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Theorem coahom 15673
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 2402 . . 3  |-  (comp `  C )  =  (comp `  C )
61, 2, 3, 4, 5coaval 15671 . 2  |-  ( ph  ->  ( G  .x.  F
)  =  <. X ,  Z ,  ( ( 2nd `  G ) (
<. X ,  Y >. (comp `  C ) Z ) ( 2nd `  F
) ) >. )
7 eqid 2402 . . 3  |-  ( Base `  C )  =  (
Base `  C )
82homarcl 15631 . . . 4  |-  ( F  e.  ( X H Y )  ->  C  e.  Cat )
93, 8syl 17 . . 3  |-  ( ph  ->  C  e.  Cat )
10 eqid 2402 . . 3  |-  ( Hom  `  C )  =  ( Hom  `  C )
112, 7homarcl2 15638 . . . . 5  |-  ( F  e.  ( X H Y )  ->  ( X  e.  ( Base `  C )  /\  Y  e.  ( Base `  C
) ) )
123, 11syl 17 . . . 4  |-  ( ph  ->  ( X  e.  (
Base `  C )  /\  Y  e.  ( Base `  C ) ) )
1312simpld 457 . . 3  |-  ( ph  ->  X  e.  ( Base `  C ) )
142, 7homarcl2 15638 . . . . 5  |-  ( G  e.  ( Y H Z )  ->  ( Y  e.  ( Base `  C )  /\  Z  e.  ( Base `  C
) ) )
154, 14syl 17 . . . 4  |-  ( ph  ->  ( Y  e.  (
Base `  C )  /\  Z  e.  ( Base `  C ) ) )
1615simprd 461 . . 3  |-  ( ph  ->  Z  e.  ( Base `  C ) )
1712simprd 461 . . . 4  |-  ( ph  ->  Y  e.  ( Base `  C ) )
182, 10homahom 15642 . . . . 5  |-  ( F  e.  ( X H Y )  ->  ( 2nd `  F )  e.  ( X ( Hom  `  C ) Y ) )
193, 18syl 17 . . . 4  |-  ( ph  ->  ( 2nd `  F
)  e.  ( X ( Hom  `  C
) Y ) )
202, 10homahom 15642 . . . . 5  |-  ( G  e.  ( Y H Z )  ->  ( 2nd `  G )  e.  ( Y ( Hom  `  C ) Z ) )
214, 20syl 17 . . . 4  |-  ( ph  ->  ( 2nd `  G
)  e.  ( Y ( Hom  `  C
) Z ) )
227, 10, 5, 9, 13, 17, 16, 19, 21catcocl 15299 . . 3  |-  ( ph  ->  ( ( 2nd `  G
) ( <. X ,  Y >. (comp `  C
) Z ) ( 2nd `  F ) )  e.  ( X ( Hom  `  C
) Z ) )
232, 7, 9, 10, 13, 16, 22elhomai2 15637 . 2  |-  ( ph  -> 
<. X ,  Z , 
( ( 2nd `  G
) ( <. X ,  Y >. (comp `  C
) Z ) ( 2nd `  F ) ) >.  e.  ( X H Z ) )
246, 23eqeltrd 2490 1  |-  ( ph  ->  ( G  .x.  F
)  e.  ( X H Z ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 367    = wceq 1405    e. wcel 1842   <.cop 3978   <.cotp 3980   ` cfv 5569  (class class class)co 6278   2ndc2nd 6783   Basecbs 14841   Hom chom 14920  compcco 14921   Catccat 15278  Homachoma 15626  compaccoa 15657
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-8 1844  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-rep 4507  ax-sep 4517  ax-nul 4525  ax-pow 4572  ax-pr 4630  ax-un 6574
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 976  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-ral 2759  df-rex 2760  df-reu 2761  df-rab 2763  df-v 3061  df-sbc 3278  df-csb 3374  df-dif 3417  df-un 3419  df-in 3421  df-ss 3428  df-nul 3739  df-if 3886  df-pw 3957  df-sn 3973  df-pr 3975  df-op 3979  df-ot 3981  df-uni 4192  df-iun 4273  df-br 4396  df-opab 4454  df-mpt 4455  df-id 4738  df-xp 4829  df-rel 4830  df-cnv 4831  df-co 4832  df-dm 4833  df-rn 4834  df-res 4835  df-ima 4836  df-iota 5533  df-fun 5571  df-fn 5572  df-f 5573  df-f1 5574  df-fo 5575  df-f1o 5576  df-fv 5577  df-ov 6281  df-oprab 6282  df-mpt2 6283  df-1st 6784  df-2nd 6785  df-cat 15282  df-doma 15627  df-coda 15628  df-homa 15629  df-arw 15630  df-coa 15659
This theorem is referenced by:  coapm  15674  arwass  15677
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