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Theorem ghmgrp1 16083
Description: A group homomorphism is only defined when the domain is a group. (Contributed by Stefan O'Rear, 31-Dec-2014.)
Assertion
Ref Expression
ghmgrp1  |-  ( F  e.  ( S  GrpHom  T )  ->  S  e.  Grp )

Proof of Theorem ghmgrp1
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqid 2467 . . . 4  |-  ( Base `  S )  =  (
Base `  S )
2 eqid 2467 . . . 4  |-  ( Base `  T )  =  (
Base `  T )
3 eqid 2467 . . . 4  |-  ( +g  `  S )  =  ( +g  `  S )
4 eqid 2467 . . . 4  |-  ( +g  `  T )  =  ( +g  `  T )
51, 2, 3, 4isghm 16081 . . 3  |-  ( F  e.  ( S  GrpHom  T )  <->  ( ( S  e.  Grp  /\  T  e.  Grp )  /\  ( F : ( Base `  S
) --> ( Base `  T
)  /\  A. y  e.  ( Base `  S
) A. x  e.  ( Base `  S
) ( F `  ( y ( +g  `  S ) x ) )  =  ( ( F `  y ) ( +g  `  T
) ( F `  x ) ) ) ) )
65simplbi 460 . 2  |-  ( F  e.  ( S  GrpHom  T )  ->  ( S  e.  Grp  /\  T  e. 
Grp ) )
76simpld 459 1  |-  ( F  e.  ( S  GrpHom  T )  ->  S  e.  Grp )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1379    e. wcel 1767   A.wral 2814   -->wf 5584   ` cfv 5588  (class class class)co 6285   Basecbs 14493   +g cplusg 14558   Grpcgrp 15730    GrpHom cghm 16078
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6577
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-reu 2821  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-id 4795  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5551  df-fun 5590  df-fn 5591  df-f 5592  df-f1 5593  df-fo 5594  df-f1o 5595  df-fv 5596  df-ov 6288  df-oprab 6289  df-mpt2 6290  df-ghm 16079
This theorem is referenced by:  ghmid  16087  ghminv  16088  ghmsub  16089  ghmmhm  16091  ghmmulg  16093  ghmrn  16094  resghm2  16098  resghm2b  16099  ghmco  16100  ghmpreima  16102  ghmeql  16103  ghmnsgima  16104  ghmnsgpreima  16105  ghmeqker  16107  ghmf1  16109  ghmf1o  16110  ghmpropd  16118  isgim  16124  giclcl  16134  lactghmga  16243  invghm  16657  ghmplusg  16667  evl1addd  18188  evl1subd  18189  ghmcnp  20440  gicabl  30878
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