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Theorem ghmco 15757
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 15748 . . 3  |-  ( F  e.  ( T  GrpHom  U )  ->  F  e.  ( T MndHom  U ) )
2 ghmmhm 15748 . . 3  |-  ( G  e.  ( S  GrpHom  T )  ->  G  e.  ( S MndHom  T ) )
3 mhmco 15481 . . 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 15740 . . 3  |-  ( G  e.  ( S  GrpHom  T )  ->  S  e.  Grp )
6 ghmgrp2 15741 . . 3  |-  ( F  e.  ( T  GrpHom  U )  ->  U  e.  Grp )
7 ghmmhmb 15749 . . 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 2515 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 1369    e. wcel 1756    o. ccom 4839  (class class class)co 6086   Grpcgrp 15402   MndHom cmhm 15454    GrpHom cghm 15735
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 2419  ax-rep 4398  ax-sep 4408  ax-nul 4416  ax-pow 4465  ax-pr 4526  ax-un 6367
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 2256  df-mo 2257  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2715  df-rex 2716  df-reu 2717  df-rmo 2718  df-rab 2719  df-v 2969  df-sbc 3182  df-csb 3284  df-dif 3326  df-un 3328  df-in 3330  df-ss 3337  df-nul 3633  df-if 3787  df-pw 3857  df-sn 3873  df-pr 3875  df-op 3879  df-uni 4087  df-iun 4168  df-br 4288  df-opab 4346  df-mpt 4347  df-id 4631  df-xp 4841  df-rel 4842  df-cnv 4843  df-co 4844  df-dm 4845  df-rn 4846  df-res 4847  df-ima 4848  df-iota 5376  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-riota 6047  df-ov 6089  df-oprab 6090  df-mpt2 6091  df-map 7208  df-0g 14372  df-mnd 15407  df-mhm 15456  df-grp 15536  df-ghm 15736
This theorem is referenced by:  gimco  15787  rhmco  16809  lmhmco  17104  lmhmvsca  17106  frgpcyg  17986  nmoco  20296  nghmco  20297
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