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Theorem subrgunit 17426
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 17423 . . . 4  |-  ( A  e.  (SubRing `  R
)  ->  V  C_  U
)
54sselda 3489 . . 3  |-  ( ( A  e.  (SubRing `  R
)  /\  X  e.  V )  ->  X  e.  U )
6 eqid 2443 . . . . . 6  |-  ( Base `  S )  =  (
Base `  S )
76, 3unitcl 17287 . . . . 5  |-  ( X  e.  V  ->  X  e.  ( Base `  S
) )
87adantl 466 . . . 4  |-  ( ( A  e.  (SubRing `  R
)  /\  X  e.  V )  ->  X  e.  ( Base `  S
) )
91subrgbas 17417 . . . . 5  |-  ( A  e.  (SubRing `  R
)  ->  A  =  ( Base `  S )
)
109adantr 465 . . . 4  |-  ( ( A  e.  (SubRing `  R
)  /\  X  e.  V )  ->  A  =  ( Base `  S
) )
118, 10eleqtrrd 2534 . . 3  |-  ( ( A  e.  (SubRing `  R
)  /\  X  e.  V )  ->  X  e.  A )
121subrgring 17411 . . . . 5  |-  ( A  e.  (SubRing `  R
)  ->  S  e.  Ring )
13 eqid 2443 . . . . . 6  |-  ( invr `  S )  =  (
invr `  S )
143, 13, 6ringinvcl 17304 . . . . 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 17424 . . . 4  |-  ( ( A  e.  (SubRing `  R
)  /\  X  e.  V )  ->  (
I `  X )  =  ( ( invr `  S ) `  X
) )
1815, 17, 103eltr4d 2546 . . 3  |-  ( ( A  e.  (SubRing `  R
)  /\  X  e.  V )  ->  (
I `  X )  e.  A )
195, 11, 183jca 1177 . 2  |-  ( ( A  e.  (SubRing `  R
)  /\  X  e.  V )  ->  ( X  e.  U  /\  X  e.  A  /\  ( I `  X
)  e.  A ) )
20 simpr2 1004 . . . . . 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 2533 . . . . 5  |-  ( ( A  e.  (SubRing `  R
)  /\  ( X  e.  U  /\  X  e.  A  /\  ( I `
 X )  e.  A ) )  ->  X  e.  ( Base `  S ) )
23 simpr3 1005 . . . . . 6  |-  ( ( A  e.  (SubRing `  R
)  /\  ( X  e.  U  /\  X  e.  A  /\  ( I `
 X )  e.  A ) )  -> 
( I `  X
)  e.  A )
2423, 21eleqtrd 2533 . . . . 5  |-  ( ( A  e.  (SubRing `  R
)  /\  ( X  e.  U  /\  X  e.  A  /\  ( I `
 X )  e.  A ) )  -> 
( I `  X
)  e.  ( Base `  S ) )
25 eqid 2443 . . . . . 6  |-  ( ||r `  S
)  =  ( ||r `  S
)
26 eqid 2443 . . . . . 6  |-  ( .r
`  S )  =  ( .r `  S
)
276, 25, 26dvdsrmul 17276 . . . . 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 17413 . . . . . . 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 1003 . . . . . 6  |-  ( ( A  e.  (SubRing `  R
)  /\  ( X  e.  U  /\  X  e.  A  /\  ( I `
 X )  e.  A ) )  ->  X  e.  U )
32 eqid 2443 . . . . . . 7  |-  ( .r
`  R )  =  ( .r `  R
)
33 eqid 2443 . . . . . . 7  |-  ( 1r
`  R )  =  ( 1r `  R
)
342, 16, 32, 33unitlinv 17305 . . . . . 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 14732 . . . . . . 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 6298 . . . . 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 17418 . . . . . 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 2492 . . . 4  |-  ( ( A  e.  (SubRing `  R
)  /\  ( X  e.  U  /\  X  e.  A  /\  ( I `
 X )  e.  A ) )  -> 
( ( I `  X ) ( .r
`  S ) X )  =  ( 1r
`  S ) )
4228, 41breqtrd 4461 . . 3  |-  ( ( A  e.  (SubRing `  R
)  /\  ( X  e.  U  /\  X  e.  A  /\  ( I `
 X )  e.  A ) )  ->  X ( ||r `
 S ) ( 1r `  S ) )
43 eqid 2443 . . . . . . 7  |-  (oppr `  S
)  =  (oppr `  S
)
4443, 6opprbas 17257 . . . . . 6  |-  ( Base `  S )  =  (
Base `  (oppr
`  S ) )
45 eqid 2443 . . . . . 6  |-  ( ||r `  (oppr `  S
) )  =  (
||r `  (oppr
`  S ) )
46 eqid 2443 . . . . . 6  |-  ( .r
`  (oppr
`  S ) )  =  ( .r `  (oppr `  S ) )
4744, 45, 46dvdsrmul 17276 . . . . 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 17254 . . . . 5  |-  ( ( I `  X ) ( .r `  (oppr `  S
) ) X )  =  ( X ( .r `  S ) ( I `  X
) )
502, 16, 32, 33unitrinv 17306 . . . . . . 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 6298 . . . . . 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 2492 . . . . 5  |-  ( ( A  e.  (SubRing `  R
)  /\  ( X  e.  U  /\  X  e.  A  /\  ( I `
 X )  e.  A ) )  -> 
( X ( .r
`  S ) ( I `  X ) )  =  ( 1r
`  S ) )
5449, 53syl5eq 2496 . . . 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 4461 . . 3  |-  ( ( A  e.  (SubRing `  R
)  /\  ( X  e.  U  /\  X  e.  A  /\  ( I `
 X )  e.  A ) )  ->  X ( ||r `
 (oppr
`  S ) ) ( 1r `  S
) )
56 eqid 2443 . . . 4  |-  ( 1r
`  S )  =  ( 1r `  S
)
573, 56, 25, 43, 45isunit 17285 . . 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 832 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 974    = wceq 1383    e. wcel 1804   class class class wbr 4437   ` cfv 5578  (class class class)co 6281   Basecbs 14614   ↾s cress 14615   .rcmulr 14680   1rcur 17132   Ringcrg 17177  opprcoppr 17250   ||rcdsr 17266  Unitcui 17267   invrcinvr 17299  SubRingcsubrg 17404
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1605  ax-4 1618  ax-5 1691  ax-6 1734  ax-7 1776  ax-8 1806  ax-9 1808  ax-10 1823  ax-11 1828  ax-12 1840  ax-13 1985  ax-ext 2421  ax-rep 4548  ax-sep 4558  ax-nul 4566  ax-pow 4615  ax-pr 4676  ax-un 6577  ax-cnex 9551  ax-resscn 9552  ax-1cn 9553  ax-icn 9554  ax-addcl 9555  ax-addrcl 9556  ax-mulcl 9557  ax-mulrcl 9558  ax-mulcom 9559  ax-addass 9560  ax-mulass 9561  ax-distr 9562  ax-i2m1 9563  ax-1ne0 9564  ax-1rid 9565  ax-rnegex 9566  ax-rrecex 9567  ax-cnre 9568  ax-pre-lttri 9569  ax-pre-lttrn 9570  ax-pre-ltadd 9571  ax-pre-mulgt0 9572
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 975  df-3an 976  df-tru 1386  df-ex 1600  df-nf 1604  df-sb 1727  df-eu 2272  df-mo 2273  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2593  df-ne 2640  df-nel 2641  df-ral 2798  df-rex 2799  df-reu 2800  df-rmo 2801  df-rab 2802  df-v 3097  df-sbc 3314  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-pss 3477  df-nul 3771  df-if 3927  df-pw 3999  df-sn 4015  df-pr 4017  df-tp 4019  df-op 4021  df-uni 4235  df-iun 4317  df-br 4438  df-opab 4496  df-mpt 4497  df-tr 4531  df-eprel 4781  df-id 4785  df-po 4790  df-so 4791  df-fr 4828  df-we 4830  df-ord 4871  df-on 4872  df-lim 4873  df-suc 4874  df-xp 4995  df-rel 4996  df-cnv 4997  df-co 4998  df-dm 4999  df-rn 5000  df-res 5001  df-ima 5002  df-iota 5541  df-fun 5580  df-fn 5581  df-f 5582  df-f1 5583  df-fo 5584  df-f1o 5585  df-fv 5586  df-riota 6242  df-ov 6284  df-oprab 6285  df-mpt2 6286  df-om 6686  df-tpos 6957  df-recs 7044  df-rdg 7078  df-er 7313  df-en 7519  df-dom 7520  df-sdom 7521  df-pnf 9633  df-mnf 9634  df-xr 9635  df-ltxr 9636  df-le 9637  df-sub 9812  df-neg 9813  df-nn 10544  df-2 10601  df-3 10602  df-ndx 14617  df-slot 14618  df-base 14619  df-sets 14620  df-ress 14621  df-plusg 14692  df-mulr 14693  df-0g 14821  df-mgm 15851  df-sgrp 15890  df-mnd 15900  df-grp 16036  df-minusg 16037  df-subg 16177  df-mgp 17121  df-ur 17133  df-ring 17179  df-oppr 17251  df-dvdsr 17269  df-unit 17270  df-invr 17300  df-subrg 17406
This theorem is referenced by:  issubdrg  17433  gzrngunit  18462  zringunit  18498  zrngunit  18499  cphreccllem  21603
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