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Theorem rrgeq0i 16354
Description: Property of a left-regular element. (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
rrgeq0i  |-  ( ( X  e.  E  /\  Y  e.  B )  ->  ( ( X  .x.  Y )  =  .0. 
->  Y  =  .0.  ) )

Proof of Theorem rrgeq0i
Dummy variable  y is distinct from all other variables.
StepHypRef Expression
1 rrgval.e . . . 4  |-  E  =  (RLReg `  R )
2 rrgval.b . . . 4  |-  B  =  ( Base `  R
)
3 rrgval.t . . . 4  |-  .x.  =  ( .r `  R )
4 rrgval.z . . . 4  |-  .0.  =  ( 0g `  R )
51, 2, 3, 4isrrg 16353 . . 3  |-  ( X  e.  E  <->  ( X  e.  B  /\  A. y  e.  B  ( ( X  .x.  y )  =  .0.  ->  y  =  .0.  ) ) )
65simprbi 452 . 2  |-  ( X  e.  E  ->  A. y  e.  B  ( ( X  .x.  y )  =  .0.  ->  y  =  .0.  ) )
7 oveq2 6092 . . . . 5  |-  ( y  =  Y  ->  ( X  .x.  y )  =  ( X  .x.  Y
) )
87eqeq1d 2446 . . . 4  |-  ( y  =  Y  ->  (
( X  .x.  y
)  =  .0.  <->  ( X  .x.  Y )  =  .0.  ) )
9 eqeq1 2444 . . . 4  |-  ( y  =  Y  ->  (
y  =  .0.  <->  Y  =  .0.  ) )
108, 9imbi12d 313 . . 3  |-  ( y  =  Y  ->  (
( ( X  .x.  y )  =  .0. 
->  y  =  .0.  ) 
<->  ( ( X  .x.  Y )  =  .0. 
->  Y  =  .0.  ) ) )
1110rspcv 3050 . 2  |-  ( Y  e.  B  ->  ( A. y  e.  B  ( ( X  .x.  y )  =  .0. 
->  y  =  .0.  )  ->  ( ( X 
.x.  Y )  =  .0.  ->  Y  =  .0.  ) ) )
126, 11mpan9 457 1  |-  ( ( X  e.  E  /\  Y  e.  B )  ->  ( ( X  .x.  Y )  =  .0. 
->  Y  =  .0.  ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 360    = wceq 1653    e. wcel 1726   A.wral 2707   ` cfv 5457  (class class class)co 6084   Basecbs 13474   .rcmulr 13535   0gc0g 13728  RLRegcrlreg 16344
This theorem is referenced by:  rrgeq0  16355  znrrg  16851  deg1mul2  20042
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-sep 4333  ax-nul 4341  ax-pow 4380  ax-pr 4406
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-rab 2716  df-v 2960  df-sbc 3164  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-br 4216  df-opab 4270  df-mpt 4271  df-id 4501  df-xp 4887  df-rel 4888  df-cnv 4889  df-co 4890  df-dm 4891  df-iota 5421  df-fun 5459  df-fv 5465  df-ov 6087  df-rlreg 16348
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