317	. . . . . . . 8 |- ( ( O e. Ring /\ x e. ( Base ` R ) /\ y e. ( Base ` R ) ) -> ( x ( .r ` (oppr ` O ) ) y ) = ( y ( .r ` O ) x ) )
21	20	3adant1l 1257	. . . . . . 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 1048	. . . . . . . 8 |- ( ( ( R e. V /\ O e. Ring ) /\ x e. ( Base ` R ) /\ y e. ( Base ` R ) ) -> R e. V )
23		simp3 1026	. . . . . . . 8 |- ( ( ( R e. V /\ O e. Ring ) /\ x e. ( Base ` R ) /\ y e. ( Base ` R ) ) -> y e. ( Base ` R ) )
24		simp2 1025	. . . . . . . 8 |- ( ( ( R e. V /\ O e. Ring ) /\ x e. ( Base ` R ) /\ y e. ( Base ` R ) ) -> x e. ( Base ` R ) )
25		eqid 2234	. . . . . . . . 9 |- ( .r ` R ) = ( .r ` R )
26	7, 25, 1, 18	opprmulg 14317	. . . . . . . 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 1274	. . . . . . 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 2268	. . . . . 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 1231	. . . . 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 14284	. . . 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 1005   = wceq 1398   e. wcel 2205   ` cfv 5357  (class class class)co 6058   Basecbs 13299   +g cplusg 13377   .rcmulr 13378   Ringcrg 14242  opp_rcoppr 14313

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 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2207  ax-14 2208  ax-ext 2216  ax-sep 4233  ax-nul 4241  ax-pow 4292  ax-pr 4327  ax-un 4559  ax-setind 4664  ax-cnex 8234  ax-resscn 8235  ax-1cn 8236  ax-1re 8237  ax-icn 8238  ax-addcl 8239  ax-addrcl 8240  ax-mulcl 8241  ax-addcom 8243  ax-addass 8245  ax-i2m1 8248  ax-0lt1 8249  ax-0id 8251  ax-rnegex 8252  ax-pre-ltirr 8255  ax-pre-lttrn 8257  ax-pre-ltadd 8259

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ne 2415  df-nel 2510  df-ral 2527  df-rex 2528  df-reu 2529  df-rmo 2530  df-rab 2531  df-v 2817  df-sbc 3046  df-csb 3142  df-dif 3216  df-un 3218  df-in 3220  df-ss 3227  df-nul 3513  df-pw 3676  df-sn 3700  df-pr 3701  df-op 3703  df-uni 3920  df-int 3955  df-br 4115  df-opab 4177  df-mpt 4178  df-id 4419  df-xp 4760  df-rel 4761  df-cnv 4762  df-co 4763  df-dm 4764  df-rn 4765  df-res 4766  df-ima 4767  df-iota 5317  df-fun 5359  df-fn 5360  df-fv 5365  df-riota 6011  df-ov 6061  df-oprab 6062  df-mpo 6063  df-tpos 6489  df-pnf 8326  df-mnf 8327  df-ltxr 8329  df-inn 9258  df-2 9316  df-3 9317  df-ndx 13302  df-slot 13303  df-base 13305  df-sets 13306  df-plusg 13390  df-mulr 13391  df-0g 13558  df-mgm 13622  df-sgrp 13668  df-mnd 13681  df-grp 13761  df-mgp 14163  df-ur 14206  df-ring 14244  df-oppr 14314

This theorem is referenced by:  opprringb  14327  rhmopp  14424  opprnzrbg  14433