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Theorem subrgunit 15563
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 15560 . . . 4  |-  ( A  e.  (SubRing `  R
)  ->  V  C_  U
)
54sselda 3180 . . 3  |-  ( ( A  e.  (SubRing `  R
)  /\  X  e.  V )  ->  X  e.  U )
6 eqid 2283 . . . . . 6  |-  ( Base `  S )  =  (
Base `  S )
76, 3unitcl 15441 . . . . 5  |-  ( X  e.  V  ->  X  e.  ( Base `  S
) )
87adantl 452 . . . 4  |-  ( ( A  e.  (SubRing `  R
)  /\  X  e.  V )  ->  X  e.  ( Base `  S
) )
91subrgbas 15554 . . . . 5  |-  ( A  e.  (SubRing `  R
)  ->  A  =  ( Base `  S )
)
109adantr 451 . . . 4  |-  ( ( A  e.  (SubRing `  R
)  /\  X  e.  V )  ->  A  =  ( Base `  S
) )
118, 10eleqtrrd 2360 . . 3  |-  ( ( A  e.  (SubRing `  R
)  /\  X  e.  V )  ->  X  e.  A )
12 subrgunit.4 . . . . 5  |-  I  =  ( invr `  R
)
13 eqid 2283 . . . . 5  |-  ( invr `  S )  =  (
invr `  S )
141, 12, 3, 13subrginv 15561 . . . 4  |-  ( ( A  e.  (SubRing `  R
)  /\  X  e.  V )  ->  (
I `  X )  =  ( ( invr `  S ) `  X
) )
151subrgrng 15548 . . . . . 6  |-  ( A  e.  (SubRing `  R
)  ->  S  e.  Ring )
163, 13, 6rnginvcl 15458 . . . . . 6  |-  ( ( S  e.  Ring  /\  X  e.  V )  ->  (
( invr `  S ) `  X )  e.  (
Base `  S )
)
1715, 16sylan 457 . . . . 5  |-  ( ( A  e.  (SubRing `  R
)  /\  X  e.  V )  ->  (
( invr `  S ) `  X )  e.  (
Base `  S )
)
1817, 10eleqtrrd 2360 . . . 4  |-  ( ( A  e.  (SubRing `  R
)  /\  X  e.  V )  ->  (
( invr `  S ) `  X )  e.  A
)
1914, 18eqeltrd 2357 . . 3  |-  ( ( A  e.  (SubRing `  R
)  /\  X  e.  V )  ->  (
I `  X )  e.  A )
205, 11, 193jca 1132 . 2  |-  ( ( A  e.  (SubRing `  R
)  /\  X  e.  V )  ->  ( X  e.  U  /\  X  e.  A  /\  ( I `  X
)  e.  A ) )
21 simpr2 962 . . . . . 6  |-  ( ( A  e.  (SubRing `  R
)  /\  ( X  e.  U  /\  X  e.  A  /\  ( I `
 X )  e.  A ) )  ->  X  e.  A )
229adantr 451 . . . . . 6  |-  ( ( A  e.  (SubRing `  R
)  /\  ( X  e.  U  /\  X  e.  A  /\  ( I `
 X )  e.  A ) )  ->  A  =  ( Base `  S ) )
2321, 22eleqtrd 2359 . . . . 5  |-  ( ( A  e.  (SubRing `  R
)  /\  ( X  e.  U  /\  X  e.  A  /\  ( I `
 X )  e.  A ) )  ->  X  e.  ( Base `  S ) )
24 simpr3 963 . . . . . 6  |-  ( ( A  e.  (SubRing `  R
)  /\  ( X  e.  U  /\  X  e.  A  /\  ( I `
 X )  e.  A ) )  -> 
( I `  X
)  e.  A )
2524, 22eleqtrd 2359 . . . . 5  |-  ( ( A  e.  (SubRing `  R
)  /\  ( X  e.  U  /\  X  e.  A  /\  ( I `
 X )  e.  A ) )  -> 
( I `  X
)  e.  ( Base `  S ) )
26 eqid 2283 . . . . . 6  |-  ( ||r `  S
)  =  ( ||r `  S
)
27 eqid 2283 . . . . . 6  |-  ( .r
`  S )  =  ( .r `  S
)
286, 26, 27dvdsrmul 15430 . . . . 5  |-  ( ( X  e.  ( Base `  S )  /\  (
I `  X )  e.  ( Base `  S
) )  ->  X
( ||r `
 S ) ( ( I `  X
) ( .r `  S ) X ) )
2923, 25, 28syl2anc 642 . . . 4  |-  ( ( A  e.  (SubRing `  R
)  /\  ( X  e.  U  /\  X  e.  A  /\  ( I `
 X )  e.  A ) )  ->  X ( ||r `
 S ) ( ( I `  X
) ( .r `  S ) X ) )
30 subrgrcl 15550 . . . . . . 7  |-  ( A  e.  (SubRing `  R
)  ->  R  e.  Ring )
3130adantr 451 . . . . . 6  |-  ( ( A  e.  (SubRing `  R
)  /\  ( X  e.  U  /\  X  e.  A  /\  ( I `
 X )  e.  A ) )  ->  R  e.  Ring )
32 simpr1 961 . . . . . 6  |-  ( ( A  e.  (SubRing `  R
)  /\  ( X  e.  U  /\  X  e.  A  /\  ( I `
 X )  e.  A ) )  ->  X  e.  U )
33 eqid 2283 . . . . . . 7  |-  ( .r
`  R )  =  ( .r `  R
)
34 eqid 2283 . . . . . . 7  |-  ( 1r
`  R )  =  ( 1r `  R
)
352, 12, 33, 34unitlinv 15459 . . . . . 6  |-  ( ( R  e.  Ring  /\  X  e.  U )  ->  (
( I `  X
) ( .r `  R ) X )  =  ( 1r `  R ) )
3631, 32, 35syl2anc 642 . . . . 5  |-  ( ( A  e.  (SubRing `  R
)  /\  ( X  e.  U  /\  X  e.  A  /\  ( I `
 X )  e.  A ) )  -> 
( ( I `  X ) ( .r
`  R ) X )  =  ( 1r
`  R ) )
371, 33ressmulr 13261 . . . . . . 7  |-  ( A  e.  (SubRing `  R
)  ->  ( .r `  R )  =  ( .r `  S ) )
3837adantr 451 . . . . . 6  |-  ( ( A  e.  (SubRing `  R
)  /\  ( X  e.  U  /\  X  e.  A  /\  ( I `
 X )  e.  A ) )  -> 
( .r `  R
)  =  ( .r
`  S ) )
3938oveqd 5875 . . . . 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 ) )
401, 34subrg1 15555 . . . . . 6  |-  ( A  e.  (SubRing `  R
)  ->  ( 1r `  R )  =  ( 1r `  S ) )
4140adantr 451 . . . . 5  |-  ( ( A  e.  (SubRing `  R
)  /\  ( X  e.  U  /\  X  e.  A  /\  ( I `
 X )  e.  A ) )  -> 
( 1r `  R
)  =  ( 1r
`  S ) )
4236, 39, 413eqtr3d 2323 . . . 4  |-  ( ( A  e.  (SubRing `  R
)  /\  ( X  e.  U  /\  X  e.  A  /\  ( I `
 X )  e.  A ) )  -> 
( ( I `  X ) ( .r
`  S ) X )  =  ( 1r
`  S ) )
4329, 42breqtrd 4047 . . 3  |-  ( ( A  e.  (SubRing `  R
)  /\  ( X  e.  U  /\  X  e.  A  /\  ( I `
 X )  e.  A ) )  ->  X ( ||r `
 S ) ( 1r `  S ) )
44 eqid 2283 . . . . . . 7  |-  (oppr `  S
)  =  (oppr `  S
)
4544, 6opprbas 15411 . . . . . 6  |-  ( Base `  S )  =  (
Base `  (oppr
`  S ) )
46 eqid 2283 . . . . . 6  |-  ( ||r `  (oppr `  S
) )  =  (
||r `  (oppr
`  S ) )
47 eqid 2283 . . . . . 6  |-  ( .r
`  (oppr
`  S ) )  =  ( .r `  (oppr `  S ) )
4845, 46, 47dvdsrmul 15430 . . . . 5  |-  ( ( X  e.  ( Base `  S )  /\  (
I `  X )  e.  ( Base `  S
) )  ->  X
( ||r `
 (oppr
`  S ) ) ( ( I `  X ) ( .r
`  (oppr
`  S ) ) X ) )
4923, 25, 48syl2anc 642 . . . 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 ) )
506, 27, 44, 47opprmul 15408 . . . . 5  |-  ( ( I `  X ) ( .r `  (oppr `  S
) ) X )  =  ( X ( .r `  S ) ( I `  X
) )
512, 12, 33, 34unitrinv 15460 . . . . . . 7  |-  ( ( R  e.  Ring  /\  X  e.  U )  ->  ( X ( .r `  R ) ( I `
 X ) )  =  ( 1r `  R ) )
5231, 32, 51syl2anc 642 . . . . . 6  |-  ( ( A  e.  (SubRing `  R
)  /\  ( X  e.  U  /\  X  e.  A  /\  ( I `
 X )  e.  A ) )  -> 
( X ( .r
`  R ) ( I `  X ) )  =  ( 1r
`  R ) )
5338oveqd 5875 . . . . . 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 ) ) )
5452, 53, 413eqtr3d 2323 . . . . 5  |-  ( ( A  e.  (SubRing `  R
)  /\  ( X  e.  U  /\  X  e.  A  /\  ( I `
 X )  e.  A ) )  -> 
( X ( .r
`  S ) ( I `  X ) )  =  ( 1r
`  S ) )
5550, 54syl5eq 2327 . . . 4  |-  ( ( A  e.  (SubRing `  R
)  /\  ( X  e.  U  /\  X  e.  A  /\  ( I `
 X )  e.  A ) )  -> 
( ( I `  X ) ( .r
`  (oppr
`  S ) ) X )  =  ( 1r `  S ) )
5649, 55breqtrd 4047 . . 3  |-  ( ( A  e.  (SubRing `  R
)  /\  ( X  e.  U  /\  X  e.  A  /\  ( I `
 X )  e.  A ) )  ->  X ( ||r `
 (oppr
`  S ) ) ( 1r `  S
) )
57 eqid 2283 . . . 4  |-  ( 1r
`  S )  =  ( 1r `  S
)
583, 57, 26, 44, 46isunit 15439 . . 3  |-  ( X  e.  V  <->  ( X
( ||r `
 S ) ( 1r `  S )  /\  X ( ||r `  (oppr `  S
) ) ( 1r
`  S ) ) )
5943, 56, 58sylanbrc 645 . 2  |-  ( ( A  e.  (SubRing `  R
)  /\  ( X  e.  U  /\  X  e.  A  /\  ( I `
 X )  e.  A ) )  ->  X  e.  V )
6020, 59impbida 805 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    <-> wb 176    /\ wa 358    /\ w3a 934    = wceq 1623    e. wcel 1684   class class class wbr 4023   ` cfv 5255  (class class class)co 5858   Basecbs 13148   ↾s cress 13149   .rcmulr 13209   Ringcrg 15337   1rcur 15339  opprcoppr 15404   ||rcdsr 15420  Unitcui 15421   invrcinvr 15453  SubRingcsubrg 15541
This theorem is referenced by:  issubdrg  15570  gzrngunit  16437  zrngunit  16438  cphreccllem  18614
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-cnex 8793  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813  ax-pre-mulgt0 8814
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-tpos 6234  df-riota 6304  df-recs 6388  df-rdg 6423  df-er 6660  df-en 6864  df-dom 6865  df-sdom 6866  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-sub 9039  df-neg 9040  df-nn 9747  df-2 9804  df-3 9805  df-ndx 13151  df-slot 13152  df-base 13153  df-sets 13154  df-ress 13155  df-plusg 13221  df-mulr 13222  df-0g 13404  df-mnd 14367  df-grp 14489  df-minusg 14490  df-subg 14618  df-mgp 15326  df-rng 15340  df-ur 15342  df-oppr 15405  df-dvdsr 15423  df-unit 15424  df-invr 15454  df-subrg 15543
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