Theorem isdrng2 20711

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 20701	. 2 ⊢ (𝑅 ∈ DivRing ↔ (𝑅 ∈ Ring ∧ (Unit‘𝑅) = (𝐵 ∖ { 0 })))
5		oveq2 7368	. . . . . . 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 20354	. . . . . 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 20346	. . . . . . . . 9 ⊢ (𝑥 ∈ (Unit‘𝑅) → 𝑥 ∈ 𝐵)
14	13	adantl 481	. . . . . . . 8 ⊢ (((𝑅 ∈ Ring ∧ 𝐺 ∈ Grp) ∧ 𝑥 ∈ (Unit‘𝑅)) → 𝑥 ∈ 𝐵)
15		difss 4077	. . . . . . . . . . . . . . 15 ⊢ (𝐵 ∖ { 0 }) ⊆ 𝐵
16		eqid 2737	. . . . . . . . . . . . . . . . 17 ⊢ (mulGrp‘𝑅) = (mulGrp‘𝑅)
17	16, 1	mgpbas 20117	. . . . . . . . . . . . . . . 16 ⊢ 𝐵 = (Base‘(mulGrp‘𝑅))
18	7, 17	ressbas2 17199	. . . . . . . . . . . . . . 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 18932	. . . . . . . . . . . . 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 4730	. . . . . . . . . . . 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 3902	. . . . . . . . . . 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 20380	. . . . . . . . . . 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 20375	. . . . . . . . . . . 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 20266	. . . . . . . . . . 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 7368	. . . . . . . . . 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 4730	. . . . . . . 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 3928	. . . . 5 ⊢ ((𝑅 ∈ Ring ∧ 𝐺 ∈ Grp) → (Unit‘𝑅) ⊆ (𝐵 ∖ { 0 }))
45		eldifi 4072	. . . . . . . . 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 18954	. . . . . . . . . 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 3902	. . . . . . . 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 20335	. . . . . . . 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 6848	. . . . . . . . . . 11 ⊢ 𝐵 ∈ V
55		difexg 5266	. . . . . . . . . . 11 ⊢ (𝐵 ∈ V → (𝐵 ∖ { 0 }) ∈ V)
56	16, 30	mgpplusg 20116	. . . . . . . . . . . 12 ⊢ (.r‘𝑅) = (+g‘(mulGrp‘𝑅))
57	7, 56	ressplusg 17245	. . . . . . . . . . 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 18956	. . . . . . . . 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 20237	. . . . . . . . . . . 12 ⊢ (𝑅 ∈ Ring → (1r‘𝑅) ∈ 𝐵)
63	1, 30, 61	ringlidm 20241	. . . . . . . . . . . 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 20345	. . . . . . . . . . . . 13 ⊢ (𝑅 ∈ Ring → (1r‘𝑅) ∈ (Unit‘𝑅))
68	67	adantr 480	. . . . . . . . . . . 12 ⊢ ((𝑅 ∈ Ring ∧ 𝐺 ∈ Grp) → (1r‘𝑅) ∈ (Unit‘𝑅))
69	44, 68	sseldd 3923	. . . . . . . . . . 11 ⊢ ((𝑅 ∈ Ring ∧ 𝐺 ∈ Grp) → (1r‘𝑅) ∈ (𝐵 ∖ { 0 }))
70	19, 58, 20	grpid 18942	. . . . . . . . . . 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 5112	. . . . . 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 20314	. . . . . . . . 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 20335	. . . . . . . 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 20311	. . . . . . . 8 ⊢ (((invg‘𝐺)‘𝑥)(.r‘(oppr‘𝑅))𝑥) = (𝑥(.r‘𝑅)((invg‘𝐺)‘𝑥))
83	19, 58, 20, 47	grprinv 18957	. . . . . . . . . 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 5112	. . . . . 6 ⊢ (((𝑅 ∈ Ring ∧ 𝐺 ∈ Grp) ∧ 𝑥 ∈ (𝐵 ∖ { 0 })) → 𝑥(∥r‘(oppr‘𝑅))(1r‘𝑅))
88	2, 61, 51, 76, 78	isunit 20344	. . . . . 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 3944	. . . 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 3430   ∖ cdif 3887   ⊆ wss 3890  {csn 4568   class class class wbr 5086  ‘cfv 6492  (class class class)co 7360  Basecbs 17170   ↾_s cress 17191  +_gcplusg 17211  ._rcmulr 17212  0_gc0g 17393  Grpcgrp 18900  inv_gcminusg 18901  mulGrpcmgp 20112  1_rcur 20153  Ringcrg 20205  opp_rcoppr 20307  ∥_rcdsr 20325  Unitcui 20326  /_rcdvr 20371  DivRingcdr 20697

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 5212  ax-sep 5231  ax-nul 5241  ax-pow 5302  ax-pr 5370  ax-un 7682  ax-cnex 11085  ax-resscn 11086  ax-1cn 11087  ax-icn 11088  ax-addcl 11089  ax-addrcl 11090  ax-mulcl 11091  ax-mulrcl 11092  ax-mulcom 11093  ax-addass 11094  ax-mulass 11095  ax-distr 11096  ax-i2m1 11097  ax-1ne0 11098  ax-1rid 11099  ax-rnegex 11100  ax-rrecex 11101  ax-cnre 11102  ax-pre-lttri 11103  ax-pre-lttrn 11104  ax-pre-ltadd 11105  ax-pre-mulgt0 11106

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 3343  df-reu 3344  df-rab 3391  df-v 3432  df-sbc 3730  df-csb 3839  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-pss 3910  df-nul 4275  df-if 4468  df-pw 4544  df-sn 4569  df-pr 4571  df-op 4575  df-uni 4852  df-iun 4936  df-br 5087  df-opab 5149  df-mpt 5168  df-tr 5194  df-id 5519  df-eprel 5524  df-po 5532  df-so 5533  df-fr 5577  df-we 5579  df-xp 5630  df-rel 5631  df-cnv 5632  df-co 5633  df-dm 5634  df-rn 5635  df-res 5636  df-ima 5637  df-pred 6259  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6494  df-fn 6495  df-f 6496  df-f1 6497  df-fo 6498  df-f1o 6499  df-fv 6500  df-riota 7317  df-ov 7363  df-oprab 7364  df-mpo 7365  df-om 7811  df-1st 7935  df-2nd 7936  df-tpos 8169  df-frecs 8224  df-wrecs 8255  df-recs 8304  df-rdg 8342  df-er 8636  df-en 8887  df-dom 8888  df-sdom 8889  df-pnf 11172  df-mnf 11173  df-xr 11174  df-ltxr 11175  df-le 11176  df-sub 11370  df-neg 11371  df-nn 12166  df-2 12235  df-3 12236  df-sets 17125  df-slot 17143  df-ndx 17155  df-base 17171  df-ress 17192  df-plusg 17224  df-mulr 17225  df-0g 17395  df-mgm 18599  df-sgrp 18678  df-mnd 18694  df-grp 18903  df-minusg 18904  df-cmn 19748  df-abl 19749  df-mgp 20113  df-rng 20125  df-ur 20154  df-ring 20207  df-oppr 20308  df-dvdsr 20328  df-unit 20329  df-invr 20359  df-dvr 20372  df-drng 20699

This theorem is referenced by:  drngmgp  20713  isdrngd  20733  isdrngdOLD  20735  subdrgint  20771