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Theorem ghmmhmb 15763
Description: Group homomorphisms 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 15762 . . 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 753 . . . . 5  |-  ( ( ( S  e.  Grp  /\  T  e.  Grp )  /\  f  e.  ( S MndHom  T ) )  ->  S  e.  Grp )
7 simplr 754 . . . . 5  |-  ( ( ( S  e.  Grp  /\  T  e.  Grp )  /\  f  e.  ( S MndHom  T ) )  ->  T  e.  Grp )
82, 3mhmf 15474 . . . . . 6  |-  ( f  e.  ( S MndHom  T
)  ->  f :
( Base `  S ) --> ( Base `  T )
)
98adantl 466 . . . . 5  |-  ( ( ( S  e.  Grp  /\  T  e.  Grp )  /\  f  e.  ( S MndHom  T ) )  -> 
f : ( Base `  S ) --> ( Base `  T ) )
102, 4, 5mhmlin 15476 . . . . . . 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 1188 . . . . . 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 713 . . . . 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 15761 . . . 4  |-  ( ( ( S  e.  Grp  /\  T  e.  Grp )  /\  f  e.  ( S MndHom  T ) )  -> 
f  e.  ( S 
GrpHom  T ) )
1413ex 434 . . 3  |-  ( ( S  e.  Grp  /\  T  e.  Grp )  ->  ( f  e.  ( S MndHom  T )  -> 
f  e.  ( S 
GrpHom  T ) ) )
151, 14impbid2 204 . 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 setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1369    e. wcel 1756   -->wf 5419   ` cfv 5423  (class class class)co 6096   Basecbs 14179   +g cplusg 14243   Grpcgrp 15415   MndHom cmhm 15467    GrpHom cghm 15749
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4408  ax-sep 4418  ax-nul 4426  ax-pow 4475  ax-pr 4536  ax-un 6377
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2573  df-ne 2613  df-ral 2725  df-rex 2726  df-reu 2727  df-rmo 2728  df-rab 2729  df-v 2979  df-sbc 3192  df-csb 3294  df-dif 3336  df-un 3338  df-in 3340  df-ss 3347  df-nul 3643  df-if 3797  df-pw 3867  df-sn 3883  df-pr 3885  df-op 3889  df-uni 4097  df-iun 4178  df-br 4298  df-opab 4356  df-mpt 4357  df-id 4641  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 5386  df-fun 5425  df-fn 5426  df-f 5427  df-f1 5428  df-fo 5429  df-f1o 5430  df-fv 5431  df-riota 6057  df-ov 6099  df-oprab 6100  df-mpt2 6101  df-map 7221  df-0g 14385  df-mnd 15420  df-mhm 15469  df-grp 15550  df-ghm 15750
This theorem is referenced by:  0ghm  15766  resghm2  15769  resghm2b  15770  ghmco  15771  pwsdiagghm  15779  ghmpropd  15789  pwsco1rhm  16831  pwsco2rhm  16832  dchrghm  22600
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