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Theorem ghmmhmb 14694
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 14693 . . 3  |-  ( f  e.  ( S  GrpHom  T )  ->  f  e.  ( S MndHom  T ) )
2 eqid 2283 . . . . 5  |-  ( Base `  S )  =  (
Base `  S )
3 eqid 2283 . . . . 5  |-  ( Base `  T )  =  (
Base `  T )
4 eqid 2283 . . . . 5  |-  ( +g  `  S )  =  ( +g  `  S )
5 eqid 2283 . . . . 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 14420 . . . . . 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 14422 . . . . . . 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 14692 . . . 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 2281 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 1623    e. wcel 1684   -->wf 5251   ` cfv 5255  (class class class)co 5858   Basecbs 13148   +g cplusg 13208   Grpcgrp 14362   MndHom cmhm 14413    GrpHom cghm 14680
This theorem is referenced by:  0ghm  14697  resghm2  14700  resghm2b  14701  ghmco  14702  pwsdiagghm  14710  ghmpropd  14720  pwsco1rhm  15510  pwsco2rhm  15511  dchrghm  20495
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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