Theorem subrguss 13317

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 2178	. . . . . . . 8 |- ( A e. (SubRing ` R ) -> ( 1r ` S ) = ( 1r ` S ) )
4		eqidd 2178	. . . . . . . 8 |- ( A e. (SubRing ` R ) -> ( ||r ` S ) = ( ||r ` S ) )
5		eqidd 2178	. . . . . . . 8 |- ( A e. (SubRing ` R ) -> (oppr ` S ) = (oppr ` S ) )
6		eqidd 2178	. . . . . . . 8 |- ( A e. (SubRing ` R ) -> ( ||r ` (oppr ` S ) ) = ( ||r ` (oppr ` S ) ) )
7		subrguss.1	. . . . . . . . . 10 |- S = ( R ↾s A )
8	7	subrgring 13305	. . . . . . . . 9 |- ( A e. (SubRing ` R ) -> S e. Ring )
9		ringsrg 13177	. . . . . . . . 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 13228	. . . . . . 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 2177	. . . . . . . 8 |- ( 1r ` R ) = ( 1r ` R )
14	7, 13	subrg1 13312	. . . . . . 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 4031	. . . . 5 |- ( ( A e. (SubRing ` R ) /\ x e. V ) -> x ( ||r ` S ) ( 1r ` R ) )
17		eqid 2177	. . . . . . . 8 |- ( ||r ` R ) = ( ||r ` R )
18		eqid 2177	. . . . . . . 8 |- ( ||r ` S ) = ( ||r ` S )
19	7, 17, 18	subrgdvds 13316	. . . . . . 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 4046	. . . . 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 13307	. . . . . . . 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 2177	. . . . . . . 8 |- (oppr ` R ) = (oppr ` R )
26		eqid 2177	. . . . . . . 8 |- ( Base ` R ) = ( Base ` R )
27	25, 26	opprbasg 13200	. . . . . . 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 2178	. . . . . 6 |- ( ( A e. (SubRing ` R ) /\ x e. V ) -> ( ||r ` (oppr ` R ) ) = ( ||r ` (oppr ` R ) ) )
30	25	opprring 13202	. . . . . . 7 |- ( R e. Ring -> (oppr ` R ) e. Ring )
31		ringsrg 13177	. . . . . . 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 2178	. . . . . 6 |- ( ( A e. (SubRing ` R ) /\ x e. V ) -> ( .r ` (oppr ` R ) ) = ( .r ` (oppr ` R ) ) )
34	7	subrgbas 13311	. . . . . . . . 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 13303	. . . . . . . . 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 3192	. . . . . . 7 |- ( ( A e. (SubRing ` R ) /\ x e. V ) -> ( Base ` S ) C_ ( Base ` R ) )
39		eqidd 2178	. . . . . . . 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 13230	. . . . . . 7 |- ( ( A e. (SubRing ` R ) /\ x e. V ) -> x e. ( Base ` S ) )
44	38, 43	sseldd 3156	. . . . . 6 |- ( ( A e. (SubRing ` R ) /\ x e. V ) -> x e. ( Base ` R ) )
45		eqid 2177	. . . . . . . . 9 |- ( invr ` S ) = ( invr ` S )
46		eqid 2177	. . . . . . . . 9 |- ( Base ` S ) = ( Base ` S )
47	1, 45, 46	ringinvcl 13247	. . . . . . . 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 3156	. . . . . 6 |- ( ( A e. (SubRing ` R ) /\ x e. V ) -> ( ( invr ` S ) ` x ) e. ( Base ` R ) )
50	28, 29, 32, 33, 44, 49	dvdsrmuld 13218	. . . . 5 |- ( ( A e. (SubRing ` R ) /\ x e. V ) -> x ( ||r ` (oppr ` R ) ) ( ( ( invr ` S ) ` x ) ( .r ` (oppr ` R ) ) x ) )
51	1, 45	unitinvcl 13245	. . . . . . . 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 2177	. . . . . . . 8 |- ( .r ` R ) = ( .r ` R )
54		eqid 2177	. . . . . . . 8 |- ( .r ` (oppr ` R ) ) = ( .r ` (oppr ` R ) )
55	26, 53, 25, 54	opprmulg 13196	. . . . . . 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 1238	. . . . . 6 |- ( ( A e. (SubRing ` R ) /\ x e. V ) -> ( ( ( invr ` S ) ` x ) ( .r ` (oppr ` R ) ) x ) = ( x ( .r ` R ) ( ( invr ` S ) ` x ) ) )
57		eqid 2177	. . . . . . . . 9 |- ( .r ` S ) = ( .r ` S )
58		eqid 2177	. . . . . . . . 9 |- ( 1r ` S ) = ( 1r ` S )
59	1, 45, 57, 58	unitrinv 13249	. . . . . . . 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 12597	. . . . . . . . . 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 5891	. . . . . . 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 2220	. . . . . 6 |- ( ( A e. (SubRing ` R ) /\ x e. V ) -> ( x ( .r ` R ) ( ( invr ` S ) ` x ) ) = ( 1r ` R ) )
66	56, 65	eqtrd 2210	. . . . 5 |- ( ( A e. (SubRing ` R ) /\ x e. V ) -> ( ( ( invr ` S ) ` x ) ( .r ` (oppr ` R ) ) x ) = ( 1r ` R ) )
67	50, 66	breqtrd 4029	. . . 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 2178	. . . . . 6 |- ( A e. (SubRing ` R ) -> ( 1r ` R ) = ( 1r ` R ) )
71		eqidd 2178	. . . . . 6 |- ( A e. (SubRing ` R ) -> ( ||r ` R ) = ( ||r ` R ) )
72		eqidd 2178	. . . . . 6 |- ( A e. (SubRing ` R ) -> (oppr ` R ) = (oppr ` R ) )
73		eqidd 2178	. . . . . 6 |- ( A e. (SubRing ` R ) -> ( ||r ` (oppr ` R ) ) = ( ||r ` (oppr ` R ) ) )
74		ringsrg 13177	. . . . . . 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 13228	. . . . 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 944	. . 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 3161	1 |- ( A e. (SubRing ` R ) -> V C_ U )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1353   e. wcel 2148   C_ wss 3129   class class class wbr 4003   ` cfv 5216  (class class class)co 5874   Basecbs 12456   ↾_s cress 12457   .rcmulr 12531   1rcur 13095  SRingcsrg 13099   Ringcrg 13132  opp_rcoppr 13192   ||_rcdsr 13208  Unitcui 13209   invrcinvr 13242  SubRingcsubrg 13298

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 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-coll 4118  ax-sep 4121  ax-nul 4129  ax-pow 4174  ax-pr 4209  ax-un 4433  ax-setind 4536  ax-cnex 7901  ax-resscn 7902  ax-1cn 7903  ax-1re 7904  ax-icn 7905  ax-addcl 7906  ax-addrcl 7907  ax-mulcl 7908  ax-addcom 7910  ax-addass 7912  ax-i2m1 7915  ax-0lt1 7916  ax-0id 7918  ax-rnegex 7919  ax-pre-ltirr 7922  ax-pre-lttrn 7924  ax-pre-ltadd 7926

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-nel 2443  df-ral 2460  df-rex 2461  df-reu 2462  df-rmo 2463  df-rab 2464  df-v 2739  df-sbc 2963  df-csb 3058  df-dif 3131  df-un 3133  df-in 3135  df-ss 3142  df-nul 3423  df-pw 3577  df-sn 3598  df-pr 3599  df-op 3601  df-uni 3810  df-int 3845  df-iun 3888  df-br 4004  df-opab 4065  df-mpt 4066  df-id 4293  df-xp 4632  df-rel 4633  df-cnv 4634  df-co 4635  df-dm 4636  df-rn 4637  df-res 4638  df-ima 4639  df-iota 5178  df-fun 5218  df-fn 5219  df-f 5220  df-f1 5221  df-fo 5222  df-f1o 5223  df-fv 5224  df-riota 5830  df-ov 5877  df-oprab 5878  df-mpo 5879  df-tpos 6245  df-pnf 7992  df-mnf 7993  df-ltxr 7995  df-inn 8918  df-2 8976  df-3 8977  df-ndx 12459  df-slot 12460  df-base 12462  df-sets 12463  df-iress 12464  df-plusg 12543  df-mulr 12544  df-0g 12697  df-mgm 12729  df-sgrp 12762  df-mnd 12772  df-grp 12834  df-minusg 12835  df-subg 12983  df-cmn 13043  df-abl 13044  df-mgp 13084  df-ur 13096  df-srg 13100  df-ring 13134  df-oppr 13193  df-dvdsr 13211  df-unit 13212  df-invr 13243  df-subrg 13300

This theorem is referenced by:  subrginv  13318  subrgdv  13319  subrgunit  13320  subrgugrp  13321