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Theorem isrrg 16045
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 5881 . . . . 5  |-  ( x  =  X  ->  (
x  .x.  y )  =  ( X  .x.  y ) )
21eqeq1d 2304 . . . 4  |-  ( x  =  X  ->  (
( x  .x.  y
)  =  .0.  <->  ( X  .x.  y )  =  .0.  ) )
32imbi1d 308 . . 3  |-  ( x  =  X  ->  (
( ( x  .x.  y )  =  .0. 
->  y  =  .0.  ) 
<->  ( ( X  .x.  y )  =  .0. 
->  y  =  .0.  ) ) )
43ralbidv 2576 . 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 16044 . 2  |-  E  =  { x  e.  B  |  A. y  e.  B  ( ( x  .x.  y )  =  .0. 
->  y  =  .0.  ) }
104, 9elrab2 2938 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 176    /\ wa 358    = wceq 1632    e. wcel 1696   A.wral 2556   ` cfv 5271  (class class class)co 5874   Basecbs 13164   .rcmulr 13225   0gc0g 13416  RLRegcrlreg 16036
This theorem is referenced by:  rrgeq0i  16046  unitrrg  16050  isdomn2  16056
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
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-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  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-iota 5235  df-fun 5273  df-fv 5279  df-ov 5877  df-rlreg 16040
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