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Theorem isunit 17107
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 5892 . . . 4  |-  ( X  e.  (Unit `  R
)  ->  R  e.  dom Unit )
2 unit.1 . . . 4  |-  U  =  (Unit `  R )
31, 2eleq2s 2575 . . 3  |-  ( X  e.  U  ->  R  e.  dom Unit )
4 elex 3122 . . 3  |-  ( R  e.  dom Unit  ->  R  e. 
_V )
53, 4syl 16 . 2  |-  ( X  e.  U  ->  R  e.  _V )
6 df-br 4448 . . . 4  |-  ( X 
.||  .1.  <->  <. X ,  .1.  >.  e.  .||  )
7 elfvdm 5892 . . . . . 6  |-  ( <. X ,  .1.  >.  e.  (
||r `  R )  ->  R  e.  dom  ||r )
8 unit.3 . . . . . 6  |-  .||  =  (
||r `  R )
97, 8eleq2s 2575 . . . . 5  |-  ( <. X ,  .1.  >.  e.  .||  ->  R  e.  dom  ||r )
10 elex 3122 . . . . 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 5866 . . . . . . . . . 10  |-  ( r  =  R  ->  ( ||r `  r )  =  (
||r `  R ) )
1514, 8syl6eqr 2526 . . . . . . . . 9  |-  ( r  =  R  ->  ( ||r `  r )  =  .||  )
16 fveq2 5866 . . . . . . . . . . . 12  |-  ( r  =  R  ->  (oppr `  r
)  =  (oppr `  R
) )
17 unit.4 . . . . . . . . . . . 12  |-  S  =  (oppr
`  R )
1816, 17syl6eqr 2526 . . . . . . . . . . 11  |-  ( r  =  R  ->  (oppr `  r
)  =  S )
1918fveq2d 5870 . . . . . . . . . 10  |-  ( r  =  R  ->  ( ||r `  (oppr
`  r ) )  =  ( ||r `
 S ) )
20 unit.5 . . . . . . . . . 10  |-  E  =  ( ||r `
 S )
2119, 20syl6eqr 2526 . . . . . . . . 9  |-  ( r  =  R  ->  ( ||r `  (oppr
`  r ) )  =  E )
2215, 21ineq12d 3701 . . . . . . . 8  |-  ( r  =  R  ->  (
( ||r `
 r )  i^i  ( ||r `
 (oppr
`  r ) ) )  =  (  .||  i^i  E ) )
2322cnveqd 5178 . . . . . . 7  |-  ( r  =  R  ->  `' ( ( ||r `
 r )  i^i  ( ||r `
 (oppr
`  r ) ) )  =  `' ( 
.||  i^i  E )
)
24 fveq2 5866 . . . . . . . . 9  |-  ( r  =  R  ->  ( 1r `  r )  =  ( 1r `  R
) )
25 unit.2 . . . . . . . . 9  |-  .1.  =  ( 1r `  R )
2624, 25syl6eqr 2526 . . . . . . . 8  |-  ( r  =  R  ->  ( 1r `  r )  =  .1.  )
2726sneqd 4039 . . . . . . 7  |-  ( r  =  R  ->  { ( 1r `  r ) }  =  {  .1.  } )
2823, 27imaeq12d 5338 . . . . . 6  |-  ( r  =  R  ->  ( `' ( ( ||r `  r
)  i^i  ( ||r `  (oppr `  r
) ) ) " { ( 1r `  r ) } )  =  ( `' ( 
.||  i^i  E ) " {  .1.  } ) )
29 df-unit 17092 . . . . . 6  |- Unit  =  ( r  e.  _V  |->  ( `' ( ( ||r `  r
)  i^i  ( ||r `  (oppr `  r
) ) ) " { ( 1r `  r ) } ) )
30 fvex 5876 . . . . . . . . . 10  |-  ( ||r `  R
)  e.  _V
318, 30eqeltri 2551 . . . . . . . . 9  |-  .||  e.  _V
3231inex1 4588 . . . . . . . 8  |-  (  .||  i^i  E )  e.  _V
3332cnvex 6731 . . . . . . 7  |-  `' ( 
.||  i^i  E )  e.  _V
34 imaexg 6721 . . . . . . 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 5950 . . . . 5  |-  ( R  e.  _V  ->  (Unit `  R )  =  ( `' (  .||  i^i  E
) " {  .1.  } ) )
372, 36syl5eq 2520 . . . 4  |-  ( R  e.  _V  ->  U  =  ( `' ( 
.||  i^i  E ) " {  .1.  } ) )
3837eleq2d 2537 . . 3  |-  ( R  e.  _V  ->  ( X  e.  U  <->  X  e.  ( `' (  .||  i^i  E
) " {  .1.  } ) ) )
39 inss1 3718 . . . . . 6  |-  (  .||  i^i  E )  C_  .||
408reldvdsr 17094 . . . . . 6  |-  Rel  .||
41 relss 5090 . . . . . 6  |-  ( ( 
.||  i^i  E )  C_  .||  ->  ( Rel  .||  ->  Rel  (  .||  i^i  E ) ) )
4239, 40, 41mp2 9 . . . . 5  |-  Rel  (  .|| 
i^i  E )
43 eliniseg2 5376 . . . . 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 4496 . . . 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 1379    e. wcel 1767   _Vcvv 3113    i^i cin 3475    C_ wss 3476   {csn 4027   <.cop 4033   class class class wbr 4447   `'ccnv 4998   dom cdm 4999   "cima 5002   Rel wrel 5004   ` cfv 5588   1rcur 16955  opprcoppr 17072   ||rcdsr 17088  Unitcui 17089
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-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6576
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  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-ral 2819  df-rex 2820  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-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-br 4448  df-opab 4506  df-mpt 4507  df-id 4795  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-fv 5596  df-dvdsr 17091  df-unit 17092
This theorem is referenced by:  1unit  17108  unitcl  17109  opprunit  17111  crngunit  17112  unitmulcl  17114  unitgrp  17117  unitnegcl  17131  unitpropd  17147  isdrng2  17206  subrguss  17244  subrgunit  17247  fidomndrng  17755  invrvald  18973  elrhmunit  27501
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