Theorem opprringbg 13323

Description: Bidirectional form of opprring 13322. (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 13322	. 2 |- ( R e. Ring -> O e. Ring )
3		eqid 2187	. . . . . 6 |- (oppr ` O ) = (oppr ` O )
4	3	opprring 13322	. . . . 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 2188	. . . . 5 |- ( ( R e. V /\ O e. Ring ) -> ( Base ` R ) = ( Base ` R ) )
7		eqid 2187	. . . . . . 7 |- ( Base ` R ) = ( Base ` R )
8	1, 7	opprbasg 13318	. . . . . 6 |- ( R e. V -> ( Base ` R ) = ( Base ` O ) )
9		eqid 2187	. . . . . . 7 |- ( Base ` O ) = ( Base ` O )
10	3, 9	opprbasg 13318	. . . . . 6 |- ( O e. Ring -> ( Base ` O ) = ( Base ` (oppr ` O ) ) )
11	8, 10	sylan9eq 2240	. . . . 5 |- ( ( R e. V /\ O e. Ring ) -> ( Base ` R ) = ( Base ` (oppr ` O ) ) )
12		eqid 2187	. . . . . . . 8 |- ( +g ` R ) = ( +g ` R )
13	1, 12	oppraddg 13319	. . . . . . 7 |- ( R e. V -> ( +g ` R ) = ( +g ` O ) )
14		eqid 2187	. . . . . . . 8 |- ( +g ` O ) = ( +g ` O )
15	3, 14	oppraddg 13319	. . . . . . 7 |- ( O e. Ring -> ( +g ` O ) = ( +g ` (oppr ` O ) ) )
16	13, 15	sylan9eq 2240	. . . . . 6 |- ( ( R e. V /\ O e. Ring ) -> ( +g ` R ) = ( +g ` (oppr ` O ) ) )
17	16	oveqdr 5916	. . . . 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 2187	. . . . . . . . 9 |- ( .r ` O ) = ( .r ` O )
19		eqid 2187	. . . . . . . . 9 |- ( .r ` (oppr ` O ) ) = ( .r ` (oppr ` O ) )
20	9, 18, 3, 19	opprmulg 13314	. . . . . . . 8 |- ( ( O e. Ring /\ x e. ( Base ` R ) /\ y e. ( Base ` R ) ) -> ( x ( .r ` (oppr ` O ) ) y ) = ( y ( .r ` O ) x ) )
21	20	3adant1l 1231	. . . . . . 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 1022	. . . . . . . 8 |- ( ( ( R e. V /\ O e. Ring ) /\ x e. ( Base ` R ) /\ y e. ( Base ` R ) ) -> R e. V )
23		simp3 1000	. . . . . . . 8 |- ( ( ( R e. V /\ O e. Ring ) /\ x e. ( Base ` R ) /\ y e. ( Base ` R ) ) -> y e. ( Base ` R ) )
24		simp2 999	. . . . . . . 8 |- ( ( ( R e. V /\ O e. Ring ) /\ x e. ( Base ` R ) /\ y e. ( Base ` R ) ) -> x e. ( Base ` R ) )
25		eqid 2187	. . . . . . . . 9 |- ( .r ` R ) = ( .r ` R )
26	7, 25, 1, 18	opprmulg 13314	. . . . . . . 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 1248	. . . . . . 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 2221	. . . . . 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 1205	. . . . 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 13285	. . . 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 979   = wceq 1363   e. wcel 2158   ` cfv 5228  (class class class)co 5888   Basecbs 12475   +g cplusg 12550   .rcmulr 12551   Ringcrg 13243  opp_rcoppr 13310

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 1457  ax-7 1458  ax-gen 1459  ax-ie1 1503  ax-ie2 1504  ax-8 1514  ax-10 1515  ax-11 1516  ax-i12 1517  ax-bndl 1519  ax-4 1520  ax-17 1536  ax-i9 1540  ax-ial 1544  ax-i5r 1545  ax-13 2160  ax-14 2161  ax-ext 2169  ax-sep 4133  ax-nul 4141  ax-pow 4186  ax-pr 4221  ax-un 4445  ax-setind 4548  ax-cnex 7915  ax-resscn 7916  ax-1cn 7917  ax-1re 7918  ax-icn 7919  ax-addcl 7920  ax-addrcl 7921  ax-mulcl 7922  ax-addcom 7924  ax-addass 7926  ax-i2m1 7929  ax-0lt1 7930  ax-0id 7932  ax-rnegex 7933  ax-pre-ltirr 7936  ax-pre-lttrn 7938  ax-pre-ltadd 7940

This theorem depends on definitions:  df-bi 117  df-3an 981  df-tru 1366  df-fal 1369  df-nf 1471  df-sb 1773  df-eu 2039  df-mo 2040  df-clab 2174  df-cleq 2180  df-clel 2183  df-nfc 2318  df-ne 2358  df-nel 2453  df-ral 2470  df-rex 2471  df-reu 2472  df-rmo 2473  df-rab 2474  df-v 2751  df-sbc 2975  df-csb 3070  df-dif 3143  df-un 3145  df-in 3147  df-ss 3154  df-nul 3435  df-pw 3589  df-sn 3610  df-pr 3611  df-op 3613  df-uni 3822  df-int 3857  df-br 4016  df-opab 4077  df-mpt 4078  df-id 4305  df-xp 4644  df-rel 4645  df-cnv 4646  df-co 4647  df-dm 4648  df-rn 4649  df-res 4650  df-ima 4651  df-iota 5190  df-fun 5230  df-fn 5231  df-fv 5236  df-riota 5844  df-ov 5891  df-oprab 5892  df-mpo 5893  df-tpos 6259  df-pnf 8007  df-mnf 8008  df-ltxr 8010  df-inn 8933  df-2 8991  df-3 8992  df-ndx 12478  df-slot 12479  df-base 12481  df-sets 12482  df-plusg 12563  df-mulr 12564  df-0g 12724  df-mgm 12793  df-sgrp 12826  df-mnd 12839  df-grp 12901  df-mgp 13171  df-ur 13207  df-ring 13245  df-oppr 13311

This theorem is referenced by: (None)