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Theorem isunit 15455
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 5570 . . . 4  |-  ( X  e.  (Unit `  R
)  ->  R  e.  dom Unit )
2 unit.1 . . . 4  |-  U  =  (Unit `  R )
31, 2eleq2s 2388 . . 3  |-  ( X  e.  U  ->  R  e.  dom Unit )
4 elex 2809 . . 3  |-  ( R  e.  dom Unit  ->  R  e. 
_V )
53, 4syl 15 . 2  |-  ( X  e.  U  ->  R  e.  _V )
6 df-br 4040 . . . 4  |-  ( X 
.||  .1.  <->  <. X ,  .1.  >.  e.  .||  )
7 elfvdm 5570 . . . . . 6  |-  ( <. X ,  .1.  >.  e.  (
||r `  R )  ->  R  e.  dom  ||r )
8 unit.3 . . . . . 6  |-  .||  =  (
||r `  R )
97, 8eleq2s 2388 . . . . 5  |-  ( <. X ,  .1.  >.  e.  .||  ->  R  e.  dom  ||r )
10 elex 2809 . . . . 5  |-  ( R  e.  dom  ||r  ->  R  e. 
_V )
119, 10syl 15 . . . 4  |-  ( <. X ,  .1.  >.  e.  .||  ->  R  e.  _V )
126, 11sylbi 187 . . 3  |-  ( X 
.||  .1.  ->  R  e. 
_V )
1312adantr 451 . 2  |-  ( ( X  .||  .1.  /\  X E  .1.  )  ->  R  e.  _V )
14 fveq2 5541 . . . . . . . . . . 11  |-  ( r  =  R  ->  ( ||r `  r )  =  (
||r `  R ) )
1514, 8syl6eqr 2346 . . . . . . . . . 10  |-  ( r  =  R  ->  ( ||r `  r )  =  .||  )
16 fveq2 5541 . . . . . . . . . . . . 13  |-  ( r  =  R  ->  (oppr `  r
)  =  (oppr `  R
) )
17 unit.4 . . . . . . . . . . . . 13  |-  S  =  (oppr
`  R )
1816, 17syl6eqr 2346 . . . . . . . . . . . 12  |-  ( r  =  R  ->  (oppr `  r
)  =  S )
1918fveq2d 5545 . . . . . . . . . . 11  |-  ( r  =  R  ->  ( ||r `  (oppr
`  r ) )  =  ( ||r `
 S ) )
20 unit.5 . . . . . . . . . . 11  |-  E  =  ( ||r `
 S )
2119, 20syl6eqr 2346 . . . . . . . . . 10  |-  ( r  =  R  ->  ( ||r `  (oppr
`  r ) )  =  E )
2215, 21ineq12d 3384 . . . . . . . . 9  |-  ( r  =  R  ->  (
( ||r `
 r )  i^i  ( ||r `
 (oppr
`  r ) ) )  =  (  .||  i^i  E ) )
2322cnveqd 4873 . . . . . . . 8  |-  ( r  =  R  ->  `' ( ( ||r `
 r )  i^i  ( ||r `
 (oppr
`  r ) ) )  =  `' ( 
.||  i^i  E )
)
2423imaeq1d 5027 . . . . . . 7  |-  ( r  =  R  ->  ( `' ( ( ||r `  r
)  i^i  ( ||r `  (oppr `  r
) ) ) " { ( 1r `  r ) } )  =  ( `' ( 
.||  i^i  E ) " { ( 1r `  r ) } ) )
25 fveq2 5541 . . . . . . . . . 10  |-  ( r  =  R  ->  ( 1r `  r )  =  ( 1r `  R
) )
26 unit.2 . . . . . . . . . 10  |-  .1.  =  ( 1r `  R )
2725, 26syl6eqr 2346 . . . . . . . . 9  |-  ( r  =  R  ->  ( 1r `  r )  =  .1.  )
2827sneqd 3666 . . . . . . . 8  |-  ( r  =  R  ->  { ( 1r `  r ) }  =  {  .1.  } )
2928imaeq2d 5028 . . . . . . 7  |-  ( r  =  R  ->  ( `' (  .||  i^i  E
) " { ( 1r `  r ) } )  =  ( `' (  .||  i^i  E
) " {  .1.  } ) )
3024, 29eqtrd 2328 . . . . . 6  |-  ( r  =  R  ->  ( `' ( ( ||r `  r
)  i^i  ( ||r `  (oppr `  r
) ) ) " { ( 1r `  r ) } )  =  ( `' ( 
.||  i^i  E ) " {  .1.  } ) )
31 df-unit 15440 . . . . . 6  |- Unit  =  ( r  e.  _V  |->  ( `' ( ( ||r `  r
)  i^i  ( ||r `  (oppr `  r
) ) ) " { ( 1r `  r ) } ) )
32 fvex 5555 . . . . . . . . . 10  |-  ( ||r `  R
)  e.  _V
338, 32eqeltri 2366 . . . . . . . . 9  |-  .||  e.  _V
3433inex1 4171 . . . . . . . 8  |-  (  .||  i^i  E )  e.  _V
3534cnvex 5225 . . . . . . 7  |-  `' ( 
.||  i^i  E )  e.  _V
36 imaexg 5042 . . . . . . 7  |-  ( `' (  .||  i^i  E )  e.  _V  ->  ( `' (  .||  i^i  E
) " {  .1.  } )  e.  _V )
3735, 36ax-mp 8 . . . . . 6  |-  ( `' (  .||  i^i  E )
" {  .1.  }
)  e.  _V
3830, 31, 37fvmpt 5618 . . . . 5  |-  ( R  e.  _V  ->  (Unit `  R )  =  ( `' (  .||  i^i  E
) " {  .1.  } ) )
392, 38syl5eq 2340 . . . 4  |-  ( R  e.  _V  ->  U  =  ( `' ( 
.||  i^i  E ) " {  .1.  } ) )
4039eleq2d 2363 . . 3  |-  ( R  e.  _V  ->  ( X  e.  U  <->  X  e.  ( `' (  .||  i^i  E
) " {  .1.  } ) ) )
41 inss1 3402 . . . . . 6  |-  (  .||  i^i  E )  C_  .||
428reldvdsr 15442 . . . . . 6  |-  Rel  .||
43 relss 4791 . . . . . 6  |-  ( ( 
.||  i^i  E )  C_  .||  ->  ( Rel  .||  ->  Rel  (  .||  i^i  E ) ) )
4441, 42, 43mp2 17 . . . . 5  |-  Rel  (  .|| 
i^i  E )
45 eliniseg2 5069 . . . . 5  |-  ( Rel  (  .||  i^i  E )  ->  ( X  e.  ( `' (  .||  i^i  E ) " {  .1.  } )  <->  X (  .|| 
i^i  E )  .1.  ) )
4644, 45ax-mp 8 . . . 4  |-  ( X  e.  ( `' ( 
.||  i^i  E ) " {  .1.  } )  <-> 
X (  .||  i^i  E
)  .1.  )
47 brin 4086 . . . 4  |-  ( X (  .||  i^i  E )  .1.  <->  ( X  .||  .1.  /\  X E  .1.  ) )
4846, 47bitri 240 . . 3  |-  ( X  e.  ( `' ( 
.||  i^i  E ) " {  .1.  } )  <-> 
( X  .||  .1.  /\  X E  .1.  )
)
4940, 48syl6bb 252 . 2  |-  ( R  e.  _V  ->  ( X  e.  U  <->  ( X  .|| 
.1.  /\  X E  .1.  ) ) )
505, 13, 49pm5.21nii 342 1  |-  ( X  e.  U  <->  ( X  .|| 
.1.  /\  X E  .1.  ) )
Colors of variables: wff set class
Syntax hints:    <-> wb 176    /\ wa 358    = wceq 1632    e. wcel 1696   _Vcvv 2801    i^i cin 3164    C_ wss 3165   {csn 3653   <.cop 3656   class class class wbr 4039   `'ccnv 4704   dom cdm 4705   "cima 4708   Rel wrel 4710   ` cfv 5271   1rcur 15355  opprcoppr 15420   ||rcdsr 15436  Unitcui 15437
This theorem is referenced by:  1unit  15456  unitcl  15457  opprunit  15459  crngunit  15460  unitmulcl  15462  unitgrp  15465  unitnegcl  15479  unitpropd  15495  isdrng2  15538  subrguss  15576  subrgunit  15579  fidomndrng  16064
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fv 5279  df-dvdsr 15439  df-unit 15440
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