Theorem opprrngbg 13955

Description: A set is a non-unital ring if and only if its opposite is a non-unital ring. Bidirectional form of opprrng 13954. (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 13954	. 2 |- ( R e. Rng -> O e. Rng )
3		eqid 2207	. . . 4 |- (oppr ` O ) = (oppr ` O )
4	3	opprrng 13954	. . 3 |- ( O e. Rng -> (oppr ` O ) e. Rng )
5		eqid 2207	. . . . 5 |- ( Base ` R ) = ( Base ` R )
6	5	a1i 9	. . . 4 |- ( R e. V -> ( Base ` R ) = ( Base ` R ) )
7	1, 5	opprbasg 13952	. . . . 5 |- ( R e. V -> ( Base ` R ) = ( Base ` O ) )
8	1	opprex 13950	. . . . . 6 |- ( R e. V -> O e. _V )
9		eqid 2207	. . . . . . 7 |- ( Base ` O ) = ( Base ` O )
10	3, 9	opprbasg 13952	. . . . . 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 2240	. . . 4 |- ( R e. V -> ( Base ` R ) = ( Base ` (oppr ` O ) ) )
13		eqid 2207	. . . . . . 7 |- ( +g ` R ) = ( +g ` R )
14	1, 13	oppraddg 13953	. . . . . 6 |- ( R e. V -> ( +g ` R ) = ( +g ` O ) )
15		eqid 2207	. . . . . . . 8 |- ( +g ` O ) = ( +g ` O )
16	3, 15	oppraddg 13953	. . . . . . 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 2240	. . . . 5 |- ( R e. V -> ( +g ` R ) = ( +g ` (oppr ` O ) ) )
19	18	oveqdr 5995	. . . 4 |- ( ( R e. V /\ ( x e. ( Base ` R ) /\ y e. ( Base ` R ) ) ) -> ( x ( +g ` R ) y ) = ( x ( +g ` (oppr ` O ) ) y ) )
20		vex 2779	. . . . . . . 8 |- x e. _V
21	20	a1i 9	. . . . . . 7 |- ( R e. V -> x e. _V )
22		vex 2779	. . . . . . . 8 |- y e. _V
23	22	a1i 9	. . . . . . 7 |- ( R e. V -> y e. _V )
24		eqid 2207	. . . . . . . 8 |- ( .r ` O ) = ( .r ` O )
25		eqid 2207	. . . . . . . 8 |- ( .r ` (oppr ` O ) ) = ( .r ` (oppr ` O ) )
26	9, 24, 3, 25	opprmulg 13948	. . . . . . 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 1250	. . . . . 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 531	. . . . . 6 |- ( ( R e. V /\ ( x e. ( Base ` R ) /\ y e. ( Base ` R ) ) ) -> y e. ( Base ` R ) )
31		simprl 529	. . . . . 6 |- ( ( R e. V /\ ( x e. ( Base ` R ) /\ y e. ( Base ` R ) ) ) -> x e. ( Base ` R ) )
32		eqid 2207	. . . . . . 7 |- ( .r ` R ) = ( .r ` R )
33	5, 32, 1, 24	opprmulg 13948	. . . . . 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 1250	. . . . 5 |- ( ( R e. V /\ ( x e. ( Base ` R ) /\ y e. ( Base ` R ) ) ) -> ( y ( .r ` O ) x ) = ( x ( .r ` R ) y ) )
35	28, 34	eqtr2d 2241	. . . 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 13832	. . 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 1373   e. wcel 2178   _Vcvv 2776   ` cfv 5290  (class class class)co 5967   Basecbs 12947   +g cplusg 13024   .rcmulr 13025  Rngcrng 13809  opp_rcoppr 13944

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 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-13 2180  ax-14 2181  ax-ext 2189  ax-sep 4178  ax-nul 4186  ax-pow 4234  ax-pr 4269  ax-un 4498  ax-setind 4603  ax-cnex 8051  ax-resscn 8052  ax-1cn 8053  ax-1re 8054  ax-icn 8055  ax-addcl 8056  ax-addrcl 8057  ax-mulcl 8058  ax-addcom 8060  ax-addass 8062  ax-i2m1 8065  ax-0lt1 8066  ax-0id 8068  ax-rnegex 8069  ax-pre-ltirr 8072  ax-pre-lttrn 8074  ax-pre-ltadd 8076

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-fal 1379  df-nf 1485  df-sb 1787  df-eu 2058  df-mo 2059  df-clab 2194  df-cleq 2200  df-clel 2203  df-nfc 2339  df-ne 2379  df-nel 2474  df-ral 2491  df-rex 2492  df-rab 2495  df-v 2778  df-sbc 3006  df-csb 3102  df-dif 3176  df-un 3178  df-in 3180  df-ss 3187  df-nul 3469  df-pw 3628  df-sn 3649  df-pr 3650  df-op 3652  df-uni 3865  df-int 3900  df-br 4060  df-opab 4122  df-mpt 4123  df-id 4358  df-xp 4699  df-rel 4700  df-cnv 4701  df-co 4702  df-dm 4703  df-rn 4704  df-res 4705  df-ima 4706  df-iota 5251  df-fun 5292  df-fn 5293  df-fv 5298  df-riota 5922  df-ov 5970  df-oprab 5971  df-mpo 5972  df-tpos 6354  df-pnf 8144  df-mnf 8145  df-ltxr 8147  df-inn 9072  df-2 9130  df-3 9131  df-ndx 12950  df-slot 12951  df-base 12953  df-sets 12954  df-plusg 13037  df-mulr 13038  df-0g 13205  df-mgm 13303  df-sgrp 13349  df-mnd 13364  df-grp 13450  df-cmn 13737  df-abl 13738  df-mgp 13798  df-rng 13810  df-oppr 13945

This theorem is referenced by:  opprsubrngg  14088