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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 = (0g𝑅)
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.
StepHypRef Expression
1 isdrng2.b . . 3 𝐵 = (Base‘𝑅)
2 eqid 2610 . . 3 (Unit‘𝑅) = (Unit‘𝑅)
3 isdrng2.z . . 3 0 = (0g𝑅)
41, 2, 3isdrng 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 })))
65adantl 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 }))
86, 7syl6eqr 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))
102, 9unitgrp 18490 . . . . . 6 (𝑅 ∈ Ring → ((mulGrp‘𝑅) ↾s (Unit‘𝑅)) ∈ Grp)
1110adantr 480 . . . . 5 ((𝑅 ∈ Ring ∧ (Unit‘𝑅) = (𝐵 ∖ { 0 })) → ((mulGrp‘𝑅) ↾s (Unit‘𝑅)) ∈ Grp)
128, 11eqeltrrd 2689 . . . 4 ((𝑅 ∈ Ring ∧ (Unit‘𝑅) = (𝐵 ∖ { 0 })) → 𝐺 ∈ Grp)
131, 2unitcl 18482 . . . . . . . . 9 (𝑥 ∈ (Unit‘𝑅) → 𝑥𝐵)
1413adantl 481 . . . . . . . 8 (((𝑅 ∈ Ring ∧ 𝐺 ∈ Grp) ∧ 𝑥 ∈ (Unit‘𝑅)) → 𝑥𝐵)
15 difss 3699 . . . . . . . . . . . . . . 15 (𝐵 ∖ { 0 }) ⊆ 𝐵
16 eqid 2610 . . . . . . . . . . . . . . . . 17 (mulGrp‘𝑅) = (mulGrp‘𝑅)
1716, 1mgpbas 18318 . . . . . . . . . . . . . . . 16 𝐵 = (Base‘(mulGrp‘𝑅))
187, 17ressbas2 15758 . . . . . . . . . . . . . . 15 ((𝐵 ∖ { 0 }) ⊆ 𝐵 → (𝐵 ∖ { 0 }) = (Base‘𝐺))
1915, 18ax-mp 5 . . . . . . . . . . . . . 14 (𝐵 ∖ { 0 }) = (Base‘𝐺)
20 eqid 2610 . . . . . . . . . . . . . 14 (0g𝐺) = (0g𝐺)
2119, 20grpidcl 17273 . . . . . . . . . . . . 13 (𝐺 ∈ Grp → (0g𝐺) ∈ (𝐵 ∖ { 0 }))
2221ad2antlr 759 . . . . . . . . . . . 12 (((𝑅 ∈ Ring ∧ 𝐺 ∈ Grp) ∧ 𝑥 ∈ (Unit‘𝑅)) → (0g𝐺) ∈ (𝐵 ∖ { 0 }))
23 eldifsn 4260 . . . . . . . . . . . 12 ((0g𝐺) ∈ (𝐵 ∖ { 0 }) ↔ ((0g𝐺) ∈ 𝐵 ∧ (0g𝐺) ≠ 0 ))
2422, 23sylib 207 . . . . . . . . . . 11 (((𝑅 ∈ Ring ∧ 𝐺 ∈ Grp) ∧ 𝑥 ∈ (Unit‘𝑅)) → ((0g𝐺) ∈ 𝐵 ∧ (0g𝐺) ≠ 0 ))
2524simprd 478 . . . . . . . . . 10 (((𝑅 ∈ Ring ∧ 𝐺 ∈ Grp) ∧ 𝑥 ∈ (Unit‘𝑅)) → (0g𝐺) ≠ 0 )
26 simpll 786 . . . . . . . . . . 11 (((𝑅 ∈ Ring ∧ 𝐺 ∈ Grp) ∧ 𝑥 ∈ (Unit‘𝑅)) → 𝑅 ∈ Ring)
2722eldifad 3552 . . . . . . . . . . 11 (((𝑅 ∈ Ring ∧ 𝐺 ∈ Grp) ∧ 𝑥 ∈ (Unit‘𝑅)) → (0g𝐺) ∈ 𝐵)
28 simpr 476 . . . . . . . . . . 11 (((𝑅 ∈ Ring ∧ 𝐺 ∈ Grp) ∧ 𝑥 ∈ (Unit‘𝑅)) → 𝑥 ∈ (Unit‘𝑅))
29 eqid 2610 . . . . . . . . . . . 12 (/r𝑅) = (/r𝑅)
30 eqid 2610 . . . . . . . . . . . 12 (.r𝑅) = (.r𝑅)
311, 2, 29, 30dvrcan1 18514 . . . . . . . . . . 11 ((𝑅 ∈ Ring ∧ (0g𝐺) ∈ 𝐵𝑥 ∈ (Unit‘𝑅)) → (((0g𝐺)(/r𝑅)𝑥)(.r𝑅)𝑥) = (0g𝐺))
3226, 27, 28, 31syl3anc 1318 . . . . . . . . . 10 (((𝑅 ∈ Ring ∧ 𝐺 ∈ Grp) ∧ 𝑥 ∈ (Unit‘𝑅)) → (((0g𝐺)(/r𝑅)𝑥)(.r𝑅)𝑥) = (0g𝐺))
331, 2, 29dvrcl 18509 . . . . . . . . . . . 12 ((𝑅 ∈ Ring ∧ (0g𝐺) ∈ 𝐵𝑥 ∈ (Unit‘𝑅)) → ((0g𝐺)(/r𝑅)𝑥) ∈ 𝐵)
3426, 27, 28, 33syl3anc 1318 . . . . . . . . . . 11 (((𝑅 ∈ Ring ∧ 𝐺 ∈ Grp) ∧ 𝑥 ∈ (Unit‘𝑅)) → ((0g𝐺)(/r𝑅)𝑥) ∈ 𝐵)
351, 30, 3ringrz 18411 . . . . . . . . . . 11 ((𝑅 ∈ Ring ∧ ((0g𝐺)(/r𝑅)𝑥) ∈ 𝐵) → (((0g𝐺)(/r𝑅)𝑥)(.r𝑅) 0 ) = 0 )
3626, 34, 35syl2anc 691 . . . . . . . . . 10 (((𝑅 ∈ Ring ∧ 𝐺 ∈ Grp) ∧ 𝑥 ∈ (Unit‘𝑅)) → (((0g𝐺)(/r𝑅)𝑥)(.r𝑅) 0 ) = 0 )
3725, 32, 363netr4d 2859 . . . . . . . . 9 (((𝑅 ∈ Ring ∧ 𝐺 ∈ Grp) ∧ 𝑥 ∈ (Unit‘𝑅)) → (((0g𝐺)(/r𝑅)𝑥)(.r𝑅)𝑥) ≠ (((0g𝐺)(/r𝑅)𝑥)(.r𝑅) 0 ))
38 oveq2 6557 . . . . . . . . . 10 (𝑥 = 0 → (((0g𝐺)(/r𝑅)𝑥)(.r𝑅)𝑥) = (((0g𝐺)(/r𝑅)𝑥)(.r𝑅) 0 ))
3938necon3i 2814 . . . . . . . . 9 ((((0g𝐺)(/r𝑅)𝑥)(.r𝑅)𝑥) ≠ (((0g𝐺)(/r𝑅)𝑥)(.r𝑅) 0 ) → 𝑥0 )
4037, 39syl 17 . . . . . . . 8 (((𝑅 ∈ Ring ∧ 𝐺 ∈ Grp) ∧ 𝑥 ∈ (Unit‘𝑅)) → 𝑥0 )
41 eldifsn 4260 . . . . . . . 8 (𝑥 ∈ (𝐵 ∖ { 0 }) ↔ (𝑥𝐵𝑥0 ))
4214, 40, 41sylanbrc 695 . . . . . . 7 (((𝑅 ∈ Ring ∧ 𝐺 ∈ Grp) ∧ 𝑥 ∈ (Unit‘𝑅)) → 𝑥 ∈ (𝐵 ∖ { 0 }))
4342ex 449 . . . . . 6 ((𝑅 ∈ Ring ∧ 𝐺 ∈ Grp) → (𝑥 ∈ (Unit‘𝑅) → 𝑥 ∈ (𝐵 ∖ { 0 })))
4443ssrdv 3574 . . . . 5 ((𝑅 ∈ Ring ∧ 𝐺 ∈ Grp) → (Unit‘𝑅) ⊆ (𝐵 ∖ { 0 }))
45 eldifi 3694 . . . . . . . . . . 11 (𝑥 ∈ (𝐵 ∖ { 0 }) → 𝑥𝐵)
4645adantl 481 . . . . . . . . . 10 (((𝑅 ∈ Ring ∧ 𝐺 ∈ Grp) ∧ 𝑥 ∈ (𝐵 ∖ { 0 })) → 𝑥𝐵)
47 eqid 2610 . . . . . . . . . . . . 13 (invg𝐺) = (invg𝐺)
4819, 47grpinvcl 17290 . . . . . . . . . . . 12 ((𝐺 ∈ Grp ∧ 𝑥 ∈ (𝐵 ∖ { 0 })) → ((invg𝐺)‘𝑥) ∈ (𝐵 ∖ { 0 }))
4948adantll 746 . . . . . . . . . . 11 (((𝑅 ∈ Ring ∧ 𝐺 ∈ Grp) ∧ 𝑥 ∈ (𝐵 ∖ { 0 })) → ((invg𝐺)‘𝑥) ∈ (𝐵 ∖ { 0 }))
5049eldifad 3552 . . . . . . . . . 10 (((𝑅 ∈ Ring ∧ 𝐺 ∈ Grp) ∧ 𝑥 ∈ (𝐵 ∖ { 0 })) → ((invg𝐺)‘𝑥) ∈ 𝐵)
51 eqid 2610 . . . . . . . . . . 11 (∥r𝑅) = (∥r𝑅)
521, 51, 30dvdsrmul 18471 . . . . . . . . . 10 ((𝑥𝐵 ∧ ((invg𝐺)‘𝑥) ∈ 𝐵) → 𝑥(∥r𝑅)(((invg𝐺)‘𝑥)(.r𝑅)𝑥))
5346, 50, 52syl2anc 691 . . . . . . . . 9 (((𝑅 ∈ Ring ∧ 𝐺 ∈ Grp) ∧ 𝑥 ∈ (𝐵 ∖ { 0 })) → 𝑥(∥r𝑅)(((invg𝐺)‘𝑥)(.r𝑅)𝑥))
54 fvex 6113 . . . . . . . . . . . . . 14 (Base‘𝑅) ∈ V
551, 54eqeltri 2684 . . . . . . . . . . . . 13 𝐵 ∈ V
56 difexg 4735 . . . . . . . . . . . . 13 (𝐵 ∈ V → (𝐵 ∖ { 0 }) ∈ V)
5716, 30mgpplusg 18316 . . . . . . . . . . . . . 14 (.r𝑅) = (+g‘(mulGrp‘𝑅))
587, 57ressplusg 15818 . . . . . . . . . . . . 13 ((𝐵 ∖ { 0 }) ∈ V → (.r𝑅) = (+g𝐺))
5955, 56, 58mp2b 10 . . . . . . . . . . . 12 (.r𝑅) = (+g𝐺)
6019, 59, 20, 47grplinv 17291 . . . . . . . . . . 11 ((𝐺 ∈ Grp ∧ 𝑥 ∈ (𝐵 ∖ { 0 })) → (((invg𝐺)‘𝑥)(.r𝑅)𝑥) = (0g𝐺))
6160adantll 746 . . . . . . . . . 10 (((𝑅 ∈ Ring ∧ 𝐺 ∈ Grp) ∧ 𝑥 ∈ (𝐵 ∖ { 0 })) → (((invg𝐺)‘𝑥)(.r𝑅)𝑥) = (0g𝐺))
62 eqid 2610 . . . . . . . . . . . . . . 15 (1r𝑅) = (1r𝑅)
631, 62ringidcl 18391 . . . . . . . . . . . . . 14 (𝑅 ∈ Ring → (1r𝑅) ∈ 𝐵)
641, 30, 62ringlidm 18394 . . . . . . . . . . . . . 14 ((𝑅 ∈ Ring ∧ (1r𝑅) ∈ 𝐵) → ((1r𝑅)(.r𝑅)(1r𝑅)) = (1r𝑅))
6563, 64mpdan 699 . . . . . . . . . . . . 13 (𝑅 ∈ Ring → ((1r𝑅)(.r𝑅)(1r𝑅)) = (1r𝑅))
6665adantr 480 . . . . . . . . . . . 12 ((𝑅 ∈ Ring ∧ 𝐺 ∈ Grp) → ((1r𝑅)(.r𝑅)(1r𝑅)) = (1r𝑅))
67 simpr 476 . . . . . . . . . . . . 13 ((𝑅 ∈ Ring ∧ 𝐺 ∈ Grp) → 𝐺 ∈ Grp)
682, 621unit 18481 . . . . . . . . . . . . . . 15 (𝑅 ∈ Ring → (1r𝑅) ∈ (Unit‘𝑅))
6968adantr 480 . . . . . . . . . . . . . 14 ((𝑅 ∈ Ring ∧ 𝐺 ∈ Grp) → (1r𝑅) ∈ (Unit‘𝑅))
7044, 69sseldd 3569 . . . . . . . . . . . . 13 ((𝑅 ∈ Ring ∧ 𝐺 ∈ Grp) → (1r𝑅) ∈ (𝐵 ∖ { 0 }))
7119, 59, 20grpid 17280 . . . . . . . . . . . . 13 ((𝐺 ∈ Grp ∧ (1r𝑅) ∈ (𝐵 ∖ { 0 })) → (((1r𝑅)(.r𝑅)(1r𝑅)) = (1r𝑅) ↔ (0g𝐺) = (1r𝑅)))
7267, 70, 71syl2anc 691 . . . . . . . . . . . 12 ((𝑅 ∈ Ring ∧ 𝐺 ∈ Grp) → (((1r𝑅)(.r𝑅)(1r𝑅)) = (1r𝑅) ↔ (0g𝐺) = (1r𝑅)))
7366, 72mpbid 221 . . . . . . . . . . 11 ((𝑅 ∈ Ring ∧ 𝐺 ∈ Grp) → (0g𝐺) = (1r𝑅))
7473adantr 480 . . . . . . . . . 10 (((𝑅 ∈ Ring ∧ 𝐺 ∈ Grp) ∧ 𝑥 ∈ (𝐵 ∖ { 0 })) → (0g𝐺) = (1r𝑅))
7561, 74eqtrd 2644 . . . . . . . . 9 (((𝑅 ∈ Ring ∧ 𝐺 ∈ Grp) ∧ 𝑥 ∈ (𝐵 ∖ { 0 })) → (((invg𝐺)‘𝑥)(.r𝑅)𝑥) = (1r𝑅))
7653, 75breqtrd 4609 . . . . . . . 8 (((𝑅 ∈ Ring ∧ 𝐺 ∈ Grp) ∧ 𝑥 ∈ (𝐵 ∖ { 0 })) → 𝑥(∥r𝑅)(1r𝑅))
77 eqid 2610 . . . . . . . . . . . 12 (oppr𝑅) = (oppr𝑅)
7877, 1opprbas 18452 . . . . . . . . . . 11 𝐵 = (Base‘(oppr𝑅))
79 eqid 2610 . . . . . . . . . . 11 (∥r‘(oppr𝑅)) = (∥r‘(oppr𝑅))
80 eqid 2610 . . . . . . . . . . 11 (.r‘(oppr𝑅)) = (.r‘(oppr𝑅))
8178, 79, 80dvdsrmul 18471 . . . . . . . . . 10 ((𝑥𝐵 ∧ ((invg𝐺)‘𝑥) ∈ 𝐵) → 𝑥(∥r‘(oppr𝑅))(((invg𝐺)‘𝑥)(.r‘(oppr𝑅))𝑥))
8246, 50, 81syl2anc 691 . . . . . . . . 9 (((𝑅 ∈ Ring ∧ 𝐺 ∈ Grp) ∧ 𝑥 ∈ (𝐵 ∖ { 0 })) → 𝑥(∥r‘(oppr𝑅))(((invg𝐺)‘𝑥)(.r‘(oppr𝑅))𝑥))
831, 30, 77, 80opprmul 18449 . . . . . . . . . 10 (((invg𝐺)‘𝑥)(.r‘(oppr𝑅))𝑥) = (𝑥(.r𝑅)((invg𝐺)‘𝑥))
8419, 59, 20, 47grprinv 17292 . . . . . . . . . . . 12 ((𝐺 ∈ Grp ∧ 𝑥 ∈ (𝐵 ∖ { 0 })) → (𝑥(.r𝑅)((invg𝐺)‘𝑥)) = (0g𝐺))
8584adantll 746 . . . . . . . . . . 11 (((𝑅 ∈ Ring ∧ 𝐺 ∈ Grp) ∧ 𝑥 ∈ (𝐵 ∖ { 0 })) → (𝑥(.r𝑅)((invg𝐺)‘𝑥)) = (0g𝐺))
8685, 74eqtrd 2644 . . . . . . . . . 10 (((𝑅 ∈ Ring ∧ 𝐺 ∈ Grp) ∧ 𝑥 ∈ (𝐵 ∖ { 0 })) → (𝑥(.r𝑅)((invg𝐺)‘𝑥)) = (1r𝑅))
8783, 86syl5eq 2656 . . . . . . . . 9 (((𝑅 ∈ Ring ∧ 𝐺 ∈ Grp) ∧ 𝑥 ∈ (𝐵 ∖ { 0 })) → (((invg𝐺)‘𝑥)(.r‘(oppr𝑅))𝑥) = (1r𝑅))
8882, 87breqtrd 4609 . . . . . . . 8 (((𝑅 ∈ Ring ∧ 𝐺 ∈ Grp) ∧ 𝑥 ∈ (𝐵 ∖ { 0 })) → 𝑥(∥r‘(oppr𝑅))(1r𝑅))
892, 62, 51, 77, 79isunit 18480 . . . . . . . 8 (𝑥 ∈ (Unit‘𝑅) ↔ (𝑥(∥r𝑅)(1r𝑅) ∧ 𝑥(∥r‘(oppr𝑅))(1r𝑅)))
9076, 88, 89sylanbrc 695 . . . . . . 7 (((𝑅 ∈ Ring ∧ 𝐺 ∈ Grp) ∧ 𝑥 ∈ (𝐵 ∖ { 0 })) → 𝑥 ∈ (Unit‘𝑅))
9190ex 449 . . . . . 6 ((𝑅 ∈ Ring ∧ 𝐺 ∈ Grp) → (𝑥 ∈ (𝐵 ∖ { 0 }) → 𝑥 ∈ (Unit‘𝑅)))
9291ssrdv 3574 . . . . 5 ((𝑅 ∈ Ring ∧ 𝐺 ∈ Grp) → (𝐵 ∖ { 0 }) ⊆ (Unit‘𝑅))
9344, 92eqssd 3585 . . . 4 ((𝑅 ∈ Ring ∧ 𝐺 ∈ Grp) → (Unit‘𝑅) = (𝐵 ∖ { 0 }))
9412, 93impbida 873 . . 3 (𝑅 ∈ Ring → ((Unit‘𝑅) = (𝐵 ∖ { 0 }) ↔ 𝐺 ∈ Grp))
9594pm5.32i 667 . 2 ((𝑅 ∈ Ring ∧ (Unit‘𝑅) = (𝐵 ∖ { 0 })) ↔ (𝑅 ∈ Ring ∧ 𝐺 ∈ Grp))
964, 95bitri 263 1 (𝑅 ∈ DivRing ↔ (𝑅 ∈ Ring ∧ 𝐺 ∈ Grp))
 Colors of variables: wff setvar class 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  0gc0g 15923  Grpcgrp 17245  invgcminusg 17246  mulGrpcmgp 18312  1rcur 18324  Ringcrg 18370  opprcoppr 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
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