Theorem subrgugrp 13736

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 13732	. . 3 ⊢ (𝐴 ∈ (SubRing‘𝑅) → 𝑉 ⊆ 𝑈)
5		subrgrcl 13722	. . . 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 13543	. . . . 5 ⊢ (𝑅 ∈ Ring → 𝑅 ∈ SRing)
10	6, 8, 9	unitgrpbasd 13611	. . . 4 ⊢ (𝑅 ∈ Ring → 𝑈 = (Base‘𝐺))
11	5, 10	syl 14	. . 3 ⊢ (𝐴 ∈ (SubRing‘𝑅) → 𝑈 = (Base‘𝐺))
12	4, 11	sseqtrd 3217	. 2 ⊢ (𝐴 ∈ (SubRing‘𝑅) → 𝑉 ⊆ (Base‘𝐺))
13	1	subrgring 13720	. . 3 ⊢ (𝐴 ∈ (SubRing‘𝑅) → 𝑆 ∈ Ring)
14		eqid 2193	. . . 4 ⊢ (1r‘𝑆) = (1r‘𝑆)
15	3, 14	1unit 13603	. . 3 ⊢ (𝑆 ∈ Ring → (1r‘𝑆) ∈ 𝑉)
16		elex2 2776	. . 3 ⊢ ((1r‘𝑆) ∈ 𝑉 → ∃𝑤 𝑤 ∈ 𝑉)
17	13, 15, 16	3syl 17	. 2 ⊢ (𝐴 ∈ (SubRing‘𝑅) → ∃𝑤 𝑤 ∈ 𝑉)
18		eqid 2193	. . . . . . . . . . . 12 ⊢ (.r‘𝑅) = (.r‘𝑅)
19	1, 18	ressmulrg 12762	. . . . . . . . . . 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 1020	. . . . . . . . 9 ⊢ ((𝐴 ∈ (SubRing‘𝑅) ∧ 𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉) → (.r‘𝑅) = (.r‘𝑆))
22	21	oveqd 5935	. . . . . . . 8 ⊢ ((𝐴 ∈ (SubRing‘𝑅) ∧ 𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉) → (𝑥(.r‘𝑅)𝑦) = (𝑥(.r‘𝑆)𝑦))
23		eqid 2193	. . . . . . . . . 10 ⊢ (.r‘𝑆) = (.r‘𝑆)
24	3, 23	unitmulcl 13609	. . . . . . . . 9 ⊢ ((𝑆 ∈ Ring ∧ 𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉) → (𝑥(.r‘𝑆)𝑦) ∈ 𝑉)
25	13, 24	syl3an1 1282	. . . . . . . 8 ⊢ ((𝐴 ∈ (SubRing‘𝑅) ∧ 𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉) → (𝑥(.r‘𝑆)𝑦) ∈ 𝑉)
26	22, 25	eqeltrd 2270	. . . . . . 7 ⊢ ((𝐴 ∈ (SubRing‘𝑅) ∧ 𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉) → (𝑥(.r‘𝑅)𝑦) ∈ 𝑉)
27	26	3expa 1205	. . . . . 6 ⊢ (((𝐴 ∈ (SubRing‘𝑅) ∧ 𝑥 ∈ 𝑉) ∧ 𝑦 ∈ 𝑉) → (𝑥(.r‘𝑅)𝑦) ∈ 𝑉)
28	27	ralrimiva 2567	. . . . 5 ⊢ ((𝐴 ∈ (SubRing‘𝑅) ∧ 𝑥 ∈ 𝑉) → ∀𝑦 ∈ 𝑉 (𝑥(.r‘𝑅)𝑦) ∈ 𝑉)
29		eqid 2193	. . . . . . 7 ⊢ (invr‘𝑅) = (invr‘𝑅)
30		eqid 2193	. . . . . . 7 ⊢ (invr‘𝑆) = (invr‘𝑆)
31	1, 29, 3, 30	subrginv 13733	. . . . . 6 ⊢ ((𝐴 ∈ (SubRing‘𝑅) ∧ 𝑥 ∈ 𝑉) → ((invr‘𝑅)‘𝑥) = ((invr‘𝑆)‘𝑥))
32	3, 30	unitinvcl 13619	. . . . . . 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 2270	. . . . 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 2567	. . 3 ⊢ (𝐴 ∈ (SubRing‘𝑅) → ∀𝑥 ∈ 𝑉 (∀𝑦 ∈ 𝑉 (𝑥(.r‘𝑅)𝑦) ∈ 𝑉 ∧ ((invr‘𝑅)‘𝑥) ∈ 𝑉))
37		eqid 2193	. . . . . . . . . . 11 ⊢ (mulGrp‘𝑅) = (mulGrp‘𝑅)
38	37, 18	mgpplusgg 13420	. . . . . . . . . 10 ⊢ (𝑅 ∈ Ring → (.r‘𝑅) = (+g‘(mulGrp‘𝑅)))
39		basfn 12676	. . . . . . . . . . . 12 ⊢ Base Fn V
40		elex 2771	. . . . . . . . . . . 12 ⊢ (𝑅 ∈ Ring → 𝑅 ∈ V)
41		funfvex 5571	. . . . . . . . . . . . 13 ⊢ ((Fun Base ∧ 𝑅 ∈ dom Base) → (Base‘𝑅) ∈ V)
42	41	funfni 5354	. . . . . . . . . . . 12 ⊢ ((Base Fn V ∧ 𝑅 ∈ V) → (Base‘𝑅) ∈ V)
43	39, 40, 42	sylancr 414	. . . . . . . . . . 11 ⊢ (𝑅 ∈ Ring → (Base‘𝑅) ∈ V)
44		eqidd 2194	. . . . . . . . . . . 12 ⊢ (𝑅 ∈ Ring → (Base‘𝑅) = (Base‘𝑅))
45	44, 6, 9	unitssd 13605	. . . . . . . . . . 11 ⊢ (𝑅 ∈ Ring → 𝑈 ⊆ (Base‘𝑅))
46	43, 45	ssexd 4169	. . . . . . . . . 10 ⊢ (𝑅 ∈ Ring → 𝑈 ∈ V)
47	37	ringmgp 13498	. . . . . . . . . 10 ⊢ (𝑅 ∈ Ring → (mulGrp‘𝑅) ∈ Mnd)
48	8, 38, 46, 47	ressplusgd 12746	. . . . . . . . 9 ⊢ (𝑅 ∈ Ring → (.r‘𝑅) = (+g‘𝐺))
49	5, 48	syl 14	. . . . . . . 8 ⊢ (𝐴 ∈ (SubRing‘𝑅) → (.r‘𝑅) = (+g‘𝐺))
50	49	oveqd 5935	. . . . . . 7 ⊢ (𝐴 ∈ (SubRing‘𝑅) → (𝑥(.r‘𝑅)𝑦) = (𝑥(+g‘𝐺)𝑦))
51	50	eleq1d 2262	. . . . . 6 ⊢ (𝐴 ∈ (SubRing‘𝑅) → ((𝑥(.r‘𝑅)𝑦) ∈ 𝑉 ↔ (𝑥(+g‘𝐺)𝑦) ∈ 𝑉))
52	51	ralbidv 2494	. . . . 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 2194	. . . . . . . 8 ⊢ (𝐴 ∈ (SubRing‘𝑅) → (invr‘𝑅) = (invr‘𝑅))
56	53, 54, 55, 5	invrfvald 13618	. . . . . . 7 ⊢ (𝐴 ∈ (SubRing‘𝑅) → (invr‘𝑅) = (invg‘𝐺))
57	56	fveq1d 5556	. . . . . 6 ⊢ (𝐴 ∈ (SubRing‘𝑅) → ((invr‘𝑅)‘𝑥) = ((invg‘𝐺)‘𝑥))
58	57	eleq1d 2262	. . . . 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 2494	. . 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 13612	. . 3 ⊢ (𝑅 ∈ Ring → 𝐺 ∈ Grp)
63		eqid 2193	. . . 4 ⊢ (Base‘𝐺) = (Base‘𝐺)
64		eqid 2193	. . . 4 ⊢ (+g‘𝐺) = (+g‘𝐺)
65		eqid 2193	. . . 4 ⊢ (invg‘𝐺) = (invg‘𝐺)
66	63, 64, 65	issubg2m 13259	. . 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 1182	1 ⊢ (𝐴 ∈ (SubRing‘𝑅) → 𝑉 ∈ (SubGrp‘𝐺))

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105   ∧ w3a 980   = wceq 1364  ∃wex 1503   ∈ wcel 2164  ∀wral 2472  Vcvv 2760   ⊆ wss 3153   Fn wfn 5249  ‘cfv 5254  (class class class)co 5918  Basecbs 12618   ↾_s cress 12619  +_gcplusg 12695  ._rcmulr 12696  Mndcmnd 12997  Grpcgrp 13072  inv_gcminusg 13073  SubGrpcsubg 13237  mulGrpcmgp 13416  1_rcur 13455  Ringcrg 13492  Unitcui 13583  inv_rcinvr 13616  SubRingcsubrg 13713

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 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2166  ax-14 2167  ax-ext 2175  ax-coll 4144  ax-sep 4147  ax-nul 4155  ax-pow 4203  ax-pr 4238  ax-un 4464  ax-setind 4569  ax-cnex 7963  ax-resscn 7964  ax-1cn 7965  ax-1re 7966  ax-icn 7967  ax-addcl 7968  ax-addrcl 7969  ax-mulcl 7970  ax-addcom 7972  ax-addass 7974  ax-i2m1 7977  ax-0lt1 7978  ax-0id 7980  ax-rnegex 7981  ax-pre-ltirr 7984  ax-pre-lttrn 7986  ax-pre-ltadd 7988

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ne 2365  df-nel 2460  df-ral 2477  df-rex 2478  df-reu 2479  df-rmo 2480  df-rab 2481  df-v 2762  df-sbc 2986  df-csb 3081  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3447  df-pw 3603  df-sn 3624  df-pr 3625  df-op 3627  df-uni 3836  df-int 3871  df-iun 3914  df-br 4030  df-opab 4091  df-mpt 4092  df-id 4324  df-xp 4665  df-rel 4666  df-cnv 4667  df-co 4668  df-dm 4669  df-rn 4670  df-res 4671  df-ima 4672  df-iota 5215  df-fun 5256  df-fn 5257  df-f 5258  df-f1 5259  df-fo 5260  df-f1o 5261  df-fv 5262  df-riota 5873  df-ov 5921  df-oprab 5922  df-mpo 5923  df-tpos 6298  df-pnf 8056  df-mnf 8057  df-ltxr 8059  df-inn 8983  df-2 9041  df-3 9042  df-ndx 12621  df-slot 12622  df-base 12624  df-sets 12625  df-iress 12626  df-plusg 12708  df-mulr 12709  df-0g 12869  df-mgm 12939  df-sgrp 12985  df-mnd 12998  df-grp 13075  df-minusg 13076  df-subg 13240  df-cmn 13356  df-abl 13357  df-mgp 13417  df-ur 13456  df-srg 13460  df-ring 13494  df-oppr 13564  df-dvdsr 13585  df-unit 13586  df-invr 13617  df-subrg 13715

This theorem is referenced by: (None)