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Theorem arwass 15255
Description: Associativity of composition in a category using arrow notation. (Contributed by Mario Carneiro, 11-Jan-2017.)
Hypotheses
Ref Expression
arwlid.h  |-  H  =  (Homa
`  C )
arwlid.o  |-  .x.  =  (compa `  C )
arwlid.a  |-  .1.  =  (Ida `  C )
arwlid.f  |-  ( ph  ->  F  e.  ( X H Y ) )
arwass.g  |-  ( ph  ->  G  e.  ( Y H Z ) )
arwass.k  |-  ( ph  ->  K  e.  ( Z H W ) )
Assertion
Ref Expression
arwass  |-  ( ph  ->  ( ( K  .x.  G )  .x.  F
)  =  ( K 
.x.  ( G  .x.  F ) ) )

Proof of Theorem arwass
StepHypRef Expression
1 eqid 2467 . . . . 5  |-  ( Base `  C )  =  (
Base `  C )
2 eqid 2467 . . . . 5  |-  ( Hom  `  C )  =  ( Hom  `  C )
3 eqid 2467 . . . . 5  |-  (comp `  C )  =  (comp `  C )
4 arwlid.f . . . . . 6  |-  ( ph  ->  F  e.  ( X H Y ) )
5 arwlid.h . . . . . . 7  |-  H  =  (Homa
`  C )
65homarcl 15209 . . . . . 6  |-  ( F  e.  ( X H Y )  ->  C  e.  Cat )
74, 6syl 16 . . . . 5  |-  ( ph  ->  C  e.  Cat )
85, 1homarcl2 15216 . . . . . . 7  |-  ( F  e.  ( X H Y )  ->  ( X  e.  ( Base `  C )  /\  Y  e.  ( Base `  C
) ) )
94, 8syl 16 . . . . . 6  |-  ( ph  ->  ( X  e.  (
Base `  C )  /\  Y  e.  ( Base `  C ) ) )
109simpld 459 . . . . 5  |-  ( ph  ->  X  e.  ( Base `  C ) )
119simprd 463 . . . . 5  |-  ( ph  ->  Y  e.  ( Base `  C ) )
12 arwass.k . . . . . . 7  |-  ( ph  ->  K  e.  ( Z H W ) )
135, 1homarcl2 15216 . . . . . . 7  |-  ( K  e.  ( Z H W )  ->  ( Z  e.  ( Base `  C )  /\  W  e.  ( Base `  C
) ) )
1412, 13syl 16 . . . . . 6  |-  ( ph  ->  ( Z  e.  (
Base `  C )  /\  W  e.  ( Base `  C ) ) )
1514simpld 459 . . . . 5  |-  ( ph  ->  Z  e.  ( Base `  C ) )
165, 2homahom 15220 . . . . . 6  |-  ( F  e.  ( X H Y )  ->  ( 2nd `  F )  e.  ( X ( Hom  `  C ) Y ) )
174, 16syl 16 . . . . 5  |-  ( ph  ->  ( 2nd `  F
)  e.  ( X ( Hom  `  C
) Y ) )
18 arwass.g . . . . . 6  |-  ( ph  ->  G  e.  ( Y H Z ) )
195, 2homahom 15220 . . . . . 6  |-  ( G  e.  ( Y H Z )  ->  ( 2nd `  G )  e.  ( Y ( Hom  `  C ) Z ) )
2018, 19syl 16 . . . . 5  |-  ( ph  ->  ( 2nd `  G
)  e.  ( Y ( Hom  `  C
) Z ) )
2114simprd 463 . . . . 5  |-  ( ph  ->  W  e.  ( Base `  C ) )
225, 2homahom 15220 . . . . . 6  |-  ( K  e.  ( Z H W )  ->  ( 2nd `  K )  e.  ( Z ( Hom  `  C ) W ) )
2312, 22syl 16 . . . . 5  |-  ( ph  ->  ( 2nd `  K
)  e.  ( Z ( Hom  `  C
) W ) )
241, 2, 3, 7, 10, 11, 15, 17, 20, 21, 23catass 14937 . . . 4  |-  ( ph  ->  ( ( ( 2nd `  K ) ( <. Y ,  Z >. (comp `  C ) W ) ( 2nd `  G
) ) ( <. X ,  Y >. (comp `  C ) W ) ( 2nd `  F
) )  =  ( ( 2nd `  K
) ( <. X ,  Z >. (comp `  C
) W ) ( ( 2nd `  G
) ( <. X ,  Y >. (comp `  C
) Z ) ( 2nd `  F ) ) ) )
25 arwlid.o . . . . . 6  |-  .x.  =  (compa `  C )
2625, 5, 18, 12, 3coa2 15250 . . . . 5  |-  ( ph  ->  ( 2nd `  ( K  .x.  G ) )  =  ( ( 2nd `  K ) ( <. Y ,  Z >. (comp `  C ) W ) ( 2nd `  G
) ) )
2726oveq1d 6297 . . . 4  |-  ( ph  ->  ( ( 2nd `  ( K  .x.  G ) ) ( <. X ,  Y >. (comp `  C ) W ) ( 2nd `  F ) )  =  ( ( ( 2nd `  K ) ( <. Y ,  Z >. (comp `  C ) W ) ( 2nd `  G
) ) ( <. X ,  Y >. (comp `  C ) W ) ( 2nd `  F
) ) )
2825, 5, 4, 18, 3coa2 15250 . . . . 5  |-  ( ph  ->  ( 2nd `  ( G  .x.  F ) )  =  ( ( 2nd `  G ) ( <. X ,  Y >. (comp `  C ) Z ) ( 2nd `  F
) ) )
2928oveq2d 6298 . . . 4  |-  ( ph  ->  ( ( 2nd `  K
) ( <. X ,  Z >. (comp `  C
) W ) ( 2nd `  ( G 
.x.  F ) ) )  =  ( ( 2nd `  K ) ( <. X ,  Z >. (comp `  C ) W ) ( ( 2nd `  G ) ( <. X ,  Y >. (comp `  C ) Z ) ( 2nd `  F ) ) ) )
3024, 27, 293eqtr4d 2518 . . 3  |-  ( ph  ->  ( ( 2nd `  ( K  .x.  G ) ) ( <. X ,  Y >. (comp `  C ) W ) ( 2nd `  F ) )  =  ( ( 2nd `  K
) ( <. X ,  Z >. (comp `  C
) W ) ( 2nd `  ( G 
.x.  F ) ) ) )
3130oteq3d 4227 . 2  |-  ( ph  -> 
<. X ,  W , 
( ( 2nd `  ( K  .x.  G ) ) ( <. X ,  Y >. (comp `  C ) W ) ( 2nd `  F ) ) >.  =  <. X ,  W ,  ( ( 2nd `  K ) ( <. X ,  Z >. (comp `  C ) W ) ( 2nd `  ( G  .x.  F ) ) ) >. )
3225, 5, 18, 12coahom 15251 . . 3  |-  ( ph  ->  ( K  .x.  G
)  e.  ( Y H W ) )
3325, 5, 4, 32, 3coaval 15249 . 2  |-  ( ph  ->  ( ( K  .x.  G )  .x.  F
)  =  <. X ,  W ,  ( ( 2nd `  ( K  .x.  G ) ) (
<. X ,  Y >. (comp `  C ) W ) ( 2nd `  F
) ) >. )
3425, 5, 4, 18coahom 15251 . . 3  |-  ( ph  ->  ( G  .x.  F
)  e.  ( X H Z ) )
3525, 5, 34, 12, 3coaval 15249 . 2  |-  ( ph  ->  ( K  .x.  ( G  .x.  F ) )  =  <. X ,  W ,  ( ( 2nd `  K ) ( <. X ,  Z >. (comp `  C ) W ) ( 2nd `  ( G  .x.  F ) ) ) >. )
3631, 33, 353eqtr4d 2518 1  |-  ( ph  ->  ( ( K  .x.  G )  .x.  F
)  =  ( K 
.x.  ( G  .x.  F ) ) )
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  Idacida 15234  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: (None)
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