Theorem isdrng2 20688

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 2737	. . 3 ⊢ (Unit‘𝑅) = (Unit‘𝑅)
3		isdrng2.z	. . 3 ⊢ 0 = (0g‘𝑅)
4	1, 2, 3	isdrng 20678	. 2 ⊢ (𝑅 ∈ DivRing ↔ (𝑅 ∈ Ring ∧ (Unit‘𝑅) = (𝐵 ∖ { 0 })))
5		oveq2 7376	. . . . . . 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	eqtr4di 2790	. . . . 5 ⊢ ((𝑅 ∈ Ring ∧ (Unit‘𝑅) = (𝐵 ∖ { 0 })) → ((mulGrp‘𝑅) ↾s (Unit‘𝑅)) = 𝐺)
9		eqid 2737	. . . . . . 7 ⊢ ((mulGrp‘𝑅) ↾s (Unit‘𝑅)) = ((mulGrp‘𝑅) ↾s (Unit‘𝑅))
10	2, 9	unitgrp 20331	. . . . . 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 2838	. . . 4 ⊢ ((𝑅 ∈ Ring ∧ (Unit‘𝑅) = (𝐵 ∖ { 0 })) → 𝐺 ∈ Grp)
13	1, 2	unitcl 20323	. . . . . . . . 9 ⊢ (𝑥 ∈ (Unit‘𝑅) → 𝑥 ∈ 𝐵)
14	13	adantl 481	. . . . . . . 8 ⊢ (((𝑅 ∈ Ring ∧ 𝐺 ∈ Grp) ∧ 𝑥 ∈ (Unit‘𝑅)) → 𝑥 ∈ 𝐵)
15		difss 4090	. . . . . . . . . . . . . . 15 ⊢ (𝐵 ∖ { 0 }) ⊆ 𝐵
16		eqid 2737	. . . . . . . . . . . . . . . . 17 ⊢ (mulGrp‘𝑅) = (mulGrp‘𝑅)
17	16, 1	mgpbas 20092	. . . . . . . . . . . . . . . 16 ⊢ 𝐵 = (Base‘(mulGrp‘𝑅))
18	7, 17	ressbas2 17177	. . . . . . . . . . . . . . 15 ⊢ ((𝐵 ∖ { 0 }) ⊆ 𝐵 → (𝐵 ∖ { 0 }) = (Base‘𝐺))
19	15, 18	ax-mp 5	. . . . . . . . . . . . . 14 ⊢ (𝐵 ∖ { 0 }) = (Base‘𝐺)
20		eqid 2737	. . . . . . . . . . . . . 14 ⊢ (0g‘𝐺) = (0g‘𝐺)
21	19, 20	grpidcl 18907	. . . . . . . . . . . . 13 ⊢ (𝐺 ∈ Grp → (0g‘𝐺) ∈ (𝐵 ∖ { 0 }))
22	21	ad2antlr 728	. . . . . . . . . . . 12 ⊢ (((𝑅 ∈ Ring ∧ 𝐺 ∈ Grp) ∧ 𝑥 ∈ (Unit‘𝑅)) → (0g‘𝐺) ∈ (𝐵 ∖ { 0 }))
23		eldifsn 4744	. . . . . . . . . . . 12 ⊢ ((0g‘𝐺) ∈ (𝐵 ∖ { 0 }) ↔ ((0g‘𝐺) ∈ 𝐵 ∧ (0g‘𝐺) ≠ 0 ))
24	22, 23	sylib 218	. . . . . . . . . . 11 ⊢ (((𝑅 ∈ Ring ∧ 𝐺 ∈ Grp) ∧ 𝑥 ∈ (Unit‘𝑅)) → ((0g‘𝐺) ∈ 𝐵 ∧ (0g‘𝐺) ≠ 0 ))
25	24	simprd 495	. . . . . . . . . 10 ⊢ (((𝑅 ∈ Ring ∧ 𝐺 ∈ Grp) ∧ 𝑥 ∈ (Unit‘𝑅)) → (0g‘𝐺) ≠ 0 )
26		simpll 767	. . . . . . . . . . 11 ⊢ (((𝑅 ∈ Ring ∧ 𝐺 ∈ Grp) ∧ 𝑥 ∈ (Unit‘𝑅)) → 𝑅 ∈ Ring)
27	22	eldifad 3915	. . . . . . . . . . 11 ⊢ (((𝑅 ∈ Ring ∧ 𝐺 ∈ Grp) ∧ 𝑥 ∈ (Unit‘𝑅)) → (0g‘𝐺) ∈ 𝐵)
28		simpr 484	. . . . . . . . . . 11 ⊢ (((𝑅 ∈ Ring ∧ 𝐺 ∈ Grp) ∧ 𝑥 ∈ (Unit‘𝑅)) → 𝑥 ∈ (Unit‘𝑅))
29		eqid 2737	. . . . . . . . . . . 12 ⊢ (/r‘𝑅) = (/r‘𝑅)
30		eqid 2737	. . . . . . . . . . . 12 ⊢ (.r‘𝑅) = (.r‘𝑅)
31	1, 2, 29, 30	dvrcan1 20357	. . . . . . . . . . 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 1374	. . . . . . . . . 10 ⊢ (((𝑅 ∈ Ring ∧ 𝐺 ∈ Grp) ∧ 𝑥 ∈ (Unit‘𝑅)) → (((0g‘𝐺)(/r‘𝑅)𝑥)(.r‘𝑅)𝑥) = (0g‘𝐺))
33	1, 2, 29	dvrcl 20352	. . . . . . . . . . . 12 ⊢ ((𝑅 ∈ Ring ∧ (0g‘𝐺) ∈ 𝐵 ∧ 𝑥 ∈ (Unit‘𝑅)) → ((0g‘𝐺)(/r‘𝑅)𝑥) ∈ 𝐵)
34	26, 27, 28, 33	syl3anc 1374	. . . . . . . . . . 11 ⊢ (((𝑅 ∈ Ring ∧ 𝐺 ∈ Grp) ∧ 𝑥 ∈ (Unit‘𝑅)) → ((0g‘𝐺)(/r‘𝑅)𝑥) ∈ 𝐵)
35	1, 30, 3	ringrz 20241	. . . . . . . . . . 11 ⊢ ((𝑅 ∈ Ring ∧ ((0g‘𝐺)(/r‘𝑅)𝑥) ∈ 𝐵) → (((0g‘𝐺)(/r‘𝑅)𝑥)(.r‘𝑅) 0 ) = 0 )
36	26, 34, 35	syl2anc 585	. . . . . . . . . 10 ⊢ (((𝑅 ∈ Ring ∧ 𝐺 ∈ Grp) ∧ 𝑥 ∈ (Unit‘𝑅)) → (((0g‘𝐺)(/r‘𝑅)𝑥)(.r‘𝑅) 0 ) = 0 )
37	25, 32, 36	3netr4d 3010	. . . . . . . . 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 7376	. . . . . . . . . 10 ⊢ (𝑥 = 0 → (((0g‘𝐺)(/r‘𝑅)𝑥)(.r‘𝑅)𝑥) = (((0g‘𝐺)(/r‘𝑅)𝑥)(.r‘𝑅) 0 ))
39	38	necon3i 2965	. . . . . . . . 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 4744	. . . . . . . 8 ⊢ (𝑥 ∈ (𝐵 ∖ { 0 }) ↔ (𝑥 ∈ 𝐵 ∧ 𝑥 ≠ 0 ))
42	14, 40, 41	sylanbrc 584	. . . . . . 7 ⊢ (((𝑅 ∈ Ring ∧ 𝐺 ∈ Grp) ∧ 𝑥 ∈ (Unit‘𝑅)) → 𝑥 ∈ (𝐵 ∖ { 0 }))
43	42	ex 412	. . . . . 6 ⊢ ((𝑅 ∈ Ring ∧ 𝐺 ∈ Grp) → (𝑥 ∈ (Unit‘𝑅) → 𝑥 ∈ (𝐵 ∖ { 0 })))
44	43	ssrdv 3941	. . . . 5 ⊢ ((𝑅 ∈ Ring ∧ 𝐺 ∈ Grp) → (Unit‘𝑅) ⊆ (𝐵 ∖ { 0 }))
45		eldifi 4085	. . . . . . . . 9 ⊢ (𝑥 ∈ (𝐵 ∖ { 0 }) → 𝑥 ∈ 𝐵)
46	45	adantl 481	. . . . . . . 8 ⊢ (((𝑅 ∈ Ring ∧ 𝐺 ∈ Grp) ∧ 𝑥 ∈ (𝐵 ∖ { 0 })) → 𝑥 ∈ 𝐵)
47		eqid 2737	. . . . . . . . . . 11 ⊢ (invg‘𝐺) = (invg‘𝐺)
48	19, 47	grpinvcl 18929	. . . . . . . . . 10 ⊢ ((𝐺 ∈ Grp ∧ 𝑥 ∈ (𝐵 ∖ { 0 })) → ((invg‘𝐺)‘𝑥) ∈ (𝐵 ∖ { 0 }))
49	48	adantll 715	. . . . . . . . 9 ⊢ (((𝑅 ∈ Ring ∧ 𝐺 ∈ Grp) ∧ 𝑥 ∈ (𝐵 ∖ { 0 })) → ((invg‘𝐺)‘𝑥) ∈ (𝐵 ∖ { 0 }))
50	49	eldifad 3915	. . . . . . . 8 ⊢ (((𝑅 ∈ Ring ∧ 𝐺 ∈ Grp) ∧ 𝑥 ∈ (𝐵 ∖ { 0 })) → ((invg‘𝐺)‘𝑥) ∈ 𝐵)
51		eqid 2737	. . . . . . . . 9 ⊢ (∥r‘𝑅) = (∥r‘𝑅)
52	1, 51, 30	dvdsrmul 20312	. . . . . . . 8 ⊢ ((𝑥 ∈ 𝐵 ∧ ((invg‘𝐺)‘𝑥) ∈ 𝐵) → 𝑥(∥r‘𝑅)(((invg‘𝐺)‘𝑥)(.r‘𝑅)𝑥))
53	46, 50, 52	syl2anc 585	. . . . . . 7 ⊢ (((𝑅 ∈ Ring ∧ 𝐺 ∈ Grp) ∧ 𝑥 ∈ (𝐵 ∖ { 0 })) → 𝑥(∥r‘𝑅)(((invg‘𝐺)‘𝑥)(.r‘𝑅)𝑥))
54	1	fvexi 6856	. . . . . . . . . . 11 ⊢ 𝐵 ∈ V
55		difexg 5276	. . . . . . . . . . 11 ⊢ (𝐵 ∈ V → (𝐵 ∖ { 0 }) ∈ V)
56	16, 30	mgpplusg 20091	. . . . . . . . . . . 12 ⊢ (.r‘𝑅) = (+g‘(mulGrp‘𝑅))
57	7, 56	ressplusg 17223	. . . . . . . . . . 11 ⊢ ((𝐵 ∖ { 0 }) ∈ V → (.r‘𝑅) = (+g‘𝐺))
58	54, 55, 57	mp2b 10	. . . . . . . . . 10 ⊢ (.r‘𝑅) = (+g‘𝐺)
59	19, 58, 20, 47	grplinv 18931	. . . . . . . . 9 ⊢ ((𝐺 ∈ Grp ∧ 𝑥 ∈ (𝐵 ∖ { 0 })) → (((invg‘𝐺)‘𝑥)(.r‘𝑅)𝑥) = (0g‘𝐺))
60	59	adantll 715	. . . . . . . 8 ⊢ (((𝑅 ∈ Ring ∧ 𝐺 ∈ Grp) ∧ 𝑥 ∈ (𝐵 ∖ { 0 })) → (((invg‘𝐺)‘𝑥)(.r‘𝑅)𝑥) = (0g‘𝐺))
61		eqid 2737	. . . . . . . . . . . . 13 ⊢ (1r‘𝑅) = (1r‘𝑅)
62	1, 61	ringidcl 20212	. . . . . . . . . . . 12 ⊢ (𝑅 ∈ Ring → (1r‘𝑅) ∈ 𝐵)
63	1, 30, 61	ringlidm 20216	. . . . . . . . . . . 12 ⊢ ((𝑅 ∈ Ring ∧ (1r‘𝑅) ∈ 𝐵) → ((1r‘𝑅)(.r‘𝑅)(1r‘𝑅)) = (1r‘𝑅))
64	62, 63	mpdan 688	. . . . . . . . . . 11 ⊢ (𝑅 ∈ Ring → ((1r‘𝑅)(.r‘𝑅)(1r‘𝑅)) = (1r‘𝑅))
65	64	adantr 480	. . . . . . . . . 10 ⊢ ((𝑅 ∈ Ring ∧ 𝐺 ∈ Grp) → ((1r‘𝑅)(.r‘𝑅)(1r‘𝑅)) = (1r‘𝑅))
66		simpr 484	. . . . . . . . . . 11 ⊢ ((𝑅 ∈ Ring ∧ 𝐺 ∈ Grp) → 𝐺 ∈ Grp)
67	2, 61	1unit 20322	. . . . . . . . . . . . 13 ⊢ (𝑅 ∈ Ring → (1r‘𝑅) ∈ (Unit‘𝑅))
68	67	adantr 480	. . . . . . . . . . . 12 ⊢ ((𝑅 ∈ Ring ∧ 𝐺 ∈ Grp) → (1r‘𝑅) ∈ (Unit‘𝑅))
69	44, 68	sseldd 3936	. . . . . . . . . . 11 ⊢ ((𝑅 ∈ Ring ∧ 𝐺 ∈ Grp) → (1r‘𝑅) ∈ (𝐵 ∖ { 0 }))
70	19, 58, 20	grpid 18917	. . . . . . . . . . 11 ⊢ ((𝐺 ∈ Grp ∧ (1r‘𝑅) ∈ (𝐵 ∖ { 0 })) → (((1r‘𝑅)(.r‘𝑅)(1r‘𝑅)) = (1r‘𝑅) ↔ (0g‘𝐺) = (1r‘𝑅)))
71	66, 69, 70	syl2anc 585	. . . . . . . . . 10 ⊢ ((𝑅 ∈ Ring ∧ 𝐺 ∈ Grp) → (((1r‘𝑅)(.r‘𝑅)(1r‘𝑅)) = (1r‘𝑅) ↔ (0g‘𝐺) = (1r‘𝑅)))
72	65, 71	mpbid 232	. . . . . . . . 9 ⊢ ((𝑅 ∈ Ring ∧ 𝐺 ∈ Grp) → (0g‘𝐺) = (1r‘𝑅))
73	72	adantr 480	. . . . . . . 8 ⊢ (((𝑅 ∈ Ring ∧ 𝐺 ∈ Grp) ∧ 𝑥 ∈ (𝐵 ∖ { 0 })) → (0g‘𝐺) = (1r‘𝑅))
74	60, 73	eqtrd 2772	. . . . . . 7 ⊢ (((𝑅 ∈ Ring ∧ 𝐺 ∈ Grp) ∧ 𝑥 ∈ (𝐵 ∖ { 0 })) → (((invg‘𝐺)‘𝑥)(.r‘𝑅)𝑥) = (1r‘𝑅))
75	53, 74	breqtrd 5126	. . . . . 6 ⊢ (((𝑅 ∈ Ring ∧ 𝐺 ∈ Grp) ∧ 𝑥 ∈ (𝐵 ∖ { 0 })) → 𝑥(∥r‘𝑅)(1r‘𝑅))
76		eqid 2737	. . . . . . . . . 10 ⊢ (oppr‘𝑅) = (oppr‘𝑅)
77	76, 1	opprbas 20291	. . . . . . . . 9 ⊢ 𝐵 = (Base‘(oppr‘𝑅))
78		eqid 2737	. . . . . . . . 9 ⊢ (∥r‘(oppr‘𝑅)) = (∥r‘(oppr‘𝑅))
79		eqid 2737	. . . . . . . . 9 ⊢ (.r‘(oppr‘𝑅)) = (.r‘(oppr‘𝑅))
80	77, 78, 79	dvdsrmul 20312	. . . . . . . 8 ⊢ ((𝑥 ∈ 𝐵 ∧ ((invg‘𝐺)‘𝑥) ∈ 𝐵) → 𝑥(∥r‘(oppr‘𝑅))(((invg‘𝐺)‘𝑥)(.r‘(oppr‘𝑅))𝑥))
81	46, 50, 80	syl2anc 585	. . . . . . 7 ⊢ (((𝑅 ∈ Ring ∧ 𝐺 ∈ Grp) ∧ 𝑥 ∈ (𝐵 ∖ { 0 })) → 𝑥(∥r‘(oppr‘𝑅))(((invg‘𝐺)‘𝑥)(.r‘(oppr‘𝑅))𝑥))
82	1, 30, 76, 79	opprmul 20288	. . . . . . . 8 ⊢ (((invg‘𝐺)‘𝑥)(.r‘(oppr‘𝑅))𝑥) = (𝑥(.r‘𝑅)((invg‘𝐺)‘𝑥))
83	19, 58, 20, 47	grprinv 18932	. . . . . . . . . 10 ⊢ ((𝐺 ∈ Grp ∧ 𝑥 ∈ (𝐵 ∖ { 0 })) → (𝑥(.r‘𝑅)((invg‘𝐺)‘𝑥)) = (0g‘𝐺))
84	83	adantll 715	. . . . . . . . 9 ⊢ (((𝑅 ∈ Ring ∧ 𝐺 ∈ Grp) ∧ 𝑥 ∈ (𝐵 ∖ { 0 })) → (𝑥(.r‘𝑅)((invg‘𝐺)‘𝑥)) = (0g‘𝐺))
85	84, 73	eqtrd 2772	. . . . . . . 8 ⊢ (((𝑅 ∈ Ring ∧ 𝐺 ∈ Grp) ∧ 𝑥 ∈ (𝐵 ∖ { 0 })) → (𝑥(.r‘𝑅)((invg‘𝐺)‘𝑥)) = (1r‘𝑅))
86	82, 85	eqtrid 2784	. . . . . . 7 ⊢ (((𝑅 ∈ Ring ∧ 𝐺 ∈ Grp) ∧ 𝑥 ∈ (𝐵 ∖ { 0 })) → (((invg‘𝐺)‘𝑥)(.r‘(oppr‘𝑅))𝑥) = (1r‘𝑅))
87	81, 86	breqtrd 5126	. . . . . 6 ⊢ (((𝑅 ∈ Ring ∧ 𝐺 ∈ Grp) ∧ 𝑥 ∈ (𝐵 ∖ { 0 })) → 𝑥(∥r‘(oppr‘𝑅))(1r‘𝑅))
88	2, 61, 51, 76, 78	isunit 20321	. . . . . 6 ⊢ (𝑥 ∈ (Unit‘𝑅) ↔ (𝑥(∥r‘𝑅)(1r‘𝑅) ∧ 𝑥(∥r‘(oppr‘𝑅))(1r‘𝑅)))
89	75, 87, 88	sylanbrc 584	. . . . 5 ⊢ (((𝑅 ∈ Ring ∧ 𝐺 ∈ Grp) ∧ 𝑥 ∈ (𝐵 ∖ { 0 })) → 𝑥 ∈ (Unit‘𝑅))
90	44, 89	eqelssd 3957	. . . 4 ⊢ ((𝑅 ∈ Ring ∧ 𝐺 ∈ Grp) → (Unit‘𝑅) = (𝐵 ∖ { 0 }))
91	12, 90	impbida 801	. . 3 ⊢ (𝑅 ∈ Ring → ((Unit‘𝑅) = (𝐵 ∖ { 0 }) ↔ 𝐺 ∈ Grp))
92	91	pm5.32i 574	. 2 ⊢ ((𝑅 ∈ Ring ∧ (Unit‘𝑅) = (𝐵 ∖ { 0 })) ↔ (𝑅 ∈ Ring ∧ 𝐺 ∈ Grp))
93	4, 92	bitri 275	1 ⊢ (𝑅 ∈ DivRing ↔ (𝑅 ∈ Ring ∧ 𝐺 ∈ Grp))

Syntax hints:   ↔ wb 206   ∧ wa 395   = wceq 1542   ∈ wcel 2114   ≠ wne 2933  Vcvv 3442   ∖ cdif 3900   ⊆ wss 3903  {csn 4582   class class class wbr 5100  ‘cfv 6500  (class class class)co 7368  Basecbs 17148   ↾_s cress 17169  +_gcplusg 17189  ._rcmulr 17190  0_gc0g 17371  Grpcgrp 18875  inv_gcminusg 18876  mulGrpcmgp 20087  1_rcur 20128  Ringcrg 20180  opp_rcoppr 20284  ∥_rcdsr 20302  Unitcui 20303  /_rcdvr 20348  DivRingcdr 20674

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-rep 5226  ax-sep 5243  ax-nul 5253  ax-pow 5312  ax-pr 5379  ax-un 7690  ax-cnex 11094  ax-resscn 11095  ax-1cn 11096  ax-icn 11097  ax-addcl 11098  ax-addrcl 11099  ax-mulcl 11100  ax-mulrcl 11101  ax-mulcom 11102  ax-addass 11103  ax-mulass 11104  ax-distr 11105  ax-i2m1 11106  ax-1ne0 11107  ax-1rid 11108  ax-rnegex 11109  ax-rrecex 11110  ax-cnre 11111  ax-pre-lttri 11112  ax-pre-lttrn 11113  ax-pre-ltadd 11114  ax-pre-mulgt0 11115

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-nel 3038  df-ral 3053  df-rex 3063  df-rmo 3352  df-reu 3353  df-rab 3402  df-v 3444  df-sbc 3743  df-csb 3852  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-pss 3923  df-nul 4288  df-if 4482  df-pw 4558  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-iun 4950  df-br 5101  df-opab 5163  df-mpt 5182  df-tr 5208  df-id 5527  df-eprel 5532  df-po 5540  df-so 5541  df-fr 5585  df-we 5587  df-xp 5638  df-rel 5639  df-cnv 5640  df-co 5641  df-dm 5642  df-rn 5643  df-res 5644  df-ima 5645  df-pred 6267  df-ord 6328  df-on 6329  df-lim 6330  df-suc 6331  df-iota 6456  df-fun 6502  df-fn 6503  df-f 6504  df-f1 6505  df-fo 6506  df-f1o 6507  df-fv 6508  df-riota 7325  df-ov 7371  df-oprab 7372  df-mpo 7373  df-om 7819  df-1st 7943  df-2nd 7944  df-tpos 8178  df-frecs 8233  df-wrecs 8264  df-recs 8313  df-rdg 8351  df-er 8645  df-en 8896  df-dom 8897  df-sdom 8898  df-pnf 11180  df-mnf 11181  df-xr 11182  df-ltxr 11183  df-le 11184  df-sub 11378  df-neg 11379  df-nn 12158  df-2 12220  df-3 12221  df-sets 17103  df-slot 17121  df-ndx 17133  df-base 17149  df-ress 17170  df-plusg 17202  df-mulr 17203  df-0g 17373  df-mgm 18577  df-sgrp 18656  df-mnd 18672  df-grp 18878  df-minusg 18879  df-cmn 19723  df-abl 19724  df-mgp 20088  df-rng 20100  df-ur 20129  df-ring 20182  df-oppr 20285  df-dvdsr 20305  df-unit 20306  df-invr 20336  df-dvr 20349  df-drng 20676

This theorem is referenced by:  drngmgp  20690  isdrngd  20710  isdrngdOLD  20712  subdrgint  20748