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Theorem unitgrp 20096
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 20095 . . 3 𝑈 = (Base‘𝐺)
43a1i 11 . 2 (𝑅 ∈ Ring → 𝑈 = (Base‘𝐺))
51fvexi 6856 . . 3 𝑈 ∈ V
6 eqid 2736 . . . . 5 (mulGrp‘𝑅) = (mulGrp‘𝑅)
7 eqid 2736 . . . . 5 (.r𝑅) = (.r𝑅)
86, 7mgpplusg 19900 . . . 4 (.r𝑅) = (+g‘(mulGrp‘𝑅))
92, 8ressplusg 17171 . . 3 (𝑈 ∈ V → (.r𝑅) = (+g𝐺))
105, 9mp1i 13 . 2 (𝑅 ∈ Ring → (.r𝑅) = (+g𝐺))
111, 7unitmulcl 20093 . 2 ((𝑅 ∈ Ring ∧ 𝑥𝑈𝑦𝑈) → (𝑥(.r𝑅)𝑦) ∈ 𝑈)
12 eqid 2736 . . . . 5 (Base‘𝑅) = (Base‘𝑅)
1312, 1unitcl 20088 . . . 4 (𝑥𝑈𝑥 ∈ (Base‘𝑅))
1412, 1unitcl 20088 . . . 4 (𝑦𝑈𝑦 ∈ (Base‘𝑅))
1512, 1unitcl 20088 . . . 4 (𝑧𝑈𝑧 ∈ (Base‘𝑅))
1613, 14, 153anim123i 1151 . . 3 ((𝑥𝑈𝑦𝑈𝑧𝑈) → (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅) ∧ 𝑧 ∈ (Base‘𝑅)))
1712, 7ringass 19984 . . 3 ((𝑅 ∈ Ring ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅) ∧ 𝑧 ∈ (Base‘𝑅))) → ((𝑥(.r𝑅)𝑦)(.r𝑅)𝑧) = (𝑥(.r𝑅)(𝑦(.r𝑅)𝑧)))
1816, 17sylan2 593 . 2 ((𝑅 ∈ Ring ∧ (𝑥𝑈𝑦𝑈𝑧𝑈)) → ((𝑥(.r𝑅)𝑦)(.r𝑅)𝑧) = (𝑥(.r𝑅)(𝑦(.r𝑅)𝑧)))
19 eqid 2736 . . 3 (1r𝑅) = (1r𝑅)
201, 191unit 20087 . 2 (𝑅 ∈ Ring → (1r𝑅) ∈ 𝑈)
2112, 7, 19ringlidm 19992 . . 3 ((𝑅 ∈ Ring ∧ 𝑥 ∈ (Base‘𝑅)) → ((1r𝑅)(.r𝑅)𝑥) = 𝑥)
2213, 21sylan2 593 . 2 ((𝑅 ∈ Ring ∧ 𝑥𝑈) → ((1r𝑅)(.r𝑅)𝑥) = 𝑥)
23 simpr 485 . . . 4 ((𝑅 ∈ Ring ∧ 𝑥𝑈) → 𝑥𝑈)
24 eqid 2736 . . . . 5 (∥r𝑅) = (∥r𝑅)
25 eqid 2736 . . . . 5 (oppr𝑅) = (oppr𝑅)
26 eqid 2736 . . . . 5 (∥r‘(oppr𝑅)) = (∥r‘(oppr𝑅))
271, 19, 24, 25, 26isunit 20086 . . . 4 (𝑥𝑈 ↔ (𝑥(∥r𝑅)(1r𝑅) ∧ 𝑥(∥r‘(oppr𝑅))(1r𝑅)))
2823, 27sylib 217 . . 3 ((𝑅 ∈ Ring ∧ 𝑥𝑈) → (𝑥(∥r𝑅)(1r𝑅) ∧ 𝑥(∥r‘(oppr𝑅))(1r𝑅)))
2913adantl 482 . . . . . 6 ((𝑅 ∈ Ring ∧ 𝑥𝑈) → 𝑥 ∈ (Base‘𝑅))
3012, 24, 7dvdsr2 20076 . . . . . 6 (𝑥 ∈ (Base‘𝑅) → (𝑥(∥r𝑅)(1r𝑅) ↔ ∃𝑦 ∈ (Base‘𝑅)(𝑦(.r𝑅)𝑥) = (1r𝑅)))
3129, 30syl 17 . . . . 5 ((𝑅 ∈ Ring ∧ 𝑥𝑈) → (𝑥(∥r𝑅)(1r𝑅) ↔ ∃𝑦 ∈ (Base‘𝑅)(𝑦(.r𝑅)𝑥) = (1r𝑅)))
3225, 12opprbas 20056 . . . . . . 7 (Base‘𝑅) = (Base‘(oppr𝑅))
33 eqid 2736 . . . . . . 7 (.r‘(oppr𝑅)) = (.r‘(oppr𝑅))
3432, 26, 33dvdsr2 20076 . . . . . 6 (𝑥 ∈ (Base‘𝑅) → (𝑥(∥r‘(oppr𝑅))(1r𝑅) ↔ ∃𝑚 ∈ (Base‘𝑅)(𝑚(.r‘(oppr𝑅))𝑥) = (1r𝑅)))
3529, 34syl 17 . . . . 5 ((𝑅 ∈ Ring ∧ 𝑥𝑈) → (𝑥(∥r‘(oppr𝑅))(1r𝑅) ↔ ∃𝑚 ∈ (Base‘𝑅)(𝑚(.r‘(oppr𝑅))𝑥) = (1r𝑅)))
3631, 35anbi12d 631 . . . 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))))
37 reeanv 3217 . . . . 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)))
38 simprl 769 . . . . . . . . . . . 12 ((((𝑅 ∈ Ring ∧ 𝑥𝑈) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ (𝑚 ∈ (Base‘𝑅) ∧ ((𝑦(.r𝑅)𝑥) = (1r𝑅) ∧ (𝑚(.r‘(oppr𝑅))𝑥) = (1r𝑅)))) → 𝑚 ∈ (Base‘𝑅))
3929ad2antrr 724 . . . . . . . . . . . 12 ((((𝑅 ∈ Ring ∧ 𝑥𝑈) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ (𝑚 ∈ (Base‘𝑅) ∧ ((𝑦(.r𝑅)𝑥) = (1r𝑅) ∧ (𝑚(.r‘(oppr𝑅))𝑥) = (1r𝑅)))) → 𝑥 ∈ (Base‘𝑅))
4012, 24, 7dvdsrmul 20077 . . . . . . . . . . . 12 ((𝑚 ∈ (Base‘𝑅) ∧ 𝑥 ∈ (Base‘𝑅)) → 𝑚(∥r𝑅)(𝑥(.r𝑅)𝑚))
4138, 39, 40syl2anc 584 . . . . . . . . . . 11 ((((𝑅 ∈ Ring ∧ 𝑥𝑈) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ (𝑚 ∈ (Base‘𝑅) ∧ ((𝑦(.r𝑅)𝑥) = (1r𝑅) ∧ (𝑚(.r‘(oppr𝑅))𝑥) = (1r𝑅)))) → 𝑚(∥r𝑅)(𝑥(.r𝑅)𝑚))
42 simplll 773 . . . . . . . . . . . . . 14 ((((𝑅 ∈ Ring ∧ 𝑥𝑈) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ (𝑚 ∈ (Base‘𝑅) ∧ ((𝑦(.r𝑅)𝑥) = (1r𝑅) ∧ (𝑚(.r‘(oppr𝑅))𝑥) = (1r𝑅)))) → 𝑅 ∈ Ring)
43 simplr 767 . . . . . . . . . . . . . 14 ((((𝑅 ∈ Ring ∧ 𝑥𝑈) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ (𝑚 ∈ (Base‘𝑅) ∧ ((𝑦(.r𝑅)𝑥) = (1r𝑅) ∧ (𝑚(.r‘(oppr𝑅))𝑥) = (1r𝑅)))) → 𝑦 ∈ (Base‘𝑅))
4412, 7ringass 19984 . . . . . . . . . . . . . 14 ((𝑅 ∈ Ring ∧ (𝑦 ∈ (Base‘𝑅) ∧ 𝑥 ∈ (Base‘𝑅) ∧ 𝑚 ∈ (Base‘𝑅))) → ((𝑦(.r𝑅)𝑥)(.r𝑅)𝑚) = (𝑦(.r𝑅)(𝑥(.r𝑅)𝑚)))
4542, 43, 39, 38, 44syl13anc 1372 . . . . . . . . . . . . 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)))
46 simprrl 779 . . . . . . . . . . . . . 14 ((((𝑅 ∈ Ring ∧ 𝑥𝑈) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ (𝑚 ∈ (Base‘𝑅) ∧ ((𝑦(.r𝑅)𝑥) = (1r𝑅) ∧ (𝑚(.r‘(oppr𝑅))𝑥) = (1r𝑅)))) → (𝑦(.r𝑅)𝑥) = (1r𝑅))
4746oveq1d 7372 . . . . . . . . . . . . 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))
4812, 7, 25, 33opprmul 20052 . . . . . . . . . . . . . . 15 (𝑚(.r‘(oppr𝑅))𝑥) = (𝑥(.r𝑅)𝑚)
49 simprrr 780 . . . . . . . . . . . . . . 15 ((((𝑅 ∈ Ring ∧ 𝑥𝑈) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ (𝑚 ∈ (Base‘𝑅) ∧ ((𝑦(.r𝑅)𝑥) = (1r𝑅) ∧ (𝑚(.r‘(oppr𝑅))𝑥) = (1r𝑅)))) → (𝑚(.r‘(oppr𝑅))𝑥) = (1r𝑅))
5048, 49eqtr3id 2790 . . . . . . . . . . . . . 14 ((((𝑅 ∈ Ring ∧ 𝑥𝑈) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ (𝑚 ∈ (Base‘𝑅) ∧ ((𝑦(.r𝑅)𝑥) = (1r𝑅) ∧ (𝑚(.r‘(oppr𝑅))𝑥) = (1r𝑅)))) → (𝑥(.r𝑅)𝑚) = (1r𝑅))
5150oveq2d 7373 . . . . . . . . . . . . 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)))
5245, 47, 513eqtr3d 2784 . . . . . . . . . . . 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)))
5312, 7, 19ringlidm 19992 . . . . . . . . . . . . 13 ((𝑅 ∈ Ring ∧ 𝑚 ∈ (Base‘𝑅)) → ((1r𝑅)(.r𝑅)𝑚) = 𝑚)
5442, 38, 53syl2anc 584 . . . . . . . . . . . 12 ((((𝑅 ∈ Ring ∧ 𝑥𝑈) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ (𝑚 ∈ (Base‘𝑅) ∧ ((𝑦(.r𝑅)𝑥) = (1r𝑅) ∧ (𝑚(.r‘(oppr𝑅))𝑥) = (1r𝑅)))) → ((1r𝑅)(.r𝑅)𝑚) = 𝑚)
5512, 7, 19ringridm 19993 . . . . . . . . . . . . 13 ((𝑅 ∈ Ring ∧ 𝑦 ∈ (Base‘𝑅)) → (𝑦(.r𝑅)(1r𝑅)) = 𝑦)
5642, 43, 55syl2anc 584 . . . . . . . . . . . 12 ((((𝑅 ∈ Ring ∧ 𝑥𝑈) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ (𝑚 ∈ (Base‘𝑅) ∧ ((𝑦(.r𝑅)𝑥) = (1r𝑅) ∧ (𝑚(.r‘(oppr𝑅))𝑥) = (1r𝑅)))) → (𝑦(.r𝑅)(1r𝑅)) = 𝑦)
5752, 54, 563eqtr3d 2784 . . . . . . . . . . 11 ((((𝑅 ∈ Ring ∧ 𝑥𝑈) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ (𝑚 ∈ (Base‘𝑅) ∧ ((𝑦(.r𝑅)𝑥) = (1r𝑅) ∧ (𝑚(.r‘(oppr𝑅))𝑥) = (1r𝑅)))) → 𝑚 = 𝑦)
5841, 57, 503brtr3d 5136 . . . . . . . . . 10 ((((𝑅 ∈ Ring ∧ 𝑥𝑈) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ (𝑚 ∈ (Base‘𝑅) ∧ ((𝑦(.r𝑅)𝑥) = (1r𝑅) ∧ (𝑚(.r‘(oppr𝑅))𝑥) = (1r𝑅)))) → 𝑦(∥r𝑅)(1r𝑅))
5932, 26, 33dvdsrmul 20077 . . . . . . . . . . . 12 ((𝑦 ∈ (Base‘𝑅) ∧ 𝑥 ∈ (Base‘𝑅)) → 𝑦(∥r‘(oppr𝑅))(𝑥(.r‘(oppr𝑅))𝑦))
6043, 39, 59syl2anc 584 . . . . . . . . . . 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))
6112, 7, 25, 33opprmul 20052 . . . . . . . . . . . 12 (𝑥(.r‘(oppr𝑅))𝑦) = (𝑦(.r𝑅)𝑥)
6261, 46eqtrid 2788 . . . . . . . . . . 11 ((((𝑅 ∈ Ring ∧ 𝑥𝑈) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ (𝑚 ∈ (Base‘𝑅) ∧ ((𝑦(.r𝑅)𝑥) = (1r𝑅) ∧ (𝑚(.r‘(oppr𝑅))𝑥) = (1r𝑅)))) → (𝑥(.r‘(oppr𝑅))𝑦) = (1r𝑅))
6360, 62breqtrd 5131 . . . . . . . . . 10 ((((𝑅 ∈ Ring ∧ 𝑥𝑈) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ (𝑚 ∈ (Base‘𝑅) ∧ ((𝑦(.r𝑅)𝑥) = (1r𝑅) ∧ (𝑚(.r‘(oppr𝑅))𝑥) = (1r𝑅)))) → 𝑦(∥r‘(oppr𝑅))(1r𝑅))
641, 19, 24, 25, 26isunit 20086 . . . . . . . . . 10 (𝑦𝑈 ↔ (𝑦(∥r𝑅)(1r𝑅) ∧ 𝑦(∥r‘(oppr𝑅))(1r𝑅)))
6558, 63, 64sylanbrc 583 . . . . . . . . 9 ((((𝑅 ∈ Ring ∧ 𝑥𝑈) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ (𝑚 ∈ (Base‘𝑅) ∧ ((𝑦(.r𝑅)𝑥) = (1r𝑅) ∧ (𝑚(.r‘(oppr𝑅))𝑥) = (1r𝑅)))) → 𝑦𝑈)
6665, 46jca 512 . . . . . . . 8 ((((𝑅 ∈ Ring ∧ 𝑥𝑈) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ (𝑚 ∈ (Base‘𝑅) ∧ ((𝑦(.r𝑅)𝑥) = (1r𝑅) ∧ (𝑚(.r‘(oppr𝑅))𝑥) = (1r𝑅)))) → (𝑦𝑈 ∧ (𝑦(.r𝑅)𝑥) = (1r𝑅)))
6766rexlimdvaa 3153 . . . . . . 7 (((𝑅 ∈ Ring ∧ 𝑥𝑈) ∧ 𝑦 ∈ (Base‘𝑅)) → (∃𝑚 ∈ (Base‘𝑅)((𝑦(.r𝑅)𝑥) = (1r𝑅) ∧ (𝑚(.r‘(oppr𝑅))𝑥) = (1r𝑅)) → (𝑦𝑈 ∧ (𝑦(.r𝑅)𝑥) = (1r𝑅))))
6867expimpd 454 . . . . . 6 ((𝑅 ∈ Ring ∧ 𝑥𝑈) → ((𝑦 ∈ (Base‘𝑅) ∧ ∃𝑚 ∈ (Base‘𝑅)((𝑦(.r𝑅)𝑥) = (1r𝑅) ∧ (𝑚(.r‘(oppr𝑅))𝑥) = (1r𝑅))) → (𝑦𝑈 ∧ (𝑦(.r𝑅)𝑥) = (1r𝑅))))
6968reximdv2 3161 . . . . 5 ((𝑅 ∈ Ring ∧ 𝑥𝑈) → (∃𝑦 ∈ (Base‘𝑅)∃𝑚 ∈ (Base‘𝑅)((𝑦(.r𝑅)𝑥) = (1r𝑅) ∧ (𝑚(.r‘(oppr𝑅))𝑥) = (1r𝑅)) → ∃𝑦𝑈 (𝑦(.r𝑅)𝑥) = (1r𝑅)))
7037, 69biimtrrid 242 . . . 4 ((𝑅 ∈ Ring ∧ 𝑥𝑈) → ((∃𝑦 ∈ (Base‘𝑅)(𝑦(.r𝑅)𝑥) = (1r𝑅) ∧ ∃𝑚 ∈ (Base‘𝑅)(𝑚(.r‘(oppr𝑅))𝑥) = (1r𝑅)) → ∃𝑦𝑈 (𝑦(.r𝑅)𝑥) = (1r𝑅)))
7136, 70sylbid 239 . . 3 ((𝑅 ∈ Ring ∧ 𝑥𝑈) → ((𝑥(∥r𝑅)(1r𝑅) ∧ 𝑥(∥r‘(oppr𝑅))(1r𝑅)) → ∃𝑦𝑈 (𝑦(.r𝑅)𝑥) = (1r𝑅)))
7228, 71mpd 15 . 2 ((𝑅 ∈ Ring ∧ 𝑥𝑈) → ∃𝑦𝑈 (𝑦(.r𝑅)𝑥) = (1r𝑅))
734, 10, 11, 18, 20, 22, 72isgrpde 18771 1 (𝑅 ∈ Ring → 𝐺 ∈ Grp)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 396  w3a 1087   = wceq 1541  wcel 2106  wrex 3073  Vcvv 3445   class class class wbr 5105  cfv 6496  (class class class)co 7357  Basecbs 17083  s cress 17112  +gcplusg 17133  .rcmulr 17134  Grpcgrp 18748  mulGrpcmgp 19896  1rcur 19913  Ringcrg 19964  opprcoppr 20048  rcdsr 20067  Unitcui 20068
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-rep 5242  ax-sep 5256  ax-nul 5263  ax-pow 5320  ax-pr 5384  ax-un 7672  ax-cnex 11107  ax-resscn 11108  ax-1cn 11109  ax-icn 11110  ax-addcl 11111  ax-addrcl 11112  ax-mulcl 11113  ax-mulrcl 11114  ax-mulcom 11115  ax-addass 11116  ax-mulass 11117  ax-distr 11118  ax-i2m1 11119  ax-1ne0 11120  ax-1rid 11121  ax-rnegex 11122  ax-rrecex 11123  ax-cnre 11124  ax-pre-lttri 11125  ax-pre-lttrn 11126  ax-pre-ltadd 11127  ax-pre-mulgt0 11128
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-nel 3050  df-ral 3065  df-rex 3074  df-rmo 3353  df-reu 3354  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-pss 3929  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-op 4593  df-uni 4866  df-iun 4956  df-br 5106  df-opab 5168  df-mpt 5189  df-tr 5223  df-id 5531  df-eprel 5537  df-po 5545  df-so 5546  df-fr 5588  df-we 5590  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-pred 6253  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-f1 6501  df-fo 6502  df-f1o 6503  df-fv 6504  df-riota 7313  df-ov 7360  df-oprab 7361  df-mpo 7362  df-om 7803  df-2nd 7922  df-tpos 8157  df-frecs 8212  df-wrecs 8243  df-recs 8317  df-rdg 8356  df-er 8648  df-en 8884  df-dom 8885  df-sdom 8886  df-pnf 11191  df-mnf 11192  df-xr 11193  df-ltxr 11194  df-le 11195  df-sub 11387  df-neg 11388  df-nn 12154  df-2 12216  df-3 12217  df-sets 17036  df-slot 17054  df-ndx 17066  df-base 17084  df-ress 17113  df-plusg 17146  df-mulr 17147  df-0g 17323  df-mgm 18497  df-sgrp 18546  df-mnd 18557  df-grp 18751  df-mgp 19897  df-ur 19914  df-ring 19966  df-oppr 20049  df-dvdsr 20070  df-unit 20071
This theorem is referenced by:  unitabl  20097  unitsubm  20099  unitinvcl  20103  unitinvinv  20104  unitlinv  20106  unitrinv  20107  rhmunitinv  20184  isdrng2  20198  subrgugrp  20241  expghm  20896  invrvald  22025  nrginvrcn  24056  nrgtdrg  24057  dchrfi  26603  dchrghm  26604  dchrabs  26608  dchrptlem1  26612  dchrptlem2  26613  dchrptlem3  26614  dchrsum2  26616  rdivmuldivd  32071  dvrcan5  32073  idomodle  41509  proot1mul  41512  proot1hash  41513  proot1ex  41514
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