Theorem opprrngbg 14356

Description: A set is a non-unital ring if and only if its opposite is a non-unital ring. Bidirectional form of opprrng 14355. (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 14355	. 2 |- ( R e. Rng -> O e. Rng )
3		eqid 2238	. . . 4 |- (oppr ` O ) = (oppr ` O )
4	3	opprrng 14355	. . 3 |- ( O e. Rng -> (oppr ` O ) e. Rng )
5		eqid 2238	. . . . 5 |- ( Base ` R ) = ( Base ` R )
6	5	a1i 9	. . . 4 |- ( R e. V -> ( Base ` R ) = ( Base ` R ) )
7	1, 5	opprbasg 14353	. . . . 5 |- ( R e. V -> ( Base ` R ) = ( Base ` O ) )
8	1	opprex 14351	. . . . . 6 |- ( R e. V -> O e. _V )
9		eqid 2238	. . . . . . 7 |- ( Base ` O ) = ( Base ` O )
10	3, 9	opprbasg 14353	. . . . . 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 2271	. . . 4 |- ( R e. V -> ( Base ` R ) = ( Base ` (oppr ` O ) ) )
13		eqid 2238	. . . . . . 7 |- ( +g ` R ) = ( +g ` R )
14	1, 13	oppraddg 14354	. . . . . 6 |- ( R e. V -> ( +g ` R ) = ( +g ` O ) )
15		eqid 2238	. . . . . . . 8 |- ( +g ` O ) = ( +g ` O )
16	3, 15	oppraddg 14354	. . . . . . 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 2271	. . . . 5 |- ( R e. V -> ( +g ` R ) = ( +g ` (oppr ` O ) ) )
19	18	oveqdr 6103	. . . 4 |- ( ( R e. V /\ ( x e. ( Base ` R ) /\ y e. ( Base ` R ) ) ) -> ( x ( +g ` R ) y ) = ( x ( +g ` (oppr ` O ) ) y ) )
20		vex 2824	. . . . . . . 8 |- x e. _V
21	20	a1i 9	. . . . . . 7 |- ( R e. V -> x e. _V )
22		vex 2824	. . . . . . . 8 |- y e. _V
23	22	a1i 9	. . . . . . 7 |- ( R e. V -> y e. _V )
24		eqid 2238	. . . . . . . 8 |- ( .r ` O ) = ( .r ` O )
25		eqid 2238	. . . . . . . 8 |- ( .r ` (oppr ` O ) ) = ( .r ` (oppr ` O ) )
26	9, 24, 3, 25	opprmulg 14349	. . . . . . 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 1278	. . . . . 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 537	. . . . . 6 |- ( ( R e. V /\ ( x e. ( Base ` R ) /\ y e. ( Base ` R ) ) ) -> y e. ( Base ` R ) )
31		simprl 535	. . . . . 6 |- ( ( R e. V /\ ( x e. ( Base ` R ) /\ y e. ( Base ` R ) ) ) -> x e. ( Base ` R ) )
32		eqid 2238	. . . . . . 7 |- ( .r ` R ) = ( .r ` R )
33	5, 32, 1, 24	opprmulg 14349	. . . . . 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 1278	. . . . 5 |- ( ( R e. V /\ ( x e. ( Base ` R ) /\ y e. ( Base ` R ) ) ) -> ( y ( .r ` O ) x ) = ( x ( .r ` R ) y ) )
35	28, 34	eqtr2d 2272	. . . 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 14229	. . 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 1402   e. wcel 2209   _Vcvv 2821   ` cfv 5372  (class class class)co 6075   Basecbs 13330   +g cplusg 13408   .rcmulr 13409  Rngcrng 14206  opp_rcoppr 14345

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 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 4244  ax-nul 4254  ax-pow 4306  ax-pr 4341  ax-un 4573  ax-setind 4679  ax-cnex 8260  ax-resscn 8261  ax-1cn 8262  ax-1re 8263  ax-icn 8264  ax-addcl 8265  ax-addrcl 8266  ax-mulcl 8267  ax-addcom 8269  ax-addass 8271  ax-i2m1 8274  ax-0lt1 8275  ax-0id 8277  ax-rnegex 8278  ax-pre-ltirr 8281  ax-pre-lttrn 8283  ax-pre-ltadd 8285

This theorem 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-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 3687  df-sn 3711  df-pr 3712  df-op 3714  df-uni 3931  df-int 3966  df-br 4126  df-opab 4188  df-mpt 4189  df-id 4433  df-xp 4775  df-rel 4776  df-cnv 4777  df-co 4778  df-dm 4779  df-rn 4780  df-res 4781  df-ima 4782  df-iota 5332  df-fun 5374  df-fn 5375  df-fv 5380  df-riota 6028  df-ov 6078  df-oprab 6079  df-mpo 6080  df-tpos 6506  df-pnf 8352  df-mnf 8353  df-ltxr 8355  df-inn 9284  df-2 9342  df-3 9343  df-ndx 13333  df-slot 13334  df-base 13336  df-sets 13337  df-plusg 13421  df-mulr 13422  df-0g 13589  df-mgm 13653  df-sgrp 13694  df-mnd 13707  df-grp 13785  df-cmn 14066  df-abl 14067  df-mgp 14195  df-rng 14207  df-oppr 14346

This theorem is referenced by:  opprsubrngg  14492