Theorem subrgunit 13298

Description: An element of a ring is a unit of a subring iff it is a unit of the parent ring and both it and its inverse are in the subring. (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 )
subrgunit.4	|- I = ( invr ` R )

Assertion

Ref	Expression
subrgunit	|- ( A e. (SubRing ` R ) -> ( X e. V <-> ( X e. U /\ X e. A /\ ( I ` X ) e. A ) ) )

Proof of Theorem subrgunit

Step	Hyp	Ref	Expression
1		subrgugrp.1	. . . . 5 |- S = ( R ↾s A )
2		subrgugrp.2	. . . . 5 |- U = (Unit ` R )
3		subrgugrp.3	. . . . 5 |- V = (Unit ` S )
4	1, 2, 3	subrguss 13295	. . . 4 |- ( A e. (SubRing ` R ) -> V C_ U )
5	4	sselda 3155	. . 3 |- ( ( A e. (SubRing ` R ) /\ X e. V ) -> X e. U )
6	1	subrgbas 13289	. . . . 5 |- ( A e. (SubRing ` R ) -> A = ( Base ` S ) )
7	6	adantr 276	. . . 4 |- ( ( A e. (SubRing ` R ) /\ X e. V ) -> A = ( Base ` S ) )
8	3	a1i 9	. . . 4 |- ( ( A e. (SubRing ` R ) /\ X e. V ) -> V = (Unit ` S ) )
9	1	subrgring 13283	. . . . . 6 |- ( A e. (SubRing ` R ) -> S e. Ring )
10		ringsrg 13155	. . . . . 6 |- ( S e. Ring -> S e. SRing )
11	9, 10	syl 14	. . . . 5 |- ( A e. (SubRing ` R ) -> S e. SRing )
12	11	adantr 276	. . . 4 |- ( ( A e. (SubRing ` R ) /\ X e. V ) -> S e. SRing )
13		simpr 110	. . . 4 |- ( ( A e. (SubRing ` R ) /\ X e. V ) -> X e. V )
14	7, 8, 12, 13	unitcld 13208	. . 3 |- ( ( A e. (SubRing ` R ) /\ X e. V ) -> X e. A )
15		eqid 2177	. . . . . 6 |- ( invr ` S ) = ( invr ` S )
16		eqid 2177	. . . . . 6 |- ( Base ` S ) = ( Base ` S )
17	3, 15, 16	ringinvcl 13225	. . . . 5 |- ( ( S e. Ring /\ X e. V ) -> ( ( invr ` S ) ` X ) e. ( Base ` S ) )
18	9, 17	sylan 283	. . . 4 |- ( ( A e. (SubRing ` R ) /\ X e. V ) -> ( ( invr ` S ) ` X ) e. ( Base ` S ) )
19		subrgunit.4	. . . . 5 |- I = ( invr ` R )
20	1, 19, 3, 15	subrginv 13296	. . . 4 |- ( ( A e. (SubRing ` R ) /\ X e. V ) -> ( I ` X ) = ( ( invr ` S ) ` X ) )
21	18, 20, 7	3eltr4d 2261	. . 3 |- ( ( A e. (SubRing ` R ) /\ X e. V ) -> ( I ` X ) e. A )
22	5, 14, 21	3jca 1177	. 2 |- ( ( A e. (SubRing ` R ) /\ X e. V ) -> ( X e. U /\ X e. A /\ ( I ` X ) e. A ) )
23		eqidd 2178	. . . . 5 |- ( ( A e. (SubRing ` R ) /\ ( X e. U /\ X e. A /\ ( I ` X ) e. A ) ) -> ( Base ` S ) = ( Base ` S ) )
24		eqidd 2178	. . . . 5 |- ( ( A e. (SubRing ` R ) /\ ( X e. U /\ X e. A /\ ( I ` X ) e. A ) ) -> ( ||r ` S ) = ( ||r ` S ) )
25	11	adantr 276	. . . . 5 |- ( ( A e. (SubRing ` R ) /\ ( X e. U /\ X e. A /\ ( I ` X ) e. A ) ) -> S e. SRing )
26		eqidd 2178	. . . . 5 |- ( ( A e. (SubRing ` R ) /\ ( X e. U /\ X e. A /\ ( I ` X ) e. A ) ) -> ( .r ` S ) = ( .r ` S ) )
27		simpr2 1004	. . . . . 6 |- ( ( A e. (SubRing ` R ) /\ ( X e. U /\ X e. A /\ ( I ` X ) e. A ) ) -> X e. A )
28	6	adantr 276	. . . . . 6 |- ( ( A e. (SubRing ` R ) /\ ( X e. U /\ X e. A /\ ( I ` X ) e. A ) ) -> A = ( Base ` S ) )
29	27, 28	eleqtrd 2256	. . . . 5 |- ( ( A e. (SubRing ` R ) /\ ( X e. U /\ X e. A /\ ( I ` X ) e. A ) ) -> X e. ( Base ` S ) )
30		simpr3 1005	. . . . . 6 |- ( ( A e. (SubRing ` R ) /\ ( X e. U /\ X e. A /\ ( I ` X ) e. A ) ) -> ( I ` X ) e. A )
31	30, 28	eleqtrd 2256	. . . . 5 |- ( ( A e. (SubRing ` R ) /\ ( X e. U /\ X e. A /\ ( I ` X ) e. A ) ) -> ( I ` X ) e. ( Base ` S ) )
32	23, 24, 25, 26, 29, 31	dvdsrmuld 13196	. . . 4 |- ( ( A e. (SubRing ` R ) /\ ( X e. U /\ X e. A /\ ( I ` X ) e. A ) ) -> X ( ||r ` S ) ( ( I ` X ) ( .r ` S ) X ) )
33		subrgrcl 13285	. . . . . 6 |- ( A e. (SubRing ` R ) -> R e. Ring )
34		simpr1 1003	. . . . . 6 |- ( ( A e. (SubRing ` R ) /\ ( X e. U /\ X e. A /\ ( I ` X ) e. A ) ) -> X e. U )
35		eqid 2177	. . . . . . 7 |- ( .r ` R ) = ( .r ` R )
36		eqid 2177	. . . . . . 7 |- ( 1r ` R ) = ( 1r ` R )
37	2, 19, 35, 36	unitlinv 13226	. . . . . 6 |- ( ( R e. Ring /\ X e. U ) -> ( ( I ` X ) ( .r ` R ) X ) = ( 1r ` R ) )
38	33, 34, 37	syl2an2r 595	. . . . 5 |- ( ( A e. (SubRing ` R ) /\ ( X e. U /\ X e. A /\ ( I ` X ) e. A ) ) -> ( ( I ` X ) ( .r ` R ) X ) = ( 1r ` R ) )
39	1, 35	ressmulrg 12595	. . . . . . . 8 |- ( ( A e. (SubRing ` R ) /\ R e. Ring ) -> ( .r ` R ) = ( .r ` S ) )
40	33, 39	mpdan 421	. . . . . . 7 |- ( A e. (SubRing ` R ) -> ( .r ` R ) = ( .r ` S ) )
41	40	adantr 276	. . . . . 6 |- ( ( A e. (SubRing ` R ) /\ ( X e. U /\ X e. A /\ ( I ` X ) e. A ) ) -> ( .r ` R ) = ( .r ` S ) )
42	41	oveqd 5889	. . . . 5 |- ( ( A e. (SubRing ` R ) /\ ( X e. U /\ X e. A /\ ( I ` X ) e. A ) ) -> ( ( I ` X ) ( .r ` R ) X ) = ( ( I ` X ) ( .r ` S ) X ) )
43	1, 36	subrg1 13290	. . . . . 6 |- ( A e. (SubRing ` R ) -> ( 1r ` R ) = ( 1r ` S ) )
44	43	adantr 276	. . . . 5 |- ( ( A e. (SubRing ` R ) /\ ( X e. U /\ X e. A /\ ( I ` X ) e. A ) ) -> ( 1r ` R ) = ( 1r ` S ) )
45	38, 42, 44	3eqtr3d 2218	. . . 4 |- ( ( A e. (SubRing ` R ) /\ ( X e. U /\ X e. A /\ ( I ` X ) e. A ) ) -> ( ( I ` X ) ( .r ` S ) X ) = ( 1r ` S ) )
46	32, 45	breqtrd 4028	. . 3 |- ( ( A e. (SubRing ` R ) /\ ( X e. U /\ X e. A /\ ( I ` X ) e. A ) ) -> X ( ||r ` S ) ( 1r ` S ) )
47	9	adantr 276	. . . . . 6 |- ( ( A e. (SubRing ` R ) /\ ( X e. U /\ X e. A /\ ( I ` X ) e. A ) ) -> S e. Ring )
48		eqid 2177	. . . . . . 7 |- (oppr ` S ) = (oppr ` S )
49	48, 16	opprbasg 13178	. . . . . 6 |- ( S e. Ring -> ( Base ` S ) = ( Base ` (oppr ` S ) ) )
50	47, 49	syl 14	. . . . 5 |- ( ( A e. (SubRing ` R ) /\ ( X e. U /\ X e. A /\ ( I ` X ) e. A ) ) -> ( Base ` S ) = ( Base ` (oppr ` S ) ) )
51		eqidd 2178	. . . . 5 |- ( ( A e. (SubRing ` R ) /\ ( X e. U /\ X e. A /\ ( I ` X ) e. A ) ) -> ( ||r ` (oppr ` S ) ) = ( ||r ` (oppr ` S ) ) )
52	48	opprring 13180	. . . . . 6 |- ( S e. Ring -> (oppr ` S ) e. Ring )
53		ringsrg 13155	. . . . . 6 |- ( (oppr ` S ) e. Ring -> (oppr ` S ) e. SRing )
54	47, 52, 53	3syl 17	. . . . 5 |- ( ( A e. (SubRing ` R ) /\ ( X e. U /\ X e. A /\ ( I ` X ) e. A ) ) -> (oppr ` S ) e. SRing )
55		eqidd 2178	. . . . 5 |- ( ( A e. (SubRing ` R ) /\ ( X e. U /\ X e. A /\ ( I ` X ) e. A ) ) -> ( .r ` (oppr ` S ) ) = ( .r ` (oppr ` S ) ) )
56	50, 51, 54, 55, 29, 31	dvdsrmuld 13196	. . . 4 |- ( ( A e. (SubRing ` R ) /\ ( X e. U /\ X e. A /\ ( I ` X ) e. A ) ) -> X ( ||r ` (oppr ` S ) ) ( ( I ` X ) ( .r ` (oppr ` S ) ) X ) )
57		eqid 2177	. . . . . . 7 |- ( .r ` S ) = ( .r ` S )
58		eqid 2177	. . . . . . 7 |- ( .r ` (oppr ` S ) ) = ( .r ` (oppr ` S ) )
59	16, 57, 48, 58	opprmulg 13174	. . . . . 6 |- ( ( S e. Ring /\ ( I ` X ) e. ( Base ` S ) /\ X e. ( Base ` S ) ) -> ( ( I ` X ) ( .r ` (oppr ` S ) ) X ) = ( X ( .r ` S ) ( I ` X ) ) )
60	47, 31, 29, 59	syl3anc 1238	. . . . 5 |- ( ( A e. (SubRing ` R ) /\ ( X e. U /\ X e. A /\ ( I ` X ) e. A ) ) -> ( ( I ` X ) ( .r ` (oppr ` S ) ) X ) = ( X ( .r ` S ) ( I ` X ) ) )
61	2, 19, 35, 36	unitrinv 13227	. . . . . . 7 |- ( ( R e. Ring /\ X e. U ) -> ( X ( .r ` R ) ( I ` X ) ) = ( 1r ` R ) )
62	33, 34, 61	syl2an2r 595	. . . . . 6 |- ( ( A e. (SubRing ` R ) /\ ( X e. U /\ X e. A /\ ( I ` X ) e. A ) ) -> ( X ( .r ` R ) ( I ` X ) ) = ( 1r ` R ) )
63	41	oveqd 5889	. . . . . 6 |- ( ( A e. (SubRing ` R ) /\ ( X e. U /\ X e. A /\ ( I ` X ) e. A ) ) -> ( X ( .r ` R ) ( I ` X ) ) = ( X ( .r ` S ) ( I ` X ) ) )
64	62, 63, 44	3eqtr3d 2218	. . . . 5 |- ( ( A e. (SubRing ` R ) /\ ( X e. U /\ X e. A /\ ( I ` X ) e. A ) ) -> ( X ( .r ` S ) ( I ` X ) ) = ( 1r ` S ) )
65	60, 64	eqtrd 2210	. . . 4 |- ( ( A e. (SubRing ` R ) /\ ( X e. U /\ X e. A /\ ( I ` X ) e. A ) ) -> ( ( I ` X ) ( .r ` (oppr ` S ) ) X ) = ( 1r ` S ) )
66	56, 65	breqtrd 4028	. . 3 |- ( ( A e. (SubRing ` R ) /\ ( X e. U /\ X e. A /\ ( I ` X ) e. A ) ) -> X ( ||r ` (oppr ` S ) ) ( 1r ` S ) )
67	3	a1i 9	. . . . 5 |- ( A e. (SubRing ` R ) -> V = (Unit ` S ) )
68		eqidd 2178	. . . . 5 |- ( A e. (SubRing ` R ) -> ( 1r ` S ) = ( 1r ` S ) )
69		eqidd 2178	. . . . 5 |- ( A e. (SubRing ` R ) -> ( ||r ` S ) = ( ||r ` S ) )
70		eqidd 2178	. . . . 5 |- ( A e. (SubRing ` R ) -> (oppr ` S ) = (oppr ` S ) )
71		eqidd 2178	. . . . 5 |- ( A e. (SubRing ` R ) -> ( ||r ` (oppr ` S ) ) = ( ||r ` (oppr ` S ) ) )
72	67, 68, 69, 70, 71, 11	isunitd 13206	. . . 4 |- ( A e. (SubRing ` R ) -> ( X e. V <-> ( X ( ||r ` S ) ( 1r ` S ) /\ X ( ||r ` (oppr ` S ) ) ( 1r ` S ) ) ) )
73	72	adantr 276	. . 3 |- ( ( A e. (SubRing ` R ) /\ ( X e. U /\ X e. A /\ ( I ` X ) e. A ) ) -> ( X e. V <-> ( X ( ||r ` S ) ( 1r ` S ) /\ X ( ||r ` (oppr ` S ) ) ( 1r ` S ) ) ) )
74	46, 66, 73	mpbir2and 944	. 2 |- ( ( A e. (SubRing ` R ) /\ ( X e. U /\ X e. A /\ ( I ` X ) e. A ) ) -> X e. V )
75	22, 74	impbida 596	1 |- ( A e. (SubRing ` R ) -> ( X e. V <-> ( X e. U /\ X e. A /\ ( I ` X ) e. A ) ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   /\ w3a 978   = wceq 1353   e. wcel 2148   class class class wbr 4002   ` cfv 5215  (class class class)co 5872   Basecbs 12454   ↾_s cress 12455   .rcmulr 12529   1rcur 13073  SRingcsrg 13077   Ringcrg 13110  opp_rcoppr 13170   ||_rcdsr 13186  Unitcui 13187   invrcinvr 13220  SubRingcsubrg 13276

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 4117  ax-sep 4120  ax-nul 4128  ax-pow 4173  ax-pr 4208  ax-un 4432  ax-setind 4535  ax-cnex 7899  ax-resscn 7900  ax-1cn 7901  ax-1re 7902  ax-icn 7903  ax-addcl 7904  ax-addrcl 7905  ax-mulcl 7906  ax-addcom 7908  ax-addass 7910  ax-i2m1 7913  ax-0lt1 7914  ax-0id 7916  ax-rnegex 7917  ax-pre-ltirr 7920  ax-pre-lttrn 7922  ax-pre-ltadd 7924

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 4003  df-opab 4064  df-mpt 4065  df-id 4292  df-xp 4631  df-rel 4632  df-cnv 4633  df-co 4634  df-dm 4635  df-rn 4636  df-res 4637  df-ima 4638  df-iota 5177  df-fun 5217  df-fn 5218  df-f 5219  df-f1 5220  df-fo 5221  df-f1o 5222  df-fv 5223  df-riota 5828  df-ov 5875  df-oprab 5876  df-mpo 5877  df-tpos 6243  df-pnf 7990  df-mnf 7991  df-ltxr 7993  df-inn 8916  df-2 8974  df-3 8975  df-ndx 12457  df-slot 12458  df-base 12460  df-sets 12461  df-iress 12462  df-plusg 12541  df-mulr 12542  df-0g 12695  df-mgm 12707  df-sgrp 12740  df-mnd 12750  df-grp 12812  df-minusg 12813  df-subg 12961  df-cmn 13021  df-abl 13022  df-mgp 13062  df-ur 13074  df-srg 13078  df-ring 13112  df-oppr 13171  df-dvdsr 13189  df-unit 13190  df-invr 13221  df-subrg 13278

This theorem is referenced by: (None)