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Theorem ghmmhm 14693
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 14685 . . . 4  |-  ( F  e.  ( S  GrpHom  T )  ->  S  e.  Grp )
2 grpmnd 14494 . . . 4  |-  ( S  e.  Grp  ->  S  e.  Mnd )
31, 2syl 15 . . 3  |-  ( F  e.  ( S  GrpHom  T )  ->  S  e.  Mnd )
4 ghmgrp2 14686 . . . 4  |-  ( F  e.  ( S  GrpHom  T )  ->  T  e.  Grp )
5 grpmnd 14494 . . . 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 2283 . . . 4  |-  ( Base `  S )  =  (
Base `  S )
9 eqid 2283 . . . 4  |-  ( Base `  T )  =  (
Base `  T )
108, 9ghmf 14687 . . 3  |-  ( F  e.  ( S  GrpHom  T )  ->  F :
( Base `  S ) --> ( Base `  T )
)
11 eqid 2283 . . . . . 6  |-  ( +g  `  S )  =  ( +g  `  S )
12 eqid 2283 . . . . . 6  |-  ( +g  `  T )  =  ( +g  `  T )
138, 11, 12ghmlin 14688 . . . . 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 2635 . . 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 2283 . . . 4  |-  ( 0g
`  S )  =  ( 0g `  S
)
17 eqid 2283 . . . 4  |-  ( 0g
`  T )  =  ( 0g `  T
)
1816, 17ghmid 14689 . . 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 14417 . 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 1623    e. wcel 1684   A.wral 2543   -->wf 5251   ` cfv 5255  (class class class)co 5858   Basecbs 13148   +g cplusg 13208   0gc0g 13400   Mndcmnd 14361   Grpcgrp 14362   MndHom cmhm 14413    GrpHom cghm 14680
This theorem is referenced by:  ghmmhmb  14694  ghmmulg  14695  resghm2  14700  ghmco  14702  ghmeql  14705  frgpup3lem  15086  gsummulglem  15213  gsumzinv  15217  gsuminv  15218  gsummulc1  15390  gsummulc2  15391  pwsco2rhm  15511  evlslem2  16249  tsmsinv  17830  plypf1  19594  amgmlem  20284  lgseisenlem4  20591  gsumvsmul  26764  symgtrinv  27413  mendrng  27500
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-riota 6304  df-map 6774  df-0g 13404  df-mnd 14367  df-mhm 14415  df-grp 14489  df-ghm 14681
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