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Theorem unitgrp 20274

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 20273	. . 3 ⊢ 𝑈 = (Base‘𝐺)
4	3	a1i 11	. 2 ⊢ (𝑅 ∈ Ring → 𝑈 = (Base‘𝐺))
5	1	fvexi 6904	. . 3 ⊢ 𝑈 ∈ V
6		eqid 2730	. . . . 5 ⊢ (mulGrp‘𝑅) = (mulGrp‘𝑅)
7		eqid 2730	. . . . 5 ⊢ (._r‘𝑅) = (._r‘𝑅)
8	6, 7	mgpplusg 20032	. . . 4 ⊢ (._r‘𝑅) = (+_g‘(mulGrp‘𝑅))
9	2, 8	ressplusg 17239	. . 3 ⊢ (𝑈 ∈ V → (._r‘𝑅) = (+_g‘𝐺))
10	5, 9	mp1i 13	. 2 ⊢ (𝑅 ∈ Ring → (._r‘𝑅) = (+_g‘𝐺))
11	1, 7	unitmulcl 20271	. 2 ⊢ ((𝑅 ∈ Ring ∧ 𝑥 ∈ 𝑈 ∧ 𝑦 ∈ 𝑈) → (𝑥(._r‘𝑅)𝑦) ∈ 𝑈)
12		eqid 2730	. . . . 5 ⊢ (Base‘𝑅) = (Base‘𝑅)
13	12, 1	unitcl 20266	. . . 4 ⊢ (𝑥 ∈ 𝑈 → 𝑥 ∈ (Base‘𝑅))
14	12, 1	unitcl 20266	. . . 4 ⊢ (𝑦 ∈ 𝑈 → 𝑦 ∈ (Base‘𝑅))
15	12, 1	unitcl 20266	. . . 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 20147	. . 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 591	. 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 2730	. . 3 ⊢ (1_r‘𝑅) = (1_r‘𝑅)
20	1, 19	1unit 20265	. 2 ⊢ (𝑅 ∈ Ring → (1_r‘𝑅) ∈ 𝑈)
21	12, 7, 19	ringlidm 20157	. . 3 ⊢ ((𝑅 ∈ Ring ∧ 𝑥 ∈ (Base‘𝑅)) → ((1_r‘𝑅)(._r‘𝑅)𝑥) = 𝑥)
22	13, 21	sylan2 591	. 2 ⊢ ((𝑅 ∈ Ring ∧ 𝑥 ∈ 𝑈) → ((1_r‘𝑅)(._r‘𝑅)𝑥) = 𝑥)
23		simpr 483	. . . 4 ⊢ ((𝑅 ∈ Ring ∧ 𝑥 ∈ 𝑈) → 𝑥 ∈ 𝑈)
24		eqid 2730	. . . . 5 ⊢ (∥_r‘𝑅) = (∥_r‘𝑅)
25		eqid 2730	. . . . 5 ⊢ (opp_r‘𝑅) = (opp_r‘𝑅)
26		eqid 2730	. . . . 5 ⊢ (∥_r‘(opp_r‘𝑅)) = (∥_r‘(opp_r‘𝑅))
27	1, 19, 24, 25, 26	isunit 20264	. . . 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 480	. . . . . 6 ⊢ ((𝑅 ∈ Ring ∧ 𝑥 ∈ 𝑈) → 𝑥 ∈ (Base‘𝑅))
30	12, 24, 7	dvdsr2 20254	. . . . . 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 20232	. . . . . . 7 ⊢ (Base‘𝑅) = (Base‘(opp_r‘𝑅))
33		eqid 2730	. . . . . . 7 ⊢ (._r‘(opp_r‘𝑅)) = (._r‘(opp_r‘𝑅))
34	32, 26, 33	dvdsr2 20254	. . . . . 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 629	. . . 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 3224	. . . . 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 767	. . . . . . . . . . . 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 722	. . . . . . . . . . . 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 20255	. . . . . . . . . . . 12 ⊢ ((𝑚 ∈ (Base‘𝑅) ∧ 𝑥 ∈ (Base‘𝑅)) → 𝑚(∥_r‘𝑅)(𝑥(._r‘𝑅)𝑚))
41	38, 39, 40	syl2anc 582	. . . . . . . . . . 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 771	. . . . . . . . . . . . . 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 765	. . . . . . . . . . . . . 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 20147	. . . . . . . . . . . . . 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 777	. . . . . . . . . . . . . 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 7426	. . . . . . . . . . . . 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 20228	. . . . . . . . . . . . . . 15 ⊢ (𝑚(._r‘(opp_r‘𝑅))𝑥) = (𝑥(._r‘𝑅)𝑚)
49		simprrr 778	. . . . . . . . . . . . . . 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 2784	. . . . . . . . . . . . . 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 7427	. . . . . . . . . . . . 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 2778	. . . . . . . . . . . 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 20157	. . . . . . . . . . . . 13 ⊢ ((𝑅 ∈ Ring ∧ 𝑚 ∈ (Base‘𝑅)) → ((1_r‘𝑅)(._r‘𝑅)𝑚) = 𝑚)
54	42, 38, 53	syl2anc 582	. . . . . . . . . . . 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 20158	. . . . . . . . . . . . 13 ⊢ ((𝑅 ∈ Ring ∧ 𝑦 ∈ (Base‘𝑅)) → (𝑦(._r‘𝑅)(1_r‘𝑅)) = 𝑦)
56	42, 43, 55	syl2anc 582	. . . . . . . . . . . 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 2778	. . . . . . . . . . 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 5178	. . . . . . . . . 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 20255	. . . . . . . . . . . 12 ⊢ ((𝑦 ∈ (Base‘𝑅) ∧ 𝑥 ∈ (Base‘𝑅)) → 𝑦(∥_r‘(opp_r‘𝑅))(𝑥(._r‘(opp_r‘𝑅))𝑦))
60	43, 39, 59	syl2anc 582	. . . . . . . . . . 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 20228	. . . . . . . . . . . 12 ⊢ (𝑥(._r‘(opp_r‘𝑅))𝑦) = (𝑦(._r‘𝑅)𝑥)
62	61, 46	eqtrid 2782	. . . . . . . . . . 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 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‘(opp_r‘𝑅))(1_r‘𝑅))
64	1, 19, 24, 25, 26	isunit 20264	. . . . . . . . . 10 ⊢ (𝑦 ∈ 𝑈 ↔ (𝑦(∥_r‘𝑅)(1_r‘𝑅) ∧ 𝑦(∥_r‘(opp_r‘𝑅))(1_r‘𝑅)))
65	58, 63, 64	sylanbrc 581	. . . . . . . . 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 510	. . . . . . . 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 3154	. . . . . . 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 452	. . . . . 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 3162	. . . . 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 18879	1 ⊢ (𝑅 ∈ Ring → 𝐺 ∈ Grp)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 205 ∧ wa 394 ∧ w3a 1085 = wceq 1539 ∈ wcel 2104 ∃wrex 3068 Vcvv 3472 class class class wbr 5147 ‘cfv 6542 (class class class)co 7411 Basecbs 17148 ↾_s cress 17177 +_gcplusg 17201 ._rcmulr 17202 Grpcgrp 18855 mulGrpcmgp 20028 1_rcur 20075 Ringcrg 20127 opp_rcoppr 20224 ∥_rcdsr 20245 Unitcui 20246

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1911 ax-6 1969 ax-7 2009 ax-8 2106 ax-9 2114 ax-10 2135 ax-11 2152 ax-12 2169 ax-ext 2701 ax-rep 5284 ax-sep 5298 ax-nul 5305 ax-pow 5362 ax-pr 5426 ax-un 7727 ax-cnex 11168 ax-resscn 11169 ax-1cn 11170 ax-icn 11171 ax-addcl 11172 ax-addrcl 11173 ax-mulcl 11174 ax-mulrcl 11175 ax-mulcom 11176 ax-addass 11177 ax-mulass 11178 ax-distr 11179 ax-i2m1 11180 ax-1ne0 11181 ax-1rid 11182 ax-rnegex 11183 ax-rrecex 11184 ax-cnre 11185 ax-pre-lttri 11186 ax-pre-lttrn 11187 ax-pre-ltadd 11188 ax-pre-mulgt0 11189

This theorem depends on definitions: df-bi 206 df-an 395 df-or 844 df-3or 1086 df-3an 1087 df-tru 1542 df-fal 1552 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2532 df-eu 2561 df-clab 2708 df-cleq 2722 df-clel 2808 df-nfc 2883 df-ne 2939 df-nel 3045 df-ral 3060 df-rex 3069 df-rmo 3374 df-reu 3375 df-rab 3431 df-v 3474 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4322 df-if 4528 df-pw 4603 df-sn 4628 df-pr 4630 df-op 4634 df-uni 4908 df-iun 4998 df-br 5148 df-opab 5210 df-mpt 5231 df-tr 5265 df-id 5573 df-eprel 5579 df-po 5587 df-so 5588 df-fr 5630 df-we 5632 df-xp 5681 df-rel 5682 df-cnv 5683 df-co 5684 df-dm 5685 df-rn 5686 df-res 5687 df-ima 5688 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 7367 df-ov 7414 df-oprab 7415 df-mpo 7416 df-om 7858 df-2nd 7978 df-tpos 8213 df-frecs 8268 df-wrecs 8299 df-recs 8373 df-rdg 8412 df-er 8705 df-en 8942 df-dom 8943 df-sdom 8944 df-pnf 11254 df-mnf 11255 df-xr 11256 df-ltxr 11257 df-le 11258 df-sub 11450 df-neg 11451 df-nn 12217 df-2 12279 df-3 12280 df-sets 17101 df-slot 17119 df-ndx 17131 df-base 17149 df-ress 17178 df-plusg 17214 df-mulr 17215 df-0g 17391 df-mgm 18565 df-sgrp 18644 df-mnd 18660 df-grp 18858 df-minusg 18859 df-cmn 19691 df-abl 19692 df-mgp 20029 df-rng 20047 df-ur 20076 df-ring 20129 df-oppr 20225 df-dvdsr 20248 df-unit 20249

This theorem is referenced by: unitabl 20275 unitsubm 20277 unitinvcl 20281 unitinvinv 20282 unitlinv 20284 unitrinv 20285 rdivmuldivd 20304 rhmunitinv 20402 subrgugrp 20481 isdrng2 20514 expghm 21246 invrvald 22398 nrginvrcn 24429 nrgtdrg 24430 dchrfi 26994 dchrghm 26995 dchrabs 26999 dchrptlem1 27003 dchrptlem2 27004 dchrptlem3 27005 dchrsum2 27007 dvrcan5 32655 idomodle 42240 proot1mul 42243 proot1hash 42244 proot1ex 42245

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