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Theorem ghmmhmb 14980
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 14979 . . 3  |-  ( f  e.  ( S  GrpHom  T )  ->  f  e.  ( S MndHom  T ) )
2 eqid 2412 . . . . 5  |-  ( Base `  S )  =  (
Base `  S )
3 eqid 2412 . . . . 5  |-  ( Base `  T )  =  (
Base `  T )
4 eqid 2412 . . . . 5  |-  ( +g  `  S )  =  ( +g  `  S )
5 eqid 2412 . . . . 5  |-  ( +g  `  T )  =  ( +g  `  T )
6 simpll 731 . . . . 5  |-  ( ( ( S  e.  Grp  /\  T  e.  Grp )  /\  f  e.  ( S MndHom  T ) )  ->  S  e.  Grp )
7 simplr 732 . . . . 5  |-  ( ( ( S  e.  Grp  /\  T  e.  Grp )  /\  f  e.  ( S MndHom  T ) )  ->  T  e.  Grp )
82, 3mhmf 14706 . . . . . 6  |-  ( f  e.  ( S MndHom  T
)  ->  f :
( Base `  S ) --> ( Base `  T )
)
98adantl 453 . . . . 5  |-  ( ( ( S  e.  Grp  /\  T  e.  Grp )  /\  f  e.  ( S MndHom  T ) )  -> 
f : ( Base `  S ) --> ( Base `  T ) )
102, 4, 5mhmlin 14708 . . . . . . 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 1154 . . . . . 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 695 . . . . 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 14978 . . . 4  |-  ( ( ( S  e.  Grp  /\  T  e.  Grp )  /\  f  e.  ( S MndHom  T ) )  -> 
f  e.  ( S 
GrpHom  T ) )
1413ex 424 . . 3  |-  ( ( S  e.  Grp  /\  T  e.  Grp )  ->  ( f  e.  ( S MndHom  T )  -> 
f  e.  ( S 
GrpHom  T ) ) )
151, 14impbid2 196 . 2  |-  ( ( S  e.  Grp  /\  T  e.  Grp )  ->  ( f  e.  ( S  GrpHom  T )  <->  f  e.  ( S MndHom  T ) ) )
1615eqrdv 2410 1  |-  ( ( S  e.  Grp  /\  T  e.  Grp )  ->  ( S  GrpHom  T )  =  ( S MndHom  T
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1649    e. wcel 1721   -->wf 5417   ` cfv 5421  (class class class)co 6048   Basecbs 13432   +g cplusg 13492   Grpcgrp 14648   MndHom cmhm 14699    GrpHom cghm 14966
This theorem is referenced by:  0ghm  14983  resghm2  14986  resghm2b  14987  ghmco  14988  pwsdiagghm  14996  ghmpropd  15006  pwsco1rhm  15796  pwsco2rhm  15797  dchrghm  21001
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2393  ax-rep 4288  ax-sep 4298  ax-nul 4306  ax-pow 4345  ax-pr 4371  ax-un 4668
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2266  df-mo 2267  df-clab 2399  df-cleq 2405  df-clel 2408  df-nfc 2537  df-ne 2577  df-ral 2679  df-rex 2680  df-reu 2681  df-rmo 2682  df-rab 2683  df-v 2926  df-sbc 3130  df-csb 3220  df-dif 3291  df-un 3293  df-in 3295  df-ss 3302  df-nul 3597  df-if 3708  df-pw 3769  df-sn 3788  df-pr 3789  df-op 3791  df-uni 3984  df-iun 4063  df-br 4181  df-opab 4235  df-mpt 4236  df-id 4466  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-iota 5385  df-fun 5423  df-fn 5424  df-f 5425  df-f1 5426  df-fo 5427  df-f1o 5428  df-fv 5429  df-ov 6051  df-oprab 6052  df-mpt2 6053  df-riota 6516  df-map 6987  df-0g 13690  df-mnd 14653  df-mhm 14701  df-grp 14775  df-ghm 14967
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