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Theorem isunit 15754
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 5749 . . . 4  |-  ( X  e.  (Unit `  R
)  ->  R  e.  dom Unit )
2 unit.1 . . . 4  |-  U  =  (Unit `  R )
31, 2eleq2s 2527 . . 3  |-  ( X  e.  U  ->  R  e.  dom Unit )
4 elex 2956 . . 3  |-  ( R  e.  dom Unit  ->  R  e. 
_V )
53, 4syl 16 . 2  |-  ( X  e.  U  ->  R  e.  _V )
6 df-br 4205 . . . 4  |-  ( X 
.||  .1.  <->  <. X ,  .1.  >.  e.  .||  )
7 elfvdm 5749 . . . . . 6  |-  ( <. X ,  .1.  >.  e.  (
||r `  R )  ->  R  e.  dom  ||r )
8 unit.3 . . . . . 6  |-  .||  =  (
||r `  R )
97, 8eleq2s 2527 . . . . 5  |-  ( <. X ,  .1.  >.  e.  .||  ->  R  e.  dom  ||r )
10 elex 2956 . . . . 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 5720 . . . . . . . . . 10  |-  ( r  =  R  ->  ( ||r `  r )  =  (
||r `  R ) )
1514, 8syl6eqr 2485 . . . . . . . . 9  |-  ( r  =  R  ->  ( ||r `  r )  =  .||  )
16 fveq2 5720 . . . . . . . . . . . 12  |-  ( r  =  R  ->  (oppr `  r
)  =  (oppr `  R
) )
17 unit.4 . . . . . . . . . . . 12  |-  S  =  (oppr
`  R )
1816, 17syl6eqr 2485 . . . . . . . . . . 11  |-  ( r  =  R  ->  (oppr `  r
)  =  S )
1918fveq2d 5724 . . . . . . . . . 10  |-  ( r  =  R  ->  ( ||r `  (oppr
`  r ) )  =  ( ||r `
 S ) )
20 unit.5 . . . . . . . . . 10  |-  E  =  ( ||r `
 S )
2119, 20syl6eqr 2485 . . . . . . . . 9  |-  ( r  =  R  ->  ( ||r `  (oppr
`  r ) )  =  E )
2215, 21ineq12d 3535 . . . . . . . 8  |-  ( r  =  R  ->  (
( ||r `
 r )  i^i  ( ||r `
 (oppr
`  r ) ) )  =  (  .||  i^i  E ) )
2322cnveqd 5040 . . . . . . 7  |-  ( r  =  R  ->  `' ( ( ||r `
 r )  i^i  ( ||r `
 (oppr
`  r ) ) )  =  `' ( 
.||  i^i  E )
)
24 fveq2 5720 . . . . . . . . 9  |-  ( r  =  R  ->  ( 1r `  r )  =  ( 1r `  R
) )
25 unit.2 . . . . . . . . 9  |-  .1.  =  ( 1r `  R )
2624, 25syl6eqr 2485 . . . . . . . 8  |-  ( r  =  R  ->  ( 1r `  r )  =  .1.  )
2726sneqd 3819 . . . . . . 7  |-  ( r  =  R  ->  { ( 1r `  r ) }  =  {  .1.  } )
2823, 27imaeq12d 5196 . . . . . 6  |-  ( r  =  R  ->  ( `' ( ( ||r `  r
)  i^i  ( ||r `  (oppr `  r
) ) ) " { ( 1r `  r ) } )  =  ( `' ( 
.||  i^i  E ) " {  .1.  } ) )
29 df-unit 15739 . . . . . 6  |- Unit  =  ( r  e.  _V  |->  ( `' ( ( ||r `  r
)  i^i  ( ||r `  (oppr `  r
) ) ) " { ( 1r `  r ) } ) )
30 fvex 5734 . . . . . . . . . 10  |-  ( ||r `  R
)  e.  _V
318, 30eqeltri 2505 . . . . . . . . 9  |-  .||  e.  _V
3231inex1 4336 . . . . . . . 8  |-  (  .||  i^i  E )  e.  _V
3332cnvex 5398 . . . . . . 7  |-  `' ( 
.||  i^i  E )  e.  _V
34 imaexg 5209 . . . . . . 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 5798 . . . . 5  |-  ( R  e.  _V  ->  (Unit `  R )  =  ( `' (  .||  i^i  E
) " {  .1.  } ) )
372, 36syl5eq 2479 . . . 4  |-  ( R  e.  _V  ->  U  =  ( `' ( 
.||  i^i  E ) " {  .1.  } ) )
3837eleq2d 2502 . . 3  |-  ( R  e.  _V  ->  ( X  e.  U  <->  X  e.  ( `' (  .||  i^i  E
) " {  .1.  } ) ) )
39 inss1 3553 . . . . . 6  |-  (  .||  i^i  E )  C_  .||
408reldvdsr 15741 . . . . . 6  |-  Rel  .||
41 relss 4955 . . . . . 6  |-  ( ( 
.||  i^i  E )  C_  .||  ->  ( Rel  .||  ->  Rel  (  .||  i^i  E ) ) )
4239, 40, 41mp2 9 . . . . 5  |-  Rel  (  .|| 
i^i  E )
43 eliniseg2 5236 . . . . 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 4251 . . . 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 1652    e. wcel 1725   _Vcvv 2948    i^i cin 3311    C_ wss 3312   {csn 3806   <.cop 3809   class class class wbr 4204   `'ccnv 4869   dom cdm 4870   "cima 4873   Rel wrel 4875   ` cfv 5446   1rcur 15654  opprcoppr 15719   ||rcdsr 15735  Unitcui 15736
This theorem is referenced by:  1unit  15755  unitcl  15756  opprunit  15758  crngunit  15759  unitmulcl  15761  unitgrp  15764  unitnegcl  15778  unitpropd  15794  isdrng2  15837  subrguss  15875  subrgunit  15878  fidomndrng  16359  elrhmunit  24250
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-br 4205  df-opab 4259  df-mpt 4260  df-id 4490  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fv 5454  df-dvdsr 15738  df-unit 15739
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