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Theorem ghmmhmb 14793
Description: Group homorphisms and monoid homomorphisms coincide. (Thus,  GrpHom is somewhat redundant, although its stronger reverse closure properties are sometimes useful.) (Contributed by Stefan O'Rear, 7-Mar-2015.)
Assertion
Ref Expression
ghmmhmb  |-  ( ( S  e.  Grp  /\  T  e.  Grp )  ->  ( S  GrpHom  T )  =  ( S MndHom  T
) )

Proof of Theorem ghmmhmb
Dummy variables  x  y  f are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ghmmhm 14792 . . 3  |-  ( f  e.  ( S  GrpHom  T )  ->  f  e.  ( S MndHom  T ) )
2 eqid 2358 . . . . 5  |-  ( Base `  S )  =  (
Base `  S )
3 eqid 2358 . . . . 5  |-  ( Base `  T )  =  (
Base `  T )
4 eqid 2358 . . . . 5  |-  ( +g  `  S )  =  ( +g  `  S )
5 eqid 2358 . . . . 5  |-  ( +g  `  T )  =  ( +g  `  T )
6 simpll 730 . . . . 5  |-  ( ( ( S  e.  Grp  /\  T  e.  Grp )  /\  f  e.  ( S MndHom  T ) )  ->  S  e.  Grp )
7 simplr 731 . . . . 5  |-  ( ( ( S  e.  Grp  /\  T  e.  Grp )  /\  f  e.  ( S MndHom  T ) )  ->  T  e.  Grp )
82, 3mhmf 14519 . . . . . 6  |-  ( f  e.  ( S MndHom  T
)  ->  f :
( Base `  S ) --> ( Base `  T )
)
98adantl 452 . . . . 5  |-  ( ( ( S  e.  Grp  /\  T  e.  Grp )  /\  f  e.  ( S MndHom  T ) )  -> 
f : ( Base `  S ) --> ( Base `  T ) )
102, 4, 5mhmlin 14521 . . . . . . 7  |-  ( ( f  e.  ( S MndHom  T )  /\  x  e.  ( Base `  S
)  /\  y  e.  ( Base `  S )
)  ->  ( f `  ( x ( +g  `  S ) y ) )  =  ( ( f `  x ) ( +g  `  T
) ( f `  y ) ) )
11103expb 1152 . . . . . 6  |-  ( ( f  e.  ( S MndHom  T )  /\  (
x  e.  ( Base `  S )  /\  y  e.  ( Base `  S
) ) )  -> 
( f `  (
x ( +g  `  S
) y ) )  =  ( ( f `
 x ) ( +g  `  T ) ( f `  y
) ) )
1211adantll 694 . . . . 5  |-  ( ( ( ( S  e. 
Grp  /\  T  e.  Grp )  /\  f  e.  ( S MndHom  T ) )  /\  ( x  e.  ( Base `  S
)  /\  y  e.  ( Base `  S )
) )  ->  (
f `  ( x
( +g  `  S ) y ) )  =  ( ( f `  x ) ( +g  `  T ) ( f `
 y ) ) )
132, 3, 4, 5, 6, 7, 9, 12isghmd 14791 . . . 4  |-  ( ( ( S  e.  Grp  /\  T  e.  Grp )  /\  f  e.  ( S MndHom  T ) )  -> 
f  e.  ( S 
GrpHom  T ) )
1413ex 423 . . 3  |-  ( ( S  e.  Grp  /\  T  e.  Grp )  ->  ( f  e.  ( S MndHom  T )  -> 
f  e.  ( S 
GrpHom  T ) ) )
151, 14impbid2 195 . 2  |-  ( ( S  e.  Grp  /\  T  e.  Grp )  ->  ( f  e.  ( S  GrpHom  T )  <->  f  e.  ( S MndHom  T ) ) )
1615eqrdv 2356 1  |-  ( ( S  e.  Grp  /\  T  e.  Grp )  ->  ( S  GrpHom  T )  =  ( S MndHom  T
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1642    e. wcel 1710   -->wf 5333   ` cfv 5337  (class class class)co 5945   Basecbs 13245   +g cplusg 13305   Grpcgrp 14461   MndHom cmhm 14512    GrpHom cghm 14779
This theorem is referenced by:  0ghm  14796  resghm2  14799  resghm2b  14800  ghmco  14801  pwsdiagghm  14809  ghmpropd  14819  pwsco1rhm  15609  pwsco2rhm  15610  dchrghm  20607
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-13 1712  ax-14 1714  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1930  ax-ext 2339  ax-rep 4212  ax-sep 4222  ax-nul 4230  ax-pow 4269  ax-pr 4295  ax-un 4594
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2213  df-mo 2214  df-clab 2345  df-cleq 2351  df-clel 2354  df-nfc 2483  df-ne 2523  df-ral 2624  df-rex 2625  df-reu 2626  df-rmo 2627  df-rab 2628  df-v 2866  df-sbc 3068  df-csb 3158  df-dif 3231  df-un 3233  df-in 3235  df-ss 3242  df-nul 3532  df-if 3642  df-pw 3703  df-sn 3722  df-pr 3723  df-op 3725  df-uni 3909  df-iun 3988  df-br 4105  df-opab 4159  df-mpt 4160  df-id 4391  df-xp 4777  df-rel 4778  df-cnv 4779  df-co 4780  df-dm 4781  df-rn 4782  df-res 4783  df-ima 4784  df-iota 5301  df-fun 5339  df-fn 5340  df-f 5341  df-f1 5342  df-fo 5343  df-f1o 5344  df-fv 5345  df-ov 5948  df-oprab 5949  df-mpt2 5950  df-riota 6391  df-map 6862  df-0g 13503  df-mnd 14466  df-mhm 14514  df-grp 14588  df-ghm 14780
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