Theorem unitgrp 14088

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 14018	. . 3 ⊢ (𝑅 ∈ Ring → 𝑅 ∈ SRing)
6	2, 4, 5	unitgrpbasd 14087	. 2 ⊢ (𝑅 ∈ Ring → 𝑈 = (Base‘𝐺))
7		eqid 2229	. . . 4 ⊢ (mulGrp‘𝑅) = (mulGrp‘𝑅)
8		eqid 2229	. . . 4 ⊢ (.r‘𝑅) = (.r‘𝑅)
9	7, 8	mgpplusgg 13895	. . 3 ⊢ (𝑅 ∈ Ring → (.r‘𝑅) = (+g‘(mulGrp‘𝑅)))
10		basfn 13099	. . . . 5 ⊢ Base Fn V
11		elex 2811	. . . . 5 ⊢ (𝑅 ∈ Ring → 𝑅 ∈ V)
12		funfvex 5646	. . . . . 6 ⊢ ((Fun Base ∧ 𝑅 ∈ dom Base) → (Base‘𝑅) ∈ V)
13	12	funfni 5423	. . . . 5 ⊢ ((Base Fn V ∧ 𝑅 ∈ V) → (Base‘𝑅) ∈ V)
14	10, 11, 13	sylancr 414	. . . 4 ⊢ (𝑅 ∈ Ring → (Base‘𝑅) ∈ V)
15		eqidd 2230	. . . . 5 ⊢ (𝑅 ∈ Ring → (Base‘𝑅) = (Base‘𝑅))
16	15, 2, 5	unitssd 14081	. . . 4 ⊢ (𝑅 ∈ Ring → 𝑈 ⊆ (Base‘𝑅))
17	14, 16	ssexd 4224	. . 3 ⊢ (𝑅 ∈ Ring → 𝑈 ∈ V)
18	7	mgpex 13896	. . 3 ⊢ (𝑅 ∈ Ring → (mulGrp‘𝑅) ∈ V)
19	4, 9, 17, 18	ressplusgd 13170	. 2 ⊢ (𝑅 ∈ Ring → (.r‘𝑅) = (+g‘𝐺))
20	1, 8	unitmulcl 14085	. 2 ⊢ ((𝑅 ∈ Ring ∧ 𝑥 ∈ 𝑈 ∧ 𝑦 ∈ 𝑈) → (𝑥(.r‘𝑅)𝑦) ∈ 𝑈)
21		eqidd 2230	. . . . 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 1027	. . . . 5 ⊢ ((𝑅 ∈ Ring ∧ (𝑥 ∈ 𝑈 ∧ 𝑦 ∈ 𝑈 ∧ 𝑧 ∈ 𝑈)) → 𝑥 ∈ 𝑈)
25	21, 22, 23, 24	unitcld 14080	. . . 4 ⊢ ((𝑅 ∈ Ring ∧ (𝑥 ∈ 𝑈 ∧ 𝑦 ∈ 𝑈 ∧ 𝑧 ∈ 𝑈)) → 𝑥 ∈ (Base‘𝑅))
26		simpr2 1028	. . . . 5 ⊢ ((𝑅 ∈ Ring ∧ (𝑥 ∈ 𝑈 ∧ 𝑦 ∈ 𝑈 ∧ 𝑧 ∈ 𝑈)) → 𝑦 ∈ 𝑈)
27	21, 22, 23, 26	unitcld 14080	. . . 4 ⊢ ((𝑅 ∈ Ring ∧ (𝑥 ∈ 𝑈 ∧ 𝑦 ∈ 𝑈 ∧ 𝑧 ∈ 𝑈)) → 𝑦 ∈ (Base‘𝑅))
28		simpr3 1029	. . . . 5 ⊢ ((𝑅 ∈ Ring ∧ (𝑥 ∈ 𝑈 ∧ 𝑦 ∈ 𝑈 ∧ 𝑧 ∈ 𝑈)) → 𝑧 ∈ 𝑈)
29	21, 22, 23, 28	unitcld 14080	. . . 4 ⊢ ((𝑅 ∈ Ring ∧ (𝑥 ∈ 𝑈 ∧ 𝑦 ∈ 𝑈 ∧ 𝑧 ∈ 𝑈)) → 𝑧 ∈ (Base‘𝑅))
30	25, 27, 29	3jca 1201	. . 3 ⊢ ((𝑅 ∈ Ring ∧ (𝑥 ∈ 𝑈 ∧ 𝑦 ∈ 𝑈 ∧ 𝑧 ∈ 𝑈)) → (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅) ∧ 𝑧 ∈ (Base‘𝑅)))
31		eqid 2229	. . . 4 ⊢ (Base‘𝑅) = (Base‘𝑅)
32	31, 8	ringass 13987	. . 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 2229	. . 3 ⊢ (1r‘𝑅) = (1r‘𝑅)
35	1, 34	1unit 14079	. 2 ⊢ (𝑅 ∈ Ring → (1r‘𝑅) ∈ 𝑈)
36		eqidd 2230	. . . 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 14080	. . 3 ⊢ ((𝑅 ∈ Ring ∧ 𝑥 ∈ 𝑈) → 𝑥 ∈ (Base‘𝑅))
41	31, 8, 34	ringlidm 13994	. . 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 2230	. . . . 5 ⊢ ((𝑅 ∈ Ring ∧ 𝑥 ∈ 𝑈) → (1r‘𝑅) = (1r‘𝑅))
44		eqidd 2230	. . . . 5 ⊢ ((𝑅 ∈ Ring ∧ 𝑥 ∈ 𝑈) → (∥r‘𝑅) = (∥r‘𝑅))
45		eqidd 2230	. . . . 5 ⊢ ((𝑅 ∈ Ring ∧ 𝑥 ∈ 𝑈) → (oppr‘𝑅) = (oppr‘𝑅))
46		eqidd 2230	. . . . 5 ⊢ ((𝑅 ∈ Ring ∧ 𝑥 ∈ 𝑈) → (∥r‘(oppr‘𝑅)) = (∥r‘(oppr‘𝑅)))
47	37, 43, 44, 45, 46, 38	isunitd 14078	. . . 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 2230	. . . . . 6 ⊢ ((𝑅 ∈ Ring ∧ 𝑥 ∈ 𝑈) → (.r‘𝑅) = (.r‘𝑅))
50	36, 44, 38, 49, 40	dvdsr2d 14067	. . . . 5 ⊢ ((𝑅 ∈ Ring ∧ 𝑥 ∈ 𝑈) → (𝑥(∥r‘𝑅)(1r‘𝑅) ↔ ∃𝑦 ∈ (Base‘𝑅)(𝑦(.r‘𝑅)𝑥) = (1r‘𝑅)))
51		eqid 2229	. . . . . . . 8 ⊢ (oppr‘𝑅) = (oppr‘𝑅)
52	51, 31	opprbasg 14046	. . . . . . 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 14050	. . . . . . . 8 ⊢ (𝑅 ∈ Ring → (oppr‘𝑅) ∈ Ring)
55		ringsrg 14018	. . . . . . . 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 2230	. . . . . 6 ⊢ ((𝑅 ∈ Ring ∧ 𝑥 ∈ 𝑈) → (.r‘(oppr‘𝑅)) = (.r‘(oppr‘𝑅)))
59	53, 46, 57, 58, 40	dvdsr2d 14067	. . . . 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 2701	. . . . 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 2230	. . . . . . . . . . . 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 2230	. . . . . . . . . . . 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 2230	. . . . . . . . . . . 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 14068	. . . . . . . . . . 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 13987	. . . . . . . . . . . . . 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 1273	. . . . . . . . . . . . 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 6022	. . . . . . . . . . . . 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 2229	. . . . . . . . . . . . . . . . 17 ⊢ (.r‘(oppr‘𝑅)) = (.r‘(oppr‘𝑅))
77	31, 8, 51, 76	opprmulg 14042	. . . . . . . . . . . . . . . 16 ⊢ ((𝑅 ∈ Ring ∧ 𝑚 ∈ (Base‘𝑅) ∧ 𝑥 ∈ 𝑈) → (𝑚(.r‘(oppr‘𝑅))𝑥) = (𝑥(.r‘𝑅)𝑚))
78	69, 66, 75, 77	syl3anc 1271	. . . . . . . . . . . . . . 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 2264	. . . . . . . . . . . . . 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 6023	. . . . . . . . . . . . 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 2270	. . . . . . . . . . . 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 13994	. . . . . . . . . . . . 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 13995	. . . . . . . . . . . . 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 2270	. . . . . . . . . . 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 4114	. . . . . . . . . 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 2230	. . . . . . . . . . . 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 2230	. . . . . . . . . . . 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 14068	. . . . . . . . . . 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 14042	. . . . . . . . . . . . 13 ⊢ ((𝑅 ∈ Ring ∧ 𝑥 ∈ 𝑈 ∧ 𝑦 ∈ (Base‘𝑅)) → (𝑥(.r‘(oppr‘𝑅))𝑦) = (𝑦(.r‘𝑅)𝑥))
95	69, 75, 70, 94	syl3anc 1271	. . . . . . . . . . . 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 2262	. . . . . . . . . . 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 4109	. . . . . . . . . 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 2230	. . . . . . . . . . 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 2230	. . . . . . . . . . 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 14078	. . . . . . . . . 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 950	. . . . . . . . 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 2649	. . . . . . 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 2629	. . . . 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 13563	1 ⊢ (𝑅 ∈ Ring → 𝐺 ∈ Grp)

Syntax hints:   → wi 4   ∧ wa 104   ∧ w3a 1002   = wceq 1395   ∈ wcel 2200  ∃wrex 2509  Vcvv 2799   class class class wbr 4083   Fn wfn 5313  ‘cfv 5318  (class class class)co 6007  Basecbs 13040   ↾_s cress 13041  ._rcmulr 13119  Grpcgrp 13541  mulGrpcmgp 13891  1_rcur 13930  SRingcsrg 13934  Ringcrg 13967  opp_rcoppr 14038  ∥_rcdsr 14057  Unitcui 14058

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 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-coll 4199  ax-sep 4202  ax-nul 4210  ax-pow 4258  ax-pr 4293  ax-un 4524  ax-setind 4629  ax-cnex 8098  ax-resscn 8099  ax-1cn 8100  ax-1re 8101  ax-icn 8102  ax-addcl 8103  ax-addrcl 8104  ax-mulcl 8105  ax-addcom 8107  ax-addass 8109  ax-i2m1 8112  ax-0lt1 8113  ax-0id 8115  ax-rnegex 8116  ax-pre-ltirr 8119  ax-pre-lttrn 8121  ax-pre-ltadd 8123

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-nel 2496  df-ral 2513  df-rex 2514  df-reu 2515  df-rmo 2516  df-rab 2517  df-v 2801  df-sbc 3029  df-csb 3125  df-dif 3199  df-un 3201  df-in 3203  df-ss 3210  df-nul 3492  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3889  df-int 3924  df-iun 3967  df-br 4084  df-opab 4146  df-mpt 4147  df-id 4384  df-xp 4725  df-rel 4726  df-cnv 4727  df-co 4728  df-dm 4729  df-rn 4730  df-res 4731  df-ima 4732  df-iota 5278  df-fun 5320  df-fn 5321  df-f 5322  df-f1 5323  df-fo 5324  df-f1o 5325  df-fv 5326  df-riota 5960  df-ov 6010  df-oprab 6011  df-mpo 6012  df-tpos 6397  df-pnf 8191  df-mnf 8192  df-ltxr 8194  df-inn 9119  df-2 9177  df-3 9178  df-ndx 13043  df-slot 13044  df-base 13046  df-sets 13047  df-iress 13048  df-plusg 13131  df-mulr 13132  df-0g 13299  df-mgm 13397  df-sgrp 13443  df-mnd 13458  df-grp 13544  df-minusg 13545  df-cmn 13831  df-abl 13832  df-mgp 13892  df-ur 13931  df-srg 13935  df-ring 13969  df-oppr 14039  df-dvdsr 14060  df-unit 14061

This theorem is referenced by:  unitabl  14089  unitsubm  14091  invrfvald  14094  unitinvcl  14095  unitinvinv  14096  unitlinv  14098  unitrinv  14099  rdivmuldivd  14116  rhmunitinv  14150  subrgugrp  14212  expghmap  14579