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Theorem isrrg 16277
Description: Membership in the set of left-regular elements. (Contributed by Stefan O'Rear, 22-Mar-2015.)
Hypotheses
Ref Expression
rrgval.e  |-  E  =  (RLReg `  R )
rrgval.b  |-  B  =  ( Base `  R
)
rrgval.t  |-  .x.  =  ( .r `  R )
rrgval.z  |-  .0.  =  ( 0g `  R )
Assertion
Ref Expression
isrrg  |-  ( X  e.  E  <->  ( X  e.  B  /\  A. y  e.  B  ( ( X  .x.  y )  =  .0.  ->  y  =  .0.  ) ) )
Distinct variable groups:    y, B    y, R    y, X
Allowed substitution hints:    .x. ( y)    E( y)    .0. ( y)

Proof of Theorem isrrg
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 oveq1 6029 . . . . 5  |-  ( x  =  X  ->  (
x  .x.  y )  =  ( X  .x.  y ) )
21eqeq1d 2397 . . . 4  |-  ( x  =  X  ->  (
( x  .x.  y
)  =  .0.  <->  ( X  .x.  y )  =  .0.  ) )
32imbi1d 309 . . 3  |-  ( x  =  X  ->  (
( ( x  .x.  y )  =  .0. 
->  y  =  .0.  ) 
<->  ( ( X  .x.  y )  =  .0. 
->  y  =  .0.  ) ) )
43ralbidv 2671 . 2  |-  ( x  =  X  ->  ( A. y  e.  B  ( ( x  .x.  y )  =  .0. 
->  y  =  .0.  ) 
<-> 
A. y  e.  B  ( ( X  .x.  y )  =  .0. 
->  y  =  .0.  ) ) )
5 rrgval.e . . 3  |-  E  =  (RLReg `  R )
6 rrgval.b . . 3  |-  B  =  ( Base `  R
)
7 rrgval.t . . 3  |-  .x.  =  ( .r `  R )
8 rrgval.z . . 3  |-  .0.  =  ( 0g `  R )
95, 6, 7, 8rrgval 16276 . 2  |-  E  =  { x  e.  B  |  A. y  e.  B  ( ( x  .x.  y )  =  .0. 
->  y  =  .0.  ) }
104, 9elrab2 3039 1  |-  ( X  e.  E  <->  ( X  e.  B  /\  A. y  e.  B  ( ( X  .x.  y )  =  .0.  ->  y  =  .0.  ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    = wceq 1649    e. wcel 1717   A.wral 2651   ` cfv 5396  (class class class)co 6022   Basecbs 13398   .rcmulr 13459   0gc0g 13652  RLRegcrlreg 16268
This theorem is referenced by:  rrgeq0i  16278  unitrrg  16282  isdomn2  16288
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2370  ax-sep 4273  ax-nul 4281  ax-pow 4320  ax-pr 4346
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2244  df-mo 2245  df-clab 2376  df-cleq 2382  df-clel 2385  df-nfc 2514  df-ne 2554  df-ral 2656  df-rex 2657  df-rab 2660  df-v 2903  df-sbc 3107  df-dif 3268  df-un 3270  df-in 3272  df-ss 3279  df-nul 3574  df-if 3685  df-sn 3765  df-pr 3766  df-op 3768  df-uni 3960  df-br 4156  df-opab 4210  df-mpt 4211  df-id 4441  df-xp 4826  df-rel 4827  df-cnv 4828  df-co 4829  df-dm 4830  df-iota 5360  df-fun 5398  df-fv 5404  df-ov 6025  df-rlreg 16272
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