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Theorem crngunit 15769
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 2438 . . . . . . . . . . 11  |-  ( Base `  R )  =  (
Base `  R )
2 eqid 2438 . . . . . . . . . . 11  |-  ( .r
`  R )  =  ( .r `  R
)
3 eqid 2438 . . . . . . . . . . 11  |-  (oppr `  R
)  =  (oppr `  R
)
4 eqid 2438 . . . . . . . . . . 11  |-  ( .r
`  (oppr
`  R ) )  =  ( .r `  (oppr `  R ) )
51, 2, 3, 4crngoppr 15734 . . . . . . . . . 10  |-  ( ( R  e.  CRing  /\  y  e.  ( Base `  R
)  /\  X  e.  ( Base `  R )
)  ->  ( y
( .r `  R
) X )  =  ( y ( .r
`  (oppr
`  R ) ) X ) )
653expa 1154 . . . . . . . . 9  |-  ( ( ( R  e.  CRing  /\  y  e.  ( Base `  R ) )  /\  X  e.  ( Base `  R ) )  -> 
( y ( .r
`  R ) X )  =  ( y ( .r `  (oppr `  R
) ) X ) )
76eqcomd 2443 . . . . . . . 8  |-  ( ( ( R  e.  CRing  /\  y  e.  ( Base `  R ) )  /\  X  e.  ( Base `  R ) )  -> 
( y ( .r
`  (oppr
`  R ) ) X )  =  ( y ( .r `  R ) X ) )
87an32s 781 . . . . . . 7  |-  ( ( ( R  e.  CRing  /\  X  e.  ( Base `  R ) )  /\  y  e.  ( Base `  R ) )  -> 
( y ( .r
`  (oppr
`  R ) ) X )  =  ( y ( .r `  R ) X ) )
98eqeq1d 2446 . . . . . 6  |-  ( ( ( R  e.  CRing  /\  X  e.  ( Base `  R ) )  /\  y  e.  ( Base `  R ) )  -> 
( ( y ( .r `  (oppr `  R
) ) X )  =  .1.  <->  ( y
( .r `  R
) X )  =  .1.  ) )
109rexbidva 2724 . . . . 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 624 . . . 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 15736 . . . . 5  |-  ( Base `  R )  =  (
Base `  (oppr
`  R ) )
13 eqid 2438 . . . . 5  |-  ( ||r `  (oppr `  R
) )  =  (
||r `  (oppr
`  R ) )
1412, 13, 4dvdsr 15753 . . . 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 15753 . . . 4  |-  ( X 
.||  .1.  <->  ( X  e.  ( Base `  R
)  /\  E. y  e.  ( Base `  R
) ( y ( .r `  R ) X )  =  .1.  ) )
1711, 14, 163bitr4g 281 . . 3  |-  ( R  e.  CRing  ->  ( X
( ||r `
 (oppr
`  R ) )  .1.  <->  X  .||  .1.  )
)
1817anbi2d 686 . 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 15764 . 2  |-  ( X  e.  U  <->  ( X  .|| 
.1.  /\  X ( ||r `  (oppr
`  R ) )  .1.  ) )
22 pm4.24 626 . 2  |-  ( X 
.||  .1.  <->  ( X  .||  .1.  /\  X  .||  .1.  )
)
2318, 21, 223bitr4g 281 1  |-  ( R  e.  CRing  ->  ( X  e.  U  <->  X  .||  .1.  )
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 178    /\ wa 360    = wceq 1653    e. wcel 1726   E.wrex 2708   class class class wbr 4214   ` cfv 5456  (class class class)co 6083   Basecbs 13471   .rcmulr 13532   CRingccrg 15663   1rcur 15664  opprcoppr 15729   ||rcdsr 15745  Unitcui 15746
This theorem is referenced by:  dvdsunit  15770  znunit  16846
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-rep 4322  ax-sep 4332  ax-nul 4340  ax-pow 4379  ax-pr 4405  ax-un 4703  ax-cnex 9048  ax-resscn 9049  ax-1cn 9050  ax-icn 9051  ax-addcl 9052  ax-addrcl 9053  ax-mulcl 9054  ax-mulrcl 9055  ax-mulcom 9056  ax-addass 9057  ax-mulass 9058  ax-distr 9059  ax-i2m1 9060  ax-1ne0 9061  ax-1rid 9062  ax-rnegex 9063  ax-rrecex 9064  ax-cnre 9065  ax-pre-lttri 9066  ax-pre-lttrn 9067  ax-pre-ltadd 9068  ax-pre-mulgt0 9069
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-nel 2604  df-ral 2712  df-rex 2713  df-reu 2714  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-pss 3338  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-tp 3824  df-op 3825  df-uni 4018  df-iun 4097  df-br 4215  df-opab 4269  df-mpt 4270  df-tr 4305  df-eprel 4496  df-id 4500  df-po 4505  df-so 4506  df-fr 4543  df-we 4545  df-ord 4586  df-on 4587  df-lim 4588  df-suc 4589  df-om 4848  df-xp 4886  df-rel 4887  df-cnv 4888  df-co 4889  df-dm 4890  df-rn 4891  df-res 4892  df-ima 4893  df-iota 5420  df-fun 5458  df-fn 5459  df-f 5460  df-f1 5461  df-fo 5462  df-f1o 5463  df-fv 5464  df-ov 6086  df-oprab 6087  df-mpt2 6088  df-tpos 6481  df-riota 6551  df-recs 6635  df-rdg 6670  df-er 6907  df-en 7112  df-dom 7113  df-sdom 7114  df-pnf 9124  df-mnf 9125  df-xr 9126  df-ltxr 9127  df-le 9128  df-sub 9295  df-neg 9296  df-nn 10003  df-2 10060  df-3 10061  df-ndx 13474  df-slot 13475  df-base 13476  df-sets 13477  df-plusg 13544  df-mulr 13545  df-cmn 15416  df-mgp 15651  df-cring 15666  df-oppr 15730  df-dvdsr 15748  df-unit 15749
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