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Theorem isrrg 16029
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 5865 . . . . 5  |-  ( x  =  X  ->  (
x  .x.  y )  =  ( X  .x.  y ) )
21eqeq1d 2291 . . . 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 2563 . 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 16028 . 2  |-  E  =  { x  e.  B  |  A. y  e.  B  ( ( x  .x.  y )  =  .0. 
->  y  =  .0.  ) }
104, 9elrab2 2925 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 1623    e. wcel 1684   A.wral 2543   ` cfv 5255  (class class class)co 5858   Basecbs 13148   .rcmulr 13209   0gc0g 13400  RLRegcrlreg 16020
This theorem is referenced by:  rrgeq0i  16030  unitrrg  16034  isdomn2  16040
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-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  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-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-iota 5219  df-fun 5257  df-fv 5263  df-ov 5861  df-rlreg 16024
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