Theorem unitgrp 13612

Description: The group of units is a group under multiplication. (Contributed by Mario Carneiro, 2-Dec-2014.)

Hypotheses

Ref	Expression
unitgrp.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		unitgrp.1	. . . 4 ⊢ 𝑈 = (Unit‘𝑅)
2	1	a1i 9	. . 3 ⊢ (𝑅 ∈ Ring → 𝑈 = (Unit‘𝑅))
3		unitgrp.2	. . . 4 ⊢ 𝐺 = ((mulGrp‘𝑅) ↾s 𝑈)
4	3	a1i 9	. . 3 ⊢ (𝑅 ∈ Ring → 𝐺 = ((mulGrp‘𝑅) ↾s 𝑈))
5		ringsrg 13543	. . 3 ⊢ (𝑅 ∈ Ring → 𝑅 ∈ SRing)
6	2, 4, 5	unitgrpbasd 13611	. 2 ⊢ (𝑅 ∈ Ring → 𝑈 = (Base‘𝐺))
7		eqid 2193	. . . 4 ⊢ (mulGrp‘𝑅) = (mulGrp‘𝑅)
8		eqid 2193	. . . 4 ⊢ (.r‘𝑅) = (.r‘𝑅)
9	7, 8	mgpplusgg 13420	. . 3 ⊢ (𝑅 ∈ Ring → (.r‘𝑅) = (+g‘(mulGrp‘𝑅)))
10		basfn 12676	. . . . 5 ⊢ Base Fn V
11		elex 2771	. . . . 5 ⊢ (𝑅 ∈ Ring → 𝑅 ∈ V)
12		funfvex 5571	. . . . . 6 ⊢ ((Fun Base ∧ 𝑅 ∈ dom Base) → (Base‘𝑅) ∈ V)
13	12	funfni 5354	. . . . 5 ⊢ ((Base Fn V ∧ 𝑅 ∈ V) → (Base‘𝑅) ∈ V)
14	10, 11, 13	sylancr 414	. . . 4 ⊢ (𝑅 ∈ Ring → (Base‘𝑅) ∈ V)
15		eqidd 2194	. . . . 5 ⊢ (𝑅 ∈ Ring → (Base‘𝑅) = (Base‘𝑅))
16	15, 2, 5	unitssd 13605	. . . 4 ⊢ (𝑅 ∈ Ring → 𝑈 ⊆ (Base‘𝑅))
17	14, 16	ssexd 4169	. . 3 ⊢ (𝑅 ∈ Ring → 𝑈 ∈ V)
18	7	mgpex 13421	. . 3 ⊢ (𝑅 ∈ Ring → (mulGrp‘𝑅) ∈ V)
19	4, 9, 17, 18	ressplusgd 12746	. 2 ⊢ (𝑅 ∈ Ring → (.r‘𝑅) = (+g‘𝐺))
20	1, 8	unitmulcl 13609	. 2 ⊢ ((𝑅 ∈ Ring ∧ 𝑥 ∈ 𝑈 ∧ 𝑦 ∈ 𝑈) → (𝑥(.r‘𝑅)𝑦) ∈ 𝑈)
21		eqidd 2194	. . . . 5 ⊢ ((𝑅 ∈ Ring ∧ (𝑥 ∈ 𝑈 ∧ 𝑦 ∈ 𝑈 ∧ 𝑧 ∈ 𝑈)) → (Base‘𝑅) = (Base‘𝑅))
22	1	a1i 9	. . . . 5 ⊢ ((𝑅 ∈ Ring ∧ (𝑥 ∈ 𝑈 ∧ 𝑦 ∈ 𝑈 ∧ 𝑧 ∈ 𝑈)) → 𝑈 = (Unit‘𝑅))
23	5	adantr 276	. . . . 5 ⊢ ((𝑅 ∈ Ring ∧ (𝑥 ∈ 𝑈 ∧ 𝑦 ∈ 𝑈 ∧ 𝑧 ∈ 𝑈)) → 𝑅 ∈ SRing)
24		simpr1 1005	. . . . 5 ⊢ ((𝑅 ∈ Ring ∧ (𝑥 ∈ 𝑈 ∧ 𝑦 ∈ 𝑈 ∧ 𝑧 ∈ 𝑈)) → 𝑥 ∈ 𝑈)
25	21, 22, 23, 24	unitcld 13604	. . . 4 ⊢ ((𝑅 ∈ Ring ∧ (𝑥 ∈ 𝑈 ∧ 𝑦 ∈ 𝑈 ∧ 𝑧 ∈ 𝑈)) → 𝑥 ∈ (Base‘𝑅))
26		simpr2 1006	. . . . 5 ⊢ ((𝑅 ∈ Ring ∧ (𝑥 ∈ 𝑈 ∧ 𝑦 ∈ 𝑈 ∧ 𝑧 ∈ 𝑈)) → 𝑦 ∈ 𝑈)
27	21, 22, 23, 26	unitcld 13604	. . . 4 ⊢ ((𝑅 ∈ Ring ∧ (𝑥 ∈ 𝑈 ∧ 𝑦 ∈ 𝑈 ∧ 𝑧 ∈ 𝑈)) → 𝑦 ∈ (Base‘𝑅))
28		simpr3 1007	. . . . 5 ⊢ ((𝑅 ∈ Ring ∧ (𝑥 ∈ 𝑈 ∧ 𝑦 ∈ 𝑈 ∧ 𝑧 ∈ 𝑈)) → 𝑧 ∈ 𝑈)
29	21, 22, 23, 28	unitcld 13604	. . . 4 ⊢ ((𝑅 ∈ Ring ∧ (𝑥 ∈ 𝑈 ∧ 𝑦 ∈ 𝑈 ∧ 𝑧 ∈ 𝑈)) → 𝑧 ∈ (Base‘𝑅))
30	25, 27, 29	3jca 1179	. . 3 ⊢ ((𝑅 ∈ Ring ∧ (𝑥 ∈ 𝑈 ∧ 𝑦 ∈ 𝑈 ∧ 𝑧 ∈ 𝑈)) → (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅) ∧ 𝑧 ∈ (Base‘𝑅)))
31		eqid 2193	. . . 4 ⊢ (Base‘𝑅) = (Base‘𝑅)
32	31, 8	ringass 13512	. . 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)))
33	30, 32	syldan 282	. 2 ⊢ ((𝑅 ∈ Ring ∧ (𝑥 ∈ 𝑈 ∧ 𝑦 ∈ 𝑈 ∧ 𝑧 ∈ 𝑈)) → ((𝑥(.r‘𝑅)𝑦)(.r‘𝑅)𝑧) = (𝑥(.r‘𝑅)(𝑦(.r‘𝑅)𝑧)))
34		eqid 2193	. . 3 ⊢ (1r‘𝑅) = (1r‘𝑅)
35	1, 34	1unit 13603	. 2 ⊢ (𝑅 ∈ Ring → (1r‘𝑅) ∈ 𝑈)
36		eqidd 2194	. . . 4 ⊢ ((𝑅 ∈ Ring ∧ 𝑥 ∈ 𝑈) → (Base‘𝑅) = (Base‘𝑅))
37	1	a1i 9	. . . 4 ⊢ ((𝑅 ∈ Ring ∧ 𝑥 ∈ 𝑈) → 𝑈 = (Unit‘𝑅))
38	5	adantr 276	. . . 4 ⊢ ((𝑅 ∈ Ring ∧ 𝑥 ∈ 𝑈) → 𝑅 ∈ SRing)
39		simpr 110	. . . 4 ⊢ ((𝑅 ∈ Ring ∧ 𝑥 ∈ 𝑈) → 𝑥 ∈ 𝑈)
40	36, 37, 38, 39	unitcld 13604	. . 3 ⊢ ((𝑅 ∈ Ring ∧ 𝑥 ∈ 𝑈) → 𝑥 ∈ (Base‘𝑅))
41	31, 8, 34	ringlidm 13519	. . 3 ⊢ ((𝑅 ∈ Ring ∧ 𝑥 ∈ (Base‘𝑅)) → ((1r‘𝑅)(.r‘𝑅)𝑥) = 𝑥)
42	40, 41	syldan 282	. 2 ⊢ ((𝑅 ∈ Ring ∧ 𝑥 ∈ 𝑈) → ((1r‘𝑅)(.r‘𝑅)𝑥) = 𝑥)
43		eqidd 2194	. . . . 5 ⊢ ((𝑅 ∈ Ring ∧ 𝑥 ∈ 𝑈) → (1r‘𝑅) = (1r‘𝑅))
44		eqidd 2194	. . . . 5 ⊢ ((𝑅 ∈ Ring ∧ 𝑥 ∈ 𝑈) → (∥r‘𝑅) = (∥r‘𝑅))
45		eqidd 2194	. . . . 5 ⊢ ((𝑅 ∈ Ring ∧ 𝑥 ∈ 𝑈) → (oppr‘𝑅) = (oppr‘𝑅))
46		eqidd 2194	. . . . 5 ⊢ ((𝑅 ∈ Ring ∧ 𝑥 ∈ 𝑈) → (∥r‘(oppr‘𝑅)) = (∥r‘(oppr‘𝑅)))
47	37, 43, 44, 45, 46, 38	isunitd 13602	. . . 4 ⊢ ((𝑅 ∈ Ring ∧ 𝑥 ∈ 𝑈) → (𝑥 ∈ 𝑈 ↔ (𝑥(∥r‘𝑅)(1r‘𝑅) ∧ 𝑥(∥r‘(oppr‘𝑅))(1r‘𝑅))))
48	39, 47	mpbid 147	. . 3 ⊢ ((𝑅 ∈ Ring ∧ 𝑥 ∈ 𝑈) → (𝑥(∥r‘𝑅)(1r‘𝑅) ∧ 𝑥(∥r‘(oppr‘𝑅))(1r‘𝑅)))
49		eqidd 2194	. . . . . 6 ⊢ ((𝑅 ∈ Ring ∧ 𝑥 ∈ 𝑈) → (.r‘𝑅) = (.r‘𝑅))
50	36, 44, 38, 49, 40	dvdsr2d 13591	. . . . 5 ⊢ ((𝑅 ∈ Ring ∧ 𝑥 ∈ 𝑈) → (𝑥(∥r‘𝑅)(1r‘𝑅) ↔ ∃𝑦 ∈ (Base‘𝑅)(𝑦(.r‘𝑅)𝑥) = (1r‘𝑅)))
51		eqid 2193	. . . . . . . 8 ⊢ (oppr‘𝑅) = (oppr‘𝑅)
52	51, 31	opprbasg 13571	. . . . . . 7 ⊢ (𝑅 ∈ Ring → (Base‘𝑅) = (Base‘(oppr‘𝑅)))
53	52	adantr 276	. . . . . 6 ⊢ ((𝑅 ∈ Ring ∧ 𝑥 ∈ 𝑈) → (Base‘𝑅) = (Base‘(oppr‘𝑅)))
54	51	opprring 13575	. . . . . . . 8 ⊢ (𝑅 ∈ Ring → (oppr‘𝑅) ∈ Ring)
55		ringsrg 13543	. . . . . . . 8 ⊢ ((oppr‘𝑅) ∈ Ring → (oppr‘𝑅) ∈ SRing)
56	54, 55	syl 14	. . . . . . 7 ⊢ (𝑅 ∈ Ring → (oppr‘𝑅) ∈ SRing)
57	56	adantr 276	. . . . . 6 ⊢ ((𝑅 ∈ Ring ∧ 𝑥 ∈ 𝑈) → (oppr‘𝑅) ∈ SRing)
58		eqidd 2194	. . . . . 6 ⊢ ((𝑅 ∈ Ring ∧ 𝑥 ∈ 𝑈) → (.r‘(oppr‘𝑅)) = (.r‘(oppr‘𝑅)))
59	53, 46, 57, 58, 40	dvdsr2d 13591	. . . . 5 ⊢ ((𝑅 ∈ Ring ∧ 𝑥 ∈ 𝑈) → (𝑥(∥r‘(oppr‘𝑅))(1r‘𝑅) ↔ ∃𝑚 ∈ (Base‘𝑅)(𝑚(.r‘(oppr‘𝑅))𝑥) = (1r‘𝑅)))
60	50, 59	anbi12d 473	. . . 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))))
61		reeanv 2664	. . . . 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)))
62		eqidd 2194	. . . . . . . . . . . 12 ⊢ ((((𝑅 ∈ Ring ∧ 𝑥 ∈ 𝑈) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ (𝑚 ∈ (Base‘𝑅) ∧ ((𝑦(.r‘𝑅)𝑥) = (1r‘𝑅) ∧ (𝑚(.r‘(oppr‘𝑅))𝑥) = (1r‘𝑅)))) → (Base‘𝑅) = (Base‘𝑅))
63		eqidd 2194	. . . . . . . . . . . 12 ⊢ ((((𝑅 ∈ Ring ∧ 𝑥 ∈ 𝑈) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ (𝑚 ∈ (Base‘𝑅) ∧ ((𝑦(.r‘𝑅)𝑥) = (1r‘𝑅) ∧ (𝑚(.r‘(oppr‘𝑅))𝑥) = (1r‘𝑅)))) → (∥r‘𝑅) = (∥r‘𝑅))
64	38	ad2antrr 488	. . . . . . . . . . . 12 ⊢ ((((𝑅 ∈ Ring ∧ 𝑥 ∈ 𝑈) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ (𝑚 ∈ (Base‘𝑅) ∧ ((𝑦(.r‘𝑅)𝑥) = (1r‘𝑅) ∧ (𝑚(.r‘(oppr‘𝑅))𝑥) = (1r‘𝑅)))) → 𝑅 ∈ SRing)
65		eqidd 2194	. . . . . . . . . . . 12 ⊢ ((((𝑅 ∈ Ring ∧ 𝑥 ∈ 𝑈) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ (𝑚 ∈ (Base‘𝑅) ∧ ((𝑦(.r‘𝑅)𝑥) = (1r‘𝑅) ∧ (𝑚(.r‘(oppr‘𝑅))𝑥) = (1r‘𝑅)))) → (.r‘𝑅) = (.r‘𝑅))
66		simprl 529	. . . . . . . . . . . 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))
67	40	ad2antrr 488	. . . . . . . . . . . 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))
68	62, 63, 64, 65, 66, 67	dvdsrmuld 13592	. . . . . . . . . . 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))
69		simplll 533	. . . . . . . . . . . . . 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)
70		simplr 528	. . . . . . . . . . . . . 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))
71	31, 8	ringass 13512	. . . . . . . . . . . . . 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)))
72	69, 70, 67, 66, 71	syl13anc 1251	. . . . . . . . . . . . 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)))
73		simprrl 539	. . . . . . . . . . . . . 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))
74	73	oveq1d 5933	. . . . . . . . . . . . 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))
75	39	ad2antrr 488	. . . . . . . . . . . . . . . 16 ⊢ ((((𝑅 ∈ Ring ∧ 𝑥 ∈ 𝑈) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ (𝑚 ∈ (Base‘𝑅) ∧ ((𝑦(.r‘𝑅)𝑥) = (1r‘𝑅) ∧ (𝑚(.r‘(oppr‘𝑅))𝑥) = (1r‘𝑅)))) → 𝑥 ∈ 𝑈)
76		eqid 2193	. . . . . . . . . . . . . . . . 17 ⊢ (.r‘(oppr‘𝑅)) = (.r‘(oppr‘𝑅))
77	31, 8, 51, 76	opprmulg 13567	. . . . . . . . . . . . . . . 16 ⊢ ((𝑅 ∈ Ring ∧ 𝑚 ∈ (Base‘𝑅) ∧ 𝑥 ∈ 𝑈) → (𝑚(.r‘(oppr‘𝑅))𝑥) = (𝑥(.r‘𝑅)𝑚))
78	69, 66, 75, 77	syl3anc 1249	. . . . . . . . . . . . . . 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) = (𝑥(.r‘𝑅)𝑚))
79		simprrr 540	. . . . . . . . . . . . . . 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))
80	78, 79	eqtr3d 2228	. . . . . . . . . . . . . 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))
81	80	oveq2d 5934	. . . . . . . . . . . . 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)))
82	72, 74, 81	3eqtr3d 2234	. . . . . . . . . . . 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)))
83	31, 8, 34	ringlidm 13519	. . . . . . . . . . . . 13 ⊢ ((𝑅 ∈ Ring ∧ 𝑚 ∈ (Base‘𝑅)) → ((1r‘𝑅)(.r‘𝑅)𝑚) = 𝑚)
84	69, 66, 83	syl2anc 411	. . . . . . . . . . . 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)
85	31, 8, 34	ringridm 13520	. . . . . . . . . . . . 13 ⊢ ((𝑅 ∈ Ring ∧ 𝑦 ∈ (Base‘𝑅)) → (𝑦(.r‘𝑅)(1r‘𝑅)) = 𝑦)
86	69, 70, 85	syl2anc 411	. . . . . . . . . . . 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)
87	82, 84, 86	3eqtr3d 2234	. . . . . . . . . . 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)
88	68, 87, 80	3brtr3d 4060	. . . . . . . . . 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))
89	69, 52	syl 14	. . . . . . . . . . . 12 ⊢ ((((𝑅 ∈ Ring ∧ 𝑥 ∈ 𝑈) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ (𝑚 ∈ (Base‘𝑅) ∧ ((𝑦(.r‘𝑅)𝑥) = (1r‘𝑅) ∧ (𝑚(.r‘(oppr‘𝑅))𝑥) = (1r‘𝑅)))) → (Base‘𝑅) = (Base‘(oppr‘𝑅)))
90		eqidd 2194	. . . . . . . . . . . 12 ⊢ ((((𝑅 ∈ Ring ∧ 𝑥 ∈ 𝑈) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ (𝑚 ∈ (Base‘𝑅) ∧ ((𝑦(.r‘𝑅)𝑥) = (1r‘𝑅) ∧ (𝑚(.r‘(oppr‘𝑅))𝑥) = (1r‘𝑅)))) → (∥r‘(oppr‘𝑅)) = (∥r‘(oppr‘𝑅)))
91	69, 56	syl 14	. . . . . . . . . . . 12 ⊢ ((((𝑅 ∈ Ring ∧ 𝑥 ∈ 𝑈) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ (𝑚 ∈ (Base‘𝑅) ∧ ((𝑦(.r‘𝑅)𝑥) = (1r‘𝑅) ∧ (𝑚(.r‘(oppr‘𝑅))𝑥) = (1r‘𝑅)))) → (oppr‘𝑅) ∈ SRing)
92		eqidd 2194	. . . . . . . . . . . 12 ⊢ ((((𝑅 ∈ Ring ∧ 𝑥 ∈ 𝑈) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ (𝑚 ∈ (Base‘𝑅) ∧ ((𝑦(.r‘𝑅)𝑥) = (1r‘𝑅) ∧ (𝑚(.r‘(oppr‘𝑅))𝑥) = (1r‘𝑅)))) → (.r‘(oppr‘𝑅)) = (.r‘(oppr‘𝑅)))
93	89, 90, 91, 92, 70, 67	dvdsrmuld 13592	. . . . . . . . . . 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))
94	31, 8, 51, 76	opprmulg 13567	. . . . . . . . . . . . 13 ⊢ ((𝑅 ∈ Ring ∧ 𝑥 ∈ 𝑈 ∧ 𝑦 ∈ (Base‘𝑅)) → (𝑥(.r‘(oppr‘𝑅))𝑦) = (𝑦(.r‘𝑅)𝑥))
95	69, 75, 70, 94	syl3anc 1249	. . . . . . . . . . . 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(.r‘(oppr‘𝑅))𝑦) = (𝑦(.r‘𝑅)𝑥))
96	95, 73	eqtrd 2226	. . . . . . . . . . 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))
97	93, 96	breqtrd 4055	. . . . . . . . . 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))
98	1	a1i 9	. . . . . . . . . . 11 ⊢ ((((𝑅 ∈ Ring ∧ 𝑥 ∈ 𝑈) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ (𝑚 ∈ (Base‘𝑅) ∧ ((𝑦(.r‘𝑅)𝑥) = (1r‘𝑅) ∧ (𝑚(.r‘(oppr‘𝑅))𝑥) = (1r‘𝑅)))) → 𝑈 = (Unit‘𝑅))
99		eqidd 2194	. . . . . . . . . . 11 ⊢ ((((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) = (1r‘𝑅))
100		eqidd 2194	. . . . . . . . . . 11 ⊢ ((((𝑅 ∈ Ring ∧ 𝑥 ∈ 𝑈) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ (𝑚 ∈ (Base‘𝑅) ∧ ((𝑦(.r‘𝑅)𝑥) = (1r‘𝑅) ∧ (𝑚(.r‘(oppr‘𝑅))𝑥) = (1r‘𝑅)))) → (oppr‘𝑅) = (oppr‘𝑅))
101	98, 99, 63, 100, 90, 64	isunitd 13602	. . . . . . . . . 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 ∈ 𝑈 ↔ (𝑦(∥r‘𝑅)(1r‘𝑅) ∧ 𝑦(∥r‘(oppr‘𝑅))(1r‘𝑅))))
102	88, 97, 101	mpbir2and 946	. . . . . . . . 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)
103	102, 73	jca 306	. . . . . . . 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)))
104	103	rexlimdvaa 2612	. . . . . . 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))))
105	104	expimpd 363	. . . . . 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))))
106	105	reximdv2 2593	. . . . 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)))
107	61, 106	biimtrrid 153	. . . 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)))
108	60, 107	sylbid 150	. . 3 ⊢ ((𝑅 ∈ Ring ∧ 𝑥 ∈ 𝑈) → ((𝑥(∥r‘𝑅)(1r‘𝑅) ∧ 𝑥(∥r‘(oppr‘𝑅))(1r‘𝑅)) → ∃𝑦 ∈ 𝑈 (𝑦(.r‘𝑅)𝑥) = (1r‘𝑅)))
109	48, 108	mpd 13	. 2 ⊢ ((𝑅 ∈ Ring ∧ 𝑥 ∈ 𝑈) → ∃𝑦 ∈ 𝑈 (𝑦(.r‘𝑅)𝑥) = (1r‘𝑅))
110	6, 19, 20, 33, 35, 42, 109	isgrpde 13094	1 ⊢ (𝑅 ∈ Ring → 𝐺 ∈ Grp)

Syntax hints:   → wi 4   ∧ wa 104   ∧ w3a 980   = wceq 1364   ∈ wcel 2164  ∃wrex 2473  Vcvv 2760   class class class wbr 4029   Fn wfn 5249  ‘cfv 5254  (class class class)co 5918  Basecbs 12618   ↾_s cress 12619  ._rcmulr 12696  Grpcgrp 13072  mulGrpcmgp 13416  1_rcur 13455  SRingcsrg 13459  Ringcrg 13492  opp_rcoppr 13563  ∥_rcdsr 13582  Unitcui 13583

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2166  ax-14 2167  ax-ext 2175  ax-coll 4144  ax-sep 4147  ax-nul 4155  ax-pow 4203  ax-pr 4238  ax-un 4464  ax-setind 4569  ax-cnex 7963  ax-resscn 7964  ax-1cn 7965  ax-1re 7966  ax-icn 7967  ax-addcl 7968  ax-addrcl 7969  ax-mulcl 7970  ax-addcom 7972  ax-addass 7974  ax-i2m1 7977  ax-0lt1 7978  ax-0id 7980  ax-rnegex 7981  ax-pre-ltirr 7984  ax-pre-lttrn 7986  ax-pre-ltadd 7988

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ne 2365  df-nel 2460  df-ral 2477  df-rex 2478  df-reu 2479  df-rmo 2480  df-rab 2481  df-v 2762  df-sbc 2986  df-csb 3081  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3447  df-pw 3603  df-sn 3624  df-pr 3625  df-op 3627  df-uni 3836  df-int 3871  df-iun 3914  df-br 4030  df-opab 4091  df-mpt 4092  df-id 4324  df-xp 4665  df-rel 4666  df-cnv 4667  df-co 4668  df-dm 4669  df-rn 4670  df-res 4671  df-ima 4672  df-iota 5215  df-fun 5256  df-fn 5257  df-f 5258  df-f1 5259  df-fo 5260  df-f1o 5261  df-fv 5262  df-riota 5873  df-ov 5921  df-oprab 5922  df-mpo 5923  df-tpos 6298  df-pnf 8056  df-mnf 8057  df-ltxr 8059  df-inn 8983  df-2 9041  df-3 9042  df-ndx 12621  df-slot 12622  df-base 12624  df-sets 12625  df-iress 12626  df-plusg 12708  df-mulr 12709  df-0g 12869  df-mgm 12939  df-sgrp 12985  df-mnd 12998  df-grp 13075  df-minusg 13076  df-cmn 13356  df-abl 13357  df-mgp 13417  df-ur 13456  df-srg 13460  df-ring 13494  df-oppr 13564  df-dvdsr 13585  df-unit 13586

This theorem is referenced by:  unitabl  13613  unitsubm  13615  invrfvald  13618  unitinvcl  13619  unitinvinv  13620  unitlinv  13622  unitrinv  13623  rdivmuldivd  13640  rhmunitinv  13674  subrgugrp  13736  expghmap  14095