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Theorem coahom 14930
Description: The composition of two composable arrows is an arrow. (Contributed by Mario Carneiro, 11-Jan-2017.)
Hypotheses
Ref Expression
homdmcoa.o  |-  .x.  =  (compa `  C )
homdmcoa.h  |-  H  =  (Homa
`  C )
homdmcoa.f  |-  ( ph  ->  F  e.  ( X H Y ) )
homdmcoa.g  |-  ( ph  ->  G  e.  ( Y H Z ) )
Assertion
Ref Expression
coahom  |-  ( ph  ->  ( G  .x.  F
)  e.  ( X H Z ) )

Proof of Theorem coahom
StepHypRef Expression
1 homdmcoa.o . . 3  |-  .x.  =  (compa `  C )
2 homdmcoa.h . . 3  |-  H  =  (Homa
`  C )
3 homdmcoa.f . . 3  |-  ( ph  ->  F  e.  ( X H Y ) )
4 homdmcoa.g . . 3  |-  ( ph  ->  G  e.  ( Y H Z ) )
5 eqid 2438 . . 3  |-  (comp `  C )  =  (comp `  C )
61, 2, 3, 4, 5coaval 14928 . 2  |-  ( ph  ->  ( G  .x.  F
)  =  <. X ,  Z ,  ( ( 2nd `  G ) (
<. X ,  Y >. (comp `  C ) Z ) ( 2nd `  F
) ) >. )
7 eqid 2438 . . 3  |-  ( Base `  C )  =  (
Base `  C )
82homarcl 14888 . . . 4  |-  ( F  e.  ( X H Y )  ->  C  e.  Cat )
93, 8syl 16 . . 3  |-  ( ph  ->  C  e.  Cat )
10 eqid 2438 . . 3  |-  ( Hom  `  C )  =  ( Hom  `  C )
112, 7homarcl2 14895 . . . . 5  |-  ( F  e.  ( X H Y )  ->  ( X  e.  ( Base `  C )  /\  Y  e.  ( Base `  C
) ) )
123, 11syl 16 . . . 4  |-  ( ph  ->  ( X  e.  (
Base `  C )  /\  Y  e.  ( Base `  C ) ) )
1312simpld 459 . . 3  |-  ( ph  ->  X  e.  ( Base `  C ) )
142, 7homarcl2 14895 . . . . 5  |-  ( G  e.  ( Y H Z )  ->  ( Y  e.  ( Base `  C )  /\  Z  e.  ( Base `  C
) ) )
154, 14syl 16 . . . 4  |-  ( ph  ->  ( Y  e.  (
Base `  C )  /\  Z  e.  ( Base `  C ) ) )
1615simprd 463 . . 3  |-  ( ph  ->  Z  e.  ( Base `  C ) )
1712simprd 463 . . . 4  |-  ( ph  ->  Y  e.  ( Base `  C ) )
182, 10homahom 14899 . . . . 5  |-  ( F  e.  ( X H Y )  ->  ( 2nd `  F )  e.  ( X ( Hom  `  C ) Y ) )
193, 18syl 16 . . . 4  |-  ( ph  ->  ( 2nd `  F
)  e.  ( X ( Hom  `  C
) Y ) )
202, 10homahom 14899 . . . . 5  |-  ( G  e.  ( Y H Z )  ->  ( 2nd `  G )  e.  ( Y ( Hom  `  C ) Z ) )
214, 20syl 16 . . . 4  |-  ( ph  ->  ( 2nd `  G
)  e.  ( Y ( Hom  `  C
) Z ) )
227, 10, 5, 9, 13, 17, 16, 19, 21catcocl 14615 . . 3  |-  ( ph  ->  ( ( 2nd `  G
) ( <. X ,  Y >. (comp `  C
) Z ) ( 2nd `  F ) )  e.  ( X ( Hom  `  C
) Z ) )
232, 7, 9, 10, 13, 16, 22elhomai2 14894 . 2  |-  ( ph  -> 
<. X ,  Z , 
( ( 2nd `  G
) ( <. X ,  Y >. (comp `  C
) Z ) ( 2nd `  F ) ) >.  e.  ( X H Z ) )
246, 23eqeltrd 2512 1  |-  ( ph  ->  ( G  .x.  F
)  e.  ( X H Z ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1369    e. wcel 1756   <.cop 3878   <.cotp 3880   ` cfv 5413  (class class class)co 6086   2ndc2nd 6571   Basecbs 14166   Hom chom 14241  compcco 14242   Catccat 14594  Homachoma 14883  compaccoa 14914
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-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-ot 3881  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-ov 6089  df-oprab 6090  df-mpt2 6091  df-1st 6572  df-2nd 6573  df-cat 14598  df-doma 14884  df-coda 14885  df-homa 14886  df-arw 14887  df-coa 14916
This theorem is referenced by:  coapm  14931  arwass  14934
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