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Theorem coahom 13902
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 2283 . . 3  |-  (comp `  C )  =  (comp `  C )
61, 2, 3, 4, 5coaval 13900 . 2  |-  ( ph  ->  ( G  .x.  F
)  =  <. X ,  Z ,  ( ( 2nd `  G ) (
<. X ,  Y >. (comp `  C ) Z ) ( 2nd `  F
) ) >. )
7 eqid 2283 . . 3  |-  ( Base `  C )  =  (
Base `  C )
82homarcl 13860 . . . 4  |-  ( F  e.  ( X H Y )  ->  C  e.  Cat )
93, 8syl 15 . . 3  |-  ( ph  ->  C  e.  Cat )
10 eqid 2283 . . 3  |-  (  Hom  `  C )  =  (  Hom  `  C )
112, 7homarcl2 13867 . . . . 5  |-  ( F  e.  ( X H Y )  ->  ( X  e.  ( Base `  C )  /\  Y  e.  ( Base `  C
) ) )
123, 11syl 15 . . . 4  |-  ( ph  ->  ( X  e.  (
Base `  C )  /\  Y  e.  ( Base `  C ) ) )
1312simpld 445 . . 3  |-  ( ph  ->  X  e.  ( Base `  C ) )
142, 7homarcl2 13867 . . . . 5  |-  ( G  e.  ( Y H Z )  ->  ( Y  e.  ( Base `  C )  /\  Z  e.  ( Base `  C
) ) )
154, 14syl 15 . . . 4  |-  ( ph  ->  ( Y  e.  (
Base `  C )  /\  Z  e.  ( Base `  C ) ) )
1615simprd 449 . . 3  |-  ( ph  ->  Z  e.  ( Base `  C ) )
1712simprd 449 . . . 4  |-  ( ph  ->  Y  e.  ( Base `  C ) )
182, 10homahom 13871 . . . . 5  |-  ( F  e.  ( X H Y )  ->  ( 2nd `  F )  e.  ( X (  Hom  `  C ) Y ) )
193, 18syl 15 . . . 4  |-  ( ph  ->  ( 2nd `  F
)  e.  ( X (  Hom  `  C
) Y ) )
202, 10homahom 13871 . . . . 5  |-  ( G  e.  ( Y H Z )  ->  ( 2nd `  G )  e.  ( Y (  Hom  `  C ) Z ) )
214, 20syl 15 . . . 4  |-  ( ph  ->  ( 2nd `  G
)  e.  ( Y (  Hom  `  C
) Z ) )
227, 10, 5, 9, 13, 17, 16, 19, 21catcocl 13587 . . 3  |-  ( ph  ->  ( ( 2nd `  G
) ( <. X ,  Y >. (comp `  C
) Z ) ( 2nd `  F ) )  e.  ( X (  Hom  `  C
) Z ) )
232, 7, 9, 10, 13, 16, 22elhomai2 13866 . 2  |-  ( ph  -> 
<. X ,  Z , 
( ( 2nd `  G
) ( <. X ,  Y >. (comp `  C
) Z ) ( 2nd `  F ) ) >.  e.  ( X H Z ) )
246, 23eqeltrd 2357 1  |-  ( ph  ->  ( G  .x.  F
)  e.  ( X H Z ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1623    e. wcel 1684   <.cop 3643   <.cotp 3644   ` cfv 5255  (class class class)co 5858   2ndc2nd 6121   Basecbs 13148    Hom chom 13219  compcco 13220   Catccat 13566  Homachoma 13855  compaccoa 13886
This theorem is referenced by:  coapm  13903  arwass  13906
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-reu 2550  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-ot 3650  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-cat 13570  df-doma 13856  df-coda 13857  df-homa 13858  df-arw 13859  df-coa 13888
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