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Theorem coahom 15042
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 2451 . . 3  |-  (comp `  C )  =  (comp `  C )
61, 2, 3, 4, 5coaval 15040 . 2  |-  ( ph  ->  ( G  .x.  F
)  =  <. X ,  Z ,  ( ( 2nd `  G ) (
<. X ,  Y >. (comp `  C ) Z ) ( 2nd `  F
) ) >. )
7 eqid 2451 . . 3  |-  ( Base `  C )  =  (
Base `  C )
82homarcl 15000 . . . 4  |-  ( F  e.  ( X H Y )  ->  C  e.  Cat )
93, 8syl 16 . . 3  |-  ( ph  ->  C  e.  Cat )
10 eqid 2451 . . 3  |-  ( Hom  `  C )  =  ( Hom  `  C )
112, 7homarcl2 15007 . . . . 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 15007 . . . . 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 15011 . . . . 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 15011 . . . . 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 14727 . . 3  |-  ( ph  ->  ( ( 2nd `  G
) ( <. X ,  Y >. (comp `  C
) Z ) ( 2nd `  F ) )  e.  ( X ( Hom  `  C
) Z ) )
232, 7, 9, 10, 13, 16, 22elhomai2 15006 . 2  |-  ( ph  -> 
<. X ,  Z , 
( ( 2nd `  G
) ( <. X ,  Y >. (comp `  C
) Z ) ( 2nd `  F ) ) >.  e.  ( X H Z ) )
246, 23eqeltrd 2539 1  |-  ( ph  ->  ( G  .x.  F
)  e.  ( X H Z ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1370    e. wcel 1758   <.cop 3983   <.cotp 3985   ` cfv 5518  (class class class)co 6192   2ndc2nd 6678   Basecbs 14278   Hom chom 14353  compcco 14354   Catccat 14706  Homachoma 14995  compaccoa 15026
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 1952  ax-ext 2430  ax-rep 4503  ax-sep 4513  ax-nul 4521  ax-pow 4570  ax-pr 4631  ax-un 6474
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 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ne 2646  df-ral 2800  df-rex 2801  df-reu 2802  df-rab 2804  df-v 3072  df-sbc 3287  df-csb 3389  df-dif 3431  df-un 3433  df-in 3435  df-ss 3442  df-nul 3738  df-if 3892  df-pw 3962  df-sn 3978  df-pr 3980  df-op 3984  df-ot 3986  df-uni 4192  df-iun 4273  df-br 4393  df-opab 4451  df-mpt 4452  df-id 4736  df-xp 4946  df-rel 4947  df-cnv 4948  df-co 4949  df-dm 4950  df-rn 4951  df-res 4952  df-ima 4953  df-iota 5481  df-fun 5520  df-fn 5521  df-f 5522  df-f1 5523  df-fo 5524  df-f1o 5525  df-fv 5526  df-ov 6195  df-oprab 6196  df-mpt2 6197  df-1st 6679  df-2nd 6680  df-cat 14710  df-doma 14996  df-coda 14997  df-homa 14998  df-arw 14999  df-coa 15028
This theorem is referenced by:  coapm  15043  arwass  15046
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