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Theorem srgrz 16744
Description: The zero of a semiring is a right-absorbing element. (Contributed by Thierry Arnoux, 1-Apr-2018.)
Hypotheses
Ref Expression
srgz.b  |-  B  =  ( Base `  R
)
srgz.t  |-  .x.  =  ( .r `  R )
srgz.z  |-  .0.  =  ( 0g `  R )
Assertion
Ref Expression
srgrz  |-  ( ( R  e. SRing  /\  X  e.  B )  ->  ( X  .x.  .0.  )  =  .0.  )

Proof of Theorem srgrz
Dummy variables  x  y  z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 srgz.b . . . . . . . 8  |-  B  =  ( Base `  R
)
2 eqid 2452 . . . . . . . 8  |-  (mulGrp `  R )  =  (mulGrp `  R )
3 eqid 2452 . . . . . . . 8  |-  ( +g  `  R )  =  ( +g  `  R )
4 srgz.t . . . . . . . 8  |-  .x.  =  ( .r `  R )
5 srgz.z . . . . . . . 8  |-  .0.  =  ( 0g `  R )
61, 2, 3, 4, 5issrg 16726 . . . . . . 7  |-  ( R  e. SRing 
<->  ( R  e. CMnd  /\  (mulGrp `  R )  e. 
Mnd  /\  A. x  e.  B  ( A. y  e.  B  A. z  e.  B  (
( x  .x.  (
y ( +g  `  R
) z ) )  =  ( ( x 
.x.  y ) ( +g  `  R ) ( x  .x.  z
) )  /\  (
( x ( +g  `  R ) y ) 
.x.  z )  =  ( ( x  .x.  z ) ( +g  `  R ) ( y 
.x.  z ) ) )  /\  ( (  .0.  .x.  x )  =  .0.  /\  ( x 
.x.  .0.  )  =  .0.  ) ) ) )
76simp3bi 1005 . . . . . 6  |-  ( R  e. SRing  ->  A. x  e.  B  ( A. y  e.  B  A. z  e.  B  ( ( x  .x.  ( y ( +g  `  R ) z ) )  =  ( ( x  .x.  y ) ( +g  `  R
) ( x  .x.  z ) )  /\  ( ( x ( +g  `  R ) y )  .x.  z
)  =  ( ( x  .x.  z ) ( +g  `  R
) ( y  .x.  z ) ) )  /\  ( (  .0. 
.x.  x )  =  .0.  /\  ( x 
.x.  .0.  )  =  .0.  ) ) )
87r19.21bi 2914 . . . . 5  |-  ( ( R  e. SRing  /\  x  e.  B )  ->  ( A. y  e.  B  A. z  e.  B  ( ( x  .x.  ( y ( +g  `  R ) z ) )  =  ( ( x  .x.  y ) ( +g  `  R
) ( x  .x.  z ) )  /\  ( ( x ( +g  `  R ) y )  .x.  z
)  =  ( ( x  .x.  z ) ( +g  `  R
) ( y  .x.  z ) ) )  /\  ( (  .0. 
.x.  x )  =  .0.  /\  ( x 
.x.  .0.  )  =  .0.  ) ) )
98simprd 463 . . . 4  |-  ( ( R  e. SRing  /\  x  e.  B )  ->  (
(  .0.  .x.  x
)  =  .0.  /\  ( x  .x.  .0.  )  =  .0.  ) )
109simprd 463 . . 3  |-  ( ( R  e. SRing  /\  x  e.  B )  ->  (
x  .x.  .0.  )  =  .0.  )
1110ralrimiva 2827 . 2  |-  ( R  e. SRing  ->  A. x  e.  B  ( x  .x.  .0.  )  =  .0.  )
12 oveq1 6202 . . . 4  |-  ( x  =  X  ->  (
x  .x.  .0.  )  =  ( X  .x.  .0.  ) )
1312eqeq1d 2454 . . 3  |-  ( x  =  X  ->  (
( x  .x.  .0.  )  =  .0.  <->  ( X  .x.  .0.  )  =  .0.  ) )
1413rspcv 3169 . 2  |-  ( X  e.  B  ->  ( A. x  e.  B  ( x  .x.  .0.  )  =  .0.  ->  ( X  .x.  .0.  )  =  .0.  ) )
1511, 14mpan9 469 1  |-  ( ( R  e. SRing  /\  X  e.  B )  ->  ( X  .x.  .0.  )  =  .0.  )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1370    e. wcel 1758   A.wral 2796   ` cfv 5521  (class class class)co 6195   Basecbs 14287   +g cplusg 14352   .rcmulr 14353   0gc0g 14492   Mndcmnd 15523  CMndccmn 16393  mulGrpcmgp 16708  SRingcsrg 16724
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1954  ax-ext 2431  ax-nul 4524
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2265  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2602  df-ne 2647  df-ral 2801  df-rex 2802  df-rab 2805  df-v 3074  df-sbc 3289  df-dif 3434  df-un 3436  df-in 3438  df-ss 3445  df-nul 3741  df-if 3895  df-sn 3981  df-pr 3983  df-op 3987  df-uni 4195  df-br 4396  df-iota 5484  df-fv 5529  df-ov 6198  df-srg 16725
This theorem is referenced by:  srgisid  16746  srglmhm  16751  slmdvs0  26381
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