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Theorem ghmco 15884
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 15875 . . 3  |-  ( F  e.  ( T  GrpHom  U )  ->  F  e.  ( T MndHom  U ) )
2 ghmmhm 15875 . . 3  |-  ( G  e.  ( S  GrpHom  T )  ->  G  e.  ( S MndHom  T ) )
3 mhmco 15608 . . 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 15867 . . 3  |-  ( G  e.  ( S  GrpHom  T )  ->  S  e.  Grp )
6 ghmgrp2 15868 . . 3  |-  ( F  e.  ( T  GrpHom  U )  ->  U  e.  Grp )
7 ghmmhmb 15876 . . 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 2545 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 1370    e. wcel 1758    o. ccom 4951  (class class class)co 6199   Grpcgrp 15528   MndHom cmhm 15580    GrpHom cghm 15862
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-rep 4510  ax-sep 4520  ax-nul 4528  ax-pow 4577  ax-pr 4638  ax-un 6481
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2649  df-ral 2803  df-rex 2804  df-reu 2805  df-rmo 2806  df-rab 2807  df-v 3078  df-sbc 3293  df-csb 3395  df-dif 3438  df-un 3440  df-in 3442  df-ss 3449  df-nul 3745  df-if 3899  df-pw 3969  df-sn 3985  df-pr 3987  df-op 3991  df-uni 4199  df-iun 4280  df-br 4400  df-opab 4458  df-mpt 4459  df-id 4743  df-xp 4953  df-rel 4954  df-cnv 4955  df-co 4956  df-dm 4957  df-rn 4958  df-res 4959  df-ima 4960  df-iota 5488  df-fun 5527  df-fn 5528  df-f 5529  df-f1 5530  df-fo 5531  df-f1o 5532  df-fv 5533  df-riota 6160  df-ov 6202  df-oprab 6203  df-mpt2 6204  df-map 7325  df-0g 14498  df-mnd 15533  df-mhm 15582  df-grp 15663  df-ghm 15863
This theorem is referenced by:  gimco  15914  rhmco  16947  lmhmco  17246  lmhmvsca  17248  frgpcyg  18130  nmoco  20447  nghmco  20448
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