Theorem subrgugrp 14253

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 14249	. . 3 ⊢ (𝐴 ∈ (SubRing‘𝑅) → 𝑉 ⊆ 𝑈)
5		subrgrcl 14239	. . . 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 14059	. . . . 5 ⊢ (𝑅 ∈ Ring → 𝑅 ∈ SRing)
10	6, 8, 9	unitgrpbasd 14128	. . . 4 ⊢ (𝑅 ∈ Ring → 𝑈 = (Base‘𝐺))
11	5, 10	syl 14	. . 3 ⊢ (𝐴 ∈ (SubRing‘𝑅) → 𝑈 = (Base‘𝐺))
12	4, 11	sseqtrd 3265	. 2 ⊢ (𝐴 ∈ (SubRing‘𝑅) → 𝑉 ⊆ (Base‘𝐺))
13	1	subrgring 14237	. . 3 ⊢ (𝐴 ∈ (SubRing‘𝑅) → 𝑆 ∈ Ring)
14		eqid 2231	. . . 4 ⊢ (1r‘𝑆) = (1r‘𝑆)
15	3, 14	1unit 14120	. . 3 ⊢ (𝑆 ∈ Ring → (1r‘𝑆) ∈ 𝑉)
16		elex2 2819	. . 3 ⊢ ((1r‘𝑆) ∈ 𝑉 → ∃𝑤 𝑤 ∈ 𝑉)
17	13, 15, 16	3syl 17	. 2 ⊢ (𝐴 ∈ (SubRing‘𝑅) → ∃𝑤 𝑤 ∈ 𝑉)
18		eqid 2231	. . . . . . . . . . . 12 ⊢ (.r‘𝑅) = (.r‘𝑅)
19	1, 18	ressmulrg 13227	. . . . . . . . . . 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 1044	. . . . . . . . 9 ⊢ ((𝐴 ∈ (SubRing‘𝑅) ∧ 𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉) → (.r‘𝑅) = (.r‘𝑆))
22	21	oveqd 6034	. . . . . . . 8 ⊢ ((𝐴 ∈ (SubRing‘𝑅) ∧ 𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉) → (𝑥(.r‘𝑅)𝑦) = (𝑥(.r‘𝑆)𝑦))
23		eqid 2231	. . . . . . . . . 10 ⊢ (.r‘𝑆) = (.r‘𝑆)
24	3, 23	unitmulcl 14126	. . . . . . . . 9 ⊢ ((𝑆 ∈ Ring ∧ 𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉) → (𝑥(.r‘𝑆)𝑦) ∈ 𝑉)
25	13, 24	syl3an1 1306	. . . . . . . 8 ⊢ ((𝐴 ∈ (SubRing‘𝑅) ∧ 𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉) → (𝑥(.r‘𝑆)𝑦) ∈ 𝑉)
26	22, 25	eqeltrd 2308	. . . . . . 7 ⊢ ((𝐴 ∈ (SubRing‘𝑅) ∧ 𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉) → (𝑥(.r‘𝑅)𝑦) ∈ 𝑉)
27	26	3expa 1229	. . . . . 6 ⊢ (((𝐴 ∈ (SubRing‘𝑅) ∧ 𝑥 ∈ 𝑉) ∧ 𝑦 ∈ 𝑉) → (𝑥(.r‘𝑅)𝑦) ∈ 𝑉)
28	27	ralrimiva 2605	. . . . 5 ⊢ ((𝐴 ∈ (SubRing‘𝑅) ∧ 𝑥 ∈ 𝑉) → ∀𝑦 ∈ 𝑉 (𝑥(.r‘𝑅)𝑦) ∈ 𝑉)
29		eqid 2231	. . . . . . 7 ⊢ (invr‘𝑅) = (invr‘𝑅)
30		eqid 2231	. . . . . . 7 ⊢ (invr‘𝑆) = (invr‘𝑆)
31	1, 29, 3, 30	subrginv 14250	. . . . . 6 ⊢ ((𝐴 ∈ (SubRing‘𝑅) ∧ 𝑥 ∈ 𝑉) → ((invr‘𝑅)‘𝑥) = ((invr‘𝑆)‘𝑥))
32	3, 30	unitinvcl 14136	. . . . . . 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 2308	. . . . 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 2605	. . 3 ⊢ (𝐴 ∈ (SubRing‘𝑅) → ∀𝑥 ∈ 𝑉 (∀𝑦 ∈ 𝑉 (𝑥(.r‘𝑅)𝑦) ∈ 𝑉 ∧ ((invr‘𝑅)‘𝑥) ∈ 𝑉))
37		eqid 2231	. . . . . . . . . . 11 ⊢ (mulGrp‘𝑅) = (mulGrp‘𝑅)
38	37, 18	mgpplusgg 13936	. . . . . . . . . 10 ⊢ (𝑅 ∈ Ring → (.r‘𝑅) = (+g‘(mulGrp‘𝑅)))
39		basfn 13140	. . . . . . . . . . . 12 ⊢ Base Fn V
40		elex 2814	. . . . . . . . . . . 12 ⊢ (𝑅 ∈ Ring → 𝑅 ∈ V)
41		funfvex 5656	. . . . . . . . . . . . 13 ⊢ ((Fun Base ∧ 𝑅 ∈ dom Base) → (Base‘𝑅) ∈ V)
42	41	funfni 5432	. . . . . . . . . . . 12 ⊢ ((Base Fn V ∧ 𝑅 ∈ V) → (Base‘𝑅) ∈ V)
43	39, 40, 42	sylancr 414	. . . . . . . . . . 11 ⊢ (𝑅 ∈ Ring → (Base‘𝑅) ∈ V)
44		eqidd 2232	. . . . . . . . . . . 12 ⊢ (𝑅 ∈ Ring → (Base‘𝑅) = (Base‘𝑅))
45	44, 6, 9	unitssd 14122	. . . . . . . . . . 11 ⊢ (𝑅 ∈ Ring → 𝑈 ⊆ (Base‘𝑅))
46	43, 45	ssexd 4229	. . . . . . . . . 10 ⊢ (𝑅 ∈ Ring → 𝑈 ∈ V)
47	37	ringmgp 14014	. . . . . . . . . 10 ⊢ (𝑅 ∈ Ring → (mulGrp‘𝑅) ∈ Mnd)
48	8, 38, 46, 47	ressplusgd 13211	. . . . . . . . 9 ⊢ (𝑅 ∈ Ring → (.r‘𝑅) = (+g‘𝐺))
49	5, 48	syl 14	. . . . . . . 8 ⊢ (𝐴 ∈ (SubRing‘𝑅) → (.r‘𝑅) = (+g‘𝐺))
50	49	oveqd 6034	. . . . . . 7 ⊢ (𝐴 ∈ (SubRing‘𝑅) → (𝑥(.r‘𝑅)𝑦) = (𝑥(+g‘𝐺)𝑦))
51	50	eleq1d 2300	. . . . . 6 ⊢ (𝐴 ∈ (SubRing‘𝑅) → ((𝑥(.r‘𝑅)𝑦) ∈ 𝑉 ↔ (𝑥(+g‘𝐺)𝑦) ∈ 𝑉))
52	51	ralbidv 2532	. . . . 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 2232	. . . . . . . 8 ⊢ (𝐴 ∈ (SubRing‘𝑅) → (invr‘𝑅) = (invr‘𝑅))
56	53, 54, 55, 5	invrfvald 14135	. . . . . . 7 ⊢ (𝐴 ∈ (SubRing‘𝑅) → (invr‘𝑅) = (invg‘𝐺))
57	56	fveq1d 5641	. . . . . 6 ⊢ (𝐴 ∈ (SubRing‘𝑅) → ((invr‘𝑅)‘𝑥) = ((invg‘𝐺)‘𝑥))
58	57	eleq1d 2300	. . . . 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 2532	. . 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 14129	. . 3 ⊢ (𝑅 ∈ Ring → 𝐺 ∈ Grp)
63		eqid 2231	. . . 4 ⊢ (Base‘𝐺) = (Base‘𝐺)
64		eqid 2231	. . . 4 ⊢ (+g‘𝐺) = (+g‘𝐺)
65		eqid 2231	. . . 4 ⊢ (invg‘𝐺) = (invg‘𝐺)
66	63, 64, 65	issubg2m 13775	. . 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 1206	1 ⊢ (𝐴 ∈ (SubRing‘𝑅) → 𝑉 ∈ (SubGrp‘𝐺))

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105   ∧ w3a 1004   = wceq 1397  ∃wex 1540   ∈ wcel 2202  ∀wral 2510  Vcvv 2802   ⊆ wss 3200   Fn wfn 5321  ‘cfv 5326  (class class class)co 6017  Basecbs 13081   ↾_s cress 13082  +_gcplusg 13159  ._rcmulr 13160  Mndcmnd 13498  Grpcgrp 13582  inv_gcminusg 13583  SubGrpcsubg 13753  mulGrpcmgp 13932  1_rcur 13971  Ringcrg 14008  Unitcui 14099  inv_rcinvr 14133  SubRingcsubrg 14230

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 8122  ax-resscn 8123  ax-1cn 8124  ax-1re 8125  ax-icn 8126  ax-addcl 8127  ax-addrcl 8128  ax-mulcl 8129  ax-addcom 8131  ax-addass 8133  ax-i2m1 8136  ax-0lt1 8137  ax-0id 8139  ax-rnegex 8140  ax-pre-ltirr 8143  ax-pre-lttrn 8145  ax-pre-ltadd 8147

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 5970  df-ov 6020  df-oprab 6021  df-mpo 6022  df-tpos 6410  df-pnf 8215  df-mnf 8216  df-ltxr 8218  df-inn 9143  df-2 9201  df-3 9202  df-ndx 13084  df-slot 13085  df-base 13087  df-sets 13088  df-iress 13089  df-plusg 13172  df-mulr 13173  df-0g 13340  df-mgm 13438  df-sgrp 13484  df-mnd 13499  df-grp 13585  df-minusg 13586  df-subg 13756  df-cmn 13872  df-abl 13873  df-mgp 13933  df-ur 13972  df-srg 13976  df-ring 14010  df-oppr 14080  df-dvdsr 14101  df-unit 14102  df-invr 14134  df-subrg 14232

This theorem is referenced by: (None)