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Theorem coahom 15251
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 2467 . . 3  |-  (comp `  C )  =  (comp `  C )
61, 2, 3, 4, 5coaval 15249 . 2  |-  ( ph  ->  ( G  .x.  F
)  =  <. X ,  Z ,  ( ( 2nd `  G ) (
<. X ,  Y >. (comp `  C ) Z ) ( 2nd `  F
) ) >. )
7 eqid 2467 . . 3  |-  ( Base `  C )  =  (
Base `  C )
82homarcl 15209 . . . 4  |-  ( F  e.  ( X H Y )  ->  C  e.  Cat )
93, 8syl 16 . . 3  |-  ( ph  ->  C  e.  Cat )
10 eqid 2467 . . 3  |-  ( Hom  `  C )  =  ( Hom  `  C )
112, 7homarcl2 15216 . . . . 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 15216 . . . . 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 15220 . . . . 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 15220 . . . . 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 14936 . . 3  |-  ( ph  ->  ( ( 2nd `  G
) ( <. X ,  Y >. (comp `  C
) Z ) ( 2nd `  F ) )  e.  ( X ( Hom  `  C
) Z ) )
232, 7, 9, 10, 13, 16, 22elhomai2 15215 . 2  |-  ( ph  -> 
<. X ,  Z , 
( ( 2nd `  G
) ( <. X ,  Y >. (comp `  C
) Z ) ( 2nd `  F ) ) >.  e.  ( X H Z ) )
246, 23eqeltrd 2555 1  |-  ( ph  ->  ( G  .x.  F
)  e.  ( X H Z ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1379    e. wcel 1767   <.cop 4033   <.cotp 4035   ` cfv 5586  (class class class)co 6282   2ndc2nd 6780   Basecbs 14486   Hom chom 14562  compcco 14563   Catccat 14915  Homachoma 15204  compaccoa 15235
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-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-ot 4036  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-ov 6285  df-oprab 6286  df-mpt2 6287  df-1st 6781  df-2nd 6782  df-cat 14919  df-doma 15205  df-coda 15206  df-homa 15207  df-arw 15208  df-coa 15237
This theorem is referenced by:  coapm  15252  arwass  15255
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