Theorem subrguss 14253

Description: A unit of a subring is a unit of the parent ring. (Contributed by Mario Carneiro, 4-Dec-2014.)

Hypotheses

Ref	Expression
subrguss.1	⊢ 𝑆 = (𝑅 ↾s 𝐴)
subrguss.2	⊢ 𝑈 = (Unit‘𝑅)
subrguss.3	⊢ 𝑉 = (Unit‘𝑆)

Assertion

Ref	Expression
subrguss	⊢ (𝐴 ∈ (SubRing‘𝑅) → 𝑉 ⊆ 𝑈)

Proof of Theorem subrguss

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		subrguss.3	. . . . . . . . 9 ⊢ 𝑉 = (Unit‘𝑆)
2	1	a1i 9	. . . . . . . 8 ⊢ (𝐴 ∈ (SubRing‘𝑅) → 𝑉 = (Unit‘𝑆))
3		eqidd 2232	. . . . . . . 8 ⊢ (𝐴 ∈ (SubRing‘𝑅) → (1r‘𝑆) = (1r‘𝑆))
4		eqidd 2232	. . . . . . . 8 ⊢ (𝐴 ∈ (SubRing‘𝑅) → (∥r‘𝑆) = (∥r‘𝑆))
5		eqidd 2232	. . . . . . . 8 ⊢ (𝐴 ∈ (SubRing‘𝑅) → (oppr‘𝑆) = (oppr‘𝑆))
6		eqidd 2232	. . . . . . . 8 ⊢ (𝐴 ∈ (SubRing‘𝑅) → (∥r‘(oppr‘𝑆)) = (∥r‘(oppr‘𝑆)))
7		subrguss.1	. . . . . . . . . 10 ⊢ 𝑆 = (𝑅 ↾s 𝐴)
8	7	subrgring 14241	. . . . . . . . 9 ⊢ (𝐴 ∈ (SubRing‘𝑅) → 𝑆 ∈ Ring)
9		ringsrg 14063	. . . . . . . . 9 ⊢ (𝑆 ∈ Ring → 𝑆 ∈ SRing)
10	8, 9	syl 14	. . . . . . . 8 ⊢ (𝐴 ∈ (SubRing‘𝑅) → 𝑆 ∈ SRing)
11	2, 3, 4, 5, 6, 10	isunitd 14123	. . . . . . 7 ⊢ (𝐴 ∈ (SubRing‘𝑅) → (𝑥 ∈ 𝑉 ↔ (𝑥(∥r‘𝑆)(1r‘𝑆) ∧ 𝑥(∥r‘(oppr‘𝑆))(1r‘𝑆))))
12	11	simprbda 383	. . . . . 6 ⊢ ((𝐴 ∈ (SubRing‘𝑅) ∧ 𝑥 ∈ 𝑉) → 𝑥(∥r‘𝑆)(1r‘𝑆))
13		eqid 2231	. . . . . . . 8 ⊢ (1r‘𝑅) = (1r‘𝑅)
14	7, 13	subrg1 14248	. . . . . . 7 ⊢ (𝐴 ∈ (SubRing‘𝑅) → (1r‘𝑅) = (1r‘𝑆))
15	14	adantr 276	. . . . . 6 ⊢ ((𝐴 ∈ (SubRing‘𝑅) ∧ 𝑥 ∈ 𝑉) → (1r‘𝑅) = (1r‘𝑆))
16	12, 15	breqtrrd 4116	. . . . 5 ⊢ ((𝐴 ∈ (SubRing‘𝑅) ∧ 𝑥 ∈ 𝑉) → 𝑥(∥r‘𝑆)(1r‘𝑅))
17		eqid 2231	. . . . . . . 8 ⊢ (∥r‘𝑅) = (∥r‘𝑅)
18		eqid 2231	. . . . . . . 8 ⊢ (∥r‘𝑆) = (∥r‘𝑆)
19	7, 17, 18	subrgdvds 14252	. . . . . . 7 ⊢ (𝐴 ∈ (SubRing‘𝑅) → (∥r‘𝑆) ⊆ (∥r‘𝑅))
20	19	adantr 276	. . . . . 6 ⊢ ((𝐴 ∈ (SubRing‘𝑅) ∧ 𝑥 ∈ 𝑉) → (∥r‘𝑆) ⊆ (∥r‘𝑅))
21	20	ssbrd 4131	. . . . 5 ⊢ ((𝐴 ∈ (SubRing‘𝑅) ∧ 𝑥 ∈ 𝑉) → (𝑥(∥r‘𝑆)(1r‘𝑅) → 𝑥(∥r‘𝑅)(1r‘𝑅)))
22	16, 21	mpd 13	. . . 4 ⊢ ((𝐴 ∈ (SubRing‘𝑅) ∧ 𝑥 ∈ 𝑉) → 𝑥(∥r‘𝑅)(1r‘𝑅))
23		subrgrcl 14243	. . . . . . . 8 ⊢ (𝐴 ∈ (SubRing‘𝑅) → 𝑅 ∈ Ring)
24	23	adantr 276	. . . . . . 7 ⊢ ((𝐴 ∈ (SubRing‘𝑅) ∧ 𝑥 ∈ 𝑉) → 𝑅 ∈ Ring)
25		eqid 2231	. . . . . . . 8 ⊢ (oppr‘𝑅) = (oppr‘𝑅)
26		eqid 2231	. . . . . . . 8 ⊢ (Base‘𝑅) = (Base‘𝑅)
27	25, 26	opprbasg 14091	. . . . . . 7 ⊢ (𝑅 ∈ Ring → (Base‘𝑅) = (Base‘(oppr‘𝑅)))
28	24, 27	syl 14	. . . . . 6 ⊢ ((𝐴 ∈ (SubRing‘𝑅) ∧ 𝑥 ∈ 𝑉) → (Base‘𝑅) = (Base‘(oppr‘𝑅)))
29		eqidd 2232	. . . . . 6 ⊢ ((𝐴 ∈ (SubRing‘𝑅) ∧ 𝑥 ∈ 𝑉) → (∥r‘(oppr‘𝑅)) = (∥r‘(oppr‘𝑅)))
30	25	opprring 14095	. . . . . . 7 ⊢ (𝑅 ∈ Ring → (oppr‘𝑅) ∈ Ring)
31		ringsrg 14063	. . . . . . 7 ⊢ ((oppr‘𝑅) ∈ Ring → (oppr‘𝑅) ∈ SRing)
32	24, 30, 31	3syl 17	. . . . . 6 ⊢ ((𝐴 ∈ (SubRing‘𝑅) ∧ 𝑥 ∈ 𝑉) → (oppr‘𝑅) ∈ SRing)
33		eqidd 2232	. . . . . 6 ⊢ ((𝐴 ∈ (SubRing‘𝑅) ∧ 𝑥 ∈ 𝑉) → (.r‘(oppr‘𝑅)) = (.r‘(oppr‘𝑅)))
34	7	subrgbas 14247	. . . . . . . . 9 ⊢ (𝐴 ∈ (SubRing‘𝑅) → 𝐴 = (Base‘𝑆))
35	34	adantr 276	. . . . . . . 8 ⊢ ((𝐴 ∈ (SubRing‘𝑅) ∧ 𝑥 ∈ 𝑉) → 𝐴 = (Base‘𝑆))
36	26	subrgss 14239	. . . . . . . . 9 ⊢ (𝐴 ∈ (SubRing‘𝑅) → 𝐴 ⊆ (Base‘𝑅))
37	36	adantr 276	. . . . . . . 8 ⊢ ((𝐴 ∈ (SubRing‘𝑅) ∧ 𝑥 ∈ 𝑉) → 𝐴 ⊆ (Base‘𝑅))
38	35, 37	eqsstrrd 3264	. . . . . . 7 ⊢ ((𝐴 ∈ (SubRing‘𝑅) ∧ 𝑥 ∈ 𝑉) → (Base‘𝑆) ⊆ (Base‘𝑅))
39		eqidd 2232	. . . . . . . 8 ⊢ ((𝐴 ∈ (SubRing‘𝑅) ∧ 𝑥 ∈ 𝑉) → (Base‘𝑆) = (Base‘𝑆))
40	1	a1i 9	. . . . . . . 8 ⊢ ((𝐴 ∈ (SubRing‘𝑅) ∧ 𝑥 ∈ 𝑉) → 𝑉 = (Unit‘𝑆))
41	10	adantr 276	. . . . . . . 8 ⊢ ((𝐴 ∈ (SubRing‘𝑅) ∧ 𝑥 ∈ 𝑉) → 𝑆 ∈ SRing)
42		simpr 110	. . . . . . . 8 ⊢ ((𝐴 ∈ (SubRing‘𝑅) ∧ 𝑥 ∈ 𝑉) → 𝑥 ∈ 𝑉)
43	39, 40, 41, 42	unitcld 14125	. . . . . . 7 ⊢ ((𝐴 ∈ (SubRing‘𝑅) ∧ 𝑥 ∈ 𝑉) → 𝑥 ∈ (Base‘𝑆))
44	38, 43	sseldd 3228	. . . . . 6 ⊢ ((𝐴 ∈ (SubRing‘𝑅) ∧ 𝑥 ∈ 𝑉) → 𝑥 ∈ (Base‘𝑅))
45		eqid 2231	. . . . . . . . 9 ⊢ (invr‘𝑆) = (invr‘𝑆)
46		eqid 2231	. . . . . . . . 9 ⊢ (Base‘𝑆) = (Base‘𝑆)
47	1, 45, 46	ringinvcl 14142	. . . . . . . 8 ⊢ ((𝑆 ∈ Ring ∧ 𝑥 ∈ 𝑉) → ((invr‘𝑆)‘𝑥) ∈ (Base‘𝑆))
48	8, 47	sylan 283	. . . . . . 7 ⊢ ((𝐴 ∈ (SubRing‘𝑅) ∧ 𝑥 ∈ 𝑉) → ((invr‘𝑆)‘𝑥) ∈ (Base‘𝑆))
49	38, 48	sseldd 3228	. . . . . 6 ⊢ ((𝐴 ∈ (SubRing‘𝑅) ∧ 𝑥 ∈ 𝑉) → ((invr‘𝑆)‘𝑥) ∈ (Base‘𝑅))
50	28, 29, 32, 33, 44, 49	dvdsrmuld 14113	. . . . 5 ⊢ ((𝐴 ∈ (SubRing‘𝑅) ∧ 𝑥 ∈ 𝑉) → 𝑥(∥r‘(oppr‘𝑅))(((invr‘𝑆)‘𝑥)(.r‘(oppr‘𝑅))𝑥))
51	1, 45	unitinvcl 14140	. . . . . . . 8 ⊢ ((𝑆 ∈ Ring ∧ 𝑥 ∈ 𝑉) → ((invr‘𝑆)‘𝑥) ∈ 𝑉)
52	8, 51	sylan 283	. . . . . . 7 ⊢ ((𝐴 ∈ (SubRing‘𝑅) ∧ 𝑥 ∈ 𝑉) → ((invr‘𝑆)‘𝑥) ∈ 𝑉)
53		eqid 2231	. . . . . . . 8 ⊢ (.r‘𝑅) = (.r‘𝑅)
54		eqid 2231	. . . . . . . 8 ⊢ (.r‘(oppr‘𝑅)) = (.r‘(oppr‘𝑅))
55	26, 53, 25, 54	opprmulg 14087	. . . . . . 7 ⊢ ((𝑅 ∈ Ring ∧ ((invr‘𝑆)‘𝑥) ∈ 𝑉 ∧ 𝑥 ∈ 𝑉) → (((invr‘𝑆)‘𝑥)(.r‘(oppr‘𝑅))𝑥) = (𝑥(.r‘𝑅)((invr‘𝑆)‘𝑥)))
56	24, 52, 42, 55	syl3anc 1273	. . . . . 6 ⊢ ((𝐴 ∈ (SubRing‘𝑅) ∧ 𝑥 ∈ 𝑉) → (((invr‘𝑆)‘𝑥)(.r‘(oppr‘𝑅))𝑥) = (𝑥(.r‘𝑅)((invr‘𝑆)‘𝑥)))
57		eqid 2231	. . . . . . . . 9 ⊢ (.r‘𝑆) = (.r‘𝑆)
58		eqid 2231	. . . . . . . . 9 ⊢ (1r‘𝑆) = (1r‘𝑆)
59	1, 45, 57, 58	unitrinv 14144	. . . . . . . 8 ⊢ ((𝑆 ∈ Ring ∧ 𝑥 ∈ 𝑉) → (𝑥(.r‘𝑆)((invr‘𝑆)‘𝑥)) = (1r‘𝑆))
60	8, 59	sylan 283	. . . . . . 7 ⊢ ((𝐴 ∈ (SubRing‘𝑅) ∧ 𝑥 ∈ 𝑉) → (𝑥(.r‘𝑆)((invr‘𝑆)‘𝑥)) = (1r‘𝑆))
61	7, 53	ressmulrg 13230	. . . . . . . . . 10 ⊢ ((𝐴 ∈ (SubRing‘𝑅) ∧ 𝑅 ∈ Ring) → (.r‘𝑅) = (.r‘𝑆))
62	23, 61	mpdan 421	. . . . . . . . 9 ⊢ (𝐴 ∈ (SubRing‘𝑅) → (.r‘𝑅) = (.r‘𝑆))
63	62	adantr 276	. . . . . . . 8 ⊢ ((𝐴 ∈ (SubRing‘𝑅) ∧ 𝑥 ∈ 𝑉) → (.r‘𝑅) = (.r‘𝑆))
64	63	oveqd 6035	. . . . . . 7 ⊢ ((𝐴 ∈ (SubRing‘𝑅) ∧ 𝑥 ∈ 𝑉) → (𝑥(.r‘𝑅)((invr‘𝑆)‘𝑥)) = (𝑥(.r‘𝑆)((invr‘𝑆)‘𝑥)))
65	60, 64, 15	3eqtr4d 2274	. . . . . 6 ⊢ ((𝐴 ∈ (SubRing‘𝑅) ∧ 𝑥 ∈ 𝑉) → (𝑥(.r‘𝑅)((invr‘𝑆)‘𝑥)) = (1r‘𝑅))
66	56, 65	eqtrd 2264	. . . . 5 ⊢ ((𝐴 ∈ (SubRing‘𝑅) ∧ 𝑥 ∈ 𝑉) → (((invr‘𝑆)‘𝑥)(.r‘(oppr‘𝑅))𝑥) = (1r‘𝑅))
67	50, 66	breqtrd 4114	. . . 4 ⊢ ((𝐴 ∈ (SubRing‘𝑅) ∧ 𝑥 ∈ 𝑉) → 𝑥(∥r‘(oppr‘𝑅))(1r‘𝑅))
68		subrguss.2	. . . . . . 7 ⊢ 𝑈 = (Unit‘𝑅)
69	68	a1i 9	. . . . . 6 ⊢ (𝐴 ∈ (SubRing‘𝑅) → 𝑈 = (Unit‘𝑅))
70		eqidd 2232	. . . . . 6 ⊢ (𝐴 ∈ (SubRing‘𝑅) → (1r‘𝑅) = (1r‘𝑅))
71		eqidd 2232	. . . . . 6 ⊢ (𝐴 ∈ (SubRing‘𝑅) → (∥r‘𝑅) = (∥r‘𝑅))
72		eqidd 2232	. . . . . 6 ⊢ (𝐴 ∈ (SubRing‘𝑅) → (oppr‘𝑅) = (oppr‘𝑅))
73		eqidd 2232	. . . . . 6 ⊢ (𝐴 ∈ (SubRing‘𝑅) → (∥r‘(oppr‘𝑅)) = (∥r‘(oppr‘𝑅)))
74		ringsrg 14063	. . . . . . 7 ⊢ (𝑅 ∈ Ring → 𝑅 ∈ SRing)
75	23, 74	syl 14	. . . . . 6 ⊢ (𝐴 ∈ (SubRing‘𝑅) → 𝑅 ∈ SRing)
76	69, 70, 71, 72, 73, 75	isunitd 14123	. . . . 5 ⊢ (𝐴 ∈ (SubRing‘𝑅) → (𝑥 ∈ 𝑈 ↔ (𝑥(∥r‘𝑅)(1r‘𝑅) ∧ 𝑥(∥r‘(oppr‘𝑅))(1r‘𝑅))))
77	76	adantr 276	. . . 4 ⊢ ((𝐴 ∈ (SubRing‘𝑅) ∧ 𝑥 ∈ 𝑉) → (𝑥 ∈ 𝑈 ↔ (𝑥(∥r‘𝑅)(1r‘𝑅) ∧ 𝑥(∥r‘(oppr‘𝑅))(1r‘𝑅))))
78	22, 67, 77	mpbir2and 952	. . 3 ⊢ ((𝐴 ∈ (SubRing‘𝑅) ∧ 𝑥 ∈ 𝑉) → 𝑥 ∈ 𝑈)
79	78	ex 115	. 2 ⊢ (𝐴 ∈ (SubRing‘𝑅) → (𝑥 ∈ 𝑉 → 𝑥 ∈ 𝑈))
80	79	ssrdv 3233	1 ⊢ (𝐴 ∈ (SubRing‘𝑅) → 𝑉 ⊆ 𝑈)

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105   = wceq 1397   ∈ wcel 2202   ⊆ wss 3200   class class class wbr 4088  ‘cfv 5326  (class class class)co 6018  Basecbs 13084   ↾_s cress 13085  ._rcmulr 13163  1_rcur 13975  SRingcsrg 13979  Ringcrg 14012  opp_rcoppr 14083  ∥_rcdsr 14102  Unitcui 14103  inv_rcinvr 14137  SubRingcsubrg 14234

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 619  ax-in2 620  ax-io 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-13 2204  ax-14 2205  ax-ext 2213  ax-coll 4204  ax-sep 4207  ax-nul 4215  ax-pow 4264  ax-pr 4299  ax-un 4530  ax-setind 4635  ax-cnex 8123  ax-resscn 8124  ax-1cn 8125  ax-1re 8126  ax-icn 8127  ax-addcl 8128  ax-addrcl 8129  ax-mulcl 8130  ax-addcom 8132  ax-addass 8134  ax-i2m1 8137  ax-0lt1 8138  ax-0id 8140  ax-rnegex 8141  ax-pre-ltirr 8144  ax-pre-lttrn 8146  ax-pre-ltadd 8148

This theorem depends on definitions:  df-bi 117  df-3an 1006  df-tru 1400  df-fal 1403  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ne 2403  df-nel 2498  df-ral 2515  df-rex 2516  df-reu 2517  df-rmo 2518  df-rab 2519  df-v 2804  df-sbc 3032  df-csb 3128  df-dif 3202  df-un 3204  df-in 3206  df-ss 3213  df-nul 3495  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-uni 3894  df-int 3929  df-iun 3972  df-br 4089  df-opab 4151  df-mpt 4152  df-id 4390  df-xp 4731  df-rel 4732  df-cnv 4733  df-co 4734  df-dm 4735  df-rn 4736  df-res 4737  df-ima 4738  df-iota 5286  df-fun 5328  df-fn 5329  df-f 5330  df-f1 5331  df-fo 5332  df-f1o 5333  df-fv 5334  df-riota 5971  df-ov 6021  df-oprab 6022  df-mpo 6023  df-tpos 6411  df-pnf 8216  df-mnf 8217  df-ltxr 8219  df-inn 9144  df-2 9202  df-3 9203  df-ndx 13087  df-slot 13088  df-base 13090  df-sets 13091  df-iress 13092  df-plusg 13175  df-mulr 13176  df-0g 13343  df-mgm 13441  df-sgrp 13487  df-mnd 13502  df-grp 13588  df-minusg 13589  df-subg 13759  df-cmn 13875  df-abl 13876  df-mgp 13937  df-ur 13976  df-srg 13980  df-ring 14014  df-oppr 14084  df-dvdsr 14105  df-unit 14106  df-invr 14138  df-subrg 14236

This theorem is referenced by:  subrginv  14254  subrgdv  14255  subrgunit  14256  subrgugrp  14257