Theorem subrguss 14370

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 2233	. . . . . . . 8 ⊢ (𝐴 ∈ (SubRing‘𝑅) → (1r‘𝑆) = (1r‘𝑆))
4		eqidd 2233	. . . . . . . 8 ⊢ (𝐴 ∈ (SubRing‘𝑅) → (∥r‘𝑆) = (∥r‘𝑆))
5		eqidd 2233	. . . . . . . 8 ⊢ (𝐴 ∈ (SubRing‘𝑅) → (oppr‘𝑆) = (oppr‘𝑆))
6		eqidd 2233	. . . . . . . 8 ⊢ (𝐴 ∈ (SubRing‘𝑅) → (∥r‘(oppr‘𝑆)) = (∥r‘(oppr‘𝑆)))
7		subrguss.1	. . . . . . . . . 10 ⊢ 𝑆 = (𝑅 ↾s 𝐴)
8	7	subrgring 14358	. . . . . . . . 9 ⊢ (𝐴 ∈ (SubRing‘𝑅) → 𝑆 ∈ Ring)
9		ringsrg 14180	. . . . . . . . 9 ⊢ (𝑆 ∈ Ring → 𝑆 ∈ SRing)
10	8, 9	syl 14	. . . . . . . 8 ⊢ (𝐴 ∈ (SubRing‘𝑅) → 𝑆 ∈ SRing)
11	2, 3, 4, 5, 6, 10	isunitd 14240	. . . . . . 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 2232	. . . . . . . 8 ⊢ (1r‘𝑅) = (1r‘𝑅)
14	7, 13	subrg1 14365	. . . . . . 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 4136	. . . . 5 ⊢ ((𝐴 ∈ (SubRing‘𝑅) ∧ 𝑥 ∈ 𝑉) → 𝑥(∥r‘𝑆)(1r‘𝑅))
17		eqid 2232	. . . . . . . 8 ⊢ (∥r‘𝑅) = (∥r‘𝑅)
18		eqid 2232	. . . . . . . 8 ⊢ (∥r‘𝑆) = (∥r‘𝑆)
19	7, 17, 18	subrgdvds 14369	. . . . . . 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 4151	. . . . 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 14360	. . . . . . . 8 ⊢ (𝐴 ∈ (SubRing‘𝑅) → 𝑅 ∈ Ring)
24	23	adantr 276	. . . . . . 7 ⊢ ((𝐴 ∈ (SubRing‘𝑅) ∧ 𝑥 ∈ 𝑉) → 𝑅 ∈ Ring)
25		eqid 2232	. . . . . . . 8 ⊢ (oppr‘𝑅) = (oppr‘𝑅)
26		eqid 2232	. . . . . . . 8 ⊢ (Base‘𝑅) = (Base‘𝑅)
27	25, 26	opprbasg 14208	. . . . . . 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 2233	. . . . . 6 ⊢ ((𝐴 ∈ (SubRing‘𝑅) ∧ 𝑥 ∈ 𝑉) → (∥r‘(oppr‘𝑅)) = (∥r‘(oppr‘𝑅)))
30	25	opprring 14212	. . . . . . 7 ⊢ (𝑅 ∈ Ring → (oppr‘𝑅) ∈ Ring)
31		ringsrg 14180	. . . . . . 7 ⊢ ((oppr‘𝑅) ∈ Ring → (oppr‘𝑅) ∈ SRing)
32	24, 30, 31	3syl 17	. . . . . 6 ⊢ ((𝐴 ∈ (SubRing‘𝑅) ∧ 𝑥 ∈ 𝑉) → (oppr‘𝑅) ∈ SRing)
33		eqidd 2233	. . . . . 6 ⊢ ((𝐴 ∈ (SubRing‘𝑅) ∧ 𝑥 ∈ 𝑉) → (.r‘(oppr‘𝑅)) = (.r‘(oppr‘𝑅)))
34	7	subrgbas 14364	. . . . . . . . 9 ⊢ (𝐴 ∈ (SubRing‘𝑅) → 𝐴 = (Base‘𝑆))
35	34	adantr 276	. . . . . . . 8 ⊢ ((𝐴 ∈ (SubRing‘𝑅) ∧ 𝑥 ∈ 𝑉) → 𝐴 = (Base‘𝑆))
36	26	subrgss 14356	. . . . . . . . 9 ⊢ (𝐴 ∈ (SubRing‘𝑅) → 𝐴 ⊆ (Base‘𝑅))
37	36	adantr 276	. . . . . . . 8 ⊢ ((𝐴 ∈ (SubRing‘𝑅) ∧ 𝑥 ∈ 𝑉) → 𝐴 ⊆ (Base‘𝑅))
38	35, 37	eqsstrrd 3274	. . . . . . 7 ⊢ ((𝐴 ∈ (SubRing‘𝑅) ∧ 𝑥 ∈ 𝑉) → (Base‘𝑆) ⊆ (Base‘𝑅))
39		eqidd 2233	. . . . . . . 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 14242	. . . . . . 7 ⊢ ((𝐴 ∈ (SubRing‘𝑅) ∧ 𝑥 ∈ 𝑉) → 𝑥 ∈ (Base‘𝑆))
44	38, 43	sseldd 3238	. . . . . 6 ⊢ ((𝐴 ∈ (SubRing‘𝑅) ∧ 𝑥 ∈ 𝑉) → 𝑥 ∈ (Base‘𝑅))
45		eqid 2232	. . . . . . . . 9 ⊢ (invr‘𝑆) = (invr‘𝑆)
46		eqid 2232	. . . . . . . . 9 ⊢ (Base‘𝑆) = (Base‘𝑆)
47	1, 45, 46	ringinvcl 14259	. . . . . . . 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 3238	. . . . . 6 ⊢ ((𝐴 ∈ (SubRing‘𝑅) ∧ 𝑥 ∈ 𝑉) → ((invr‘𝑆)‘𝑥) ∈ (Base‘𝑅))
50	28, 29, 32, 33, 44, 49	dvdsrmuld 14230	. . . . 5 ⊢ ((𝐴 ∈ (SubRing‘𝑅) ∧ 𝑥 ∈ 𝑉) → 𝑥(∥r‘(oppr‘𝑅))(((invr‘𝑆)‘𝑥)(.r‘(oppr‘𝑅))𝑥))
51	1, 45	unitinvcl 14257	. . . . . . . 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 2232	. . . . . . . 8 ⊢ (.r‘𝑅) = (.r‘𝑅)
54		eqid 2232	. . . . . . . 8 ⊢ (.r‘(oppr‘𝑅)) = (.r‘(oppr‘𝑅))
55	26, 53, 25, 54	opprmulg 14204	. . . . . . 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 1274	. . . . . 6 ⊢ ((𝐴 ∈ (SubRing‘𝑅) ∧ 𝑥 ∈ 𝑉) → (((invr‘𝑆)‘𝑥)(.r‘(oppr‘𝑅))𝑥) = (𝑥(.r‘𝑅)((invr‘𝑆)‘𝑥)))
57		eqid 2232	. . . . . . . . 9 ⊢ (.r‘𝑆) = (.r‘𝑆)
58		eqid 2232	. . . . . . . . 9 ⊢ (1r‘𝑆) = (1r‘𝑆)
59	1, 45, 57, 58	unitrinv 14261	. . . . . . . 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 13347	. . . . . . . . . 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 6066	. . . . . . 7 ⊢ ((𝐴 ∈ (SubRing‘𝑅) ∧ 𝑥 ∈ 𝑉) → (𝑥(.r‘𝑅)((invr‘𝑆)‘𝑥)) = (𝑥(.r‘𝑆)((invr‘𝑆)‘𝑥)))
65	60, 64, 15	3eqtr4d 2275	. . . . . 6 ⊢ ((𝐴 ∈ (SubRing‘𝑅) ∧ 𝑥 ∈ 𝑉) → (𝑥(.r‘𝑅)((invr‘𝑆)‘𝑥)) = (1r‘𝑅))
66	56, 65	eqtrd 2265	. . . . 5 ⊢ ((𝐴 ∈ (SubRing‘𝑅) ∧ 𝑥 ∈ 𝑉) → (((invr‘𝑆)‘𝑥)(.r‘(oppr‘𝑅))𝑥) = (1r‘𝑅))
67	50, 66	breqtrd 4134	. . . 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 2233	. . . . . 6 ⊢ (𝐴 ∈ (SubRing‘𝑅) → (1r‘𝑅) = (1r‘𝑅))
71		eqidd 2233	. . . . . 6 ⊢ (𝐴 ∈ (SubRing‘𝑅) → (∥r‘𝑅) = (∥r‘𝑅))
72		eqidd 2233	. . . . . 6 ⊢ (𝐴 ∈ (SubRing‘𝑅) → (oppr‘𝑅) = (oppr‘𝑅))
73		eqidd 2233	. . . . . 6 ⊢ (𝐴 ∈ (SubRing‘𝑅) → (∥r‘(oppr‘𝑅)) = (∥r‘(oppr‘𝑅)))
74		ringsrg 14180	. . . . . . 7 ⊢ (𝑅 ∈ Ring → 𝑅 ∈ SRing)
75	23, 74	syl 14	. . . . . 6 ⊢ (𝐴 ∈ (SubRing‘𝑅) → 𝑅 ∈ SRing)
76	69, 70, 71, 72, 73, 75	isunitd 14240	. . . . 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 953	. . 3 ⊢ ((𝐴 ∈ (SubRing‘𝑅) ∧ 𝑥 ∈ 𝑉) → 𝑥 ∈ 𝑈)
79	78	ex 115	. 2 ⊢ (𝐴 ∈ (SubRing‘𝑅) → (𝑥 ∈ 𝑉 → 𝑥 ∈ 𝑈))
80	79	ssrdv 3243	1 ⊢ (𝐴 ∈ (SubRing‘𝑅) → 𝑉 ⊆ 𝑈)

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105   = wceq 1398   ∈ wcel 2203   ⊆ wss 3210   class class class wbr 4108  ‘cfv 5351  (class class class)co 6049  Basecbs 13201   ↾_s cress 13202  ._rcmulr 13280  1_rcur 14092  SRingcsrg 14096  Ringcrg 14129  opp_rcoppr 14200  ∥_rcdsr 14219  Unitcui 14220  inv_rcinvr 14254  SubRingcsubrg 14351

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2205  ax-14 2206  ax-ext 2214  ax-coll 4224  ax-sep 4227  ax-nul 4235  ax-pow 4286  ax-pr 4321  ax-un 4553  ax-setind 4658  ax-cnex 8214  ax-resscn 8215  ax-1cn 8216  ax-1re 8217  ax-icn 8218  ax-addcl 8219  ax-addrcl 8220  ax-mulcl 8221  ax-addcom 8223  ax-addass 8225  ax-i2m1 8228  ax-0lt1 8229  ax-0id 8231  ax-rnegex 8232  ax-pre-ltirr 8235  ax-pre-lttrn 8237  ax-pre-ltadd 8239

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2083  df-mo 2084  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ne 2413  df-nel 2508  df-ral 2525  df-rex 2526  df-reu 2527  df-rmo 2528  df-rab 2529  df-v 2814  df-sbc 3042  df-csb 3138  df-dif 3212  df-un 3214  df-in 3216  df-ss 3223  df-nul 3508  df-pw 3670  df-sn 3694  df-pr 3695  df-op 3697  df-uni 3914  df-int 3949  df-iun 3992  df-br 4109  df-opab 4171  df-mpt 4172  df-id 4413  df-xp 4754  df-rel 4755  df-cnv 4756  df-co 4757  df-dm 4758  df-rn 4759  df-res 4760  df-ima 4761  df-iota 5311  df-fun 5353  df-fn 5354  df-f 5355  df-f1 5356  df-fo 5357  df-f1o 5358  df-fv 5359  df-riota 6002  df-ov 6052  df-oprab 6053  df-mpo 6054  df-tpos 6475  df-pnf 8306  df-mnf 8307  df-ltxr 8309  df-inn 9234  df-2 9292  df-3 9293  df-ndx 13204  df-slot 13205  df-base 13207  df-sets 13208  df-iress 13209  df-plusg 13292  df-mulr 13293  df-0g 13460  df-mgm 13558  df-sgrp 13604  df-mnd 13619  df-grp 13705  df-minusg 13706  df-subg 13876  df-cmn 13992  df-abl 13993  df-mgp 14054  df-ur 14093  df-srg 14097  df-ring 14131  df-oppr 14201  df-dvdsr 14222  df-unit 14223  df-invr 14255  df-subrg 14353

This theorem is referenced by:  subrginv  14371  subrgdv  14372  subrgunit  14373  subrgugrp  14374