Theorem subrgugrp 13739

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 e. (SubRing ` R ) -> V e. (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 13735	. . 3 |- ( A e. (SubRing ` R ) -> V C_ U )
5		subrgrcl 13725	. . . 4 |- ( A e. (SubRing ` R ) -> R e. Ring )
6	2	a1i 9	. . . . 5 |- ( R e. Ring -> U = (Unit ` R ) )
7		subrgugrp.4	. . . . . 6 |- G = ( (mulGrp ` R ) ↾s U )
8	7	a1i 9	. . . . 5 |- ( R e. Ring -> G = ( (mulGrp ` R ) ↾s U ) )
9		ringsrg 13546	. . . . 5 |- ( R e. Ring -> R e. SRing )
10	6, 8, 9	unitgrpbasd 13614	. . . 4 |- ( R e. Ring -> U = ( Base ` G ) )
11	5, 10	syl 14	. . 3 |- ( A e. (SubRing ` R ) -> U = ( Base ` G ) )
12	4, 11	sseqtrd 3218	. 2 |- ( A e. (SubRing ` R ) -> V C_ ( Base ` G ) )
13	1	subrgring 13723	. . 3 |- ( A e. (SubRing ` R ) -> S e. Ring )
14		eqid 2193	. . . 4 |- ( 1r ` S ) = ( 1r ` S )
15	3, 14	1unit 13606	. . 3 |- ( S e. Ring -> ( 1r ` S ) e. V )
16		elex2 2776	. . 3 |- ( ( 1r ` S ) e. V -> E. w w e. V )
17	13, 15, 16	3syl 17	. 2 |- ( A e. (SubRing ` R ) -> E. w w e. V )
18		eqid 2193	. . . . . . . . . . . 12 |- ( .r ` R ) = ( .r ` R )
19	1, 18	ressmulrg 12765	. . . . . . . . . . 11 |- ( ( A e. (SubRing ` R ) /\ R e. Ring ) -> ( .r ` R ) = ( .r ` S ) )
20	5, 19	mpdan 421	. . . . . . . . . 10 |- ( A e. (SubRing ` R ) -> ( .r ` R ) = ( .r ` S ) )
21	20	3ad2ant1 1020	. . . . . . . . 9 |- ( ( A e. (SubRing ` R ) /\ x e. V /\ y e. V ) -> ( .r ` R ) = ( .r ` S ) )
22	21	oveqd 5936	. . . . . . . 8 |- ( ( A e. (SubRing ` R ) /\ x e. V /\ y e. V ) -> ( x ( .r ` R ) y ) = ( x ( .r ` S ) y ) )
23		eqid 2193	. . . . . . . . . 10 |- ( .r ` S ) = ( .r ` S )
24	3, 23	unitmulcl 13612	. . . . . . . . 9 |- ( ( S e. Ring /\ x e. V /\ y e. V ) -> ( x ( .r ` S ) y ) e. V )
25	13, 24	syl3an1 1282	. . . . . . . 8 |- ( ( A e. (SubRing ` R ) /\ x e. V /\ y e. V ) -> ( x ( .r ` S ) y ) e. V )
26	22, 25	eqeltrd 2270	. . . . . . 7 |- ( ( A e. (SubRing ` R ) /\ x e. V /\ y e. V ) -> ( x ( .r ` R ) y ) e. V )
27	26	3expa 1205	. . . . . 6 |- ( ( ( A e. (SubRing ` R ) /\ x e. V ) /\ y e. V ) -> ( x ( .r ` R ) y ) e. V )
28	27	ralrimiva 2567	. . . . 5 |- ( ( A e. (SubRing ` R ) /\ x e. V ) -> A. y e. V ( x ( .r ` R ) y ) e. V )
29		eqid 2193	. . . . . . 7 |- ( invr ` R ) = ( invr ` R )
30		eqid 2193	. . . . . . 7 |- ( invr ` S ) = ( invr ` S )
31	1, 29, 3, 30	subrginv 13736	. . . . . 6 |- ( ( A e. (SubRing ` R ) /\ x e. V ) -> ( ( invr ` R ) ` x ) = ( ( invr ` S ) ` x ) )
32	3, 30	unitinvcl 13622	. . . . . . 7 |- ( ( S e. Ring /\ x e. V ) -> ( ( invr ` S ) ` x ) e. V )
33	13, 32	sylan 283	. . . . . 6 |- ( ( A e. (SubRing ` R ) /\ x e. V ) -> ( ( invr ` S ) ` x ) e. V )
34	31, 33	eqeltrd 2270	. . . . 5 |- ( ( A e. (SubRing ` R ) /\ x e. V ) -> ( ( invr ` R ) ` x ) e. V )
35	28, 34	jca 306	. . . 4 |- ( ( A e. (SubRing ` R ) /\ x e. V ) -> ( A. y e. V ( x ( .r ` R ) y ) e. V /\ ( ( invr ` R ) ` x ) e. V ) )
36	35	ralrimiva 2567	. . 3 |- ( A e. (SubRing ` R ) -> A. x e. V ( A. y e. V ( x ( .r ` R ) y ) e. V /\ ( ( invr ` R ) ` x ) e. V ) )
37		eqid 2193	. . . . . . . . . . 11 |- (mulGrp ` R ) = (mulGrp ` R )
38	37, 18	mgpplusgg 13423	. . . . . . . . . 10 |- ( R e. Ring -> ( .r ` R ) = ( +g ` (mulGrp ` R ) ) )
39		basfn 12679	. . . . . . . . . . . 12 |- Base Fn _V
40		elex 2771	. . . . . . . . . . . 12 |- ( R e. Ring -> R e. _V )
41		funfvex 5572	. . . . . . . . . . . . 13 |- ( ( Fun Base /\ R e. dom Base ) -> ( Base ` R ) e. _V )
42	41	funfni 5355	. . . . . . . . . . . 12 |- ( ( Base Fn _V /\ R e. _V ) -> ( Base ` R ) e. _V )
43	39, 40, 42	sylancr 414	. . . . . . . . . . 11 |- ( R e. Ring -> ( Base ` R ) e. _V )
44		eqidd 2194	. . . . . . . . . . . 12 |- ( R e. Ring -> ( Base ` R ) = ( Base ` R ) )
45	44, 6, 9	unitssd 13608	. . . . . . . . . . 11 |- ( R e. Ring -> U C_ ( Base ` R ) )
46	43, 45	ssexd 4170	. . . . . . . . . 10 |- ( R e. Ring -> U e. _V )
47	37	ringmgp 13501	. . . . . . . . . 10 |- ( R e. Ring -> (mulGrp ` R ) e. Mnd )
48	8, 38, 46, 47	ressplusgd 12749	. . . . . . . . 9 |- ( R e. Ring -> ( .r ` R ) = ( +g ` G ) )
49	5, 48	syl 14	. . . . . . . 8 |- ( A e. (SubRing ` R ) -> ( .r ` R ) = ( +g ` G ) )
50	49	oveqd 5936	. . . . . . 7 |- ( A e. (SubRing ` R ) -> ( x ( .r ` R ) y ) = ( x ( +g ` G ) y ) )
51	50	eleq1d 2262	. . . . . 6 |- ( A e. (SubRing ` R ) -> ( ( x ( .r ` R ) y ) e. V <-> ( x ( +g ` G ) y ) e. V ) )
52	51	ralbidv 2494	. . . . 5 |- ( A e. (SubRing ` R ) -> ( A. y e. V ( x ( .r ` R ) y ) e. V <-> A. y e. V ( x ( +g ` G ) y ) e. V ) )
53	2	a1i 9	. . . . . . . 8 |- ( A e. (SubRing ` R ) -> U = (Unit ` R ) )
54	7	a1i 9	. . . . . . . 8 |- ( A e. (SubRing ` R ) -> G = ( (mulGrp ` R ) ↾s U ) )
55		eqidd 2194	. . . . . . . 8 |- ( A e. (SubRing ` R ) -> ( invr ` R ) = ( invr ` R ) )
56	53, 54, 55, 5	invrfvald 13621	. . . . . . 7 |- ( A e. (SubRing ` R ) -> ( invr ` R ) = ( invg ` G ) )
57	56	fveq1d 5557	. . . . . 6 |- ( A e. (SubRing ` R ) -> ( ( invr ` R ) ` x ) = ( ( invg ` G ) ` x ) )
58	57	eleq1d 2262	. . . . 5 |- ( A e. (SubRing ` R ) -> ( ( ( invr ` R ) ` x ) e. V <-> ( ( invg ` G ) ` x ) e. V ) )
59	52, 58	anbi12d 473	. . . 4 |- ( A e. (SubRing ` R ) -> ( ( A. y e. V ( x ( .r ` R ) y ) e. V /\ ( ( invr ` R ) ` x ) e. V ) <-> ( A. y e. V ( x ( +g ` G ) y ) e. V /\ ( ( invg ` G ) ` x ) e. V ) ) )
60	59	ralbidv 2494	. . 3 |- ( A e. (SubRing ` R ) -> ( A. x e. V ( A. y e. V ( x ( .r ` R ) y ) e. V /\ ( ( invr ` R ) ` x ) e. V ) <-> A. x e. V ( A. y e. V ( x ( +g ` G ) y ) e. V /\ ( ( invg ` G ) ` x ) e. V ) ) )
61	36, 60	mpbid 147	. 2 |- ( A e. (SubRing ` R ) -> A. x e. V ( A. y e. V ( x ( +g ` G ) y ) e. V /\ ( ( invg ` G ) ` x ) e. V ) )
62	2, 7	unitgrp 13615	. . 3 |- ( R e. Ring -> G e. 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 13262	. . 3 |- ( G e. Grp -> ( V e. (SubGrp ` G ) <-> ( V C_ ( Base ` G ) /\ E. w w e. V /\ A. x e. V ( A. y e. V ( x ( +g ` G ) y ) e. V /\ ( ( invg ` G ) ` x ) e. V ) ) ) )
67	5, 62, 66	3syl 17	. 2 |- ( A e. (SubRing ` R ) -> ( V e. (SubGrp ` G ) <-> ( V C_ ( Base ` G ) /\ E. w w e. V /\ A. x e. V ( A. y e. V ( x ( +g ` G ) y ) e. V /\ ( ( invg ` G ) ` x ) e. V ) ) ) )
68	12, 17, 61, 67	mpbir3and 1182	1 |- ( A e. (SubRing ` R ) -> V e. (SubGrp ` G ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   /\ w3a 980   = wceq 1364   E.wex 1503   e. wcel 2164   A.wral 2472   _Vcvv 2760   C_ wss 3154   Fn wfn 5250   ` cfv 5255  (class class class)co 5919   Basecbs 12621   ↾_s cress 12622   +g cplusg 12698   .rcmulr 12699   Mndcmnd 13000   Grpcgrp 13075   invgcminusg 13076  SubGrpcsubg 13240  mulGrpcmgp 13419   1rcur 13458   Ringcrg 13495  Unitcui 13586   invrcinvr 13619  SubRingcsubrg 13716

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 4145  ax-sep 4148  ax-nul 4156  ax-pow 4204  ax-pr 4239  ax-un 4465  ax-setind 4570  ax-cnex 7965  ax-resscn 7966  ax-1cn 7967  ax-1re 7968  ax-icn 7969  ax-addcl 7970  ax-addrcl 7971  ax-mulcl 7972  ax-addcom 7974  ax-addass 7976  ax-i2m1 7979  ax-0lt1 7980  ax-0id 7982  ax-rnegex 7983  ax-pre-ltirr 7986  ax-pre-lttrn 7988  ax-pre-ltadd 7990

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 2987  df-csb 3082  df-dif 3156  df-un 3158  df-in 3160  df-ss 3167  df-nul 3448  df-pw 3604  df-sn 3625  df-pr 3626  df-op 3628  df-uni 3837  df-int 3872  df-iun 3915  df-br 4031  df-opab 4092  df-mpt 4093  df-id 4325  df-xp 4666  df-rel 4667  df-cnv 4668  df-co 4669  df-dm 4670  df-rn 4671  df-res 4672  df-ima 4673  df-iota 5216  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-riota 5874  df-ov 5922  df-oprab 5923  df-mpo 5924  df-tpos 6300  df-pnf 8058  df-mnf 8059  df-ltxr 8061  df-inn 8985  df-2 9043  df-3 9044  df-ndx 12624  df-slot 12625  df-base 12627  df-sets 12628  df-iress 12629  df-plusg 12711  df-mulr 12712  df-0g 12872  df-mgm 12942  df-sgrp 12988  df-mnd 13001  df-grp 13078  df-minusg 13079  df-subg 13243  df-cmn 13359  df-abl 13360  df-mgp 13420  df-ur 13459  df-srg 13463  df-ring 13497  df-oppr 13567  df-dvdsr 13588  df-unit 13589  df-invr 13620  df-subrg 13718

This theorem is referenced by: (None)