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Theorem ghmmhm 15748
Description: A group homomorphism is a monoid homomorphism. (Contributed by Stefan O'Rear, 7-Mar-2015.)
Assertion
Ref Expression
ghmmhm  |-  ( F  e.  ( S  GrpHom  T )  ->  F  e.  ( S MndHom  T ) )

Proof of Theorem ghmmhm
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ghmgrp1 15740 . . . 4  |-  ( F  e.  ( S  GrpHom  T )  ->  S  e.  Grp )
2 grpmnd 15541 . . . 4  |-  ( S  e.  Grp  ->  S  e.  Mnd )
31, 2syl 16 . . 3  |-  ( F  e.  ( S  GrpHom  T )  ->  S  e.  Mnd )
4 ghmgrp2 15741 . . . 4  |-  ( F  e.  ( S  GrpHom  T )  ->  T  e.  Grp )
5 grpmnd 15541 . . . 4  |-  ( T  e.  Grp  ->  T  e.  Mnd )
64, 5syl 16 . . 3  |-  ( F  e.  ( S  GrpHom  T )  ->  T  e.  Mnd )
73, 6jca 532 . 2  |-  ( F  e.  ( S  GrpHom  T )  ->  ( S  e.  Mnd  /\  T  e. 
Mnd ) )
8 eqid 2438 . . . 4  |-  ( Base `  S )  =  (
Base `  S )
9 eqid 2438 . . . 4  |-  ( Base `  T )  =  (
Base `  T )
108, 9ghmf 15742 . . 3  |-  ( F  e.  ( S  GrpHom  T )  ->  F :
( Base `  S ) --> ( Base `  T )
)
11 eqid 2438 . . . . . 6  |-  ( +g  `  S )  =  ( +g  `  S )
12 eqid 2438 . . . . . 6  |-  ( +g  `  T )  =  ( +g  `  T )
138, 11, 12ghmlin 15743 . . . . 5  |-  ( ( F  e.  ( S 
GrpHom  T )  /\  x  e.  ( Base `  S
)  /\  y  e.  ( Base `  S )
)  ->  ( F `  ( x ( +g  `  S ) y ) )  =  ( ( F `  x ) ( +g  `  T
) ( F `  y ) ) )
14133expb 1188 . . . 4  |-  ( ( F  e.  ( S 
GrpHom  T )  /\  (
x  e.  ( Base `  S )  /\  y  e.  ( Base `  S
) ) )  -> 
( F `  (
x ( +g  `  S
) y ) )  =  ( ( F `
 x ) ( +g  `  T ) ( F `  y
) ) )
1514ralrimivva 2803 . . 3  |-  ( F  e.  ( S  GrpHom  T )  ->  A. x  e.  ( Base `  S
) A. y  e.  ( Base `  S
) ( F `  ( x ( +g  `  S ) y ) )  =  ( ( F `  x ) ( +g  `  T
) ( F `  y ) ) )
16 eqid 2438 . . . 4  |-  ( 0g
`  S )  =  ( 0g `  S
)
17 eqid 2438 . . . 4  |-  ( 0g
`  T )  =  ( 0g `  T
)
1816, 17ghmid 15744 . . 3  |-  ( F  e.  ( S  GrpHom  T )  ->  ( F `  ( 0g `  S
) )  =  ( 0g `  T ) )
1910, 15, 183jca 1168 . 2  |-  ( F  e.  ( S  GrpHom  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 `  ( 0g
`  S ) )  =  ( 0g `  T ) ) )
208, 9, 11, 12, 16, 17ismhm 15458 . 2  |-  ( 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 `  ( 0g
`  S ) )  =  ( 0g `  T ) ) ) )
217, 19, 20sylanbrc 664 1  |-  ( F  e.  ( S  GrpHom  T )  ->  F  e.  ( S MndHom  T ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    /\ w3a 965    = wceq 1369    e. wcel 1756   A.wral 2710   -->wf 5409   ` cfv 5413  (class class class)co 6086   Basecbs 14166   +g cplusg 14230   0gc0g 14370   Mndcmnd 15401   Grpcgrp 15402   MndHom cmhm 15454    GrpHom cghm 15735
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2419  ax-rep 4398  ax-sep 4408  ax-nul 4416  ax-pow 4465  ax-pr 4526  ax-un 6367
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2256  df-mo 2257  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2715  df-rex 2716  df-reu 2717  df-rmo 2718  df-rab 2719  df-v 2969  df-sbc 3182  df-csb 3284  df-dif 3326  df-un 3328  df-in 3330  df-ss 3337  df-nul 3633  df-if 3787  df-pw 3857  df-sn 3873  df-pr 3875  df-op 3879  df-uni 4087  df-iun 4168  df-br 4288  df-opab 4346  df-mpt 4347  df-id 4631  df-xp 4841  df-rel 4842  df-cnv 4843  df-co 4844  df-dm 4845  df-rn 4846  df-res 4847  df-ima 4848  df-iota 5376  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-riota 6047  df-ov 6089  df-oprab 6090  df-mpt2 6091  df-map 7208  df-0g 14372  df-mnd 15407  df-mhm 15456  df-grp 15536  df-ghm 15736
This theorem is referenced by:  ghmmhmb  15749  ghmmulg  15750  resghm2  15755  ghmco  15757  ghmeql  15760  symgtrinv  15969  frgpup3lem  16265  gsummulglem  16427  gsumzinv  16432  gsumzinvOLD  16433  gsuminv  16434  gsuminvOLD  16436  gsummulc1  16683  gsummulc2  16684  gsummulc1OLD  16685  gsummulc2OLD  16686  pwsco2rhm  16809  gsumvsmul  16987  gsumvsmulOLD  16988  evlslem2  17572  evls1gsumadd  17734  zrhpsgnmhm  17989  tsmsinv  19697  plypf1  21655  amgmlem  22358  lgseisenlem4  22666  mendrng  29502
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