Theorem subrgugrp 13796

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 13792	. . 3 |- ( A e. (SubRing ` R ) -> V C_ U )
5		subrgrcl 13782	. . . 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 13603	. . . . 5 |- ( R e. Ring -> R e. SRing )
10	6, 8, 9	unitgrpbasd 13671	. . . 4 |- ( R e. Ring -> U = ( Base ` G ) )
11	5, 10	syl 14	. . 3 |- ( A e. (SubRing ` R ) -> U = ( Base ` G ) )
12	4, 11	sseqtrd 3221	. 2 |- ( A e. (SubRing ` R ) -> V C_ ( Base ` G ) )
13	1	subrgring 13780	. . 3 |- ( A e. (SubRing ` R ) -> S e. Ring )
14		eqid 2196	. . . 4 |- ( 1r ` S ) = ( 1r ` S )
15	3, 14	1unit 13663	. . 3 |- ( S e. Ring -> ( 1r ` S ) e. V )
16		elex2 2779	. . 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 2196	. . . . . . . . . . . 12 |- ( .r ` R ) = ( .r ` R )
19	1, 18	ressmulrg 12822	. . . . . . . . . . 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 5939	. . . . . . . 8 |- ( ( A e. (SubRing ` R ) /\ x e. V /\ y e. V ) -> ( x ( .r ` R ) y ) = ( x ( .r ` S ) y ) )
23		eqid 2196	. . . . . . . . . 10 |- ( .r ` S ) = ( .r ` S )
24	3, 23	unitmulcl 13669	. . . . . . . . 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 2273	. . . . . . 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 2570	. . . . 5 |- ( ( A e. (SubRing ` R ) /\ x e. V ) -> A. y e. V ( x ( .r ` R ) y ) e. V )
29		eqid 2196	. . . . . . 7 |- ( invr ` R ) = ( invr ` R )
30		eqid 2196	. . . . . . 7 |- ( invr ` S ) = ( invr ` S )
31	1, 29, 3, 30	subrginv 13793	. . . . . 6 |- ( ( A e. (SubRing ` R ) /\ x e. V ) -> ( ( invr ` R ) ` x ) = ( ( invr ` S ) ` x ) )
32	3, 30	unitinvcl 13679	. . . . . . 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 2273	. . . . 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 2570	. . 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 2196	. . . . . . . . . . 11 |- (mulGrp ` R ) = (mulGrp ` R )
38	37, 18	mgpplusgg 13480	. . . . . . . . . 10 |- ( R e. Ring -> ( .r ` R ) = ( +g ` (mulGrp ` R ) ) )
39		basfn 12736	. . . . . . . . . . . 12 |- Base Fn _V
40		elex 2774	. . . . . . . . . . . 12 |- ( R e. Ring -> R e. _V )
41		funfvex 5575	. . . . . . . . . . . . 13 |- ( ( Fun Base /\ R e. dom Base ) -> ( Base ` R ) e. _V )
42	41	funfni 5358	. . . . . . . . . . . 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 2197	. . . . . . . . . . . 12 |- ( R e. Ring -> ( Base ` R ) = ( Base ` R ) )
45	44, 6, 9	unitssd 13665	. . . . . . . . . . 11 |- ( R e. Ring -> U C_ ( Base ` R ) )
46	43, 45	ssexd 4173	. . . . . . . . . 10 |- ( R e. Ring -> U e. _V )
47	37	ringmgp 13558	. . . . . . . . . 10 |- ( R e. Ring -> (mulGrp ` R ) e. Mnd )
48	8, 38, 46, 47	ressplusgd 12806	. . . . . . . . 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 5939	. . . . . . 7 |- ( A e. (SubRing ` R ) -> ( x ( .r ` R ) y ) = ( x ( +g ` G ) y ) )
51	50	eleq1d 2265	. . . . . 6 |- ( A e. (SubRing ` R ) -> ( ( x ( .r ` R ) y ) e. V <-> ( x ( +g ` G ) y ) e. V ) )
52	51	ralbidv 2497	. . . . 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 2197	. . . . . . . 8 |- ( A e. (SubRing ` R ) -> ( invr ` R ) = ( invr ` R ) )
56	53, 54, 55, 5	invrfvald 13678	. . . . . . 7 |- ( A e. (SubRing ` R ) -> ( invr ` R ) = ( invg ` G ) )
57	56	fveq1d 5560	. . . . . 6 |- ( A e. (SubRing ` R ) -> ( ( invr ` R ) ` x ) = ( ( invg ` G ) ` x ) )
58	57	eleq1d 2265	. . . . 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 2497	. . 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 13672	. . 3 |- ( R e. Ring -> G e. Grp )
63		eqid 2196	. . . 4 |- ( Base ` G ) = ( Base ` G )
64		eqid 2196	. . . 4 |- ( +g ` G ) = ( +g ` G )
65		eqid 2196	. . . 4 |- ( invg ` G ) = ( invg ` G )
66	63, 64, 65	issubg2m 13319	. . 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 1506   e. wcel 2167   A.wral 2475   _Vcvv 2763   C_ wss 3157   Fn wfn 5253   ` cfv 5258  (class class class)co 5922   Basecbs 12678   ↾_s cress 12679   +g cplusg 12755   .rcmulr 12756   Mndcmnd 13057   Grpcgrp 13132   invgcminusg 13133  SubGrpcsubg 13297  mulGrpcmgp 13476   1rcur 13515   Ringcrg 13552  Unitcui 13643   invrcinvr 13676  SubRingcsubrg 13773

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 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-13 2169  ax-14 2170  ax-ext 2178  ax-coll 4148  ax-sep 4151  ax-nul 4159  ax-pow 4207  ax-pr 4242  ax-un 4468  ax-setind 4573  ax-cnex 7970  ax-resscn 7971  ax-1cn 7972  ax-1re 7973  ax-icn 7974  ax-addcl 7975  ax-addrcl 7976  ax-mulcl 7977  ax-addcom 7979  ax-addass 7981  ax-i2m1 7984  ax-0lt1 7985  ax-0id 7987  ax-rnegex 7988  ax-pre-ltirr 7991  ax-pre-lttrn 7993  ax-pre-ltadd 7995

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ne 2368  df-nel 2463  df-ral 2480  df-rex 2481  df-reu 2482  df-rmo 2483  df-rab 2484  df-v 2765  df-sbc 2990  df-csb 3085  df-dif 3159  df-un 3161  df-in 3163  df-ss 3170  df-nul 3451  df-pw 3607  df-sn 3628  df-pr 3629  df-op 3631  df-uni 3840  df-int 3875  df-iun 3918  df-br 4034  df-opab 4095  df-mpt 4096  df-id 4328  df-xp 4669  df-rel 4670  df-cnv 4671  df-co 4672  df-dm 4673  df-rn 4674  df-res 4675  df-ima 4676  df-iota 5219  df-fun 5260  df-fn 5261  df-f 5262  df-f1 5263  df-fo 5264  df-f1o 5265  df-fv 5266  df-riota 5877  df-ov 5925  df-oprab 5926  df-mpo 5927  df-tpos 6303  df-pnf 8063  df-mnf 8064  df-ltxr 8066  df-inn 8991  df-2 9049  df-3 9050  df-ndx 12681  df-slot 12682  df-base 12684  df-sets 12685  df-iress 12686  df-plusg 12768  df-mulr 12769  df-0g 12929  df-mgm 12999  df-sgrp 13045  df-mnd 13058  df-grp 13135  df-minusg 13136  df-subg 13300  df-cmn 13416  df-abl 13417  df-mgp 13477  df-ur 13516  df-srg 13520  df-ring 13554  df-oppr 13624  df-dvdsr 13645  df-unit 13646  df-invr 13677  df-subrg 13775

This theorem is referenced by: (None)