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Theorem rrgeq0i 18063
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 18062 . . 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 6304 . . . . 5  |-  ( y  =  Y  ->  ( X  .x.  y )  =  ( X  .x.  Y
) )
87eqeq1d 2459 . . . 4  |-  ( y  =  Y  ->  (
( X  .x.  y
)  =  .0.  <->  ( X  .x.  Y )  =  .0.  ) )
9 eqeq1 2461 . . . 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 3206 . 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 1395    e. wcel 1819   A.wral 2807   ` cfv 5594  (class class class)co 6296   Basecbs 14643   .rcmulr 14712   0gc0g 14856  RLRegcrlreg 18053
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-rex 2813  df-rab 2816  df-v 3111  df-sbc 3328  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3794  df-if 3945  df-sn 4033  df-pr 4035  df-op 4039  df-uni 4252  df-br 4457  df-opab 4516  df-mpt 4517  df-id 4804  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-iota 5557  df-fun 5596  df-fv 5602  df-ov 6299  df-rlreg 18057
This theorem is referenced by:  rrgeq0  18064  znrrg  18730  deg1mul2  22640
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