Theorem subrgugrp 14002

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 13998	. . 3 |- ( A e. (SubRing ` R ) -> V C_ U )
5		subrgrcl 13988	. . . 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 13809	. . . . 5 |- ( R e. Ring -> R e. SRing )
10	6, 8, 9	unitgrpbasd 13877	. . . 4 |- ( R e. Ring -> U = ( Base ` G ) )
11	5, 10	syl 14	. . 3 |- ( A e. (SubRing ` R ) -> U = ( Base ` G ) )
12	4, 11	sseqtrd 3231	. 2 |- ( A e. (SubRing ` R ) -> V C_ ( Base ` G ) )
13	1	subrgring 13986	. . 3 |- ( A e. (SubRing ` R ) -> S e. Ring )
14		eqid 2205	. . . 4 |- ( 1r ` S ) = ( 1r ` S )
15	3, 14	1unit 13869	. . 3 |- ( S e. Ring -> ( 1r ` S ) e. V )
16		elex2 2788	. . 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 2205	. . . . . . . . . . . 12 |- ( .r ` R ) = ( .r ` R )
19	1, 18	ressmulrg 12977	. . . . . . . . . . 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 1021	. . . . . . . . 9 |- ( ( A e. (SubRing ` R ) /\ x e. V /\ y e. V ) -> ( .r ` R ) = ( .r ` S ) )
22	21	oveqd 5961	. . . . . . . 8 |- ( ( A e. (SubRing ` R ) /\ x e. V /\ y e. V ) -> ( x ( .r ` R ) y ) = ( x ( .r ` S ) y ) )
23		eqid 2205	. . . . . . . . . 10 |- ( .r ` S ) = ( .r ` S )
24	3, 23	unitmulcl 13875	. . . . . . . . 9 |- ( ( S e. Ring /\ x e. V /\ y e. V ) -> ( x ( .r ` S ) y ) e. V )
25	13, 24	syl3an1 1283	. . . . . . . 8 |- ( ( A e. (SubRing ` R ) /\ x e. V /\ y e. V ) -> ( x ( .r ` S ) y ) e. V )
26	22, 25	eqeltrd 2282	. . . . . . 7 |- ( ( A e. (SubRing ` R ) /\ x e. V /\ y e. V ) -> ( x ( .r ` R ) y ) e. V )
27	26	3expa 1206	. . . . . 6 |- ( ( ( A e. (SubRing ` R ) /\ x e. V ) /\ y e. V ) -> ( x ( .r ` R ) y ) e. V )
28	27	ralrimiva 2579	. . . . 5 |- ( ( A e. (SubRing ` R ) /\ x e. V ) -> A. y e. V ( x ( .r ` R ) y ) e. V )
29		eqid 2205	. . . . . . 7 |- ( invr ` R ) = ( invr ` R )
30		eqid 2205	. . . . . . 7 |- ( invr ` S ) = ( invr ` S )
31	1, 29, 3, 30	subrginv 13999	. . . . . 6 |- ( ( A e. (SubRing ` R ) /\ x e. V ) -> ( ( invr ` R ) ` x ) = ( ( invr ` S ) ` x ) )
32	3, 30	unitinvcl 13885	. . . . . . 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 2282	. . . . 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 2579	. . 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 2205	. . . . . . . . . . 11 |- (mulGrp ` R ) = (mulGrp ` R )
38	37, 18	mgpplusgg 13686	. . . . . . . . . 10 |- ( R e. Ring -> ( .r ` R ) = ( +g ` (mulGrp ` R ) ) )
39		basfn 12890	. . . . . . . . . . . 12 |- Base Fn _V
40		elex 2783	. . . . . . . . . . . 12 |- ( R e. Ring -> R e. _V )
41		funfvex 5593	. . . . . . . . . . . . 13 |- ( ( Fun Base /\ R e. dom Base ) -> ( Base ` R ) e. _V )
42	41	funfni 5376	. . . . . . . . . . . 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 2206	. . . . . . . . . . . 12 |- ( R e. Ring -> ( Base ` R ) = ( Base ` R ) )
45	44, 6, 9	unitssd 13871	. . . . . . . . . . 11 |- ( R e. Ring -> U C_ ( Base ` R ) )
46	43, 45	ssexd 4184	. . . . . . . . . 10 |- ( R e. Ring -> U e. _V )
47	37	ringmgp 13764	. . . . . . . . . 10 |- ( R e. Ring -> (mulGrp ` R ) e. Mnd )
48	8, 38, 46, 47	ressplusgd 12961	. . . . . . . . 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 5961	. . . . . . 7 |- ( A e. (SubRing ` R ) -> ( x ( .r ` R ) y ) = ( x ( +g ` G ) y ) )
51	50	eleq1d 2274	. . . . . 6 |- ( A e. (SubRing ` R ) -> ( ( x ( .r ` R ) y ) e. V <-> ( x ( +g ` G ) y ) e. V ) )
52	51	ralbidv 2506	. . . . 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 2206	. . . . . . . 8 |- ( A e. (SubRing ` R ) -> ( invr ` R ) = ( invr ` R ) )
56	53, 54, 55, 5	invrfvald 13884	. . . . . . 7 |- ( A e. (SubRing ` R ) -> ( invr ` R ) = ( invg ` G ) )
57	56	fveq1d 5578	. . . . . 6 |- ( A e. (SubRing ` R ) -> ( ( invr ` R ) ` x ) = ( ( invg ` G ) ` x ) )
58	57	eleq1d 2274	. . . . 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 2506	. . 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 13878	. . 3 |- ( R e. Ring -> G e. Grp )
63		eqid 2205	. . . 4 |- ( Base ` G ) = ( Base ` G )
64		eqid 2205	. . . 4 |- ( +g ` G ) = ( +g ` G )
65		eqid 2205	. . . 4 |- ( invg ` G ) = ( invg ` G )
66	63, 64, 65	issubg2m 13525	. . 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 1183	1 |- ( A e. (SubRing ` R ) -> V e. (SubGrp ` G ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   /\ w3a 981   = wceq 1373   E.wex 1515   e. wcel 2176   A.wral 2484   _Vcvv 2772   C_ wss 3166   Fn wfn 5266   ` cfv 5271  (class class class)co 5944   Basecbs 12832   ↾_s cress 12833   +g cplusg 12909   .rcmulr 12910   Mndcmnd 13248   Grpcgrp 13332   invgcminusg 13333  SubGrpcsubg 13503  mulGrpcmgp 13682   1rcur 13721   Ringcrg 13758  Unitcui 13849   invrcinvr 13882  SubRingcsubrg 13979

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 711  ax-5 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-13 2178  ax-14 2179  ax-ext 2187  ax-coll 4159  ax-sep 4162  ax-nul 4170  ax-pow 4218  ax-pr 4253  ax-un 4480  ax-setind 4585  ax-cnex 8016  ax-resscn 8017  ax-1cn 8018  ax-1re 8019  ax-icn 8020  ax-addcl 8021  ax-addrcl 8022  ax-mulcl 8023  ax-addcom 8025  ax-addass 8027  ax-i2m1 8030  ax-0lt1 8031  ax-0id 8033  ax-rnegex 8034  ax-pre-ltirr 8037  ax-pre-lttrn 8039  ax-pre-ltadd 8041

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-fal 1379  df-nf 1484  df-sb 1786  df-eu 2057  df-mo 2058  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-ne 2377  df-nel 2472  df-ral 2489  df-rex 2490  df-reu 2491  df-rmo 2492  df-rab 2493  df-v 2774  df-sbc 2999  df-csb 3094  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3461  df-pw 3618  df-sn 3639  df-pr 3640  df-op 3642  df-uni 3851  df-int 3886  df-iun 3929  df-br 4045  df-opab 4106  df-mpt 4107  df-id 4340  df-xp 4681  df-rel 4682  df-cnv 4683  df-co 4684  df-dm 4685  df-rn 4686  df-res 4687  df-ima 4688  df-iota 5232  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-riota 5899  df-ov 5947  df-oprab 5948  df-mpo 5949  df-tpos 6331  df-pnf 8109  df-mnf 8110  df-ltxr 8112  df-inn 9037  df-2 9095  df-3 9096  df-ndx 12835  df-slot 12836  df-base 12838  df-sets 12839  df-iress 12840  df-plusg 12922  df-mulr 12923  df-0g 13090  df-mgm 13188  df-sgrp 13234  df-mnd 13249  df-grp 13335  df-minusg 13336  df-subg 13506  df-cmn 13622  df-abl 13623  df-mgp 13683  df-ur 13722  df-srg 13726  df-ring 13760  df-oppr 13830  df-dvdsr 13851  df-unit 13852  df-invr 13883  df-subrg 13981

This theorem is referenced by: (None)