MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  subrgunit Structured version   Unicode version

Theorem subrgunit 17247
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 17244 . . . 4  |-  ( A  e.  (SubRing `  R
)  ->  V  C_  U
)
54sselda 3504 . . 3  |-  ( ( A  e.  (SubRing `  R
)  /\  X  e.  V )  ->  X  e.  U )
6 eqid 2467 . . . . . 6  |-  ( Base `  S )  =  (
Base `  S )
76, 3unitcl 17109 . . . . 5  |-  ( X  e.  V  ->  X  e.  ( Base `  S
) )
87adantl 466 . . . 4  |-  ( ( A  e.  (SubRing `  R
)  /\  X  e.  V )  ->  X  e.  ( Base `  S
) )
91subrgbas 17238 . . . . 5  |-  ( A  e.  (SubRing `  R
)  ->  A  =  ( Base `  S )
)
109adantr 465 . . . 4  |-  ( ( A  e.  (SubRing `  R
)  /\  X  e.  V )  ->  A  =  ( Base `  S
) )
118, 10eleqtrrd 2558 . . 3  |-  ( ( A  e.  (SubRing `  R
)  /\  X  e.  V )  ->  X  e.  A )
121subrgrng 17232 . . . . 5  |-  ( A  e.  (SubRing `  R
)  ->  S  e.  Ring )
13 eqid 2467 . . . . . 6  |-  ( invr `  S )  =  (
invr `  S )
143, 13, 6rnginvcl 17126 . . . . 5  |-  ( ( S  e.  Ring  /\  X  e.  V )  ->  (
( invr `  S ) `  X )  e.  (
Base `  S )
)
1512, 14sylan 471 . . . 4  |-  ( ( A  e.  (SubRing `  R
)  /\  X  e.  V )  ->  (
( invr `  S ) `  X )  e.  (
Base `  S )
)
16 subrgunit.4 . . . . 5  |-  I  =  ( invr `  R
)
171, 16, 3, 13subrginv 17245 . . . 4  |-  ( ( A  e.  (SubRing `  R
)  /\  X  e.  V )  ->  (
I `  X )  =  ( ( invr `  S ) `  X
) )
1815, 17, 103eltr4d 2570 . . 3  |-  ( ( A  e.  (SubRing `  R
)  /\  X  e.  V )  ->  (
I `  X )  e.  A )
195, 11, 183jca 1176 . 2  |-  ( ( A  e.  (SubRing `  R
)  /\  X  e.  V )  ->  ( X  e.  U  /\  X  e.  A  /\  ( I `  X
)  e.  A ) )
20 simpr2 1003 . . . . . 6  |-  ( ( A  e.  (SubRing `  R
)  /\  ( X  e.  U  /\  X  e.  A  /\  ( I `
 X )  e.  A ) )  ->  X  e.  A )
219adantr 465 . . . . . 6  |-  ( ( A  e.  (SubRing `  R
)  /\  ( X  e.  U  /\  X  e.  A  /\  ( I `
 X )  e.  A ) )  ->  A  =  ( Base `  S ) )
2220, 21eleqtrd 2557 . . . . 5  |-  ( ( A  e.  (SubRing `  R
)  /\  ( X  e.  U  /\  X  e.  A  /\  ( I `
 X )  e.  A ) )  ->  X  e.  ( Base `  S ) )
23 simpr3 1004 . . . . . 6  |-  ( ( A  e.  (SubRing `  R
)  /\  ( X  e.  U  /\  X  e.  A  /\  ( I `
 X )  e.  A ) )  -> 
( I `  X
)  e.  A )
2423, 21eleqtrd 2557 . . . . 5  |-  ( ( A  e.  (SubRing `  R
)  /\  ( X  e.  U  /\  X  e.  A  /\  ( I `
 X )  e.  A ) )  -> 
( I `  X
)  e.  ( Base `  S ) )
25 eqid 2467 . . . . . 6  |-  ( ||r `  S
)  =  ( ||r `  S
)
26 eqid 2467 . . . . . 6  |-  ( .r
`  S )  =  ( .r `  S
)
276, 25, 26dvdsrmul 17098 . . . . 5  |-  ( ( X  e.  ( Base `  S )  /\  (
I `  X )  e.  ( Base `  S
) )  ->  X
( ||r `
 S ) ( ( I `  X
) ( .r `  S ) X ) )
2822, 24, 27syl2anc 661 . . . 4  |-  ( ( A  e.  (SubRing `  R
)  /\  ( X  e.  U  /\  X  e.  A  /\  ( I `
 X )  e.  A ) )  ->  X ( ||r `
 S ) ( ( I `  X
) ( .r `  S ) X ) )
29 subrgrcl 17234 . . . . . . 7  |-  ( A  e.  (SubRing `  R
)  ->  R  e.  Ring )
3029adantr 465 . . . . . 6  |-  ( ( A  e.  (SubRing `  R
)  /\  ( X  e.  U  /\  X  e.  A  /\  ( I `
 X )  e.  A ) )  ->  R  e.  Ring )
31 simpr1 1002 . . . . . 6  |-  ( ( A  e.  (SubRing `  R
)  /\  ( X  e.  U  /\  X  e.  A  /\  ( I `
 X )  e.  A ) )  ->  X  e.  U )
32 eqid 2467 . . . . . . 7  |-  ( .r
`  R )  =  ( .r `  R
)
33 eqid 2467 . . . . . . 7  |-  ( 1r
`  R )  =  ( 1r `  R
)
342, 16, 32, 33unitlinv 17127 . . . . . 6  |-  ( ( R  e.  Ring  /\  X  e.  U )  ->  (
( I `  X
) ( .r `  R ) X )  =  ( 1r `  R ) )
3530, 31, 34syl2anc 661 . . . . 5  |-  ( ( A  e.  (SubRing `  R
)  /\  ( X  e.  U  /\  X  e.  A  /\  ( I `
 X )  e.  A ) )  -> 
( ( I `  X ) ( .r
`  R ) X )  =  ( 1r
`  R ) )
361, 32ressmulr 14608 . . . . . . 7  |-  ( A  e.  (SubRing `  R
)  ->  ( .r `  R )  =  ( .r `  S ) )
3736adantr 465 . . . . . 6  |-  ( ( A  e.  (SubRing `  R
)  /\  ( X  e.  U  /\  X  e.  A  /\  ( I `
 X )  e.  A ) )  -> 
( .r `  R
)  =  ( .r
`  S ) )
3837oveqd 6301 . . . . 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 ) )
391, 33subrg1 17239 . . . . . 6  |-  ( A  e.  (SubRing `  R
)  ->  ( 1r `  R )  =  ( 1r `  S ) )
4039adantr 465 . . . . 5  |-  ( ( A  e.  (SubRing `  R
)  /\  ( X  e.  U  /\  X  e.  A  /\  ( I `
 X )  e.  A ) )  -> 
( 1r `  R
)  =  ( 1r
`  S ) )
4135, 38, 403eqtr3d 2516 . . . 4  |-  ( ( A  e.  (SubRing `  R
)  /\  ( X  e.  U  /\  X  e.  A  /\  ( I `
 X )  e.  A ) )  -> 
( ( I `  X ) ( .r
`  S ) X )  =  ( 1r
`  S ) )
4228, 41breqtrd 4471 . . 3  |-  ( ( A  e.  (SubRing `  R
)  /\  ( X  e.  U  /\  X  e.  A  /\  ( I `
 X )  e.  A ) )  ->  X ( ||r `
 S ) ( 1r `  S ) )
43 eqid 2467 . . . . . . 7  |-  (oppr `  S
)  =  (oppr `  S
)
4443, 6opprbas 17079 . . . . . 6  |-  ( Base `  S )  =  (
Base `  (oppr
`  S ) )
45 eqid 2467 . . . . . 6  |-  ( ||r `  (oppr `  S
) )  =  (
||r `  (oppr
`  S ) )
46 eqid 2467 . . . . . 6  |-  ( .r
`  (oppr
`  S ) )  =  ( .r `  (oppr `  S ) )
4744, 45, 46dvdsrmul 17098 . . . . 5  |-  ( ( X  e.  ( Base `  S )  /\  (
I `  X )  e.  ( Base `  S
) )  ->  X
( ||r `
 (oppr
`  S ) ) ( ( I `  X ) ( .r
`  (oppr
`  S ) ) X ) )
4822, 24, 47syl2anc 661 . . . 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 ) )
496, 26, 43, 46opprmul 17076 . . . . 5  |-  ( ( I `  X ) ( .r `  (oppr `  S
) ) X )  =  ( X ( .r `  S ) ( I `  X
) )
502, 16, 32, 33unitrinv 17128 . . . . . . 7  |-  ( ( R  e.  Ring  /\  X  e.  U )  ->  ( X ( .r `  R ) ( I `
 X ) )  =  ( 1r `  R ) )
5130, 31, 50syl2anc 661 . . . . . 6  |-  ( ( A  e.  (SubRing `  R
)  /\  ( X  e.  U  /\  X  e.  A  /\  ( I `
 X )  e.  A ) )  -> 
( X ( .r
`  R ) ( I `  X ) )  =  ( 1r
`  R ) )
5237oveqd 6301 . . . . . 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 ) ) )
5351, 52, 403eqtr3d 2516 . . . . 5  |-  ( ( A  e.  (SubRing `  R
)  /\  ( X  e.  U  /\  X  e.  A  /\  ( I `
 X )  e.  A ) )  -> 
( X ( .r
`  S ) ( I `  X ) )  =  ( 1r
`  S ) )
5449, 53syl5eq 2520 . . . 4  |-  ( ( A  e.  (SubRing `  R
)  /\  ( X  e.  U  /\  X  e.  A  /\  ( I `
 X )  e.  A ) )  -> 
( ( I `  X ) ( .r
`  (oppr
`  S ) ) X )  =  ( 1r `  S ) )
5548, 54breqtrd 4471 . . 3  |-  ( ( A  e.  (SubRing `  R
)  /\  ( X  e.  U  /\  X  e.  A  /\  ( I `
 X )  e.  A ) )  ->  X ( ||r `
 (oppr
`  S ) ) ( 1r `  S
) )
56 eqid 2467 . . . 4  |-  ( 1r
`  S )  =  ( 1r `  S
)
573, 56, 25, 43, 45isunit 17107 . . 3  |-  ( X  e.  V  <->  ( X
( ||r `
 S ) ( 1r `  S )  /\  X ( ||r `  (oppr `  S
) ) ( 1r
`  S ) ) )
5842, 55, 57sylanbrc 664 . 2  |-  ( ( A  e.  (SubRing `  R
)  /\  ( X  e.  U  /\  X  e.  A  /\  ( I `
 X )  e.  A ) )  ->  X  e.  V )
5919, 58impbida 830 1  |-  ( A  e.  (SubRing `  R
)  ->  ( X  e.  V  <->  ( X  e.  U  /\  X  e.  A  /\  ( I `
 X )  e.  A ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 973    = wceq 1379    e. wcel 1767   class class class wbr 4447   ` cfv 5588  (class class class)co 6284   Basecbs 14490   ↾s cress 14491   .rcmulr 14556   1rcur 16955   Ringcrg 17000  opprcoppr 17072   ||rcdsr 17088  Unitcui 17089   invrcinvr 17121  SubRingcsubrg 17225
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6576  ax-cnex 9548  ax-resscn 9549  ax-1cn 9550  ax-icn 9551  ax-addcl 9552  ax-addrcl 9553  ax-mulcl 9554  ax-mulrcl 9555  ax-mulcom 9556  ax-addass 9557  ax-mulass 9558  ax-distr 9559  ax-i2m1 9560  ax-1ne0 9561  ax-1rid 9562  ax-rnegex 9563  ax-rrecex 9564  ax-cnre 9565  ax-pre-lttri 9566  ax-pre-lttrn 9567  ax-pre-ltadd 9568  ax-pre-mulgt0 9569
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2819  df-rex 2820  df-reu 2821  df-rmo 2822  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-tp 4032  df-op 4034  df-uni 4246  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-tr 4541  df-eprel 4791  df-id 4795  df-po 4800  df-so 4801  df-fr 4838  df-we 4840  df-ord 4881  df-on 4882  df-lim 4883  df-suc 4884  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5551  df-fun 5590  df-fn 5591  df-f 5592  df-f1 5593  df-fo 5594  df-f1o 5595  df-fv 5596  df-riota 6245  df-ov 6287  df-oprab 6288  df-mpt2 6289  df-om 6685  df-tpos 6955  df-recs 7042  df-rdg 7076  df-er 7311  df-en 7517  df-dom 7518  df-sdom 7519  df-pnf 9630  df-mnf 9631  df-xr 9632  df-ltxr 9633  df-le 9634  df-sub 9807  df-neg 9808  df-nn 10537  df-2 10594  df-3 10595  df-ndx 14493  df-slot 14494  df-base 14495  df-sets 14496  df-ress 14497  df-plusg 14568  df-mulr 14569  df-0g 14697  df-mnd 15732  df-grp 15867  df-minusg 15868  df-subg 16003  df-mgp 16944  df-ur 16956  df-rng 17002  df-oppr 17073  df-dvdsr 17091  df-unit 17092  df-invr 17122  df-subrg 17227
This theorem is referenced by:  issubdrg  17254  gzrngunit  18279  zringunit  18315  zrngunit  18316  cphreccllem  21388
  Copyright terms: Public domain W3C validator