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Theorem ghmmhm 14709
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 14701 . . . 4  |-  ( F  e.  ( S  GrpHom  T )  ->  S  e.  Grp )
2 grpmnd 14510 . . . 4  |-  ( S  e.  Grp  ->  S  e.  Mnd )
31, 2syl 15 . . 3  |-  ( F  e.  ( S  GrpHom  T )  ->  S  e.  Mnd )
4 ghmgrp2 14702 . . . 4  |-  ( F  e.  ( S  GrpHom  T )  ->  T  e.  Grp )
5 grpmnd 14510 . . . 4  |-  ( T  e.  Grp  ->  T  e.  Mnd )
64, 5syl 15 . . 3  |-  ( F  e.  ( S  GrpHom  T )  ->  T  e.  Mnd )
73, 6jca 518 . 2  |-  ( F  e.  ( S  GrpHom  T )  ->  ( S  e.  Mnd  /\  T  e. 
Mnd ) )
8 eqid 2296 . . . 4  |-  ( Base `  S )  =  (
Base `  S )
9 eqid 2296 . . . 4  |-  ( Base `  T )  =  (
Base `  T )
108, 9ghmf 14703 . . 3  |-  ( F  e.  ( S  GrpHom  T )  ->  F :
( Base `  S ) --> ( Base `  T )
)
11 eqid 2296 . . . . . 6  |-  ( +g  `  S )  =  ( +g  `  S )
12 eqid 2296 . . . . . 6  |-  ( +g  `  T )  =  ( +g  `  T )
138, 11, 12ghmlin 14704 . . . . 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 1152 . . . 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 2648 . . 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 2296 . . . 4  |-  ( 0g
`  S )  =  ( 0g `  S
)
17 eqid 2296 . . . 4  |-  ( 0g
`  T )  =  ( 0g `  T
)
1816, 17ghmid 14705 . . 3  |-  ( F  e.  ( S  GrpHom  T )  ->  ( F `  ( 0g `  S
) )  =  ( 0g `  T ) )
1910, 15, 183jca 1132 . 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 14433 . 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 645 1  |-  ( F  e.  ( S  GrpHom  T )  ->  F  e.  ( S MndHom  T ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    /\ w3a 934    = wceq 1632    e. wcel 1696   A.wral 2556   -->wf 5267   ` cfv 5271  (class class class)co 5874   Basecbs 13164   +g cplusg 13224   0gc0g 13416   Mndcmnd 14377   Grpcgrp 14378   MndHom cmhm 14429    GrpHom cghm 14696
This theorem is referenced by:  ghmmhmb  14710  ghmmulg  14711  resghm2  14716  ghmco  14718  ghmeql  14721  frgpup3lem  15102  gsummulglem  15229  gsumzinv  15233  gsuminv  15234  gsummulc1  15406  gsummulc2  15407  pwsco2rhm  15527  evlslem2  16265  tsmsinv  17846  plypf1  19610  amgmlem  20300  lgseisenlem4  20607  gsumvsmul  26867  symgtrinv  27516  mendrng  27603
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-reu 2563  df-rmo 2564  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-riota 6320  df-map 6790  df-0g 13420  df-mnd 14383  df-mhm 14431  df-grp 14505  df-ghm 14697
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