Theorem subrguss 13870

Description: A unit of a subring is a unit of the parent ring. (Contributed by Mario Carneiro, 4-Dec-2014.)

Hypotheses

Ref	Expression
subrguss.1	|- S = ( R ↾s A )
subrguss.2	|- U = (Unit ` R )
subrguss.3	|- V = (Unit ` S )

Assertion

Ref	Expression
subrguss	|- ( A e. (SubRing ` R ) -> V C_ U )

Proof of Theorem subrguss

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		subrguss.3	. . . . . . . . 9 |- V = (Unit ` S )
2	1	a1i 9	. . . . . . . 8 |- ( A e. (SubRing ` R ) -> V = (Unit ` S ) )
3		eqidd 2197	. . . . . . . 8 |- ( A e. (SubRing ` R ) -> ( 1r ` S ) = ( 1r ` S ) )
4		eqidd 2197	. . . . . . . 8 |- ( A e. (SubRing ` R ) -> ( ||r ` S ) = ( ||r ` S ) )
5		eqidd 2197	. . . . . . . 8 |- ( A e. (SubRing ` R ) -> (oppr ` S ) = (oppr ` S ) )
6		eqidd 2197	. . . . . . . 8 |- ( A e. (SubRing ` R ) -> ( ||r ` (oppr ` S ) ) = ( ||r ` (oppr ` S ) ) )
7		subrguss.1	. . . . . . . . . 10 |- S = ( R ↾s A )
8	7	subrgring 13858	. . . . . . . . 9 |- ( A e. (SubRing ` R ) -> S e. Ring )
9		ringsrg 13681	. . . . . . . . 9 |- ( S e. Ring -> S e. SRing )
10	8, 9	syl 14	. . . . . . . 8 |- ( A e. (SubRing ` R ) -> S e. SRing )
11	2, 3, 4, 5, 6, 10	isunitd 13740	. . . . . . 7 |- ( A e. (SubRing ` R ) -> ( x e. V <-> ( x ( ||r ` S ) ( 1r ` S ) /\ x ( ||r ` (oppr ` S ) ) ( 1r ` S ) ) ) )
12	11	simprbda 383	. . . . . 6 |- ( ( A e. (SubRing ` R ) /\ x e. V ) -> x ( ||r ` S ) ( 1r ` S ) )
13		eqid 2196	. . . . . . . 8 |- ( 1r ` R ) = ( 1r ` R )
14	7, 13	subrg1 13865	. . . . . . 7 |- ( A e. (SubRing ` R ) -> ( 1r ` R ) = ( 1r ` S ) )
15	14	adantr 276	. . . . . 6 |- ( ( A e. (SubRing ` R ) /\ x e. V ) -> ( 1r ` R ) = ( 1r ` S ) )
16	12, 15	breqtrrd 4062	. . . . 5 |- ( ( A e. (SubRing ` R ) /\ x e. V ) -> x ( ||r ` S ) ( 1r ` R ) )
17		eqid 2196	. . . . . . . 8 |- ( ||r ` R ) = ( ||r ` R )
18		eqid 2196	. . . . . . . 8 |- ( ||r ` S ) = ( ||r ` S )
19	7, 17, 18	subrgdvds 13869	. . . . . . 7 |- ( A e. (SubRing ` R ) -> ( ||r ` S ) C_ ( ||r ` R ) )
20	19	adantr 276	. . . . . 6 |- ( ( A e. (SubRing ` R ) /\ x e. V ) -> ( ||r ` S ) C_ ( ||r ` R ) )
21	20	ssbrd 4077	. . . . 5 |- ( ( A e. (SubRing ` R ) /\ x e. V ) -> ( x ( ||r ` S ) ( 1r ` R ) -> x ( ||r ` R ) ( 1r ` R ) ) )
22	16, 21	mpd 13	. . . 4 |- ( ( A e. (SubRing ` R ) /\ x e. V ) -> x ( ||r ` R ) ( 1r ` R ) )
23		subrgrcl 13860	. . . . . . . 8 |- ( A e. (SubRing ` R ) -> R e. Ring )
24	23	adantr 276	. . . . . . 7 |- ( ( A e. (SubRing ` R ) /\ x e. V ) -> R e. Ring )
25		eqid 2196	. . . . . . . 8 |- (oppr ` R ) = (oppr ` R )
26		eqid 2196	. . . . . . . 8 |- ( Base ` R ) = ( Base ` R )
27	25, 26	opprbasg 13709	. . . . . . 7 |- ( R e. Ring -> ( Base ` R ) = ( Base ` (oppr ` R ) ) )
28	24, 27	syl 14	. . . . . 6 |- ( ( A e. (SubRing ` R ) /\ x e. V ) -> ( Base ` R ) = ( Base ` (oppr ` R ) ) )
29		eqidd 2197	. . . . . 6 |- ( ( A e. (SubRing ` R ) /\ x e. V ) -> ( ||r ` (oppr ` R ) ) = ( ||r ` (oppr ` R ) ) )
30	25	opprring 13713	. . . . . . 7 |- ( R e. Ring -> (oppr ` R ) e. Ring )
31		ringsrg 13681	. . . . . . 7 |- ( (oppr ` R ) e. Ring -> (oppr ` R ) e. SRing )
32	24, 30, 31	3syl 17	. . . . . 6 |- ( ( A e. (SubRing ` R ) /\ x e. V ) -> (oppr ` R ) e. SRing )
33		eqidd 2197	. . . . . 6 |- ( ( A e. (SubRing ` R ) /\ x e. V ) -> ( .r ` (oppr ` R ) ) = ( .r ` (oppr ` R ) ) )
34	7	subrgbas 13864	. . . . . . . . 9 |- ( A e. (SubRing ` R ) -> A = ( Base ` S ) )
35	34	adantr 276	. . . . . . . 8 |- ( ( A e. (SubRing ` R ) /\ x e. V ) -> A = ( Base ` S ) )
36	26	subrgss 13856	. . . . . . . . 9 |- ( A e. (SubRing ` R ) -> A C_ ( Base ` R ) )
37	36	adantr 276	. . . . . . . 8 |- ( ( A e. (SubRing ` R ) /\ x e. V ) -> A C_ ( Base ` R ) )
38	35, 37	eqsstrrd 3221	. . . . . . 7 |- ( ( A e. (SubRing ` R ) /\ x e. V ) -> ( Base ` S ) C_ ( Base ` R ) )
39		eqidd 2197	. . . . . . . 8 |- ( ( A e. (SubRing ` R ) /\ x e. V ) -> ( Base ` S ) = ( Base ` S ) )
40	1	a1i 9	. . . . . . . 8 |- ( ( A e. (SubRing ` R ) /\ x e. V ) -> V = (Unit ` S ) )
41	10	adantr 276	. . . . . . . 8 |- ( ( A e. (SubRing ` R ) /\ x e. V ) -> S e. SRing )
42		simpr 110	. . . . . . . 8 |- ( ( A e. (SubRing ` R ) /\ x e. V ) -> x e. V )
43	39, 40, 41, 42	unitcld 13742	. . . . . . 7 |- ( ( A e. (SubRing ` R ) /\ x e. V ) -> x e. ( Base ` S ) )
44	38, 43	sseldd 3185	. . . . . 6 |- ( ( A e. (SubRing ` R ) /\ x e. V ) -> x e. ( Base ` R ) )
45		eqid 2196	. . . . . . . . 9 |- ( invr ` S ) = ( invr ` S )
46		eqid 2196	. . . . . . . . 9 |- ( Base ` S ) = ( Base ` S )
47	1, 45, 46	ringinvcl 13759	. . . . . . . 8 |- ( ( S e. Ring /\ x e. V ) -> ( ( invr ` S ) ` x ) e. ( Base ` S ) )
48	8, 47	sylan 283	. . . . . . 7 |- ( ( A e. (SubRing ` R ) /\ x e. V ) -> ( ( invr ` S ) ` x ) e. ( Base ` S ) )
49	38, 48	sseldd 3185	. . . . . 6 |- ( ( A e. (SubRing ` R ) /\ x e. V ) -> ( ( invr ` S ) ` x ) e. ( Base ` R ) )
50	28, 29, 32, 33, 44, 49	dvdsrmuld 13730	. . . . 5 |- ( ( A e. (SubRing ` R ) /\ x e. V ) -> x ( ||r ` (oppr ` R ) ) ( ( ( invr ` S ) ` x ) ( .r ` (oppr ` R ) ) x ) )
51	1, 45	unitinvcl 13757	. . . . . . . 8 |- ( ( S e. Ring /\ x e. V ) -> ( ( invr ` S ) ` x ) e. V )
52	8, 51	sylan 283	. . . . . . 7 |- ( ( A e. (SubRing ` R ) /\ x e. V ) -> ( ( invr ` S ) ` x ) e. V )
53		eqid 2196	. . . . . . . 8 |- ( .r ` R ) = ( .r ` R )
54		eqid 2196	. . . . . . . 8 |- ( .r ` (oppr ` R ) ) = ( .r ` (oppr ` R ) )
55	26, 53, 25, 54	opprmulg 13705	. . . . . . 7 |- ( ( R e. Ring /\ ( ( invr ` S ) ` x ) e. V /\ x e. V ) -> ( ( ( invr ` S ) ` x ) ( .r ` (oppr ` R ) ) x ) = ( x ( .r ` R ) ( ( invr ` S ) ` x ) ) )
56	24, 52, 42, 55	syl3anc 1249	. . . . . 6 |- ( ( A e. (SubRing ` R ) /\ x e. V ) -> ( ( ( invr ` S ) ` x ) ( .r ` (oppr ` R ) ) x ) = ( x ( .r ` R ) ( ( invr ` S ) ` x ) ) )
57		eqid 2196	. . . . . . . . 9 |- ( .r ` S ) = ( .r ` S )
58		eqid 2196	. . . . . . . . 9 |- ( 1r ` S ) = ( 1r ` S )
59	1, 45, 57, 58	unitrinv 13761	. . . . . . . 8 |- ( ( S e. Ring /\ x e. V ) -> ( x ( .r ` S ) ( ( invr ` S ) ` x ) ) = ( 1r ` S ) )
60	8, 59	sylan 283	. . . . . . 7 |- ( ( A e. (SubRing ` R ) /\ x e. V ) -> ( x ( .r ` S ) ( ( invr ` S ) ` x ) ) = ( 1r ` S ) )
61	7, 53	ressmulrg 12849	. . . . . . . . . 10 |- ( ( A e. (SubRing ` R ) /\ R e. Ring ) -> ( .r ` R ) = ( .r ` S ) )
62	23, 61	mpdan 421	. . . . . . . . 9 |- ( A e. (SubRing ` R ) -> ( .r ` R ) = ( .r ` S ) )
63	62	adantr 276	. . . . . . . 8 |- ( ( A e. (SubRing ` R ) /\ x e. V ) -> ( .r ` R ) = ( .r ` S ) )
64	63	oveqd 5942	. . . . . . 7 |- ( ( A e. (SubRing ` R ) /\ x e. V ) -> ( x ( .r ` R ) ( ( invr ` S ) ` x ) ) = ( x ( .r ` S ) ( ( invr ` S ) ` x ) ) )
65	60, 64, 15	3eqtr4d 2239	. . . . . 6 |- ( ( A e. (SubRing ` R ) /\ x e. V ) -> ( x ( .r ` R ) ( ( invr ` S ) ` x ) ) = ( 1r ` R ) )
66	56, 65	eqtrd 2229	. . . . 5 |- ( ( A e. (SubRing ` R ) /\ x e. V ) -> ( ( ( invr ` S ) ` x ) ( .r ` (oppr ` R ) ) x ) = ( 1r ` R ) )
67	50, 66	breqtrd 4060	. . . 4 |- ( ( A e. (SubRing ` R ) /\ x e. V ) -> x ( ||r ` (oppr ` R ) ) ( 1r ` R ) )
68		subrguss.2	. . . . . . 7 |- U = (Unit ` R )
69	68	a1i 9	. . . . . 6 |- ( A e. (SubRing ` R ) -> U = (Unit ` R ) )
70		eqidd 2197	. . . . . 6 |- ( A e. (SubRing ` R ) -> ( 1r ` R ) = ( 1r ` R ) )
71		eqidd 2197	. . . . . 6 |- ( A e. (SubRing ` R ) -> ( ||r ` R ) = ( ||r ` R ) )
72		eqidd 2197	. . . . . 6 |- ( A e. (SubRing ` R ) -> (oppr ` R ) = (oppr ` R ) )
73		eqidd 2197	. . . . . 6 |- ( A e. (SubRing ` R ) -> ( ||r ` (oppr ` R ) ) = ( ||r ` (oppr ` R ) ) )
74		ringsrg 13681	. . . . . . 7 |- ( R e. Ring -> R e. SRing )
75	23, 74	syl 14	. . . . . 6 |- ( A e. (SubRing ` R ) -> R e. SRing )
76	69, 70, 71, 72, 73, 75	isunitd 13740	. . . . 5 |- ( A e. (SubRing ` R ) -> ( x e. U <-> ( x ( ||r ` R ) ( 1r ` R ) /\ x ( ||r ` (oppr ` R ) ) ( 1r ` R ) ) ) )
77	76	adantr 276	. . . 4 |- ( ( A e. (SubRing ` R ) /\ x e. V ) -> ( x e. U <-> ( x ( ||r ` R ) ( 1r ` R ) /\ x ( ||r ` (oppr ` R ) ) ( 1r ` R ) ) ) )
78	22, 67, 77	mpbir2and 946	. . 3 |- ( ( A e. (SubRing ` R ) /\ x e. V ) -> x e. U )
79	78	ex 115	. 2 |- ( A e. (SubRing ` R ) -> ( x e. V -> x e. U ) )
80	79	ssrdv 3190	1 |- ( A e. (SubRing ` R ) -> V C_ U )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1364   e. wcel 2167   C_ wss 3157   class class class wbr 4034   ` cfv 5259  (class class class)co 5925   Basecbs 12705   ↾_s cress 12706   .rcmulr 12783   1rcur 13593  SRingcsrg 13597   Ringcrg 13630  opp_rcoppr 13701   ||_rcdsr 13720  Unitcui 13721   invrcinvr 13754  SubRingcsubrg 13851

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 4149  ax-sep 4152  ax-nul 4160  ax-pow 4208  ax-pr 4243  ax-un 4469  ax-setind 4574  ax-cnex 7989  ax-resscn 7990  ax-1cn 7991  ax-1re 7992  ax-icn 7993  ax-addcl 7994  ax-addrcl 7995  ax-mulcl 7996  ax-addcom 7998  ax-addass 8000  ax-i2m1 8003  ax-0lt1 8004  ax-0id 8006  ax-rnegex 8007  ax-pre-ltirr 8010  ax-pre-lttrn 8012  ax-pre-ltadd 8014

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 3452  df-pw 3608  df-sn 3629  df-pr 3630  df-op 3632  df-uni 3841  df-int 3876  df-iun 3919  df-br 4035  df-opab 4096  df-mpt 4097  df-id 4329  df-xp 4670  df-rel 4671  df-cnv 4672  df-co 4673  df-dm 4674  df-rn 4675  df-res 4676  df-ima 4677  df-iota 5220  df-fun 5261  df-fn 5262  df-f 5263  df-f1 5264  df-fo 5265  df-f1o 5266  df-fv 5267  df-riota 5880  df-ov 5928  df-oprab 5929  df-mpo 5930  df-tpos 6312  df-pnf 8082  df-mnf 8083  df-ltxr 8085  df-inn 9010  df-2 9068  df-3 9069  df-ndx 12708  df-slot 12709  df-base 12711  df-sets 12712  df-iress 12713  df-plusg 12795  df-mulr 12796  df-0g 12962  df-mgm 13060  df-sgrp 13106  df-mnd 13121  df-grp 13207  df-minusg 13208  df-subg 13378  df-cmn 13494  df-abl 13495  df-mgp 13555  df-ur 13594  df-srg 13598  df-ring 13632  df-oppr 13702  df-dvdsr 13723  df-unit 13724  df-invr 13755  df-subrg 13853

This theorem is referenced by:  subrginv  13871  subrgdv  13872  subrgunit  13873  subrgugrp  13874