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Theorem ghmgrp1 15770
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 2443 . . . 4  |-  ( Base `  S )  =  (
Base `  S )
2 eqid 2443 . . . 4  |-  ( Base `  T )  =  (
Base `  T )
3 eqid 2443 . . . 4  |-  ( +g  `  S )  =  ( +g  `  S )
4 eqid 2443 . . . 4  |-  ( +g  `  T )  =  ( +g  `  T )
51, 2, 3, 4isghm 15768 . . 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 1369    e. wcel 1756   A.wral 2736   -->wf 5435   ` cfv 5439  (class class class)co 6112   Basecbs 14195   +g cplusg 14259   Grpcgrp 15431    GrpHom cghm 15765
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 4424  ax-sep 4434  ax-nul 4442  ax-pow 4491  ax-pr 4552  ax-un 6393
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 2577  df-ne 2622  df-ral 2741  df-rex 2742  df-reu 2743  df-rab 2745  df-v 2995  df-sbc 3208  df-csb 3310  df-dif 3352  df-un 3354  df-in 3356  df-ss 3363  df-nul 3659  df-if 3813  df-pw 3883  df-sn 3899  df-pr 3901  df-op 3905  df-uni 4113  df-iun 4194  df-br 4314  df-opab 4372  df-mpt 4373  df-id 4657  df-xp 4867  df-rel 4868  df-cnv 4869  df-co 4870  df-dm 4871  df-rn 4872  df-res 4873  df-ima 4874  df-iota 5402  df-fun 5441  df-fn 5442  df-f 5443  df-f1 5444  df-fo 5445  df-f1o 5446  df-fv 5447  df-ov 6115  df-oprab 6116  df-mpt2 6117  df-ghm 15766
This theorem is referenced by:  ghmid  15774  ghminv  15775  ghmsub  15776  ghmmhm  15778  ghmmulg  15780  ghmrn  15781  resghm2  15785  resghm2b  15786  ghmco  15787  ghmpreima  15789  ghmeql  15790  ghmnsgima  15791  ghmnsgpreima  15792  ghmeqker  15794  ghmf1  15796  ghmf1o  15797  ghmpropd  15805  isgim  15811  giclcl  15821  lactghmga  15930  invghm  16339  ghmplusg  16349  evl1addd  17797  evl1subd  17798  ghmcnp  19707  gicabl  29480
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