Theorem subrgugrp 14385

Description: The units of a subring form a subgroup of the unit group of the original ring. (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)
subrgugrp.4	⊢ 𝐺 = ((mulGrp‘𝑅) ↾s 𝑈)

Assertion

Ref	Expression
subrgugrp	⊢ (𝐴 ∈ (SubRing‘𝑅) → 𝑉 ∈ (SubGrp‘𝐺))

Proof of Theorem subrgugrp

Dummy variables 𝑥 𝑦 𝑤 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		subrgugrp.1	. . . 4 ⊢ 𝑆 = (𝑅 ↾s 𝐴)
2		subrgugrp.2	. . . 4 ⊢ 𝑈 = (Unit‘𝑅)
3		subrgugrp.3	. . . 4 ⊢ 𝑉 = (Unit‘𝑆)
4	1, 2, 3	subrguss 14381	. . 3 ⊢ (𝐴 ∈ (SubRing‘𝑅) → 𝑉 ⊆ 𝑈)
5		subrgrcl 14371	. . . 4 ⊢ (𝐴 ∈ (SubRing‘𝑅) → 𝑅 ∈ Ring)
6	2	a1i 9	. . . . 5 ⊢ (𝑅 ∈ Ring → 𝑈 = (Unit‘𝑅))
7		subrgugrp.4	. . . . . 6 ⊢ 𝐺 = ((mulGrp‘𝑅) ↾s 𝑈)
8	7	a1i 9	. . . . 5 ⊢ (𝑅 ∈ Ring → 𝐺 = ((mulGrp‘𝑅) ↾s 𝑈))
9		ringsrg 14191	. . . . 5 ⊢ (𝑅 ∈ Ring → 𝑅 ∈ SRing)
10	6, 8, 9	unitgrpbasd 14260	. . . 4 ⊢ (𝑅 ∈ Ring → 𝑈 = (Base‘𝐺))
11	5, 10	syl 14	. . 3 ⊢ (𝐴 ∈ (SubRing‘𝑅) → 𝑈 = (Base‘𝐺))
12	4, 11	sseqtrd 3276	. 2 ⊢ (𝐴 ∈ (SubRing‘𝑅) → 𝑉 ⊆ (Base‘𝐺))
13	1	subrgring 14369	. . 3 ⊢ (𝐴 ∈ (SubRing‘𝑅) → 𝑆 ∈ Ring)
14		eqid 2232	. . . 4 ⊢ (1r‘𝑆) = (1r‘𝑆)
15	3, 14	1unit 14252	. . 3 ⊢ (𝑆 ∈ Ring → (1r‘𝑆) ∈ 𝑉)
16		elex2 2830	. . 3 ⊢ ((1r‘𝑆) ∈ 𝑉 → ∃𝑤 𝑤 ∈ 𝑉)
17	13, 15, 16	3syl 17	. 2 ⊢ (𝐴 ∈ (SubRing‘𝑅) → ∃𝑤 𝑤 ∈ 𝑉)
18		eqid 2232	. . . . . . . . . . . 12 ⊢ (.r‘𝑅) = (.r‘𝑅)
19	1, 18	ressmulrg 13358	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ (SubRing‘𝑅) ∧ 𝑅 ∈ Ring) → (.r‘𝑅) = (.r‘𝑆))
20	5, 19	mpdan 421	. . . . . . . . . 10 ⊢ (𝐴 ∈ (SubRing‘𝑅) → (.r‘𝑅) = (.r‘𝑆))
21	20	3ad2ant1 1045	. . . . . . . . 9 ⊢ ((𝐴 ∈ (SubRing‘𝑅) ∧ 𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉) → (.r‘𝑅) = (.r‘𝑆))
22	21	oveqd 6067	. . . . . . . 8 ⊢ ((𝐴 ∈ (SubRing‘𝑅) ∧ 𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉) → (𝑥(.r‘𝑅)𝑦) = (𝑥(.r‘𝑆)𝑦))
23		eqid 2232	. . . . . . . . . 10 ⊢ (.r‘𝑆) = (.r‘𝑆)
24	3, 23	unitmulcl 14258	. . . . . . . . 9 ⊢ ((𝑆 ∈ Ring ∧ 𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉) → (𝑥(.r‘𝑆)𝑦) ∈ 𝑉)
25	13, 24	syl3an1 1307	. . . . . . . 8 ⊢ ((𝐴 ∈ (SubRing‘𝑅) ∧ 𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉) → (𝑥(.r‘𝑆)𝑦) ∈ 𝑉)
26	22, 25	eqeltrd 2309	. . . . . . 7 ⊢ ((𝐴 ∈ (SubRing‘𝑅) ∧ 𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉) → (𝑥(.r‘𝑅)𝑦) ∈ 𝑉)
27	26	3expa 1230	. . . . . 6 ⊢ (((𝐴 ∈ (SubRing‘𝑅) ∧ 𝑥 ∈ 𝑉) ∧ 𝑦 ∈ 𝑉) → (𝑥(.r‘𝑅)𝑦) ∈ 𝑉)
28	27	ralrimiva 2615	. . . . 5 ⊢ ((𝐴 ∈ (SubRing‘𝑅) ∧ 𝑥 ∈ 𝑉) → ∀𝑦 ∈ 𝑉 (𝑥(.r‘𝑅)𝑦) ∈ 𝑉)
29		eqid 2232	. . . . . . 7 ⊢ (invr‘𝑅) = (invr‘𝑅)
30		eqid 2232	. . . . . . 7 ⊢ (invr‘𝑆) = (invr‘𝑆)
31	1, 29, 3, 30	subrginv 14382	. . . . . 6 ⊢ ((𝐴 ∈ (SubRing‘𝑅) ∧ 𝑥 ∈ 𝑉) → ((invr‘𝑅)‘𝑥) = ((invr‘𝑆)‘𝑥))
32	3, 30	unitinvcl 14268	. . . . . . 7 ⊢ ((𝑆 ∈ Ring ∧ 𝑥 ∈ 𝑉) → ((invr‘𝑆)‘𝑥) ∈ 𝑉)
33	13, 32	sylan 283	. . . . . 6 ⊢ ((𝐴 ∈ (SubRing‘𝑅) ∧ 𝑥 ∈ 𝑉) → ((invr‘𝑆)‘𝑥) ∈ 𝑉)
34	31, 33	eqeltrd 2309	. . . . 5 ⊢ ((𝐴 ∈ (SubRing‘𝑅) ∧ 𝑥 ∈ 𝑉) → ((invr‘𝑅)‘𝑥) ∈ 𝑉)
35	28, 34	jca 306	. . . 4 ⊢ ((𝐴 ∈ (SubRing‘𝑅) ∧ 𝑥 ∈ 𝑉) → (∀𝑦 ∈ 𝑉 (𝑥(.r‘𝑅)𝑦) ∈ 𝑉 ∧ ((invr‘𝑅)‘𝑥) ∈ 𝑉))
36	35	ralrimiva 2615	. . 3 ⊢ (𝐴 ∈ (SubRing‘𝑅) → ∀𝑥 ∈ 𝑉 (∀𝑦 ∈ 𝑉 (𝑥(.r‘𝑅)𝑦) ∈ 𝑉 ∧ ((invr‘𝑅)‘𝑥) ∈ 𝑉))
37		eqid 2232	. . . . . . . . . . 11 ⊢ (mulGrp‘𝑅) = (mulGrp‘𝑅)
38	37, 18	mgpplusgg 14068	. . . . . . . . . 10 ⊢ (𝑅 ∈ Ring → (.r‘𝑅) = (+g‘(mulGrp‘𝑅)))
39		basfn 13271	. . . . . . . . . . . 12 ⊢ Base Fn V
40		elex 2825	. . . . . . . . . . . 12 ⊢ (𝑅 ∈ Ring → 𝑅 ∈ V)
41		funfvex 5687	. . . . . . . . . . . . 13 ⊢ ((Fun Base ∧ 𝑅 ∈ dom Base) → (Base‘𝑅) ∈ V)
42	41	funfni 5458	. . . . . . . . . . . 12 ⊢ ((Base Fn V ∧ 𝑅 ∈ V) → (Base‘𝑅) ∈ V)
43	39, 40, 42	sylancr 414	. . . . . . . . . . 11 ⊢ (𝑅 ∈ Ring → (Base‘𝑅) ∈ V)
44		eqidd 2233	. . . . . . . . . . . 12 ⊢ (𝑅 ∈ Ring → (Base‘𝑅) = (Base‘𝑅))
45	44, 6, 9	unitssd 14254	. . . . . . . . . . 11 ⊢ (𝑅 ∈ Ring → 𝑈 ⊆ (Base‘𝑅))
46	43, 45	ssexd 4250	. . . . . . . . . 10 ⊢ (𝑅 ∈ Ring → 𝑈 ∈ V)
47	37	ringmgp 14146	. . . . . . . . . 10 ⊢ (𝑅 ∈ Ring → (mulGrp‘𝑅) ∈ Mnd)
48	8, 38, 46, 47	ressplusgd 13342	. . . . . . . . 9 ⊢ (𝑅 ∈ Ring → (.r‘𝑅) = (+g‘𝐺))
49	5, 48	syl 14	. . . . . . . 8 ⊢ (𝐴 ∈ (SubRing‘𝑅) → (.r‘𝑅) = (+g‘𝐺))
50	49	oveqd 6067	. . . . . . 7 ⊢ (𝐴 ∈ (SubRing‘𝑅) → (𝑥(.r‘𝑅)𝑦) = (𝑥(+g‘𝐺)𝑦))
51	50	eleq1d 2301	. . . . . 6 ⊢ (𝐴 ∈ (SubRing‘𝑅) → ((𝑥(.r‘𝑅)𝑦) ∈ 𝑉 ↔ (𝑥(+g‘𝐺)𝑦) ∈ 𝑉))
52	51	ralbidv 2542	. . . . 5 ⊢ (𝐴 ∈ (SubRing‘𝑅) → (∀𝑦 ∈ 𝑉 (𝑥(.r‘𝑅)𝑦) ∈ 𝑉 ↔ ∀𝑦 ∈ 𝑉 (𝑥(+g‘𝐺)𝑦) ∈ 𝑉))
53	2	a1i 9	. . . . . . . 8 ⊢ (𝐴 ∈ (SubRing‘𝑅) → 𝑈 = (Unit‘𝑅))
54	7	a1i 9	. . . . . . . 8 ⊢ (𝐴 ∈ (SubRing‘𝑅) → 𝐺 = ((mulGrp‘𝑅) ↾s 𝑈))
55		eqidd 2233	. . . . . . . 8 ⊢ (𝐴 ∈ (SubRing‘𝑅) → (invr‘𝑅) = (invr‘𝑅))
56	53, 54, 55, 5	invrfvald 14267	. . . . . . 7 ⊢ (𝐴 ∈ (SubRing‘𝑅) → (invr‘𝑅) = (invg‘𝐺))
57	56	fveq1d 5672	. . . . . 6 ⊢ (𝐴 ∈ (SubRing‘𝑅) → ((invr‘𝑅)‘𝑥) = ((invg‘𝐺)‘𝑥))
58	57	eleq1d 2301	. . . . 5 ⊢ (𝐴 ∈ (SubRing‘𝑅) → (((invr‘𝑅)‘𝑥) ∈ 𝑉 ↔ ((invg‘𝐺)‘𝑥) ∈ 𝑉))
59	52, 58	anbi12d 473	. . . 4 ⊢ (𝐴 ∈ (SubRing‘𝑅) → ((∀𝑦 ∈ 𝑉 (𝑥(.r‘𝑅)𝑦) ∈ 𝑉 ∧ ((invr‘𝑅)‘𝑥) ∈ 𝑉) ↔ (∀𝑦 ∈ 𝑉 (𝑥(+g‘𝐺)𝑦) ∈ 𝑉 ∧ ((invg‘𝐺)‘𝑥) ∈ 𝑉)))
60	59	ralbidv 2542	. . 3 ⊢ (𝐴 ∈ (SubRing‘𝑅) → (∀𝑥 ∈ 𝑉 (∀𝑦 ∈ 𝑉 (𝑥(.r‘𝑅)𝑦) ∈ 𝑉 ∧ ((invr‘𝑅)‘𝑥) ∈ 𝑉) ↔ ∀𝑥 ∈ 𝑉 (∀𝑦 ∈ 𝑉 (𝑥(+g‘𝐺)𝑦) ∈ 𝑉 ∧ ((invg‘𝐺)‘𝑥) ∈ 𝑉)))
61	36, 60	mpbid 147	. 2 ⊢ (𝐴 ∈ (SubRing‘𝑅) → ∀𝑥 ∈ 𝑉 (∀𝑦 ∈ 𝑉 (𝑥(+g‘𝐺)𝑦) ∈ 𝑉 ∧ ((invg‘𝐺)‘𝑥) ∈ 𝑉))
62	2, 7	unitgrp 14261	. . 3 ⊢ (𝑅 ∈ Ring → 𝐺 ∈ Grp)
63		eqid 2232	. . . 4 ⊢ (Base‘𝐺) = (Base‘𝐺)
64		eqid 2232	. . . 4 ⊢ (+g‘𝐺) = (+g‘𝐺)
65		eqid 2232	. . . 4 ⊢ (invg‘𝐺) = (invg‘𝐺)
66	63, 64, 65	issubg2m 13906	. . 3 ⊢ (𝐺 ∈ Grp → (𝑉 ∈ (SubGrp‘𝐺) ↔ (𝑉 ⊆ (Base‘𝐺) ∧ ∃𝑤 𝑤 ∈ 𝑉 ∧ ∀𝑥 ∈ 𝑉 (∀𝑦 ∈ 𝑉 (𝑥(+g‘𝐺)𝑦) ∈ 𝑉 ∧ ((invg‘𝐺)‘𝑥) ∈ 𝑉))))
67	5, 62, 66	3syl 17	. 2 ⊢ (𝐴 ∈ (SubRing‘𝑅) → (𝑉 ∈ (SubGrp‘𝐺) ↔ (𝑉 ⊆ (Base‘𝐺) ∧ ∃𝑤 𝑤 ∈ 𝑉 ∧ ∀𝑥 ∈ 𝑉 (∀𝑦 ∈ 𝑉 (𝑥(+g‘𝐺)𝑦) ∈ 𝑉 ∧ ((invg‘𝐺)‘𝑥) ∈ 𝑉))))
68	12, 17, 61, 67	mpbir3and 1207	1 ⊢ (𝐴 ∈ (SubRing‘𝑅) → 𝑉 ∈ (SubGrp‘𝐺))

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105   ∧ w3a 1005   = wceq 1398  ∃wex 1541   ∈ wcel 2203  ∀wral 2520  Vcvv 2813   ⊆ wss 3211   Fn wfn 5347  ‘cfv 5352  (class class class)co 6050  Basecbs 13212   ↾_s cress 13213  +_gcplusg 13290  ._rcmulr 13291  Mndcmnd 13629  Grpcgrp 13713  inv_gcminusg 13714  SubGrpcsubg 13884  mulGrpcmgp 14064  1_rcur 14103  Ringcrg 14140  Unitcui 14231  inv_rcinvr 14265  SubRingcsubrg 14362

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 4225  ax-sep 4228  ax-nul 4236  ax-pow 4287  ax-pr 4322  ax-un 4554  ax-setind 4659  ax-cnex 8218  ax-resscn 8219  ax-1cn 8220  ax-1re 8221  ax-icn 8222  ax-addcl 8223  ax-addrcl 8224  ax-mulcl 8225  ax-addcom 8227  ax-addass 8229  ax-i2m1 8232  ax-0lt1 8233  ax-0id 8235  ax-rnegex 8236  ax-pre-ltirr 8239  ax-pre-lttrn 8241  ax-pre-ltadd 8243

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 2815  df-sbc 3043  df-csb 3139  df-dif 3213  df-un 3215  df-in 3217  df-ss 3224  df-nul 3509  df-pw 3671  df-sn 3695  df-pr 3696  df-op 3698  df-uni 3915  df-int 3950  df-iun 3993  df-br 4110  df-opab 4172  df-mpt 4173  df-id 4414  df-xp 4755  df-rel 4756  df-cnv 4757  df-co 4758  df-dm 4759  df-rn 4760  df-res 4761  df-ima 4762  df-iota 5312  df-fun 5354  df-fn 5355  df-f 5356  df-f1 5357  df-fo 5358  df-f1o 5359  df-fv 5360  df-riota 6003  df-ov 6053  df-oprab 6054  df-mpo 6055  df-tpos 6476  df-pnf 8310  df-mnf 8311  df-ltxr 8313  df-inn 9238  df-2 9296  df-3 9297  df-ndx 13215  df-slot 13216  df-base 13218  df-sets 13219  df-iress 13220  df-plusg 13303  df-mulr 13304  df-0g 13471  df-mgm 13569  df-sgrp 13615  df-mnd 13630  df-grp 13716  df-minusg 13717  df-subg 13887  df-cmn 14003  df-abl 14004  df-mgp 14065  df-ur 14104  df-srg 14108  df-ring 14142  df-oppr 14212  df-dvdsr 14233  df-unit 14234  df-invr 14266  df-subrg 14364

This theorem is referenced by: (None)