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Theorem arwass 14184
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 2404 . . . . 5  |-  ( Base `  C )  =  (
Base `  C )
2 eqid 2404 . . . . 5  |-  (  Hom  `  C )  =  (  Hom  `  C )
3 eqid 2404 . . . . 5  |-  (comp `  C )  =  (comp `  C )
4 arwlid.f . . . . . 6  |-  ( ph  ->  F  e.  ( X H Y ) )
5 arwlid.h . . . . . . 7  |-  H  =  (Homa
`  C )
65homarcl 14138 . . . . . 6  |-  ( F  e.  ( X H Y )  ->  C  e.  Cat )
74, 6syl 16 . . . . 5  |-  ( ph  ->  C  e.  Cat )
85, 1homarcl2 14145 . . . . . . 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 446 . . . . 5  |-  ( ph  ->  X  e.  ( Base `  C ) )
119simprd 450 . . . . 5  |-  ( ph  ->  Y  e.  ( Base `  C ) )
12 arwass.k . . . . . . 7  |-  ( ph  ->  K  e.  ( Z H W ) )
135, 1homarcl2 14145 . . . . . . 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 446 . . . . 5  |-  ( ph  ->  Z  e.  ( Base `  C ) )
165, 2homahom 14149 . . . . . 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 14149 . . . . . 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 450 . . . . 5  |-  ( ph  ->  W  e.  ( Base `  C ) )
225, 2homahom 14149 . . . . . 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 13866 . . . 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 14179 . . . . 5  |-  ( ph  ->  ( 2nd `  ( K  .x.  G ) )  =  ( ( 2nd `  K ) ( <. Y ,  Z >. (comp `  C ) W ) ( 2nd `  G
) ) )
2726oveq1d 6055 . . . 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 14179 . . . . 5  |-  ( ph  ->  ( 2nd `  ( G  .x.  F ) )  =  ( ( 2nd `  G ) ( <. X ,  Y >. (comp `  C ) Z ) ( 2nd `  F
) ) )
2928oveq2d 6056 . . . 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 2446 . . 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 3958 . 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 14180 . . 3  |-  ( ph  ->  ( K  .x.  G
)  e.  ( Y H W ) )
3325, 5, 4, 32, 3coaval 14178 . 2  |-  ( ph  ->  ( ( K  .x.  G )  .x.  F
)  =  <. X ,  W ,  ( ( 2nd `  ( K  .x.  G ) ) (
<. X ,  Y >. (comp `  C ) W ) ( 2nd `  F
) ) >. )
3425, 5, 4, 18coahom 14180 . . 3  |-  ( ph  ->  ( G  .x.  F
)  e.  ( X H Z ) )
3525, 5, 34, 12, 3coaval 14178 . 2  |-  ( ph  ->  ( K  .x.  ( G  .x.  F ) )  =  <. X ,  W ,  ( ( 2nd `  K ) ( <. X ,  Z >. (comp `  C ) W ) ( 2nd `  ( G  .x.  F ) ) ) >. )
3631, 33, 353eqtr4d 2446 1  |-  ( ph  ->  ( ( K  .x.  G )  .x.  F
)  =  ( K 
.x.  ( G  .x.  F ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1649    e. wcel 1721   <.cop 3777   <.cotp 3778   ` cfv 5413  (class class class)co 6040   2ndc2nd 6307   Basecbs 13424    Hom chom 13495  compcco 13496   Catccat 13844  Homachoma 14133  Idacida 14163  compaccoa 14164
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-rep 4280  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-ral 2671  df-rex 2672  df-reu 2673  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-op 3783  df-ot 3784  df-uni 3976  df-iun 4055  df-br 4173  df-opab 4227  df-mpt 4228  df-id 4458  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-ov 6043  df-oprab 6044  df-mpt2 6045  df-1st 6308  df-2nd 6309  df-cat 13848  df-doma 14134  df-coda 14135  df-homa 14136  df-arw 14137  df-coa 14166
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