Theorem subrguss 14381

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 2233	. . . . . . . 8 |- ( A e. (SubRing ` R ) -> ( 1r ` S ) = ( 1r ` S ) )
4		eqidd 2233	. . . . . . . 8 |- ( A e. (SubRing ` R ) -> ( ||r ` S ) = ( ||r ` S ) )
5		eqidd 2233	. . . . . . . 8 |- ( A e. (SubRing ` R ) -> (oppr ` S ) = (oppr ` S ) )
6		eqidd 2233	. . . . . . . 8 |- ( A e. (SubRing ` R ) -> ( ||r ` (oppr ` S ) ) = ( ||r ` (oppr ` S ) ) )
7		subrguss.1	. . . . . . . . . 10 |- S = ( R ↾s A )
8	7	subrgring 14369	. . . . . . . . 9 |- ( A e. (SubRing ` R ) -> S e. Ring )
9		ringsrg 14191	. . . . . . . . 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 14251	. . . . . . 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 2232	. . . . . . . 8 |- ( 1r ` R ) = ( 1r ` R )
14	7, 13	subrg1 14376	. . . . . . 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 4137	. . . . 5 |- ( ( A e. (SubRing ` R ) /\ x e. V ) -> x ( ||r ` S ) ( 1r ` R ) )
17		eqid 2232	. . . . . . . 8 |- ( ||r ` R ) = ( ||r ` R )
18		eqid 2232	. . . . . . . 8 |- ( ||r ` S ) = ( ||r ` S )
19	7, 17, 18	subrgdvds 14380	. . . . . . 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 4152	. . . . 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 14371	. . . . . . . 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 2232	. . . . . . . 8 |- (oppr ` R ) = (oppr ` R )
26		eqid 2232	. . . . . . . 8 |- ( Base ` R ) = ( Base ` R )
27	25, 26	opprbasg 14219	. . . . . . 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 2233	. . . . . 6 |- ( ( A e. (SubRing ` R ) /\ x e. V ) -> ( ||r ` (oppr ` R ) ) = ( ||r ` (oppr ` R ) ) )
30	25	opprring 14223	. . . . . . 7 |- ( R e. Ring -> (oppr ` R ) e. Ring )
31		ringsrg 14191	. . . . . . 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 2233	. . . . . 6 |- ( ( A e. (SubRing ` R ) /\ x e. V ) -> ( .r ` (oppr ` R ) ) = ( .r ` (oppr ` R ) ) )
34	7	subrgbas 14375	. . . . . . . . 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 14367	. . . . . . . . 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 3275	. . . . . . 7 |- ( ( A e. (SubRing ` R ) /\ x e. V ) -> ( Base ` S ) C_ ( Base ` R ) )
39		eqidd 2233	. . . . . . . 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 14253	. . . . . . 7 |- ( ( A e. (SubRing ` R ) /\ x e. V ) -> x e. ( Base ` S ) )
44	38, 43	sseldd 3239	. . . . . 6 |- ( ( A e. (SubRing ` R ) /\ x e. V ) -> x e. ( Base ` R ) )
45		eqid 2232	. . . . . . . . 9 |- ( invr ` S ) = ( invr ` S )
46		eqid 2232	. . . . . . . . 9 |- ( Base ` S ) = ( Base ` S )
47	1, 45, 46	ringinvcl 14270	. . . . . . . 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 3239	. . . . . 6 |- ( ( A e. (SubRing ` R ) /\ x e. V ) -> ( ( invr ` S ) ` x ) e. ( Base ` R ) )
50	28, 29, 32, 33, 44, 49	dvdsrmuld 14241	. . . . 5 |- ( ( A e. (SubRing ` R ) /\ x e. V ) -> x ( ||r ` (oppr ` R ) ) ( ( ( invr ` S ) ` x ) ( .r ` (oppr ` R ) ) x ) )
51	1, 45	unitinvcl 14268	. . . . . . . 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 2232	. . . . . . . 8 |- ( .r ` R ) = ( .r ` R )
54		eqid 2232	. . . . . . . 8 |- ( .r ` (oppr ` R ) ) = ( .r ` (oppr ` R ) )
55	26, 53, 25, 54	opprmulg 14215	. . . . . . 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 1274	. . . . . 6 |- ( ( A e. (SubRing ` R ) /\ x e. V ) -> ( ( ( invr ` S ) ` x ) ( .r ` (oppr ` R ) ) x ) = ( x ( .r ` R ) ( ( invr ` S ) ` x ) ) )
57		eqid 2232	. . . . . . . . 9 |- ( .r ` S ) = ( .r ` S )
58		eqid 2232	. . . . . . . . 9 |- ( 1r ` S ) = ( 1r ` S )
59	1, 45, 57, 58	unitrinv 14272	. . . . . . . 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 13358	. . . . . . . . . 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 6067	. . . . . . 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 2275	. . . . . 6 |- ( ( A e. (SubRing ` R ) /\ x e. V ) -> ( x ( .r ` R ) ( ( invr ` S ) ` x ) ) = ( 1r ` R ) )
66	56, 65	eqtrd 2265	. . . . 5 |- ( ( A e. (SubRing ` R ) /\ x e. V ) -> ( ( ( invr ` S ) ` x ) ( .r ` (oppr ` R ) ) x ) = ( 1r ` R ) )
67	50, 66	breqtrd 4135	. . . 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 2233	. . . . . 6 |- ( A e. (SubRing ` R ) -> ( 1r ` R ) = ( 1r ` R ) )
71		eqidd 2233	. . . . . 6 |- ( A e. (SubRing ` R ) -> ( ||r ` R ) = ( ||r ` R ) )
72		eqidd 2233	. . . . . 6 |- ( A e. (SubRing ` R ) -> (oppr ` R ) = (oppr ` R ) )
73		eqidd 2233	. . . . . 6 |- ( A e. (SubRing ` R ) -> ( ||r ` (oppr ` R ) ) = ( ||r ` (oppr ` R ) ) )
74		ringsrg 14191	. . . . . . 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 14251	. . . . 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 953	. . 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 3244	1 |- ( A e. (SubRing ` R ) -> V C_ U )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1398   e. wcel 2203   C_ wss 3211   class class class wbr 4109   ` cfv 5352  (class class class)co 6050   Basecbs 13212   ↾_s cress 13213   .rcmulr 13291   1rcur 14103  SRingcsrg 14107   Ringcrg 14140  opp_rcoppr 14211   ||_rcdsr 14230  Unitcui 14231   invrcinvr 14265  SubRingcsubrg 14362

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 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2205  ax-14 2206  ax-ext 2214  ax-coll 4225  ax-sep 4228  ax-nul 4236  ax-pow 4287  ax-pr 4322  ax-un 4554  ax-setind 4659  ax-cnex 8218  ax-resscn 8219  ax-1cn 8220  ax-1re 8221  ax-icn 8222  ax-addcl 8223  ax-addrcl 8224  ax-mulcl 8225  ax-addcom 8227  ax-addass 8229  ax-i2m1 8232  ax-0lt1 8233  ax-0id 8235  ax-rnegex 8236  ax-pre-ltirr 8239  ax-pre-lttrn 8241  ax-pre-ltadd 8243

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2083  df-mo 2084  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ne 2413  df-nel 2508  df-ral 2525  df-rex 2526  df-reu 2527  df-rmo 2528  df-rab 2529  df-v 2815  df-sbc 3043  df-csb 3139  df-dif 3213  df-un 3215  df-in 3217  df-ss 3224  df-nul 3509  df-pw 3671  df-sn 3695  df-pr 3696  df-op 3698  df-uni 3915  df-int 3950  df-iun 3993  df-br 4110  df-opab 4172  df-mpt 4173  df-id 4414  df-xp 4755  df-rel 4756  df-cnv 4757  df-co 4758  df-dm 4759  df-rn 4760  df-res 4761  df-ima 4762  df-iota 5312  df-fun 5354  df-fn 5355  df-f 5356  df-f1 5357  df-fo 5358  df-f1o 5359  df-fv 5360  df-riota 6003  df-ov 6053  df-oprab 6054  df-mpo 6055  df-tpos 6476  df-pnf 8310  df-mnf 8311  df-ltxr 8313  df-inn 9238  df-2 9296  df-3 9297  df-ndx 13215  df-slot 13216  df-base 13218  df-sets 13219  df-iress 13220  df-plusg 13303  df-mulr 13304  df-0g 13471  df-mgm 13569  df-sgrp 13615  df-mnd 13630  df-grp 13716  df-minusg 13717  df-subg 13887  df-cmn 14003  df-abl 14004  df-mgp 14065  df-ur 14104  df-srg 14108  df-ring 14142  df-oppr 14212  df-dvdsr 14233  df-unit 14234  df-invr 14266  df-subrg 14364

This theorem is referenced by:  subrginv  14382  subrgdv  14383  subrgunit  14384  subrgugrp  14385