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Theorem rrgeq0i 16308
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 16307 . . 3  |-  ( X  e.  E  <->  ( X  e.  B  /\  A. y  e.  B  ( ( X  .x.  y )  =  .0.  ->  y  =  .0.  ) ) )
65simprbi 451 . 2  |-  ( X  e.  E  ->  A. y  e.  B  ( ( X  .x.  y )  =  .0.  ->  y  =  .0.  ) )
7 oveq2 6052 . . . . 5  |-  ( y  =  Y  ->  ( X  .x.  y )  =  ( X  .x.  Y
) )
87eqeq1d 2416 . . . 4  |-  ( y  =  Y  ->  (
( X  .x.  y
)  =  .0.  <->  ( X  .x.  Y )  =  .0.  ) )
9 eqeq1 2414 . . . 4  |-  ( y  =  Y  ->  (
y  =  .0.  <->  Y  =  .0.  ) )
108, 9imbi12d 312 . . 3  |-  ( y  =  Y  ->  (
( ( X  .x.  y )  =  .0. 
->  y  =  .0.  ) 
<->  ( ( X  .x.  Y )  =  .0. 
->  Y  =  .0.  ) ) )
1110rspcv 3012 . 2  |-  ( Y  e.  B  ->  ( A. y  e.  B  ( ( X  .x.  y )  =  .0. 
->  y  =  .0.  )  ->  ( ( X 
.x.  Y )  =  .0.  ->  Y  =  .0.  ) ) )
126, 11mpan9 456 1  |-  ( ( X  e.  E  /\  Y  e.  B )  ->  ( ( X  .x.  Y )  =  .0. 
->  Y  =  .0.  ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1649    e. wcel 1721   A.wral 2670   ` cfv 5417  (class class class)co 6044   Basecbs 13428   .rcmulr 13489   0gc0g 13682  RLRegcrlreg 16298
This theorem is referenced by:  rrgeq0  16309  znrrg  16805  deg1mul2  19994
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2389  ax-sep 4294  ax-nul 4302  ax-pow 4341  ax-pr 4367
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2262  df-mo 2263  df-clab 2395  df-cleq 2401  df-clel 2404  df-nfc 2533  df-ne 2573  df-ral 2675  df-rex 2676  df-rab 2679  df-v 2922  df-sbc 3126  df-dif 3287  df-un 3289  df-in 3291  df-ss 3298  df-nul 3593  df-if 3704  df-sn 3784  df-pr 3785  df-op 3787  df-uni 3980  df-br 4177  df-opab 4231  df-mpt 4232  df-id 4462  df-xp 4847  df-rel 4848  df-cnv 4849  df-co 4850  df-dm 4851  df-iota 5381  df-fun 5419  df-fv 5425  df-ov 6047  df-rlreg 16302
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