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Theorem ghmmhmb 15055
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 15054 . . 3  |-  ( f  e.  ( S  GrpHom  T )  ->  f  e.  ( S MndHom  T ) )
2 eqid 2443 . . . . 5  |-  ( Base `  S )  =  (
Base `  S )
3 eqid 2443 . . . . 5  |-  ( Base `  T )  =  (
Base `  T )
4 eqid 2443 . . . . 5  |-  ( +g  `  S )  =  ( +g  `  S )
5 eqid 2443 . . . . 5  |-  ( +g  `  T )  =  ( +g  `  T )
6 simpll 732 . . . . 5  |-  ( ( ( S  e.  Grp  /\  T  e.  Grp )  /\  f  e.  ( S MndHom  T ) )  ->  S  e.  Grp )
7 simplr 733 . . . . 5  |-  ( ( ( S  e.  Grp  /\  T  e.  Grp )  /\  f  e.  ( S MndHom  T ) )  ->  T  e.  Grp )
82, 3mhmf 14781 . . . . . 6  |-  ( f  e.  ( S MndHom  T
)  ->  f :
( Base `  S ) --> ( Base `  T )
)
98adantl 454 . . . . 5  |-  ( ( ( S  e.  Grp  /\  T  e.  Grp )  /\  f  e.  ( S MndHom  T ) )  -> 
f : ( Base `  S ) --> ( Base `  T ) )
102, 4, 5mhmlin 14783 . . . . . . 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 1155 . . . . . 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 696 . . . . 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 15053 . . . 4  |-  ( ( ( S  e.  Grp  /\  T  e.  Grp )  /\  f  e.  ( S MndHom  T ) )  -> 
f  e.  ( S 
GrpHom  T ) )
1413ex 425 . . 3  |-  ( ( S  e.  Grp  /\  T  e.  Grp )  ->  ( f  e.  ( S MndHom  T )  -> 
f  e.  ( S 
GrpHom  T ) ) )
151, 14impbid2 197 . 2  |-  ( ( S  e.  Grp  /\  T  e.  Grp )  ->  ( f  e.  ( S  GrpHom  T )  <->  f  e.  ( S MndHom  T ) ) )
1615eqrdv 2441 1  |-  ( ( S  e.  Grp  /\  T  e.  Grp )  ->  ( S  GrpHom  T )  =  ( S MndHom  T
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 360    = wceq 1654    e. wcel 1728   -->wf 5485   ` cfv 5489  (class class class)co 6117   Basecbs 13507   +g cplusg 13567   Grpcgrp 14723   MndHom cmhm 14774    GrpHom cghm 15041
This theorem is referenced by:  0ghm  15058  resghm2  15061  resghm2b  15062  ghmco  15063  pwsdiagghm  15071  ghmpropd  15081  pwsco1rhm  15871  pwsco2rhm  15872  dchrghm  21078
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1628  ax-9 1669  ax-8 1690  ax-13 1730  ax-14 1732  ax-6 1747  ax-7 1752  ax-11 1764  ax-12 1954  ax-ext 2424  ax-rep 4351  ax-sep 4361  ax-nul 4369  ax-pow 4412  ax-pr 4438  ax-un 4736
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1661  df-eu 2292  df-mo 2293  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2568  df-ne 2608  df-ral 2717  df-rex 2718  df-reu 2719  df-rmo 2720  df-rab 2721  df-v 2967  df-sbc 3171  df-csb 3271  df-dif 3312  df-un 3314  df-in 3316  df-ss 3323  df-nul 3617  df-if 3768  df-pw 3830  df-sn 3849  df-pr 3850  df-op 3852  df-uni 4045  df-iun 4124  df-br 4244  df-opab 4298  df-mpt 4299  df-id 4533  df-xp 4919  df-rel 4920  df-cnv 4921  df-co 4922  df-dm 4923  df-rn 4924  df-res 4925  df-ima 4926  df-iota 5453  df-fun 5491  df-fn 5492  df-f 5493  df-f1 5494  df-fo 5495  df-f1o 5496  df-fv 5497  df-ov 6120  df-oprab 6121  df-mpt2 6122  df-riota 6585  df-map 7056  df-0g 13765  df-mnd 14728  df-mhm 14776  df-grp 14850  df-ghm 15042
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