Theorem opprring 13841

Description: An opposite ring is a ring. (Contributed by Mario Carneiro, 1-Dec-2014.) (Revised by Mario Carneiro, 30-Aug-2015.)

Hypothesis

Ref	Expression
opprbas.1	⊢ 𝑂 = (oppr‘𝑅)

Assertion

Ref	Expression
opprring	⊢ (𝑅 ∈ Ring → 𝑂 ∈ Ring)

Proof of Theorem opprring

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 2205	. . 3 ⊢ (Base‘𝑅) = (Base‘𝑅)
3	1, 2	opprbasg 13837	. 2 ⊢ (𝑅 ∈ Ring → (Base‘𝑅) = (Base‘𝑂))
4		eqid 2205	. . 3 ⊢ (+g‘𝑅) = (+g‘𝑅)
5	1, 4	oppraddg 13838	. 2 ⊢ (𝑅 ∈ Ring → (+g‘𝑅) = (+g‘𝑂))
6		eqidd 2206	. 2 ⊢ (𝑅 ∈ Ring → (.r‘𝑂) = (.r‘𝑂))
7		ringgrp 13763	. . 3 ⊢ (𝑅 ∈ Ring → 𝑅 ∈ Grp)
8		eqidd 2206	. . . 4 ⊢ (𝑅 ∈ Ring → (Base‘𝑅) = (Base‘𝑅))
9	5	oveqdr 5972	. . . 4 ⊢ ((𝑅 ∈ Ring ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅))) → (𝑥(+g‘𝑅)𝑦) = (𝑥(+g‘𝑂)𝑦))
10	8, 3, 9	grppropd 13349	. . 3 ⊢ (𝑅 ∈ Ring → (𝑅 ∈ Grp ↔ 𝑂 ∈ Grp))
11	7, 10	mpbid 147	. 2 ⊢ (𝑅 ∈ Ring → 𝑂 ∈ Grp)
12		eqid 2205	. . . 4 ⊢ (.r‘𝑅) = (.r‘𝑅)
13		eqid 2205	. . . 4 ⊢ (.r‘𝑂) = (.r‘𝑂)
14	2, 12, 1, 13	opprmulg 13833	. . 3 ⊢ ((𝑅 ∈ Ring ∧ 𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅)) → (𝑥(.r‘𝑂)𝑦) = (𝑦(.r‘𝑅)𝑥))
15	2, 12	ringcl 13775	. . . 4 ⊢ ((𝑅 ∈ Ring ∧ 𝑦 ∈ (Base‘𝑅) ∧ 𝑥 ∈ (Base‘𝑅)) → (𝑦(.r‘𝑅)𝑥) ∈ (Base‘𝑅))
16	15	3com23 1212	. . 3 ⊢ ((𝑅 ∈ Ring ∧ 𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅)) → (𝑦(.r‘𝑅)𝑥) ∈ (Base‘𝑅))
17	14, 16	eqeltrd 2282	. 2 ⊢ ((𝑅 ∈ Ring ∧ 𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅)) → (𝑥(.r‘𝑂)𝑦) ∈ (Base‘𝑅))
18		simpl 109	. . . 4 ⊢ ((𝑅 ∈ Ring ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅) ∧ 𝑧 ∈ (Base‘𝑅))) → 𝑅 ∈ Ring)
19		simpr3 1008	. . . 4 ⊢ ((𝑅 ∈ Ring ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅) ∧ 𝑧 ∈ (Base‘𝑅))) → 𝑧 ∈ (Base‘𝑅))
20		simpr2 1007	. . . 4 ⊢ ((𝑅 ∈ Ring ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅) ∧ 𝑧 ∈ (Base‘𝑅))) → 𝑦 ∈ (Base‘𝑅))
21		simpr1 1006	. . . 4 ⊢ ((𝑅 ∈ Ring ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅) ∧ 𝑧 ∈ (Base‘𝑅))) → 𝑥 ∈ (Base‘𝑅))
22	2, 12	ringass 13778	. . . 4 ⊢ ((𝑅 ∈ Ring ∧ (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 1252	. . 3 ⊢ ((𝑅 ∈ Ring ∧ (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 13833	. . . . . 6 ⊢ ((𝑅 ∈ Ring ∧ 𝑦 ∈ (Base‘𝑅) ∧ 𝑧 ∈ (Base‘𝑅)) → (𝑦(.r‘𝑂)𝑧) = (𝑧(.r‘𝑅)𝑦))
25	24	3adant3r1 1215	. . . . 5 ⊢ ((𝑅 ∈ Ring ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅) ∧ 𝑧 ∈ (Base‘𝑅))) → (𝑦(.r‘𝑂)𝑧) = (𝑧(.r‘𝑅)𝑦))
26	25	oveq2d 5960	. . . 4 ⊢ ((𝑅 ∈ Ring ∧ (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	ringcl 13775	. . . . . 6 ⊢ ((𝑅 ∈ Ring ∧ 𝑧 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅)) → (𝑧(.r‘𝑅)𝑦) ∈ (Base‘𝑅))
28	18, 19, 20, 27	syl3anc 1250	. . . . 5 ⊢ ((𝑅 ∈ Ring ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅) ∧ 𝑧 ∈ (Base‘𝑅))) → (𝑧(.r‘𝑅)𝑦) ∈ (Base‘𝑅))
29	2, 12, 1, 13	opprmulg 13833	. . . . 5 ⊢ ((𝑅 ∈ Ring ∧ 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 1250	. . . 4 ⊢ ((𝑅 ∈ Ring ∧ (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 2238	. . 3 ⊢ ((𝑅 ∈ Ring ∧ (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	oveq1d 5959	. . . . 5 ⊢ ((𝑅 ∈ Ring ∧ 𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅)) → ((𝑥(.r‘𝑂)𝑦)(.r‘𝑂)𝑧) = ((𝑦(.r‘𝑅)𝑥)(.r‘𝑂)𝑧))
33	32	3adant3r3 1217	. . . 4 ⊢ ((𝑅 ∈ Ring ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅) ∧ 𝑧 ∈ (Base‘𝑅))) → ((𝑥(.r‘𝑂)𝑦)(.r‘𝑂)𝑧) = ((𝑦(.r‘𝑅)𝑥)(.r‘𝑂)𝑧))
34	16	3adant3r3 1217	. . . . 5 ⊢ ((𝑅 ∈ Ring ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅) ∧ 𝑧 ∈ (Base‘𝑅))) → (𝑦(.r‘𝑅)𝑥) ∈ (Base‘𝑅))
35	2, 12, 1, 13	opprmulg 13833	. . . . 5 ⊢ ((𝑅 ∈ Ring ∧ (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 1250	. . . 4 ⊢ ((𝑅 ∈ Ring ∧ (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 2238	. . 3 ⊢ ((𝑅 ∈ Ring ∧ (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 2249	. 2 ⊢ ((𝑅 ∈ Ring ∧ (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	ringdir 13781	. . . 4 ⊢ ((𝑅 ∈ Ring ∧ (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 1252	. . 3 ⊢ ((𝑅 ∈ Ring ∧ (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	ringacl 13792	. . . . 5 ⊢ ((𝑅 ∈ Ring ∧ 𝑦 ∈ (Base‘𝑅) ∧ 𝑧 ∈ (Base‘𝑅)) → (𝑦(+g‘𝑅)𝑧) ∈ (Base‘𝑅))
42	41	3adant3r1 1215	. . . 4 ⊢ ((𝑅 ∈ Ring ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅) ∧ 𝑧 ∈ (Base‘𝑅))) → (𝑦(+g‘𝑅)𝑧) ∈ (Base‘𝑅))
43	2, 12, 1, 13	opprmulg 13833	. . . 4 ⊢ ((𝑅 ∈ Ring ∧ 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 1250	. . 3 ⊢ ((𝑅 ∈ Ring ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅) ∧ 𝑧 ∈ (Base‘𝑅))) → (𝑥(.r‘𝑂)(𝑦(+g‘𝑅)𝑧)) = ((𝑦(+g‘𝑅)𝑧)(.r‘𝑅)𝑥))
45	14	3adant3r3 1217	. . . 4 ⊢ ((𝑅 ∈ Ring ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅) ∧ 𝑧 ∈ (Base‘𝑅))) → (𝑥(.r‘𝑂)𝑦) = (𝑦(.r‘𝑅)𝑥))
46	2, 12, 1, 13	opprmulg 13833	. . . . 5 ⊢ ((𝑅 ∈ Ring ∧ 𝑥 ∈ (Base‘𝑅) ∧ 𝑧 ∈ (Base‘𝑅)) → (𝑥(.r‘𝑂)𝑧) = (𝑧(.r‘𝑅)𝑥))
47	46	3adant3r2 1216	. . . 4 ⊢ ((𝑅 ∈ Ring ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅) ∧ 𝑧 ∈ (Base‘𝑅))) → (𝑥(.r‘𝑂)𝑧) = (𝑧(.r‘𝑅)𝑥))
48	45, 47	oveq12d 5962	. . 3 ⊢ ((𝑅 ∈ Ring ∧ (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)))
49	40, 44, 48	3eqtr4d 2248	. 2 ⊢ ((𝑅 ∈ Ring ∧ (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)))
50	2, 4, 12	ringdi 13780	. . . 4 ⊢ ((𝑅 ∈ Ring ∧ (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)))
51	18, 19, 21, 20, 50	syl13anc 1252	. . 3 ⊢ ((𝑅 ∈ Ring ∧ (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)))
52	2, 4	ringacl 13792	. . . . 5 ⊢ ((𝑅 ∈ Ring ∧ 𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅)) → (𝑥(+g‘𝑅)𝑦) ∈ (Base‘𝑅))
53	52	3adant3r3 1217	. . . 4 ⊢ ((𝑅 ∈ Ring ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅) ∧ 𝑧 ∈ (Base‘𝑅))) → (𝑥(+g‘𝑅)𝑦) ∈ (Base‘𝑅))
54	2, 12, 1, 13	opprmulg 13833	. . . 4 ⊢ ((𝑅 ∈ Ring ∧ (𝑥(+g‘𝑅)𝑦) ∈ (Base‘𝑅) ∧ 𝑧 ∈ (Base‘𝑅)) → ((𝑥(+g‘𝑅)𝑦)(.r‘𝑂)𝑧) = (𝑧(.r‘𝑅)(𝑥(+g‘𝑅)𝑦)))
55	18, 53, 19, 54	syl3anc 1250	. . 3 ⊢ ((𝑅 ∈ Ring ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅) ∧ 𝑧 ∈ (Base‘𝑅))) → ((𝑥(+g‘𝑅)𝑦)(.r‘𝑂)𝑧) = (𝑧(.r‘𝑅)(𝑥(+g‘𝑅)𝑦)))
56	47, 25	oveq12d 5962	. . 3 ⊢ ((𝑅 ∈ Ring ∧ (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)))
57	51, 55, 56	3eqtr4d 2248	. 2 ⊢ ((𝑅 ∈ Ring ∧ (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)))
58		eqid 2205	. . 3 ⊢ (1r‘𝑅) = (1r‘𝑅)
59	2, 58	ringidcl 13782	. 2 ⊢ (𝑅 ∈ Ring → (1r‘𝑅) ∈ (Base‘𝑅))
60		simpl 109	. . . 4 ⊢ ((𝑅 ∈ Ring ∧ 𝑥 ∈ (Base‘𝑅)) → 𝑅 ∈ Ring)
61	60, 59	syl 14	. . . 4 ⊢ ((𝑅 ∈ Ring ∧ 𝑥 ∈ (Base‘𝑅)) → (1r‘𝑅) ∈ (Base‘𝑅))
62		simpr 110	. . . 4 ⊢ ((𝑅 ∈ Ring ∧ 𝑥 ∈ (Base‘𝑅)) → 𝑥 ∈ (Base‘𝑅))
63	2, 12, 1, 13	opprmulg 13833	. . . 4 ⊢ ((𝑅 ∈ Ring ∧ (1r‘𝑅) ∈ (Base‘𝑅) ∧ 𝑥 ∈ (Base‘𝑅)) → ((1r‘𝑅)(.r‘𝑂)𝑥) = (𝑥(.r‘𝑅)(1r‘𝑅)))
64	60, 61, 62, 63	syl3anc 1250	. . 3 ⊢ ((𝑅 ∈ Ring ∧ 𝑥 ∈ (Base‘𝑅)) → ((1r‘𝑅)(.r‘𝑂)𝑥) = (𝑥(.r‘𝑅)(1r‘𝑅)))
65	2, 12, 58	ringridm 13786	. . 3 ⊢ ((𝑅 ∈ Ring ∧ 𝑥 ∈ (Base‘𝑅)) → (𝑥(.r‘𝑅)(1r‘𝑅)) = 𝑥)
66	64, 65	eqtrd 2238	. 2 ⊢ ((𝑅 ∈ Ring ∧ 𝑥 ∈ (Base‘𝑅)) → ((1r‘𝑅)(.r‘𝑂)𝑥) = 𝑥)
67	2, 12, 1, 13	opprmulg 13833	. . . 4 ⊢ ((𝑅 ∈ Ring ∧ 𝑥 ∈ (Base‘𝑅) ∧ (1r‘𝑅) ∈ (Base‘𝑅)) → (𝑥(.r‘𝑂)(1r‘𝑅)) = ((1r‘𝑅)(.r‘𝑅)𝑥))
68	60, 62, 61, 67	syl3anc 1250	. . 3 ⊢ ((𝑅 ∈ Ring ∧ 𝑥 ∈ (Base‘𝑅)) → (𝑥(.r‘𝑂)(1r‘𝑅)) = ((1r‘𝑅)(.r‘𝑅)𝑥))
69	2, 12, 58	ringlidm 13785	. . 3 ⊢ ((𝑅 ∈ Ring ∧ 𝑥 ∈ (Base‘𝑅)) → ((1r‘𝑅)(.r‘𝑅)𝑥) = 𝑥)
70	68, 69	eqtrd 2238	. 2 ⊢ ((𝑅 ∈ Ring ∧ 𝑥 ∈ (Base‘𝑅)) → (𝑥(.r‘𝑂)(1r‘𝑅)) = 𝑥)
71	3, 5, 6, 11, 17, 38, 49, 57, 59, 66, 70	isringd 13803	1 ⊢ (𝑅 ∈ Ring → 𝑂 ∈ Ring)

Syntax hints:   → wi 4   ∧ wa 104   ∧ w3a 981   = wceq 1373   ∈ wcel 2176  ‘cfv 5271  (class class class)co 5944  Basecbs 12832  +_gcplusg 12909  ._rcmulr 12910  Grpcgrp 13332  1_rcur 13721  Ringcrg 13758  opp_rcoppr 13829

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 4162  ax-nul 4170  ax-pow 4218  ax-pr 4253  ax-un 4480  ax-setind 4585  ax-cnex 8016  ax-resscn 8017  ax-1cn 8018  ax-1re 8019  ax-icn 8020  ax-addcl 8021  ax-addrcl 8022  ax-mulcl 8023  ax-addcom 8025  ax-addass 8027  ax-i2m1 8030  ax-0lt1 8031  ax-0id 8033  ax-rnegex 8034  ax-pre-ltirr 8037  ax-pre-lttrn 8039  ax-pre-ltadd 8041

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-reu 2491  df-rmo 2492  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 4045  df-opab 4106  df-mpt 4107  df-id 4340  df-xp 4681  df-rel 4682  df-cnv 4683  df-co 4684  df-dm 4685  df-rn 4686  df-res 4687  df-ima 4688  df-iota 5232  df-fun 5273  df-fn 5274  df-fv 5279  df-riota 5899  df-ov 5947  df-oprab 5948  df-mpo 5949  df-tpos 6331  df-pnf 8109  df-mnf 8110  df-ltxr 8112  df-inn 9037  df-2 9095  df-3 9096  df-ndx 12835  df-slot 12836  df-base 12838  df-sets 12839  df-plusg 12922  df-mulr 12923  df-0g 13090  df-mgm 13188  df-sgrp 13234  df-mnd 13249  df-grp 13335  df-mgp 13683  df-ur 13722  df-ring 13760  df-oppr 13830

This theorem is referenced by:  opprringbg  13842  mulgass3  13847  1unit  13869  opprunitd  13872  crngunit  13873  unitmulcl  13875  unitgrp  13878  unitnegcl  13892  unitpropdg  13910  subrguss  13998  subrgunit  14001  isridl  14266  ridl0  14272  ridl1  14273