Theorem unitgrp 18490

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.

Step	Hyp	Ref	Expression
1		unitmulcl.1	. . . 4 ⊢ 𝑈 = (Unit‘𝑅)
2		unitgrp.2	. . . 4 ⊢ 𝐺 = ((mulGrp‘𝑅) ↾s 𝑈)
3	1, 2	unitgrpbas 18489	. . 3 ⊢ 𝑈 = (Base‘𝐺)
4	3	a1i 11	. 2 ⊢ (𝑅 ∈ Ring → 𝑈 = (Base‘𝐺))
5		fvex 6113	. . . 4 ⊢ (Base‘𝐺) ∈ V
6	3, 5	eqeltri 2684	. . 3 ⊢ 𝑈 ∈ V
7		eqid 2610	. . . . 5 ⊢ (mulGrp‘𝑅) = (mulGrp‘𝑅)
8		eqid 2610	. . . . 5 ⊢ (.r‘𝑅) = (.r‘𝑅)
9	7, 8	mgpplusg 18316	. . . 4 ⊢ (.r‘𝑅) = (+g‘(mulGrp‘𝑅))
10	2, 9	ressplusg 15818	. . 3 ⊢ (𝑈 ∈ V → (.r‘𝑅) = (+g‘𝐺))
11	6, 10	mp1i 13	. 2 ⊢ (𝑅 ∈ Ring → (.r‘𝑅) = (+g‘𝐺))
12	1, 8	unitmulcl 18487	. 2 ⊢ ((𝑅 ∈ Ring ∧ 𝑥 ∈ 𝑈 ∧ 𝑦 ∈ 𝑈) → (𝑥(.r‘𝑅)𝑦) ∈ 𝑈)
13		eqid 2610	. . . . 5 ⊢ (Base‘𝑅) = (Base‘𝑅)
14	13, 1	unitcl 18482	. . . 4 ⊢ (𝑥 ∈ 𝑈 → 𝑥 ∈ (Base‘𝑅))
15	13, 1	unitcl 18482	. . . 4 ⊢ (𝑦 ∈ 𝑈 → 𝑦 ∈ (Base‘𝑅))
16	13, 1	unitcl 18482	. . . 4 ⊢ (𝑧 ∈ 𝑈 → 𝑧 ∈ (Base‘𝑅))
17	14, 15, 16	3anim123i 1240	. . 3 ⊢ ((𝑥 ∈ 𝑈 ∧ 𝑦 ∈ 𝑈 ∧ 𝑧 ∈ 𝑈) → (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅) ∧ 𝑧 ∈ (Base‘𝑅)))
18	13, 8	ringass 18387	. . 3 ⊢ ((𝑅 ∈ Ring ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅) ∧ 𝑧 ∈ (Base‘𝑅))) → ((𝑥(.r‘𝑅)𝑦)(.r‘𝑅)𝑧) = (𝑥(.r‘𝑅)(𝑦(.r‘𝑅)𝑧)))
19	17, 18	sylan2 490	. 2 ⊢ ((𝑅 ∈ Ring ∧ (𝑥 ∈ 𝑈 ∧ 𝑦 ∈ 𝑈 ∧ 𝑧 ∈ 𝑈)) → ((𝑥(.r‘𝑅)𝑦)(.r‘𝑅)𝑧) = (𝑥(.r‘𝑅)(𝑦(.r‘𝑅)𝑧)))
20		eqid 2610	. . 3 ⊢ (1r‘𝑅) = (1r‘𝑅)
21	1, 20	1unit 18481	. 2 ⊢ (𝑅 ∈ Ring → (1r‘𝑅) ∈ 𝑈)
22	13, 8, 20	ringlidm 18394	. . 3 ⊢ ((𝑅 ∈ Ring ∧ 𝑥 ∈ (Base‘𝑅)) → ((1r‘𝑅)(.r‘𝑅)𝑥) = 𝑥)
23	14, 22	sylan2 490	. 2 ⊢ ((𝑅 ∈ Ring ∧ 𝑥 ∈ 𝑈) → ((1r‘𝑅)(.r‘𝑅)𝑥) = 𝑥)
24		simpr 476	. . . 4 ⊢ ((𝑅 ∈ Ring ∧ 𝑥 ∈ 𝑈) → 𝑥 ∈ 𝑈)
25		eqid 2610	. . . . 5 ⊢ (∥r‘𝑅) = (∥r‘𝑅)
26		eqid 2610	. . . . 5 ⊢ (oppr‘𝑅) = (oppr‘𝑅)
27		eqid 2610	. . . . 5 ⊢ (∥r‘(oppr‘𝑅)) = (∥r‘(oppr‘𝑅))
28	1, 20, 25, 26, 27	isunit 18480	. . . 4 ⊢ (𝑥 ∈ 𝑈 ↔ (𝑥(∥r‘𝑅)(1r‘𝑅) ∧ 𝑥(∥r‘(oppr‘𝑅))(1r‘𝑅)))
29	24, 28	sylib 207	. . 3 ⊢ ((𝑅 ∈ Ring ∧ 𝑥 ∈ 𝑈) → (𝑥(∥r‘𝑅)(1r‘𝑅) ∧ 𝑥(∥r‘(oppr‘𝑅))(1r‘𝑅)))
30	14	adantl 481	. . . . . 6 ⊢ ((𝑅 ∈ Ring ∧ 𝑥 ∈ 𝑈) → 𝑥 ∈ (Base‘𝑅))
31	13, 25, 8	dvdsr2 18470	. . . . . 6 ⊢ (𝑥 ∈ (Base‘𝑅) → (𝑥(∥r‘𝑅)(1r‘𝑅) ↔ ∃𝑦 ∈ (Base‘𝑅)(𝑦(.r‘𝑅)𝑥) = (1r‘𝑅)))
32	30, 31	syl 17	. . . . 5 ⊢ ((𝑅 ∈ Ring ∧ 𝑥 ∈ 𝑈) → (𝑥(∥r‘𝑅)(1r‘𝑅) ↔ ∃𝑦 ∈ (Base‘𝑅)(𝑦(.r‘𝑅)𝑥) = (1r‘𝑅)))
33	26, 13	opprbas 18452	. . . . . . 7 ⊢ (Base‘𝑅) = (Base‘(oppr‘𝑅))
34		eqid 2610	. . . . . . 7 ⊢ (.r‘(oppr‘𝑅)) = (.r‘(oppr‘𝑅))
35	33, 27, 34	dvdsr2 18470	. . . . . 6 ⊢ (𝑥 ∈ (Base‘𝑅) → (𝑥(∥r‘(oppr‘𝑅))(1r‘𝑅) ↔ ∃𝑚 ∈ (Base‘𝑅)(𝑚(.r‘(oppr‘𝑅))𝑥) = (1r‘𝑅)))
36	30, 35	syl 17	. . . . 5 ⊢ ((𝑅 ∈ Ring ∧ 𝑥 ∈ 𝑈) → (𝑥(∥r‘(oppr‘𝑅))(1r‘𝑅) ↔ ∃𝑚 ∈ (Base‘𝑅)(𝑚(.r‘(oppr‘𝑅))𝑥) = (1r‘𝑅)))
37	32, 36	anbi12d 743	. . . 4 ⊢ ((𝑅 ∈ Ring ∧ 𝑥 ∈ 𝑈) → ((𝑥(∥r‘𝑅)(1r‘𝑅) ∧ 𝑥(∥r‘(oppr‘𝑅))(1r‘𝑅)) ↔ (∃𝑦 ∈ (Base‘𝑅)(𝑦(.r‘𝑅)𝑥) = (1r‘𝑅) ∧ ∃𝑚 ∈ (Base‘𝑅)(𝑚(.r‘(oppr‘𝑅))𝑥) = (1r‘𝑅))))
38		reeanv 3086	. . . . 5 ⊢ (∃𝑦 ∈ (Base‘𝑅)∃𝑚 ∈ (Base‘𝑅)((𝑦(.r‘𝑅)𝑥) = (1r‘𝑅) ∧ (𝑚(.r‘(oppr‘𝑅))𝑥) = (1r‘𝑅)) ↔ (∃𝑦 ∈ (Base‘𝑅)(𝑦(.r‘𝑅)𝑥) = (1r‘𝑅) ∧ ∃𝑚 ∈ (Base‘𝑅)(𝑚(.r‘(oppr‘𝑅))𝑥) = (1r‘𝑅)))
39		simprl 790	. . . . . . . . . . . 12 ⊢ ((((𝑅 ∈ Ring ∧ 𝑥 ∈ 𝑈) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ (𝑚 ∈ (Base‘𝑅) ∧ ((𝑦(.r‘𝑅)𝑥) = (1r‘𝑅) ∧ (𝑚(.r‘(oppr‘𝑅))𝑥) = (1r‘𝑅)))) → 𝑚 ∈ (Base‘𝑅))
40	30	ad2antrr 758	. . . . . . . . . . . 12 ⊢ ((((𝑅 ∈ Ring ∧ 𝑥 ∈ 𝑈) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ (𝑚 ∈ (Base‘𝑅) ∧ ((𝑦(.r‘𝑅)𝑥) = (1r‘𝑅) ∧ (𝑚(.r‘(oppr‘𝑅))𝑥) = (1r‘𝑅)))) → 𝑥 ∈ (Base‘𝑅))
41	13, 25, 8	dvdsrmul 18471	. . . . . . . . . . . 12 ⊢ ((𝑚 ∈ (Base‘𝑅) ∧ 𝑥 ∈ (Base‘𝑅)) → 𝑚(∥r‘𝑅)(𝑥(.r‘𝑅)𝑚))
42	39, 40, 41	syl2anc 691	. . . . . . . . . . 11 ⊢ ((((𝑅 ∈ Ring ∧ 𝑥 ∈ 𝑈) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ (𝑚 ∈ (Base‘𝑅) ∧ ((𝑦(.r‘𝑅)𝑥) = (1r‘𝑅) ∧ (𝑚(.r‘(oppr‘𝑅))𝑥) = (1r‘𝑅)))) → 𝑚(∥r‘𝑅)(𝑥(.r‘𝑅)𝑚))
43		simplll 794	. . . . . . . . . . . . . 14 ⊢ ((((𝑅 ∈ Ring ∧ 𝑥 ∈ 𝑈) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ (𝑚 ∈ (Base‘𝑅) ∧ ((𝑦(.r‘𝑅)𝑥) = (1r‘𝑅) ∧ (𝑚(.r‘(oppr‘𝑅))𝑥) = (1r‘𝑅)))) → 𝑅 ∈ Ring)
44		simplr 788	. . . . . . . . . . . . . 14 ⊢ ((((𝑅 ∈ Ring ∧ 𝑥 ∈ 𝑈) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ (𝑚 ∈ (Base‘𝑅) ∧ ((𝑦(.r‘𝑅)𝑥) = (1r‘𝑅) ∧ (𝑚(.r‘(oppr‘𝑅))𝑥) = (1r‘𝑅)))) → 𝑦 ∈ (Base‘𝑅))
45	13, 8	ringass 18387	. . . . . . . . . . . . . 14 ⊢ ((𝑅 ∈ Ring ∧ (𝑦 ∈ (Base‘𝑅) ∧ 𝑥 ∈ (Base‘𝑅) ∧ 𝑚 ∈ (Base‘𝑅))) → ((𝑦(.r‘𝑅)𝑥)(.r‘𝑅)𝑚) = (𝑦(.r‘𝑅)(𝑥(.r‘𝑅)𝑚)))
46	43, 44, 40, 39, 45	syl13anc 1320	. . . . . . . . . . . . 13 ⊢ ((((𝑅 ∈ Ring ∧ 𝑥 ∈ 𝑈) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ (𝑚 ∈ (Base‘𝑅) ∧ ((𝑦(.r‘𝑅)𝑥) = (1r‘𝑅) ∧ (𝑚(.r‘(oppr‘𝑅))𝑥) = (1r‘𝑅)))) → ((𝑦(.r‘𝑅)𝑥)(.r‘𝑅)𝑚) = (𝑦(.r‘𝑅)(𝑥(.r‘𝑅)𝑚)))
47		simprrl 800	. . . . . . . . . . . . . 14 ⊢ ((((𝑅 ∈ Ring ∧ 𝑥 ∈ 𝑈) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ (𝑚 ∈ (Base‘𝑅) ∧ ((𝑦(.r‘𝑅)𝑥) = (1r‘𝑅) ∧ (𝑚(.r‘(oppr‘𝑅))𝑥) = (1r‘𝑅)))) → (𝑦(.r‘𝑅)𝑥) = (1r‘𝑅))
48	47	oveq1d 6564	. . . . . . . . . . . . 13 ⊢ ((((𝑅 ∈ Ring ∧ 𝑥 ∈ 𝑈) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ (𝑚 ∈ (Base‘𝑅) ∧ ((𝑦(.r‘𝑅)𝑥) = (1r‘𝑅) ∧ (𝑚(.r‘(oppr‘𝑅))𝑥) = (1r‘𝑅)))) → ((𝑦(.r‘𝑅)𝑥)(.r‘𝑅)𝑚) = ((1r‘𝑅)(.r‘𝑅)𝑚))
49	13, 8, 26, 34	opprmul 18449	. . . . . . . . . . . . . . 15 ⊢ (𝑚(.r‘(oppr‘𝑅))𝑥) = (𝑥(.r‘𝑅)𝑚)
50		simprrr 801	. . . . . . . . . . . . . . 15 ⊢ ((((𝑅 ∈ Ring ∧ 𝑥 ∈ 𝑈) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ (𝑚 ∈ (Base‘𝑅) ∧ ((𝑦(.r‘𝑅)𝑥) = (1r‘𝑅) ∧ (𝑚(.r‘(oppr‘𝑅))𝑥) = (1r‘𝑅)))) → (𝑚(.r‘(oppr‘𝑅))𝑥) = (1r‘𝑅))
51	49, 50	syl5eqr 2658	. . . . . . . . . . . . . 14 ⊢ ((((𝑅 ∈ Ring ∧ 𝑥 ∈ 𝑈) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ (𝑚 ∈ (Base‘𝑅) ∧ ((𝑦(.r‘𝑅)𝑥) = (1r‘𝑅) ∧ (𝑚(.r‘(oppr‘𝑅))𝑥) = (1r‘𝑅)))) → (𝑥(.r‘𝑅)𝑚) = (1r‘𝑅))
52	51	oveq2d 6565	. . . . . . . . . . . . 13 ⊢ ((((𝑅 ∈ Ring ∧ 𝑥 ∈ 𝑈) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ (𝑚 ∈ (Base‘𝑅) ∧ ((𝑦(.r‘𝑅)𝑥) = (1r‘𝑅) ∧ (𝑚(.r‘(oppr‘𝑅))𝑥) = (1r‘𝑅)))) → (𝑦(.r‘𝑅)(𝑥(.r‘𝑅)𝑚)) = (𝑦(.r‘𝑅)(1r‘𝑅)))
53	46, 48, 52	3eqtr3d 2652	. . . . . . . . . . . 12 ⊢ ((((𝑅 ∈ Ring ∧ 𝑥 ∈ 𝑈) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ (𝑚 ∈ (Base‘𝑅) ∧ ((𝑦(.r‘𝑅)𝑥) = (1r‘𝑅) ∧ (𝑚(.r‘(oppr‘𝑅))𝑥) = (1r‘𝑅)))) → ((1r‘𝑅)(.r‘𝑅)𝑚) = (𝑦(.r‘𝑅)(1r‘𝑅)))
54	13, 8, 20	ringlidm 18394	. . . . . . . . . . . . 13 ⊢ ((𝑅 ∈ Ring ∧ 𝑚 ∈ (Base‘𝑅)) → ((1r‘𝑅)(.r‘𝑅)𝑚) = 𝑚)
55	43, 39, 54	syl2anc 691	. . . . . . . . . . . 12 ⊢ ((((𝑅 ∈ Ring ∧ 𝑥 ∈ 𝑈) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ (𝑚 ∈ (Base‘𝑅) ∧ ((𝑦(.r‘𝑅)𝑥) = (1r‘𝑅) ∧ (𝑚(.r‘(oppr‘𝑅))𝑥) = (1r‘𝑅)))) → ((1r‘𝑅)(.r‘𝑅)𝑚) = 𝑚)
56	13, 8, 20	ringridm 18395	. . . . . . . . . . . . 13 ⊢ ((𝑅 ∈ Ring ∧ 𝑦 ∈ (Base‘𝑅)) → (𝑦(.r‘𝑅)(1r‘𝑅)) = 𝑦)
57	43, 44, 56	syl2anc 691	. . . . . . . . . . . 12 ⊢ ((((𝑅 ∈ Ring ∧ 𝑥 ∈ 𝑈) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ (𝑚 ∈ (Base‘𝑅) ∧ ((𝑦(.r‘𝑅)𝑥) = (1r‘𝑅) ∧ (𝑚(.r‘(oppr‘𝑅))𝑥) = (1r‘𝑅)))) → (𝑦(.r‘𝑅)(1r‘𝑅)) = 𝑦)
58	53, 55, 57	3eqtr3d 2652	. . . . . . . . . . 11 ⊢ ((((𝑅 ∈ Ring ∧ 𝑥 ∈ 𝑈) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ (𝑚 ∈ (Base‘𝑅) ∧ ((𝑦(.r‘𝑅)𝑥) = (1r‘𝑅) ∧ (𝑚(.r‘(oppr‘𝑅))𝑥) = (1r‘𝑅)))) → 𝑚 = 𝑦)
59	42, 58, 51	3brtr3d 4614	. . . . . . . . . 10 ⊢ ((((𝑅 ∈ Ring ∧ 𝑥 ∈ 𝑈) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ (𝑚 ∈ (Base‘𝑅) ∧ ((𝑦(.r‘𝑅)𝑥) = (1r‘𝑅) ∧ (𝑚(.r‘(oppr‘𝑅))𝑥) = (1r‘𝑅)))) → 𝑦(∥r‘𝑅)(1r‘𝑅))
60	33, 27, 34	dvdsrmul 18471	. . . . . . . . . . . 12 ⊢ ((𝑦 ∈ (Base‘𝑅) ∧ 𝑥 ∈ (Base‘𝑅)) → 𝑦(∥r‘(oppr‘𝑅))(𝑥(.r‘(oppr‘𝑅))𝑦))
61	44, 40, 60	syl2anc 691	. . . . . . . . . . 11 ⊢ ((((𝑅 ∈ Ring ∧ 𝑥 ∈ 𝑈) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ (𝑚 ∈ (Base‘𝑅) ∧ ((𝑦(.r‘𝑅)𝑥) = (1r‘𝑅) ∧ (𝑚(.r‘(oppr‘𝑅))𝑥) = (1r‘𝑅)))) → 𝑦(∥r‘(oppr‘𝑅))(𝑥(.r‘(oppr‘𝑅))𝑦))
62	13, 8, 26, 34	opprmul 18449	. . . . . . . . . . . 12 ⊢ (𝑥(.r‘(oppr‘𝑅))𝑦) = (𝑦(.r‘𝑅)𝑥)
63	62, 47	syl5eq 2656	. . . . . . . . . . 11 ⊢ ((((𝑅 ∈ Ring ∧ 𝑥 ∈ 𝑈) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ (𝑚 ∈ (Base‘𝑅) ∧ ((𝑦(.r‘𝑅)𝑥) = (1r‘𝑅) ∧ (𝑚(.r‘(oppr‘𝑅))𝑥) = (1r‘𝑅)))) → (𝑥(.r‘(oppr‘𝑅))𝑦) = (1r‘𝑅))
64	61, 63	breqtrd 4609	. . . . . . . . . 10 ⊢ ((((𝑅 ∈ Ring ∧ 𝑥 ∈ 𝑈) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ (𝑚 ∈ (Base‘𝑅) ∧ ((𝑦(.r‘𝑅)𝑥) = (1r‘𝑅) ∧ (𝑚(.r‘(oppr‘𝑅))𝑥) = (1r‘𝑅)))) → 𝑦(∥r‘(oppr‘𝑅))(1r‘𝑅))
65	1, 20, 25, 26, 27	isunit 18480	. . . . . . . . . 10 ⊢ (𝑦 ∈ 𝑈 ↔ (𝑦(∥r‘𝑅)(1r‘𝑅) ∧ 𝑦(∥r‘(oppr‘𝑅))(1r‘𝑅)))
66	59, 64, 65	sylanbrc 695	. . . . . . . . 9 ⊢ ((((𝑅 ∈ Ring ∧ 𝑥 ∈ 𝑈) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ (𝑚 ∈ (Base‘𝑅) ∧ ((𝑦(.r‘𝑅)𝑥) = (1r‘𝑅) ∧ (𝑚(.r‘(oppr‘𝑅))𝑥) = (1r‘𝑅)))) → 𝑦 ∈ 𝑈)
67	66, 47	jca 553	. . . . . . . 8 ⊢ ((((𝑅 ∈ Ring ∧ 𝑥 ∈ 𝑈) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ (𝑚 ∈ (Base‘𝑅) ∧ ((𝑦(.r‘𝑅)𝑥) = (1r‘𝑅) ∧ (𝑚(.r‘(oppr‘𝑅))𝑥) = (1r‘𝑅)))) → (𝑦 ∈ 𝑈 ∧ (𝑦(.r‘𝑅)𝑥) = (1r‘𝑅)))
68	67	rexlimdvaa 3014	. . . . . . 7 ⊢ (((𝑅 ∈ Ring ∧ 𝑥 ∈ 𝑈) ∧ 𝑦 ∈ (Base‘𝑅)) → (∃𝑚 ∈ (Base‘𝑅)((𝑦(.r‘𝑅)𝑥) = (1r‘𝑅) ∧ (𝑚(.r‘(oppr‘𝑅))𝑥) = (1r‘𝑅)) → (𝑦 ∈ 𝑈 ∧ (𝑦(.r‘𝑅)𝑥) = (1r‘𝑅))))
69	68	expimpd 627	. . . . . 6 ⊢ ((𝑅 ∈ Ring ∧ 𝑥 ∈ 𝑈) → ((𝑦 ∈ (Base‘𝑅) ∧ ∃𝑚 ∈ (Base‘𝑅)((𝑦(.r‘𝑅)𝑥) = (1r‘𝑅) ∧ (𝑚(.r‘(oppr‘𝑅))𝑥) = (1r‘𝑅))) → (𝑦 ∈ 𝑈 ∧ (𝑦(.r‘𝑅)𝑥) = (1r‘𝑅))))
70	69	reximdv2 2997	. . . . 5 ⊢ ((𝑅 ∈ Ring ∧ 𝑥 ∈ 𝑈) → (∃𝑦 ∈ (Base‘𝑅)∃𝑚 ∈ (Base‘𝑅)((𝑦(.r‘𝑅)𝑥) = (1r‘𝑅) ∧ (𝑚(.r‘(oppr‘𝑅))𝑥) = (1r‘𝑅)) → ∃𝑦 ∈ 𝑈 (𝑦(.r‘𝑅)𝑥) = (1r‘𝑅)))
71	38, 70	syl5bir 232	. . . 4 ⊢ ((𝑅 ∈ Ring ∧ 𝑥 ∈ 𝑈) → ((∃𝑦 ∈ (Base‘𝑅)(𝑦(.r‘𝑅)𝑥) = (1r‘𝑅) ∧ ∃𝑚 ∈ (Base‘𝑅)(𝑚(.r‘(oppr‘𝑅))𝑥) = (1r‘𝑅)) → ∃𝑦 ∈ 𝑈 (𝑦(.r‘𝑅)𝑥) = (1r‘𝑅)))
72	37, 71	sylbid 229	. . 3 ⊢ ((𝑅 ∈ Ring ∧ 𝑥 ∈ 𝑈) → ((𝑥(∥r‘𝑅)(1r‘𝑅) ∧ 𝑥(∥r‘(oppr‘𝑅))(1r‘𝑅)) → ∃𝑦 ∈ 𝑈 (𝑦(.r‘𝑅)𝑥) = (1r‘𝑅)))
73	29, 72	mpd 15	. 2 ⊢ ((𝑅 ∈ Ring ∧ 𝑥 ∈ 𝑈) → ∃𝑦 ∈ 𝑈 (𝑦(.r‘𝑅)𝑥) = (1r‘𝑅))
74	4, 11, 12, 19, 21, 23, 73	isgrpde 17266	1 ⊢ (𝑅 ∈ Ring → 𝐺 ∈ Grp)

Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383   ∧ w3a 1031   = wceq 1475   ∈ wcel 1977  ∃wrex 2897  Vcvv 3173   class class class wbr 4583  ‘cfv 5804  (class class class)co 6549  Basecbs 15695   ↾_s cress 15696  +_gcplusg 15768  ._rcmulr 15769  Grpcgrp 17245  mulGrpcmgp 18312  1_rcur 18324  Ringcrg 18370  opp_rcoppr 18445  ∥_rcdsr 18461  Unitcui 18462

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-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-mgp 18313  df-ur 18325  df-ring 18372  df-oppr 18446  df-dvdsr 18464  df-unit 18465

This theorem is referenced by:  unitabl  18491  unitsubm  18493  unitinvcl  18497  unitinvinv  18498  unitlinv  18500  unitrinv  18501  isdrng2  18580  subrgugrp  18622  expghm  19663  invrvald  20301  nrginvrcn  22306  nrgtdrg  22307  dchrfi  24780  dchrghm  24781  dchrabs  24785  dchrptlem1  24789  dchrptlem2  24790  dchrptlem3  24791  dchrsum2  24793  rdivmuldivd  29122  dvrcan5  29124  rhmunitinv  29153  idomodle  36793  proot1mul  36796  proot1hash  36797  proot1ex  36798