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Theorem rrgeq0i 16129
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 16128 . . 3  |-  ( X  e.  E  <->  ( X  e.  B  /\  A. y  e.  B  ( ( X  .x.  y )  =  .0.  ->  y  =  .0.  ) ) )
65simprbi 450 . 2  |-  ( X  e.  E  ->  A. y  e.  B  ( ( X  .x.  y )  =  .0.  ->  y  =  .0.  ) )
7 oveq2 5953 . . . . 5  |-  ( y  =  Y  ->  ( X  .x.  y )  =  ( X  .x.  Y
) )
87eqeq1d 2366 . . . 4  |-  ( y  =  Y  ->  (
( X  .x.  y
)  =  .0.  <->  ( X  .x.  Y )  =  .0.  ) )
9 eqeq1 2364 . . . 4  |-  ( y  =  Y  ->  (
y  =  .0.  <->  Y  =  .0.  ) )
108, 9imbi12d 311 . . 3  |-  ( y  =  Y  ->  (
( ( X  .x.  y )  =  .0. 
->  y  =  .0.  ) 
<->  ( ( X  .x.  Y )  =  .0. 
->  Y  =  .0.  ) ) )
1110rspcv 2956 . 2  |-  ( Y  e.  B  ->  ( A. y  e.  B  ( ( X  .x.  y )  =  .0. 
->  y  =  .0.  )  ->  ( ( X 
.x.  Y )  =  .0.  ->  Y  =  .0.  ) ) )
126, 11mpan9 455 1  |-  ( ( X  e.  E  /\  Y  e.  B )  ->  ( ( X  .x.  Y )  =  .0. 
->  Y  =  .0.  ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1642    e. wcel 1710   A.wral 2619   ` cfv 5337  (class class class)co 5945   Basecbs 13245   .rcmulr 13306   0gc0g 13499  RLRegcrlreg 16119
This theorem is referenced by:  rrgeq0  16130  znrrg  16625  deg1mul2  19604
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-13 1712  ax-14 1714  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1930  ax-ext 2339  ax-sep 4222  ax-nul 4230  ax-pow 4269  ax-pr 4295
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2213  df-mo 2214  df-clab 2345  df-cleq 2351  df-clel 2354  df-nfc 2483  df-ne 2523  df-ral 2624  df-rex 2625  df-rab 2628  df-v 2866  df-sbc 3068  df-dif 3231  df-un 3233  df-in 3235  df-ss 3242  df-nul 3532  df-if 3642  df-sn 3722  df-pr 3723  df-op 3725  df-uni 3909  df-br 4105  df-opab 4159  df-mpt 4160  df-id 4391  df-xp 4777  df-rel 4778  df-cnv 4779  df-co 4780  df-dm 4781  df-iota 5301  df-fun 5339  df-fv 5345  df-ov 5948  df-rlreg 16123
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