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Theorem ghmmhm 15868
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 15860 . . . 4  |-  ( F  e.  ( S  GrpHom  T )  ->  S  e.  Grp )
2 grpmnd 15661 . . . 4  |-  ( S  e.  Grp  ->  S  e.  Mnd )
31, 2syl 16 . . 3  |-  ( F  e.  ( S  GrpHom  T )  ->  S  e.  Mnd )
4 ghmgrp2 15861 . . . 4  |-  ( F  e.  ( S  GrpHom  T )  ->  T  e.  Grp )
5 grpmnd 15661 . . . 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 2451 . . . 4  |-  ( Base `  S )  =  (
Base `  S )
9 eqid 2451 . . . 4  |-  ( Base `  T )  =  (
Base `  T )
108, 9ghmf 15862 . . 3  |-  ( F  e.  ( S  GrpHom  T )  ->  F :
( Base `  S ) --> ( Base `  T )
)
11 eqid 2451 . . . . . 6  |-  ( +g  `  S )  =  ( +g  `  S )
12 eqid 2451 . . . . . 6  |-  ( +g  `  T )  =  ( +g  `  T )
138, 11, 12ghmlin 15863 . . . . 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 1189 . . . 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 2907 . . 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 2451 . . . 4  |-  ( 0g
`  S )  =  ( 0g `  S
)
17 eqid 2451 . . . 4  |-  ( 0g
`  T )  =  ( 0g `  T
)
1816, 17ghmid 15864 . . 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 15577 . 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 1370    e. wcel 1758   A.wral 2795   -->wf 5515   ` cfv 5519  (class class class)co 6193   Basecbs 14285   +g cplusg 14349   0gc0g 14489   Mndcmnd 15520   Grpcgrp 15521   MndHom cmhm 15573    GrpHom cghm 15855
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 1952  ax-ext 2430  ax-rep 4504  ax-sep 4514  ax-nul 4522  ax-pow 4571  ax-pr 4632  ax-un 6475
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 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ne 2646  df-ral 2800  df-rex 2801  df-reu 2802  df-rmo 2803  df-rab 2804  df-v 3073  df-sbc 3288  df-csb 3390  df-dif 3432  df-un 3434  df-in 3436  df-ss 3443  df-nul 3739  df-if 3893  df-pw 3963  df-sn 3979  df-pr 3981  df-op 3985  df-uni 4193  df-iun 4274  df-br 4394  df-opab 4452  df-mpt 4453  df-id 4737  df-xp 4947  df-rel 4948  df-cnv 4949  df-co 4950  df-dm 4951  df-rn 4952  df-res 4953  df-ima 4954  df-iota 5482  df-fun 5521  df-fn 5522  df-f 5523  df-f1 5524  df-fo 5525  df-f1o 5526  df-fv 5527  df-riota 6154  df-ov 6196  df-oprab 6197  df-mpt2 6198  df-map 7319  df-0g 14491  df-mnd 15526  df-mhm 15575  df-grp 15656  df-ghm 15856
This theorem is referenced by:  ghmmhmb  15869  ghmmulg  15870  resghm2  15875  ghmco  15877  ghmeql  15880  symgtrinv  16089  frgpup3lem  16387  gsummulglem  16551  gsumzinv  16556  gsumzinvOLD  16557  gsuminv  16558  gsuminvOLD  16560  gsummulc1  16810  gsummulc2  16811  gsummulc1OLD  16812  gsummulc2OLD  16813  pwsco2rhm  16942  gsumvsmul  17124  gsumvsmulOLD  17125  evlslem2  17713  evls1gsumadd  17877  zrhpsgnmhm  18132  tsmsinv  19847  plypf1  21806  amgmlem  22509  lgseisenlem4  22817  mendrng  29690  mat2pmatmul  31197  pmat2matp  31282  cayhamlem4  31346
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