Theorem opprrng 13633

Description: An opposite non-unital ring is a non-unital ring. (Contributed by AV, 15-Feb-2025.)

Hypothesis

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

Assertion

Ref	Expression
opprrng	|- ( R e. Rng -> O e. Rng )

Proof of Theorem opprrng

Dummy variables x y z are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		opprbas.1	. . 3 |- O = (oppr ` R )
2		eqid 2196	. . 3 |- ( Base ` R ) = ( Base ` R )
3	1, 2	opprbasg 13631	. 2 |- ( R e. Rng -> ( Base ` R ) = ( Base ` O ) )
4		eqid 2196	. . 3 |- ( +g ` R ) = ( +g ` R )
5	1, 4	oppraddg 13632	. 2 |- ( R e. Rng -> ( +g ` R ) = ( +g ` O ) )
6		eqidd 2197	. 2 |- ( R e. Rng -> ( .r ` O ) = ( .r ` O ) )
7		rngabl 13491	. . 3 |- ( R e. Rng -> R e. Abel )
8		eqidd 2197	. . . 4 |- ( R e. Rng -> ( Base ` R ) = ( Base ` R ) )
9	5	oveqdr 5950	. . . 4 |- ( ( R e. Rng /\ ( x e. ( Base ` R ) /\ y e. ( Base ` R ) ) ) -> ( x ( +g ` R ) y ) = ( x ( +g ` O ) y ) )
10	8, 3, 9	ablpropd 13426	. . 3 |- ( R e. Rng -> ( R e. Abel <-> O e. Abel ) )
11	7, 10	mpbid 147	. 2 |- ( R e. Rng -> O e. Abel )
12		eqid 2196	. . . 4 |- ( .r ` R ) = ( .r ` R )
13		eqid 2196	. . . 4 |- ( .r ` O ) = ( .r ` O )
14	2, 12, 1, 13	opprmulg 13627	. . 3 |- ( ( R e. Rng /\ x e. ( Base ` R ) /\ y e. ( Base ` R ) ) -> ( x ( .r ` O ) y ) = ( y ( .r ` R ) x ) )
15	2, 12	rngcl 13500	. . . 4 |- ( ( R e. Rng /\ y e. ( Base ` R ) /\ x e. ( Base ` R ) ) -> ( y ( .r ` R ) x ) e. ( Base ` R ) )
16	15	3com23 1211	. . 3 |- ( ( R e. Rng /\ x e. ( Base ` R ) /\ y e. ( Base ` R ) ) -> ( y ( .r ` R ) x ) e. ( Base ` R ) )
17	14, 16	eqeltrd 2273	. 2 |- ( ( R e. Rng /\ x e. ( Base ` R ) /\ y e. ( Base ` R ) ) -> ( x ( .r ` O ) y ) e. ( Base ` R ) )
18		simpl 109	. . . 4 |- ( ( R e. Rng /\ ( x e. ( Base ` R ) /\ y e. ( Base ` R ) /\ z e. ( Base ` R ) ) ) -> R e. Rng )
19		simpr3 1007	. . . 4 |- ( ( R e. Rng /\ ( x e. ( Base ` R ) /\ y e. ( Base ` R ) /\ z e. ( Base ` R ) ) ) -> z e. ( Base ` R ) )
20		simpr2 1006	. . . 4 |- ( ( R e. Rng /\ ( x e. ( Base ` R ) /\ y e. ( Base ` R ) /\ z e. ( Base ` R ) ) ) -> y e. ( Base ` R ) )
21		simpr1 1005	. . . 4 |- ( ( R e. Rng /\ ( x e. ( Base ` R ) /\ y e. ( Base ` R ) /\ z e. ( Base ` R ) ) ) -> x e. ( Base ` R ) )
22	2, 12	rngass 13495	. . . 4 |- ( ( R e. Rng /\ ( z e. ( Base ` R ) /\ y e. ( Base ` R ) /\ x e. ( Base ` R ) ) ) -> ( ( z ( .r ` R ) y ) ( .r ` R ) x ) = ( z ( .r ` R ) ( y ( .r ` R ) x ) ) )
23	18, 19, 20, 21, 22	syl13anc 1251	. . 3 |- ( ( R e. Rng /\ ( x e. ( Base ` R ) /\ y e. ( Base ` R ) /\ z e. ( Base ` R ) ) ) -> ( ( z ( .r ` R ) y ) ( .r ` R ) x ) = ( z ( .r ` R ) ( y ( .r ` R ) x ) ) )
24	2, 12, 1, 13	opprmulg 13627	. . . . . 6 |- ( ( R e. Rng /\ y e. ( Base ` R ) /\ z e. ( Base ` R ) ) -> ( y ( .r ` O ) z ) = ( z ( .r ` R ) y ) )
25	24	3adant3r1 1214	. . . . 5 |- ( ( R e. Rng /\ ( x e. ( Base ` R ) /\ y e. ( Base ` R ) /\ z e. ( Base ` R ) ) ) -> ( y ( .r ` O ) z ) = ( z ( .r ` R ) y ) )
26	25	oveq2d 5938	. . . 4 |- ( ( R e. Rng /\ ( x e. ( Base ` R ) /\ y e. ( Base ` R ) /\ z e. ( Base ` R ) ) ) -> ( x ( .r ` O ) ( y ( .r ` O ) z ) ) = ( x ( .r ` O ) ( z ( .r ` R ) y ) ) )
27	2, 12	rngcl 13500	. . . . . 6 |- ( ( R e. Rng /\ z e. ( Base ` R ) /\ y e. ( Base ` R ) ) -> ( z ( .r ` R ) y ) e. ( Base ` R ) )
28	18, 19, 20, 27	syl3anc 1249	. . . . 5 |- ( ( R e. Rng /\ ( x e. ( Base ` R ) /\ y e. ( Base ` R ) /\ z e. ( Base ` R ) ) ) -> ( z ( .r ` R ) y ) e. ( Base ` R ) )
29	2, 12, 1, 13	opprmulg 13627	. . . . 5 |- ( ( R e. Rng /\ x e. ( Base ` R ) /\ ( z ( .r ` R ) y ) e. ( Base ` R ) ) -> ( x ( .r ` O ) ( z ( .r ` R ) y ) ) = ( ( z ( .r ` R ) y ) ( .r ` R ) x ) )
30	18, 21, 28, 29	syl3anc 1249	. . . 4 |- ( ( R e. Rng /\ ( x e. ( Base ` R ) /\ y e. ( Base ` R ) /\ z e. ( Base ` R ) ) ) -> ( x ( .r ` O ) ( z ( .r ` R ) y ) ) = ( ( z ( .r ` R ) y ) ( .r ` R ) x ) )
31	26, 30	eqtrd 2229	. . 3 |- ( ( R e. Rng /\ ( x e. ( Base ` R ) /\ y e. ( Base ` R ) /\ z e. ( Base ` R ) ) ) -> ( x ( .r ` O ) ( y ( .r ` O ) z ) ) = ( ( z ( .r ` R ) y ) ( .r ` R ) x ) )
32	14	3adant3r3 1216	. . . . 5 |- ( ( R e. Rng /\ ( x e. ( Base ` R ) /\ y e. ( Base ` R ) /\ z e. ( Base ` R ) ) ) -> ( x ( .r ` O ) y ) = ( y ( .r ` R ) x ) )
33	32	oveq1d 5937	. . . 4 |- ( ( R e. Rng /\ ( x e. ( Base ` R ) /\ y e. ( Base ` R ) /\ z e. ( Base ` R ) ) ) -> ( ( x ( .r ` O ) y ) ( .r ` O ) z ) = ( ( y ( .r ` R ) x ) ( .r ` O ) z ) )
34	18, 20, 21, 15	syl3anc 1249	. . . . 5 |- ( ( R e. Rng /\ ( x e. ( Base ` R ) /\ y e. ( Base ` R ) /\ z e. ( Base ` R ) ) ) -> ( y ( .r ` R ) x ) e. ( Base ` R ) )
35	2, 12, 1, 13	opprmulg 13627	. . . . 5 |- ( ( R e. Rng /\ ( y ( .r ` R ) x ) e. ( Base ` R ) /\ z e. ( Base ` R ) ) -> ( ( y ( .r ` R ) x ) ( .r ` O ) z ) = ( z ( .r ` R ) ( y ( .r ` R ) x ) ) )
36	18, 34, 19, 35	syl3anc 1249	. . . 4 |- ( ( R e. Rng /\ ( x e. ( Base ` R ) /\ y e. ( Base ` R ) /\ z e. ( Base ` R ) ) ) -> ( ( y ( .r ` R ) x ) ( .r ` O ) z ) = ( z ( .r ` R ) ( y ( .r ` R ) x ) ) )
37	33, 36	eqtrd 2229	. . 3 |- ( ( R e. Rng /\ ( x e. ( Base ` R ) /\ y e. ( Base ` R ) /\ z e. ( Base ` R ) ) ) -> ( ( x ( .r ` O ) y ) ( .r ` O ) z ) = ( z ( .r ` R ) ( y ( .r ` R ) x ) ) )
38	23, 31, 37	3eqtr4rd 2240	. 2 |- ( ( R e. Rng /\ ( x e. ( Base ` R ) /\ y e. ( Base ` R ) /\ z e. ( Base ` R ) ) ) -> ( ( x ( .r ` O ) y ) ( .r ` O ) z ) = ( x ( .r ` O ) ( y ( .r ` O ) z ) ) )
39	2, 4, 12	rngdir 13497	. . . 4 |- ( ( R e. Rng /\ ( y e. ( Base ` R ) /\ z e. ( Base ` R ) /\ x e. ( Base ` R ) ) ) -> ( ( y ( +g ` R ) z ) ( .r ` R ) x ) = ( ( y ( .r ` R ) x ) ( +g ` R ) ( z ( .r ` R ) x ) ) )
40	18, 20, 19, 21, 39	syl13anc 1251	. . 3 |- ( ( R e. Rng /\ ( x e. ( Base ` R ) /\ y e. ( Base ` R ) /\ z e. ( Base ` R ) ) ) -> ( ( y ( +g ` R ) z ) ( .r ` R ) x ) = ( ( y ( .r ` R ) x ) ( +g ` R ) ( z ( .r ` R ) x ) ) )
41	2, 4	rngacl 13498	. . . . 5 |- ( ( R e. Rng /\ y e. ( Base ` R ) /\ z e. ( Base ` R ) ) -> ( y ( +g ` R ) z ) e. ( Base ` R ) )
42	41	3adant3r1 1214	. . . 4 |- ( ( R e. Rng /\ ( x e. ( Base ` R ) /\ y e. ( Base ` R ) /\ z e. ( Base ` R ) ) ) -> ( y ( +g ` R ) z ) e. ( Base ` R ) )
43	2, 12, 1, 13	opprmulg 13627	. . . 4 |- ( ( R e. Rng /\ x e. ( Base ` R ) /\ ( y ( +g ` R ) z ) e. ( Base ` R ) ) -> ( x ( .r ` O ) ( y ( +g ` R ) z ) ) = ( ( y ( +g ` R ) z ) ( .r ` R ) x ) )
44	18, 21, 42, 43	syl3anc 1249	. . 3 |- ( ( R e. Rng /\ ( x e. ( Base ` R ) /\ y e. ( Base ` R ) /\ z e. ( Base ` R ) ) ) -> ( x ( .r ` O ) ( y ( +g ` R ) z ) ) = ( ( y ( +g ` R ) z ) ( .r ` R ) x ) )
45	2, 12, 1, 13	opprmulg 13627	. . . . 5 |- ( ( R e. Rng /\ x e. ( Base ` R ) /\ z e. ( Base ` R ) ) -> ( x ( .r ` O ) z ) = ( z ( .r ` R ) x ) )
46	18, 21, 19, 45	syl3anc 1249	. . . 4 |- ( ( R e. Rng /\ ( x e. ( Base ` R ) /\ y e. ( Base ` R ) /\ z e. ( Base ` R ) ) ) -> ( x ( .r ` O ) z ) = ( z ( .r ` R ) x ) )
47	32, 46	oveq12d 5940	. . 3 |- ( ( R e. Rng /\ ( x e. ( Base ` R ) /\ y e. ( Base ` R ) /\ z e. ( Base ` R ) ) ) -> ( ( x ( .r ` O ) y ) ( +g ` R ) ( x ( .r ` O ) z ) ) = ( ( y ( .r ` R ) x ) ( +g ` R ) ( z ( .r ` R ) x ) ) )
48	40, 44, 47	3eqtr4d 2239	. 2 |- ( ( R e. Rng /\ ( x e. ( Base ` R ) /\ y e. ( Base ` R ) /\ z e. ( Base ` R ) ) ) -> ( x ( .r ` O ) ( y ( +g ` R ) z ) ) = ( ( x ( .r ` O ) y ) ( +g ` R ) ( x ( .r ` O ) z ) ) )
49	2, 4, 12	rngdi 13496	. . . 4 |- ( ( R e. Rng /\ ( z e. ( Base ` R ) /\ x e. ( Base ` R ) /\ y e. ( Base ` R ) ) ) -> ( z ( .r ` R ) ( x ( +g ` R ) y ) ) = ( ( z ( .r ` R ) x ) ( +g ` R ) ( z ( .r ` R ) y ) ) )
50	18, 19, 21, 20, 49	syl13anc 1251	. . 3 |- ( ( R e. Rng /\ ( x e. ( Base ` R ) /\ y e. ( Base ` R ) /\ z e. ( Base ` R ) ) ) -> ( z ( .r ` R ) ( x ( +g ` R ) y ) ) = ( ( z ( .r ` R ) x ) ( +g ` R ) ( z ( .r ` R ) y ) ) )
51	2, 4	rngacl 13498	. . . . 5 |- ( ( R e. Rng /\ x e. ( Base ` R ) /\ y e. ( Base ` R ) ) -> ( x ( +g ` R ) y ) e. ( Base ` R ) )
52	51	3adant3r3 1216	. . . 4 |- ( ( R e. Rng /\ ( x e. ( Base ` R ) /\ y e. ( Base ` R ) /\ z e. ( Base ` R ) ) ) -> ( x ( +g ` R ) y ) e. ( Base ` R ) )
53	2, 12, 1, 13	opprmulg 13627	. . . 4 |- ( ( R e. Rng /\ ( x ( +g ` R ) y ) e. ( Base ` R ) /\ z e. ( Base ` R ) ) -> ( ( x ( +g ` R ) y ) ( .r ` O ) z ) = ( z ( .r ` R ) ( x ( +g ` R ) y ) ) )
54	18, 52, 19, 53	syl3anc 1249	. . 3 |- ( ( R e. Rng /\ ( x e. ( Base ` R ) /\ y e. ( Base ` R ) /\ z e. ( Base ` R ) ) ) -> ( ( x ( +g ` R ) y ) ( .r ` O ) z ) = ( z ( .r ` R ) ( x ( +g ` R ) y ) ) )
55	46, 25	oveq12d 5940	. . 3 |- ( ( R e. Rng /\ ( x e. ( Base ` R ) /\ y e. ( Base ` R ) /\ z e. ( Base ` R ) ) ) -> ( ( x ( .r ` O ) z ) ( +g ` R ) ( y ( .r ` O ) z ) ) = ( ( z ( .r ` R ) x ) ( +g ` R ) ( z ( .r ` R ) y ) ) )
56	50, 54, 55	3eqtr4d 2239	. 2 |- ( ( R e. Rng /\ ( x e. ( Base ` R ) /\ y e. ( Base ` R ) /\ z e. ( Base ` R ) ) ) -> ( ( x ( +g ` R ) y ) ( .r ` O ) z ) = ( ( x ( .r ` O ) z ) ( +g ` R ) ( y ( .r ` O ) z ) ) )
57	3, 5, 6, 11, 17, 38, 48, 56	isrngd 13509	1 |- ( R e. Rng -> O e. Rng )

Syntax hints:   -> wi 4   /\ wa 104   /\ w3a 980   = wceq 1364   e. wcel 2167   ` cfv 5258  (class class class)co 5922   Basecbs 12678   +g cplusg 12755   .rcmulr 12756   Abelcabl 13415  Rngcrng 13488  opp_rcoppr 13623

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 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-13 2169  ax-14 2170  ax-ext 2178  ax-sep 4151  ax-nul 4159  ax-pow 4207  ax-pr 4242  ax-un 4468  ax-setind 4573  ax-cnex 7970  ax-resscn 7971  ax-1cn 7972  ax-1re 7973  ax-icn 7974  ax-addcl 7975  ax-addrcl 7976  ax-mulcl 7977  ax-addcom 7979  ax-addass 7981  ax-i2m1 7984  ax-0lt1 7985  ax-0id 7987  ax-rnegex 7988  ax-pre-ltirr 7991  ax-pre-lttrn 7993  ax-pre-ltadd 7995

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ne 2368  df-nel 2463  df-ral 2480  df-rex 2481  df-rab 2484  df-v 2765  df-sbc 2990  df-csb 3085  df-dif 3159  df-un 3161  df-in 3163  df-ss 3170  df-nul 3451  df-pw 3607  df-sn 3628  df-pr 3629  df-op 3631  df-uni 3840  df-int 3875  df-br 4034  df-opab 4095  df-mpt 4096  df-id 4328  df-xp 4669  df-rel 4670  df-cnv 4671  df-co 4672  df-dm 4673  df-rn 4674  df-res 4675  df-ima 4676  df-iota 5219  df-fun 5260  df-fn 5261  df-fv 5266  df-riota 5877  df-ov 5925  df-oprab 5926  df-mpo 5927  df-tpos 6303  df-pnf 8063  df-mnf 8064  df-ltxr 8066  df-inn 8991  df-2 9049  df-3 9050  df-ndx 12681  df-slot 12682  df-base 12684  df-sets 12685  df-plusg 12768  df-mulr 12769  df-0g 12929  df-mgm 12999  df-sgrp 13045  df-mnd 13058  df-grp 13135  df-cmn 13416  df-abl 13417  df-mgp 13477  df-rng 13489  df-oppr 13624

This theorem is referenced by:  opprrngbg  13634  opprsubrngg  13767  isridlrng  14038  2idlcpblrng  14079