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Theorem isunit 15690
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 5698 . . . 4  |-  ( X  e.  (Unit `  R
)  ->  R  e.  dom Unit )
2 unit.1 . . . 4  |-  U  =  (Unit `  R )
31, 2eleq2s 2480 . . 3  |-  ( X  e.  U  ->  R  e.  dom Unit )
4 elex 2908 . . 3  |-  ( R  e.  dom Unit  ->  R  e. 
_V )
53, 4syl 16 . 2  |-  ( X  e.  U  ->  R  e.  _V )
6 df-br 4155 . . . 4  |-  ( X 
.||  .1.  <->  <. X ,  .1.  >.  e.  .||  )
7 elfvdm 5698 . . . . . 6  |-  ( <. X ,  .1.  >.  e.  (
||r `  R )  ->  R  e.  dom  ||r )
8 unit.3 . . . . . 6  |-  .||  =  (
||r `  R )
97, 8eleq2s 2480 . . . . 5  |-  ( <. X ,  .1.  >.  e.  .||  ->  R  e.  dom  ||r )
10 elex 2908 . . . . 5  |-  ( R  e.  dom  ||r  ->  R  e. 
_V )
119, 10syl 16 . . . 4  |-  ( <. X ,  .1.  >.  e.  .||  ->  R  e.  _V )
126, 11sylbi 188 . . 3  |-  ( X 
.||  .1.  ->  R  e. 
_V )
1312adantr 452 . 2  |-  ( ( X  .||  .1.  /\  X E  .1.  )  ->  R  e.  _V )
14 fveq2 5669 . . . . . . . . . 10  |-  ( r  =  R  ->  ( ||r `  r )  =  (
||r `  R ) )
1514, 8syl6eqr 2438 . . . . . . . . 9  |-  ( r  =  R  ->  ( ||r `  r )  =  .||  )
16 fveq2 5669 . . . . . . . . . . . 12  |-  ( r  =  R  ->  (oppr `  r
)  =  (oppr `  R
) )
17 unit.4 . . . . . . . . . . . 12  |-  S  =  (oppr
`  R )
1816, 17syl6eqr 2438 . . . . . . . . . . 11  |-  ( r  =  R  ->  (oppr `  r
)  =  S )
1918fveq2d 5673 . . . . . . . . . 10  |-  ( r  =  R  ->  ( ||r `  (oppr
`  r ) )  =  ( ||r `
 S ) )
20 unit.5 . . . . . . . . . 10  |-  E  =  ( ||r `
 S )
2119, 20syl6eqr 2438 . . . . . . . . 9  |-  ( r  =  R  ->  ( ||r `  (oppr
`  r ) )  =  E )
2215, 21ineq12d 3487 . . . . . . . 8  |-  ( r  =  R  ->  (
( ||r `
 r )  i^i  ( ||r `
 (oppr
`  r ) ) )  =  (  .||  i^i  E ) )
2322cnveqd 4989 . . . . . . 7  |-  ( r  =  R  ->  `' ( ( ||r `
 r )  i^i  ( ||r `
 (oppr
`  r ) ) )  =  `' ( 
.||  i^i  E )
)
24 fveq2 5669 . . . . . . . . 9  |-  ( r  =  R  ->  ( 1r `  r )  =  ( 1r `  R
) )
25 unit.2 . . . . . . . . 9  |-  .1.  =  ( 1r `  R )
2624, 25syl6eqr 2438 . . . . . . . 8  |-  ( r  =  R  ->  ( 1r `  r )  =  .1.  )
2726sneqd 3771 . . . . . . 7  |-  ( r  =  R  ->  { ( 1r `  r ) }  =  {  .1.  } )
2823, 27imaeq12d 5145 . . . . . 6  |-  ( r  =  R  ->  ( `' ( ( ||r `  r
)  i^i  ( ||r `  (oppr `  r
) ) ) " { ( 1r `  r ) } )  =  ( `' ( 
.||  i^i  E ) " {  .1.  } ) )
29 df-unit 15675 . . . . . 6  |- Unit  =  ( r  e.  _V  |->  ( `' ( ( ||r `  r
)  i^i  ( ||r `  (oppr `  r
) ) ) " { ( 1r `  r ) } ) )
30 fvex 5683 . . . . . . . . . 10  |-  ( ||r `  R
)  e.  _V
318, 30eqeltri 2458 . . . . . . . . 9  |-  .||  e.  _V
3231inex1 4286 . . . . . . . 8  |-  (  .||  i^i  E )  e.  _V
3332cnvex 5347 . . . . . . 7  |-  `' ( 
.||  i^i  E )  e.  _V
34 imaexg 5158 . . . . . . 7  |-  ( `' (  .||  i^i  E )  e.  _V  ->  ( `' (  .||  i^i  E
) " {  .1.  } )  e.  _V )
3533, 34ax-mp 8 . . . . . 6  |-  ( `' (  .||  i^i  E )
" {  .1.  }
)  e.  _V
3628, 29, 35fvmpt 5746 . . . . 5  |-  ( R  e.  _V  ->  (Unit `  R )  =  ( `' (  .||  i^i  E
) " {  .1.  } ) )
372, 36syl5eq 2432 . . . 4  |-  ( R  e.  _V  ->  U  =  ( `' ( 
.||  i^i  E ) " {  .1.  } ) )
3837eleq2d 2455 . . 3  |-  ( R  e.  _V  ->  ( X  e.  U  <->  X  e.  ( `' (  .||  i^i  E
) " {  .1.  } ) ) )
39 inss1 3505 . . . . . 6  |-  (  .||  i^i  E )  C_  .||
408reldvdsr 15677 . . . . . 6  |-  Rel  .||
41 relss 4904 . . . . . 6  |-  ( ( 
.||  i^i  E )  C_  .||  ->  ( Rel  .||  ->  Rel  (  .||  i^i  E ) ) )
4239, 40, 41mp2 9 . . . . 5  |-  Rel  (  .|| 
i^i  E )
43 eliniseg2 5185 . . . . 5  |-  ( Rel  (  .||  i^i  E )  ->  ( X  e.  ( `' (  .||  i^i  E ) " {  .1.  } )  <->  X (  .|| 
i^i  E )  .1.  ) )
4442, 43ax-mp 8 . . . 4  |-  ( X  e.  ( `' ( 
.||  i^i  E ) " {  .1.  } )  <-> 
X (  .||  i^i  E
)  .1.  )
45 brin 4201 . . . 4  |-  ( X (  .||  i^i  E )  .1.  <->  ( X  .||  .1.  /\  X E  .1.  ) )
4644, 45bitri 241 . . 3  |-  ( X  e.  ( `' ( 
.||  i^i  E ) " {  .1.  } )  <-> 
( X  .||  .1.  /\  X E  .1.  )
)
4738, 46syl6bb 253 . 2  |-  ( R  e.  _V  ->  ( X  e.  U  <->  ( X  .|| 
.1.  /\  X E  .1.  ) ) )
485, 13, 47pm5.21nii 343 1  |-  ( X  e.  U  <->  ( X  .|| 
.1.  /\  X E  .1.  ) )
Colors of variables: wff set class
Syntax hints:    <-> wb 177    /\ wa 359    = wceq 1649    e. wcel 1717   _Vcvv 2900    i^i cin 3263    C_ wss 3264   {csn 3758   <.cop 3761   class class class wbr 4154   `'ccnv 4818   dom cdm 4819   "cima 4822   Rel wrel 4824   ` cfv 5395   1rcur 15590  opprcoppr 15655   ||rcdsr 15671  Unitcui 15672
This theorem is referenced by:  1unit  15691  unitcl  15692  opprunit  15694  crngunit  15695  unitmulcl  15697  unitgrp  15700  unitnegcl  15714  unitpropd  15730  isdrng2  15773  subrguss  15811  subrgunit  15814  fidomndrng  16295  elrhmunit  24075
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2369  ax-sep 4272  ax-nul 4280  ax-pow 4319  ax-pr 4345  ax-un 4642
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2243  df-mo 2244  df-clab 2375  df-cleq 2381  df-clel 2384  df-nfc 2513  df-ne 2553  df-ral 2655  df-rex 2656  df-rab 2659  df-v 2902  df-sbc 3106  df-csb 3196  df-dif 3267  df-un 3269  df-in 3271  df-ss 3278  df-nul 3573  df-if 3684  df-pw 3745  df-sn 3764  df-pr 3765  df-op 3767  df-uni 3959  df-br 4155  df-opab 4209  df-mpt 4210  df-id 4440  df-xp 4825  df-rel 4826  df-cnv 4827  df-co 4828  df-dm 4829  df-rn 4830  df-res 4831  df-ima 4832  df-iota 5359  df-fun 5397  df-fv 5403  df-dvdsr 15674  df-unit 15675
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