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Theorem unitgrp 13283

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 13222	. . 3 ⊢ (𝑅 ∈ Ring → 𝑅 ∈ SRing)
6	2, 4, 5	unitgrpbasd 13282	. 2 ⊢ (𝑅 ∈ Ring → 𝑈 = (Base‘𝐺))
7		eqid 2177	. . . 4 ⊢ (mulGrp‘𝑅) = (mulGrp‘𝑅)
8		eqid 2177	. . . 4 ⊢ (._r‘𝑅) = (._r‘𝑅)
9	7, 8	mgpplusgg 13132	. . 3 ⊢ (𝑅 ∈ Ring → (._r‘𝑅) = (+_g‘(mulGrp‘𝑅)))
10		basfn 12519	. . . . 5 ⊢ Base Fn V
11		elex 2748	. . . . 5 ⊢ (𝑅 ∈ Ring → 𝑅 ∈ V)
12		funfvex 5532	. . . . . 6 ⊢ ((Fun Base ∧ 𝑅 ∈ dom Base) → (Base‘𝑅) ∈ V)
13	12	funfni 5316	. . . . 5 ⊢ ((Base Fn V ∧ 𝑅 ∈ V) → (Base‘𝑅) ∈ V)
14	10, 11, 13	sylancr 414	. . . 4 ⊢ (𝑅 ∈ Ring → (Base‘𝑅) ∈ V)
15		eqidd 2178	. . . . 5 ⊢ (𝑅 ∈ Ring → (Base‘𝑅) = (Base‘𝑅))
16	15, 2, 5	unitssd 13276	. . . 4 ⊢ (𝑅 ∈ Ring → 𝑈 ⊆ (Base‘𝑅))
17	14, 16	ssexd 4143	. . 3 ⊢ (𝑅 ∈ Ring → 𝑈 ∈ V)
18	7	mgpex 13133	. . 3 ⊢ (𝑅 ∈ Ring → (mulGrp‘𝑅) ∈ V)
19	4, 9, 17, 18	ressplusgd 12586	. 2 ⊢ (𝑅 ∈ Ring → (._r‘𝑅) = (+_g‘𝐺))
20	1, 8	unitmulcl 13280	. 2 ⊢ ((𝑅 ∈ Ring ∧ 𝑥 ∈ 𝑈 ∧ 𝑦 ∈ 𝑈) → (𝑥(._r‘𝑅)𝑦) ∈ 𝑈)
21		eqidd 2178	. . . . 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 1003	. . . . 5 ⊢ ((𝑅 ∈ Ring ∧ (𝑥 ∈ 𝑈 ∧ 𝑦 ∈ 𝑈 ∧ 𝑧 ∈ 𝑈)) → 𝑥 ∈ 𝑈)
25	21, 22, 23, 24	unitcld 13275	. . . 4 ⊢ ((𝑅 ∈ Ring ∧ (𝑥 ∈ 𝑈 ∧ 𝑦 ∈ 𝑈 ∧ 𝑧 ∈ 𝑈)) → 𝑥 ∈ (Base‘𝑅))
26		simpr2 1004	. . . . 5 ⊢ ((𝑅 ∈ Ring ∧ (𝑥 ∈ 𝑈 ∧ 𝑦 ∈ 𝑈 ∧ 𝑧 ∈ 𝑈)) → 𝑦 ∈ 𝑈)
27	21, 22, 23, 26	unitcld 13275	. . . 4 ⊢ ((𝑅 ∈ Ring ∧ (𝑥 ∈ 𝑈 ∧ 𝑦 ∈ 𝑈 ∧ 𝑧 ∈ 𝑈)) → 𝑦 ∈ (Base‘𝑅))
28		simpr3 1005	. . . . 5 ⊢ ((𝑅 ∈ Ring ∧ (𝑥 ∈ 𝑈 ∧ 𝑦 ∈ 𝑈 ∧ 𝑧 ∈ 𝑈)) → 𝑧 ∈ 𝑈)
29	21, 22, 23, 28	unitcld 13275	. . . 4 ⊢ ((𝑅 ∈ Ring ∧ (𝑥 ∈ 𝑈 ∧ 𝑦 ∈ 𝑈 ∧ 𝑧 ∈ 𝑈)) → 𝑧 ∈ (Base‘𝑅))
30	25, 27, 29	3jca 1177	. . 3 ⊢ ((𝑅 ∈ Ring ∧ (𝑥 ∈ 𝑈 ∧ 𝑦 ∈ 𝑈 ∧ 𝑧 ∈ 𝑈)) → (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅) ∧ 𝑧 ∈ (Base‘𝑅)))
31		eqid 2177	. . . 4 ⊢ (Base‘𝑅) = (Base‘𝑅)
32	31, 8	ringass 13197	. . 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 2177	. . 3 ⊢ (1_r‘𝑅) = (1_r‘𝑅)
35	1, 34	1unit 13274	. 2 ⊢ (𝑅 ∈ Ring → (1_r‘𝑅) ∈ 𝑈)
36		eqidd 2178	. . . 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 13275	. . 3 ⊢ ((𝑅 ∈ Ring ∧ 𝑥 ∈ 𝑈) → 𝑥 ∈ (Base‘𝑅))
41	31, 8, 34	ringlidm 13204	. . 3 ⊢ ((𝑅 ∈ Ring ∧ 𝑥 ∈ (Base‘𝑅)) → ((1_r‘𝑅)(._r‘𝑅)𝑥) = 𝑥)
42	40, 41	syldan 282	. 2 ⊢ ((𝑅 ∈ Ring ∧ 𝑥 ∈ 𝑈) → ((1_r‘𝑅)(._r‘𝑅)𝑥) = 𝑥)
43		eqidd 2178	. . . . 5 ⊢ ((𝑅 ∈ Ring ∧ 𝑥 ∈ 𝑈) → (1_r‘𝑅) = (1_r‘𝑅))
44		eqidd 2178	. . . . 5 ⊢ ((𝑅 ∈ Ring ∧ 𝑥 ∈ 𝑈) → (∥_r‘𝑅) = (∥_r‘𝑅))
45		eqidd 2178	. . . . 5 ⊢ ((𝑅 ∈ Ring ∧ 𝑥 ∈ 𝑈) → (opp_r‘𝑅) = (opp_r‘𝑅))
46		eqidd 2178	. . . . 5 ⊢ ((𝑅 ∈ Ring ∧ 𝑥 ∈ 𝑈) → (∥_r‘(opp_r‘𝑅)) = (∥_r‘(opp_r‘𝑅)))
47	37, 43, 44, 45, 46, 38	isunitd 13273	. . . 4 ⊢ ((𝑅 ∈ Ring ∧ 𝑥 ∈ 𝑈) → (𝑥 ∈ 𝑈 ↔ (𝑥(∥_r‘𝑅)(1_r‘𝑅) ∧ 𝑥(∥_r‘(opp_r‘𝑅))(1_r‘𝑅))))
48	39, 47	mpbid 147	. . 3 ⊢ ((𝑅 ∈ Ring ∧ 𝑥 ∈ 𝑈) → (𝑥(∥_r‘𝑅)(1_r‘𝑅) ∧ 𝑥(∥_r‘(opp_r‘𝑅))(1_r‘𝑅)))
49		eqidd 2178	. . . . . 6 ⊢ ((𝑅 ∈ Ring ∧ 𝑥 ∈ 𝑈) → (._r‘𝑅) = (._r‘𝑅))
50	36, 44, 38, 49, 40	dvdsr2d 13262	. . . . 5 ⊢ ((𝑅 ∈ Ring ∧ 𝑥 ∈ 𝑈) → (𝑥(∥_r‘𝑅)(1_r‘𝑅) ↔ ∃𝑦 ∈ (Base‘𝑅)(𝑦(._r‘𝑅)𝑥) = (1_r‘𝑅)))
51		eqid 2177	. . . . . . . 8 ⊢ (opp_r‘𝑅) = (opp_r‘𝑅)
52	51, 31	opprbasg 13245	. . . . . . 7 ⊢ (𝑅 ∈ Ring → (Base‘𝑅) = (Base‘(opp_r‘𝑅)))
53	52	adantr 276	. . . . . 6 ⊢ ((𝑅 ∈ Ring ∧ 𝑥 ∈ 𝑈) → (Base‘𝑅) = (Base‘(opp_r‘𝑅)))
54	51	opprring 13247	. . . . . . . 8 ⊢ (𝑅 ∈ Ring → (opp_r‘𝑅) ∈ Ring)
55		ringsrg 13222	. . . . . . . 8 ⊢ ((opp_r‘𝑅) ∈ Ring → (opp_r‘𝑅) ∈ SRing)
56	54, 55	syl 14	. . . . . . 7 ⊢ (𝑅 ∈ Ring → (opp_r‘𝑅) ∈ SRing)
57	56	adantr 276	. . . . . 6 ⊢ ((𝑅 ∈ Ring ∧ 𝑥 ∈ 𝑈) → (opp_r‘𝑅) ∈ SRing)
58		eqidd 2178	. . . . . 6 ⊢ ((𝑅 ∈ Ring ∧ 𝑥 ∈ 𝑈) → (._r‘(opp_r‘𝑅)) = (._r‘(opp_r‘𝑅)))
59	53, 46, 57, 58, 40	dvdsr2d 13262	. . . . 5 ⊢ ((𝑅 ∈ Ring ∧ 𝑥 ∈ 𝑈) → (𝑥(∥_r‘(opp_r‘𝑅))(1_r‘𝑅) ↔ ∃𝑚 ∈ (Base‘𝑅)(𝑚(._r‘(opp_r‘𝑅))𝑥) = (1_r‘𝑅)))
60	50, 59	anbi12d 473	. . . 4 ⊢ ((𝑅 ∈ Ring ∧ 𝑥 ∈ 𝑈) → ((𝑥(∥_r‘𝑅)(1_r‘𝑅) ∧ 𝑥(∥_r‘(opp_r‘𝑅))(1_r‘𝑅)) ↔ (∃𝑦 ∈ (Base‘𝑅)(𝑦(._r‘𝑅)𝑥) = (1_r‘𝑅) ∧ ∃𝑚 ∈ (Base‘𝑅)(𝑚(._r‘(opp_r‘𝑅))𝑥) = (1_r‘𝑅))))
61		reeanv 2646	. . . . 5 ⊢ (∃𝑦 ∈ (Base‘𝑅)∃𝑚 ∈ (Base‘𝑅)((𝑦(._r‘𝑅)𝑥) = (1_r‘𝑅) ∧ (𝑚(._r‘(opp_r‘𝑅))𝑥) = (1_r‘𝑅)) ↔ (∃𝑦 ∈ (Base‘𝑅)(𝑦(._r‘𝑅)𝑥) = (1_r‘𝑅) ∧ ∃𝑚 ∈ (Base‘𝑅)(𝑚(._r‘(opp_r‘𝑅))𝑥) = (1_r‘𝑅)))
62		eqidd 2178	. . . . . . . . . . . 12 ⊢ ((((𝑅 ∈ Ring ∧ 𝑥 ∈ 𝑈) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ (𝑚 ∈ (Base‘𝑅) ∧ ((𝑦(._r‘𝑅)𝑥) = (1_r‘𝑅) ∧ (𝑚(._r‘(opp_r‘𝑅))𝑥) = (1_r‘𝑅)))) → (Base‘𝑅) = (Base‘𝑅))
63		eqidd 2178	. . . . . . . . . . . 12 ⊢ ((((𝑅 ∈ Ring ∧ 𝑥 ∈ 𝑈) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ (𝑚 ∈ (Base‘𝑅) ∧ ((𝑦(._r‘𝑅)𝑥) = (1_r‘𝑅) ∧ (𝑚(._r‘(opp_r‘𝑅))𝑥) = (1_r‘𝑅)))) → (∥_r‘𝑅) = (∥_r‘𝑅))
64	38	ad2antrr 488	. . . . . . . . . . . 12 ⊢ ((((𝑅 ∈ Ring ∧ 𝑥 ∈ 𝑈) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ (𝑚 ∈ (Base‘𝑅) ∧ ((𝑦(._r‘𝑅)𝑥) = (1_r‘𝑅) ∧ (𝑚(._r‘(opp_r‘𝑅))𝑥) = (1_r‘𝑅)))) → 𝑅 ∈ SRing)
65		eqidd 2178	. . . . . . . . . . . 12 ⊢ ((((𝑅 ∈ Ring ∧ 𝑥 ∈ 𝑈) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ (𝑚 ∈ (Base‘𝑅) ∧ ((𝑦(._r‘𝑅)𝑥) = (1_r‘𝑅) ∧ (𝑚(._r‘(opp_r‘𝑅))𝑥) = (1_r‘𝑅)))) → (._r‘𝑅) = (._r‘𝑅))
66		simprl 529	. . . . . . . . . . . 12 ⊢ ((((𝑅 ∈ Ring ∧ 𝑥 ∈ 𝑈) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ (𝑚 ∈ (Base‘𝑅) ∧ ((𝑦(._r‘𝑅)𝑥) = (1_r‘𝑅) ∧ (𝑚(._r‘(opp_r‘𝑅))𝑥) = (1_r‘𝑅)))) → 𝑚 ∈ (Base‘𝑅))
67	40	ad2antrr 488	. . . . . . . . . . . 12 ⊢ ((((𝑅 ∈ Ring ∧ 𝑥 ∈ 𝑈) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ (𝑚 ∈ (Base‘𝑅) ∧ ((𝑦(._r‘𝑅)𝑥) = (1_r‘𝑅) ∧ (𝑚(._r‘(opp_r‘𝑅))𝑥) = (1_r‘𝑅)))) → 𝑥 ∈ (Base‘𝑅))
68	62, 63, 64, 65, 66, 67	dvdsrmuld 13263	. . . . . . . . . . 11 ⊢ ((((𝑅 ∈ Ring ∧ 𝑥 ∈ 𝑈) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ (𝑚 ∈ (Base‘𝑅) ∧ ((𝑦(._r‘𝑅)𝑥) = (1_r‘𝑅) ∧ (𝑚(._r‘(opp_r‘𝑅))𝑥) = (1_r‘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) = (1_r‘𝑅) ∧ (𝑚(._r‘(opp_r‘𝑅))𝑥) = (1_r‘𝑅)))) → 𝑅 ∈ Ring)
70		simplr 528	. . . . . . . . . . . . . 14 ⊢ ((((𝑅 ∈ Ring ∧ 𝑥 ∈ 𝑈) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ (𝑚 ∈ (Base‘𝑅) ∧ ((𝑦(._r‘𝑅)𝑥) = (1_r‘𝑅) ∧ (𝑚(._r‘(opp_r‘𝑅))𝑥) = (1_r‘𝑅)))) → 𝑦 ∈ (Base‘𝑅))
71	31, 8	ringass 13197	. . . . . . . . . . . . . 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 1240	. . . . . . . . . . . . 13 ⊢ ((((𝑅 ∈ Ring ∧ 𝑥 ∈ 𝑈) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ (𝑚 ∈ (Base‘𝑅) ∧ ((𝑦(._r‘𝑅)𝑥) = (1_r‘𝑅) ∧ (𝑚(._r‘(opp_r‘𝑅))𝑥) = (1_r‘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) = (1_r‘𝑅) ∧ (𝑚(._r‘(opp_r‘𝑅))𝑥) = (1_r‘𝑅)))) → (𝑦(._r‘𝑅)𝑥) = (1_r‘𝑅))
74	73	oveq1d 5889	. . . . . . . . . . . . 13 ⊢ ((((𝑅 ∈ Ring ∧ 𝑥 ∈ 𝑈) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ (𝑚 ∈ (Base‘𝑅) ∧ ((𝑦(._r‘𝑅)𝑥) = (1_r‘𝑅) ∧ (𝑚(._r‘(opp_r‘𝑅))𝑥) = (1_r‘𝑅)))) → ((𝑦(._r‘𝑅)𝑥)(._r‘𝑅)𝑚) = ((1_r‘𝑅)(._r‘𝑅)𝑚))
75	39	ad2antrr 488	. . . . . . . . . . . . . . . 16 ⊢ ((((𝑅 ∈ Ring ∧ 𝑥 ∈ 𝑈) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ (𝑚 ∈ (Base‘𝑅) ∧ ((𝑦(._r‘𝑅)𝑥) = (1_r‘𝑅) ∧ (𝑚(._r‘(opp_r‘𝑅))𝑥) = (1_r‘𝑅)))) → 𝑥 ∈ 𝑈)
76		eqid 2177	. . . . . . . . . . . . . . . . 17 ⊢ (._r‘(opp_r‘𝑅)) = (._r‘(opp_r‘𝑅))
77	31, 8, 51, 76	opprmulg 13241	. . . . . . . . . . . . . . . 16 ⊢ ((𝑅 ∈ Ring ∧ 𝑚 ∈ (Base‘𝑅) ∧ 𝑥 ∈ 𝑈) → (𝑚(._r‘(opp_r‘𝑅))𝑥) = (𝑥(._r‘𝑅)𝑚))
78	69, 66, 75, 77	syl3anc 1238	. . . . . . . . . . . . . . 15 ⊢ ((((𝑅 ∈ Ring ∧ 𝑥 ∈ 𝑈) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ (𝑚 ∈ (Base‘𝑅) ∧ ((𝑦(._r‘𝑅)𝑥) = (1_r‘𝑅) ∧ (𝑚(._r‘(opp_r‘𝑅))𝑥) = (1_r‘𝑅)))) → (𝑚(._r‘(opp_r‘𝑅))𝑥) = (𝑥(._r‘𝑅)𝑚))
79		simprrr 540	. . . . . . . . . . . . . . 15 ⊢ ((((𝑅 ∈ Ring ∧ 𝑥 ∈ 𝑈) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ (𝑚 ∈ (Base‘𝑅) ∧ ((𝑦(._r‘𝑅)𝑥) = (1_r‘𝑅) ∧ (𝑚(._r‘(opp_r‘𝑅))𝑥) = (1_r‘𝑅)))) → (𝑚(._r‘(opp_r‘𝑅))𝑥) = (1_r‘𝑅))
80	78, 79	eqtr3d 2212	. . . . . . . . . . . . . 14 ⊢ ((((𝑅 ∈ Ring ∧ 𝑥 ∈ 𝑈) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ (𝑚 ∈ (Base‘𝑅) ∧ ((𝑦(._r‘𝑅)𝑥) = (1_r‘𝑅) ∧ (𝑚(._r‘(opp_r‘𝑅))𝑥) = (1_r‘𝑅)))) → (𝑥(._r‘𝑅)𝑚) = (1_r‘𝑅))
81	80	oveq2d 5890	. . . . . . . . . . . . 13 ⊢ ((((𝑅 ∈ Ring ∧ 𝑥 ∈ 𝑈) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ (𝑚 ∈ (Base‘𝑅) ∧ ((𝑦(._r‘𝑅)𝑥) = (1_r‘𝑅) ∧ (𝑚(._r‘(opp_r‘𝑅))𝑥) = (1_r‘𝑅)))) → (𝑦(._r‘𝑅)(𝑥(._r‘𝑅)𝑚)) = (𝑦(._r‘𝑅)(1_r‘𝑅)))
82	72, 74, 81	3eqtr3d 2218	. . . . . . . . . . . 12 ⊢ ((((𝑅 ∈ Ring ∧ 𝑥 ∈ 𝑈) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ (𝑚 ∈ (Base‘𝑅) ∧ ((𝑦(._r‘𝑅)𝑥) = (1_r‘𝑅) ∧ (𝑚(._r‘(opp_r‘𝑅))𝑥) = (1_r‘𝑅)))) → ((1_r‘𝑅)(._r‘𝑅)𝑚) = (𝑦(._r‘𝑅)(1_r‘𝑅)))
83	31, 8, 34	ringlidm 13204	. . . . . . . . . . . . 13 ⊢ ((𝑅 ∈ Ring ∧ 𝑚 ∈ (Base‘𝑅)) → ((1_r‘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) = (1_r‘𝑅) ∧ (𝑚(._r‘(opp_r‘𝑅))𝑥) = (1_r‘𝑅)))) → ((1_r‘𝑅)(._r‘𝑅)𝑚) = 𝑚)
85	31, 8, 34	ringridm 13205	. . . . . . . . . . . . 13 ⊢ ((𝑅 ∈ Ring ∧ 𝑦 ∈ (Base‘𝑅)) → (𝑦(._r‘𝑅)(1_r‘𝑅)) = 𝑦)
86	69, 70, 85	syl2anc 411	. . . . . . . . . . . 12 ⊢ ((((𝑅 ∈ Ring ∧ 𝑥 ∈ 𝑈) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ (𝑚 ∈ (Base‘𝑅) ∧ ((𝑦(._r‘𝑅)𝑥) = (1_r‘𝑅) ∧ (𝑚(._r‘(opp_r‘𝑅))𝑥) = (1_r‘𝑅)))) → (𝑦(._r‘𝑅)(1_r‘𝑅)) = 𝑦)
87	82, 84, 86	3eqtr3d 2218	. . . . . . . . . . 11 ⊢ ((((𝑅 ∈ Ring ∧ 𝑥 ∈ 𝑈) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ (𝑚 ∈ (Base‘𝑅) ∧ ((𝑦(._r‘𝑅)𝑥) = (1_r‘𝑅) ∧ (𝑚(._r‘(opp_r‘𝑅))𝑥) = (1_r‘𝑅)))) → 𝑚 = 𝑦)
88	68, 87, 80	3brtr3d 4034	. . . . . . . . . 10 ⊢ ((((𝑅 ∈ Ring ∧ 𝑥 ∈ 𝑈) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ (𝑚 ∈ (Base‘𝑅) ∧ ((𝑦(._r‘𝑅)𝑥) = (1_r‘𝑅) ∧ (𝑚(._r‘(opp_r‘𝑅))𝑥) = (1_r‘𝑅)))) → 𝑦(∥_r‘𝑅)(1_r‘𝑅))
89	69, 52	syl 14	. . . . . . . . . . . 12 ⊢ ((((𝑅 ∈ Ring ∧ 𝑥 ∈ 𝑈) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ (𝑚 ∈ (Base‘𝑅) ∧ ((𝑦(._r‘𝑅)𝑥) = (1_r‘𝑅) ∧ (𝑚(._r‘(opp_r‘𝑅))𝑥) = (1_r‘𝑅)))) → (Base‘𝑅) = (Base‘(opp_r‘𝑅)))
90		eqidd 2178	. . . . . . . . . . . 12 ⊢ ((((𝑅 ∈ Ring ∧ 𝑥 ∈ 𝑈) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ (𝑚 ∈ (Base‘𝑅) ∧ ((𝑦(._r‘𝑅)𝑥) = (1_r‘𝑅) ∧ (𝑚(._r‘(opp_r‘𝑅))𝑥) = (1_r‘𝑅)))) → (∥_r‘(opp_r‘𝑅)) = (∥_r‘(opp_r‘𝑅)))
91	69, 56	syl 14	. . . . . . . . . . . 12 ⊢ ((((𝑅 ∈ Ring ∧ 𝑥 ∈ 𝑈) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ (𝑚 ∈ (Base‘𝑅) ∧ ((𝑦(._r‘𝑅)𝑥) = (1_r‘𝑅) ∧ (𝑚(._r‘(opp_r‘𝑅))𝑥) = (1_r‘𝑅)))) → (opp_r‘𝑅) ∈ SRing)
92		eqidd 2178	. . . . . . . . . . . 12 ⊢ ((((𝑅 ∈ Ring ∧ 𝑥 ∈ 𝑈) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ (𝑚 ∈ (Base‘𝑅) ∧ ((𝑦(._r‘𝑅)𝑥) = (1_r‘𝑅) ∧ (𝑚(._r‘(opp_r‘𝑅))𝑥) = (1_r‘𝑅)))) → (._r‘(opp_r‘𝑅)) = (._r‘(opp_r‘𝑅)))
93	89, 90, 91, 92, 70, 67	dvdsrmuld 13263	. . . . . . . . . . 11 ⊢ ((((𝑅 ∈ Ring ∧ 𝑥 ∈ 𝑈) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ (𝑚 ∈ (Base‘𝑅) ∧ ((𝑦(._r‘𝑅)𝑥) = (1_r‘𝑅) ∧ (𝑚(._r‘(opp_r‘𝑅))𝑥) = (1_r‘𝑅)))) → 𝑦(∥_r‘(opp_r‘𝑅))(𝑥(._r‘(opp_r‘𝑅))𝑦))
94	31, 8, 51, 76	opprmulg 13241	. . . . . . . . . . . . 13 ⊢ ((𝑅 ∈ Ring ∧ 𝑥 ∈ 𝑈 ∧ 𝑦 ∈ (Base‘𝑅)) → (𝑥(._r‘(opp_r‘𝑅))𝑦) = (𝑦(._r‘𝑅)𝑥))
95	69, 75, 70, 94	syl3anc 1238	. . . . . . . . . . . 12 ⊢ ((((𝑅 ∈ Ring ∧ 𝑥 ∈ 𝑈) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ (𝑚 ∈ (Base‘𝑅) ∧ ((𝑦(._r‘𝑅)𝑥) = (1_r‘𝑅) ∧ (𝑚(._r‘(opp_r‘𝑅))𝑥) = (1_r‘𝑅)))) → (𝑥(._r‘(opp_r‘𝑅))𝑦) = (𝑦(._r‘𝑅)𝑥))
96	95, 73	eqtrd 2210	. . . . . . . . . . 11 ⊢ ((((𝑅 ∈ Ring ∧ 𝑥 ∈ 𝑈) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ (𝑚 ∈ (Base‘𝑅) ∧ ((𝑦(._r‘𝑅)𝑥) = (1_r‘𝑅) ∧ (𝑚(._r‘(opp_r‘𝑅))𝑥) = (1_r‘𝑅)))) → (𝑥(._r‘(opp_r‘𝑅))𝑦) = (1_r‘𝑅))
97	93, 96	breqtrd 4029	. . . . . . . . . 10 ⊢ ((((𝑅 ∈ Ring ∧ 𝑥 ∈ 𝑈) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ (𝑚 ∈ (Base‘𝑅) ∧ ((𝑦(._r‘𝑅)𝑥) = (1_r‘𝑅) ∧ (𝑚(._r‘(opp_r‘𝑅))𝑥) = (1_r‘𝑅)))) → 𝑦(∥_r‘(opp_r‘𝑅))(1_r‘𝑅))
98	1	a1i 9	. . . . . . . . . . 11 ⊢ ((((𝑅 ∈ Ring ∧ 𝑥 ∈ 𝑈) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ (𝑚 ∈ (Base‘𝑅) ∧ ((𝑦(._r‘𝑅)𝑥) = (1_r‘𝑅) ∧ (𝑚(._r‘(opp_r‘𝑅))𝑥) = (1_r‘𝑅)))) → 𝑈 = (Unit‘𝑅))
99		eqidd 2178	. . . . . . . . . . 11 ⊢ ((((𝑅 ∈ Ring ∧ 𝑥 ∈ 𝑈) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ (𝑚 ∈ (Base‘𝑅) ∧ ((𝑦(._r‘𝑅)𝑥) = (1_r‘𝑅) ∧ (𝑚(._r‘(opp_r‘𝑅))𝑥) = (1_r‘𝑅)))) → (1_r‘𝑅) = (1_r‘𝑅))
100		eqidd 2178	. . . . . . . . . . 11 ⊢ ((((𝑅 ∈ Ring ∧ 𝑥 ∈ 𝑈) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ (𝑚 ∈ (Base‘𝑅) ∧ ((𝑦(._r‘𝑅)𝑥) = (1_r‘𝑅) ∧ (𝑚(._r‘(opp_r‘𝑅))𝑥) = (1_r‘𝑅)))) → (opp_r‘𝑅) = (opp_r‘𝑅))
101	98, 99, 63, 100, 90, 64	isunitd 13273	. . . . . . . . . 10 ⊢ ((((𝑅 ∈ Ring ∧ 𝑥 ∈ 𝑈) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ (𝑚 ∈ (Base‘𝑅) ∧ ((𝑦(._r‘𝑅)𝑥) = (1_r‘𝑅) ∧ (𝑚(._r‘(opp_r‘𝑅))𝑥) = (1_r‘𝑅)))) → (𝑦 ∈ 𝑈 ↔ (𝑦(∥_r‘𝑅)(1_r‘𝑅) ∧ 𝑦(∥_r‘(opp_r‘𝑅))(1_r‘𝑅))))
102	88, 97, 101	mpbir2and 944	. . . . . . . . 9 ⊢ ((((𝑅 ∈ Ring ∧ 𝑥 ∈ 𝑈) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ (𝑚 ∈ (Base‘𝑅) ∧ ((𝑦(._r‘𝑅)𝑥) = (1_r‘𝑅) ∧ (𝑚(._r‘(opp_r‘𝑅))𝑥) = (1_r‘𝑅)))) → 𝑦 ∈ 𝑈)
103	102, 73	jca 306	. . . . . . . 8 ⊢ ((((𝑅 ∈ Ring ∧ 𝑥 ∈ 𝑈) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ (𝑚 ∈ (Base‘𝑅) ∧ ((𝑦(._r‘𝑅)𝑥) = (1_r‘𝑅) ∧ (𝑚(._r‘(opp_r‘𝑅))𝑥) = (1_r‘𝑅)))) → (𝑦 ∈ 𝑈 ∧ (𝑦(._r‘𝑅)𝑥) = (1_r‘𝑅)))
104	103	rexlimdvaa 2595	. . . . . . 7 ⊢ (((𝑅 ∈ Ring ∧ 𝑥 ∈ 𝑈) ∧ 𝑦 ∈ (Base‘𝑅)) → (∃𝑚 ∈ (Base‘𝑅)((𝑦(._r‘𝑅)𝑥) = (1_r‘𝑅) ∧ (𝑚(._r‘(opp_r‘𝑅))𝑥) = (1_r‘𝑅)) → (𝑦 ∈ 𝑈 ∧ (𝑦(._r‘𝑅)𝑥) = (1_r‘𝑅))))
105	104	expimpd 363	. . . . . 6 ⊢ ((𝑅 ∈ Ring ∧ 𝑥 ∈ 𝑈) → ((𝑦 ∈ (Base‘𝑅) ∧ ∃𝑚 ∈ (Base‘𝑅)((𝑦(._r‘𝑅)𝑥) = (1_r‘𝑅) ∧ (𝑚(._r‘(opp_r‘𝑅))𝑥) = (1_r‘𝑅))) → (𝑦 ∈ 𝑈 ∧ (𝑦(._r‘𝑅)𝑥) = (1_r‘𝑅))))
106	105	reximdv2 2576	. . . . 5 ⊢ ((𝑅 ∈ Ring ∧ 𝑥 ∈ 𝑈) → (∃𝑦 ∈ (Base‘𝑅)∃𝑚 ∈ (Base‘𝑅)((𝑦(._r‘𝑅)𝑥) = (1_r‘𝑅) ∧ (𝑚(._r‘(opp_r‘𝑅))𝑥) = (1_r‘𝑅)) → ∃𝑦 ∈ 𝑈 (𝑦(._r‘𝑅)𝑥) = (1_r‘𝑅)))
107	61, 106	biimtrrid 153	. . . 4 ⊢ ((𝑅 ∈ Ring ∧ 𝑥 ∈ 𝑈) → ((∃𝑦 ∈ (Base‘𝑅)(𝑦(._r‘𝑅)𝑥) = (1_r‘𝑅) ∧ ∃𝑚 ∈ (Base‘𝑅)(𝑚(._r‘(opp_r‘𝑅))𝑥) = (1_r‘𝑅)) → ∃𝑦 ∈ 𝑈 (𝑦(._r‘𝑅)𝑥) = (1_r‘𝑅)))
108	60, 107	sylbid 150	. . 3 ⊢ ((𝑅 ∈ Ring ∧ 𝑥 ∈ 𝑈) → ((𝑥(∥_r‘𝑅)(1_r‘𝑅) ∧ 𝑥(∥_r‘(opp_r‘𝑅))(1_r‘𝑅)) → ∃𝑦 ∈ 𝑈 (𝑦(._r‘𝑅)𝑥) = (1_r‘𝑅)))
109	48, 108	mpd 13	. 2 ⊢ ((𝑅 ∈ Ring ∧ 𝑥 ∈ 𝑈) → ∃𝑦 ∈ 𝑈 (𝑦(._r‘𝑅)𝑥) = (1_r‘𝑅))
110	6, 19, 20, 33, 35, 42, 109	isgrpde 12897	1 ⊢ (𝑅 ∈ Ring → 𝐺 ∈ Grp)

Colors of variables: wff set class

Syntax hints: → wi 4 ∧ wa 104 ∧ w3a 978 = wceq 1353 ∈ wcel 2148 ∃wrex 2456 Vcvv 2737 class class class wbr 4003 Fn wfn 5211 ‘cfv 5216 (class class class)co 5874 Basecbs 12461 ↾_s cress 12462 ._rcmulr 12536 Grpcgrp 12876 mulGrpcmgp 13128 1_rcur 13140 SRingcsrg 13144 Ringcrg 13177 opp_rcoppr 13237 ∥_rcdsr 13253 Unitcui 13254

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 614 ax-in2 615 ax-io 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-13 2150 ax-14 2151 ax-ext 2159 ax-coll 4118 ax-sep 4121 ax-nul 4129 ax-pow 4174 ax-pr 4209 ax-un 4433 ax-setind 4536 ax-cnex 7901 ax-resscn 7902 ax-1cn 7903 ax-1re 7904 ax-icn 7905 ax-addcl 7906 ax-addrcl 7907 ax-mulcl 7908 ax-addcom 7910 ax-addass 7912 ax-i2m1 7915 ax-0lt1 7916 ax-0id 7918 ax-rnegex 7919 ax-pre-ltirr 7922 ax-pre-lttrn 7924 ax-pre-ltadd 7926

This theorem depends on definitions: df-bi 117 df-3an 980 df-tru 1356 df-fal 1359 df-nf 1461 df-sb 1763 df-eu 2029 df-mo 2030 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-ne 2348 df-nel 2443 df-ral 2460 df-rex 2461 df-reu 2462 df-rmo 2463 df-rab 2464 df-v 2739 df-sbc 2963 df-csb 3058 df-dif 3131 df-un 3133 df-in 3135 df-ss 3142 df-nul 3423 df-pw 3577 df-sn 3598 df-pr 3599 df-op 3601 df-uni 3810 df-int 3845 df-iun 3888 df-br 4004 df-opab 4065 df-mpt 4066 df-id 4293 df-xp 4632 df-rel 4633 df-cnv 4634 df-co 4635 df-dm 4636 df-rn 4637 df-res 4638 df-ima 4639 df-iota 5178 df-fun 5218 df-fn 5219 df-f 5220 df-f1 5221 df-fo 5222 df-f1o 5223 df-fv 5224 df-riota 5830 df-ov 5877 df-oprab 5878 df-mpo 5879 df-tpos 6245 df-pnf 7993 df-mnf 7994 df-ltxr 7996 df-inn 8919 df-2 8977 df-3 8978 df-ndx 12464 df-slot 12465 df-base 12467 df-sets 12468 df-iress 12469 df-plusg 12548 df-mulr 12549 df-0g 12706 df-mgm 12774 df-sgrp 12807 df-mnd 12817 df-grp 12879 df-minusg 12880 df-cmn 13088 df-abl 13089 df-mgp 13129 df-ur 13141 df-srg 13145 df-ring 13179 df-oppr 13238 df-dvdsr 13256 df-unit 13257

This theorem is referenced by: unitabl 13284 unitsubm 13286 invrfvald 13289 unitinvcl 13290 unitinvinv 13291 unitlinv 13293 unitrinv 13294 rdivmuldivd 13311 subrgugrp 13359

W3C validator