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Theorem ghmco 16078
Description: The composition of group homomorphisms is a homomorphism. (Contributed by Mario Carneiro, 12-Jun-2015.)
Assertion
Ref Expression
ghmco  |-  ( ( F  e.  ( T 
GrpHom  U )  /\  G  e.  ( S  GrpHom  T ) )  ->  ( F  o.  G )  e.  ( S  GrpHom  U ) )

Proof of Theorem ghmco
StepHypRef Expression
1 ghmmhm 16069 . . 3  |-  ( F  e.  ( T  GrpHom  U )  ->  F  e.  ( T MndHom  U ) )
2 ghmmhm 16069 . . 3  |-  ( G  e.  ( S  GrpHom  T )  ->  G  e.  ( S MndHom  T ) )
3 mhmco 15800 . . 3  |-  ( ( F  e.  ( T MndHom  U )  /\  G  e.  ( S MndHom  T ) )  ->  ( F  o.  G )  e.  ( S MndHom  U ) )
41, 2, 3syl2an 477 . 2  |-  ( ( F  e.  ( T 
GrpHom  U )  /\  G  e.  ( S  GrpHom  T ) )  ->  ( F  o.  G )  e.  ( S MndHom  U ) )
5 ghmgrp1 16061 . . 3  |-  ( G  e.  ( S  GrpHom  T )  ->  S  e.  Grp )
6 ghmgrp2 16062 . . 3  |-  ( F  e.  ( T  GrpHom  U )  ->  U  e.  Grp )
7 ghmmhmb 16070 . . 3  |-  ( ( S  e.  Grp  /\  U  e.  Grp )  ->  ( S  GrpHom  U )  =  ( S MndHom  U
) )
85, 6, 7syl2anr 478 . 2  |-  ( ( F  e.  ( T 
GrpHom  U )  /\  G  e.  ( S  GrpHom  T ) )  ->  ( S  GrpHom  U )  =  ( S MndHom  U ) )
94, 8eleqtrrd 2558 1  |-  ( ( F  e.  ( T 
GrpHom  U )  /\  G  e.  ( S  GrpHom  T ) )  ->  ( F  o.  G )  e.  ( S  GrpHom  U ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1379    e. wcel 1767    o. ccom 5003  (class class class)co 6282   Grpcgrp 15720   MndHom cmhm 15772    GrpHom cghm 16056
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 6574
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-rmo 2822  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 5549  df-fun 5588  df-fn 5589  df-f 5590  df-f1 5591  df-fo 5592  df-f1o 5593  df-fv 5594  df-riota 6243  df-ov 6285  df-oprab 6286  df-mpt2 6287  df-map 7419  df-0g 14690  df-mnd 15725  df-mhm 15774  df-grp 15855  df-ghm 16057
This theorem is referenced by:  gimco  16108  rhmco  17166  lmhmco  17469  lmhmvsca  17471  frgpcyg  18376  nmoco  20976  nghmco  20977
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