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Theorem isrrg 16340
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 6080 . . . . 5  |-  ( x  =  X  ->  (
x  .x.  y )  =  ( X  .x.  y ) )
21eqeq1d 2443 . . . 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 2717 . 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 16339 . 2  |-  E  =  { x  e.  B  |  A. y  e.  B  ( ( x  .x.  y )  =  .0. 
->  y  =  .0.  ) }
104, 9elrab2 3086 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 1652    e. wcel 1725   A.wral 2697   ` cfv 5446  (class class class)co 6073   Basecbs 13461   .rcmulr 13522   0gc0g 13715  RLRegcrlreg 16331
This theorem is referenced by:  rrgeq0i  16341  unitrrg  16345  isdomn2  16351
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-rab 2706  df-v 2950  df-sbc 3154  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-br 4205  df-opab 4259  df-mpt 4260  df-id 4490  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-iota 5410  df-fun 5448  df-fv 5454  df-ov 6076  df-rlreg 16335
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