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Theorem arwass 15470
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 2382 . . . . 5  |-  ( Base `  C )  =  (
Base `  C )
2 eqid 2382 . . . . 5  |-  ( Hom  `  C )  =  ( Hom  `  C )
3 eqid 2382 . . . . 5  |-  (comp `  C )  =  (comp `  C )
4 arwlid.f . . . . . 6  |-  ( ph  ->  F  e.  ( X H Y ) )
5 arwlid.h . . . . . . 7  |-  H  =  (Homa
`  C )
65homarcl 15424 . . . . . 6  |-  ( F  e.  ( X H Y )  ->  C  e.  Cat )
74, 6syl 16 . . . . 5  |-  ( ph  ->  C  e.  Cat )
85, 1homarcl2 15431 . . . . . . 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 457 . . . . 5  |-  ( ph  ->  X  e.  ( Base `  C ) )
119simprd 461 . . . . 5  |-  ( ph  ->  Y  e.  ( Base `  C ) )
12 arwass.k . . . . . . 7  |-  ( ph  ->  K  e.  ( Z H W ) )
135, 1homarcl2 15431 . . . . . . 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 457 . . . . 5  |-  ( ph  ->  Z  e.  ( Base `  C ) )
165, 2homahom 15435 . . . . . 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 15435 . . . . . 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 461 . . . . 5  |-  ( ph  ->  W  e.  ( Base `  C ) )
225, 2homahom 15435 . . . . . 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 15093 . . . 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 15465 . . . . 5  |-  ( ph  ->  ( 2nd `  ( K  .x.  G ) )  =  ( ( 2nd `  K ) ( <. Y ,  Z >. (comp `  C ) W ) ( 2nd `  G
) ) )
2726oveq1d 6211 . . . 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 15465 . . . . 5  |-  ( ph  ->  ( 2nd `  ( G  .x.  F ) )  =  ( ( 2nd `  G ) ( <. X ,  Y >. (comp `  C ) Z ) ( 2nd `  F
) ) )
2928oveq2d 6212 . . . 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 2433 . . 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 4145 . 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 15466 . . 3  |-  ( ph  ->  ( K  .x.  G
)  e.  ( Y H W ) )
3325, 5, 4, 32, 3coaval 15464 . 2  |-  ( ph  ->  ( ( K  .x.  G )  .x.  F
)  =  <. X ,  W ,  ( ( 2nd `  ( K  .x.  G ) ) (
<. X ,  Y >. (comp `  C ) W ) ( 2nd `  F
) ) >. )
3425, 5, 4, 18coahom 15466 . . 3  |-  ( ph  ->  ( G  .x.  F
)  e.  ( X H Z ) )
3525, 5, 34, 12, 3coaval 15464 . 2  |-  ( ph  ->  ( K  .x.  ( G  .x.  F ) )  =  <. X ,  W ,  ( ( 2nd `  K ) ( <. X ,  Z >. (comp `  C ) W ) ( 2nd `  ( G  .x.  F ) ) ) >. )
3631, 33, 353eqtr4d 2433 1  |-  ( ph  ->  ( ( K  .x.  G )  .x.  F
)  =  ( K 
.x.  ( G  .x.  F ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 367    = wceq 1399    e. wcel 1826   <.cop 3950   <.cotp 3952   ` cfv 5496  (class class class)co 6196   2ndc2nd 6698   Basecbs 14634   Hom chom 14713  compcco 14714   Catccat 15071  Homachoma 15419  Idacida 15449  compaccoa 15450
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1626  ax-4 1639  ax-5 1712  ax-6 1755  ax-7 1798  ax-8 1828  ax-9 1830  ax-10 1845  ax-11 1850  ax-12 1862  ax-13 2006  ax-ext 2360  ax-rep 4478  ax-sep 4488  ax-nul 4496  ax-pow 4543  ax-pr 4601  ax-un 6491
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1402  df-ex 1621  df-nf 1625  df-sb 1748  df-eu 2222  df-mo 2223  df-clab 2368  df-cleq 2374  df-clel 2377  df-nfc 2532  df-ne 2579  df-ral 2737  df-rex 2738  df-reu 2739  df-rab 2741  df-v 3036  df-sbc 3253  df-csb 3349  df-dif 3392  df-un 3394  df-in 3396  df-ss 3403  df-nul 3712  df-if 3858  df-pw 3929  df-sn 3945  df-pr 3947  df-op 3951  df-ot 3953  df-uni 4164  df-iun 4245  df-br 4368  df-opab 4426  df-mpt 4427  df-id 4709  df-xp 4919  df-rel 4920  df-cnv 4921  df-co 4922  df-dm 4923  df-rn 4924  df-res 4925  df-ima 4926  df-iota 5460  df-fun 5498  df-fn 5499  df-f 5500  df-f1 5501  df-fo 5502  df-f1o 5503  df-fv 5504  df-ov 6199  df-oprab 6200  df-mpt2 6201  df-1st 6699  df-2nd 6700  df-cat 15075  df-doma 15420  df-coda 15421  df-homa 15422  df-arw 15423  df-coa 15452
This theorem is referenced by: (None)
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