Theorem subrguss 20587

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		simpr 484	. . . . . . . 8 ⊢ ((𝐴 ∈ (SubRing‘𝑅) ∧ 𝑥 ∈ 𝑉) → 𝑥 ∈ 𝑉)
2		subrguss.3	. . . . . . . . 9 ⊢ 𝑉 = (Unit‘𝑆)
3		eqid 2737	. . . . . . . . 9 ⊢ (1r‘𝑆) = (1r‘𝑆)
4		eqid 2737	. . . . . . . . 9 ⊢ (∥r‘𝑆) = (∥r‘𝑆)
5		eqid 2737	. . . . . . . . 9 ⊢ (oppr‘𝑆) = (oppr‘𝑆)
6		eqid 2737	. . . . . . . . 9 ⊢ (∥r‘(oppr‘𝑆)) = (∥r‘(oppr‘𝑆))
7	2, 3, 4, 5, 6	isunit 20373	. . . . . . . 8 ⊢ (𝑥 ∈ 𝑉 ↔ (𝑥(∥r‘𝑆)(1r‘𝑆) ∧ 𝑥(∥r‘(oppr‘𝑆))(1r‘𝑆)))
8	1, 7	sylib 218	. . . . . . 7 ⊢ ((𝐴 ∈ (SubRing‘𝑅) ∧ 𝑥 ∈ 𝑉) → (𝑥(∥r‘𝑆)(1r‘𝑆) ∧ 𝑥(∥r‘(oppr‘𝑆))(1r‘𝑆)))
9	8	simpld 494	. . . . . 6 ⊢ ((𝐴 ∈ (SubRing‘𝑅) ∧ 𝑥 ∈ 𝑉) → 𝑥(∥r‘𝑆)(1r‘𝑆))
10		subrguss.1	. . . . . . . 8 ⊢ 𝑆 = (𝑅 ↾s 𝐴)
11		eqid 2737	. . . . . . . 8 ⊢ (1r‘𝑅) = (1r‘𝑅)
12	10, 11	subrg1 20582	. . . . . . 7 ⊢ (𝐴 ∈ (SubRing‘𝑅) → (1r‘𝑅) = (1r‘𝑆))
13	12	adantr 480	. . . . . 6 ⊢ ((𝐴 ∈ (SubRing‘𝑅) ∧ 𝑥 ∈ 𝑉) → (1r‘𝑅) = (1r‘𝑆))
14	9, 13	breqtrrd 5171	. . . . 5 ⊢ ((𝐴 ∈ (SubRing‘𝑅) ∧ 𝑥 ∈ 𝑉) → 𝑥(∥r‘𝑆)(1r‘𝑅))
15		eqid 2737	. . . . . . . 8 ⊢ (∥r‘𝑅) = (∥r‘𝑅)
16	10, 15, 4	subrgdvds 20586	. . . . . . 7 ⊢ (𝐴 ∈ (SubRing‘𝑅) → (∥r‘𝑆) ⊆ (∥r‘𝑅))
17	16	adantr 480	. . . . . 6 ⊢ ((𝐴 ∈ (SubRing‘𝑅) ∧ 𝑥 ∈ 𝑉) → (∥r‘𝑆) ⊆ (∥r‘𝑅))
18	17	ssbrd 5186	. . . . 5 ⊢ ((𝐴 ∈ (SubRing‘𝑅) ∧ 𝑥 ∈ 𝑉) → (𝑥(∥r‘𝑆)(1r‘𝑅) → 𝑥(∥r‘𝑅)(1r‘𝑅)))
19	14, 18	mpd 15	. . . 4 ⊢ ((𝐴 ∈ (SubRing‘𝑅) ∧ 𝑥 ∈ 𝑉) → 𝑥(∥r‘𝑅)(1r‘𝑅))
20	10	subrgbas 20581	. . . . . . . . 9 ⊢ (𝐴 ∈ (SubRing‘𝑅) → 𝐴 = (Base‘𝑆))
21	20	adantr 480	. . . . . . . 8 ⊢ ((𝐴 ∈ (SubRing‘𝑅) ∧ 𝑥 ∈ 𝑉) → 𝐴 = (Base‘𝑆))
22		eqid 2737	. . . . . . . . . 10 ⊢ (Base‘𝑅) = (Base‘𝑅)
23	22	subrgss 20572	. . . . . . . . 9 ⊢ (𝐴 ∈ (SubRing‘𝑅) → 𝐴 ⊆ (Base‘𝑅))
24	23	adantr 480	. . . . . . . 8 ⊢ ((𝐴 ∈ (SubRing‘𝑅) ∧ 𝑥 ∈ 𝑉) → 𝐴 ⊆ (Base‘𝑅))
25	21, 24	eqsstrrd 4019	. . . . . . 7 ⊢ ((𝐴 ∈ (SubRing‘𝑅) ∧ 𝑥 ∈ 𝑉) → (Base‘𝑆) ⊆ (Base‘𝑅))
26		eqid 2737	. . . . . . . . 9 ⊢ (Base‘𝑆) = (Base‘𝑆)
27	26, 2	unitcl 20375	. . . . . . . 8 ⊢ (𝑥 ∈ 𝑉 → 𝑥 ∈ (Base‘𝑆))
28	27	adantl 481	. . . . . . 7 ⊢ ((𝐴 ∈ (SubRing‘𝑅) ∧ 𝑥 ∈ 𝑉) → 𝑥 ∈ (Base‘𝑆))
29	25, 28	sseldd 3984	. . . . . 6 ⊢ ((𝐴 ∈ (SubRing‘𝑅) ∧ 𝑥 ∈ 𝑉) → 𝑥 ∈ (Base‘𝑅))
30	10	subrgring 20574	. . . . . . . 8 ⊢ (𝐴 ∈ (SubRing‘𝑅) → 𝑆 ∈ Ring)
31		eqid 2737	. . . . . . . . 9 ⊢ (invr‘𝑆) = (invr‘𝑆)
32	2, 31, 26	ringinvcl 20392	. . . . . . . 8 ⊢ ((𝑆 ∈ Ring ∧ 𝑥 ∈ 𝑉) → ((invr‘𝑆)‘𝑥) ∈ (Base‘𝑆))
33	30, 32	sylan 580	. . . . . . 7 ⊢ ((𝐴 ∈ (SubRing‘𝑅) ∧ 𝑥 ∈ 𝑉) → ((invr‘𝑆)‘𝑥) ∈ (Base‘𝑆))
34	25, 33	sseldd 3984	. . . . . 6 ⊢ ((𝐴 ∈ (SubRing‘𝑅) ∧ 𝑥 ∈ 𝑉) → ((invr‘𝑆)‘𝑥) ∈ (Base‘𝑅))
35		eqid 2737	. . . . . . . 8 ⊢ (oppr‘𝑅) = (oppr‘𝑅)
36	35, 22	opprbas 20341	. . . . . . 7 ⊢ (Base‘𝑅) = (Base‘(oppr‘𝑅))
37		eqid 2737	. . . . . . 7 ⊢ (∥r‘(oppr‘𝑅)) = (∥r‘(oppr‘𝑅))
38		eqid 2737	. . . . . . 7 ⊢ (.r‘(oppr‘𝑅)) = (.r‘(oppr‘𝑅))
39	36, 37, 38	dvdsrmul 20364	. . . . . 6 ⊢ ((𝑥 ∈ (Base‘𝑅) ∧ ((invr‘𝑆)‘𝑥) ∈ (Base‘𝑅)) → 𝑥(∥r‘(oppr‘𝑅))(((invr‘𝑆)‘𝑥)(.r‘(oppr‘𝑅))𝑥))
40	29, 34, 39	syl2anc 584	. . . . 5 ⊢ ((𝐴 ∈ (SubRing‘𝑅) ∧ 𝑥 ∈ 𝑉) → 𝑥(∥r‘(oppr‘𝑅))(((invr‘𝑆)‘𝑥)(.r‘(oppr‘𝑅))𝑥))
41		eqid 2737	. . . . . . 7 ⊢ (.r‘𝑅) = (.r‘𝑅)
42	22, 41, 35, 38	opprmul 20337	. . . . . 6 ⊢ (((invr‘𝑆)‘𝑥)(.r‘(oppr‘𝑅))𝑥) = (𝑥(.r‘𝑅)((invr‘𝑆)‘𝑥))
43		eqid 2737	. . . . . . . . 9 ⊢ (.r‘𝑆) = (.r‘𝑆)
44	2, 31, 43, 3	unitrinv 20394	. . . . . . . 8 ⊢ ((𝑆 ∈ Ring ∧ 𝑥 ∈ 𝑉) → (𝑥(.r‘𝑆)((invr‘𝑆)‘𝑥)) = (1r‘𝑆))
45	30, 44	sylan 580	. . . . . . 7 ⊢ ((𝐴 ∈ (SubRing‘𝑅) ∧ 𝑥 ∈ 𝑉) → (𝑥(.r‘𝑆)((invr‘𝑆)‘𝑥)) = (1r‘𝑆))
46	10, 41	ressmulr 17351	. . . . . . . . 9 ⊢ (𝐴 ∈ (SubRing‘𝑅) → (.r‘𝑅) = (.r‘𝑆))
47	46	adantr 480	. . . . . . . 8 ⊢ ((𝐴 ∈ (SubRing‘𝑅) ∧ 𝑥 ∈ 𝑉) → (.r‘𝑅) = (.r‘𝑆))
48	47	oveqd 7448	. . . . . . 7 ⊢ ((𝐴 ∈ (SubRing‘𝑅) ∧ 𝑥 ∈ 𝑉) → (𝑥(.r‘𝑅)((invr‘𝑆)‘𝑥)) = (𝑥(.r‘𝑆)((invr‘𝑆)‘𝑥)))
49	45, 48, 13	3eqtr4d 2787	. . . . . 6 ⊢ ((𝐴 ∈ (SubRing‘𝑅) ∧ 𝑥 ∈ 𝑉) → (𝑥(.r‘𝑅)((invr‘𝑆)‘𝑥)) = (1r‘𝑅))
50	42, 49	eqtrid 2789	. . . . 5 ⊢ ((𝐴 ∈ (SubRing‘𝑅) ∧ 𝑥 ∈ 𝑉) → (((invr‘𝑆)‘𝑥)(.r‘(oppr‘𝑅))𝑥) = (1r‘𝑅))
51	40, 50	breqtrd 5169	. . . 4 ⊢ ((𝐴 ∈ (SubRing‘𝑅) ∧ 𝑥 ∈ 𝑉) → 𝑥(∥r‘(oppr‘𝑅))(1r‘𝑅))
52		subrguss.2	. . . . 5 ⊢ 𝑈 = (Unit‘𝑅)
53	52, 11, 15, 35, 37	isunit 20373	. . . 4 ⊢ (𝑥 ∈ 𝑈 ↔ (𝑥(∥r‘𝑅)(1r‘𝑅) ∧ 𝑥(∥r‘(oppr‘𝑅))(1r‘𝑅)))
54	19, 51, 53	sylanbrc 583	. . 3 ⊢ ((𝐴 ∈ (SubRing‘𝑅) ∧ 𝑥 ∈ 𝑉) → 𝑥 ∈ 𝑈)
55	54	ex 412	. 2 ⊢ (𝐴 ∈ (SubRing‘𝑅) → (𝑥 ∈ 𝑉 → 𝑥 ∈ 𝑈))
56	55	ssrdv 3989	1 ⊢ (𝐴 ∈ (SubRing‘𝑅) → 𝑉 ⊆ 𝑈)

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1540   ∈ wcel 2108   ⊆ wss 3951   class class class wbr 5143  ‘cfv 6561  (class class class)co 7431  Basecbs 17247   ↾_s cress 17274  ._rcmulr 17298  1_rcur 20178  Ringcrg 20230  opp_rcoppr 20333  ∥_rcdsr 20354  Unitcui 20355  inv_rcinvr 20387  SubRingcsubrg 20569

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 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755  ax-cnex 11211  ax-resscn 11212  ax-1cn 11213  ax-icn 11214  ax-addcl 11215  ax-addrcl 11216  ax-mulcl 11217  ax-mulrcl 11218  ax-mulcom 11219  ax-addass 11220  ax-mulass 11221  ax-distr 11222  ax-i2m1 11223  ax-1ne0 11224  ax-1rid 11225  ax-rnegex 11226  ax-rrecex 11227  ax-cnre 11228  ax-pre-lttri 11229  ax-pre-lttrn 11230  ax-pre-ltadd 11231  ax-pre-mulgt0 11232

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-rmo 3380  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-tr 5260  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-we 5639  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-pred 6321  df-ord 6387  df-on 6388  df-lim 6389  df-suc 6390  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-2nd 8015  df-tpos 8251  df-frecs 8306  df-wrecs 8337  df-recs 8411  df-rdg 8450  df-er 8745  df-en 8986  df-dom 8987  df-sdom 8988  df-pnf 11297  df-mnf 11298  df-xr 11299  df-ltxr 11300  df-le 11301  df-sub 11494  df-neg 11495  df-nn 12267  df-2 12329  df-3 12330  df-sets 17201  df-slot 17219  df-ndx 17231  df-base 17248  df-ress 17275  df-plusg 17310  df-mulr 17311  df-0g 17486  df-mgm 18653  df-sgrp 18732  df-mnd 18748  df-grp 18954  df-minusg 18955  df-subg 19141  df-cmn 19800  df-abl 19801  df-mgp 20138  df-rng 20150  df-ur 20179  df-ring 20232  df-oppr 20334  df-dvdsr 20357  df-unit 20358  df-invr 20388  df-subrg 20570

This theorem is referenced by:  subrginv  20588  subrgdv  20589  subrgunit  20590  subrgugrp  20591  issubdrg  20781  zringunit  21477