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Theorem isunit 17934
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 5914 . . . 4  |-  ( X  e.  (Unit `  R
)  ->  R  e.  dom Unit )
2 unit.1 . . . 4  |-  U  =  (Unit `  R )
31, 2eleq2s 2558 . . 3  |-  ( X  e.  U  ->  R  e.  dom Unit )
4 elex 3066 . . 3  |-  ( R  e.  dom Unit  ->  R  e. 
_V )
53, 4syl 17 . 2  |-  ( X  e.  U  ->  R  e.  _V )
6 df-br 4417 . . . 4  |-  ( X 
.||  .1.  <->  <. X ,  .1.  >.  e.  .||  )
7 elfvdm 5914 . . . . . 6  |-  ( <. X ,  .1.  >.  e.  (
||r `  R )  ->  R  e.  dom  ||r )
8 unit.3 . . . . . 6  |-  .||  =  (
||r `  R )
97, 8eleq2s 2558 . . . . 5  |-  ( <. X ,  .1.  >.  e.  .||  ->  R  e.  dom  ||r )
10 elex 3066 . . . . 5  |-  ( R  e.  dom  ||r  ->  R  e. 
_V )
119, 10syl 17 . . . 4  |-  ( <. X ,  .1.  >.  e.  .||  ->  R  e.  _V )
126, 11sylbi 200 . . 3  |-  ( X 
.||  .1.  ->  R  e. 
_V )
1312adantr 471 . 2  |-  ( ( X  .||  .1.  /\  X E  .1.  )  ->  R  e.  _V )
14 fveq2 5888 . . . . . . . . . 10  |-  ( r  =  R  ->  ( ||r `  r )  =  (
||r `  R ) )
1514, 8syl6eqr 2514 . . . . . . . . 9  |-  ( r  =  R  ->  ( ||r `  r )  =  .||  )
16 fveq2 5888 . . . . . . . . . . . 12  |-  ( r  =  R  ->  (oppr `  r
)  =  (oppr `  R
) )
17 unit.4 . . . . . . . . . . . 12  |-  S  =  (oppr
`  R )
1816, 17syl6eqr 2514 . . . . . . . . . . 11  |-  ( r  =  R  ->  (oppr `  r
)  =  S )
1918fveq2d 5892 . . . . . . . . . 10  |-  ( r  =  R  ->  ( ||r `  (oppr
`  r ) )  =  ( ||r `
 S ) )
20 unit.5 . . . . . . . . . 10  |-  E  =  ( ||r `
 S )
2119, 20syl6eqr 2514 . . . . . . . . 9  |-  ( r  =  R  ->  ( ||r `  (oppr
`  r ) )  =  E )
2215, 21ineq12d 3647 . . . . . . . 8  |-  ( r  =  R  ->  (
( ||r `
 r )  i^i  ( ||r `
 (oppr
`  r ) ) )  =  (  .||  i^i  E ) )
2322cnveqd 5029 . . . . . . 7  |-  ( r  =  R  ->  `' ( ( ||r `
 r )  i^i  ( ||r `
 (oppr
`  r ) ) )  =  `' ( 
.||  i^i  E )
)
24 fveq2 5888 . . . . . . . . 9  |-  ( r  =  R  ->  ( 1r `  r )  =  ( 1r `  R
) )
25 unit.2 . . . . . . . . 9  |-  .1.  =  ( 1r `  R )
2624, 25syl6eqr 2514 . . . . . . . 8  |-  ( r  =  R  ->  ( 1r `  r )  =  .1.  )
2726sneqd 3992 . . . . . . 7  |-  ( r  =  R  ->  { ( 1r `  r ) }  =  {  .1.  } )
2823, 27imaeq12d 5188 . . . . . 6  |-  ( r  =  R  ->  ( `' ( ( ||r `  r
)  i^i  ( ||r `  (oppr `  r
) ) ) " { ( 1r `  r ) } )  =  ( `' ( 
.||  i^i  E ) " {  .1.  } ) )
29 df-unit 17919 . . . . . 6  |- Unit  =  ( r  e.  _V  |->  ( `' ( ( ||r `  r
)  i^i  ( ||r `  (oppr `  r
) ) ) " { ( 1r `  r ) } ) )
30 fvex 5898 . . . . . . . . . 10  |-  ( ||r `  R
)  e.  _V
318, 30eqeltri 2536 . . . . . . . . 9  |-  .||  e.  _V
3231inex1 4558 . . . . . . . 8  |-  (  .||  i^i  E )  e.  _V
3332cnvex 6767 . . . . . . 7  |-  `' ( 
.||  i^i  E )  e.  _V
34 imaexg 6757 . . . . . . 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 5971 . . . . 5  |-  ( R  e.  _V  ->  (Unit `  R )  =  ( `' (  .||  i^i  E
) " {  .1.  } ) )
372, 36syl5eq 2508 . . . 4  |-  ( R  e.  _V  ->  U  =  ( `' ( 
.||  i^i  E ) " {  .1.  } ) )
3837eleq2d 2525 . . 3  |-  ( R  e.  _V  ->  ( X  e.  U  <->  X  e.  ( `' (  .||  i^i  E
) " {  .1.  } ) ) )
39 inss1 3664 . . . . . 6  |-  (  .||  i^i  E )  C_  .||
408reldvdsr 17921 . . . . . 6  |-  Rel  .||
41 relss 4941 . . . . . 6  |-  ( ( 
.||  i^i  E )  C_  .||  ->  ( Rel  .||  ->  Rel  (  .||  i^i  E ) ) )
4239, 40, 41mp2 9 . . . . 5  |-  Rel  (  .|| 
i^i  E )
43 eliniseg2 5228 . . . . 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 4466 . . . 4  |-  ( X (  .||  i^i  E )  .1.  <->  ( X  .||  .1.  /\  X E  .1.  ) )
4644, 45bitri 257 . . 3  |-  ( X  e.  ( `' ( 
.||  i^i  E ) " {  .1.  } )  <-> 
( X  .||  .1.  /\  X E  .1.  )
)
4738, 46syl6bb 269 . 2  |-  ( R  e.  _V  ->  ( X  e.  U  <->  ( X  .|| 
.1.  /\  X E  .1.  ) ) )
485, 13, 47pm5.21nii 359 1  |-  ( X  e.  U  <->  ( X  .|| 
.1.  /\  X E  .1.  ) )
Colors of variables: wff setvar class
Syntax hints:    <-> wb 189    /\ wa 375    = wceq 1455    e. wcel 1898   _Vcvv 3057    i^i cin 3415    C_ wss 3416   {csn 3980   <.cop 3986   class class class wbr 4416   `'ccnv 4852   dom cdm 4853   "cima 4856   Rel wrel 4858   ` cfv 5601   1rcur 17784  opprcoppr 17899   ||rcdsr 17915  Unitcui 17916
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1680  ax-4 1693  ax-5 1769  ax-6 1816  ax-7 1862  ax-8 1900  ax-9 1907  ax-10 1926  ax-11 1931  ax-12 1944  ax-13 2102  ax-ext 2442  ax-sep 4539  ax-nul 4548  ax-pow 4595  ax-pr 4653  ax-un 6610
This theorem depends on definitions:  df-bi 190  df-or 376  df-an 377  df-3an 993  df-tru 1458  df-ex 1675  df-nf 1679  df-sb 1809  df-eu 2314  df-mo 2315  df-clab 2449  df-cleq 2455  df-clel 2458  df-nfc 2592  df-ne 2635  df-ral 2754  df-rex 2755  df-rab 2758  df-v 3059  df-sbc 3280  df-csb 3376  df-dif 3419  df-un 3421  df-in 3423  df-ss 3430  df-nul 3744  df-if 3894  df-pw 3965  df-sn 3981  df-pr 3983  df-op 3987  df-uni 4213  df-br 4417  df-opab 4476  df-mpt 4477  df-id 4768  df-xp 4859  df-rel 4860  df-cnv 4861  df-co 4862  df-dm 4863  df-rn 4864  df-res 4865  df-ima 4866  df-iota 5565  df-fun 5603  df-fv 5609  df-dvdsr 17918  df-unit 17919
This theorem is referenced by:  1unit  17935  unitcl  17936  opprunit  17938  crngunit  17939  unitmulcl  17941  unitgrp  17944  unitnegcl  17958  unitpropd  17974  isdrng2  18034  subrguss  18072  subrgunit  18075  fidomndrng  18580  invrvald  19750  elrhmunit  28632
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