Theorem opprringbg 14213

Description: Bidirectional form of opprring 14212. (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 14212	. 2 |- ( R e. Ring -> O e. Ring )
3		eqid 2232	. . . . . 6 |- (oppr ` O ) = (oppr ` O )
4	3	opprring 14212	. . . . 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 2233	. . . . 5 |- ( ( R e. V /\ O e. Ring ) -> ( Base ` R ) = ( Base ` R ) )
7		eqid 2232	. . . . . . 7 |- ( Base ` R ) = ( Base ` R )
8	1, 7	opprbasg 14208	. . . . . 6 |- ( R e. V -> ( Base ` R ) = ( Base ` O ) )
9		eqid 2232	. . . . . . 7 |- ( Base ` O ) = ( Base ` O )
10	3, 9	opprbasg 14208	. . . . . 6 |- ( O e. Ring -> ( Base ` O ) = ( Base ` (oppr ` O ) ) )
11	8, 10	sylan9eq 2285	. . . . 5 |- ( ( R e. V /\ O e. Ring ) -> ( Base ` R ) = ( Base ` (oppr ` O ) ) )
12		eqid 2232	. . . . . . . 8 |- ( +g ` R ) = ( +g ` R )
13	1, 12	oppraddg 14209	. . . . . . 7 |- ( R e. V -> ( +g ` R ) = ( +g ` O ) )
14		eqid 2232	. . . . . . . 8 |- ( +g ` O ) = ( +g ` O )
15	3, 14	oppraddg 14209	. . . . . . 7 |- ( O e. Ring -> ( +g ` O ) = ( +g ` (oppr ` O ) ) )
16	13, 15	sylan9eq 2285	. . . . . 6 |- ( ( R e. V /\ O e. Ring ) -> ( +g ` R ) = ( +g ` (oppr ` O ) ) )
17	16	oveqdr 6077	. . . . 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 2232	. . . . . . . . 9 |- ( .r ` O ) = ( .r ` O )
19		eqid 2232	. . . . . . . . 9 |- ( .r ` (oppr ` O ) ) = ( .r ` (oppr ` O ) )
20	9, 18, 3, 19	opprmulg 14204	. . . . . . . 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 2232	. . . . . . . . 9 |- ( .r ` R ) = ( .r ` R )
26	7, 25, 1, 18	opprmulg 14204	. . . . . . . 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 2266	. . . . . 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 14171	. . . 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 2203   ` cfv 5351  (class class class)co 6049   Basecbs 13201   +g cplusg 13279   .rcmulr 13280   Ringcrg 14129  opp_rcoppr 14200

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 2205  ax-14 2206  ax-ext 2214  ax-sep 4227  ax-nul 4235  ax-pow 4286  ax-pr 4321  ax-un 4553  ax-setind 4658  ax-cnex 8214  ax-resscn 8215  ax-1cn 8216  ax-1re 8217  ax-icn 8218  ax-addcl 8219  ax-addrcl 8220  ax-mulcl 8221  ax-addcom 8223  ax-addass 8225  ax-i2m1 8228  ax-0lt1 8229  ax-0id 8231  ax-rnegex 8232  ax-pre-ltirr 8235  ax-pre-lttrn 8237  ax-pre-ltadd 8239

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 2083  df-mo 2084  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ne 2413  df-nel 2508  df-ral 2525  df-rex 2526  df-reu 2527  df-rmo 2528  df-rab 2529  df-v 2814  df-sbc 3042  df-csb 3138  df-dif 3212  df-un 3214  df-in 3216  df-ss 3223  df-nul 3508  df-pw 3670  df-sn 3694  df-pr 3695  df-op 3697  df-uni 3914  df-int 3949  df-br 4109  df-opab 4171  df-mpt 4172  df-id 4413  df-xp 4754  df-rel 4755  df-cnv 4756  df-co 4757  df-dm 4758  df-rn 4759  df-res 4760  df-ima 4761  df-iota 5311  df-fun 5353  df-fn 5354  df-fv 5359  df-riota 6002  df-ov 6052  df-oprab 6053  df-mpo 6054  df-tpos 6475  df-pnf 8306  df-mnf 8307  df-ltxr 8309  df-inn 9234  df-2 9292  df-3 9293  df-ndx 13204  df-slot 13205  df-base 13207  df-sets 13208  df-plusg 13292  df-mulr 13293  df-0g 13460  df-mgm 13558  df-sgrp 13604  df-mnd 13619  df-grp 13705  df-mgp 14054  df-ur 14093  df-ring 14131  df-oppr 14201

This theorem is referenced by:  rhmopp  14310  opprnzrbg  14319