Theorem opprrngbg 13873

Description: A set is a non-unital ring if and only if its opposite is a non-unital ring. Bidirectional form of opprrng 13872. (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 13872	. 2 |- ( R e. Rng -> O e. Rng )
3		eqid 2205	. . . 4 |- (oppr ` O ) = (oppr ` O )
4	3	opprrng 13872	. . 3 |- ( O e. Rng -> (oppr ` O ) e. Rng )
5		eqid 2205	. . . . 5 |- ( Base ` R ) = ( Base ` R )
6	5	a1i 9	. . . 4 |- ( R e. V -> ( Base ` R ) = ( Base ` R ) )
7	1, 5	opprbasg 13870	. . . . 5 |- ( R e. V -> ( Base ` R ) = ( Base ` O ) )
8	1	opprex 13868	. . . . . 6 |- ( R e. V -> O e. _V )
9		eqid 2205	. . . . . . 7 |- ( Base ` O ) = ( Base ` O )
10	3, 9	opprbasg 13870	. . . . . 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 2238	. . . 4 |- ( R e. V -> ( Base ` R ) = ( Base ` (oppr ` O ) ) )
13		eqid 2205	. . . . . . 7 |- ( +g ` R ) = ( +g ` R )
14	1, 13	oppraddg 13871	. . . . . 6 |- ( R e. V -> ( +g ` R ) = ( +g ` O ) )
15		eqid 2205	. . . . . . . 8 |- ( +g ` O ) = ( +g ` O )
16	3, 15	oppraddg 13871	. . . . . . 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 2238	. . . . 5 |- ( R e. V -> ( +g ` R ) = ( +g ` (oppr ` O ) ) )
19	18	oveqdr 5974	. . . 4 |- ( ( R e. V /\ ( x e. ( Base ` R ) /\ y e. ( Base ` R ) ) ) -> ( x ( +g ` R ) y ) = ( x ( +g ` (oppr ` O ) ) y ) )
20		vex 2775	. . . . . . . 8 |- x e. _V
21	20	a1i 9	. . . . . . 7 |- ( R e. V -> x e. _V )
22		vex 2775	. . . . . . . 8 |- y e. _V
23	22	a1i 9	. . . . . . 7 |- ( R e. V -> y e. _V )
24		eqid 2205	. . . . . . . 8 |- ( .r ` O ) = ( .r ` O )
25		eqid 2205	. . . . . . . 8 |- ( .r ` (oppr ` O ) ) = ( .r ` (oppr ` O ) )
26	9, 24, 3, 25	opprmulg 13866	. . . . . . 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 2205	. . . . . . 7 |- ( .r ` R ) = ( .r ` R )
33	5, 32, 1, 24	opprmulg 13866	. . . . . 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 2239	. . . 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 13750	. . 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 2176   _Vcvv 2772   ` cfv 5272  (class class class)co 5946   Basecbs 12865   +g cplusg 12942   .rcmulr 12943  Rngcrng 13727  opp_rcoppr 13862

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 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-13 2178  ax-14 2179  ax-ext 2187  ax-sep 4163  ax-nul 4171  ax-pow 4219  ax-pr 4254  ax-un 4481  ax-setind 4586  ax-cnex 8018  ax-resscn 8019  ax-1cn 8020  ax-1re 8021  ax-icn 8022  ax-addcl 8023  ax-addrcl 8024  ax-mulcl 8025  ax-addcom 8027  ax-addass 8029  ax-i2m1 8032  ax-0lt1 8033  ax-0id 8035  ax-rnegex 8036  ax-pre-ltirr 8039  ax-pre-lttrn 8041  ax-pre-ltadd 8043

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-fal 1379  df-nf 1484  df-sb 1786  df-eu 2057  df-mo 2058  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-ne 2377  df-nel 2472  df-ral 2489  df-rex 2490  df-rab 2493  df-v 2774  df-sbc 2999  df-csb 3094  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3461  df-pw 3618  df-sn 3639  df-pr 3640  df-op 3642  df-uni 3851  df-int 3886  df-br 4046  df-opab 4107  df-mpt 4108  df-id 4341  df-xp 4682  df-rel 4683  df-cnv 4684  df-co 4685  df-dm 4686  df-rn 4687  df-res 4688  df-ima 4689  df-iota 5233  df-fun 5274  df-fn 5275  df-fv 5280  df-riota 5901  df-ov 5949  df-oprab 5950  df-mpo 5951  df-tpos 6333  df-pnf 8111  df-mnf 8112  df-ltxr 8114  df-inn 9039  df-2 9097  df-3 9098  df-ndx 12868  df-slot 12869  df-base 12871  df-sets 12872  df-plusg 12955  df-mulr 12956  df-0g 13123  df-mgm 13221  df-sgrp 13267  df-mnd 13282  df-grp 13368  df-cmn 13655  df-abl 13656  df-mgp 13716  df-rng 13728  df-oppr 13863

This theorem is referenced by:  opprsubrngg  14006