Theorem subrgunit 14045

Description: An element of a ring is a unit of a subring iff it is a unit of the parent ring and both it and its inverse are in the subring. (Contributed by Mario Carneiro, 4-Dec-2014.)

Hypotheses

Ref	Expression
subrgugrp.1	⊢ 𝑆 = (𝑅 ↾s 𝐴)
subrgugrp.2	⊢ 𝑈 = (Unit‘𝑅)
subrgugrp.3	⊢ 𝑉 = (Unit‘𝑆)
subrgunit.4	⊢ 𝐼 = (invr‘𝑅)

Assertion

Ref	Expression
subrgunit	⊢ (𝐴 ∈ (SubRing‘𝑅) → (𝑋 ∈ 𝑉 ↔ (𝑋 ∈ 𝑈 ∧ 𝑋 ∈ 𝐴 ∧ (𝐼‘𝑋) ∈ 𝐴)))

Proof of Theorem subrgunit

Step	Hyp	Ref	Expression
1		subrgugrp.1	. . . . 5 ⊢ 𝑆 = (𝑅 ↾s 𝐴)
2		subrgugrp.2	. . . . 5 ⊢ 𝑈 = (Unit‘𝑅)
3		subrgugrp.3	. . . . 5 ⊢ 𝑉 = (Unit‘𝑆)
4	1, 2, 3	subrguss 14042	. . . 4 ⊢ (𝐴 ∈ (SubRing‘𝑅) → 𝑉 ⊆ 𝑈)
5	4	sselda 3194	. . 3 ⊢ ((𝐴 ∈ (SubRing‘𝑅) ∧ 𝑋 ∈ 𝑉) → 𝑋 ∈ 𝑈)
6	1	subrgbas 14036	. . . . 5 ⊢ (𝐴 ∈ (SubRing‘𝑅) → 𝐴 = (Base‘𝑆))
7	6	adantr 276	. . . 4 ⊢ ((𝐴 ∈ (SubRing‘𝑅) ∧ 𝑋 ∈ 𝑉) → 𝐴 = (Base‘𝑆))
8	3	a1i 9	. . . 4 ⊢ ((𝐴 ∈ (SubRing‘𝑅) ∧ 𝑋 ∈ 𝑉) → 𝑉 = (Unit‘𝑆))
9	1	subrgring 14030	. . . . . 6 ⊢ (𝐴 ∈ (SubRing‘𝑅) → 𝑆 ∈ Ring)
10		ringsrg 13853	. . . . . 6 ⊢ (𝑆 ∈ Ring → 𝑆 ∈ SRing)
11	9, 10	syl 14	. . . . 5 ⊢ (𝐴 ∈ (SubRing‘𝑅) → 𝑆 ∈ SRing)
12	11	adantr 276	. . . 4 ⊢ ((𝐴 ∈ (SubRing‘𝑅) ∧ 𝑋 ∈ 𝑉) → 𝑆 ∈ SRing)
13		simpr 110	. . . 4 ⊢ ((𝐴 ∈ (SubRing‘𝑅) ∧ 𝑋 ∈ 𝑉) → 𝑋 ∈ 𝑉)
14	7, 8, 12, 13	unitcld 13914	. . 3 ⊢ ((𝐴 ∈ (SubRing‘𝑅) ∧ 𝑋 ∈ 𝑉) → 𝑋 ∈ 𝐴)
15		eqid 2206	. . . . . 6 ⊢ (invr‘𝑆) = (invr‘𝑆)
16		eqid 2206	. . . . . 6 ⊢ (Base‘𝑆) = (Base‘𝑆)
17	3, 15, 16	ringinvcl 13931	. . . . 5 ⊢ ((𝑆 ∈ Ring ∧ 𝑋 ∈ 𝑉) → ((invr‘𝑆)‘𝑋) ∈ (Base‘𝑆))
18	9, 17	sylan 283	. . . 4 ⊢ ((𝐴 ∈ (SubRing‘𝑅) ∧ 𝑋 ∈ 𝑉) → ((invr‘𝑆)‘𝑋) ∈ (Base‘𝑆))
19		subrgunit.4	. . . . 5 ⊢ 𝐼 = (invr‘𝑅)
20	1, 19, 3, 15	subrginv 14043	. . . 4 ⊢ ((𝐴 ∈ (SubRing‘𝑅) ∧ 𝑋 ∈ 𝑉) → (𝐼‘𝑋) = ((invr‘𝑆)‘𝑋))
21	18, 20, 7	3eltr4d 2290	. . 3 ⊢ ((𝐴 ∈ (SubRing‘𝑅) ∧ 𝑋 ∈ 𝑉) → (𝐼‘𝑋) ∈ 𝐴)
22	5, 14, 21	3jca 1180	. 2 ⊢ ((𝐴 ∈ (SubRing‘𝑅) ∧ 𝑋 ∈ 𝑉) → (𝑋 ∈ 𝑈 ∧ 𝑋 ∈ 𝐴 ∧ (𝐼‘𝑋) ∈ 𝐴))
23		eqidd 2207	. . . . 5 ⊢ ((𝐴 ∈ (SubRing‘𝑅) ∧ (𝑋 ∈ 𝑈 ∧ 𝑋 ∈ 𝐴 ∧ (𝐼‘𝑋) ∈ 𝐴)) → (Base‘𝑆) = (Base‘𝑆))
24		eqidd 2207	. . . . 5 ⊢ ((𝐴 ∈ (SubRing‘𝑅) ∧ (𝑋 ∈ 𝑈 ∧ 𝑋 ∈ 𝐴 ∧ (𝐼‘𝑋) ∈ 𝐴)) → (∥r‘𝑆) = (∥r‘𝑆))
25	11	adantr 276	. . . . 5 ⊢ ((𝐴 ∈ (SubRing‘𝑅) ∧ (𝑋 ∈ 𝑈 ∧ 𝑋 ∈ 𝐴 ∧ (𝐼‘𝑋) ∈ 𝐴)) → 𝑆 ∈ SRing)
26		eqidd 2207	. . . . 5 ⊢ ((𝐴 ∈ (SubRing‘𝑅) ∧ (𝑋 ∈ 𝑈 ∧ 𝑋 ∈ 𝐴 ∧ (𝐼‘𝑋) ∈ 𝐴)) → (.r‘𝑆) = (.r‘𝑆))
27		simpr2 1007	. . . . . 6 ⊢ ((𝐴 ∈ (SubRing‘𝑅) ∧ (𝑋 ∈ 𝑈 ∧ 𝑋 ∈ 𝐴 ∧ (𝐼‘𝑋) ∈ 𝐴)) → 𝑋 ∈ 𝐴)
28	6	adantr 276	. . . . . 6 ⊢ ((𝐴 ∈ (SubRing‘𝑅) ∧ (𝑋 ∈ 𝑈 ∧ 𝑋 ∈ 𝐴 ∧ (𝐼‘𝑋) ∈ 𝐴)) → 𝐴 = (Base‘𝑆))
29	27, 28	eleqtrd 2285	. . . . 5 ⊢ ((𝐴 ∈ (SubRing‘𝑅) ∧ (𝑋 ∈ 𝑈 ∧ 𝑋 ∈ 𝐴 ∧ (𝐼‘𝑋) ∈ 𝐴)) → 𝑋 ∈ (Base‘𝑆))
30		simpr3 1008	. . . . . 6 ⊢ ((𝐴 ∈ (SubRing‘𝑅) ∧ (𝑋 ∈ 𝑈 ∧ 𝑋 ∈ 𝐴 ∧ (𝐼‘𝑋) ∈ 𝐴)) → (𝐼‘𝑋) ∈ 𝐴)
31	30, 28	eleqtrd 2285	. . . . 5 ⊢ ((𝐴 ∈ (SubRing‘𝑅) ∧ (𝑋 ∈ 𝑈 ∧ 𝑋 ∈ 𝐴 ∧ (𝐼‘𝑋) ∈ 𝐴)) → (𝐼‘𝑋) ∈ (Base‘𝑆))
32	23, 24, 25, 26, 29, 31	dvdsrmuld 13902	. . . 4 ⊢ ((𝐴 ∈ (SubRing‘𝑅) ∧ (𝑋 ∈ 𝑈 ∧ 𝑋 ∈ 𝐴 ∧ (𝐼‘𝑋) ∈ 𝐴)) → 𝑋(∥r‘𝑆)((𝐼‘𝑋)(.r‘𝑆)𝑋))
33		subrgrcl 14032	. . . . . 6 ⊢ (𝐴 ∈ (SubRing‘𝑅) → 𝑅 ∈ Ring)
34		simpr1 1006	. . . . . 6 ⊢ ((𝐴 ∈ (SubRing‘𝑅) ∧ (𝑋 ∈ 𝑈 ∧ 𝑋 ∈ 𝐴 ∧ (𝐼‘𝑋) ∈ 𝐴)) → 𝑋 ∈ 𝑈)
35		eqid 2206	. . . . . . 7 ⊢ (.r‘𝑅) = (.r‘𝑅)
36		eqid 2206	. . . . . . 7 ⊢ (1r‘𝑅) = (1r‘𝑅)
37	2, 19, 35, 36	unitlinv 13932	. . . . . 6 ⊢ ((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝑈) → ((𝐼‘𝑋)(.r‘𝑅)𝑋) = (1r‘𝑅))
38	33, 34, 37	syl2an2r 595	. . . . 5 ⊢ ((𝐴 ∈ (SubRing‘𝑅) ∧ (𝑋 ∈ 𝑈 ∧ 𝑋 ∈ 𝐴 ∧ (𝐼‘𝑋) ∈ 𝐴)) → ((𝐼‘𝑋)(.r‘𝑅)𝑋) = (1r‘𝑅))
39	1, 35	ressmulrg 13021	. . . . . . . 8 ⊢ ((𝐴 ∈ (SubRing‘𝑅) ∧ 𝑅 ∈ Ring) → (.r‘𝑅) = (.r‘𝑆))
40	33, 39	mpdan 421	. . . . . . 7 ⊢ (𝐴 ∈ (SubRing‘𝑅) → (.r‘𝑅) = (.r‘𝑆))
41	40	adantr 276	. . . . . 6 ⊢ ((𝐴 ∈ (SubRing‘𝑅) ∧ (𝑋 ∈ 𝑈 ∧ 𝑋 ∈ 𝐴 ∧ (𝐼‘𝑋) ∈ 𝐴)) → (.r‘𝑅) = (.r‘𝑆))
42	41	oveqd 5968	. . . . 5 ⊢ ((𝐴 ∈ (SubRing‘𝑅) ∧ (𝑋 ∈ 𝑈 ∧ 𝑋 ∈ 𝐴 ∧ (𝐼‘𝑋) ∈ 𝐴)) → ((𝐼‘𝑋)(.r‘𝑅)𝑋) = ((𝐼‘𝑋)(.r‘𝑆)𝑋))
43	1, 36	subrg1 14037	. . . . . 6 ⊢ (𝐴 ∈ (SubRing‘𝑅) → (1r‘𝑅) = (1r‘𝑆))
44	43	adantr 276	. . . . 5 ⊢ ((𝐴 ∈ (SubRing‘𝑅) ∧ (𝑋 ∈ 𝑈 ∧ 𝑋 ∈ 𝐴 ∧ (𝐼‘𝑋) ∈ 𝐴)) → (1r‘𝑅) = (1r‘𝑆))
45	38, 42, 44	3eqtr3d 2247	. . . 4 ⊢ ((𝐴 ∈ (SubRing‘𝑅) ∧ (𝑋 ∈ 𝑈 ∧ 𝑋 ∈ 𝐴 ∧ (𝐼‘𝑋) ∈ 𝐴)) → ((𝐼‘𝑋)(.r‘𝑆)𝑋) = (1r‘𝑆))
46	32, 45	breqtrd 4073	. . 3 ⊢ ((𝐴 ∈ (SubRing‘𝑅) ∧ (𝑋 ∈ 𝑈 ∧ 𝑋 ∈ 𝐴 ∧ (𝐼‘𝑋) ∈ 𝐴)) → 𝑋(∥r‘𝑆)(1r‘𝑆))
47	9	adantr 276	. . . . . 6 ⊢ ((𝐴 ∈ (SubRing‘𝑅) ∧ (𝑋 ∈ 𝑈 ∧ 𝑋 ∈ 𝐴 ∧ (𝐼‘𝑋) ∈ 𝐴)) → 𝑆 ∈ Ring)
48		eqid 2206	. . . . . . 7 ⊢ (oppr‘𝑆) = (oppr‘𝑆)
49	48, 16	opprbasg 13881	. . . . . 6 ⊢ (𝑆 ∈ Ring → (Base‘𝑆) = (Base‘(oppr‘𝑆)))
50	47, 49	syl 14	. . . . 5 ⊢ ((𝐴 ∈ (SubRing‘𝑅) ∧ (𝑋 ∈ 𝑈 ∧ 𝑋 ∈ 𝐴 ∧ (𝐼‘𝑋) ∈ 𝐴)) → (Base‘𝑆) = (Base‘(oppr‘𝑆)))
51		eqidd 2207	. . . . 5 ⊢ ((𝐴 ∈ (SubRing‘𝑅) ∧ (𝑋 ∈ 𝑈 ∧ 𝑋 ∈ 𝐴 ∧ (𝐼‘𝑋) ∈ 𝐴)) → (∥r‘(oppr‘𝑆)) = (∥r‘(oppr‘𝑆)))
52	48	opprring 13885	. . . . . 6 ⊢ (𝑆 ∈ Ring → (oppr‘𝑆) ∈ Ring)
53		ringsrg 13853	. . . . . 6 ⊢ ((oppr‘𝑆) ∈ Ring → (oppr‘𝑆) ∈ SRing)
54	47, 52, 53	3syl 17	. . . . 5 ⊢ ((𝐴 ∈ (SubRing‘𝑅) ∧ (𝑋 ∈ 𝑈 ∧ 𝑋 ∈ 𝐴 ∧ (𝐼‘𝑋) ∈ 𝐴)) → (oppr‘𝑆) ∈ SRing)
55		eqidd 2207	. . . . 5 ⊢ ((𝐴 ∈ (SubRing‘𝑅) ∧ (𝑋 ∈ 𝑈 ∧ 𝑋 ∈ 𝐴 ∧ (𝐼‘𝑋) ∈ 𝐴)) → (.r‘(oppr‘𝑆)) = (.r‘(oppr‘𝑆)))
56	50, 51, 54, 55, 29, 31	dvdsrmuld 13902	. . . 4 ⊢ ((𝐴 ∈ (SubRing‘𝑅) ∧ (𝑋 ∈ 𝑈 ∧ 𝑋 ∈ 𝐴 ∧ (𝐼‘𝑋) ∈ 𝐴)) → 𝑋(∥r‘(oppr‘𝑆))((𝐼‘𝑋)(.r‘(oppr‘𝑆))𝑋))
57		eqid 2206	. . . . . . 7 ⊢ (.r‘𝑆) = (.r‘𝑆)
58		eqid 2206	. . . . . . 7 ⊢ (.r‘(oppr‘𝑆)) = (.r‘(oppr‘𝑆))
59	16, 57, 48, 58	opprmulg 13877	. . . . . 6 ⊢ ((𝑆 ∈ Ring ∧ (𝐼‘𝑋) ∈ (Base‘𝑆) ∧ 𝑋 ∈ (Base‘𝑆)) → ((𝐼‘𝑋)(.r‘(oppr‘𝑆))𝑋) = (𝑋(.r‘𝑆)(𝐼‘𝑋)))
60	47, 31, 29, 59	syl3anc 1250	. . . . 5 ⊢ ((𝐴 ∈ (SubRing‘𝑅) ∧ (𝑋 ∈ 𝑈 ∧ 𝑋 ∈ 𝐴 ∧ (𝐼‘𝑋) ∈ 𝐴)) → ((𝐼‘𝑋)(.r‘(oppr‘𝑆))𝑋) = (𝑋(.r‘𝑆)(𝐼‘𝑋)))
61	2, 19, 35, 36	unitrinv 13933	. . . . . . 7 ⊢ ((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝑈) → (𝑋(.r‘𝑅)(𝐼‘𝑋)) = (1r‘𝑅))
62	33, 34, 61	syl2an2r 595	. . . . . 6 ⊢ ((𝐴 ∈ (SubRing‘𝑅) ∧ (𝑋 ∈ 𝑈 ∧ 𝑋 ∈ 𝐴 ∧ (𝐼‘𝑋) ∈ 𝐴)) → (𝑋(.r‘𝑅)(𝐼‘𝑋)) = (1r‘𝑅))
63	41	oveqd 5968	. . . . . 6 ⊢ ((𝐴 ∈ (SubRing‘𝑅) ∧ (𝑋 ∈ 𝑈 ∧ 𝑋 ∈ 𝐴 ∧ (𝐼‘𝑋) ∈ 𝐴)) → (𝑋(.r‘𝑅)(𝐼‘𝑋)) = (𝑋(.r‘𝑆)(𝐼‘𝑋)))
64	62, 63, 44	3eqtr3d 2247	. . . . 5 ⊢ ((𝐴 ∈ (SubRing‘𝑅) ∧ (𝑋 ∈ 𝑈 ∧ 𝑋 ∈ 𝐴 ∧ (𝐼‘𝑋) ∈ 𝐴)) → (𝑋(.r‘𝑆)(𝐼‘𝑋)) = (1r‘𝑆))
65	60, 64	eqtrd 2239	. . . 4 ⊢ ((𝐴 ∈ (SubRing‘𝑅) ∧ (𝑋 ∈ 𝑈 ∧ 𝑋 ∈ 𝐴 ∧ (𝐼‘𝑋) ∈ 𝐴)) → ((𝐼‘𝑋)(.r‘(oppr‘𝑆))𝑋) = (1r‘𝑆))
66	56, 65	breqtrd 4073	. . 3 ⊢ ((𝐴 ∈ (SubRing‘𝑅) ∧ (𝑋 ∈ 𝑈 ∧ 𝑋 ∈ 𝐴 ∧ (𝐼‘𝑋) ∈ 𝐴)) → 𝑋(∥r‘(oppr‘𝑆))(1r‘𝑆))
67	3	a1i 9	. . . . 5 ⊢ (𝐴 ∈ (SubRing‘𝑅) → 𝑉 = (Unit‘𝑆))
68		eqidd 2207	. . . . 5 ⊢ (𝐴 ∈ (SubRing‘𝑅) → (1r‘𝑆) = (1r‘𝑆))
69		eqidd 2207	. . . . 5 ⊢ (𝐴 ∈ (SubRing‘𝑅) → (∥r‘𝑆) = (∥r‘𝑆))
70		eqidd 2207	. . . . 5 ⊢ (𝐴 ∈ (SubRing‘𝑅) → (oppr‘𝑆) = (oppr‘𝑆))
71		eqidd 2207	. . . . 5 ⊢ (𝐴 ∈ (SubRing‘𝑅) → (∥r‘(oppr‘𝑆)) = (∥r‘(oppr‘𝑆)))
72	67, 68, 69, 70, 71, 11	isunitd 13912	. . . 4 ⊢ (𝐴 ∈ (SubRing‘𝑅) → (𝑋 ∈ 𝑉 ↔ (𝑋(∥r‘𝑆)(1r‘𝑆) ∧ 𝑋(∥r‘(oppr‘𝑆))(1r‘𝑆))))
73	72	adantr 276	. . 3 ⊢ ((𝐴 ∈ (SubRing‘𝑅) ∧ (𝑋 ∈ 𝑈 ∧ 𝑋 ∈ 𝐴 ∧ (𝐼‘𝑋) ∈ 𝐴)) → (𝑋 ∈ 𝑉 ↔ (𝑋(∥r‘𝑆)(1r‘𝑆) ∧ 𝑋(∥r‘(oppr‘𝑆))(1r‘𝑆))))
74	46, 66, 73	mpbir2and 947	. 2 ⊢ ((𝐴 ∈ (SubRing‘𝑅) ∧ (𝑋 ∈ 𝑈 ∧ 𝑋 ∈ 𝐴 ∧ (𝐼‘𝑋) ∈ 𝐴)) → 𝑋 ∈ 𝑉)
75	22, 74	impbida 596	1 ⊢ (𝐴 ∈ (SubRing‘𝑅) → (𝑋 ∈ 𝑉 ↔ (𝑋 ∈ 𝑈 ∧ 𝑋 ∈ 𝐴 ∧ (𝐼‘𝑋) ∈ 𝐴)))

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105   ∧ w3a 981   = wceq 1373   ∈ wcel 2177   class class class wbr 4047  ‘cfv 5276  (class class class)co 5951  Basecbs 12876   ↾_s cress 12877  ._rcmulr 12954  1_rcur 13765  SRingcsrg 13769  Ringcrg 13802  opp_rcoppr 13873  ∥_rcdsr 13892  Unitcui 13893  inv_rcinvr 13926  SubRingcsubrg 14023

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 615  ax-in2 616  ax-io 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-13 2179  ax-14 2180  ax-ext 2188  ax-coll 4163  ax-sep 4166  ax-nul 4174  ax-pow 4222  ax-pr 4257  ax-un 4484  ax-setind 4589  ax-cnex 8023  ax-resscn 8024  ax-1cn 8025  ax-1re 8026  ax-icn 8027  ax-addcl 8028  ax-addrcl 8029  ax-mulcl 8030  ax-addcom 8032  ax-addass 8034  ax-i2m1 8037  ax-0lt1 8038  ax-0id 8040  ax-rnegex 8041  ax-pre-ltirr 8044  ax-pre-lttrn 8046  ax-pre-ltadd 8048

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-fal 1379  df-nf 1485  df-sb 1787  df-eu 2058  df-mo 2059  df-clab 2193  df-cleq 2199  df-clel 2202  df-nfc 2338  df-ne 2378  df-nel 2473  df-ral 2490  df-rex 2491  df-reu 2492  df-rmo 2493  df-rab 2494  df-v 2775  df-sbc 3000  df-csb 3095  df-dif 3169  df-un 3171  df-in 3173  df-ss 3180  df-nul 3462  df-pw 3619  df-sn 3640  df-pr 3641  df-op 3643  df-uni 3853  df-int 3888  df-iun 3931  df-br 4048  df-opab 4110  df-mpt 4111  df-id 4344  df-xp 4685  df-rel 4686  df-cnv 4687  df-co 4688  df-dm 4689  df-rn 4690  df-res 4691  df-ima 4692  df-iota 5237  df-fun 5278  df-fn 5279  df-f 5280  df-f1 5281  df-fo 5282  df-f1o 5283  df-fv 5284  df-riota 5906  df-ov 5954  df-oprab 5955  df-mpo 5956  df-tpos 6338  df-pnf 8116  df-mnf 8117  df-ltxr 8119  df-inn 9044  df-2 9102  df-3 9103  df-ndx 12879  df-slot 12880  df-base 12882  df-sets 12883  df-iress 12884  df-plusg 12966  df-mulr 12967  df-0g 13134  df-mgm 13232  df-sgrp 13278  df-mnd 13293  df-grp 13379  df-minusg 13380  df-subg 13550  df-cmn 13666  df-abl 13667  df-mgp 13727  df-ur 13766  df-srg 13770  df-ring 13804  df-oppr 13874  df-dvdsr 13895  df-unit 13896  df-invr 13927  df-subrg 14025

This theorem is referenced by: (None)