Theorem opprrng 14111

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 2230	. . 3 |- ( Base ` R ) = ( Base ` R )
3	1, 2	opprbasg 14109	. 2 |- ( R e. Rng -> ( Base ` R ) = ( Base ` O ) )
4		eqid 2230	. . 3 |- ( +g ` R ) = ( +g ` R )
5	1, 4	oppraddg 14110	. 2 |- ( R e. Rng -> ( +g ` R ) = ( +g ` O ) )
6		eqidd 2231	. 2 |- ( R e. Rng -> ( .r ` O ) = ( .r ` O ) )
7		rngabl 13969	. . 3 |- ( R e. Rng -> R e. Abel )
8		eqidd 2231	. . . 4 |- ( R e. Rng -> ( Base ` R ) = ( Base ` R ) )
9	5	oveqdr 6048	. . . 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 13903	. . 3 |- ( R e. Rng -> ( R e. Abel <-> O e. Abel ) )
11	7, 10	mpbid 147	. 2 |- ( R e. Rng -> O e. Abel )
12		eqid 2230	. . . 4 |- ( .r ` R ) = ( .r ` R )
13		eqid 2230	. . . 4 |- ( .r ` O ) = ( .r ` O )
14	2, 12, 1, 13	opprmulg 14105	. . 3 |- ( ( R e. Rng /\ x e. ( Base ` R ) /\ y e. ( Base ` R ) ) -> ( x ( .r ` O ) y ) = ( y ( .r ` R ) x ) )
15	2, 12	rngcl 13978	. . . 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 2307	. 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 13973	. . . 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 14105	. . . . . 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 6036	. . . 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 13978	. . . . . 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 14105	. . . . 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 2263	. . 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 6035	. . . 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 14105	. . . . 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 2263	. . 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 2274	. 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 13975	. . . 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 13976	. . . . 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 14105	. . . 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 14105	. . . . 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 6038	. . 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 2273	. 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 13974	. . . 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 13976	. . . . 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 14105	. . . 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 6038	. . 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 2273	. 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 13987	1 |- ( R e. Rng -> O e. Rng )

Syntax hints:   -> wi 4   /\ wa 104   /\ w3a 1004   = wceq 1397   e. wcel 2201   ` cfv 5325  (class class class)co 6020   Basecbs 13102   +g cplusg 13180   .rcmulr 13181   Abelcabl 13892  Rngcrng 13966  opp_rcoppr 14101

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 2203  ax-14 2204  ax-ext 2212  ax-sep 4206  ax-nul 4214  ax-pow 4263  ax-pr 4298  ax-un 4529  ax-setind 4634  ax-cnex 8125  ax-resscn 8126  ax-1cn 8127  ax-1re 8128  ax-icn 8129  ax-addcl 8130  ax-addrcl 8131  ax-mulcl 8132  ax-addcom 8134  ax-addass 8136  ax-i2m1 8139  ax-0lt1 8140  ax-0id 8142  ax-rnegex 8143  ax-pre-ltirr 8146  ax-pre-lttrn 8148  ax-pre-ltadd 8150

This theorem depends on definitions:  df-bi 117  df-3an 1006  df-tru 1400  df-fal 1403  df-nf 1509  df-sb 1810  df-eu 2081  df-mo 2082  df-clab 2217  df-cleq 2223  df-clel 2226  df-nfc 2362  df-ne 2402  df-nel 2497  df-ral 2514  df-rex 2515  df-rab 2518  df-v 2803  df-sbc 3031  df-csb 3127  df-dif 3201  df-un 3203  df-in 3205  df-ss 3212  df-nul 3494  df-pw 3653  df-sn 3674  df-pr 3675  df-op 3677  df-uni 3893  df-int 3928  df-br 4088  df-opab 4150  df-mpt 4151  df-id 4389  df-xp 4730  df-rel 4731  df-cnv 4732  df-co 4733  df-dm 4734  df-rn 4735  df-res 4736  df-ima 4737  df-iota 5285  df-fun 5327  df-fn 5328  df-fv 5333  df-riota 5973  df-ov 6023  df-oprab 6024  df-mpo 6025  df-tpos 6413  df-pnf 8218  df-mnf 8219  df-ltxr 8221  df-inn 9146  df-2 9204  df-3 9205  df-ndx 13105  df-slot 13106  df-base 13108  df-sets 13109  df-plusg 13193  df-mulr 13194  df-0g 13361  df-mgm 13459  df-sgrp 13505  df-mnd 13520  df-grp 13606  df-cmn 13893  df-abl 13894  df-mgp 13955  df-rng 13967  df-oppr 14102

This theorem is referenced by:  opprrngbg  14112  opprsubrngg  14246  isridlrng  14517  2idlcpblrng  14558