Theorem subrgunit 14252

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 14249	. . . 4 |- ( A e. (SubRing ` R ) -> V C_ U )
5	4	sselda 3227	. . 3 |- ( ( A e. (SubRing ` R ) /\ X e. V ) -> X e. U )
6	1	subrgbas 14243	. . . . 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 14237	. . . . . 6 |- ( A e. (SubRing ` R ) -> S e. Ring )
10		ringsrg 14059	. . . . . 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 14121	. . 3 |- ( ( A e. (SubRing ` R ) /\ X e. V ) -> X e. A )
15		eqid 2231	. . . . . 6 |- ( invr ` S ) = ( invr ` S )
16		eqid 2231	. . . . . 6 |- ( Base ` S ) = ( Base ` S )
17	3, 15, 16	ringinvcl 14138	. . . . 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 14250	. . . 4 |- ( ( A e. (SubRing ` R ) /\ X e. V ) -> ( I ` X ) = ( ( invr ` S ) ` X ) )
21	18, 20, 7	3eltr4d 2315	. . 3 |- ( ( A e. (SubRing ` R ) /\ X e. V ) -> ( I ` X ) e. A )
22	5, 14, 21	3jca 1203	. 2 |- ( ( A e. (SubRing ` R ) /\ X e. V ) -> ( X e. U /\ X e. A /\ ( I ` X ) e. A ) )
23		eqidd 2232	. . . . 5 |- ( ( A e. (SubRing ` R ) /\ ( X e. U /\ X e. A /\ ( I ` X ) e. A ) ) -> ( Base ` S ) = ( Base ` S ) )
24		eqidd 2232	. . . . 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 2232	. . . . 5 |- ( ( A e. (SubRing ` R ) /\ ( X e. U /\ X e. A /\ ( I ` X ) e. A ) ) -> ( .r ` S ) = ( .r ` S ) )
27		simpr2 1030	. . . . . 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 2310	. . . . 5 |- ( ( A e. (SubRing ` R ) /\ ( X e. U /\ X e. A /\ ( I ` X ) e. A ) ) -> X e. ( Base ` S ) )
30		simpr3 1031	. . . . . 6 |- ( ( A e. (SubRing ` R ) /\ ( X e. U /\ X e. A /\ ( I ` X ) e. A ) ) -> ( I ` X ) e. A )
31	30, 28	eleqtrd 2310	. . . . 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 14109	. . . 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 14239	. . . . . 6 |- ( A e. (SubRing ` R ) -> R e. Ring )
34		simpr1 1029	. . . . . 6 |- ( ( A e. (SubRing ` R ) /\ ( X e. U /\ X e. A /\ ( I ` X ) e. A ) ) -> X e. U )
35		eqid 2231	. . . . . . 7 |- ( .r ` R ) = ( .r ` R )
36		eqid 2231	. . . . . . 7 |- ( 1r ` R ) = ( 1r ` R )
37	2, 19, 35, 36	unitlinv 14139	. . . . . 6 |- ( ( R e. Ring /\ X e. U ) -> ( ( I ` X ) ( .r ` R ) X ) = ( 1r ` R ) )
38	33, 34, 37	syl2an2r 599	. . . . 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 13227	. . . . . . . 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 6034	. . . . 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 14244	. . . . . 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 2272	. . . 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 4114	. . 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 2231	. . . . . . 7 |- (oppr ` S ) = (oppr ` S )
49	48, 16	opprbasg 14087	. . . . . 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 2232	. . . . 5 |- ( ( A e. (SubRing ` R ) /\ ( X e. U /\ X e. A /\ ( I ` X ) e. A ) ) -> ( ||r ` (oppr ` S ) ) = ( ||r ` (oppr ` S ) ) )
52	48	opprring 14091	. . . . . 6 |- ( S e. Ring -> (oppr ` S ) e. Ring )
53		ringsrg 14059	. . . . . 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 2232	. . . . 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 14109	. . . 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 2231	. . . . . . 7 |- ( .r ` S ) = ( .r ` S )
58		eqid 2231	. . . . . . 7 |- ( .r ` (oppr ` S ) ) = ( .r ` (oppr ` S ) )
59	16, 57, 48, 58	opprmulg 14083	. . . . . 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 1273	. . . . 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 14140	. . . . . . 7 |- ( ( R e. Ring /\ X e. U ) -> ( X ( .r ` R ) ( I ` X ) ) = ( 1r ` R ) )
62	33, 34, 61	syl2an2r 599	. . . . . 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 6034	. . . . . 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 2272	. . . . 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 2264	. . . 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 4114	. . 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 2232	. . . . 5 |- ( A e. (SubRing ` R ) -> ( 1r ` S ) = ( 1r ` S ) )
69		eqidd 2232	. . . . 5 |- ( A e. (SubRing ` R ) -> ( ||r ` S ) = ( ||r ` S ) )
70		eqidd 2232	. . . . 5 |- ( A e. (SubRing ` R ) -> (oppr ` S ) = (oppr ` S ) )
71		eqidd 2232	. . . . 5 |- ( A e. (SubRing ` R ) -> ( ||r ` (oppr ` S ) ) = ( ||r ` (oppr ` S ) ) )
72	67, 68, 69, 70, 71, 11	isunitd 14119	. . . 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 952	. 2 |- ( ( A e. (SubRing ` R ) /\ ( X e. U /\ X e. A /\ ( I ` X ) e. A ) ) -> X e. V )
75	22, 74	impbida 600	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 1004   = wceq 1397   e. wcel 2202   class class class wbr 4088   ` cfv 5326  (class class class)co 6017   Basecbs 13081   ↾_s cress 13082   .rcmulr 13160   1rcur 13971  SRingcsrg 13975   Ringcrg 14008  opp_rcoppr 14079   ||_rcdsr 14098  Unitcui 14099   invrcinvr 14133  SubRingcsubrg 14230

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-13 2204  ax-14 2205  ax-ext 2213  ax-coll 4204  ax-sep 4207  ax-nul 4215  ax-pow 4264  ax-pr 4299  ax-un 4530  ax-setind 4635  ax-cnex 8122  ax-resscn 8123  ax-1cn 8124  ax-1re 8125  ax-icn 8126  ax-addcl 8127  ax-addrcl 8128  ax-mulcl 8129  ax-addcom 8131  ax-addass 8133  ax-i2m1 8136  ax-0lt1 8137  ax-0id 8139  ax-rnegex 8140  ax-pre-ltirr 8143  ax-pre-lttrn 8145  ax-pre-ltadd 8147

This theorem depends on definitions:  df-bi 117  df-3an 1006  df-tru 1400  df-fal 1403  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ne 2403  df-nel 2498  df-ral 2515  df-rex 2516  df-reu 2517  df-rmo 2518  df-rab 2519  df-v 2804  df-sbc 3032  df-csb 3128  df-dif 3202  df-un 3204  df-in 3206  df-ss 3213  df-nul 3495  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-uni 3894  df-int 3929  df-iun 3972  df-br 4089  df-opab 4151  df-mpt 4152  df-id 4390  df-xp 4731  df-rel 4732  df-cnv 4733  df-co 4734  df-dm 4735  df-rn 4736  df-res 4737  df-ima 4738  df-iota 5286  df-fun 5328  df-fn 5329  df-f 5330  df-f1 5331  df-fo 5332  df-f1o 5333  df-fv 5334  df-riota 5970  df-ov 6020  df-oprab 6021  df-mpo 6022  df-tpos 6410  df-pnf 8215  df-mnf 8216  df-ltxr 8218  df-inn 9143  df-2 9201  df-3 9202  df-ndx 13084  df-slot 13085  df-base 13087  df-sets 13088  df-iress 13089  df-plusg 13172  df-mulr 13173  df-0g 13340  df-mgm 13438  df-sgrp 13484  df-mnd 13499  df-grp 13585  df-minusg 13586  df-subg 13756  df-cmn 13872  df-abl 13873  df-mgp 13933  df-ur 13972  df-srg 13976  df-ring 14010  df-oppr 14080  df-dvdsr 14101  df-unit 14102  df-invr 14134  df-subrg 14232

This theorem is referenced by: (None)