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Theorem isunit 16771
Description: Property of being a unit of a ring. A unit is an element that left- and right-divides one. (Contributed by Mario Carneiro, 1-Dec-2014.) (Revised by Mario Carneiro, 8-Dec-2015.)
Hypotheses
Ref Expression
unit.1  |-  U  =  (Unit `  R )
unit.2  |-  .1.  =  ( 1r `  R )
unit.3  |-  .||  =  (
||r `  R )
unit.4  |-  S  =  (oppr
`  R )
unit.5  |-  E  =  ( ||r `
 S )
Assertion
Ref Expression
isunit  |-  ( X  e.  U  <->  ( X  .|| 
.1.  /\  X E  .1.  ) )

Proof of Theorem isunit
Dummy variable  r is distinct from all other variables.
StepHypRef Expression
1 elfvdm 5737 . . . 4  |-  ( X  e.  (Unit `  R
)  ->  R  e.  dom Unit )
2 unit.1 . . . 4  |-  U  =  (Unit `  R )
31, 2eleq2s 2535 . . 3  |-  ( X  e.  U  ->  R  e.  dom Unit )
4 elex 3002 . . 3  |-  ( R  e.  dom Unit  ->  R  e. 
_V )
53, 4syl 16 . 2  |-  ( X  e.  U  ->  R  e.  _V )
6 df-br 4314 . . . 4  |-  ( X 
.||  .1.  <->  <. X ,  .1.  >.  e.  .||  )
7 elfvdm 5737 . . . . . 6  |-  ( <. X ,  .1.  >.  e.  (
||r `  R )  ->  R  e.  dom  ||r )
8 unit.3 . . . . . 6  |-  .||  =  (
||r `  R )
97, 8eleq2s 2535 . . . . 5  |-  ( <. X ,  .1.  >.  e.  .||  ->  R  e.  dom  ||r )
10 elex 3002 . . . . 5  |-  ( R  e.  dom  ||r  ->  R  e. 
_V )
119, 10syl 16 . . . 4  |-  ( <. X ,  .1.  >.  e.  .||  ->  R  e.  _V )
126, 11sylbi 195 . . 3  |-  ( X 
.||  .1.  ->  R  e. 
_V )
1312adantr 465 . 2  |-  ( ( X  .||  .1.  /\  X E  .1.  )  ->  R  e.  _V )
14 fveq2 5712 . . . . . . . . . 10  |-  ( r  =  R  ->  ( ||r `  r )  =  (
||r `  R ) )
1514, 8syl6eqr 2493 . . . . . . . . 9  |-  ( r  =  R  ->  ( ||r `  r )  =  .||  )
16 fveq2 5712 . . . . . . . . . . . 12  |-  ( r  =  R  ->  (oppr `  r
)  =  (oppr `  R
) )
17 unit.4 . . . . . . . . . . . 12  |-  S  =  (oppr
`  R )
1816, 17syl6eqr 2493 . . . . . . . . . . 11  |-  ( r  =  R  ->  (oppr `  r
)  =  S )
1918fveq2d 5716 . . . . . . . . . 10  |-  ( r  =  R  ->  ( ||r `  (oppr
`  r ) )  =  ( ||r `
 S ) )
20 unit.5 . . . . . . . . . 10  |-  E  =  ( ||r `
 S )
2119, 20syl6eqr 2493 . . . . . . . . 9  |-  ( r  =  R  ->  ( ||r `  (oppr
`  r ) )  =  E )
2215, 21ineq12d 3574 . . . . . . . 8  |-  ( r  =  R  ->  (
( ||r `
 r )  i^i  ( ||r `
 (oppr
`  r ) ) )  =  (  .||  i^i  E ) )
2322cnveqd 5036 . . . . . . 7  |-  ( r  =  R  ->  `' ( ( ||r `
 r )  i^i  ( ||r `
 (oppr
`  r ) ) )  =  `' ( 
.||  i^i  E )
)
24 fveq2 5712 . . . . . . . . 9  |-  ( r  =  R  ->  ( 1r `  r )  =  ( 1r `  R
) )
25 unit.2 . . . . . . . . 9  |-  .1.  =  ( 1r `  R )
2624, 25syl6eqr 2493 . . . . . . . 8  |-  ( r  =  R  ->  ( 1r `  r )  =  .1.  )
2726sneqd 3910 . . . . . . 7  |-  ( r  =  R  ->  { ( 1r `  r ) }  =  {  .1.  } )
2823, 27imaeq12d 5191 . . . . . 6  |-  ( r  =  R  ->  ( `' ( ( ||r `  r
)  i^i  ( ||r `  (oppr `  r
) ) ) " { ( 1r `  r ) } )  =  ( `' ( 
.||  i^i  E ) " {  .1.  } ) )
29 df-unit 16756 . . . . . 6  |- Unit  =  ( r  e.  _V  |->  ( `' ( ( ||r `  r
)  i^i  ( ||r `  (oppr `  r
) ) ) " { ( 1r `  r ) } ) )
30 fvex 5722 . . . . . . . . . 10  |-  ( ||r `  R
)  e.  _V
318, 30eqeltri 2513 . . . . . . . . 9  |-  .||  e.  _V
3231inex1 4454 . . . . . . . 8  |-  (  .||  i^i  E )  e.  _V
3332cnvex 6546 . . . . . . 7  |-  `' ( 
.||  i^i  E )  e.  _V
34 imaexg 6536 . . . . . . 7  |-  ( `' (  .||  i^i  E )  e.  _V  ->  ( `' (  .||  i^i  E
) " {  .1.  } )  e.  _V )
3533, 34ax-mp 5 . . . . . 6  |-  ( `' (  .||  i^i  E )
" {  .1.  }
)  e.  _V
3628, 29, 35fvmpt 5795 . . . . 5  |-  ( R  e.  _V  ->  (Unit `  R )  =  ( `' (  .||  i^i  E
) " {  .1.  } ) )
372, 36syl5eq 2487 . . . 4  |-  ( R  e.  _V  ->  U  =  ( `' ( 
.||  i^i  E ) " {  .1.  } ) )
3837eleq2d 2510 . . 3  |-  ( R  e.  _V  ->  ( X  e.  U  <->  X  e.  ( `' (  .||  i^i  E
) " {  .1.  } ) ) )
39 inss1 3591 . . . . . 6  |-  (  .||  i^i  E )  C_  .||
408reldvdsr 16758 . . . . . 6  |-  Rel  .||
41 relss 4948 . . . . . 6  |-  ( ( 
.||  i^i  E )  C_  .||  ->  ( Rel  .||  ->  Rel  (  .||  i^i  E ) ) )
4239, 40, 41mp2 9 . . . . 5  |-  Rel  (  .|| 
i^i  E )
43 eliniseg2 5229 . . . . 5  |-  ( Rel  (  .||  i^i  E )  ->  ( X  e.  ( `' (  .||  i^i  E ) " {  .1.  } )  <->  X (  .|| 
i^i  E )  .1.  ) )
4442, 43ax-mp 5 . . . 4  |-  ( X  e.  ( `' ( 
.||  i^i  E ) " {  .1.  } )  <-> 
X (  .||  i^i  E
)  .1.  )
45 brin 4362 . . . 4  |-  ( X (  .||  i^i  E )  .1.  <->  ( X  .||  .1.  /\  X E  .1.  ) )
4644, 45bitri 249 . . 3  |-  ( X  e.  ( `' ( 
.||  i^i  E ) " {  .1.  } )  <-> 
( X  .||  .1.  /\  X E  .1.  )
)
4738, 46syl6bb 261 . 2  |-  ( R  e.  _V  ->  ( X  e.  U  <->  ( X  .|| 
.1.  /\  X E  .1.  ) ) )
485, 13, 47pm5.21nii 353 1  |-  ( X  e.  U  <->  ( X  .|| 
.1.  /\  X E  .1.  ) )
Colors of variables: wff setvar class
Syntax hints:    <-> wb 184    /\ wa 369    = wceq 1369    e. wcel 1756   _Vcvv 2993    i^i cin 3348    C_ wss 3349   {csn 3898   <.cop 3904   class class class wbr 4313   `'ccnv 4860   dom cdm 4861   "cima 4864   Rel wrel 4866   ` cfv 5439   1rcur 16625  opprcoppr 16736   ||rcdsr 16752  Unitcui 16753
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-sep 4434  ax-nul 4442  ax-pow 4491  ax-pr 4552  ax-un 6393
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2577  df-ne 2622  df-ral 2741  df-rex 2742  df-rab 2745  df-v 2995  df-sbc 3208  df-csb 3310  df-dif 3352  df-un 3354  df-in 3356  df-ss 3363  df-nul 3659  df-if 3813  df-pw 3883  df-sn 3899  df-pr 3901  df-op 3905  df-uni 4113  df-br 4314  df-opab 4372  df-mpt 4373  df-id 4657  df-xp 4867  df-rel 4868  df-cnv 4869  df-co 4870  df-dm 4871  df-rn 4872  df-res 4873  df-ima 4874  df-iota 5402  df-fun 5441  df-fv 5447  df-dvdsr 16755  df-unit 16756
This theorem is referenced by:  1unit  16772  unitcl  16773  opprunit  16775  crngunit  16776  unitmulcl  16778  unitgrp  16781  unitnegcl  16795  unitpropd  16811  isdrng2  16864  subrguss  16902  subrgunit  16905  fidomndrng  17401  invrvald  18504  elrhmunit  26310
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