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Theorem unitgrp 18713
Description: The group of units is a group under multiplication. (Contributed by Mario Carneiro, 2-Dec-2014.)
Hypotheses
Ref Expression
unitmulcl.1 𝑈 = (Unit‘𝑅)
unitgrp.2 𝐺 = ((mulGrp‘𝑅) ↾s 𝑈)
Assertion
Ref Expression
unitgrp (𝑅 ∈ Ring → 𝐺 ∈ Grp)

Proof of Theorem unitgrp
Dummy variables 𝑥 𝑦 𝑧 𝑚 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 unitmulcl.1 . . . 4 𝑈 = (Unit‘𝑅)
2 unitgrp.2 . . . 4 𝐺 = ((mulGrp‘𝑅) ↾s 𝑈)
31, 2unitgrpbas 18712 . . 3 𝑈 = (Base‘𝐺)
43a1i 11 . 2 (𝑅 ∈ Ring → 𝑈 = (Base‘𝐺))
5 fvex 6239 . . . 4 (Base‘𝐺) ∈ V
63, 5eqeltri 2726 . . 3 𝑈 ∈ V
7 eqid 2651 . . . . 5 (mulGrp‘𝑅) = (mulGrp‘𝑅)
8 eqid 2651 . . . . 5 (.r𝑅) = (.r𝑅)
97, 8mgpplusg 18539 . . . 4 (.r𝑅) = (+g‘(mulGrp‘𝑅))
102, 9ressplusg 16040 . . 3 (𝑈 ∈ V → (.r𝑅) = (+g𝐺))
116, 10mp1i 13 . 2 (𝑅 ∈ Ring → (.r𝑅) = (+g𝐺))
121, 8unitmulcl 18710 . 2 ((𝑅 ∈ Ring ∧ 𝑥𝑈𝑦𝑈) → (𝑥(.r𝑅)𝑦) ∈ 𝑈)
13 eqid 2651 . . . . 5 (Base‘𝑅) = (Base‘𝑅)
1413, 1unitcl 18705 . . . 4 (𝑥𝑈𝑥 ∈ (Base‘𝑅))
1513, 1unitcl 18705 . . . 4 (𝑦𝑈𝑦 ∈ (Base‘𝑅))
1613, 1unitcl 18705 . . . 4 (𝑧𝑈𝑧 ∈ (Base‘𝑅))
1714, 15, 163anim123i 1266 . . 3 ((𝑥𝑈𝑦𝑈𝑧𝑈) → (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅) ∧ 𝑧 ∈ (Base‘𝑅)))
1813, 8ringass 18610 . . 3 ((𝑅 ∈ Ring ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅) ∧ 𝑧 ∈ (Base‘𝑅))) → ((𝑥(.r𝑅)𝑦)(.r𝑅)𝑧) = (𝑥(.r𝑅)(𝑦(.r𝑅)𝑧)))
1917, 18sylan2 490 . 2 ((𝑅 ∈ Ring ∧ (𝑥𝑈𝑦𝑈𝑧𝑈)) → ((𝑥(.r𝑅)𝑦)(.r𝑅)𝑧) = (𝑥(.r𝑅)(𝑦(.r𝑅)𝑧)))
20 eqid 2651 . . 3 (1r𝑅) = (1r𝑅)
211, 201unit 18704 . 2 (𝑅 ∈ Ring → (1r𝑅) ∈ 𝑈)
2213, 8, 20ringlidm 18617 . . 3 ((𝑅 ∈ Ring ∧ 𝑥 ∈ (Base‘𝑅)) → ((1r𝑅)(.r𝑅)𝑥) = 𝑥)
2314, 22sylan2 490 . 2 ((𝑅 ∈ Ring ∧ 𝑥𝑈) → ((1r𝑅)(.r𝑅)𝑥) = 𝑥)
24 simpr 476 . . . 4 ((𝑅 ∈ Ring ∧ 𝑥𝑈) → 𝑥𝑈)
25 eqid 2651 . . . . 5 (∥r𝑅) = (∥r𝑅)
26 eqid 2651 . . . . 5 (oppr𝑅) = (oppr𝑅)
27 eqid 2651 . . . . 5 (∥r‘(oppr𝑅)) = (∥r‘(oppr𝑅))
281, 20, 25, 26, 27isunit 18703 . . . 4 (𝑥𝑈 ↔ (𝑥(∥r𝑅)(1r𝑅) ∧ 𝑥(∥r‘(oppr𝑅))(1r𝑅)))
2924, 28sylib 208 . . 3 ((𝑅 ∈ Ring ∧ 𝑥𝑈) → (𝑥(∥r𝑅)(1r𝑅) ∧ 𝑥(∥r‘(oppr𝑅))(1r𝑅)))
3014adantl 481 . . . . . 6 ((𝑅 ∈ Ring ∧ 𝑥𝑈) → 𝑥 ∈ (Base‘𝑅))
3113, 25, 8dvdsr2 18693 . . . . . 6 (𝑥 ∈ (Base‘𝑅) → (𝑥(∥r𝑅)(1r𝑅) ↔ ∃𝑦 ∈ (Base‘𝑅)(𝑦(.r𝑅)𝑥) = (1r𝑅)))
3230, 31syl 17 . . . . 5 ((𝑅 ∈ Ring ∧ 𝑥𝑈) → (𝑥(∥r𝑅)(1r𝑅) ↔ ∃𝑦 ∈ (Base‘𝑅)(𝑦(.r𝑅)𝑥) = (1r𝑅)))
3326, 13opprbas 18675 . . . . . . 7 (Base‘𝑅) = (Base‘(oppr𝑅))
34 eqid 2651 . . . . . . 7 (.r‘(oppr𝑅)) = (.r‘(oppr𝑅))
3533, 27, 34dvdsr2 18693 . . . . . 6 (𝑥 ∈ (Base‘𝑅) → (𝑥(∥r‘(oppr𝑅))(1r𝑅) ↔ ∃𝑚 ∈ (Base‘𝑅)(𝑚(.r‘(oppr𝑅))𝑥) = (1r𝑅)))
3630, 35syl 17 . . . . 5 ((𝑅 ∈ Ring ∧ 𝑥𝑈) → (𝑥(∥r‘(oppr𝑅))(1r𝑅) ↔ ∃𝑚 ∈ (Base‘𝑅)(𝑚(.r‘(oppr𝑅))𝑥) = (1r𝑅)))
3732, 36anbi12d 747 . . . 4 ((𝑅 ∈ Ring ∧ 𝑥𝑈) → ((𝑥(∥r𝑅)(1r𝑅) ∧ 𝑥(∥r‘(oppr𝑅))(1r𝑅)) ↔ (∃𝑦 ∈ (Base‘𝑅)(𝑦(.r𝑅)𝑥) = (1r𝑅) ∧ ∃𝑚 ∈ (Base‘𝑅)(𝑚(.r‘(oppr𝑅))𝑥) = (1r𝑅))))
38 reeanv 3136 . . . . 5 (∃𝑦 ∈ (Base‘𝑅)∃𝑚 ∈ (Base‘𝑅)((𝑦(.r𝑅)𝑥) = (1r𝑅) ∧ (𝑚(.r‘(oppr𝑅))𝑥) = (1r𝑅)) ↔ (∃𝑦 ∈ (Base‘𝑅)(𝑦(.r𝑅)𝑥) = (1r𝑅) ∧ ∃𝑚 ∈ (Base‘𝑅)(𝑚(.r‘(oppr𝑅))𝑥) = (1r𝑅)))
39 simprl 809 . . . . . . . . . . . 12 ((((𝑅 ∈ Ring ∧ 𝑥𝑈) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ (𝑚 ∈ (Base‘𝑅) ∧ ((𝑦(.r𝑅)𝑥) = (1r𝑅) ∧ (𝑚(.r‘(oppr𝑅))𝑥) = (1r𝑅)))) → 𝑚 ∈ (Base‘𝑅))
4030ad2antrr 762 . . . . . . . . . . . 12 ((((𝑅 ∈ Ring ∧ 𝑥𝑈) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ (𝑚 ∈ (Base‘𝑅) ∧ ((𝑦(.r𝑅)𝑥) = (1r𝑅) ∧ (𝑚(.r‘(oppr𝑅))𝑥) = (1r𝑅)))) → 𝑥 ∈ (Base‘𝑅))
4113, 25, 8dvdsrmul 18694 . . . . . . . . . . . 12 ((𝑚 ∈ (Base‘𝑅) ∧ 𝑥 ∈ (Base‘𝑅)) → 𝑚(∥r𝑅)(𝑥(.r𝑅)𝑚))
4239, 40, 41syl2anc 694 . . . . . . . . . . 11 ((((𝑅 ∈ Ring ∧ 𝑥𝑈) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ (𝑚 ∈ (Base‘𝑅) ∧ ((𝑦(.r𝑅)𝑥) = (1r𝑅) ∧ (𝑚(.r‘(oppr𝑅))𝑥) = (1r𝑅)))) → 𝑚(∥r𝑅)(𝑥(.r𝑅)𝑚))
43 simplll 813 . . . . . . . . . . . . . 14 ((((𝑅 ∈ Ring ∧ 𝑥𝑈) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ (𝑚 ∈ (Base‘𝑅) ∧ ((𝑦(.r𝑅)𝑥) = (1r𝑅) ∧ (𝑚(.r‘(oppr𝑅))𝑥) = (1r𝑅)))) → 𝑅 ∈ Ring)
44 simplr 807 . . . . . . . . . . . . . 14 ((((𝑅 ∈ Ring ∧ 𝑥𝑈) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ (𝑚 ∈ (Base‘𝑅) ∧ ((𝑦(.r𝑅)𝑥) = (1r𝑅) ∧ (𝑚(.r‘(oppr𝑅))𝑥) = (1r𝑅)))) → 𝑦 ∈ (Base‘𝑅))
4513, 8ringass 18610 . . . . . . . . . . . . . 14 ((𝑅 ∈ Ring ∧ (𝑦 ∈ (Base‘𝑅) ∧ 𝑥 ∈ (Base‘𝑅) ∧ 𝑚 ∈ (Base‘𝑅))) → ((𝑦(.r𝑅)𝑥)(.r𝑅)𝑚) = (𝑦(.r𝑅)(𝑥(.r𝑅)𝑚)))
4643, 44, 40, 39, 45syl13anc 1368 . . . . . . . . . . . . 13 ((((𝑅 ∈ Ring ∧ 𝑥𝑈) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ (𝑚 ∈ (Base‘𝑅) ∧ ((𝑦(.r𝑅)𝑥) = (1r𝑅) ∧ (𝑚(.r‘(oppr𝑅))𝑥) = (1r𝑅)))) → ((𝑦(.r𝑅)𝑥)(.r𝑅)𝑚) = (𝑦(.r𝑅)(𝑥(.r𝑅)𝑚)))
47 simprrl 821 . . . . . . . . . . . . . 14 ((((𝑅 ∈ Ring ∧ 𝑥𝑈) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ (𝑚 ∈ (Base‘𝑅) ∧ ((𝑦(.r𝑅)𝑥) = (1r𝑅) ∧ (𝑚(.r‘(oppr𝑅))𝑥) = (1r𝑅)))) → (𝑦(.r𝑅)𝑥) = (1r𝑅))
4847oveq1d 6705 . . . . . . . . . . . . 13 ((((𝑅 ∈ Ring ∧ 𝑥𝑈) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ (𝑚 ∈ (Base‘𝑅) ∧ ((𝑦(.r𝑅)𝑥) = (1r𝑅) ∧ (𝑚(.r‘(oppr𝑅))𝑥) = (1r𝑅)))) → ((𝑦(.r𝑅)𝑥)(.r𝑅)𝑚) = ((1r𝑅)(.r𝑅)𝑚))
4913, 8, 26, 34opprmul 18672 . . . . . . . . . . . . . . 15 (𝑚(.r‘(oppr𝑅))𝑥) = (𝑥(.r𝑅)𝑚)
50 simprrr 822 . . . . . . . . . . . . . . 15 ((((𝑅 ∈ Ring ∧ 𝑥𝑈) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ (𝑚 ∈ (Base‘𝑅) ∧ ((𝑦(.r𝑅)𝑥) = (1r𝑅) ∧ (𝑚(.r‘(oppr𝑅))𝑥) = (1r𝑅)))) → (𝑚(.r‘(oppr𝑅))𝑥) = (1r𝑅))
5149, 50syl5eqr 2699 . . . . . . . . . . . . . 14 ((((𝑅 ∈ Ring ∧ 𝑥𝑈) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ (𝑚 ∈ (Base‘𝑅) ∧ ((𝑦(.r𝑅)𝑥) = (1r𝑅) ∧ (𝑚(.r‘(oppr𝑅))𝑥) = (1r𝑅)))) → (𝑥(.r𝑅)𝑚) = (1r𝑅))
5251oveq2d 6706 . . . . . . . . . . . . 13 ((((𝑅 ∈ Ring ∧ 𝑥𝑈) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ (𝑚 ∈ (Base‘𝑅) ∧ ((𝑦(.r𝑅)𝑥) = (1r𝑅) ∧ (𝑚(.r‘(oppr𝑅))𝑥) = (1r𝑅)))) → (𝑦(.r𝑅)(𝑥(.r𝑅)𝑚)) = (𝑦(.r𝑅)(1r𝑅)))
5346, 48, 523eqtr3d 2693 . . . . . . . . . . . 12 ((((𝑅 ∈ Ring ∧ 𝑥𝑈) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ (𝑚 ∈ (Base‘𝑅) ∧ ((𝑦(.r𝑅)𝑥) = (1r𝑅) ∧ (𝑚(.r‘(oppr𝑅))𝑥) = (1r𝑅)))) → ((1r𝑅)(.r𝑅)𝑚) = (𝑦(.r𝑅)(1r𝑅)))
5413, 8, 20ringlidm 18617 . . . . . . . . . . . . 13 ((𝑅 ∈ Ring ∧ 𝑚 ∈ (Base‘𝑅)) → ((1r𝑅)(.r𝑅)𝑚) = 𝑚)
5543, 39, 54syl2anc 694 . . . . . . . . . . . 12 ((((𝑅 ∈ Ring ∧ 𝑥𝑈) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ (𝑚 ∈ (Base‘𝑅) ∧ ((𝑦(.r𝑅)𝑥) = (1r𝑅) ∧ (𝑚(.r‘(oppr𝑅))𝑥) = (1r𝑅)))) → ((1r𝑅)(.r𝑅)𝑚) = 𝑚)
5613, 8, 20ringridm 18618 . . . . . . . . . . . . 13 ((𝑅 ∈ Ring ∧ 𝑦 ∈ (Base‘𝑅)) → (𝑦(.r𝑅)(1r𝑅)) = 𝑦)
5743, 44, 56syl2anc 694 . . . . . . . . . . . 12 ((((𝑅 ∈ Ring ∧ 𝑥𝑈) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ (𝑚 ∈ (Base‘𝑅) ∧ ((𝑦(.r𝑅)𝑥) = (1r𝑅) ∧ (𝑚(.r‘(oppr𝑅))𝑥) = (1r𝑅)))) → (𝑦(.r𝑅)(1r𝑅)) = 𝑦)
5853, 55, 573eqtr3d 2693 . . . . . . . . . . 11 ((((𝑅 ∈ Ring ∧ 𝑥𝑈) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ (𝑚 ∈ (Base‘𝑅) ∧ ((𝑦(.r𝑅)𝑥) = (1r𝑅) ∧ (𝑚(.r‘(oppr𝑅))𝑥) = (1r𝑅)))) → 𝑚 = 𝑦)
5942, 58, 513brtr3d 4716 . . . . . . . . . 10 ((((𝑅 ∈ Ring ∧ 𝑥𝑈) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ (𝑚 ∈ (Base‘𝑅) ∧ ((𝑦(.r𝑅)𝑥) = (1r𝑅) ∧ (𝑚(.r‘(oppr𝑅))𝑥) = (1r𝑅)))) → 𝑦(∥r𝑅)(1r𝑅))
6033, 27, 34dvdsrmul 18694 . . . . . . . . . . . 12 ((𝑦 ∈ (Base‘𝑅) ∧ 𝑥 ∈ (Base‘𝑅)) → 𝑦(∥r‘(oppr𝑅))(𝑥(.r‘(oppr𝑅))𝑦))
6144, 40, 60syl2anc 694 . . . . . . . . . . 11 ((((𝑅 ∈ Ring ∧ 𝑥𝑈) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ (𝑚 ∈ (Base‘𝑅) ∧ ((𝑦(.r𝑅)𝑥) = (1r𝑅) ∧ (𝑚(.r‘(oppr𝑅))𝑥) = (1r𝑅)))) → 𝑦(∥r‘(oppr𝑅))(𝑥(.r‘(oppr𝑅))𝑦))
6213, 8, 26, 34opprmul 18672 . . . . . . . . . . . 12 (𝑥(.r‘(oppr𝑅))𝑦) = (𝑦(.r𝑅)𝑥)
6362, 47syl5eq 2697 . . . . . . . . . . 11 ((((𝑅 ∈ Ring ∧ 𝑥𝑈) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ (𝑚 ∈ (Base‘𝑅) ∧ ((𝑦(.r𝑅)𝑥) = (1r𝑅) ∧ (𝑚(.r‘(oppr𝑅))𝑥) = (1r𝑅)))) → (𝑥(.r‘(oppr𝑅))𝑦) = (1r𝑅))
6461, 63breqtrd 4711 . . . . . . . . . 10 ((((𝑅 ∈ Ring ∧ 𝑥𝑈) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ (𝑚 ∈ (Base‘𝑅) ∧ ((𝑦(.r𝑅)𝑥) = (1r𝑅) ∧ (𝑚(.r‘(oppr𝑅))𝑥) = (1r𝑅)))) → 𝑦(∥r‘(oppr𝑅))(1r𝑅))
651, 20, 25, 26, 27isunit 18703 . . . . . . . . . 10 (𝑦𝑈 ↔ (𝑦(∥r𝑅)(1r𝑅) ∧ 𝑦(∥r‘(oppr𝑅))(1r𝑅)))
6659, 64, 65sylanbrc 699 . . . . . . . . 9 ((((𝑅 ∈ Ring ∧ 𝑥𝑈) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ (𝑚 ∈ (Base‘𝑅) ∧ ((𝑦(.r𝑅)𝑥) = (1r𝑅) ∧ (𝑚(.r‘(oppr𝑅))𝑥) = (1r𝑅)))) → 𝑦𝑈)
6766, 47jca 553 . . . . . . . 8 ((((𝑅 ∈ Ring ∧ 𝑥𝑈) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ (𝑚 ∈ (Base‘𝑅) ∧ ((𝑦(.r𝑅)𝑥) = (1r𝑅) ∧ (𝑚(.r‘(oppr𝑅))𝑥) = (1r𝑅)))) → (𝑦𝑈 ∧ (𝑦(.r𝑅)𝑥) = (1r𝑅)))
6867rexlimdvaa 3061 . . . . . . 7 (((𝑅 ∈ Ring ∧ 𝑥𝑈) ∧ 𝑦 ∈ (Base‘𝑅)) → (∃𝑚 ∈ (Base‘𝑅)((𝑦(.r𝑅)𝑥) = (1r𝑅) ∧ (𝑚(.r‘(oppr𝑅))𝑥) = (1r𝑅)) → (𝑦𝑈 ∧ (𝑦(.r𝑅)𝑥) = (1r𝑅))))
6968expimpd 628 . . . . . 6 ((𝑅 ∈ Ring ∧ 𝑥𝑈) → ((𝑦 ∈ (Base‘𝑅) ∧ ∃𝑚 ∈ (Base‘𝑅)((𝑦(.r𝑅)𝑥) = (1r𝑅) ∧ (𝑚(.r‘(oppr𝑅))𝑥) = (1r𝑅))) → (𝑦𝑈 ∧ (𝑦(.r𝑅)𝑥) = (1r𝑅))))
7069reximdv2 3043 . . . . 5 ((𝑅 ∈ Ring ∧ 𝑥𝑈) → (∃𝑦 ∈ (Base‘𝑅)∃𝑚 ∈ (Base‘𝑅)((𝑦(.r𝑅)𝑥) = (1r𝑅) ∧ (𝑚(.r‘(oppr𝑅))𝑥) = (1r𝑅)) → ∃𝑦𝑈 (𝑦(.r𝑅)𝑥) = (1r𝑅)))
7138, 70syl5bir 233 . . . 4 ((𝑅 ∈ Ring ∧ 𝑥𝑈) → ((∃𝑦 ∈ (Base‘𝑅)(𝑦(.r𝑅)𝑥) = (1r𝑅) ∧ ∃𝑚 ∈ (Base‘𝑅)(𝑚(.r‘(oppr𝑅))𝑥) = (1r𝑅)) → ∃𝑦𝑈 (𝑦(.r𝑅)𝑥) = (1r𝑅)))
7237, 71sylbid 230 . . 3 ((𝑅 ∈ Ring ∧ 𝑥𝑈) → ((𝑥(∥r𝑅)(1r𝑅) ∧ 𝑥(∥r‘(oppr𝑅))(1r𝑅)) → ∃𝑦𝑈 (𝑦(.r𝑅)𝑥) = (1r𝑅)))
7329, 72mpd 15 . 2 ((𝑅 ∈ Ring ∧ 𝑥𝑈) → ∃𝑦𝑈 (𝑦(.r𝑅)𝑥) = (1r𝑅))
744, 11, 12, 19, 21, 23, 73isgrpde 17490 1 (𝑅 ∈ Ring → 𝐺 ∈ Grp)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 383  w3a 1054   = wceq 1523  wcel 2030  wrex 2942  Vcvv 3231   class class class wbr 4685  cfv 5926  (class class class)co 6690  Basecbs 15904  s cress 15905  +gcplusg 15988  .rcmulr 15989  Grpcgrp 17469  mulGrpcmgp 18535  1rcur 18547  Ringcrg 18593  opprcoppr 18668  rcdsr 18684  Unitcui 18685
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-rep 4804  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991  ax-cnex 10030  ax-resscn 10031  ax-1cn 10032  ax-icn 10033  ax-addcl 10034  ax-addrcl 10035  ax-mulcl 10036  ax-mulrcl 10037  ax-mulcom 10038  ax-addass 10039  ax-mulass 10040  ax-distr 10041  ax-i2m1 10042  ax-1ne0 10043  ax-1rid 10044  ax-rnegex 10045  ax-rrecex 10046  ax-cnre 10047  ax-pre-lttri 10048  ax-pre-lttrn 10049  ax-pre-ltadd 10050  ax-pre-mulgt0 10051
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1055  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-nel 2927  df-ral 2946  df-rex 2947  df-reu 2948  df-rmo 2949  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-pss 3623  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-tp 4215  df-op 4217  df-uni 4469  df-iun 4554  df-br 4686  df-opab 4746  df-mpt 4763  df-tr 4786  df-id 5053  df-eprel 5058  df-po 5064  df-so 5065  df-fr 5102  df-we 5104  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-pred 5718  df-ord 5764  df-on 5765  df-lim 5766  df-suc 5767  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-riota 6651  df-ov 6693  df-oprab 6694  df-mpt2 6695  df-om 7108  df-tpos 7397  df-wrecs 7452  df-recs 7513  df-rdg 7551  df-er 7787  df-en 7998  df-dom 7999  df-sdom 8000  df-pnf 10114  df-mnf 10115  df-xr 10116  df-ltxr 10117  df-le 10118  df-sub 10306  df-neg 10307  df-nn 11059  df-2 11117  df-3 11118  df-ndx 15907  df-slot 15908  df-base 15910  df-sets 15911  df-ress 15912  df-plusg 16001  df-mulr 16002  df-0g 16149  df-mgm 17289  df-sgrp 17331  df-mnd 17342  df-grp 17472  df-mgp 18536  df-ur 18548  df-ring 18595  df-oppr 18669  df-dvdsr 18687  df-unit 18688
This theorem is referenced by:  unitabl  18714  unitsubm  18716  unitinvcl  18720  unitinvinv  18721  unitlinv  18723  unitrinv  18724  isdrng2  18805  subrgugrp  18847  expghm  19892  invrvald  20530  nrginvrcn  22543  nrgtdrg  22544  dchrfi  25025  dchrghm  25026  dchrabs  25030  dchrptlem1  25034  dchrptlem2  25035  dchrptlem3  25036  dchrsum2  25038  rdivmuldivd  29919  dvrcan5  29921  rhmunitinv  29950  idomodle  38091  proot1mul  38094  proot1hash  38095  proot1ex  38096
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