Theorem subrgunit 13795

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 13792	. . . 4 |- ( A e. (SubRing ` R ) -> V C_ U )
5	4	sselda 3183	. . 3 |- ( ( A e. (SubRing ` R ) /\ X e. V ) -> X e. U )
6	1	subrgbas 13786	. . . . 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 13780	. . . . . 6 |- ( A e. (SubRing ` R ) -> S e. Ring )
10		ringsrg 13603	. . . . . 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 13664	. . 3 |- ( ( A e. (SubRing ` R ) /\ X e. V ) -> X e. A )
15		eqid 2196	. . . . . 6 |- ( invr ` S ) = ( invr ` S )
16		eqid 2196	. . . . . 6 |- ( Base ` S ) = ( Base ` S )
17	3, 15, 16	ringinvcl 13681	. . . . 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 13793	. . . 4 |- ( ( A e. (SubRing ` R ) /\ X e. V ) -> ( I ` X ) = ( ( invr ` S ) ` X ) )
21	18, 20, 7	3eltr4d 2280	. . 3 |- ( ( A e. (SubRing ` R ) /\ X e. V ) -> ( I ` X ) e. A )
22	5, 14, 21	3jca 1179	. 2 |- ( ( A e. (SubRing ` R ) /\ X e. V ) -> ( X e. U /\ X e. A /\ ( I ` X ) e. A ) )
23		eqidd 2197	. . . . 5 |- ( ( A e. (SubRing ` R ) /\ ( X e. U /\ X e. A /\ ( I ` X ) e. A ) ) -> ( Base ` S ) = ( Base ` S ) )
24		eqidd 2197	. . . . 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 2197	. . . . 5 |- ( ( A e. (SubRing ` R ) /\ ( X e. U /\ X e. A /\ ( I ` X ) e. A ) ) -> ( .r ` S ) = ( .r ` S ) )
27		simpr2 1006	. . . . . 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 2275	. . . . 5 |- ( ( A e. (SubRing ` R ) /\ ( X e. U /\ X e. A /\ ( I ` X ) e. A ) ) -> X e. ( Base ` S ) )
30		simpr3 1007	. . . . . 6 |- ( ( A e. (SubRing ` R ) /\ ( X e. U /\ X e. A /\ ( I ` X ) e. A ) ) -> ( I ` X ) e. A )
31	30, 28	eleqtrd 2275	. . . . 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 13652	. . . 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 13782	. . . . . 6 |- ( A e. (SubRing ` R ) -> R e. Ring )
34		simpr1 1005	. . . . . 6 |- ( ( A e. (SubRing ` R ) /\ ( X e. U /\ X e. A /\ ( I ` X ) e. A ) ) -> X e. U )
35		eqid 2196	. . . . . . 7 |- ( .r ` R ) = ( .r ` R )
36		eqid 2196	. . . . . . 7 |- ( 1r ` R ) = ( 1r ` R )
37	2, 19, 35, 36	unitlinv 13682	. . . . . 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 12822	. . . . . . . 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 5939	. . . . 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 13787	. . . . . 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 2237	. . . 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 4059	. . 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 2196	. . . . . . 7 |- (oppr ` S ) = (oppr ` S )
49	48, 16	opprbasg 13631	. . . . . 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 2197	. . . . 5 |- ( ( A e. (SubRing ` R ) /\ ( X e. U /\ X e. A /\ ( I ` X ) e. A ) ) -> ( ||r ` (oppr ` S ) ) = ( ||r ` (oppr ` S ) ) )
52	48	opprring 13635	. . . . . 6 |- ( S e. Ring -> (oppr ` S ) e. Ring )
53		ringsrg 13603	. . . . . 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 2197	. . . . 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 13652	. . . 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 2196	. . . . . . 7 |- ( .r ` S ) = ( .r ` S )
58		eqid 2196	. . . . . . 7 |- ( .r ` (oppr ` S ) ) = ( .r ` (oppr ` S ) )
59	16, 57, 48, 58	opprmulg 13627	. . . . . 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 1249	. . . . 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 13683	. . . . . . 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 5939	. . . . . 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 2237	. . . . 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 2229	. . . 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 4059	. . 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 2197	. . . . 5 |- ( A e. (SubRing ` R ) -> ( 1r ` S ) = ( 1r ` S ) )
69		eqidd 2197	. . . . 5 |- ( A e. (SubRing ` R ) -> ( ||r ` S ) = ( ||r ` S ) )
70		eqidd 2197	. . . . 5 |- ( A e. (SubRing ` R ) -> (oppr ` S ) = (oppr ` S ) )
71		eqidd 2197	. . . . 5 |- ( A e. (SubRing ` R ) -> ( ||r ` (oppr ` S ) ) = ( ||r ` (oppr ` S ) ) )
72	67, 68, 69, 70, 71, 11	isunitd 13662	. . . 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 946	. 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 980   = wceq 1364   e. wcel 2167   class class class wbr 4033   ` cfv 5258  (class class class)co 5922   Basecbs 12678   ↾_s cress 12679   .rcmulr 12756   1rcur 13515  SRingcsrg 13519   Ringcrg 13552  opp_rcoppr 13623   ||_rcdsr 13642  Unitcui 13643   invrcinvr 13676  SubRingcsubrg 13773

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 615  ax-in2 616  ax-io 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-13 2169  ax-14 2170  ax-ext 2178  ax-coll 4148  ax-sep 4151  ax-nul 4159  ax-pow 4207  ax-pr 4242  ax-un 4468  ax-setind 4573  ax-cnex 7970  ax-resscn 7971  ax-1cn 7972  ax-1re 7973  ax-icn 7974  ax-addcl 7975  ax-addrcl 7976  ax-mulcl 7977  ax-addcom 7979  ax-addass 7981  ax-i2m1 7984  ax-0lt1 7985  ax-0id 7987  ax-rnegex 7988  ax-pre-ltirr 7991  ax-pre-lttrn 7993  ax-pre-ltadd 7995

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ne 2368  df-nel 2463  df-ral 2480  df-rex 2481  df-reu 2482  df-rmo 2483  df-rab 2484  df-v 2765  df-sbc 2990  df-csb 3085  df-dif 3159  df-un 3161  df-in 3163  df-ss 3170  df-nul 3451  df-pw 3607  df-sn 3628  df-pr 3629  df-op 3631  df-uni 3840  df-int 3875  df-iun 3918  df-br 4034  df-opab 4095  df-mpt 4096  df-id 4328  df-xp 4669  df-rel 4670  df-cnv 4671  df-co 4672  df-dm 4673  df-rn 4674  df-res 4675  df-ima 4676  df-iota 5219  df-fun 5260  df-fn 5261  df-f 5262  df-f1 5263  df-fo 5264  df-f1o 5265  df-fv 5266  df-riota 5877  df-ov 5925  df-oprab 5926  df-mpo 5927  df-tpos 6303  df-pnf 8063  df-mnf 8064  df-ltxr 8066  df-inn 8991  df-2 9049  df-3 9050  df-ndx 12681  df-slot 12682  df-base 12684  df-sets 12685  df-iress 12686  df-plusg 12768  df-mulr 12769  df-0g 12929  df-mgm 12999  df-sgrp 13045  df-mnd 13058  df-grp 13135  df-minusg 13136  df-subg 13300  df-cmn 13416  df-abl 13417  df-mgp 13477  df-ur 13516  df-srg 13520  df-ring 13554  df-oppr 13624  df-dvdsr 13645  df-unit 13646  df-invr 13677  df-subrg 13775

This theorem is referenced by: (None)