Theorem opprrngbg 14239

Description: A set is a non-unital ring if and only if its opposite is a non-unital ring. Bidirectional form of opprrng 14238. (Contributed by AV, 15-Feb-2025.)

Hypothesis

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

Assertion

Ref	Expression
opprrngbg	|- ( R e. V -> ( R e. Rng <-> O e. Rng ) )

Proof of Theorem opprrngbg

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	opprrng 14238	. 2 |- ( R e. Rng -> O e. Rng )
3		eqid 2234	. . . 4 |- (oppr ` O ) = (oppr ` O )
4	3	opprrng 14238	. . 3 |- ( O e. Rng -> (oppr ` O ) e. Rng )
5		eqid 2234	. . . . 5 |- ( Base ` R ) = ( Base ` R )
6	5	a1i 9	. . . 4 |- ( R e. V -> ( Base ` R ) = ( Base ` R ) )
7	1, 5	opprbasg 14236	. . . . 5 |- ( R e. V -> ( Base ` R ) = ( Base ` O ) )
8	1	opprex 14234	. . . . . 6 |- ( R e. V -> O e. _V )
9		eqid 2234	. . . . . . 7 |- ( Base ` O ) = ( Base ` O )
10	3, 9	opprbasg 14236	. . . . . 6 |- ( O e. _V -> ( Base ` O ) = ( Base ` (oppr ` O ) ) )
11	8, 10	syl 14	. . . . 5 |- ( R e. V -> ( Base ` O ) = ( Base ` (oppr ` O ) ) )
12	7, 11	eqtrd 2267	. . . 4 |- ( R e. V -> ( Base ` R ) = ( Base ` (oppr ` O ) ) )
13		eqid 2234	. . . . . . 7 |- ( +g ` R ) = ( +g ` R )
14	1, 13	oppraddg 14237	. . . . . 6 |- ( R e. V -> ( +g ` R ) = ( +g ` O ) )
15		eqid 2234	. . . . . . . 8 |- ( +g ` O ) = ( +g ` O )
16	3, 15	oppraddg 14237	. . . . . . 7 |- ( O e. _V -> ( +g ` O ) = ( +g ` (oppr ` O ) ) )
17	8, 16	syl 14	. . . . . 6 |- ( R e. V -> ( +g ` O ) = ( +g ` (oppr ` O ) ) )
18	14, 17	eqtrd 2267	. . . . 5 |- ( R e. V -> ( +g ` R ) = ( +g ` (oppr ` O ) ) )
19	18	oveqdr 6080	. . . 4 |- ( ( R e. V /\ ( x e. ( Base ` R ) /\ y e. ( Base ` R ) ) ) -> ( x ( +g ` R ) y ) = ( x ( +g ` (oppr ` O ) ) y ) )
20		vex 2818	. . . . . . . 8 |- x e. _V
21	20	a1i 9	. . . . . . 7 |- ( R e. V -> x e. _V )
22		vex 2818	. . . . . . . 8 |- y e. _V
23	22	a1i 9	. . . . . . 7 |- ( R e. V -> y e. _V )
24		eqid 2234	. . . . . . . 8 |- ( .r ` O ) = ( .r ` O )
25		eqid 2234	. . . . . . . 8 |- ( .r ` (oppr ` O ) ) = ( .r ` (oppr ` O ) )
26	9, 24, 3, 25	opprmulg 14232	. . . . . . 7 |- ( ( O e. _V /\ x e. _V /\ y e. _V ) -> ( x ( .r ` (oppr ` O ) ) y ) = ( y ( .r ` O ) x ) )
27	8, 21, 23, 26	syl3anc 1274	. . . . . 6 |- ( R e. V -> ( x ( .r ` (oppr ` O ) ) y ) = ( y ( .r ` O ) x ) )
28	27	adantr 276	. . . . 5 |- ( ( R e. V /\ ( x e. ( Base ` R ) /\ y e. ( Base ` R ) ) ) -> ( x ( .r ` (oppr ` O ) ) y ) = ( y ( .r ` O ) x ) )
29		simpl 109	. . . . . 6 |- ( ( R e. V /\ ( x e. ( Base ` R ) /\ y e. ( Base ` R ) ) ) -> R e. V )
30		simprr 533	. . . . . 6 |- ( ( R e. V /\ ( x e. ( Base ` R ) /\ y e. ( Base ` R ) ) ) -> y e. ( Base ` R ) )
31		simprl 531	. . . . . 6 |- ( ( R e. V /\ ( x e. ( Base ` R ) /\ y e. ( Base ` R ) ) ) -> x e. ( Base ` R ) )
32		eqid 2234	. . . . . . 7 |- ( .r ` R ) = ( .r ` R )
33	5, 32, 1, 24	opprmulg 14232	. . . . . 6 |- ( ( R e. V /\ y e. ( Base ` R ) /\ x e. ( Base ` R ) ) -> ( y ( .r ` O ) x ) = ( x ( .r ` R ) y ) )
34	29, 30, 31, 33	syl3anc 1274	. . . . 5 |- ( ( R e. V /\ ( x e. ( Base ` R ) /\ y e. ( Base ` R ) ) ) -> ( y ( .r ` O ) x ) = ( x ( .r ` R ) y ) )
35	28, 34	eqtr2d 2268	. . . 4 |- ( ( R e. V /\ ( x e. ( Base ` R ) /\ y e. ( Base ` R ) ) ) -> ( x ( .r ` R ) y ) = ( x ( .r ` (oppr ` O ) ) y ) )
36	6, 12, 19, 35	rngpropd 14116	. . 3 |- ( R e. V -> ( R e. Rng <-> (oppr ` O ) e. Rng ) )
37	4, 36	imbitrrid 156	. 2 |- ( R e. V -> ( O e. Rng -> R e. Rng ) )
38	2, 37	impbid2 143	1 |- ( R e. V -> ( R e. Rng <-> O e. Rng ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1398   e. wcel 2205   _Vcvv 2815   ` cfv 5354  (class class class)co 6052   Basecbs 13229   +g cplusg 13307   .rcmulr 13308  Rngcrng 14093  opp_rcoppr 14228

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 4230  ax-nul 4238  ax-pow 4289  ax-pr 4324  ax-un 4556  ax-setind 4661  ax-cnex 8220  ax-resscn 8221  ax-1cn 8222  ax-1re 8223  ax-icn 8224  ax-addcl 8225  ax-addrcl 8226  ax-mulcl 8227  ax-addcom 8229  ax-addass 8231  ax-i2m1 8234  ax-0lt1 8235  ax-0id 8237  ax-rnegex 8238  ax-pre-ltirr 8241  ax-pre-lttrn 8243  ax-pre-ltadd 8245

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-rab 2531  df-v 2817  df-sbc 3045  df-csb 3141  df-dif 3215  df-un 3217  df-in 3219  df-ss 3226  df-nul 3511  df-pw 3673  df-sn 3697  df-pr 3698  df-op 3700  df-uni 3917  df-int 3952  df-br 4112  df-opab 4174  df-mpt 4175  df-id 4416  df-xp 4757  df-rel 4758  df-cnv 4759  df-co 4760  df-dm 4761  df-rn 4762  df-res 4763  df-ima 4764  df-iota 5314  df-fun 5356  df-fn 5357  df-fv 5362  df-riota 6005  df-ov 6055  df-oprab 6056  df-mpo 6057  df-tpos 6478  df-pnf 8312  df-mnf 8313  df-ltxr 8315  df-inn 9240  df-2 9298  df-3 9299  df-ndx 13232  df-slot 13233  df-base 13235  df-sets 13236  df-plusg 13320  df-mulr 13321  df-0g 13488  df-mgm 13586  df-sgrp 13632  df-mnd 13647  df-grp 13733  df-cmn 14020  df-abl 14021  df-mgp 14082  df-rng 14094  df-oppr 14229

This theorem is referenced by:  opprsubrngg  14373