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Theorem crngunit 17182
Description: Property of being a unit in a commutative ring. (Contributed by Mario Carneiro, 18-Apr-2016.)
Hypotheses
Ref Expression
crngunit.1  |-  U  =  (Unit `  R )
crngunit.2  |-  .1.  =  ( 1r `  R )
crngunit.3  |-  .||  =  (
||r `  R )
Assertion
Ref Expression
crngunit  |-  ( R  e.  CRing  ->  ( X  e.  U  <->  X  .||  .1.  )
)

Proof of Theorem crngunit
Dummy variable  y is distinct from all other variables.
StepHypRef Expression
1 eqid 2441 . . . . . . . . . . 11  |-  ( Base `  R )  =  (
Base `  R )
2 eqid 2441 . . . . . . . . . . 11  |-  ( .r
`  R )  =  ( .r `  R
)
3 eqid 2441 . . . . . . . . . . 11  |-  (oppr `  R
)  =  (oppr `  R
)
4 eqid 2441 . . . . . . . . . . 11  |-  ( .r
`  (oppr
`  R ) )  =  ( .r `  (oppr `  R ) )
51, 2, 3, 4crngoppr 17147 . . . . . . . . . 10  |-  ( ( R  e.  CRing  /\  y  e.  ( Base `  R
)  /\  X  e.  ( Base `  R )
)  ->  ( y
( .r `  R
) X )  =  ( y ( .r
`  (oppr
`  R ) ) X ) )
653expa 1195 . . . . . . . . 9  |-  ( ( ( R  e.  CRing  /\  y  e.  ( Base `  R ) )  /\  X  e.  ( Base `  R ) )  -> 
( y ( .r
`  R ) X )  =  ( y ( .r `  (oppr `  R
) ) X ) )
76eqcomd 2449 . . . . . . . 8  |-  ( ( ( R  e.  CRing  /\  y  e.  ( Base `  R ) )  /\  X  e.  ( Base `  R ) )  -> 
( y ( .r
`  (oppr
`  R ) ) X )  =  ( y ( .r `  R ) X ) )
87an32s 802 . . . . . . 7  |-  ( ( ( R  e.  CRing  /\  X  e.  ( Base `  R ) )  /\  y  e.  ( Base `  R ) )  -> 
( y ( .r
`  (oppr
`  R ) ) X )  =  ( y ( .r `  R ) X ) )
98eqeq1d 2443 . . . . . 6  |-  ( ( ( R  e.  CRing  /\  X  e.  ( Base `  R ) )  /\  y  e.  ( Base `  R ) )  -> 
( ( y ( .r `  (oppr `  R
) ) X )  =  .1.  <->  ( y
( .r `  R
) X )  =  .1.  ) )
109rexbidva 2949 . . . . 5  |-  ( ( R  e.  CRing  /\  X  e.  ( Base `  R
) )  ->  ( E. y  e.  ( Base `  R ) ( y ( .r `  (oppr `  R ) ) X )  =  .1.  <->  E. y  e.  ( Base `  R
) ( y ( .r `  R ) X )  =  .1.  ) )
1110pm5.32da 641 . . . 4  |-  ( R  e.  CRing  ->  ( ( X  e.  ( Base `  R )  /\  E. y  e.  ( Base `  R ) ( y ( .r `  (oppr `  R
) ) X )  =  .1.  )  <->  ( X  e.  ( Base `  R
)  /\  E. y  e.  ( Base `  R
) ( y ( .r `  R ) X )  =  .1.  ) ) )
123, 1opprbas 17149 . . . . 5  |-  ( Base `  R )  =  (
Base `  (oppr
`  R ) )
13 eqid 2441 . . . . 5  |-  ( ||r `  (oppr `  R
) )  =  (
||r `  (oppr
`  R ) )
1412, 13, 4dvdsr 17166 . . . 4  |-  ( X ( ||r `
 (oppr
`  R ) )  .1.  <->  ( X  e.  ( Base `  R
)  /\  E. y  e.  ( Base `  R
) ( y ( .r `  (oppr `  R
) ) X )  =  .1.  ) )
15 crngunit.3 . . . . 5  |-  .||  =  (
||r `  R )
161, 15, 2dvdsr 17166 . . . 4  |-  ( X 
.||  .1.  <->  ( X  e.  ( Base `  R
)  /\  E. y  e.  ( Base `  R
) ( y ( .r `  R ) X )  =  .1.  ) )
1711, 14, 163bitr4g 288 . . 3  |-  ( R  e.  CRing  ->  ( X
( ||r `
 (oppr
`  R ) )  .1.  <->  X  .||  .1.  )
)
1817anbi2d 703 . 2  |-  ( R  e.  CRing  ->  ( ( X  .||  .1.  /\  X
( ||r `
 (oppr
`  R ) )  .1.  )  <->  ( X  .|| 
.1.  /\  X  .||  .1.  )
) )
19 crngunit.1 . . 3  |-  U  =  (Unit `  R )
20 crngunit.2 . . 3  |-  .1.  =  ( 1r `  R )
2119, 20, 15, 3, 13isunit 17177 . 2  |-  ( X  e.  U  <->  ( X  .|| 
.1.  /\  X ( ||r `  (oppr
`  R ) )  .1.  ) )
22 pm4.24 643 . 2  |-  ( X 
.||  .1.  <->  ( X  .||  .1.  /\  X  .||  .1.  )
)
2318, 21, 223bitr4g 288 1  |-  ( R  e.  CRing  ->  ( X  e.  U  <->  X  .||  .1.  )
)
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1381    e. wcel 1802   E.wrex 2792   class class class wbr 4434   ` cfv 5575  (class class class)co 6278   Basecbs 14506   .rcmulr 14572   1rcur 17024   CRingccrg 17070  opprcoppr 17142   ||rcdsr 17158  Unitcui 17159
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1603  ax-4 1616  ax-5 1689  ax-6 1732  ax-7 1774  ax-8 1804  ax-9 1806  ax-10 1821  ax-11 1826  ax-12 1838  ax-13 1983  ax-ext 2419  ax-rep 4545  ax-sep 4555  ax-nul 4563  ax-pow 4612  ax-pr 4673  ax-un 6574  ax-cnex 9548  ax-resscn 9549  ax-1cn 9550  ax-icn 9551  ax-addcl 9552  ax-addrcl 9553  ax-mulcl 9554  ax-mulrcl 9555  ax-mulcom 9556  ax-addass 9557  ax-mulass 9558  ax-distr 9559  ax-i2m1 9560  ax-1ne0 9561  ax-1rid 9562  ax-rnegex 9563  ax-rrecex 9564  ax-cnre 9565  ax-pre-lttri 9566  ax-pre-lttrn 9567  ax-pre-ltadd 9568  ax-pre-mulgt0 9569
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 973  df-3an 974  df-tru 1384  df-ex 1598  df-nf 1602  df-sb 1725  df-eu 2270  df-mo 2271  df-clab 2427  df-cleq 2433  df-clel 2436  df-nfc 2591  df-ne 2638  df-nel 2639  df-ral 2796  df-rex 2797  df-reu 2798  df-rab 2800  df-v 3095  df-sbc 3312  df-csb 3419  df-dif 3462  df-un 3464  df-in 3466  df-ss 3473  df-pss 3475  df-nul 3769  df-if 3924  df-pw 3996  df-sn 4012  df-pr 4014  df-tp 4016  df-op 4018  df-uni 4232  df-iun 4314  df-br 4435  df-opab 4493  df-mpt 4494  df-tr 4528  df-eprel 4778  df-id 4782  df-po 4787  df-so 4788  df-fr 4825  df-we 4827  df-ord 4868  df-on 4869  df-lim 4870  df-suc 4871  df-xp 4992  df-rel 4993  df-cnv 4994  df-co 4995  df-dm 4996  df-rn 4997  df-res 4998  df-ima 4999  df-iota 5538  df-fun 5577  df-fn 5578  df-f 5579  df-f1 5580  df-fo 5581  df-f1o 5582  df-fv 5583  df-riota 6239  df-ov 6281  df-oprab 6282  df-mpt2 6283  df-om 6683  df-tpos 6954  df-recs 7041  df-rdg 7075  df-er 7310  df-en 7516  df-dom 7517  df-sdom 7518  df-pnf 9630  df-mnf 9631  df-xr 9632  df-ltxr 9633  df-le 9634  df-sub 9809  df-neg 9810  df-nn 10540  df-2 10597  df-3 10598  df-ndx 14509  df-slot 14510  df-base 14511  df-sets 14512  df-plusg 14584  df-mulr 14585  df-cmn 16671  df-mgp 17013  df-cring 17072  df-oppr 17143  df-dvdsr 17161  df-unit 17162
This theorem is referenced by:  dvdsunit  17183  znunit  18472
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