Theorem subrgugrp 18799

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 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		subrgugrp.1	. . 3 |- S = ( R ↾s A )
2		subrgugrp.2	. . 3 |- U = (Unit ` R )
3		subrgugrp.3	. . 3 |- V = (Unit ` S )
4	1, 2, 3	subrguss 18795	. 2 |- ( A e. (SubRing ` R ) -> V C_ U )
5	1	subrgring 18783	. . 3 |- ( A e. (SubRing ` R ) -> S e. Ring )
6		eqid 2622	. . . 4 |- ( 1r ` S ) = ( 1r ` S )
7	3, 6	1unit 18658	. . 3 |- ( S e. Ring -> ( 1r ` S ) e. V )
8		ne0i 3921	. . 3 |- ( ( 1r ` S ) e. V -> V =/= (/) )
9	5, 7, 8	3syl 18	. 2 |- ( A e. (SubRing ` R ) -> V =/= (/) )
10		eqid 2622	. . . . . . . . . 10 |- ( .r ` R ) = ( .r ` R )
11	1, 10	ressmulr 16006	. . . . . . . . 9 |- ( A e. (SubRing ` R ) -> ( .r ` R ) = ( .r ` S ) )
12	11	3ad2ant1 1082	. . . . . . . 8 |- ( ( A e. (SubRing ` R ) /\ x e. V /\ y e. V ) -> ( .r ` R ) = ( .r ` S ) )
13	12	oveqd 6667	. . . . . . 7 |- ( ( A e. (SubRing ` R ) /\ x e. V /\ y e. V ) -> ( x ( .r ` R ) y ) = ( x ( .r ` S ) y ) )
14		eqid 2622	. . . . . . . . 9 |- ( .r ` S ) = ( .r ` S )
15	3, 14	unitmulcl 18664	. . . . . . . 8 |- ( ( S e. Ring /\ x e. V /\ y e. V ) -> ( x ( .r ` S ) y ) e. V )
16	5, 15	syl3an1 1359	. . . . . . 7 |- ( ( A e. (SubRing ` R ) /\ x e. V /\ y e. V ) -> ( x ( .r ` S ) y ) e. V )
17	13, 16	eqeltrd 2701	. . . . . 6 |- ( ( A e. (SubRing ` R ) /\ x e. V /\ y e. V ) -> ( x ( .r ` R ) y ) e. V )
18	17	3expa 1265	. . . . 5 |- ( ( ( A e. (SubRing ` R ) /\ x e. V ) /\ y e. V ) -> ( x ( .r ` R ) y ) e. V )
19	18	ralrimiva 2966	. . . 4 |- ( ( A e. (SubRing ` R ) /\ x e. V ) -> A. y e. V ( x ( .r ` R ) y ) e. V )
20		eqid 2622	. . . . . 6 |- ( invr ` R ) = ( invr ` R )
21		eqid 2622	. . . . . 6 |- ( invr ` S ) = ( invr ` S )
22	1, 20, 3, 21	subrginv 18796	. . . . 5 |- ( ( A e. (SubRing ` R ) /\ x e. V ) -> ( ( invr ` R ) ` x ) = ( ( invr ` S ) ` x ) )
23	3, 21	unitinvcl 18674	. . . . . 6 |- ( ( S e. Ring /\ x e. V ) -> ( ( invr ` S ) ` x ) e. V )
24	5, 23	sylan 488	. . . . 5 |- ( ( A e. (SubRing ` R ) /\ x e. V ) -> ( ( invr ` S ) ` x ) e. V )
25	22, 24	eqeltrd 2701	. . . 4 |- ( ( A e. (SubRing ` R ) /\ x e. V ) -> ( ( invr ` R ) ` x ) e. V )
26	19, 25	jca 554	. . 3 |- ( ( A e. (SubRing ` R ) /\ x e. V ) -> ( A. y e. V ( x ( .r ` R ) y ) e. V /\ ( ( invr ` R ) ` x ) e. V ) )
27	26	ralrimiva 2966	. 2 |- ( A e. (SubRing ` R ) -> A. x e. V ( A. y e. V ( x ( .r ` R ) y ) e. V /\ ( ( invr ` R ) ` x ) e. V ) )
28		subrgrcl 18785	. . 3 |- ( A e. (SubRing ` R ) -> R e. Ring )
29		subrgugrp.4	. . . 4 |- G = ( (mulGrp ` R ) ↾s U )
30	2, 29	unitgrp 18667	. . 3 |- ( R e. Ring -> G e. Grp )
31	2, 29	unitgrpbas 18666	. . . 4 |- U = ( Base ` G )
32		fvex 6201	. . . . . 6 |- (Unit ` R ) e. _V
33	2, 32	eqeltri 2697	. . . . 5 |- U e. _V
34		eqid 2622	. . . . . . 7 |- (mulGrp ` R ) = (mulGrp ` R )
35	34, 10	mgpplusg 18493	. . . . . 6 |- ( .r ` R ) = ( +g ` (mulGrp ` R ) )
36	29, 35	ressplusg 15993	. . . . 5 |- ( U e. _V -> ( .r ` R ) = ( +g ` G ) )
37	33, 36	ax-mp 5	. . . 4 |- ( .r ` R ) = ( +g ` G )
38	2, 29, 20	invrfval 18673	. . . 4 |- ( invr ` R ) = ( invg ` G )
39	31, 37, 38	issubg2 17609	. . 3 |- ( G e. Grp -> ( V e. (SubGrp ` G ) <-> ( V C_ U /\ V =/= (/) /\ A. x e. V ( A. y e. V ( x ( .r ` R ) y ) e. V /\ ( ( invr ` R ) ` x ) e. V ) ) ) )
40	28, 30, 39	3syl 18	. 2 |- ( A e. (SubRing ` R ) -> ( V e. (SubGrp ` G ) <-> ( V C_ U /\ V =/= (/) /\ A. x e. V ( A. y e. V ( x ( .r ` R ) y ) e. V /\ ( ( invr ` R ) ` x ) e. V ) ) ) )
41	4, 9, 27, 40	mpbir3and 1245	1 |- ( A e. (SubRing ` R ) -> V e. (SubGrp ` G ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   =/= wne 2794   A.wral 2912   _Vcvv 3200   C_ wss 3574   (/)c0 3915   ` cfv 5888  (class class class)co 6650   ↾_s cress 15858   +g cplusg 15941   .rcmulr 15942   Grpcgrp 17422  SubGrpcsubg 17588  mulGrpcmgp 18489   1rcur 18501   Ringcrg 18547  Unitcui 18639   invrcinvr 18671  SubRingcsubrg 18776

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-tpos 7352  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-er 7742  df-en 7956  df-dom 7957  df-sdom 7958  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-nn 11021  df-2 11079  df-3 11080  df-ndx 15860  df-slot 15861  df-base 15863  df-sets 15864  df-ress 15865  df-plusg 15954  df-mulr 15955  df-0g 16102  df-mgm 17242  df-sgrp 17284  df-mnd 17295  df-grp 17425  df-minusg 17426  df-subg 17591  df-mgp 18490  df-ur 18502  df-ring 18549  df-oppr 18623  df-dvdsr 18641  df-unit 18642  df-invr 18672  df-subrg 18778

This theorem is referenced by: (None)