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Theorem ghmgrp1 16143
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 16141 . . 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 1383    e. wcel 1804   A.wral 2793   -->wf 5574   ` cfv 5578  (class class class)co 6281   Basecbs 14509   +g cplusg 14574   Grpcgrp 15927    GrpHom cghm 16138
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1605  ax-4 1618  ax-5 1691  ax-6 1734  ax-7 1776  ax-8 1806  ax-9 1808  ax-10 1823  ax-11 1828  ax-12 1840  ax-13 1985  ax-ext 2421  ax-rep 4548  ax-sep 4558  ax-nul 4566  ax-pow 4615  ax-pr 4676  ax-un 6577
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 976  df-tru 1386  df-ex 1600  df-nf 1604  df-sb 1727  df-eu 2272  df-mo 2273  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2593  df-ne 2640  df-ral 2798  df-rex 2799  df-reu 2800  df-rab 2802  df-v 3097  df-sbc 3314  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3771  df-if 3927  df-pw 3999  df-sn 4015  df-pr 4017  df-op 4021  df-uni 4235  df-iun 4317  df-br 4438  df-opab 4496  df-mpt 4497  df-id 4785  df-xp 4995  df-rel 4996  df-cnv 4997  df-co 4998  df-dm 4999  df-rn 5000  df-res 5001  df-ima 5002  df-iota 5541  df-fun 5580  df-fn 5581  df-f 5582  df-f1 5583  df-fo 5584  df-f1o 5585  df-fv 5586  df-ov 6284  df-oprab 6285  df-mpt2 6286  df-ghm 16139
This theorem is referenced by:  ghmid  16147  ghminv  16148  ghmsub  16149  ghmmhm  16151  ghmmulg  16153  ghmrn  16154  resghm2  16158  resghm2b  16159  ghmco  16160  ghmpreima  16162  ghmeql  16163  ghmnsgima  16164  ghmnsgpreima  16165  ghmeqker  16167  ghmf1  16169  ghmf1o  16170  ghmpropd  16178  isgim  16184  giclcl  16194  lactghmga  16303  invghm  16716  ghmplusg  16726  evl1addd  18251  evl1subd  18252  ghmcnp  20486  gicabl  31022
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