Theorem subrgunit 14547

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 14544	. . . 4 |- ( A e. (SubRing ` R ) -> V C_ U )
5	4	sselda 3248	. . 3 |- ( ( A e. (SubRing ` R ) /\ X e. V ) -> X e. U )
6	1	subrgbas 14538	. . . . 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 14532	. . . . . 6 |- ( A e. (SubRing ` R ) -> S e. Ring )
10		ringsrg 14352	. . . . . 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 14415	. . 3 |- ( ( A e. (SubRing ` R ) /\ X e. V ) -> X e. A )
15		eqid 2238	. . . . . 6 |- ( invr ` S ) = ( invr ` S )
16		eqid 2238	. . . . . 6 |- ( Base ` S ) = ( Base ` S )
17	3, 15, 16	ringinvcl 14432	. . . . 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 14545	. . . 4 |- ( ( A e. (SubRing ` R ) /\ X e. V ) -> ( I ` X ) = ( ( invr ` S ) ` X ) )
21	18, 20, 7	3eltr4d 2322	. . 3 |- ( ( A e. (SubRing ` R ) /\ X e. V ) -> ( I ` X ) e. A )
22	5, 14, 21	3jca 1208	. 2 |- ( ( A e. (SubRing ` R ) /\ X e. V ) -> ( X e. U /\ X e. A /\ ( I ` X ) e. A ) )
23		eqidd 2239	. . . . 5 |- ( ( A e. (SubRing ` R ) /\ ( X e. U /\ X e. A /\ ( I ` X ) e. A ) ) -> ( Base ` S ) = ( Base ` S ) )
24		eqidd 2239	. . . . 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 2239	. . . . 5 |- ( ( A e. (SubRing ` R ) /\ ( X e. U /\ X e. A /\ ( I ` X ) e. A ) ) -> ( .r ` S ) = ( .r ` S ) )
27		simpr2 1035	. . . . . 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 2317	. . . . 5 |- ( ( A e. (SubRing ` R ) /\ ( X e. U /\ X e. A /\ ( I ` X ) e. A ) ) -> X e. ( Base ` S ) )
30		simpr3 1036	. . . . . 6 |- ( ( A e. (SubRing ` R ) /\ ( X e. U /\ X e. A /\ ( I ` X ) e. A ) ) -> ( I ` X ) e. A )
31	30, 28	eleqtrd 2317	. . . . 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 14403	. . . 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 14534	. . . . . 6 |- ( A e. (SubRing ` R ) -> R e. Ring )
34		simpr1 1034	. . . . . 6 |- ( ( A e. (SubRing ` R ) /\ ( X e. U /\ X e. A /\ ( I ` X ) e. A ) ) -> X e. U )
35		eqid 2238	. . . . . . 7 |- ( .r ` R ) = ( .r ` R )
36		eqid 2238	. . . . . . 7 |- ( 1r ` R ) = ( 1r ` R )
37	2, 19, 35, 36	unitlinv 14433	. . . . . 6 |- ( ( R e. Ring /\ X e. U ) -> ( ( I ` X ) ( .r ` R ) X ) = ( 1r ` R ) )
38	33, 34, 37	syl2an2r 603	. . . . 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 13499	. . . . . . . 8 |- ( ( A e. (SubRing ` R ) /\ R e. Ring ) -> ( .r ` R ) = ( .r ` S ) )
40	33, 39	mpdan 425	. . . . . . 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 6102	. . . . 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 14539	. . . . . 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 2279	. . . 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 4156	. . 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 2238	. . . . . . 7 |- (oppr ` S ) = (oppr ` S )
49	48, 16	opprbasg 14380	. . . . . 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 2239	. . . . 5 |- ( ( A e. (SubRing ` R ) /\ ( X e. U /\ X e. A /\ ( I ` X ) e. A ) ) -> ( ||r ` (oppr ` S ) ) = ( ||r ` (oppr ` S ) ) )
52	48	opprring 14384	. . . . . 6 |- ( S e. Ring -> (oppr ` S ) e. Ring )
53		ringsrg 14352	. . . . . 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 2239	. . . . 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 14403	. . . 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 2238	. . . . . . 7 |- ( .r ` S ) = ( .r ` S )
58		eqid 2238	. . . . . . 7 |- ( .r ` (oppr ` S ) ) = ( .r ` (oppr ` S ) )
59	16, 57, 48, 58	opprmulg 14376	. . . . . 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 1278	. . . . 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 14434	. . . . . . 7 |- ( ( R e. Ring /\ X e. U ) -> ( X ( .r ` R ) ( I ` X ) ) = ( 1r ` R ) )
62	33, 34, 61	syl2an2r 603	. . . . . 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 6102	. . . . . 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 2279	. . . . 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 2271	. . . 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 4156	. . 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 2239	. . . . 5 |- ( A e. (SubRing ` R ) -> ( 1r ` S ) = ( 1r ` S ) )
69		eqidd 2239	. . . . 5 |- ( A e. (SubRing ` R ) -> ( ||r ` S ) = ( ||r ` S ) )
70		eqidd 2239	. . . . 5 |- ( A e. (SubRing ` R ) -> (oppr ` S ) = (oppr ` S ) )
71		eqidd 2239	. . . . 5 |- ( A e. (SubRing ` R ) -> ( ||r ` (oppr ` S ) ) = ( ||r ` (oppr ` S ) ) )
72	67, 68, 69, 70, 71, 11	isunitd 14413	. . . 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 957	. 2 |- ( ( A e. (SubRing ` R ) /\ ( X e. U /\ X e. A /\ ( I ` X ) e. A ) ) -> X e. V )
75	22, 74	impbida 604	1 |- ( A e. (SubRing ` R ) -> ( X e. V <-> ( X e. U /\ X e. A /\ ( I ` X ) e. A ) ) )

This proof depends on syntax axioms:   -> wi 4   /\ wa 104   <-> wb 105   /\ w3a 1009   = wceq 1402   e. wcel 2209   class class class wbr 4130   ` cfv 5377  (class class class)co 6085   Basecbs 13352   ↾_s cress 13353   .rcmulr 13432   1rcur 14262  SRingcsrg 14267   Ringcrg 14300  opp_rcoppr 14372   ||_rcdsr 14392  Unitcui 14393   invrcinvr 14427  SubRingcsubrg 14525

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-coll 4246  ax-sep 4249  ax-nul 4259  ax-pow 4311  ax-pr 4346  ax-un 4578  ax-setind 4684  ax-cnex 8270  ax-resscn 8271  ax-1cn 8272  ax-1re 8273  ax-icn 8274  ax-addcl 8275  ax-addrcl 8276  ax-mulcl 8277  ax-addcom 8279  ax-addass 8281  ax-i2m1 8284  ax-0lt1 8285  ax-0id 8287  ax-rnegex 8288  ax-pre-ltirr 8291  ax-pre-lttrn 8293  ax-pre-ltadd 8295

This proof depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-nel 2516  df-ral 2533  df-rex 2534  df-reu 2535  df-rmo 2536  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-pw 3690  df-sn 3715  df-pr 3716  df-op 3718  df-uni 3936  df-int 3971  df-iun 4014  df-br 4131  df-opab 4193  df-mpt 4194  df-id 4438  df-xp 4780  df-rel 4781  df-cnv 4782  df-co 4783  df-dm 4784  df-rn 4785  df-res 4786  df-ima 4787  df-iota 5337  df-fun 5379  df-fn 5380  df-f 5381  df-f1 5382  df-fo 5383  df-f1o 5384  df-fv 5385  df-riota 6038  df-ov 6088  df-oprab 6089  df-mpo 6090  df-tpos 6516  df-pnf 8362  df-mnf 8363  df-ltxr 8365  df-inn 9305  df-2 9363  df-3 9364  df-ndx 13355  df-slot 13356  df-base 13358  df-sets 13359  df-iress 13360  df-plusg 13444  df-mulr 13445  df-0g 13612  df-mgm 13676  df-sgrp 13717  df-mnd 13730  df-grp 13808  df-minusg 13809  df-subg 13973  df-cmn 14089  df-abl 14090  df-mgp 14218  df-ur 14263  df-srg 14268  df-ring 14302  df-oppr 14373  df-dvdsr 14395  df-unit 14396  df-invr 14428  df-subrg 14527

This theorem is used by: (None)