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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  =  ( Rs  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
StepHypRef Expression
1 subrgugrp.1 . . . . 5  |-  S  =  ( Rs  A )
2 subrgugrp.2 . . . . 5  |-  U  =  (Unit `  R )
3 subrgugrp.3 . . . . 5  |-  V  =  (Unit `  S )
41, 2, 3subrguss 13295 . . . 4  |-  ( A  e.  (SubRing `  R
)  ->  V  C_  U
)
54sselda 3155 . . 3  |-  ( ( A  e.  (SubRing `  R
)  /\  X  e.  V )  ->  X  e.  U )
61subrgbas 13289 . . . . 5  |-  ( A  e.  (SubRing `  R
)  ->  A  =  ( Base `  S )
)
76adantr 276 . . . 4  |-  ( ( A  e.  (SubRing `  R
)  /\  X  e.  V )  ->  A  =  ( Base `  S
) )
83a1i 9 . . . 4  |-  ( ( A  e.  (SubRing `  R
)  /\  X  e.  V )  ->  V  =  (Unit `  S )
)
91subrgring 13283 . . . . . 6  |-  ( A  e.  (SubRing `  R
)  ->  S  e.  Ring )
10 ringsrg 13155 . . . . . 6  |-  ( S  e.  Ring  ->  S  e. SRing
)
119, 10syl 14 . . . . 5  |-  ( A  e.  (SubRing `  R
)  ->  S  e. SRing )
1211adantr 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 )
147, 8, 12, 13unitcld 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 )
173, 15, 16ringinvcl 13225 . . . . 5  |-  ( ( S  e.  Ring  /\  X  e.  V )  ->  (
( invr `  S ) `  X )  e.  (
Base `  S )
)
189, 17sylan 283 . . . 4  |-  ( ( A  e.  (SubRing `  R
)  /\  X  e.  V )  ->  (
( invr `  S ) `  X )  e.  (
Base `  S )
)
19 subrgunit.4 . . . . 5  |-  I  =  ( invr `  R
)
201, 19, 3, 15subrginv 13296 . . . 4  |-  ( ( A  e.  (SubRing `  R
)  /\  X  e.  V )  ->  (
I `  X )  =  ( ( invr `  S ) `  X
) )
2118, 20, 73eltr4d 2261 . . 3  |-  ( ( A  e.  (SubRing `  R
)  /\  X  e.  V )  ->  (
I `  X )  e.  A )
225, 14, 213jca 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 ) )
2511adantr 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 )
286adantr 276 . . . . . 6  |-  ( ( A  e.  (SubRing `  R
)  /\  ( X  e.  U  /\  X  e.  A  /\  ( I `
 X )  e.  A ) )  ->  A  =  ( Base `  S ) )
2927, 28eleqtrd 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 )
3130, 28eleqtrd 2256 . . . . 5  |-  ( ( A  e.  (SubRing `  R
)  /\  ( X  e.  U  /\  X  e.  A  /\  ( I `
 X )  e.  A ) )  -> 
( I `  X
)  e.  ( Base `  S ) )
3223, 24, 25, 26, 29, 31dvdsrmuld 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
)
372, 19, 35, 36unitlinv 13226 . . . . . 6  |-  ( ( R  e.  Ring  /\  X  e.  U )  ->  (
( I `  X
) ( .r `  R ) X )  =  ( 1r `  R ) )
3833, 34, 37syl2an2r 595 . . . . 5  |-  ( ( A  e.  (SubRing `  R
)  /\  ( X  e.  U  /\  X  e.  A  /\  ( I `
 X )  e.  A ) )  -> 
( ( I `  X ) ( .r
`  R ) X )  =  ( 1r
`  R ) )
391, 35ressmulrg 12595 . . . . . . . 8  |-  ( ( A  e.  (SubRing `  R
)  /\  R  e.  Ring )  ->  ( .r `  R )  =  ( .r `  S ) )
4033, 39mpdan 421 . . . . . . 7  |-  ( A  e.  (SubRing `  R
)  ->  ( .r `  R )  =  ( .r `  S ) )
4140adantr 276 . . . . . 6  |-  ( ( A  e.  (SubRing `  R
)  /\  ( X  e.  U  /\  X  e.  A  /\  ( I `
 X )  e.  A ) )  -> 
( .r `  R
)  =  ( .r
`  S ) )
4241oveqd 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 ) )
431, 36subrg1 13290 . . . . . 6  |-  ( A  e.  (SubRing `  R
)  ->  ( 1r `  R )  =  ( 1r `  S ) )
4443adantr 276 . . . . 5  |-  ( ( A  e.  (SubRing `  R
)  /\  ( X  e.  U  /\  X  e.  A  /\  ( I `
 X )  e.  A ) )  -> 
( 1r `  R
)  =  ( 1r
`  S ) )
4538, 42, 443eqtr3d 2218 . . . 4  |-  ( ( A  e.  (SubRing `  R
)  /\  ( X  e.  U  /\  X  e.  A  /\  ( I `
 X )  e.  A ) )  -> 
( ( I `  X ) ( .r
`  S ) X )  =  ( 1r
`  S ) )
4632, 45breqtrd 4028 . . 3  |-  ( ( A  e.  (SubRing `  R
)  /\  ( X  e.  U  /\  X  e.  A  /\  ( I `
 X )  e.  A ) )  ->  X ( ||r `
 S ) ( 1r `  S ) )
479adantr 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
)
4948, 16opprbasg 13178 . . . . . 6  |-  ( S  e.  Ring  ->  ( Base `  S )  =  (
Base `  (oppr
`  S ) ) )
5047, 49syl 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 ) ) )
5248opprring 13180 . . . . . 6  |-  ( S  e.  Ring  ->  (oppr `  S
)  e.  Ring )
53 ringsrg 13155 . . . . . 6  |-  ( (oppr `  S )  e.  Ring  -> 
(oppr `  S )  e. SRing )
5447, 52, 533syl 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
) ) )
5650, 51, 54, 55, 29, 31dvdsrmuld 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 ) )
5916, 57, 48, 58opprmulg 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
) ) )
6047, 31, 29, 59syl3anc 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 ) ) )
612, 19, 35, 36unitrinv 13227 . . . . . . 7  |-  ( ( R  e.  Ring  /\  X  e.  U )  ->  ( X ( .r `  R ) ( I `
 X ) )  =  ( 1r `  R ) )
6233, 34, 61syl2an2r 595 . . . . . 6  |-  ( ( A  e.  (SubRing `  R
)  /\  ( X  e.  U  /\  X  e.  A  /\  ( I `
 X )  e.  A ) )  -> 
( X ( .r
`  R ) ( I `  X ) )  =  ( 1r
`  R ) )
6341oveqd 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 ) ) )
6462, 63, 443eqtr3d 2218 . . . . 5  |-  ( ( A  e.  (SubRing `  R
)  /\  ( X  e.  U  /\  X  e.  A  /\  ( I `
 X )  e.  A ) )  -> 
( X ( .r
`  S ) ( I `  X ) )  =  ( 1r
`  S ) )
6560, 64eqtrd 2210 . . . 4  |-  ( ( A  e.  (SubRing `  R
)  /\  ( X  e.  U  /\  X  e.  A  /\  ( I `
 X )  e.  A ) )  -> 
( ( I `  X ) ( .r
`  (oppr
`  S ) ) X )  =  ( 1r `  S ) )
6656, 65breqtrd 4028 . . 3  |-  ( ( A  e.  (SubRing `  R
)  /\  ( X  e.  U  /\  X  e.  A  /\  ( I `
 X )  e.  A ) )  ->  X ( ||r `
 (oppr
`  S ) ) ( 1r `  S
) )
673a1i 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 ) ) )
7267, 68, 69, 70, 71, 11isunitd 13206 . . . 4  |-  ( A  e.  (SubRing `  R
)  ->  ( X  e.  V  <->  ( X (
||r `  S ) ( 1r
`  S )  /\  X ( ||r `
 (oppr
`  S ) ) ( 1r `  S
) ) ) )
7372adantr 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 ) ) ) )
7446, 66, 73mpbir2and 944 . 2  |-  ( ( A  e.  (SubRing `  R
)  /\  ( X  e.  U  /\  X  e.  A  /\  ( I `
 X )  e.  A ) )  ->  X  e.  V )
7522, 74impbida 596 1  |-  ( A  e.  (SubRing `  R
)  ->  ( X  e.  V  <->  ( X  e.  U  /\  X  e.  A  /\  ( I `
 X )  e.  A ) ) )
Colors of variables: wff set class
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  opprcoppr 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)
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