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Theorem ghmmhm 14971
Description: A group homorphism is a monoid homorphism. (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 14963 . . . 4  |-  ( F  e.  ( S  GrpHom  T )  ->  S  e.  Grp )
2 grpmnd 14772 . . . 4  |-  ( S  e.  Grp  ->  S  e.  Mnd )
31, 2syl 16 . . 3  |-  ( F  e.  ( S  GrpHom  T )  ->  S  e.  Mnd )
4 ghmgrp2 14964 . . . 4  |-  ( F  e.  ( S  GrpHom  T )  ->  T  e.  Grp )
5 grpmnd 14772 . . . 4  |-  ( T  e.  Grp  ->  T  e.  Mnd )
64, 5syl 16 . . 3  |-  ( F  e.  ( S  GrpHom  T )  ->  T  e.  Mnd )
73, 6jca 519 . 2  |-  ( F  e.  ( S  GrpHom  T )  ->  ( S  e.  Mnd  /\  T  e. 
Mnd ) )
8 eqid 2404 . . . 4  |-  ( Base `  S )  =  (
Base `  S )
9 eqid 2404 . . . 4  |-  ( Base `  T )  =  (
Base `  T )
108, 9ghmf 14965 . . 3  |-  ( F  e.  ( S  GrpHom  T )  ->  F :
( Base `  S ) --> ( Base `  T )
)
11 eqid 2404 . . . . . 6  |-  ( +g  `  S )  =  ( +g  `  S )
12 eqid 2404 . . . . . 6  |-  ( +g  `  T )  =  ( +g  `  T )
138, 11, 12ghmlin 14966 . . . . 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 1154 . . . 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 2758 . . 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 2404 . . . 4  |-  ( 0g
`  S )  =  ( 0g `  S
)
17 eqid 2404 . . . 4  |-  ( 0g
`  T )  =  ( 0g `  T
)
1816, 17ghmid 14967 . . 3  |-  ( F  e.  ( S  GrpHom  T )  ->  ( F `  ( 0g `  S
) )  =  ( 0g `  T ) )
1910, 15, 183jca 1134 . 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 14695 . 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 646 1  |-  ( F  e.  ( S  GrpHom  T )  ->  F  e.  ( S MndHom  T ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    /\ w3a 936    = wceq 1649    e. wcel 1721   A.wral 2666   -->wf 5409   ` cfv 5413  (class class class)co 6040   Basecbs 13424   +g cplusg 13484   0gc0g 13678   Mndcmnd 14639   Grpcgrp 14640   MndHom cmhm 14691    GrpHom cghm 14958
This theorem is referenced by:  ghmmhmb  14972  ghmmulg  14973  resghm2  14978  ghmco  14980  ghmeql  14983  frgpup3lem  15364  gsummulglem  15491  gsumzinv  15495  gsuminv  15496  gsummulc1  15668  gsummulc2  15669  pwsco2rhm  15789  evlslem2  16523  tsmsinv  18130  plypf1  20084  amgmlem  20781  lgseisenlem4  21089  gsumvsmul  26635  symgtrinv  27281  mendrng  27368
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 2385  ax-rep 4280  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660
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 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-ral 2671  df-rex 2672  df-reu 2673  df-rmo 2674  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-op 3783  df-uni 3976  df-iun 4055  df-br 4173  df-opab 4227  df-mpt 4228  df-id 4458  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-ov 6043  df-oprab 6044  df-mpt2 6045  df-riota 6508  df-map 6979  df-0g 13682  df-mnd 14645  df-mhm 14693  df-grp 14767  df-ghm 14959
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