Theorem isdrng2 18580

Description: A division ring can equivalently be defined as a ring such that the nonzero elements form a group under multiplication (from which it follows that this is the same group as the group of units). (Contributed by Mario Carneiro, 2-Dec-2014.)

Hypotheses

Ref	Expression
isdrng2.b	⊢ 𝐵 = (Base‘𝑅)
isdrng2.z	⊢ 0 = (0g‘𝑅)
isdrng2.g	⊢ 𝐺 = ((mulGrp‘𝑅) ↾s (𝐵 ∖ { 0 }))

Assertion

Ref	Expression
isdrng2	⊢ (𝑅 ∈ DivRing ↔ (𝑅 ∈ Ring ∧ 𝐺 ∈ Grp))

Proof of Theorem isdrng2

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		isdrng2.b	. . 3 ⊢ 𝐵 = (Base‘𝑅)
2		eqid 2610	. . 3 ⊢ (Unit‘𝑅) = (Unit‘𝑅)
3		isdrng2.z	. . 3 ⊢ 0 = (0g‘𝑅)
4	1, 2, 3	isdrng 18574	. 2 ⊢ (𝑅 ∈ DivRing ↔ (𝑅 ∈ Ring ∧ (Unit‘𝑅) = (𝐵 ∖ { 0 })))
5		oveq2 6557	. . . . . . 7 ⊢ ((Unit‘𝑅) = (𝐵 ∖ { 0 }) → ((mulGrp‘𝑅) ↾s (Unit‘𝑅)) = ((mulGrp‘𝑅) ↾s (𝐵 ∖ { 0 })))
6	5	adantl 481	. . . . . 6 ⊢ ((𝑅 ∈ Ring ∧ (Unit‘𝑅) = (𝐵 ∖ { 0 })) → ((mulGrp‘𝑅) ↾s (Unit‘𝑅)) = ((mulGrp‘𝑅) ↾s (𝐵 ∖ { 0 })))
7		isdrng2.g	. . . . . 6 ⊢ 𝐺 = ((mulGrp‘𝑅) ↾s (𝐵 ∖ { 0 }))
8	6, 7	syl6eqr 2662	. . . . 5 ⊢ ((𝑅 ∈ Ring ∧ (Unit‘𝑅) = (𝐵 ∖ { 0 })) → ((mulGrp‘𝑅) ↾s (Unit‘𝑅)) = 𝐺)
9		eqid 2610	. . . . . . 7 ⊢ ((mulGrp‘𝑅) ↾s (Unit‘𝑅)) = ((mulGrp‘𝑅) ↾s (Unit‘𝑅))
10	2, 9	unitgrp 18490	. . . . . 6 ⊢ (𝑅 ∈ Ring → ((mulGrp‘𝑅) ↾s (Unit‘𝑅)) ∈ Grp)
11	10	adantr 480	. . . . 5 ⊢ ((𝑅 ∈ Ring ∧ (Unit‘𝑅) = (𝐵 ∖ { 0 })) → ((mulGrp‘𝑅) ↾s (Unit‘𝑅)) ∈ Grp)
12	8, 11	eqeltrrd 2689	. . . 4 ⊢ ((𝑅 ∈ Ring ∧ (Unit‘𝑅) = (𝐵 ∖ { 0 })) → 𝐺 ∈ Grp)
13	1, 2	unitcl 18482	. . . . . . . . 9 ⊢ (𝑥 ∈ (Unit‘𝑅) → 𝑥 ∈ 𝐵)
14	13	adantl 481	. . . . . . . 8 ⊢ (((𝑅 ∈ Ring ∧ 𝐺 ∈ Grp) ∧ 𝑥 ∈ (Unit‘𝑅)) → 𝑥 ∈ 𝐵)
15		difss 3699	. . . . . . . . . . . . . . 15 ⊢ (𝐵 ∖ { 0 }) ⊆ 𝐵
16		eqid 2610	. . . . . . . . . . . . . . . . 17 ⊢ (mulGrp‘𝑅) = (mulGrp‘𝑅)
17	16, 1	mgpbas 18318	. . . . . . . . . . . . . . . 16 ⊢ 𝐵 = (Base‘(mulGrp‘𝑅))
18	7, 17	ressbas2 15758	. . . . . . . . . . . . . . 15 ⊢ ((𝐵 ∖ { 0 }) ⊆ 𝐵 → (𝐵 ∖ { 0 }) = (Base‘𝐺))
19	15, 18	ax-mp 5	. . . . . . . . . . . . . 14 ⊢ (𝐵 ∖ { 0 }) = (Base‘𝐺)
20		eqid 2610	. . . . . . . . . . . . . 14 ⊢ (0g‘𝐺) = (0g‘𝐺)
21	19, 20	grpidcl 17273	. . . . . . . . . . . . 13 ⊢ (𝐺 ∈ Grp → (0g‘𝐺) ∈ (𝐵 ∖ { 0 }))
22	21	ad2antlr 759	. . . . . . . . . . . 12 ⊢ (((𝑅 ∈ Ring ∧ 𝐺 ∈ Grp) ∧ 𝑥 ∈ (Unit‘𝑅)) → (0g‘𝐺) ∈ (𝐵 ∖ { 0 }))
23		eldifsn 4260	. . . . . . . . . . . 12 ⊢ ((0g‘𝐺) ∈ (𝐵 ∖ { 0 }) ↔ ((0g‘𝐺) ∈ 𝐵 ∧ (0g‘𝐺) ≠ 0 ))
24	22, 23	sylib 207	. . . . . . . . . . 11 ⊢ (((𝑅 ∈ Ring ∧ 𝐺 ∈ Grp) ∧ 𝑥 ∈ (Unit‘𝑅)) → ((0g‘𝐺) ∈ 𝐵 ∧ (0g‘𝐺) ≠ 0 ))
25	24	simprd 478	. . . . . . . . . 10 ⊢ (((𝑅 ∈ Ring ∧ 𝐺 ∈ Grp) ∧ 𝑥 ∈ (Unit‘𝑅)) → (0g‘𝐺) ≠ 0 )
26		simpll 786	. . . . . . . . . . 11 ⊢ (((𝑅 ∈ Ring ∧ 𝐺 ∈ Grp) ∧ 𝑥 ∈ (Unit‘𝑅)) → 𝑅 ∈ Ring)
27	22	eldifad 3552	. . . . . . . . . . 11 ⊢ (((𝑅 ∈ Ring ∧ 𝐺 ∈ Grp) ∧ 𝑥 ∈ (Unit‘𝑅)) → (0g‘𝐺) ∈ 𝐵)
28		simpr 476	. . . . . . . . . . 11 ⊢ (((𝑅 ∈ Ring ∧ 𝐺 ∈ Grp) ∧ 𝑥 ∈ (Unit‘𝑅)) → 𝑥 ∈ (Unit‘𝑅))
29		eqid 2610	. . . . . . . . . . . 12 ⊢ (/r‘𝑅) = (/r‘𝑅)
30		eqid 2610	. . . . . . . . . . . 12 ⊢ (.r‘𝑅) = (.r‘𝑅)
31	1, 2, 29, 30	dvrcan1 18514	. . . . . . . . . . 11 ⊢ ((𝑅 ∈ Ring ∧ (0g‘𝐺) ∈ 𝐵 ∧ 𝑥 ∈ (Unit‘𝑅)) → (((0g‘𝐺)(/r‘𝑅)𝑥)(.r‘𝑅)𝑥) = (0g‘𝐺))
32	26, 27, 28, 31	syl3anc 1318	. . . . . . . . . 10 ⊢ (((𝑅 ∈ Ring ∧ 𝐺 ∈ Grp) ∧ 𝑥 ∈ (Unit‘𝑅)) → (((0g‘𝐺)(/r‘𝑅)𝑥)(.r‘𝑅)𝑥) = (0g‘𝐺))
33	1, 2, 29	dvrcl 18509	. . . . . . . . . . . 12 ⊢ ((𝑅 ∈ Ring ∧ (0g‘𝐺) ∈ 𝐵 ∧ 𝑥 ∈ (Unit‘𝑅)) → ((0g‘𝐺)(/r‘𝑅)𝑥) ∈ 𝐵)
34	26, 27, 28, 33	syl3anc 1318	. . . . . . . . . . 11 ⊢ (((𝑅 ∈ Ring ∧ 𝐺 ∈ Grp) ∧ 𝑥 ∈ (Unit‘𝑅)) → ((0g‘𝐺)(/r‘𝑅)𝑥) ∈ 𝐵)
35	1, 30, 3	ringrz 18411	. . . . . . . . . . 11 ⊢ ((𝑅 ∈ Ring ∧ ((0g‘𝐺)(/r‘𝑅)𝑥) ∈ 𝐵) → (((0g‘𝐺)(/r‘𝑅)𝑥)(.r‘𝑅) 0 ) = 0 )
36	26, 34, 35	syl2anc 691	. . . . . . . . . 10 ⊢ (((𝑅 ∈ Ring ∧ 𝐺 ∈ Grp) ∧ 𝑥 ∈ (Unit‘𝑅)) → (((0g‘𝐺)(/r‘𝑅)𝑥)(.r‘𝑅) 0 ) = 0 )
37	25, 32, 36	3netr4d 2859	. . . . . . . . 9 ⊢ (((𝑅 ∈ Ring ∧ 𝐺 ∈ Grp) ∧ 𝑥 ∈ (Unit‘𝑅)) → (((0g‘𝐺)(/r‘𝑅)𝑥)(.r‘𝑅)𝑥) ≠ (((0g‘𝐺)(/r‘𝑅)𝑥)(.r‘𝑅) 0 ))
38		oveq2 6557	. . . . . . . . . 10 ⊢ (𝑥 = 0 → (((0g‘𝐺)(/r‘𝑅)𝑥)(.r‘𝑅)𝑥) = (((0g‘𝐺)(/r‘𝑅)𝑥)(.r‘𝑅) 0 ))
39	38	necon3i 2814	. . . . . . . . 9 ⊢ ((((0g‘𝐺)(/r‘𝑅)𝑥)(.r‘𝑅)𝑥) ≠ (((0g‘𝐺)(/r‘𝑅)𝑥)(.r‘𝑅) 0 ) → 𝑥 ≠ 0 )
40	37, 39	syl 17	. . . . . . . 8 ⊢ (((𝑅 ∈ Ring ∧ 𝐺 ∈ Grp) ∧ 𝑥 ∈ (Unit‘𝑅)) → 𝑥 ≠ 0 )
41		eldifsn 4260	. . . . . . . 8 ⊢ (𝑥 ∈ (𝐵 ∖ { 0 }) ↔ (𝑥 ∈ 𝐵 ∧ 𝑥 ≠ 0 ))
42	14, 40, 41	sylanbrc 695	. . . . . . 7 ⊢ (((𝑅 ∈ Ring ∧ 𝐺 ∈ Grp) ∧ 𝑥 ∈ (Unit‘𝑅)) → 𝑥 ∈ (𝐵 ∖ { 0 }))
43	42	ex 449	. . . . . 6 ⊢ ((𝑅 ∈ Ring ∧ 𝐺 ∈ Grp) → (𝑥 ∈ (Unit‘𝑅) → 𝑥 ∈ (𝐵 ∖ { 0 })))
44	43	ssrdv 3574	. . . . 5 ⊢ ((𝑅 ∈ Ring ∧ 𝐺 ∈ Grp) → (Unit‘𝑅) ⊆ (𝐵 ∖ { 0 }))
45		eldifi 3694	. . . . . . . . . . 11 ⊢ (𝑥 ∈ (𝐵 ∖ { 0 }) → 𝑥 ∈ 𝐵)
46	45	adantl 481	. . . . . . . . . 10 ⊢ (((𝑅 ∈ Ring ∧ 𝐺 ∈ Grp) ∧ 𝑥 ∈ (𝐵 ∖ { 0 })) → 𝑥 ∈ 𝐵)
47		eqid 2610	. . . . . . . . . . . . 13 ⊢ (invg‘𝐺) = (invg‘𝐺)
48	19, 47	grpinvcl 17290	. . . . . . . . . . . 12 ⊢ ((𝐺 ∈ Grp ∧ 𝑥 ∈ (𝐵 ∖ { 0 })) → ((invg‘𝐺)‘𝑥) ∈ (𝐵 ∖ { 0 }))
49	48	adantll 746	. . . . . . . . . . 11 ⊢ (((𝑅 ∈ Ring ∧ 𝐺 ∈ Grp) ∧ 𝑥 ∈ (𝐵 ∖ { 0 })) → ((invg‘𝐺)‘𝑥) ∈ (𝐵 ∖ { 0 }))
50	49	eldifad 3552	. . . . . . . . . 10 ⊢ (((𝑅 ∈ Ring ∧ 𝐺 ∈ Grp) ∧ 𝑥 ∈ (𝐵 ∖ { 0 })) → ((invg‘𝐺)‘𝑥) ∈ 𝐵)
51		eqid 2610	. . . . . . . . . . 11 ⊢ (∥r‘𝑅) = (∥r‘𝑅)
52	1, 51, 30	dvdsrmul 18471	. . . . . . . . . 10 ⊢ ((𝑥 ∈ 𝐵 ∧ ((invg‘𝐺)‘𝑥) ∈ 𝐵) → 𝑥(∥r‘𝑅)(((invg‘𝐺)‘𝑥)(.r‘𝑅)𝑥))
53	46, 50, 52	syl2anc 691	. . . . . . . . 9 ⊢ (((𝑅 ∈ Ring ∧ 𝐺 ∈ Grp) ∧ 𝑥 ∈ (𝐵 ∖ { 0 })) → 𝑥(∥r‘𝑅)(((invg‘𝐺)‘𝑥)(.r‘𝑅)𝑥))
54		fvex 6113	. . . . . . . . . . . . . 14 ⊢ (Base‘𝑅) ∈ V
55	1, 54	eqeltri 2684	. . . . . . . . . . . . 13 ⊢ 𝐵 ∈ V
56		difexg 4735	. . . . . . . . . . . . 13 ⊢ (𝐵 ∈ V → (𝐵 ∖ { 0 }) ∈ V)
57	16, 30	mgpplusg 18316	. . . . . . . . . . . . . 14 ⊢ (.r‘𝑅) = (+g‘(mulGrp‘𝑅))
58	7, 57	ressplusg 15818	. . . . . . . . . . . . 13 ⊢ ((𝐵 ∖ { 0 }) ∈ V → (.r‘𝑅) = (+g‘𝐺))
59	55, 56, 58	mp2b 10	. . . . . . . . . . . 12 ⊢ (.r‘𝑅) = (+g‘𝐺)
60	19, 59, 20, 47	grplinv 17291	. . . . . . . . . . 11 ⊢ ((𝐺 ∈ Grp ∧ 𝑥 ∈ (𝐵 ∖ { 0 })) → (((invg‘𝐺)‘𝑥)(.r‘𝑅)𝑥) = (0g‘𝐺))
61	60	adantll 746	. . . . . . . . . 10 ⊢ (((𝑅 ∈ Ring ∧ 𝐺 ∈ Grp) ∧ 𝑥 ∈ (𝐵 ∖ { 0 })) → (((invg‘𝐺)‘𝑥)(.r‘𝑅)𝑥) = (0g‘𝐺))
62		eqid 2610	. . . . . . . . . . . . . . 15 ⊢ (1r‘𝑅) = (1r‘𝑅)
63	1, 62	ringidcl 18391	. . . . . . . . . . . . . 14 ⊢ (𝑅 ∈ Ring → (1r‘𝑅) ∈ 𝐵)
64	1, 30, 62	ringlidm 18394	. . . . . . . . . . . . . 14 ⊢ ((𝑅 ∈ Ring ∧ (1r‘𝑅) ∈ 𝐵) → ((1r‘𝑅)(.r‘𝑅)(1r‘𝑅)) = (1r‘𝑅))
65	63, 64	mpdan 699	. . . . . . . . . . . . 13 ⊢ (𝑅 ∈ Ring → ((1r‘𝑅)(.r‘𝑅)(1r‘𝑅)) = (1r‘𝑅))
66	65	adantr 480	. . . . . . . . . . . 12 ⊢ ((𝑅 ∈ Ring ∧ 𝐺 ∈ Grp) → ((1r‘𝑅)(.r‘𝑅)(1r‘𝑅)) = (1r‘𝑅))
67		simpr 476	. . . . . . . . . . . . 13 ⊢ ((𝑅 ∈ Ring ∧ 𝐺 ∈ Grp) → 𝐺 ∈ Grp)
68	2, 62	1unit 18481	. . . . . . . . . . . . . . 15 ⊢ (𝑅 ∈ Ring → (1r‘𝑅) ∈ (Unit‘𝑅))
69	68	adantr 480	. . . . . . . . . . . . . 14 ⊢ ((𝑅 ∈ Ring ∧ 𝐺 ∈ Grp) → (1r‘𝑅) ∈ (Unit‘𝑅))
70	44, 69	sseldd 3569	. . . . . . . . . . . . 13 ⊢ ((𝑅 ∈ Ring ∧ 𝐺 ∈ Grp) → (1r‘𝑅) ∈ (𝐵 ∖ { 0 }))
71	19, 59, 20	grpid 17280	. . . . . . . . . . . . 13 ⊢ ((𝐺 ∈ Grp ∧ (1r‘𝑅) ∈ (𝐵 ∖ { 0 })) → (((1r‘𝑅)(.r‘𝑅)(1r‘𝑅)) = (1r‘𝑅) ↔ (0g‘𝐺) = (1r‘𝑅)))
72	67, 70, 71	syl2anc 691	. . . . . . . . . . . 12 ⊢ ((𝑅 ∈ Ring ∧ 𝐺 ∈ Grp) → (((1r‘𝑅)(.r‘𝑅)(1r‘𝑅)) = (1r‘𝑅) ↔ (0g‘𝐺) = (1r‘𝑅)))
73	66, 72	mpbid 221	. . . . . . . . . . 11 ⊢ ((𝑅 ∈ Ring ∧ 𝐺 ∈ Grp) → (0g‘𝐺) = (1r‘𝑅))
74	73	adantr 480	. . . . . . . . . 10 ⊢ (((𝑅 ∈ Ring ∧ 𝐺 ∈ Grp) ∧ 𝑥 ∈ (𝐵 ∖ { 0 })) → (0g‘𝐺) = (1r‘𝑅))
75	61, 74	eqtrd 2644	. . . . . . . . 9 ⊢ (((𝑅 ∈ Ring ∧ 𝐺 ∈ Grp) ∧ 𝑥 ∈ (𝐵 ∖ { 0 })) → (((invg‘𝐺)‘𝑥)(.r‘𝑅)𝑥) = (1r‘𝑅))
76	53, 75	breqtrd 4609	. . . . . . . 8 ⊢ (((𝑅 ∈ Ring ∧ 𝐺 ∈ Grp) ∧ 𝑥 ∈ (𝐵 ∖ { 0 })) → 𝑥(∥r‘𝑅)(1r‘𝑅))
77		eqid 2610	. . . . . . . . . . . 12 ⊢ (oppr‘𝑅) = (oppr‘𝑅)
78	77, 1	opprbas 18452	. . . . . . . . . . 11 ⊢ 𝐵 = (Base‘(oppr‘𝑅))
79		eqid 2610	. . . . . . . . . . 11 ⊢ (∥r‘(oppr‘𝑅)) = (∥r‘(oppr‘𝑅))
80		eqid 2610	. . . . . . . . . . 11 ⊢ (.r‘(oppr‘𝑅)) = (.r‘(oppr‘𝑅))
81	78, 79, 80	dvdsrmul 18471	. . . . . . . . . 10 ⊢ ((𝑥 ∈ 𝐵 ∧ ((invg‘𝐺)‘𝑥) ∈ 𝐵) → 𝑥(∥r‘(oppr‘𝑅))(((invg‘𝐺)‘𝑥)(.r‘(oppr‘𝑅))𝑥))
82	46, 50, 81	syl2anc 691	. . . . . . . . 9 ⊢ (((𝑅 ∈ Ring ∧ 𝐺 ∈ Grp) ∧ 𝑥 ∈ (𝐵 ∖ { 0 })) → 𝑥(∥r‘(oppr‘𝑅))(((invg‘𝐺)‘𝑥)(.r‘(oppr‘𝑅))𝑥))
83	1, 30, 77, 80	opprmul 18449	. . . . . . . . . 10 ⊢ (((invg‘𝐺)‘𝑥)(.r‘(oppr‘𝑅))𝑥) = (𝑥(.r‘𝑅)((invg‘𝐺)‘𝑥))
84	19, 59, 20, 47	grprinv 17292	. . . . . . . . . . . 12 ⊢ ((𝐺 ∈ Grp ∧ 𝑥 ∈ (𝐵 ∖ { 0 })) → (𝑥(.r‘𝑅)((invg‘𝐺)‘𝑥)) = (0g‘𝐺))
85	84	adantll 746	. . . . . . . . . . 11 ⊢ (((𝑅 ∈ Ring ∧ 𝐺 ∈ Grp) ∧ 𝑥 ∈ (𝐵 ∖ { 0 })) → (𝑥(.r‘𝑅)((invg‘𝐺)‘𝑥)) = (0g‘𝐺))
86	85, 74	eqtrd 2644	. . . . . . . . . 10 ⊢ (((𝑅 ∈ Ring ∧ 𝐺 ∈ Grp) ∧ 𝑥 ∈ (𝐵 ∖ { 0 })) → (𝑥(.r‘𝑅)((invg‘𝐺)‘𝑥)) = (1r‘𝑅))
87	83, 86	syl5eq 2656	. . . . . . . . 9 ⊢ (((𝑅 ∈ Ring ∧ 𝐺 ∈ Grp) ∧ 𝑥 ∈ (𝐵 ∖ { 0 })) → (((invg‘𝐺)‘𝑥)(.r‘(oppr‘𝑅))𝑥) = (1r‘𝑅))
88	82, 87	breqtrd 4609	. . . . . . . 8 ⊢ (((𝑅 ∈ Ring ∧ 𝐺 ∈ Grp) ∧ 𝑥 ∈ (𝐵 ∖ { 0 })) → 𝑥(∥r‘(oppr‘𝑅))(1r‘𝑅))
89	2, 62, 51, 77, 79	isunit 18480	. . . . . . . 8 ⊢ (𝑥 ∈ (Unit‘𝑅) ↔ (𝑥(∥r‘𝑅)(1r‘𝑅) ∧ 𝑥(∥r‘(oppr‘𝑅))(1r‘𝑅)))
90	76, 88, 89	sylanbrc 695	. . . . . . 7 ⊢ (((𝑅 ∈ Ring ∧ 𝐺 ∈ Grp) ∧ 𝑥 ∈ (𝐵 ∖ { 0 })) → 𝑥 ∈ (Unit‘𝑅))
91	90	ex 449	. . . . . 6 ⊢ ((𝑅 ∈ Ring ∧ 𝐺 ∈ Grp) → (𝑥 ∈ (𝐵 ∖ { 0 }) → 𝑥 ∈ (Unit‘𝑅)))
92	91	ssrdv 3574	. . . . 5 ⊢ ((𝑅 ∈ Ring ∧ 𝐺 ∈ Grp) → (𝐵 ∖ { 0 }) ⊆ (Unit‘𝑅))
93	44, 92	eqssd 3585	. . . 4 ⊢ ((𝑅 ∈ Ring ∧ 𝐺 ∈ Grp) → (Unit‘𝑅) = (𝐵 ∖ { 0 }))
94	12, 93	impbida 873	. . 3 ⊢ (𝑅 ∈ Ring → ((Unit‘𝑅) = (𝐵 ∖ { 0 }) ↔ 𝐺 ∈ Grp))
95	94	pm5.32i 667	. 2 ⊢ ((𝑅 ∈ Ring ∧ (Unit‘𝑅) = (𝐵 ∖ { 0 })) ↔ (𝑅 ∈ Ring ∧ 𝐺 ∈ Grp))
96	4, 95	bitri 263	1 ⊢ (𝑅 ∈ DivRing ↔ (𝑅 ∈ Ring ∧ 𝐺 ∈ Grp))

Syntax hints:   ↔ wb 195   ∧ wa 383   = wceq 1475   ∈ wcel 1977   ≠ wne 2780  Vcvv 3173   ∖ cdif 3537   ⊆ wss 3540  {csn 4125   class class class wbr 4583  ‘cfv 5804  (class class class)co 6549  Basecbs 15695   ↾_s cress 15696  +_gcplusg 15768  ._rcmulr 15769  0_gc0g 15923  Grpcgrp 17245  inv_gcminusg 17246  mulGrpcmgp 18312  1_rcur 18324  Ringcrg 18370  opp_rcoppr 18445  ∥_rcdsr 18461  Unitcui 18462  /_rcdvr 18505  DivRingcdr 18570

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-cnex 9871  ax-resscn 9872  ax-1cn 9873  ax-icn 9874  ax-addcl 9875  ax-addrcl 9876  ax-mulcl 9877  ax-mulrcl 9878  ax-mulcom 9879  ax-addass 9880  ax-mulass 9881  ax-distr 9882  ax-i2m1 9883  ax-1ne0 9884  ax-1rid 9885  ax-rnegex 9886  ax-rrecex 9887  ax-cnre 9888  ax-pre-lttri 9889  ax-pre-lttrn 9890  ax-pre-ltadd 9891  ax-pre-mulgt0 9892

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-nel 2783  df-ral 2901  df-rex 2902  df-reu 2903  df-rmo 2904  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-riota 6511  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-om 6958  df-1st 7059  df-2nd 7060  df-tpos 7239  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-er 7629  df-en 7842  df-dom 7843  df-sdom 7844  df-pnf 9955  df-mnf 9956  df-xr 9957  df-ltxr 9958  df-le 9959  df-sub 10147  df-neg 10148  df-nn 10898  df-2 10956  df-3 10957  df-ndx 15698  df-slot 15699  df-base 15700  df-sets 15701  df-ress 15702  df-plusg 15781  df-mulr 15782  df-0g 15925  df-mgm 17065  df-sgrp 17107  df-mnd 17118  df-grp 17248  df-minusg 17249  df-mgp 18313  df-ur 18325  df-ring 18372  df-oppr 18446  df-dvdsr 18464  df-unit 18465  df-invr 18495  df-dvr 18506  df-drng 18572

This theorem is referenced by:  drngmgp  18582  isdrngd  18595