Theorem subrgunit 13795

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 13792	. . . 4 ⊢ (𝐴 ∈ (SubRing‘𝑅) → 𝑉 ⊆ 𝑈)
5	4	sselda 3183	. . 3 ⊢ ((𝐴 ∈ (SubRing‘𝑅) ∧ 𝑋 ∈ 𝑉) → 𝑋 ∈ 𝑈)
6	1	subrgbas 13786	. . . . 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 13780	. . . . . 6 ⊢ (𝐴 ∈ (SubRing‘𝑅) → 𝑆 ∈ Ring)
10		ringsrg 13603	. . . . . 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 13664	. . 3 ⊢ ((𝐴 ∈ (SubRing‘𝑅) ∧ 𝑋 ∈ 𝑉) → 𝑋 ∈ 𝐴)
15		eqid 2196	. . . . . 6 ⊢ (invr‘𝑆) = (invr‘𝑆)
16		eqid 2196	. . . . . 6 ⊢ (Base‘𝑆) = (Base‘𝑆)
17	3, 15, 16	ringinvcl 13681	. . . . 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 13793	. . . 4 ⊢ ((𝐴 ∈ (SubRing‘𝑅) ∧ 𝑋 ∈ 𝑉) → (𝐼‘𝑋) = ((invr‘𝑆)‘𝑋))
21	18, 20, 7	3eltr4d 2280	. . 3 ⊢ ((𝐴 ∈ (SubRing‘𝑅) ∧ 𝑋 ∈ 𝑉) → (𝐼‘𝑋) ∈ 𝐴)
22	5, 14, 21	3jca 1179	. 2 ⊢ ((𝐴 ∈ (SubRing‘𝑅) ∧ 𝑋 ∈ 𝑉) → (𝑋 ∈ 𝑈 ∧ 𝑋 ∈ 𝐴 ∧ (𝐼‘𝑋) ∈ 𝐴))
23		eqidd 2197	. . . . 5 ⊢ ((𝐴 ∈ (SubRing‘𝑅) ∧ (𝑋 ∈ 𝑈 ∧ 𝑋 ∈ 𝐴 ∧ (𝐼‘𝑋) ∈ 𝐴)) → (Base‘𝑆) = (Base‘𝑆))
24		eqidd 2197	. . . . 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 2197	. . . . 5 ⊢ ((𝐴 ∈ (SubRing‘𝑅) ∧ (𝑋 ∈ 𝑈 ∧ 𝑋 ∈ 𝐴 ∧ (𝐼‘𝑋) ∈ 𝐴)) → (.r‘𝑆) = (.r‘𝑆))
27		simpr2 1006	. . . . . 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 2275	. . . . 5 ⊢ ((𝐴 ∈ (SubRing‘𝑅) ∧ (𝑋 ∈ 𝑈 ∧ 𝑋 ∈ 𝐴 ∧ (𝐼‘𝑋) ∈ 𝐴)) → 𝑋 ∈ (Base‘𝑆))
30		simpr3 1007	. . . . . 6 ⊢ ((𝐴 ∈ (SubRing‘𝑅) ∧ (𝑋 ∈ 𝑈 ∧ 𝑋 ∈ 𝐴 ∧ (𝐼‘𝑋) ∈ 𝐴)) → (𝐼‘𝑋) ∈ 𝐴)
31	30, 28	eleqtrd 2275	. . . . 5 ⊢ ((𝐴 ∈ (SubRing‘𝑅) ∧ (𝑋 ∈ 𝑈 ∧ 𝑋 ∈ 𝐴 ∧ (𝐼‘𝑋) ∈ 𝐴)) → (𝐼‘𝑋) ∈ (Base‘𝑆))
32	23, 24, 25, 26, 29, 31	dvdsrmuld 13652	. . . 4 ⊢ ((𝐴 ∈ (SubRing‘𝑅) ∧ (𝑋 ∈ 𝑈 ∧ 𝑋 ∈ 𝐴 ∧ (𝐼‘𝑋) ∈ 𝐴)) → 𝑋(∥r‘𝑆)((𝐼‘𝑋)(.r‘𝑆)𝑋))
33		subrgrcl 13782	. . . . . 6 ⊢ (𝐴 ∈ (SubRing‘𝑅) → 𝑅 ∈ Ring)
34		simpr1 1005	. . . . . 6 ⊢ ((𝐴 ∈ (SubRing‘𝑅) ∧ (𝑋 ∈ 𝑈 ∧ 𝑋 ∈ 𝐴 ∧ (𝐼‘𝑋) ∈ 𝐴)) → 𝑋 ∈ 𝑈)
35		eqid 2196	. . . . . . 7 ⊢ (.r‘𝑅) = (.r‘𝑅)
36		eqid 2196	. . . . . . 7 ⊢ (1r‘𝑅) = (1r‘𝑅)
37	2, 19, 35, 36	unitlinv 13682	. . . . . 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 12822	. . . . . . . 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 5939	. . . . 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 13787	. . . . . 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 2237	. . . 4 ⊢ ((𝐴 ∈ (SubRing‘𝑅) ∧ (𝑋 ∈ 𝑈 ∧ 𝑋 ∈ 𝐴 ∧ (𝐼‘𝑋) ∈ 𝐴)) → ((𝐼‘𝑋)(.r‘𝑆)𝑋) = (1r‘𝑆))
46	32, 45	breqtrd 4059	. . 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 2196	. . . . . . 7 ⊢ (oppr‘𝑆) = (oppr‘𝑆)
49	48, 16	opprbasg 13631	. . . . . 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 2197	. . . . 5 ⊢ ((𝐴 ∈ (SubRing‘𝑅) ∧ (𝑋 ∈ 𝑈 ∧ 𝑋 ∈ 𝐴 ∧ (𝐼‘𝑋) ∈ 𝐴)) → (∥r‘(oppr‘𝑆)) = (∥r‘(oppr‘𝑆)))
52	48	opprring 13635	. . . . . 6 ⊢ (𝑆 ∈ Ring → (oppr‘𝑆) ∈ Ring)
53		ringsrg 13603	. . . . . 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 2197	. . . . 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 13652	. . . 4 ⊢ ((𝐴 ∈ (SubRing‘𝑅) ∧ (𝑋 ∈ 𝑈 ∧ 𝑋 ∈ 𝐴 ∧ (𝐼‘𝑋) ∈ 𝐴)) → 𝑋(∥r‘(oppr‘𝑆))((𝐼‘𝑋)(.r‘(oppr‘𝑆))𝑋))
57		eqid 2196	. . . . . . 7 ⊢ (.r‘𝑆) = (.r‘𝑆)
58		eqid 2196	. . . . . . 7 ⊢ (.r‘(oppr‘𝑆)) = (.r‘(oppr‘𝑆))
59	16, 57, 48, 58	opprmulg 13627	. . . . . 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 1249	. . . . 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 13683	. . . . . . 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 5939	. . . . . 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 2237	. . . . 5 ⊢ ((𝐴 ∈ (SubRing‘𝑅) ∧ (𝑋 ∈ 𝑈 ∧ 𝑋 ∈ 𝐴 ∧ (𝐼‘𝑋) ∈ 𝐴)) → (𝑋(.r‘𝑆)(𝐼‘𝑋)) = (1r‘𝑆))
65	60, 64	eqtrd 2229	. . . 4 ⊢ ((𝐴 ∈ (SubRing‘𝑅) ∧ (𝑋 ∈ 𝑈 ∧ 𝑋 ∈ 𝐴 ∧ (𝐼‘𝑋) ∈ 𝐴)) → ((𝐼‘𝑋)(.r‘(oppr‘𝑆))𝑋) = (1r‘𝑆))
66	56, 65	breqtrd 4059	. . 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 2197	. . . . 5 ⊢ (𝐴 ∈ (SubRing‘𝑅) → (1r‘𝑆) = (1r‘𝑆))
69		eqidd 2197	. . . . 5 ⊢ (𝐴 ∈ (SubRing‘𝑅) → (∥r‘𝑆) = (∥r‘𝑆))
70		eqidd 2197	. . . . 5 ⊢ (𝐴 ∈ (SubRing‘𝑅) → (oppr‘𝑆) = (oppr‘𝑆))
71		eqidd 2197	. . . . 5 ⊢ (𝐴 ∈ (SubRing‘𝑅) → (∥r‘(oppr‘𝑆)) = (∥r‘(oppr‘𝑆)))
72	67, 68, 69, 70, 71, 11	isunitd 13662	. . . 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 946	. 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 980   = wceq 1364   ∈ wcel 2167   class class class wbr 4033  ‘cfv 5258  (class class class)co 5922  Basecbs 12678   ↾_s cress 12679  ._rcmulr 12756  1_rcur 13515  SRingcsrg 13519  Ringcrg 13552  opp_rcoppr 13623  ∥_rcdsr 13642  Unitcui 13643  inv_rcinvr 13676  SubRingcsubrg 13773

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 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-13 2169  ax-14 2170  ax-ext 2178  ax-coll 4148  ax-sep 4151  ax-nul 4159  ax-pow 4207  ax-pr 4242  ax-un 4468  ax-setind 4573  ax-cnex 7970  ax-resscn 7971  ax-1cn 7972  ax-1re 7973  ax-icn 7974  ax-addcl 7975  ax-addrcl 7976  ax-mulcl 7977  ax-addcom 7979  ax-addass 7981  ax-i2m1 7984  ax-0lt1 7985  ax-0id 7987  ax-rnegex 7988  ax-pre-ltirr 7991  ax-pre-lttrn 7993  ax-pre-ltadd 7995

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ne 2368  df-nel 2463  df-ral 2480  df-rex 2481  df-reu 2482  df-rmo 2483  df-rab 2484  df-v 2765  df-sbc 2990  df-csb 3085  df-dif 3159  df-un 3161  df-in 3163  df-ss 3170  df-nul 3451  df-pw 3607  df-sn 3628  df-pr 3629  df-op 3631  df-uni 3840  df-int 3875  df-iun 3918  df-br 4034  df-opab 4095  df-mpt 4096  df-id 4328  df-xp 4669  df-rel 4670  df-cnv 4671  df-co 4672  df-dm 4673  df-rn 4674  df-res 4675  df-ima 4676  df-iota 5219  df-fun 5260  df-fn 5261  df-f 5262  df-f1 5263  df-fo 5264  df-f1o 5265  df-fv 5266  df-riota 5877  df-ov 5925  df-oprab 5926  df-mpo 5927  df-tpos 6303  df-pnf 8063  df-mnf 8064  df-ltxr 8066  df-inn 8991  df-2 9049  df-3 9050  df-ndx 12681  df-slot 12682  df-base 12684  df-sets 12685  df-iress 12686  df-plusg 12768  df-mulr 12769  df-0g 12929  df-mgm 12999  df-sgrp 13045  df-mnd 13058  df-grp 13135  df-minusg 13136  df-subg 13300  df-cmn 13416  df-abl 13417  df-mgp 13477  df-ur 13516  df-srg 13520  df-ring 13554  df-oppr 13624  df-dvdsr 13645  df-unit 13646  df-invr 13677  df-subrg 13775

This theorem is referenced by: (None)