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Theorem cofuass 14921
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 5467 . . . 4  |-  ( ( ( 1st `  K
)  o.  ( 1st `  H ) )  o.  ( 1st `  G
) )  =  ( ( 1st `  K
)  o.  ( ( 1st `  H )  o.  ( 1st `  G
) ) )
2 eqid 2454 . . . . . 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 14915 . . . . 5  |-  ( ph  ->  ( 1st `  ( K  o.func 
H ) )  =  ( ( 1st `  K
)  o.  ( 1st `  H ) ) )
65coeq1d 5112 . . . 4  |-  ( ph  ->  ( ( 1st `  ( K  o.func 
H ) )  o.  ( 1st `  G
) )  =  ( ( ( 1st `  K
)  o.  ( 1st `  H ) )  o.  ( 1st `  G
) ) )
7 eqid 2454 . . . . . 6  |-  ( Base `  C )  =  (
Base `  C )
8 cofuass.g . . . . . 6  |-  ( ph  ->  G  e.  ( C 
Func  D ) )
97, 8, 3cofu1st 14915 . . . . 5  |-  ( ph  ->  ( 1st `  ( H  o.func 
G ) )  =  ( ( 1st `  H
)  o.  ( 1st `  G ) ) )
109coeq2d 5113 . . . 4  |-  ( ph  ->  ( ( 1st `  K
)  o.  ( 1st `  ( H  o.func  G )
) )  =  ( ( 1st `  K
)  o.  ( ( 1st `  H )  o.  ( 1st `  G
) ) ) )
111, 6, 103eqtr4a 2521 . . 3  |-  ( ph  ->  ( ( 1st `  ( K  o.func 
H ) )  o.  ( 1st `  G
) )  =  ( ( 1st `  K
)  o.  ( 1st `  ( H  o.func  G )
) ) )
12 coass 5467 . . . . 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 1009 . . . . . . 7  |-  ( (
ph  /\  x  e.  ( Base `  C )  /\  y  e.  ( Base `  C ) )  ->  H  e.  ( D  Func  E )
)
1443ad2ant1 1009 . . . . . . 7  |-  ( (
ph  /\  x  e.  ( Base `  C )  /\  y  e.  ( Base `  C ) )  ->  K  e.  ( E  Func  F )
)
15 relfunc 14894 . . . . . . . . . . 11  |-  Rel  ( C  Func  D )
16 1st2ndbr 6736 . . . . . . . . . . 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 1009 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  ( Base `  C )  /\  y  e.  ( Base `  C ) )  ->  ( 1st `  G
) ( C  Func  D ) ( 2nd `  G
) )
197, 2, 18funcf1 14898 . . . . . . . 8  |-  ( (
ph  /\  x  e.  ( Base `  C )  /\  y  e.  ( Base `  C ) )  ->  ( 1st `  G
) : ( Base `  C ) --> ( Base `  D ) )
20 simp2 989 . . . . . . . 8  |-  ( (
ph  /\  x  e.  ( Base `  C )  /\  y  e.  ( Base `  C ) )  ->  x  e.  (
Base `  C )
)
2119, 20ffvelrnd 5956 . . . . . . 7  |-  ( (
ph  /\  x  e.  ( Base `  C )  /\  y  e.  ( Base `  C ) )  ->  ( ( 1st `  G ) `  x
)  e.  ( Base `  D ) )
22 simp3 990 . . . . . . . 8  |-  ( (
ph  /\  x  e.  ( Base `  C )  /\  y  e.  ( Base `  C ) )  ->  y  e.  (
Base `  C )
)
2319, 22ffvelrnd 5956 . . . . . . 7  |-  ( (
ph  /\  x  e.  ( Base `  C )  /\  y  e.  ( Base `  C ) )  ->  ( ( 1st `  G ) `  y
)  e.  ( Base `  D ) )
242, 13, 14, 21, 23cofu2nd 14917 . . . . . 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 5112 . . . . 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 1009 . . . . . . . 8  |-  ( (
ph  /\  x  e.  ( Base `  C )  /\  y  e.  ( Base `  C ) )  ->  G  e.  ( C  Func  D )
)
277, 26, 13, 20cofu1 14916 . . . . . . 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 14916 . . . . . . 7  |-  ( (
ph  /\  x  e.  ( Base `  C )  /\  y  e.  ( Base `  C ) )  ->  ( ( 1st `  ( H  o.func  G )
) `  y )  =  ( ( 1st `  H ) `  (
( 1st `  G
) `  y )
) )
2927, 28oveq12d 6221 . . . . . 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 14917 . . . . . 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 5115 . . . . 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 2521 . . . 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 6262 . . 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 4178 . 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 14920 . . 3  |-  ( ph  ->  ( K  o.func  H )  e.  ( D  Func  F
) )
367, 8, 35cofuval 14914 . 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 14920 . . 3  |-  ( ph  ->  ( H  o.func  G )  e.  ( C  Func  E
) )
387, 37, 4cofuval 14914 . 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 2505 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 965    = wceq 1370    e. wcel 1758   <.cop 3994   class class class wbr 4403    o. ccom 4955   Rel wrel 4956   ` cfv 5529  (class class class)co 6203    |-> cmpt2 6205   1stc1st 6688   2ndc2nd 6689   Basecbs 14295    Func cfunc 14886    o.func ccofu 14888
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-rep 4514  ax-sep 4524  ax-nul 4532  ax-pow 4581  ax-pr 4642  ax-un 6485
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-ral 2804  df-rex 2805  df-reu 2806  df-rmo 2807  df-rab 2808  df-v 3080  df-sbc 3295  df-csb 3399  df-dif 3442  df-un 3444  df-in 3446  df-ss 3453  df-nul 3749  df-if 3903  df-pw 3973  df-sn 3989  df-pr 3991  df-op 3995  df-uni 4203  df-iun 4284  df-br 4404  df-opab 4462  df-mpt 4463  df-id 4747  df-xp 4957  df-rel 4958  df-cnv 4959  df-co 4960  df-dm 4961  df-rn 4962  df-res 4963  df-ima 4964  df-iota 5492  df-fun 5531  df-fn 5532  df-f 5533  df-f1 5534  df-fo 5535  df-f1o 5536  df-fv 5537  df-riota 6164  df-ov 6206  df-oprab 6207  df-mpt2 6208  df-1st 6690  df-2nd 6691  df-map 7329  df-ixp 7377  df-cat 14728  df-cid 14729  df-func 14890  df-cofu 14892
This theorem is referenced by:  catccatid  15092
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