MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  ghmgrp1 Unicode version

Theorem ghmgrp1 14937
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 2389 . . . 4  |-  ( Base `  S )  =  (
Base `  S )
2 eqid 2389 . . . 4  |-  ( Base `  T )  =  (
Base `  T )
3 eqid 2389 . . . 4  |-  ( +g  `  S )  =  ( +g  `  S )
4 eqid 2389 . . . 4  |-  ( +g  `  T )  =  ( +g  `  T )
51, 2, 3, 4isghm 14935 . . 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 447 . 2  |-  ( F  e.  ( S  GrpHom  T )  ->  ( S  e.  Grp  /\  T  e. 
Grp ) )
76simpld 446 1  |-  ( F  e.  ( S  GrpHom  T )  ->  S  e.  Grp )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1649    e. wcel 1717   A.wral 2651   -->wf 5392   ` cfv 5396  (class class class)co 6022   Basecbs 13398   +g cplusg 13458   Grpcgrp 14614    GrpHom cghm 14932
This theorem is referenced by:  ghmid  14941  ghminv  14942  ghmsub  14943  ghmmhm  14945  ghmmulg  14947  ghmrn  14948  resghm2  14952  resghm2b  14953  ghmco  14954  ghmpreima  14956  ghmeql  14957  ghmnsgima  14958  ghmnsgpreima  14959  ghmeqker  14961  ghmf1  14963  ghmf1o  14964  ghmpropd  14972  isgim  14978  giclcl  14988  lactghmga  15036  invghm  15382  ghmplusg  15390  ghmcnp  18067  evl1addd  19823  evl1subd  19824  gicabl  26934
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 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2370  ax-rep 4263  ax-sep 4273  ax-nul 4281  ax-pow 4320  ax-pr 4346  ax-un 4643
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 2244  df-mo 2245  df-clab 2376  df-cleq 2382  df-clel 2385  df-nfc 2514  df-ne 2554  df-ral 2656  df-rex 2657  df-reu 2658  df-rab 2660  df-v 2903  df-sbc 3107  df-csb 3197  df-dif 3268  df-un 3270  df-in 3272  df-ss 3279  df-nul 3574  df-if 3685  df-pw 3746  df-sn 3765  df-pr 3766  df-op 3768  df-uni 3960  df-iun 4039  df-br 4156  df-opab 4210  df-mpt 4211  df-id 4441  df-xp 4826  df-rel 4827  df-cnv 4828  df-co 4829  df-dm 4830  df-rn 4831  df-res 4832  df-ima 4833  df-iota 5360  df-fun 5398  df-fn 5399  df-f 5400  df-f1 5401  df-fo 5402  df-f1o 5403  df-fv 5404  df-ov 6025  df-oprab 6026  df-mpt2 6027  df-ghm 14933
  Copyright terms: Public domain W3C validator