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Theorem unitgrp 20311

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 20310	. . 3 ⊢ 𝑈 = (Base‘𝐺)
4	3	a1i 11	. 2 ⊢ (𝑅 ∈ Ring → 𝑈 = (Base‘𝐺))
5	1	fvexi 6905	. . 3 ⊢ 𝑈 ∈ V
6		eqid 2727	. . . . 5 ⊢ (mulGrp‘𝑅) = (mulGrp‘𝑅)
7		eqid 2727	. . . . 5 ⊢ (._r‘𝑅) = (._r‘𝑅)
8	6, 7	mgpplusg 20069	. . . 4 ⊢ (._r‘𝑅) = (+_g‘(mulGrp‘𝑅))
9	2, 8	ressplusg 17262	. . 3 ⊢ (𝑈 ∈ V → (._r‘𝑅) = (+_g‘𝐺))
10	5, 9	mp1i 13	. 2 ⊢ (𝑅 ∈ Ring → (._r‘𝑅) = (+_g‘𝐺))
11	1, 7	unitmulcl 20308	. 2 ⊢ ((𝑅 ∈ Ring ∧ 𝑥 ∈ 𝑈 ∧ 𝑦 ∈ 𝑈) → (𝑥(._r‘𝑅)𝑦) ∈ 𝑈)
12		eqid 2727	. . . . 5 ⊢ (Base‘𝑅) = (Base‘𝑅)
13	12, 1	unitcl 20303	. . . 4 ⊢ (𝑥 ∈ 𝑈 → 𝑥 ∈ (Base‘𝑅))
14	12, 1	unitcl 20303	. . . 4 ⊢ (𝑦 ∈ 𝑈 → 𝑦 ∈ (Base‘𝑅))
15	12, 1	unitcl 20303	. . . 4 ⊢ (𝑧 ∈ 𝑈 → 𝑧 ∈ (Base‘𝑅))
16	13, 14, 15	3anim123i 1149	. . 3 ⊢ ((𝑥 ∈ 𝑈 ∧ 𝑦 ∈ 𝑈 ∧ 𝑧 ∈ 𝑈) → (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅) ∧ 𝑧 ∈ (Base‘𝑅)))
17	12, 7	ringass 20184	. . 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)))
18	16, 17	sylan2 592	. 2 ⊢ ((𝑅 ∈ Ring ∧ (𝑥 ∈ 𝑈 ∧ 𝑦 ∈ 𝑈 ∧ 𝑧 ∈ 𝑈)) → ((𝑥(._r‘𝑅)𝑦)(._r‘𝑅)𝑧) = (𝑥(._r‘𝑅)(𝑦(._r‘𝑅)𝑧)))
19		eqid 2727	. . 3 ⊢ (1_r‘𝑅) = (1_r‘𝑅)
20	1, 19	1unit 20302	. 2 ⊢ (𝑅 ∈ Ring → (1_r‘𝑅) ∈ 𝑈)
21	12, 7, 19	ringlidm 20194	. . 3 ⊢ ((𝑅 ∈ Ring ∧ 𝑥 ∈ (Base‘𝑅)) → ((1_r‘𝑅)(._r‘𝑅)𝑥) = 𝑥)
22	13, 21	sylan2 592	. 2 ⊢ ((𝑅 ∈ Ring ∧ 𝑥 ∈ 𝑈) → ((1_r‘𝑅)(._r‘𝑅)𝑥) = 𝑥)
23		simpr 484	. . . 4 ⊢ ((𝑅 ∈ Ring ∧ 𝑥 ∈ 𝑈) → 𝑥 ∈ 𝑈)
24		eqid 2727	. . . . 5 ⊢ (∥_r‘𝑅) = (∥_r‘𝑅)
25		eqid 2727	. . . . 5 ⊢ (opp_r‘𝑅) = (opp_r‘𝑅)
26		eqid 2727	. . . . 5 ⊢ (∥_r‘(opp_r‘𝑅)) = (∥_r‘(opp_r‘𝑅))
27	1, 19, 24, 25, 26	isunit 20301	. . . 4 ⊢ (𝑥 ∈ 𝑈 ↔ (𝑥(∥_r‘𝑅)(1_r‘𝑅) ∧ 𝑥(∥_r‘(opp_r‘𝑅))(1_r‘𝑅)))
28	23, 27	sylib 217	. . 3 ⊢ ((𝑅 ∈ Ring ∧ 𝑥 ∈ 𝑈) → (𝑥(∥_r‘𝑅)(1_r‘𝑅) ∧ 𝑥(∥_r‘(opp_r‘𝑅))(1_r‘𝑅)))
29	13	adantl 481	. . . . . 6 ⊢ ((𝑅 ∈ Ring ∧ 𝑥 ∈ 𝑈) → 𝑥 ∈ (Base‘𝑅))
30	12, 24, 7	dvdsr2 20291	. . . . . 6 ⊢ (𝑥 ∈ (Base‘𝑅) → (𝑥(∥_r‘𝑅)(1_r‘𝑅) ↔ ∃𝑦 ∈ (Base‘𝑅)(𝑦(._r‘𝑅)𝑥) = (1_r‘𝑅)))
31	29, 30	syl 17	. . . . 5 ⊢ ((𝑅 ∈ Ring ∧ 𝑥 ∈ 𝑈) → (𝑥(∥_r‘𝑅)(1_r‘𝑅) ↔ ∃𝑦 ∈ (Base‘𝑅)(𝑦(._r‘𝑅)𝑥) = (1_r‘𝑅)))
32	25, 12	opprbas 20269	. . . . . . 7 ⊢ (Base‘𝑅) = (Base‘(opp_r‘𝑅))
33		eqid 2727	. . . . . . 7 ⊢ (._r‘(opp_r‘𝑅)) = (._r‘(opp_r‘𝑅))
34	32, 26, 33	dvdsr2 20291	. . . . . 6 ⊢ (𝑥 ∈ (Base‘𝑅) → (𝑥(∥_r‘(opp_r‘𝑅))(1_r‘𝑅) ↔ ∃𝑚 ∈ (Base‘𝑅)(𝑚(._r‘(opp_r‘𝑅))𝑥) = (1_r‘𝑅)))
35	29, 34	syl 17	. . . . 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)))
36	31, 35	anbi12d 630	. . . 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))))
37		reeanv 3221	. . . . 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)))
38		simprl 770	. . . . . . . . . . . 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))
39	29	ad2antrr 725	. . . . . . . . . . . 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))
40	12, 24, 7	dvdsrmul 20292	. . . . . . . . . . . 12 ⊢ ((𝑚 ∈ (Base‘𝑅) ∧ 𝑥 ∈ (Base‘𝑅)) → 𝑚(∥_r‘𝑅)(𝑥(._r‘𝑅)𝑚))
41	38, 39, 40	syl2anc 583	. . . . . . . . . . 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))
42		simplll 774	. . . . . . . . . . . . . 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)
43		simplr 768	. . . . . . . . . . . . . 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))
44	12, 7	ringass 20184	. . . . . . . . . . . . . 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)))
45	42, 43, 39, 38, 44	syl13anc 1370	. . . . . . . . . . . . 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)))
46		simprrl 780	. . . . . . . . . . . . . 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))
47	46	oveq1d 7429	. . . . . . . . . . . . 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))
48	12, 7, 25, 33	opprmul 20265	. . . . . . . . . . . . . . 15 ⊢ (𝑚(._r‘(opp_r‘𝑅))𝑥) = (𝑥(._r‘𝑅)𝑚)
49		simprrr 781	. . . . . . . . . . . . . . 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))
50	48, 49	eqtr3id 2781	. . . . . . . . . . . . . 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))
51	50	oveq2d 7430	. . . . . . . . . . . . 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)))
52	45, 47, 51	3eqtr3d 2775	. . . . . . . . . . . 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)))
53	12, 7, 19	ringlidm 20194	. . . . . . . . . . . . 13 ⊢ ((𝑅 ∈ Ring ∧ 𝑚 ∈ (Base‘𝑅)) → ((1_r‘𝑅)(._r‘𝑅)𝑚) = 𝑚)
54	42, 38, 53	syl2anc 583	. . . . . . . . . . . 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)
55	12, 7, 19	ringridm 20195	. . . . . . . . . . . . 13 ⊢ ((𝑅 ∈ Ring ∧ 𝑦 ∈ (Base‘𝑅)) → (𝑦(._r‘𝑅)(1_r‘𝑅)) = 𝑦)
56	42, 43, 55	syl2anc 583	. . . . . . . . . . . 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)
57	52, 54, 56	3eqtr3d 2775	. . . . . . . . . . 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)
58	41, 57, 50	3brtr3d 5173	. . . . . . . . . 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))
59	32, 26, 33	dvdsrmul 20292	. . . . . . . . . . . 12 ⊢ ((𝑦 ∈ (Base‘𝑅) ∧ 𝑥 ∈ (Base‘𝑅)) → 𝑦(∥_r‘(opp_r‘𝑅))(𝑥(._r‘(opp_r‘𝑅))𝑦))
60	43, 39, 59	syl2anc 583	. . . . . . . . . . 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))
61	12, 7, 25, 33	opprmul 20265	. . . . . . . . . . . 12 ⊢ (𝑥(._r‘(opp_r‘𝑅))𝑦) = (𝑦(._r‘𝑅)𝑥)
62	61, 46	eqtrid 2779	. . . . . . . . . . 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))
63	60, 62	breqtrd 5168	. . . . . . . . . 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))
64	1, 19, 24, 25, 26	isunit 20301	. . . . . . . . . 10 ⊢ (𝑦 ∈ 𝑈 ↔ (𝑦(∥_r‘𝑅)(1_r‘𝑅) ∧ 𝑦(∥_r‘(opp_r‘𝑅))(1_r‘𝑅)))
65	58, 63, 64	sylanbrc 582	. . . . . . . . 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)
66	65, 46	jca 511	. . . . . . . 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)))
67	66	rexlimdvaa 3151	. . . . . . 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))))
68	67	expimpd 453	. . . . . 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))))
69	68	reximdv2 3159	. . . . 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)))
70	37, 69	biimtrrid 242	. . . 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)))
71	36, 70	sylbid 239	. . 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)))
72	28, 71	mpd 15	. 2 ⊢ ((𝑅 ∈ Ring ∧ 𝑥 ∈ 𝑈) → ∃𝑦 ∈ 𝑈 (𝑦(._r‘𝑅)𝑥) = (1_r‘𝑅))
73	4, 10, 11, 18, 20, 22, 72	isgrpde 18905	1 ⊢ (𝑅 ∈ Ring → 𝐺 ∈ Grp)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 205 ∧ wa 395 ∧ w3a 1085 = wceq 1534 ∈ wcel 2099 ∃wrex 3065 Vcvv 3469 class class class wbr 5142 ‘cfv 6542 (class class class)co 7414 Basecbs 17171 ↾_s cress 17200 +_gcplusg 17224 ._rcmulr 17225 Grpcgrp 18881 mulGrpcmgp 20065 1_rcur 20112 Ringcrg 20164 opp_rcoppr 20261 ∥_rcdsr 20282 Unitcui 20283

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2164 ax-ext 2698 ax-rep 5279 ax-sep 5293 ax-nul 5300 ax-pow 5359 ax-pr 5423 ax-un 7734 ax-cnex 11186 ax-resscn 11187 ax-1cn 11188 ax-icn 11189 ax-addcl 11190 ax-addrcl 11191 ax-mulcl 11192 ax-mulrcl 11193 ax-mulcom 11194 ax-addass 11195 ax-mulass 11196 ax-distr 11197 ax-i2m1 11198 ax-1ne0 11199 ax-1rid 11200 ax-rnegex 11201 ax-rrecex 11202 ax-cnre 11203 ax-pre-lttri 11204 ax-pre-lttrn 11205 ax-pre-ltadd 11206 ax-pre-mulgt0 11207

This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3or 1086 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2705 df-cleq 2719 df-clel 2805 df-nfc 2880 df-ne 2936 df-nel 3042 df-ral 3057 df-rex 3066 df-rmo 3371 df-reu 3372 df-rab 3428 df-v 3471 df-sbc 3775 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-pss 3963 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4904 df-iun 4993 df-br 5143 df-opab 5205 df-mpt 5226 df-tr 5260 df-id 5570 df-eprel 5576 df-po 5584 df-so 5585 df-fr 5627 df-we 5629 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-pred 6299 df-ord 6366 df-on 6367 df-lim 6368 df-suc 6369 df-iota 6494 df-fun 6544 df-fn 6545 df-f 6546 df-f1 6547 df-fo 6548 df-f1o 6549 df-fv 6550 df-riota 7370 df-ov 7417 df-oprab 7418 df-mpo 7419 df-om 7865 df-2nd 7988 df-tpos 8225 df-frecs 8280 df-wrecs 8311 df-recs 8385 df-rdg 8424 df-er 8718 df-en 8956 df-dom 8957 df-sdom 8958 df-pnf 11272 df-mnf 11273 df-xr 11274 df-ltxr 11275 df-le 11276 df-sub 11468 df-neg 11469 df-nn 12235 df-2 12297 df-3 12298 df-sets 17124 df-slot 17142 df-ndx 17154 df-base 17172 df-ress 17201 df-plusg 17237 df-mulr 17238 df-0g 17414 df-mgm 18591 df-sgrp 18670 df-mnd 18686 df-grp 18884 df-minusg 18885 df-cmn 19728 df-abl 19729 df-mgp 20066 df-rng 20084 df-ur 20113 df-ring 20166 df-oppr 20262 df-dvdsr 20285 df-unit 20286

This theorem is referenced by: unitabl 20312 unitsubm 20314 unitinvcl 20318 unitinvinv 20319 unitlinv 20321 unitrinv 20322 rdivmuldivd 20341 rhmunitinv 20439 subrgugrp 20519 isdrng2 20627 expghm 21388 invrvald 22565 nrginvrcn 24596 nrgtdrg 24597 dchrfi 27175 dchrghm 27176 dchrabs 27180 dchrptlem1 27184 dchrptlem2 27185 dchrptlem3 27186 dchrsum2 27188 dvrcan5 32921 idomodle 42541 proot1mul 42544 proot1hash 42545 proot1ex 42546

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