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Theorem arwass 13906
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 2283 . . . . 5  |-  ( Base `  C )  =  (
Base `  C )
2 eqid 2283 . . . . 5  |-  (  Hom  `  C )  =  (  Hom  `  C )
3 eqid 2283 . . . . 5  |-  (comp `  C )  =  (comp `  C )
4 arwlid.f . . . . . 6  |-  ( ph  ->  F  e.  ( X H Y ) )
5 arwlid.h . . . . . . 7  |-  H  =  (Homa
`  C )
65homarcl 13860 . . . . . 6  |-  ( F  e.  ( X H Y )  ->  C  e.  Cat )
74, 6syl 15 . . . . 5  |-  ( ph  ->  C  e.  Cat )
85, 1homarcl2 13867 . . . . . . 7  |-  ( F  e.  ( X H Y )  ->  ( X  e.  ( Base `  C )  /\  Y  e.  ( Base `  C
) ) )
94, 8syl 15 . . . . . 6  |-  ( ph  ->  ( X  e.  (
Base `  C )  /\  Y  e.  ( Base `  C ) ) )
109simpld 445 . . . . 5  |-  ( ph  ->  X  e.  ( Base `  C ) )
119simprd 449 . . . . 5  |-  ( ph  ->  Y  e.  ( Base `  C ) )
12 arwass.k . . . . . . 7  |-  ( ph  ->  K  e.  ( Z H W ) )
135, 1homarcl2 13867 . . . . . . 7  |-  ( K  e.  ( Z H W )  ->  ( Z  e.  ( Base `  C )  /\  W  e.  ( Base `  C
) ) )
1412, 13syl 15 . . . . . 6  |-  ( ph  ->  ( Z  e.  (
Base `  C )  /\  W  e.  ( Base `  C ) ) )
1514simpld 445 . . . . 5  |-  ( ph  ->  Z  e.  ( Base `  C ) )
165, 2homahom 13871 . . . . . 6  |-  ( F  e.  ( X H Y )  ->  ( 2nd `  F )  e.  ( X (  Hom  `  C ) Y ) )
174, 16syl 15 . . . . 5  |-  ( ph  ->  ( 2nd `  F
)  e.  ( X (  Hom  `  C
) Y ) )
18 arwass.g . . . . . 6  |-  ( ph  ->  G  e.  ( Y H Z ) )
195, 2homahom 13871 . . . . . 6  |-  ( G  e.  ( Y H Z )  ->  ( 2nd `  G )  e.  ( Y (  Hom  `  C ) Z ) )
2018, 19syl 15 . . . . 5  |-  ( ph  ->  ( 2nd `  G
)  e.  ( Y (  Hom  `  C
) Z ) )
2114simprd 449 . . . . 5  |-  ( ph  ->  W  e.  ( Base `  C ) )
225, 2homahom 13871 . . . . . 6  |-  ( K  e.  ( Z H W )  ->  ( 2nd `  K )  e.  ( Z (  Hom  `  C ) W ) )
2312, 22syl 15 . . . . 5  |-  ( ph  ->  ( 2nd `  K
)  e.  ( Z (  Hom  `  C
) W ) )
241, 2, 3, 7, 10, 11, 15, 17, 20, 21, 23catass 13588 . . . 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 13901 . . . . 5  |-  ( ph  ->  ( 2nd `  ( K  .x.  G ) )  =  ( ( 2nd `  K ) ( <. Y ,  Z >. (comp `  C ) W ) ( 2nd `  G
) ) )
2726oveq1d 5873 . . . 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 13901 . . . . 5  |-  ( ph  ->  ( 2nd `  ( G  .x.  F ) )  =  ( ( 2nd `  G ) ( <. X ,  Y >. (comp `  C ) Z ) ( 2nd `  F
) ) )
2928oveq2d 5874 . . . 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 2325 . . 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 3810 . 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 13902 . . 3  |-  ( ph  ->  ( K  .x.  G
)  e.  ( Y H W ) )
3325, 5, 4, 32, 3coaval 13900 . 2  |-  ( ph  ->  ( ( K  .x.  G )  .x.  F
)  =  <. X ,  W ,  ( ( 2nd `  ( K  .x.  G ) ) (
<. X ,  Y >. (comp `  C ) W ) ( 2nd `  F
) ) >. )
3425, 5, 4, 18coahom 13902 . . 3  |-  ( ph  ->  ( G  .x.  F
)  e.  ( X H Z ) )
3525, 5, 34, 12, 3coaval 13900 . 2  |-  ( ph  ->  ( K  .x.  ( G  .x.  F ) )  =  <. X ,  W ,  ( ( 2nd `  K ) ( <. X ,  Z >. (comp `  C ) W ) ( 2nd `  ( G  .x.  F ) ) ) >. )
3631, 33, 353eqtr4d 2325 1  |-  ( ph  ->  ( ( K  .x.  G )  .x.  F
)  =  ( K 
.x.  ( G  .x.  F ) ) )
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  Idacida 13885  compaccoa 13886
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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