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Theorem unitgrp 20326

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 20325	. . 3 ⊢ 𝑈 = (Base‘𝐺)
4	3	a1i 11	. 2 ⊢ (𝑅 ∈ Ring → 𝑈 = (Base‘𝐺))
5	1	fvexi 6908	. . 3 ⊢ 𝑈 ∈ V
6		eqid 2725	. . . . 5 ⊢ (mulGrp‘𝑅) = (mulGrp‘𝑅)
7		eqid 2725	. . . . 5 ⊢ (._r‘𝑅) = (._r‘𝑅)
8	6, 7	mgpplusg 20082	. . . 4 ⊢ (._r‘𝑅) = (+_g‘(mulGrp‘𝑅))
9	2, 8	ressplusg 17270	. . 3 ⊢ (𝑈 ∈ V → (._r‘𝑅) = (+_g‘𝐺))
10	5, 9	mp1i 13	. 2 ⊢ (𝑅 ∈ Ring → (._r‘𝑅) = (+_g‘𝐺))
11	1, 7	unitmulcl 20323	. 2 ⊢ ((𝑅 ∈ Ring ∧ 𝑥 ∈ 𝑈 ∧ 𝑦 ∈ 𝑈) → (𝑥(._r‘𝑅)𝑦) ∈ 𝑈)
12		eqid 2725	. . . . 5 ⊢ (Base‘𝑅) = (Base‘𝑅)
13	12, 1	unitcl 20318	. . . 4 ⊢ (𝑥 ∈ 𝑈 → 𝑥 ∈ (Base‘𝑅))
14	12, 1	unitcl 20318	. . . 4 ⊢ (𝑦 ∈ 𝑈 → 𝑦 ∈ (Base‘𝑅))
15	12, 1	unitcl 20318	. . . 4 ⊢ (𝑧 ∈ 𝑈 → 𝑧 ∈ (Base‘𝑅))
16	13, 14, 15	3anim123i 1148	. . 3 ⊢ ((𝑥 ∈ 𝑈 ∧ 𝑦 ∈ 𝑈 ∧ 𝑧 ∈ 𝑈) → (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅) ∧ 𝑧 ∈ (Base‘𝑅)))
17	12, 7	ringass 20197	. . 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 2725	. . 3 ⊢ (1_r‘𝑅) = (1_r‘𝑅)
20	1, 19	1unit 20317	. 2 ⊢ (𝑅 ∈ Ring → (1_r‘𝑅) ∈ 𝑈)
21	12, 7, 19	ringlidm 20209	. . 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 2725	. . . . 5 ⊢ (∥_r‘𝑅) = (∥_r‘𝑅)
25		eqid 2725	. . . . 5 ⊢ (opp_r‘𝑅) = (opp_r‘𝑅)
26		eqid 2725	. . . . 5 ⊢ (∥_r‘(opp_r‘𝑅)) = (∥_r‘(opp_r‘𝑅))
27	1, 19, 24, 25, 26	isunit 20316	. . . 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 20306	. . . . . 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 20284	. . . . . . 7 ⊢ (Base‘𝑅) = (Base‘(opp_r‘𝑅))
33		eqid 2725	. . . . . . 7 ⊢ (._r‘(opp_r‘𝑅)) = (._r‘(opp_r‘𝑅))
34	32, 26, 33	dvdsr2 20306	. . . . . 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 3217	. . . . 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 769	. . . . . . . . . . . 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 724	. . . . . . . . . . . 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 20307	. . . . . . . . . . . 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 773	. . . . . . . . . . . . . 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 767	. . . . . . . . . . . . . 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 20197	. . . . . . . . . . . . . 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 1369	. . . . . . . . . . . . 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 779	. . . . . . . . . . . . . 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 7432	. . . . . . . . . . . . 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 20280	. . . . . . . . . . . . . . 15 ⊢ (𝑚(._r‘(opp_r‘𝑅))𝑥) = (𝑥(._r‘𝑅)𝑚)
49		simprrr 780	. . . . . . . . . . . . . . 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 2779	. . . . . . . . . . . . . 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 7433	. . . . . . . . . . . . 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 2773	. . . . . . . . . . . 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 20209	. . . . . . . . . . . . 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 20210	. . . . . . . . . . . . 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 2773	. . . . . . . . . . 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 5179	. . . . . . . . . 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 20307	. . . . . . . . . . . 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 20280	. . . . . . . . . . . 12 ⊢ (𝑥(._r‘(opp_r‘𝑅))𝑦) = (𝑦(._r‘𝑅)𝑥)
62	61, 46	eqtrid 2777	. . . . . . . . . . 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 5174	. . . . . . . . . 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 20316	. . . . . . . . . 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 3146	. . . . . . 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 3154	. . . . 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 18918	1 ⊢ (𝑅 ∈ Ring → 𝐺 ∈ Grp)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 205 ∧ wa 394 ∧ w3a 1084 = wceq 1533 ∈ wcel 2098 ∃wrex 3060 Vcvv 3463 class class class wbr 5148 ‘cfv 6547 (class class class)co 7417 Basecbs 17179 ↾_s cress 17208 +_gcplusg 17232 ._rcmulr 17233 Grpcgrp 18894 mulGrpcmgp 20078 1_rcur 20125 Ringcrg 20177 opp_rcoppr 20276 ∥_rcdsr 20297 Unitcui 20298

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2696 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5364 ax-pr 5428 ax-un 7739 ax-cnex 11194 ax-resscn 11195 ax-1cn 11196 ax-icn 11197 ax-addcl 11198 ax-addrcl 11199 ax-mulcl 11200 ax-mulrcl 11201 ax-mulcom 11202 ax-addass 11203 ax-mulass 11204 ax-distr 11205 ax-i2m1 11206 ax-1ne0 11207 ax-1rid 11208 ax-rnegex 11209 ax-rrecex 11210 ax-cnre 11211 ax-pre-lttri 11212 ax-pre-lttrn 11213 ax-pre-ltadd 11214 ax-pre-mulgt0 11215

This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2931 df-nel 3037 df-ral 3052 df-rex 3061 df-rmo 3364 df-reu 3365 df-rab 3420 df-v 3465 df-sbc 3775 df-csb 3891 df-dif 3948 df-un 3950 df-in 3952 df-ss 3962 df-pss 3965 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4909 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5575 df-eprel 5581 df-po 5589 df-so 5590 df-fr 5632 df-we 5634 df-xp 5683 df-rel 5684 df-cnv 5685 df-co 5686 df-dm 5687 df-rn 5688 df-res 5689 df-ima 5690 df-pred 6305 df-ord 6372 df-on 6373 df-lim 6374 df-suc 6375 df-iota 6499 df-fun 6549 df-fn 6550 df-f 6551 df-f1 6552 df-fo 6553 df-f1o 6554 df-fv 6555 df-riota 7373 df-ov 7420 df-oprab 7421 df-mpo 7422 df-om 7870 df-2nd 7993 df-tpos 8230 df-frecs 8285 df-wrecs 8316 df-recs 8390 df-rdg 8429 df-er 8723 df-en 8963 df-dom 8964 df-sdom 8965 df-pnf 11280 df-mnf 11281 df-xr 11282 df-ltxr 11283 df-le 11284 df-sub 11476 df-neg 11477 df-nn 12243 df-2 12305 df-3 12306 df-sets 17132 df-slot 17150 df-ndx 17162 df-base 17180 df-ress 17209 df-plusg 17245 df-mulr 17246 df-0g 17422 df-mgm 18599 df-sgrp 18678 df-mnd 18694 df-grp 18897 df-minusg 18898 df-cmn 19741 df-abl 19742 df-mgp 20079 df-rng 20097 df-ur 20126 df-ring 20179 df-oppr 20277 df-dvdsr 20300 df-unit 20301

This theorem is referenced by: unitabl 20327 unitsubm 20329 unitinvcl 20333 unitinvinv 20334 unitlinv 20336 unitrinv 20337 rdivmuldivd 20356 rhmunitinv 20454 subrgugrp 20534 isdrng2 20642 expghm 21405 invrvald 22608 nrginvrcn 24639 nrgtdrg 24640 dchrfi 27218 dchrghm 27219 dchrabs 27223 dchrptlem1 27227 dchrptlem2 27228 dchrptlem3 27229 dchrsum2 27231 dvrcan5 33000 idomodle 42684 proot1mul 42687 proot1hash 42688 proot1ex 42689

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