MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  ghmgrp2 Structured version   Unicode version

Theorem ghmgrp2 16472
Description: A group homomorphism is only defined when the codomain is a group. (Contributed by Stefan O'Rear, 31-Dec-2014.)
Assertion
Ref Expression
ghmgrp2  |-  ( F  e.  ( S  GrpHom  T )  ->  T  e.  Grp )

Proof of Theorem ghmgrp2
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqid 2454 . . . 4  |-  ( Base `  S )  =  (
Base `  S )
2 eqid 2454 . . . 4  |-  ( Base `  T )  =  (
Base `  T )
3 eqid 2454 . . . 4  |-  ( +g  `  S )  =  ( +g  `  S )
4 eqid 2454 . . . 4  |-  ( +g  `  T )  =  ( +g  `  T )
51, 2, 3, 4isghm 16469 . . 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 458 . 2  |-  ( F  e.  ( S  GrpHom  T )  ->  ( S  e.  Grp  /\  T  e. 
Grp ) )
76simprd 461 1  |-  ( F  e.  ( S  GrpHom  T )  ->  T  e.  Grp )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 367    = wceq 1398    e. wcel 1823   A.wral 2804   -->wf 5566   ` cfv 5570  (class class class)co 6270   Basecbs 14719   +g cplusg 14787   Grpcgrp 16255    GrpHom cghm 16466
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-rep 4550  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676  ax-un 6565
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-ral 2809  df-rex 2810  df-reu 2811  df-rab 2813  df-v 3108  df-sbc 3325  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3784  df-if 3930  df-pw 4001  df-sn 4017  df-pr 4019  df-op 4023  df-uni 4236  df-iun 4317  df-br 4440  df-opab 4498  df-mpt 4499  df-id 4784  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001  df-iota 5534  df-fun 5572  df-fn 5573  df-f 5574  df-f1 5575  df-fo 5576  df-f1o 5577  df-fv 5578  df-ov 6273  df-oprab 6274  df-mpt2 6275  df-ghm 16467
This theorem is referenced by:  ghmid  16475  ghminv  16476  ghmmhm  16479  ghmmulg  16481  ghmrn  16482  resghm  16485  ghmco  16488  ghmker  16494  ghmeqker  16495  ghmf1  16497  ghmf1o  16498  ghmpropd  16506  isgim  16512  gicrcl  16523  lactghmga  16631  ghmplusg  17054  ghmcyg  17100  ghmcnp  20782  abliso  27923  gicabl  31291
  Copyright terms: Public domain W3C validator