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Theorem crngunit 15444
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 2283 . . . . . . . . . . 11  |-  ( Base `  R )  =  (
Base `  R )
2 eqid 2283 . . . . . . . . . . 11  |-  ( .r
`  R )  =  ( .r `  R
)
3 eqid 2283 . . . . . . . . . . 11  |-  (oppr `  R
)  =  (oppr `  R
)
4 eqid 2283 . . . . . . . . . . 11  |-  ( .r
`  (oppr
`  R ) )  =  ( .r `  (oppr `  R ) )
51, 2, 3, 4crngoppr 15409 . . . . . . . . . 10  |-  ( ( R  e.  CRing  /\  y  e.  ( Base `  R
)  /\  X  e.  ( Base `  R )
)  ->  ( y
( .r `  R
) X )  =  ( y ( .r
`  (oppr
`  R ) ) X ) )
653expa 1151 . . . . . . . . 9  |-  ( ( ( R  e.  CRing  /\  y  e.  ( Base `  R ) )  /\  X  e.  ( Base `  R ) )  -> 
( y ( .r
`  R ) X )  =  ( y ( .r `  (oppr `  R
) ) X ) )
76eqcomd 2288 . . . . . . . 8  |-  ( ( ( R  e.  CRing  /\  y  e.  ( Base `  R ) )  /\  X  e.  ( Base `  R ) )  -> 
( y ( .r
`  (oppr
`  R ) ) X )  =  ( y ( .r `  R ) X ) )
87an32s 779 . . . . . . 7  |-  ( ( ( R  e.  CRing  /\  X  e.  ( Base `  R ) )  /\  y  e.  ( Base `  R ) )  -> 
( y ( .r
`  (oppr
`  R ) ) X )  =  ( y ( .r `  R ) X ) )
98eqeq1d 2291 . . . . . 6  |-  ( ( ( R  e.  CRing  /\  X  e.  ( Base `  R ) )  /\  y  e.  ( Base `  R ) )  -> 
( ( y ( .r `  (oppr `  R
) ) X )  =  .1.  <->  ( y
( .r `  R
) X )  =  .1.  ) )
109rexbidva 2560 . . . . 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 622 . . . 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 15411 . . . . 5  |-  ( Base `  R )  =  (
Base `  (oppr
`  R ) )
13 eqid 2283 . . . . 5  |-  ( ||r `  (oppr `  R
) )  =  (
||r `  (oppr
`  R ) )
1412, 13, 4dvdsr 15428 . . . 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 15428 . . . 4  |-  ( X 
.||  .1.  <->  ( X  e.  ( Base `  R
)  /\  E. y  e.  ( Base `  R
) ( y ( .r `  R ) X )  =  .1.  ) )
1711, 14, 163bitr4g 279 . . 3  |-  ( R  e.  CRing  ->  ( X
( ||r `
 (oppr
`  R ) )  .1.  <->  X  .||  .1.  )
)
1817anbi2d 684 . 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 15439 . 2  |-  ( X  e.  U  <->  ( X  .|| 
.1.  /\  X ( ||r `  (oppr
`  R ) )  .1.  ) )
22 pm4.24 624 . 2  |-  ( X 
.||  .1.  <->  ( X  .||  .1.  /\  X  .||  .1.  )
)
2318, 21, 223bitr4g 279 1  |-  ( R  e.  CRing  ->  ( X  e.  U  <->  X  .||  .1.  )
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    = wceq 1623    e. wcel 1684   E.wrex 2544   class class class wbr 4023   ` cfv 5255  (class class class)co 5858   Basecbs 13148   .rcmulr 13209   CRingccrg 15338   1rcur 15339  opprcoppr 15404   ||rcdsr 15420  Unitcui 15421
This theorem is referenced by:  dvdsunit  15445  znunit  16517
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-cnex 8793  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813  ax-pre-mulgt0 8814
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-tpos 6234  df-riota 6304  df-recs 6388  df-rdg 6423  df-er 6660  df-en 6864  df-dom 6865  df-sdom 6866  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-sub 9039  df-neg 9040  df-nn 9747  df-2 9804  df-3 9805  df-ndx 13151  df-slot 13152  df-base 13153  df-sets 13154  df-plusg 13221  df-mulr 13222  df-cmn 15091  df-mgp 15326  df-cring 15341  df-oppr 15405  df-dvdsr 15423  df-unit 15424
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