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Theorem rrgeq0i 17365
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 17364 . . 3  |-  ( X  e.  E  <->  ( X  e.  B  /\  A. y  e.  B  ( ( X  .x.  y )  =  .0.  ->  y  =  .0.  ) ) )
65simprbi 464 . 2  |-  ( X  e.  E  ->  A. y  e.  B  ( ( X  .x.  y )  =  .0.  ->  y  =  .0.  ) )
7 oveq2 6104 . . . . 5  |-  ( y  =  Y  ->  ( X  .x.  y )  =  ( X  .x.  Y
) )
87eqeq1d 2451 . . . 4  |-  ( y  =  Y  ->  (
( X  .x.  y
)  =  .0.  <->  ( X  .x.  Y )  =  .0.  ) )
9 eqeq1 2449 . . . 4  |-  ( y  =  Y  ->  (
y  =  .0.  <->  Y  =  .0.  ) )
108, 9imbi12d 320 . . 3  |-  ( y  =  Y  ->  (
( ( X  .x.  y )  =  .0. 
->  y  =  .0.  ) 
<->  ( ( X  .x.  Y )  =  .0. 
->  Y  =  .0.  ) ) )
1110rspcv 3074 . 2  |-  ( Y  e.  B  ->  ( A. y  e.  B  ( ( X  .x.  y )  =  .0. 
->  y  =  .0.  )  ->  ( ( X 
.x.  Y )  =  .0.  ->  Y  =  .0.  ) ) )
126, 11mpan9 469 1  |-  ( ( X  e.  E  /\  Y  e.  B )  ->  ( ( X  .x.  Y )  =  .0. 
->  Y  =  .0.  ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1369    e. wcel 1756   A.wral 2720   ` cfv 5423  (class class class)co 6096   Basecbs 14179   .rcmulr 14244   0gc0g 14383  RLRegcrlreg 17355
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-sep 4418  ax-nul 4426  ax-pow 4475  ax-pr 4536
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2573  df-ne 2613  df-ral 2725  df-rex 2726  df-rab 2729  df-v 2979  df-sbc 3192  df-dif 3336  df-un 3338  df-in 3340  df-ss 3347  df-nul 3643  df-if 3797  df-sn 3883  df-pr 3885  df-op 3889  df-uni 4097  df-br 4298  df-opab 4356  df-mpt 4357  df-id 4641  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-iota 5386  df-fun 5425  df-fv 5431  df-ov 6099  df-rlreg 17359
This theorem is referenced by:  rrgeq0  17366  znrrg  18003  deg1mul2  21591
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