Theorem subrgunit 14076

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 14073	. . . 4 |- ( A e. (SubRing ` R ) -> V C_ U )
5	4	sselda 3197	. . 3 |- ( ( A e. (SubRing ` R ) /\ X e. V ) -> X e. U )
6	1	subrgbas 14067	. . . . 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 14061	. . . . . 6 |- ( A e. (SubRing ` R ) -> S e. Ring )
10		ringsrg 13884	. . . . . 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 13945	. . 3 |- ( ( A e. (SubRing ` R ) /\ X e. V ) -> X e. A )
15		eqid 2206	. . . . . 6 |- ( invr ` S ) = ( invr ` S )
16		eqid 2206	. . . . . 6 |- ( Base ` S ) = ( Base ` S )
17	3, 15, 16	ringinvcl 13962	. . . . 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 14074	. . . 4 |- ( ( A e. (SubRing ` R ) /\ X e. V ) -> ( I ` X ) = ( ( invr ` S ) ` X ) )
21	18, 20, 7	3eltr4d 2290	. . 3 |- ( ( A e. (SubRing ` R ) /\ X e. V ) -> ( I ` X ) e. A )
22	5, 14, 21	3jca 1180	. 2 |- ( ( A e. (SubRing ` R ) /\ X e. V ) -> ( X e. U /\ X e. A /\ ( I ` X ) e. A ) )
23		eqidd 2207	. . . . 5 |- ( ( A e. (SubRing ` R ) /\ ( X e. U /\ X e. A /\ ( I ` X ) e. A ) ) -> ( Base ` S ) = ( Base ` S ) )
24		eqidd 2207	. . . . 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 2207	. . . . 5 |- ( ( A e. (SubRing ` R ) /\ ( X e. U /\ X e. A /\ ( I ` X ) e. A ) ) -> ( .r ` S ) = ( .r ` S ) )
27		simpr2 1007	. . . . . 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 2285	. . . . 5 |- ( ( A e. (SubRing ` R ) /\ ( X e. U /\ X e. A /\ ( I ` X ) e. A ) ) -> X e. ( Base ` S ) )
30		simpr3 1008	. . . . . 6 |- ( ( A e. (SubRing ` R ) /\ ( X e. U /\ X e. A /\ ( I ` X ) e. A ) ) -> ( I ` X ) e. A )
31	30, 28	eleqtrd 2285	. . . . 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 13933	. . . 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 14063	. . . . . 6 |- ( A e. (SubRing ` R ) -> R e. Ring )
34		simpr1 1006	. . . . . 6 |- ( ( A e. (SubRing ` R ) /\ ( X e. U /\ X e. A /\ ( I ` X ) e. A ) ) -> X e. U )
35		eqid 2206	. . . . . . 7 |- ( .r ` R ) = ( .r ` R )
36		eqid 2206	. . . . . . 7 |- ( 1r ` R ) = ( 1r ` R )
37	2, 19, 35, 36	unitlinv 13963	. . . . . 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 13052	. . . . . . . 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 5974	. . . . 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 14068	. . . . . 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 2247	. . . 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 4077	. . 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 2206	. . . . . . 7 |- (oppr ` S ) = (oppr ` S )
49	48, 16	opprbasg 13912	. . . . . 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 2207	. . . . 5 |- ( ( A e. (SubRing ` R ) /\ ( X e. U /\ X e. A /\ ( I ` X ) e. A ) ) -> ( ||r ` (oppr ` S ) ) = ( ||r ` (oppr ` S ) ) )
52	48	opprring 13916	. . . . . 6 |- ( S e. Ring -> (oppr ` S ) e. Ring )
53		ringsrg 13884	. . . . . 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 2207	. . . . 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 13933	. . . 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 2206	. . . . . . 7 |- ( .r ` S ) = ( .r ` S )
58		eqid 2206	. . . . . . 7 |- ( .r ` (oppr ` S ) ) = ( .r ` (oppr ` S ) )
59	16, 57, 48, 58	opprmulg 13908	. . . . . 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 1250	. . . . 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 13964	. . . . . . 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 5974	. . . . . 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 2247	. . . . 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 2239	. . . 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 4077	. . 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 2207	. . . . 5 |- ( A e. (SubRing ` R ) -> ( 1r ` S ) = ( 1r ` S ) )
69		eqidd 2207	. . . . 5 |- ( A e. (SubRing ` R ) -> ( ||r ` S ) = ( ||r ` S ) )
70		eqidd 2207	. . . . 5 |- ( A e. (SubRing ` R ) -> (oppr ` S ) = (oppr ` S ) )
71		eqidd 2207	. . . . 5 |- ( A e. (SubRing ` R ) -> ( ||r ` (oppr ` S ) ) = ( ||r ` (oppr ` S ) ) )
72	67, 68, 69, 70, 71, 11	isunitd 13943	. . . 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 947	. 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 981   = wceq 1373   e. wcel 2177   class class class wbr 4051   ` cfv 5280  (class class class)co 5957   Basecbs 12907   ↾_s cress 12908   .rcmulr 12985   1rcur 13796  SRingcsrg 13800   Ringcrg 13833  opp_rcoppr 13904   ||_rcdsr 13923  Unitcui 13924   invrcinvr 13957  SubRingcsubrg 14054

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 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-13 2179  ax-14 2180  ax-ext 2188  ax-coll 4167  ax-sep 4170  ax-nul 4178  ax-pow 4226  ax-pr 4261  ax-un 4488  ax-setind 4593  ax-cnex 8036  ax-resscn 8037  ax-1cn 8038  ax-1re 8039  ax-icn 8040  ax-addcl 8041  ax-addrcl 8042  ax-mulcl 8043  ax-addcom 8045  ax-addass 8047  ax-i2m1 8050  ax-0lt1 8051  ax-0id 8053  ax-rnegex 8054  ax-pre-ltirr 8057  ax-pre-lttrn 8059  ax-pre-ltadd 8061

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-fal 1379  df-nf 1485  df-sb 1787  df-eu 2058  df-mo 2059  df-clab 2193  df-cleq 2199  df-clel 2202  df-nfc 2338  df-ne 2378  df-nel 2473  df-ral 2490  df-rex 2491  df-reu 2492  df-rmo 2493  df-rab 2494  df-v 2775  df-sbc 3003  df-csb 3098  df-dif 3172  df-un 3174  df-in 3176  df-ss 3183  df-nul 3465  df-pw 3623  df-sn 3644  df-pr 3645  df-op 3647  df-uni 3857  df-int 3892  df-iun 3935  df-br 4052  df-opab 4114  df-mpt 4115  df-id 4348  df-xp 4689  df-rel 4690  df-cnv 4691  df-co 4692  df-dm 4693  df-rn 4694  df-res 4695  df-ima 4696  df-iota 5241  df-fun 5282  df-fn 5283  df-f 5284  df-f1 5285  df-fo 5286  df-f1o 5287  df-fv 5288  df-riota 5912  df-ov 5960  df-oprab 5961  df-mpo 5962  df-tpos 6344  df-pnf 8129  df-mnf 8130  df-ltxr 8132  df-inn 9057  df-2 9115  df-3 9116  df-ndx 12910  df-slot 12911  df-base 12913  df-sets 12914  df-iress 12915  df-plusg 12997  df-mulr 12998  df-0g 13165  df-mgm 13263  df-sgrp 13309  df-mnd 13324  df-grp 13410  df-minusg 13411  df-subg 13581  df-cmn 13697  df-abl 13698  df-mgp 13758  df-ur 13797  df-srg 13801  df-ring 13835  df-oppr 13905  df-dvdsr 13926  df-unit 13927  df-invr 13958  df-subrg 14056

This theorem is referenced by: (None)