Theorem opprrng 13709

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 ∈ Rng → 𝑂 ∈ 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 13707	. 2 ⊢ (𝑅 ∈ Rng → (Base‘𝑅) = (Base‘𝑂))
4		eqid 2196	. . 3 ⊢ (+g‘𝑅) = (+g‘𝑅)
5	1, 4	oppraddg 13708	. 2 ⊢ (𝑅 ∈ Rng → (+g‘𝑅) = (+g‘𝑂))
6		eqidd 2197	. 2 ⊢ (𝑅 ∈ Rng → (.r‘𝑂) = (.r‘𝑂))
7		rngabl 13567	. . 3 ⊢ (𝑅 ∈ Rng → 𝑅 ∈ Abel)
8		eqidd 2197	. . . 4 ⊢ (𝑅 ∈ Rng → (Base‘𝑅) = (Base‘𝑅))
9	5	oveqdr 5953	. . . 4 ⊢ ((𝑅 ∈ Rng ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅))) → (𝑥(+g‘𝑅)𝑦) = (𝑥(+g‘𝑂)𝑦))
10	8, 3, 9	ablpropd 13502	. . 3 ⊢ (𝑅 ∈ Rng → (𝑅 ∈ Abel ↔ 𝑂 ∈ Abel))
11	7, 10	mpbid 147	. 2 ⊢ (𝑅 ∈ Rng → 𝑂 ∈ Abel)
12		eqid 2196	. . . 4 ⊢ (.r‘𝑅) = (.r‘𝑅)
13		eqid 2196	. . . 4 ⊢ (.r‘𝑂) = (.r‘𝑂)
14	2, 12, 1, 13	opprmulg 13703	. . 3 ⊢ ((𝑅 ∈ Rng ∧ 𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅)) → (𝑥(.r‘𝑂)𝑦) = (𝑦(.r‘𝑅)𝑥))
15	2, 12	rngcl 13576	. . . 4 ⊢ ((𝑅 ∈ Rng ∧ 𝑦 ∈ (Base‘𝑅) ∧ 𝑥 ∈ (Base‘𝑅)) → (𝑦(.r‘𝑅)𝑥) ∈ (Base‘𝑅))
16	15	3com23 1211	. . 3 ⊢ ((𝑅 ∈ Rng ∧ 𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅)) → (𝑦(.r‘𝑅)𝑥) ∈ (Base‘𝑅))
17	14, 16	eqeltrd 2273	. 2 ⊢ ((𝑅 ∈ Rng ∧ 𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅)) → (𝑥(.r‘𝑂)𝑦) ∈ (Base‘𝑅))
18		simpl 109	. . . 4 ⊢ ((𝑅 ∈ Rng ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅) ∧ 𝑧 ∈ (Base‘𝑅))) → 𝑅 ∈ Rng)
19		simpr3 1007	. . . 4 ⊢ ((𝑅 ∈ Rng ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅) ∧ 𝑧 ∈ (Base‘𝑅))) → 𝑧 ∈ (Base‘𝑅))
20		simpr2 1006	. . . 4 ⊢ ((𝑅 ∈ Rng ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅) ∧ 𝑧 ∈ (Base‘𝑅))) → 𝑦 ∈ (Base‘𝑅))
21		simpr1 1005	. . . 4 ⊢ ((𝑅 ∈ Rng ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅) ∧ 𝑧 ∈ (Base‘𝑅))) → 𝑥 ∈ (Base‘𝑅))
22	2, 12	rngass 13571	. . . 4 ⊢ ((𝑅 ∈ Rng ∧ (𝑧 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅) ∧ 𝑥 ∈ (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 ∈ Rng ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅) ∧ 𝑧 ∈ (Base‘𝑅))) → ((𝑧(.r‘𝑅)𝑦)(.r‘𝑅)𝑥) = (𝑧(.r‘𝑅)(𝑦(.r‘𝑅)𝑥)))
24	2, 12, 1, 13	opprmulg 13703	. . . . . 6 ⊢ ((𝑅 ∈ Rng ∧ 𝑦 ∈ (Base‘𝑅) ∧ 𝑧 ∈ (Base‘𝑅)) → (𝑦(.r‘𝑂)𝑧) = (𝑧(.r‘𝑅)𝑦))
25	24	3adant3r1 1214	. . . . 5 ⊢ ((𝑅 ∈ Rng ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅) ∧ 𝑧 ∈ (Base‘𝑅))) → (𝑦(.r‘𝑂)𝑧) = (𝑧(.r‘𝑅)𝑦))
26	25	oveq2d 5941	. . . 4 ⊢ ((𝑅 ∈ Rng ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅) ∧ 𝑧 ∈ (Base‘𝑅))) → (𝑥(.r‘𝑂)(𝑦(.r‘𝑂)𝑧)) = (𝑥(.r‘𝑂)(𝑧(.r‘𝑅)𝑦)))
27	2, 12	rngcl 13576	. . . . . 6 ⊢ ((𝑅 ∈ Rng ∧ 𝑧 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅)) → (𝑧(.r‘𝑅)𝑦) ∈ (Base‘𝑅))
28	18, 19, 20, 27	syl3anc 1249	. . . . 5 ⊢ ((𝑅 ∈ Rng ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅) ∧ 𝑧 ∈ (Base‘𝑅))) → (𝑧(.r‘𝑅)𝑦) ∈ (Base‘𝑅))
29	2, 12, 1, 13	opprmulg 13703	. . . . 5 ⊢ ((𝑅 ∈ Rng ∧ 𝑥 ∈ (Base‘𝑅) ∧ (𝑧(.r‘𝑅)𝑦) ∈ (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 ∈ Rng ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅) ∧ 𝑧 ∈ (Base‘𝑅))) → (𝑥(.r‘𝑂)(𝑧(.r‘𝑅)𝑦)) = ((𝑧(.r‘𝑅)𝑦)(.r‘𝑅)𝑥))
31	26, 30	eqtrd 2229	. . 3 ⊢ ((𝑅 ∈ Rng ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅) ∧ 𝑧 ∈ (Base‘𝑅))) → (𝑥(.r‘𝑂)(𝑦(.r‘𝑂)𝑧)) = ((𝑧(.r‘𝑅)𝑦)(.r‘𝑅)𝑥))
32	14	3adant3r3 1216	. . . . 5 ⊢ ((𝑅 ∈ Rng ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅) ∧ 𝑧 ∈ (Base‘𝑅))) → (𝑥(.r‘𝑂)𝑦) = (𝑦(.r‘𝑅)𝑥))
33	32	oveq1d 5940	. . . 4 ⊢ ((𝑅 ∈ Rng ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅) ∧ 𝑧 ∈ (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 ∈ Rng ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅) ∧ 𝑧 ∈ (Base‘𝑅))) → (𝑦(.r‘𝑅)𝑥) ∈ (Base‘𝑅))
35	2, 12, 1, 13	opprmulg 13703	. . . . 5 ⊢ ((𝑅 ∈ Rng ∧ (𝑦(.r‘𝑅)𝑥) ∈ (Base‘𝑅) ∧ 𝑧 ∈ (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 ∈ Rng ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅) ∧ 𝑧 ∈ (Base‘𝑅))) → ((𝑦(.r‘𝑅)𝑥)(.r‘𝑂)𝑧) = (𝑧(.r‘𝑅)(𝑦(.r‘𝑅)𝑥)))
37	33, 36	eqtrd 2229	. . 3 ⊢ ((𝑅 ∈ Rng ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅) ∧ 𝑧 ∈ (Base‘𝑅))) → ((𝑥(.r‘𝑂)𝑦)(.r‘𝑂)𝑧) = (𝑧(.r‘𝑅)(𝑦(.r‘𝑅)𝑥)))
38	23, 31, 37	3eqtr4rd 2240	. 2 ⊢ ((𝑅 ∈ Rng ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅) ∧ 𝑧 ∈ (Base‘𝑅))) → ((𝑥(.r‘𝑂)𝑦)(.r‘𝑂)𝑧) = (𝑥(.r‘𝑂)(𝑦(.r‘𝑂)𝑧)))
39	2, 4, 12	rngdir 13573	. . . 4 ⊢ ((𝑅 ∈ Rng ∧ (𝑦 ∈ (Base‘𝑅) ∧ 𝑧 ∈ (Base‘𝑅) ∧ 𝑥 ∈ (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 ∈ Rng ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅) ∧ 𝑧 ∈ (Base‘𝑅))) → ((𝑦(+g‘𝑅)𝑧)(.r‘𝑅)𝑥) = ((𝑦(.r‘𝑅)𝑥)(+g‘𝑅)(𝑧(.r‘𝑅)𝑥)))
41	2, 4	rngacl 13574	. . . . 5 ⊢ ((𝑅 ∈ Rng ∧ 𝑦 ∈ (Base‘𝑅) ∧ 𝑧 ∈ (Base‘𝑅)) → (𝑦(+g‘𝑅)𝑧) ∈ (Base‘𝑅))
42	41	3adant3r1 1214	. . . 4 ⊢ ((𝑅 ∈ Rng ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅) ∧ 𝑧 ∈ (Base‘𝑅))) → (𝑦(+g‘𝑅)𝑧) ∈ (Base‘𝑅))
43	2, 12, 1, 13	opprmulg 13703	. . . 4 ⊢ ((𝑅 ∈ Rng ∧ 𝑥 ∈ (Base‘𝑅) ∧ (𝑦(+g‘𝑅)𝑧) ∈ (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 ∈ Rng ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅) ∧ 𝑧 ∈ (Base‘𝑅))) → (𝑥(.r‘𝑂)(𝑦(+g‘𝑅)𝑧)) = ((𝑦(+g‘𝑅)𝑧)(.r‘𝑅)𝑥))
45	2, 12, 1, 13	opprmulg 13703	. . . . 5 ⊢ ((𝑅 ∈ Rng ∧ 𝑥 ∈ (Base‘𝑅) ∧ 𝑧 ∈ (Base‘𝑅)) → (𝑥(.r‘𝑂)𝑧) = (𝑧(.r‘𝑅)𝑥))
46	18, 21, 19, 45	syl3anc 1249	. . . 4 ⊢ ((𝑅 ∈ Rng ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅) ∧ 𝑧 ∈ (Base‘𝑅))) → (𝑥(.r‘𝑂)𝑧) = (𝑧(.r‘𝑅)𝑥))
47	32, 46	oveq12d 5943	. . 3 ⊢ ((𝑅 ∈ Rng ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅) ∧ 𝑧 ∈ (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 ∈ Rng ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅) ∧ 𝑧 ∈ (Base‘𝑅))) → (𝑥(.r‘𝑂)(𝑦(+g‘𝑅)𝑧)) = ((𝑥(.r‘𝑂)𝑦)(+g‘𝑅)(𝑥(.r‘𝑂)𝑧)))
49	2, 4, 12	rngdi 13572	. . . 4 ⊢ ((𝑅 ∈ Rng ∧ (𝑧 ∈ (Base‘𝑅) ∧ 𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (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 ∈ Rng ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅) ∧ 𝑧 ∈ (Base‘𝑅))) → (𝑧(.r‘𝑅)(𝑥(+g‘𝑅)𝑦)) = ((𝑧(.r‘𝑅)𝑥)(+g‘𝑅)(𝑧(.r‘𝑅)𝑦)))
51	2, 4	rngacl 13574	. . . . 5 ⊢ ((𝑅 ∈ Rng ∧ 𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅)) → (𝑥(+g‘𝑅)𝑦) ∈ (Base‘𝑅))
52	51	3adant3r3 1216	. . . 4 ⊢ ((𝑅 ∈ Rng ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅) ∧ 𝑧 ∈ (Base‘𝑅))) → (𝑥(+g‘𝑅)𝑦) ∈ (Base‘𝑅))
53	2, 12, 1, 13	opprmulg 13703	. . . 4 ⊢ ((𝑅 ∈ Rng ∧ (𝑥(+g‘𝑅)𝑦) ∈ (Base‘𝑅) ∧ 𝑧 ∈ (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 ∈ Rng ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅) ∧ 𝑧 ∈ (Base‘𝑅))) → ((𝑥(+g‘𝑅)𝑦)(.r‘𝑂)𝑧) = (𝑧(.r‘𝑅)(𝑥(+g‘𝑅)𝑦)))
55	46, 25	oveq12d 5943	. . 3 ⊢ ((𝑅 ∈ Rng ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅) ∧ 𝑧 ∈ (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 ∈ Rng ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅) ∧ 𝑧 ∈ (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 13585	1 ⊢ (𝑅 ∈ Rng → 𝑂 ∈ Rng)

Syntax hints:   → wi 4   ∧ wa 104   ∧ w3a 980   = wceq 1364   ∈ wcel 2167  ‘cfv 5259  (class class class)co 5925  Basecbs 12703  +_gcplusg 12780  ._rcmulr 12781  Abelcabl 13491  Rngcrng 13564  opp_rcoppr 13699

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 4152  ax-nul 4160  ax-pow 4208  ax-pr 4243  ax-un 4469  ax-setind 4574  ax-cnex 7987  ax-resscn 7988  ax-1cn 7989  ax-1re 7990  ax-icn 7991  ax-addcl 7992  ax-addrcl 7993  ax-mulcl 7994  ax-addcom 7996  ax-addass 7998  ax-i2m1 8001  ax-0lt1 8002  ax-0id 8004  ax-rnegex 8005  ax-pre-ltirr 8008  ax-pre-lttrn 8010  ax-pre-ltadd 8012

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 3452  df-pw 3608  df-sn 3629  df-pr 3630  df-op 3632  df-uni 3841  df-int 3876  df-br 4035  df-opab 4096  df-mpt 4097  df-id 4329  df-xp 4670  df-rel 4671  df-cnv 4672  df-co 4673  df-dm 4674  df-rn 4675  df-res 4676  df-ima 4677  df-iota 5220  df-fun 5261  df-fn 5262  df-fv 5267  df-riota 5880  df-ov 5928  df-oprab 5929  df-mpo 5930  df-tpos 6312  df-pnf 8080  df-mnf 8081  df-ltxr 8083  df-inn 9008  df-2 9066  df-3 9067  df-ndx 12706  df-slot 12707  df-base 12709  df-sets 12710  df-plusg 12793  df-mulr 12794  df-0g 12960  df-mgm 13058  df-sgrp 13104  df-mnd 13119  df-grp 13205  df-cmn 13492  df-abl 13493  df-mgp 13553  df-rng 13565  df-oppr 13700

This theorem is referenced by:  opprrngbg  13710  opprsubrngg  13843  isridlrng  14114  2idlcpblrng  14155