Theorem subrgugrp 14521

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 14517	. . 3 |- ( A e. (SubRing ` R ) -> V C_ U )
5		subrgrcl 14507	. . . 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 14325	. . . . 5 |- ( R e. Ring -> R e. SRing )
10	6, 8, 9	unitgrpbasd 14395	. . . 4 |- ( R e. Ring -> U = ( Base ` G ) )
11	5, 10	syl 14	. . 3 |- ( A e. (SubRing ` R ) -> U = ( Base ` G ) )
12	4, 11	sseqtrd 3286	. 2 |- ( A e. (SubRing ` R ) -> V C_ ( Base ` G ) )
13	1	subrgring 14505	. . 3 |- ( A e. (SubRing ` R ) -> S e. Ring )
14		eqid 2238	. . . 4 |- ( 1r ` S ) = ( 1r ` S )
15	3, 14	1unit 14387	. . 3 |- ( S e. Ring -> ( 1r ` S ) e. V )
16		elex2 2838	. . 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 2238	. . . . . . . . . . . 12 |- ( .r ` R ) = ( .r ` R )
19	1, 18	ressmulrg 13476	. . . . . . . . . . 11 |- ( ( A e. (SubRing ` R ) /\ R e. Ring ) -> ( .r ` R ) = ( .r ` S ) )
20	5, 19	mpdan 425	. . . . . . . . . 10 |- ( A e. (SubRing ` R ) -> ( .r ` R ) = ( .r ` S ) )
21	20	3ad2ant1 1049	. . . . . . . . 9 |- ( ( A e. (SubRing ` R ) /\ x e. V /\ y e. V ) -> ( .r ` R ) = ( .r ` S ) )
22	21	oveqd 6092	. . . . . . . 8 |- ( ( A e. (SubRing ` R ) /\ x e. V /\ y e. V ) -> ( x ( .r ` R ) y ) = ( x ( .r ` S ) y ) )
23		eqid 2238	. . . . . . . . . 10 |- ( .r ` S ) = ( .r ` S )
24	3, 23	unitmulcl 14393	. . . . . . . . 9 |- ( ( S e. Ring /\ x e. V /\ y e. V ) -> ( x ( .r ` S ) y ) e. V )
25	13, 24	syl3an1 1311	. . . . . . . 8 |- ( ( A e. (SubRing ` R ) /\ x e. V /\ y e. V ) -> ( x ( .r ` S ) y ) e. V )
26	22, 25	eqeltrd 2315	. . . . . . 7 |- ( ( A e. (SubRing ` R ) /\ x e. V /\ y e. V ) -> ( x ( .r ` R ) y ) e. V )
27	26	3expa 1234	. . . . . 6 |- ( ( ( A e. (SubRing ` R ) /\ x e. V ) /\ y e. V ) -> ( x ( .r ` R ) y ) e. V )
28	27	ralrimiva 2623	. . . . 5 |- ( ( A e. (SubRing ` R ) /\ x e. V ) -> A. y e. V ( x ( .r ` R ) y ) e. V )
29		eqid 2238	. . . . . . 7 |- ( invr ` R ) = ( invr ` R )
30		eqid 2238	. . . . . . 7 |- ( invr ` S ) = ( invr ` S )
31	1, 29, 3, 30	subrginv 14518	. . . . . 6 |- ( ( A e. (SubRing ` R ) /\ x e. V ) -> ( ( invr ` R ) ` x ) = ( ( invr ` S ) ` x ) )
32	3, 30	unitinvcl 14403	. . . . . . 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 2315	. . . . 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 2623	. . 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 2238	. . . . . . . . . . 11 |- (mulGrp ` R ) = (mulGrp ` R )
38	37, 18	mgpplusgg 14198	. . . . . . . . . 10 |- ( R e. Ring -> ( .r ` R ) = ( +g ` (mulGrp ` R ) ) )
39		basfn 13389	. . . . . . . . . . . 12 |- Base Fn _V
40		elex 2833	. . . . . . . . . . . 12 |- ( R e. Ring -> R e. _V )
41		funfvex 5707	. . . . . . . . . . . . 13 |- ( ( Fun Base /\ R e. dom Base ) -> ( Base ` R ) e. _V )
42	41	funfni 5478	. . . . . . . . . . . 12 |- ( ( Base Fn _V /\ R e. _V ) -> ( Base ` R ) e. _V )
43	39, 40, 42	sylancr 418	. . . . . . . . . . 11 |- ( R e. Ring -> ( Base ` R ) e. _V )
44		eqidd 2239	. . . . . . . . . . . 12 |- ( R e. Ring -> ( Base ` R ) = ( Base ` R ) )
45	44, 6, 9	unitssd 14389	. . . . . . . . . . 11 |- ( R e. Ring -> U C_ ( Base ` R ) )
46	43, 45	ssexd 4268	. . . . . . . . . 10 |- ( R e. Ring -> U e. _V )
47	37	ringmgp 14280	. . . . . . . . . 10 |- ( R e. Ring -> (mulGrp ` R ) e. Mnd )
48	8, 38, 46, 47	ressplusgd 13460	. . . . . . . . 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 6092	. . . . . . 7 |- ( A e. (SubRing ` R ) -> ( x ( .r ` R ) y ) = ( x ( +g ` G ) y ) )
51	50	eleq1d 2307	. . . . . 6 |- ( A e. (SubRing ` R ) -> ( ( x ( .r ` R ) y ) e. V <-> ( x ( +g ` G ) y ) e. V ) )
52	51	ralbidv 2550	. . . . 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 2239	. . . . . . . 8 |- ( A e. (SubRing ` R ) -> ( invr ` R ) = ( invr ` R ) )
56	53, 54, 55, 5	invrfvald 14402	. . . . . . 7 |- ( A e. (SubRing ` R ) -> ( invr ` R ) = ( invg ` G ) )
57	56	fveq1d 5692	. . . . . 6 |- ( A e. (SubRing ` R ) -> ( ( invr ` R ) ` x ) = ( ( invg ` G ) ` x ) )
58	57	eleq1d 2307	. . . . 5 |- ( A e. (SubRing ` R ) -> ( ( ( invr ` R ) ` x ) e. V <-> ( ( invg ` G ) ` x ) e. V ) )
59	52, 58	anbi12d 477	. . . 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 2550	. . 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 14396	. . 3 |- ( R e. Ring -> G e. Grp )
63		eqid 2238	. . . 4 |- ( Base ` G ) = ( Base ` G )
64		eqid 2238	. . . 4 |- ( +g ` G ) = ( +g ` G )
65		eqid 2238	. . . 4 |- ( invg ` G ) = ( invg ` G )
66	63, 64, 65	issubg2m 13969	. . 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 1211	1 |- ( A e. (SubRing ` R ) -> V e. (SubGrp ` G ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   /\ w3a 1009   = wceq 1402   E.wex 1545   e. wcel 2209   A.wral 2528   _Vcvv 2821   C_ wss 3220   Fn wfn 5367   ` cfv 5372  (class class class)co 6075   Basecbs 13330   ↾_s cress 13331   +g cplusg 13408   .rcmulr 13409   Mndcmnd 13706   Grpcgrp 13782   invgcminusg 13783  SubGrpcsubg 13947  mulGrpcmgp 14194   1rcur 14237   Ringcrg 14274  Unitcui 14366   invrcinvr 14400  SubRingcsubrg 14498

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 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-coll 4241  ax-sep 4244  ax-nul 4254  ax-pow 4306  ax-pr 4341  ax-un 4573  ax-setind 4679  ax-cnex 8260  ax-resscn 8261  ax-1cn 8262  ax-1re 8263  ax-icn 8264  ax-addcl 8265  ax-addrcl 8266  ax-mulcl 8267  ax-addcom 8269  ax-addass 8271  ax-i2m1 8274  ax-0lt1 8275  ax-0id 8277  ax-rnegex 8278  ax-pre-ltirr 8281  ax-pre-lttrn 8283  ax-pre-ltadd 8285

This theorem depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-nel 2516  df-ral 2533  df-rex 2534  df-reu 2535  df-rmo 2536  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-pw 3687  df-sn 3711  df-pr 3712  df-op 3714  df-uni 3931  df-int 3966  df-iun 4009  df-br 4126  df-opab 4188  df-mpt 4189  df-id 4433  df-xp 4775  df-rel 4776  df-cnv 4777  df-co 4778  df-dm 4779  df-rn 4780  df-res 4781  df-ima 4782  df-iota 5332  df-fun 5374  df-fn 5375  df-f 5376  df-f1 5377  df-fo 5378  df-f1o 5379  df-fv 5380  df-riota 6028  df-ov 6078  df-oprab 6079  df-mpo 6080  df-tpos 6506  df-pnf 8352  df-mnf 8353  df-ltxr 8355  df-inn 9284  df-2 9342  df-3 9343  df-ndx 13333  df-slot 13334  df-base 13336  df-sets 13337  df-iress 13338  df-plusg 13421  df-mulr 13422  df-0g 13589  df-mgm 13653  df-sgrp 13694  df-mnd 13707  df-grp 13785  df-minusg 13786  df-subg 13950  df-cmn 14066  df-abl 14067  df-mgp 14195  df-ur 14238  df-srg 14242  df-ring 14276  df-oppr 14346  df-dvdsr 14368  df-unit 14369  df-invr 14401  df-subrg 14500

This theorem is referenced by: (None)