Theorem opprrng 14089

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 2231	. . 3 |- ( Base ` R ) = ( Base ` R )
3	1, 2	opprbasg 14087	. 2 |- ( R e. Rng -> ( Base ` R ) = ( Base ` O ) )
4		eqid 2231	. . 3 |- ( +g ` R ) = ( +g ` R )
5	1, 4	oppraddg 14088	. 2 |- ( R e. Rng -> ( +g ` R ) = ( +g ` O ) )
6		eqidd 2232	. 2 |- ( R e. Rng -> ( .r ` O ) = ( .r ` O ) )
7		rngabl 13947	. . 3 |- ( R e. Rng -> R e. Abel )
8		eqidd 2232	. . . 4 |- ( R e. Rng -> ( Base ` R ) = ( Base ` R ) )
9	5	oveqdr 6045	. . . 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 13882	. . 3 |- ( R e. Rng -> ( R e. Abel <-> O e. Abel ) )
11	7, 10	mpbid 147	. 2 |- ( R e. Rng -> O e. Abel )
12		eqid 2231	. . . 4 |- ( .r ` R ) = ( .r ` R )
13		eqid 2231	. . . 4 |- ( .r ` O ) = ( .r ` O )
14	2, 12, 1, 13	opprmulg 14083	. . 3 |- ( ( R e. Rng /\ x e. ( Base ` R ) /\ y e. ( Base ` R ) ) -> ( x ( .r ` O ) y ) = ( y ( .r ` R ) x ) )
15	2, 12	rngcl 13956	. . . 4 |- ( ( R e. Rng /\ y e. ( Base ` R ) /\ x e. ( Base ` R ) ) -> ( y ( .r ` R ) x ) e. ( Base ` R ) )
16	15	3com23 1235	. . 3 |- ( ( R e. Rng /\ x e. ( Base ` R ) /\ y e. ( Base ` R ) ) -> ( y ( .r ` R ) x ) e. ( Base ` R ) )
17	14, 16	eqeltrd 2308	. 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 1031	. . . 4 |- ( ( R e. Rng /\ ( x e. ( Base ` R ) /\ y e. ( Base ` R ) /\ z e. ( Base ` R ) ) ) -> z e. ( Base ` R ) )
20		simpr2 1030	. . . 4 |- ( ( R e. Rng /\ ( x e. ( Base ` R ) /\ y e. ( Base ` R ) /\ z e. ( Base ` R ) ) ) -> y e. ( Base ` R ) )
21		simpr1 1029	. . . 4 |- ( ( R e. Rng /\ ( x e. ( Base ` R ) /\ y e. ( Base ` R ) /\ z e. ( Base ` R ) ) ) -> x e. ( Base ` R ) )
22	2, 12	rngass 13951	. . . 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 1275	. . 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 14083	. . . . . 6 |- ( ( R e. Rng /\ y e. ( Base ` R ) /\ z e. ( Base ` R ) ) -> ( y ( .r ` O ) z ) = ( z ( .r ` R ) y ) )
25	24	3adant3r1 1238	. . . . 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 6033	. . . 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 13956	. . . . . 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 1273	. . . . 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 14083	. . . . 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 1273	. . . 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 2264	. . 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 1240	. . . . 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 6032	. . . 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 1273	. . . . 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 14083	. . . . 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 1273	. . . 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 2264	. . 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 2275	. 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 13953	. . . 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 1275	. . 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 13954	. . . . 5 |- ( ( R e. Rng /\ y e. ( Base ` R ) /\ z e. ( Base ` R ) ) -> ( y ( +g ` R ) z ) e. ( Base ` R ) )
42	41	3adant3r1 1238	. . . 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 14083	. . . 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 1273	. . 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 14083	. . . . 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 1273	. . . 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 6035	. . 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 2274	. 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 13952	. . . 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 1275	. . 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 13954	. . . . 5 |- ( ( R e. Rng /\ x e. ( Base ` R ) /\ y e. ( Base ` R ) ) -> ( x ( +g ` R ) y ) e. ( Base ` R ) )
52	51	3adant3r3 1240	. . . 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 14083	. . . 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 1273	. . 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 6035	. . 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 2274	. 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 13965	1 |- ( R e. Rng -> O e. Rng )

Syntax hints:   -> wi 4   /\ wa 104   /\ w3a 1004   = wceq 1397   e. wcel 2202   ` cfv 5326  (class class class)co 6017   Basecbs 13081   +g cplusg 13159   .rcmulr 13160   Abelcabl 13871  Rngcrng 13944  opp_rcoppr 14079

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-13 2204  ax-14 2205  ax-ext 2213  ax-sep 4207  ax-nul 4215  ax-pow 4264  ax-pr 4299  ax-un 4530  ax-setind 4635  ax-cnex 8122  ax-resscn 8123  ax-1cn 8124  ax-1re 8125  ax-icn 8126  ax-addcl 8127  ax-addrcl 8128  ax-mulcl 8129  ax-addcom 8131  ax-addass 8133  ax-i2m1 8136  ax-0lt1 8137  ax-0id 8139  ax-rnegex 8140  ax-pre-ltirr 8143  ax-pre-lttrn 8145  ax-pre-ltadd 8147

This theorem depends on definitions:  df-bi 117  df-3an 1006  df-tru 1400  df-fal 1403  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ne 2403  df-nel 2498  df-ral 2515  df-rex 2516  df-rab 2519  df-v 2804  df-sbc 3032  df-csb 3128  df-dif 3202  df-un 3204  df-in 3206  df-ss 3213  df-nul 3495  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-uni 3894  df-int 3929  df-br 4089  df-opab 4151  df-mpt 4152  df-id 4390  df-xp 4731  df-rel 4732  df-cnv 4733  df-co 4734  df-dm 4735  df-rn 4736  df-res 4737  df-ima 4738  df-iota 5286  df-fun 5328  df-fn 5329  df-fv 5334  df-riota 5970  df-ov 6020  df-oprab 6021  df-mpo 6022  df-tpos 6410  df-pnf 8215  df-mnf 8216  df-ltxr 8218  df-inn 9143  df-2 9201  df-3 9202  df-ndx 13084  df-slot 13085  df-base 13087  df-sets 13088  df-plusg 13172  df-mulr 13173  df-0g 13340  df-mgm 13438  df-sgrp 13484  df-mnd 13499  df-grp 13585  df-cmn 13872  df-abl 13873  df-mgp 13933  df-rng 13945  df-oppr 14080

This theorem is referenced by:  opprrngbg  14090  opprsubrngg  14224  isridlrng  14495  2idlcpblrng  14536