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Theorem mhm0 14739
Description: A monoid homorphism preserves zero. (Contributed by Mario Carneiro, 7-Mar-2015.)
Hypotheses
Ref Expression
mhm0.z  |-  .0.  =  ( 0g `  S )
mhm0.y  |-  Y  =  ( 0g `  T
)
Assertion
Ref Expression
mhm0  |-  ( F  e.  ( S MndHom  T
)  ->  ( F `  .0.  )  =  Y )

Proof of Theorem mhm0
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqid 2436 . . . 4  |-  ( Base `  S )  =  (
Base `  S )
2 eqid 2436 . . . 4  |-  ( Base `  T )  =  (
Base `  T )
3 eqid 2436 . . . 4  |-  ( +g  `  S )  =  ( +g  `  S )
4 eqid 2436 . . . 4  |-  ( +g  `  T )  =  ( +g  `  T )
5 mhm0.z . . . 4  |-  .0.  =  ( 0g `  S )
6 mhm0.y . . . 4  |-  Y  =  ( 0g `  T
)
71, 2, 3, 4, 5, 6ismhm 14733 . . 3  |-  ( F  e.  ( S MndHom  T
)  <->  ( ( S  e.  Mnd  /\  T  e.  Mnd )  /\  ( F : ( Base `  S
) --> ( Base `  T
)  /\  A. x  e.  ( Base `  S
) A. y  e.  ( Base `  S
) ( F `  ( x ( +g  `  S ) y ) )  =  ( ( F `  x ) ( +g  `  T
) ( F `  y ) )  /\  ( F `  .0.  )  =  Y ) ) )
87simprbi 451 . 2  |-  ( F  e.  ( S MndHom  T
)  ->  ( F : ( Base `  S
) --> ( Base `  T
)  /\  A. x  e.  ( Base `  S
) A. y  e.  ( Base `  S
) ( F `  ( x ( +g  `  S ) y ) )  =  ( ( F `  x ) ( +g  `  T
) ( F `  y ) )  /\  ( F `  .0.  )  =  Y ) )
98simp3d 971 1  |-  ( F  e.  ( S MndHom  T
)  ->  ( F `  .0.  )  =  Y )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    /\ w3a 936    = wceq 1652    e. wcel 1725   A.wral 2698   -->wf 5443   ` cfv 5447  (class class class)co 6074   Basecbs 13462   +g cplusg 13522   0gc0g 13716   Mndcmnd 14677   MndHom cmhm 14729
This theorem is referenced by:  resmhm  14752  resmhm2  14753  resmhm2b  14754  mhmco  14755  mhmima  14756  mhmeql  14757  pwsco2mhm  14763  gsumwmhm  14783  mhmmulg  14915  gsumzmhm  15526  rhm1  15824  dchrzrh1  21021  dchrmulcl  21026  dchrn0  21027  dchrinvcl  21030  dchrfi  21032  dchrabs  21037  sumdchr2  21047  rpvmasum2  21199
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-sep 4323  ax-nul 4331  ax-pow 4370  ax-pr 4396  ax-un 4694
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2703  df-rex 2704  df-rab 2707  df-v 2951  df-sbc 3155  df-dif 3316  df-un 3318  df-in 3320  df-ss 3327  df-nul 3622  df-if 3733  df-pw 3794  df-sn 3813  df-pr 3814  df-op 3816  df-uni 4009  df-br 4206  df-opab 4260  df-id 4491  df-xp 4877  df-rel 4878  df-cnv 4879  df-co 4880  df-dm 4881  df-rn 4882  df-iota 5411  df-fun 5449  df-fn 5450  df-f 5451  df-fv 5455  df-ov 6077  df-oprab 6078  df-mpt2 6079  df-map 7013  df-mhm 14731
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