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Theorem arwass 13922
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 2296 . . . . 5  |-  ( Base `  C )  =  (
Base `  C )
2 eqid 2296 . . . . 5  |-  (  Hom  `  C )  =  (  Hom  `  C )
3 eqid 2296 . . . . 5  |-  (comp `  C )  =  (comp `  C )
4 arwlid.f . . . . . 6  |-  ( ph  ->  F  e.  ( X H Y ) )
5 arwlid.h . . . . . . 7  |-  H  =  (Homa
`  C )
65homarcl 13876 . . . . . 6  |-  ( F  e.  ( X H Y )  ->  C  e.  Cat )
74, 6syl 15 . . . . 5  |-  ( ph  ->  C  e.  Cat )
85, 1homarcl2 13883 . . . . . . 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 13883 . . . . . . 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 13887 . . . . . 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 13887 . . . . . 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 13887 . . . . . 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 13604 . . . 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 13917 . . . . 5  |-  ( ph  ->  ( 2nd `  ( K  .x.  G ) )  =  ( ( 2nd `  K ) ( <. Y ,  Z >. (comp `  C ) W ) ( 2nd `  G
) ) )
2726oveq1d 5889 . . . 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 13917 . . . . 5  |-  ( ph  ->  ( 2nd `  ( G  .x.  F ) )  =  ( ( 2nd `  G ) ( <. X ,  Y >. (comp `  C ) Z ) ( 2nd `  F
) ) )
2928oveq2d 5890 . . . 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 2338 . . 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 3826 . 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 13918 . . 3  |-  ( ph  ->  ( K  .x.  G
)  e.  ( Y H W ) )
3325, 5, 4, 32, 3coaval 13916 . 2  |-  ( ph  ->  ( ( K  .x.  G )  .x.  F
)  =  <. X ,  W ,  ( ( 2nd `  ( K  .x.  G ) ) (
<. X ,  Y >. (comp `  C ) W ) ( 2nd `  F
) ) >. )
3425, 5, 4, 18coahom 13918 . . 3  |-  ( ph  ->  ( G  .x.  F
)  e.  ( X H Z ) )
3525, 5, 34, 12, 3coaval 13916 . 2  |-  ( ph  ->  ( K  .x.  ( G  .x.  F ) )  =  <. X ,  W ,  ( ( 2nd `  K ) ( <. X ,  Z >. (comp `  C ) W ) ( 2nd `  ( G  .x.  F ) ) ) >. )
3631, 33, 353eqtr4d 2338 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 1632    e. wcel 1696   <.cop 3656   <.cotp 3657   ` cfv 5271  (class class class)co 5874   2ndc2nd 6137   Basecbs 13164    Hom chom 13235  compcco 13236   Catccat 13582  Homachoma 13871  Idacida 13901  compaccoa 13902
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-reu 2563  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-op 3662  df-ot 3663  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-1st 6138  df-2nd 6139  df-cat 13586  df-doma 13872  df-coda 13873  df-homa 13874  df-arw 13875  df-coa 13904
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