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Theorem mhm0 15592
Description: A monoid homomorphism 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 2454 . . . 4  |-  ( Base `  S )  =  (
Base `  S )
2 eqid 2454 . . . 4  |-  ( Base `  T )  =  (
Base `  T )
3 eqid 2454 . . . 4  |-  ( +g  `  S )  =  ( +g  `  S )
4 eqid 2454 . . . 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 15586 . . 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 464 . 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 1002 1  |-  ( F  e.  ( S MndHom  T
)  ->  ( F `  .0.  )  =  Y )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    /\ w3a 965    = wceq 1370    e. wcel 1758   A.wral 2799   -->wf 5523   ` cfv 5527  (class class class)co 6201   Basecbs 14293   +g cplusg 14358   0gc0g 14498   Mndcmnd 15529   MndHom cmhm 15582
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-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-sep 4522  ax-nul 4530  ax-pow 4579  ax-pr 4640  ax-un 6483
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 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-ral 2804  df-rex 2805  df-rab 2808  df-v 3080  df-sbc 3295  df-dif 3440  df-un 3442  df-in 3444  df-ss 3451  df-nul 3747  df-if 3901  df-pw 3971  df-sn 3987  df-pr 3989  df-op 3993  df-uni 4201  df-br 4402  df-opab 4460  df-id 4745  df-xp 4955  df-rel 4956  df-cnv 4957  df-co 4958  df-dm 4959  df-rn 4960  df-iota 5490  df-fun 5529  df-fn 5530  df-f 5531  df-fv 5535  df-ov 6204  df-oprab 6205  df-mpt2 6206  df-map 7327  df-mhm 15584
This theorem is referenced by:  mhmf1o  15593  resmhm  15607  resmhm2  15608  resmhm2b  15609  mhmco  15610  mhmima  15611  mhmeql  15612  pwsco2mhm  15619  gsumwmhm  15643  mhmmulg  15779  gsumzmhm  16553  gsumzmhmOLD  16554  rhm1  16944  madetsumid  18474  mdetunilem7  18557  dchrzrh1  22717  dchrmulcl  22722  dchrn0  22723  dchrinvcl  22726  dchrfi  22728  dchrabs  22733  sumdchr2  22743  rpvmasum2  22895  pm2mp  31312
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