Theorem opprring 14037

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 2229	. . 3 ⊢ (Base‘𝑅) = (Base‘𝑅)
3	1, 2	opprbasg 14033	. 2 ⊢ (𝑅 ∈ Ring → (Base‘𝑅) = (Base‘𝑂))
4		eqid 2229	. . 3 ⊢ (+g‘𝑅) = (+g‘𝑅)
5	1, 4	oppraddg 14034	. 2 ⊢ (𝑅 ∈ Ring → (+g‘𝑅) = (+g‘𝑂))
6		eqidd 2230	. 2 ⊢ (𝑅 ∈ Ring → (.r‘𝑂) = (.r‘𝑂))
7		ringgrp 13959	. . 3 ⊢ (𝑅 ∈ Ring → 𝑅 ∈ Grp)
8		eqidd 2230	. . . 4 ⊢ (𝑅 ∈ Ring → (Base‘𝑅) = (Base‘𝑅))
9	5	oveqdr 6028	. . . 4 ⊢ ((𝑅 ∈ Ring ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅))) → (𝑥(+g‘𝑅)𝑦) = (𝑥(+g‘𝑂)𝑦))
10	8, 3, 9	grppropd 13545	. . 3 ⊢ (𝑅 ∈ Ring → (𝑅 ∈ Grp ↔ 𝑂 ∈ Grp))
11	7, 10	mpbid 147	. 2 ⊢ (𝑅 ∈ Ring → 𝑂 ∈ Grp)
12		eqid 2229	. . . 4 ⊢ (.r‘𝑅) = (.r‘𝑅)
13		eqid 2229	. . . 4 ⊢ (.r‘𝑂) = (.r‘𝑂)
14	2, 12, 1, 13	opprmulg 14029	. . 3 ⊢ ((𝑅 ∈ Ring ∧ 𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅)) → (𝑥(.r‘𝑂)𝑦) = (𝑦(.r‘𝑅)𝑥))
15	2, 12	ringcl 13971	. . . 4 ⊢ ((𝑅 ∈ Ring ∧ 𝑦 ∈ (Base‘𝑅) ∧ 𝑥 ∈ (Base‘𝑅)) → (𝑦(.r‘𝑅)𝑥) ∈ (Base‘𝑅))
16	15	3com23 1233	. . 3 ⊢ ((𝑅 ∈ Ring ∧ 𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅)) → (𝑦(.r‘𝑅)𝑥) ∈ (Base‘𝑅))
17	14, 16	eqeltrd 2306	. 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 1029	. . . 4 ⊢ ((𝑅 ∈ Ring ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅) ∧ 𝑧 ∈ (Base‘𝑅))) → 𝑧 ∈ (Base‘𝑅))
20		simpr2 1028	. . . 4 ⊢ ((𝑅 ∈ Ring ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅) ∧ 𝑧 ∈ (Base‘𝑅))) → 𝑦 ∈ (Base‘𝑅))
21		simpr1 1027	. . . 4 ⊢ ((𝑅 ∈ Ring ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅) ∧ 𝑧 ∈ (Base‘𝑅))) → 𝑥 ∈ (Base‘𝑅))
22	2, 12	ringass 13974	. . . 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 1273	. . 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 14029	. . . . . 6 ⊢ ((𝑅 ∈ Ring ∧ 𝑦 ∈ (Base‘𝑅) ∧ 𝑧 ∈ (Base‘𝑅)) → (𝑦(.r‘𝑂)𝑧) = (𝑧(.r‘𝑅)𝑦))
25	24	3adant3r1 1236	. . . . 5 ⊢ ((𝑅 ∈ Ring ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅) ∧ 𝑧 ∈ (Base‘𝑅))) → (𝑦(.r‘𝑂)𝑧) = (𝑧(.r‘𝑅)𝑦))
26	25	oveq2d 6016	. . . 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 13971	. . . . . 6 ⊢ ((𝑅 ∈ Ring ∧ 𝑧 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅)) → (𝑧(.r‘𝑅)𝑦) ∈ (Base‘𝑅))
28	18, 19, 20, 27	syl3anc 1271	. . . . 5 ⊢ ((𝑅 ∈ Ring ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅) ∧ 𝑧 ∈ (Base‘𝑅))) → (𝑧(.r‘𝑅)𝑦) ∈ (Base‘𝑅))
29	2, 12, 1, 13	opprmulg 14029	. . . . 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 1271	. . . 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 2262	. . 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 6015	. . . . 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 1238	. . . 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 1238	. . . . 5 ⊢ ((𝑅 ∈ Ring ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅) ∧ 𝑧 ∈ (Base‘𝑅))) → (𝑦(.r‘𝑅)𝑥) ∈ (Base‘𝑅))
35	2, 12, 1, 13	opprmulg 14029	. . . . 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 1271	. . . 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 2262	. . 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 2273	. 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 13977	. . . 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 1273	. . 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 13988	. . . . 5 ⊢ ((𝑅 ∈ Ring ∧ 𝑦 ∈ (Base‘𝑅) ∧ 𝑧 ∈ (Base‘𝑅)) → (𝑦(+g‘𝑅)𝑧) ∈ (Base‘𝑅))
42	41	3adant3r1 1236	. . . 4 ⊢ ((𝑅 ∈ Ring ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅) ∧ 𝑧 ∈ (Base‘𝑅))) → (𝑦(+g‘𝑅)𝑧) ∈ (Base‘𝑅))
43	2, 12, 1, 13	opprmulg 14029	. . . 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 1271	. . 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 1238	. . . 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 14029	. . . . 5 ⊢ ((𝑅 ∈ Ring ∧ 𝑥 ∈ (Base‘𝑅) ∧ 𝑧 ∈ (Base‘𝑅)) → (𝑥(.r‘𝑂)𝑧) = (𝑧(.r‘𝑅)𝑥))
47	46	3adant3r2 1237	. . . 4 ⊢ ((𝑅 ∈ Ring ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅) ∧ 𝑧 ∈ (Base‘𝑅))) → (𝑥(.r‘𝑂)𝑧) = (𝑧(.r‘𝑅)𝑥))
48	45, 47	oveq12d 6018	. . 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 2272	. 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 13976	. . . 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 1273	. . 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 13988	. . . . 5 ⊢ ((𝑅 ∈ Ring ∧ 𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅)) → (𝑥(+g‘𝑅)𝑦) ∈ (Base‘𝑅))
53	52	3adant3r3 1238	. . . 4 ⊢ ((𝑅 ∈ Ring ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅) ∧ 𝑧 ∈ (Base‘𝑅))) → (𝑥(+g‘𝑅)𝑦) ∈ (Base‘𝑅))
54	2, 12, 1, 13	opprmulg 14029	. . . 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 1271	. . 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 6018	. . 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 2272	. 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 2229	. . 3 ⊢ (1r‘𝑅) = (1r‘𝑅)
59	2, 58	ringidcl 13978	. 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 14029	. . . 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 1271	. . 3 ⊢ ((𝑅 ∈ Ring ∧ 𝑥 ∈ (Base‘𝑅)) → ((1r‘𝑅)(.r‘𝑂)𝑥) = (𝑥(.r‘𝑅)(1r‘𝑅)))
65	2, 12, 58	ringridm 13982	. . 3 ⊢ ((𝑅 ∈ Ring ∧ 𝑥 ∈ (Base‘𝑅)) → (𝑥(.r‘𝑅)(1r‘𝑅)) = 𝑥)
66	64, 65	eqtrd 2262	. 2 ⊢ ((𝑅 ∈ Ring ∧ 𝑥 ∈ (Base‘𝑅)) → ((1r‘𝑅)(.r‘𝑂)𝑥) = 𝑥)
67	2, 12, 1, 13	opprmulg 14029	. . . 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 1271	. . 3 ⊢ ((𝑅 ∈ Ring ∧ 𝑥 ∈ (Base‘𝑅)) → (𝑥(.r‘𝑂)(1r‘𝑅)) = ((1r‘𝑅)(.r‘𝑅)𝑥))
69	2, 12, 58	ringlidm 13981	. . 3 ⊢ ((𝑅 ∈ Ring ∧ 𝑥 ∈ (Base‘𝑅)) → ((1r‘𝑅)(.r‘𝑅)𝑥) = 𝑥)
70	68, 69	eqtrd 2262	. 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 13999	1 ⊢ (𝑅 ∈ Ring → 𝑂 ∈ Ring)

Syntax hints:   → wi 4   ∧ wa 104   ∧ w3a 1002   = wceq 1395   ∈ wcel 2200  ‘cfv 5317  (class class class)co 6000  Basecbs 13027  +_gcplusg 13105  ._rcmulr 13106  Grpcgrp 13528  1_rcur 13917  Ringcrg 13954  opp_rcoppr 14025

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 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-sep 4201  ax-nul 4209  ax-pow 4257  ax-pr 4292  ax-un 4523  ax-setind 4628  ax-cnex 8086  ax-resscn 8087  ax-1cn 8088  ax-1re 8089  ax-icn 8090  ax-addcl 8091  ax-addrcl 8092  ax-mulcl 8093  ax-addcom 8095  ax-addass 8097  ax-i2m1 8100  ax-0lt1 8101  ax-0id 8103  ax-rnegex 8104  ax-pre-ltirr 8107  ax-pre-lttrn 8109  ax-pre-ltadd 8111

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-nel 2496  df-ral 2513  df-rex 2514  df-reu 2515  df-rmo 2516  df-rab 2517  df-v 2801  df-sbc 3029  df-csb 3125  df-dif 3199  df-un 3201  df-in 3203  df-ss 3210  df-nul 3492  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3888  df-int 3923  df-br 4083  df-opab 4145  df-mpt 4146  df-id 4383  df-xp 4724  df-rel 4725  df-cnv 4726  df-co 4727  df-dm 4728  df-rn 4729  df-res 4730  df-ima 4731  df-iota 5277  df-fun 5319  df-fn 5320  df-fv 5325  df-riota 5953  df-ov 6003  df-oprab 6004  df-mpo 6005  df-tpos 6389  df-pnf 8179  df-mnf 8180  df-ltxr 8182  df-inn 9107  df-2 9165  df-3 9166  df-ndx 13030  df-slot 13031  df-base 13033  df-sets 13034  df-plusg 13118  df-mulr 13119  df-0g 13286  df-mgm 13384  df-sgrp 13430  df-mnd 13445  df-grp 13531  df-mgp 13879  df-ur 13918  df-ring 13956  df-oppr 14026

This theorem is referenced by:  opprringbg  14038  mulgass3  14043  1unit  14065  opprunitd  14068  crngunit  14069  unitmulcl  14071  unitgrp  14074  unitnegcl  14088  unitpropdg  14106  subrguss  14194  subrgunit  14197  isridl  14462  ridl0  14468  ridl1  14469