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Theorem cofuass 15133
Description: Functor composition is associative. (Contributed by Mario Carneiro, 3-Jan-2017.)
Hypotheses
Ref Expression
cofuass.g  |-  ( ph  ->  G  e.  ( C 
Func  D ) )
cofuass.h  |-  ( ph  ->  H  e.  ( D 
Func  E ) )
cofuass.k  |-  ( ph  ->  K  e.  ( E 
Func  F ) )
Assertion
Ref Expression
cofuass  |-  ( ph  ->  ( ( K  o.func  H )  o.func 
G )  =  ( K  o.func  ( H  o.func  G )
) )

Proof of Theorem cofuass
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 coass 5532 . . . 4  |-  ( ( ( 1st `  K
)  o.  ( 1st `  H ) )  o.  ( 1st `  G
) )  =  ( ( 1st `  K
)  o.  ( ( 1st `  H )  o.  ( 1st `  G
) ) )
2 eqid 2467 . . . . . 6  |-  ( Base `  D )  =  (
Base `  D )
3 cofuass.h . . . . . 6  |-  ( ph  ->  H  e.  ( D 
Func  E ) )
4 cofuass.k . . . . . 6  |-  ( ph  ->  K  e.  ( E 
Func  F ) )
52, 3, 4cofu1st 15127 . . . . 5  |-  ( ph  ->  ( 1st `  ( K  o.func 
H ) )  =  ( ( 1st `  K
)  o.  ( 1st `  H ) ) )
65coeq1d 5170 . . . 4  |-  ( ph  ->  ( ( 1st `  ( K  o.func 
H ) )  o.  ( 1st `  G
) )  =  ( ( ( 1st `  K
)  o.  ( 1st `  H ) )  o.  ( 1st `  G
) ) )
7 eqid 2467 . . . . . 6  |-  ( Base `  C )  =  (
Base `  C )
8 cofuass.g . . . . . 6  |-  ( ph  ->  G  e.  ( C 
Func  D ) )
97, 8, 3cofu1st 15127 . . . . 5  |-  ( ph  ->  ( 1st `  ( H  o.func 
G ) )  =  ( ( 1st `  H
)  o.  ( 1st `  G ) ) )
109coeq2d 5171 . . . 4  |-  ( ph  ->  ( ( 1st `  K
)  o.  ( 1st `  ( H  o.func  G )
) )  =  ( ( 1st `  K
)  o.  ( ( 1st `  H )  o.  ( 1st `  G
) ) ) )
111, 6, 103eqtr4a 2534 . . 3  |-  ( ph  ->  ( ( 1st `  ( K  o.func 
H ) )  o.  ( 1st `  G
) )  =  ( ( 1st `  K
)  o.  ( 1st `  ( H  o.func  G )
) ) )
12 coass 5532 . . . . 5  |-  ( ( ( ( ( 1st `  H ) `  (
( 1st `  G
) `  x )
) ( 2nd `  K
) ( ( 1st `  H ) `  (
( 1st `  G
) `  y )
) )  o.  (
( ( 1st `  G
) `  x )
( 2nd `  H
) ( ( 1st `  G ) `  y
) ) )  o.  ( x ( 2nd `  G ) y ) )  =  ( ( ( ( 1st `  H
) `  ( ( 1st `  G ) `  x ) ) ( 2nd `  K ) ( ( 1st `  H
) `  ( ( 1st `  G ) `  y ) ) )  o.  ( ( ( ( 1st `  G
) `  x )
( 2nd `  H
) ( ( 1st `  G ) `  y
) )  o.  (
x ( 2nd `  G
) y ) ) )
1333ad2ant1 1017 . . . . . . 7  |-  ( (
ph  /\  x  e.  ( Base `  C )  /\  y  e.  ( Base `  C ) )  ->  H  e.  ( D  Func  E )
)
1443ad2ant1 1017 . . . . . . 7  |-  ( (
ph  /\  x  e.  ( Base `  C )  /\  y  e.  ( Base `  C ) )  ->  K  e.  ( E  Func  F )
)
15 relfunc 15106 . . . . . . . . . . 11  |-  Rel  ( C  Func  D )
16 1st2ndbr 6844 . . . . . . . . . . 11  |-  ( ( Rel  ( C  Func  D )  /\  G  e.  ( C  Func  D
) )  ->  ( 1st `  G ) ( C  Func  D )
( 2nd `  G
) )
1715, 8, 16sylancr 663 . . . . . . . . . 10  |-  ( ph  ->  ( 1st `  G
) ( C  Func  D ) ( 2nd `  G
) )
18173ad2ant1 1017 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  ( Base `  C )  /\  y  e.  ( Base `  C ) )  ->  ( 1st `  G
) ( C  Func  D ) ( 2nd `  G
) )
197, 2, 18funcf1 15110 . . . . . . . 8  |-  ( (
ph  /\  x  e.  ( Base `  C )  /\  y  e.  ( Base `  C ) )  ->  ( 1st `  G
) : ( Base `  C ) --> ( Base `  D ) )
20 simp2 997 . . . . . . . 8  |-  ( (
ph  /\  x  e.  ( Base `  C )  /\  y  e.  ( Base `  C ) )  ->  x  e.  (
Base `  C )
)
2119, 20ffvelrnd 6033 . . . . . . 7  |-  ( (
ph  /\  x  e.  ( Base `  C )  /\  y  e.  ( Base `  C ) )  ->  ( ( 1st `  G ) `  x
)  e.  ( Base `  D ) )
22 simp3 998 . . . . . . . 8  |-  ( (
ph  /\  x  e.  ( Base `  C )  /\  y  e.  ( Base `  C ) )  ->  y  e.  (
Base `  C )
)
2319, 22ffvelrnd 6033 . . . . . . 7  |-  ( (
ph  /\  x  e.  ( Base `  C )  /\  y  e.  ( Base `  C ) )  ->  ( ( 1st `  G ) `  y
)  e.  ( Base `  D ) )
242, 13, 14, 21, 23cofu2nd 15129 . . . . . 6  |-  ( (
ph  /\  x  e.  ( Base `  C )  /\  y  e.  ( Base `  C ) )  ->  ( ( ( 1st `  G ) `
 x ) ( 2nd `  ( K  o.func 
H ) ) ( ( 1st `  G
) `  y )
)  =  ( ( ( ( 1st `  H
) `  ( ( 1st `  G ) `  x ) ) ( 2nd `  K ) ( ( 1st `  H
) `  ( ( 1st `  G ) `  y ) ) )  o.  ( ( ( 1st `  G ) `
 x ) ( 2nd `  H ) ( ( 1st `  G
) `  y )
) ) )
2524coeq1d 5170 . . . . 5  |-  ( (
ph  /\  x  e.  ( Base `  C )  /\  y  e.  ( Base `  C ) )  ->  ( ( ( ( 1st `  G
) `  x )
( 2nd `  ( K  o.func 
H ) ) ( ( 1st `  G
) `  y )
)  o.  ( x ( 2nd `  G
) y ) )  =  ( ( ( ( ( 1st `  H
) `  ( ( 1st `  G ) `  x ) ) ( 2nd `  K ) ( ( 1st `  H
) `  ( ( 1st `  G ) `  y ) ) )  o.  ( ( ( 1st `  G ) `
 x ) ( 2nd `  H ) ( ( 1st `  G
) `  y )
) )  o.  (
x ( 2nd `  G
) y ) ) )
2683ad2ant1 1017 . . . . . . . 8  |-  ( (
ph  /\  x  e.  ( Base `  C )  /\  y  e.  ( Base `  C ) )  ->  G  e.  ( C  Func  D )
)
277, 26, 13, 20cofu1 15128 . . . . . . 7  |-  ( (
ph  /\  x  e.  ( Base `  C )  /\  y  e.  ( Base `  C ) )  ->  ( ( 1st `  ( H  o.func  G )
) `  x )  =  ( ( 1st `  H ) `  (
( 1st `  G
) `  x )
) )
287, 26, 13, 22cofu1 15128 . . . . . . 7  |-  ( (
ph  /\  x  e.  ( Base `  C )  /\  y  e.  ( Base `  C ) )  ->  ( ( 1st `  ( H  o.func  G )
) `  y )  =  ( ( 1st `  H ) `  (
( 1st `  G
) `  y )
) )
2927, 28oveq12d 6313 . . . . . 6  |-  ( (
ph  /\  x  e.  ( Base `  C )  /\  y  e.  ( Base `  C ) )  ->  ( ( ( 1st `  ( H  o.func 
G ) ) `  x ) ( 2nd `  K ) ( ( 1st `  ( H  o.func 
G ) ) `  y ) )  =  ( ( ( 1st `  H ) `  (
( 1st `  G
) `  x )
) ( 2nd `  K
) ( ( 1st `  H ) `  (
( 1st `  G
) `  y )
) ) )
307, 26, 13, 20, 22cofu2nd 15129 . . . . . 6  |-  ( (
ph  /\  x  e.  ( Base `  C )  /\  y  e.  ( Base `  C ) )  ->  ( x ( 2nd `  ( H  o.func 
G ) ) y )  =  ( ( ( ( 1st `  G
) `  x )
( 2nd `  H
) ( ( 1st `  G ) `  y
) )  o.  (
x ( 2nd `  G
) y ) ) )
3129, 30coeq12d 5173 . . . . 5  |-  ( (
ph  /\  x  e.  ( Base `  C )  /\  y  e.  ( Base `  C ) )  ->  ( ( ( ( 1st `  ( H  o.func 
G ) ) `  x ) ( 2nd `  K ) ( ( 1st `  ( H  o.func 
G ) ) `  y ) )  o.  ( x ( 2nd `  ( H  o.func  G )
) y ) )  =  ( ( ( ( 1st `  H
) `  ( ( 1st `  G ) `  x ) ) ( 2nd `  K ) ( ( 1st `  H
) `  ( ( 1st `  G ) `  y ) ) )  o.  ( ( ( ( 1st `  G
) `  x )
( 2nd `  H
) ( ( 1st `  G ) `  y
) )  o.  (
x ( 2nd `  G
) y ) ) ) )
3212, 25, 313eqtr4a 2534 . . . 4  |-  ( (
ph  /\  x  e.  ( Base `  C )  /\  y  e.  ( Base `  C ) )  ->  ( ( ( ( 1st `  G
) `  x )
( 2nd `  ( K  o.func 
H ) ) ( ( 1st `  G
) `  y )
)  o.  ( x ( 2nd `  G
) y ) )  =  ( ( ( ( 1st `  ( H  o.func 
G ) ) `  x ) ( 2nd `  K ) ( ( 1st `  ( H  o.func 
G ) ) `  y ) )  o.  ( x ( 2nd `  ( H  o.func  G )
) y ) ) )
3332mpt2eq3dva 6356 . . 3  |-  ( ph  ->  ( x  e.  (
Base `  C ) ,  y  e.  ( Base `  C )  |->  ( ( ( ( 1st `  G ) `  x
) ( 2nd `  ( K  o.func 
H ) ) ( ( 1st `  G
) `  y )
)  o.  ( x ( 2nd `  G
) y ) ) )  =  ( x  e.  ( Base `  C
) ,  y  e.  ( Base `  C
)  |->  ( ( ( ( 1st `  ( H  o.func 
G ) ) `  x ) ( 2nd `  K ) ( ( 1st `  ( H  o.func 
G ) ) `  y ) )  o.  ( x ( 2nd `  ( H  o.func  G )
) y ) ) ) )
3411, 33opeq12d 4227 . 2  |-  ( ph  -> 
<. ( ( 1st `  ( K  o.func 
H ) )  o.  ( 1st `  G
) ) ,  ( x  e.  ( Base `  C ) ,  y  e.  ( Base `  C
)  |->  ( ( ( ( 1st `  G
) `  x )
( 2nd `  ( K  o.func 
H ) ) ( ( 1st `  G
) `  y )
)  o.  ( x ( 2nd `  G
) y ) ) ) >.  =  <. ( ( 1st `  K
)  o.  ( 1st `  ( H  o.func  G )
) ) ,  ( x  e.  ( Base `  C ) ,  y  e.  ( Base `  C
)  |->  ( ( ( ( 1st `  ( H  o.func 
G ) ) `  x ) ( 2nd `  K ) ( ( 1st `  ( H  o.func 
G ) ) `  y ) )  o.  ( x ( 2nd `  ( H  o.func  G )
) y ) ) ) >. )
353, 4cofucl 15132 . . 3  |-  ( ph  ->  ( K  o.func  H )  e.  ( D  Func  F
) )
367, 8, 35cofuval 15126 . 2  |-  ( ph  ->  ( ( K  o.func  H )  o.func 
G )  =  <. ( ( 1st `  ( K  o.func 
H ) )  o.  ( 1st `  G
) ) ,  ( x  e.  ( Base `  C ) ,  y  e.  ( Base `  C
)  |->  ( ( ( ( 1st `  G
) `  x )
( 2nd `  ( K  o.func 
H ) ) ( ( 1st `  G
) `  y )
)  o.  ( x ( 2nd `  G
) y ) ) ) >. )
378, 3cofucl 15132 . . 3  |-  ( ph  ->  ( H  o.func  G )  e.  ( C  Func  E
) )
387, 37, 4cofuval 15126 . 2  |-  ( ph  ->  ( K  o.func  ( H  o.func  G ) )  =  <. ( ( 1st `  K
)  o.  ( 1st `  ( H  o.func  G )
) ) ,  ( x  e.  ( Base `  C ) ,  y  e.  ( Base `  C
)  |->  ( ( ( ( 1st `  ( H  o.func 
G ) ) `  x ) ( 2nd `  K ) ( ( 1st `  ( H  o.func 
G ) ) `  y ) )  o.  ( x ( 2nd `  ( H  o.func  G )
) y ) ) ) >. )
3934, 36, 383eqtr4d 2518 1  |-  ( ph  ->  ( ( K  o.func  H )  o.func 
G )  =  ( K  o.func  ( H  o.func  G )
) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ w3a 973    = wceq 1379    e. wcel 1767   <.cop 4039   class class class wbr 4453    o. ccom 5009   Rel wrel 5010   ` cfv 5594  (class class class)co 6295    |-> cmpt2 6297   1stc1st 6793   2ndc2nd 6794   Basecbs 14507    Func cfunc 15098    o.func ccofu 15100
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 4564  ax-sep 4574  ax-nul 4582  ax-pow 4631  ax-pr 4692  ax-un 6587
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 2822  df-rex 2823  df-reu 2824  df-rmo 2825  df-rab 2826  df-v 3120  df-sbc 3337  df-csb 3441  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-nul 3791  df-if 3946  df-pw 4018  df-sn 4034  df-pr 4036  df-op 4040  df-uni 4252  df-iun 4333  df-br 4454  df-opab 4512  df-mpt 4513  df-id 4801  df-xp 5011  df-rel 5012  df-cnv 5013  df-co 5014  df-dm 5015  df-rn 5016  df-res 5017  df-ima 5018  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-riota 6256  df-ov 6298  df-oprab 6299  df-mpt2 6300  df-1st 6795  df-2nd 6796  df-map 7434  df-ixp 7482  df-cat 14940  df-cid 14941  df-func 15102  df-cofu 15104
This theorem is referenced by:  catccatid  15304
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