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Theorem catass 14735
Description: Associativity of composition in a category. (Contributed by Mario Carneiro, 2-Jan-2017.)
Hypotheses
Ref Expression
catcocl.b  |-  B  =  ( Base `  C
)
catcocl.h  |-  H  =  ( Hom  `  C
)
catcocl.o  |-  .x.  =  (comp `  C )
catcocl.c  |-  ( ph  ->  C  e.  Cat )
catcocl.x  |-  ( ph  ->  X  e.  B )
catcocl.y  |-  ( ph  ->  Y  e.  B )
catcocl.z  |-  ( ph  ->  Z  e.  B )
catcocl.f  |-  ( ph  ->  F  e.  ( X H Y ) )
catcocl.g  |-  ( ph  ->  G  e.  ( Y H Z ) )
catass.w  |-  ( ph  ->  W  e.  B )
catass.g  |-  ( ph  ->  K  e.  ( Z H W ) )
Assertion
Ref Expression
catass  |-  ( ph  ->  ( ( K (
<. Y ,  Z >.  .x. 
W ) G ) ( <. X ,  Y >.  .x.  W ) F )  =  ( K ( <. X ,  Z >.  .x.  W ) ( G ( <. X ,  Y >.  .x.  Z ) F ) ) )

Proof of Theorem catass
Dummy variables  f 
g  k  w  x  y  z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 catcocl.c . . 3  |-  ( ph  ->  C  e.  Cat )
2 catcocl.b . . . . 5  |-  B  =  ( Base `  C
)
3 catcocl.h . . . . 5  |-  H  =  ( Hom  `  C
)
4 catcocl.o . . . . 5  |-  .x.  =  (comp `  C )
52, 3, 4iscat 14721 . . . 4  |-  ( C  e.  Cat  ->  ( C  e.  Cat  <->  A. x  e.  B  ( E. g  e.  ( x H x ) A. y  e.  B  ( A. f  e.  (
y H x ) ( g ( <.
y ,  x >.  .x.  x ) f )  =  f  /\  A. f  e.  ( x H y ) ( f ( <. x ,  x >.  .x.  y ) g )  =  f )  /\  A. y  e.  B  A. z  e.  B  A. f  e.  ( x H y ) A. g  e.  ( y H z ) ( ( g ( <. x ,  y
>.  .x.  z ) f )  e.  ( x H z )  /\  A. w  e.  B  A. k  e.  ( z H w ) ( ( k ( <.
y ,  z >.  .x.  w ) g ) ( <. x ,  y
>.  .x.  w ) f )  =  ( k ( <. x ,  z
>.  .x.  w ) ( g ( <. x ,  y >.  .x.  z
) f ) ) ) ) ) )
65ibi 241 . . 3  |-  ( C  e.  Cat  ->  A. x  e.  B  ( E. g  e.  ( x H x ) A. y  e.  B  ( A. f  e.  (
y H x ) ( g ( <.
y ,  x >.  .x.  x ) f )  =  f  /\  A. f  e.  ( x H y ) ( f ( <. x ,  x >.  .x.  y ) g )  =  f )  /\  A. y  e.  B  A. z  e.  B  A. f  e.  ( x H y ) A. g  e.  ( y H z ) ( ( g ( <. x ,  y
>.  .x.  z ) f )  e.  ( x H z )  /\  A. w  e.  B  A. k  e.  ( z H w ) ( ( k ( <.
y ,  z >.  .x.  w ) g ) ( <. x ,  y
>.  .x.  w ) f )  =  ( k ( <. x ,  z
>.  .x.  w ) ( g ( <. x ,  y >.  .x.  z
) f ) ) ) ) )
71, 6syl 16 . 2  |-  ( ph  ->  A. x  e.  B  ( E. g  e.  ( x H x ) A. y  e.  B  ( A. f  e.  ( y H x ) ( g ( <.
y ,  x >.  .x.  x ) f )  =  f  /\  A. f  e.  ( x H y ) ( f ( <. x ,  x >.  .x.  y ) g )  =  f )  /\  A. y  e.  B  A. z  e.  B  A. f  e.  ( x H y ) A. g  e.  ( y H z ) ( ( g ( <. x ,  y
>.  .x.  z ) f )  e.  ( x H z )  /\  A. w  e.  B  A. k  e.  ( z H w ) ( ( k ( <.
y ,  z >.  .x.  w ) g ) ( <. x ,  y
>.  .x.  w ) f )  =  ( k ( <. x ,  z
>.  .x.  w ) ( g ( <. x ,  y >.  .x.  z
) f ) ) ) ) )
8 catcocl.x . . 3  |-  ( ph  ->  X  e.  B )
9 catcocl.y . . . . . 6  |-  ( ph  ->  Y  e.  B )
109adantr 465 . . . . 5  |-  ( (
ph  /\  x  =  X )  ->  Y  e.  B )
11 catcocl.z . . . . . . 7  |-  ( ph  ->  Z  e.  B )
1211ad2antrr 725 . . . . . 6  |-  ( ( ( ph  /\  x  =  X )  /\  y  =  Y )  ->  Z  e.  B )
13 catcocl.f . . . . . . . . 9  |-  ( ph  ->  F  e.  ( X H Y ) )
1413ad3antrrr 729 . . . . . . . 8  |-  ( ( ( ( ph  /\  x  =  X )  /\  y  =  Y
)  /\  z  =  Z )  ->  F  e.  ( X H Y ) )
15 simpllr 758 . . . . . . . . 9  |-  ( ( ( ( ph  /\  x  =  X )  /\  y  =  Y
)  /\  z  =  Z )  ->  x  =  X )
16 simplr 754 . . . . . . . . 9  |-  ( ( ( ( ph  /\  x  =  X )  /\  y  =  Y
)  /\  z  =  Z )  ->  y  =  Y )
1715, 16oveq12d 6211 . . . . . . . 8  |-  ( ( ( ( ph  /\  x  =  X )  /\  y  =  Y
)  /\  z  =  Z )  ->  (
x H y )  =  ( X H Y ) )
1814, 17eleqtrrd 2542 . . . . . . 7  |-  ( ( ( ( ph  /\  x  =  X )  /\  y  =  Y
)  /\  z  =  Z )  ->  F  e.  ( x H y ) )
19 catcocl.g . . . . . . . . . 10  |-  ( ph  ->  G  e.  ( Y H Z ) )
2019ad4antr 731 . . . . . . . . 9  |-  ( ( ( ( ( ph  /\  x  =  X )  /\  y  =  Y )  /\  z  =  Z )  /\  f  =  F )  ->  G  e.  ( Y H Z ) )
21 simpllr 758 . . . . . . . . . 10  |-  ( ( ( ( ( ph  /\  x  =  X )  /\  y  =  Y )  /\  z  =  Z )  /\  f  =  F )  ->  y  =  Y )
22 simplr 754 . . . . . . . . . 10  |-  ( ( ( ( ( ph  /\  x  =  X )  /\  y  =  Y )  /\  z  =  Z )  /\  f  =  F )  ->  z  =  Z )
2321, 22oveq12d 6211 . . . . . . . . 9  |-  ( ( ( ( ( ph  /\  x  =  X )  /\  y  =  Y )  /\  z  =  Z )  /\  f  =  F )  ->  (
y H z )  =  ( Y H Z ) )
2420, 23eleqtrrd 2542 . . . . . . . 8  |-  ( ( ( ( ( ph  /\  x  =  X )  /\  y  =  Y )  /\  z  =  Z )  /\  f  =  F )  ->  G  e.  ( y H z ) )
25 catass.w . . . . . . . . . . 11  |-  ( ph  ->  W  e.  B )
2625ad5antr 733 . . . . . . . . . 10  |-  ( ( ( ( ( (
ph  /\  x  =  X )  /\  y  =  Y )  /\  z  =  Z )  /\  f  =  F )  /\  g  =  G )  ->  W  e.  B )
27 catass.g . . . . . . . . . . . . 13  |-  ( ph  ->  K  e.  ( Z H W ) )
2827ad6antr 735 . . . . . . . . . . . 12  |-  ( ( ( ( ( ( ( ph  /\  x  =  X )  /\  y  =  Y )  /\  z  =  Z )  /\  f  =  F )  /\  g  =  G )  /\  w  =  W )  ->  K  e.  ( Z H W ) )
29 simp-4r 766 . . . . . . . . . . . . 13  |-  ( ( ( ( ( ( ( ph  /\  x  =  X )  /\  y  =  Y )  /\  z  =  Z )  /\  f  =  F )  /\  g  =  G )  /\  w  =  W )  ->  z  =  Z )
30 simpr 461 . . . . . . . . . . . . 13  |-  ( ( ( ( ( ( ( ph  /\  x  =  X )  /\  y  =  Y )  /\  z  =  Z )  /\  f  =  F )  /\  g  =  G )  /\  w  =  W )  ->  w  =  W )
3129, 30oveq12d 6211 . . . . . . . . . . . 12  |-  ( ( ( ( ( ( ( ph  /\  x  =  X )  /\  y  =  Y )  /\  z  =  Z )  /\  f  =  F )  /\  g  =  G )  /\  w  =  W )  ->  (
z H w )  =  ( Z H W ) )
3228, 31eleqtrrd 2542 . . . . . . . . . . 11  |-  ( ( ( ( ( ( ( ph  /\  x  =  X )  /\  y  =  Y )  /\  z  =  Z )  /\  f  =  F )  /\  g  =  G )  /\  w  =  W )  ->  K  e.  ( z H w ) )
33 simp-7r 772 . . . . . . . . . . . . . . 15  |-  ( ( ( ( ( ( ( ( ph  /\  x  =  X )  /\  y  =  Y
)  /\  z  =  Z )  /\  f  =  F )  /\  g  =  G )  /\  w  =  W )  /\  k  =  K )  ->  x  =  X )
34 simp-6r 770 . . . . . . . . . . . . . . 15  |-  ( ( ( ( ( ( ( ( ph  /\  x  =  X )  /\  y  =  Y
)  /\  z  =  Z )  /\  f  =  F )  /\  g  =  G )  /\  w  =  W )  /\  k  =  K )  ->  y  =  Y )
3533, 34opeq12d 4168 . . . . . . . . . . . . . 14  |-  ( ( ( ( ( ( ( ( ph  /\  x  =  X )  /\  y  =  Y
)  /\  z  =  Z )  /\  f  =  F )  /\  g  =  G )  /\  w  =  W )  /\  k  =  K )  ->  <. x ,  y >.  =  <. X ,  Y >. )
36 simplr 754 . . . . . . . . . . . . . 14  |-  ( ( ( ( ( ( ( ( ph  /\  x  =  X )  /\  y  =  Y
)  /\  z  =  Z )  /\  f  =  F )  /\  g  =  G )  /\  w  =  W )  /\  k  =  K )  ->  w  =  W )
3735, 36oveq12d 6211 . . . . . . . . . . . . 13  |-  ( ( ( ( ( ( ( ( ph  /\  x  =  X )  /\  y  =  Y
)  /\  z  =  Z )  /\  f  =  F )  /\  g  =  G )  /\  w  =  W )  /\  k  =  K )  ->  ( <. x ,  y >.  .x.  w )  =  (
<. X ,  Y >.  .x. 
W ) )
38 simp-5r 768 . . . . . . . . . . . . . . . 16  |-  ( ( ( ( ( ( ( ( ph  /\  x  =  X )  /\  y  =  Y
)  /\  z  =  Z )  /\  f  =  F )  /\  g  =  G )  /\  w  =  W )  /\  k  =  K )  ->  z  =  Z )
3934, 38opeq12d 4168 . . . . . . . . . . . . . . 15  |-  ( ( ( ( ( ( ( ( ph  /\  x  =  X )  /\  y  =  Y
)  /\  z  =  Z )  /\  f  =  F )  /\  g  =  G )  /\  w  =  W )  /\  k  =  K )  ->  <. y ,  z >.  =  <. Y ,  Z >. )
4039, 36oveq12d 6211 . . . . . . . . . . . . . 14  |-  ( ( ( ( ( ( ( ( ph  /\  x  =  X )  /\  y  =  Y
)  /\  z  =  Z )  /\  f  =  F )  /\  g  =  G )  /\  w  =  W )  /\  k  =  K )  ->  ( <. y ,  z >.  .x.  w )  =  (
<. Y ,  Z >.  .x. 
W ) )
41 simpr 461 . . . . . . . . . . . . . 14  |-  ( ( ( ( ( ( ( ( ph  /\  x  =  X )  /\  y  =  Y
)  /\  z  =  Z )  /\  f  =  F )  /\  g  =  G )  /\  w  =  W )  /\  k  =  K )  ->  k  =  K )
42 simpllr 758 . . . . . . . . . . . . . 14  |-  ( ( ( ( ( ( ( ( ph  /\  x  =  X )  /\  y  =  Y
)  /\  z  =  Z )  /\  f  =  F )  /\  g  =  G )  /\  w  =  W )  /\  k  =  K )  ->  g  =  G )
4340, 41, 42oveq123d 6214 . . . . . . . . . . . . 13  |-  ( ( ( ( ( ( ( ( ph  /\  x  =  X )  /\  y  =  Y
)  /\  z  =  Z )  /\  f  =  F )  /\  g  =  G )  /\  w  =  W )  /\  k  =  K )  ->  (
k ( <. y ,  z >.  .x.  w
) g )  =  ( K ( <. Y ,  Z >.  .x. 
W ) G ) )
44 simp-4r 766 . . . . . . . . . . . . 13  |-  ( ( ( ( ( ( ( ( ph  /\  x  =  X )  /\  y  =  Y
)  /\  z  =  Z )  /\  f  =  F )  /\  g  =  G )  /\  w  =  W )  /\  k  =  K )  ->  f  =  F )
4537, 43, 44oveq123d 6214 . . . . . . . . . . . 12  |-  ( ( ( ( ( ( ( ( ph  /\  x  =  X )  /\  y  =  Y
)  /\  z  =  Z )  /\  f  =  F )  /\  g  =  G )  /\  w  =  W )  /\  k  =  K )  ->  (
( k ( <.
y ,  z >.  .x.  w ) g ) ( <. x ,  y
>.  .x.  w ) f )  =  ( ( K ( <. Y ,  Z >.  .x.  W ) G ) ( <. X ,  Y >.  .x. 
W ) F ) )
4633, 38opeq12d 4168 . . . . . . . . . . . . . 14  |-  ( ( ( ( ( ( ( ( ph  /\  x  =  X )  /\  y  =  Y
)  /\  z  =  Z )  /\  f  =  F )  /\  g  =  G )  /\  w  =  W )  /\  k  =  K )  ->  <. x ,  z >.  =  <. X ,  Z >. )
4746, 36oveq12d 6211 . . . . . . . . . . . . 13  |-  ( ( ( ( ( ( ( ( ph  /\  x  =  X )  /\  y  =  Y
)  /\  z  =  Z )  /\  f  =  F )  /\  g  =  G )  /\  w  =  W )  /\  k  =  K )  ->  ( <. x ,  z >.  .x.  w )  =  (
<. X ,  Z >.  .x. 
W ) )
4835, 38oveq12d 6211 . . . . . . . . . . . . . 14  |-  ( ( ( ( ( ( ( ( ph  /\  x  =  X )  /\  y  =  Y
)  /\  z  =  Z )  /\  f  =  F )  /\  g  =  G )  /\  w  =  W )  /\  k  =  K )  ->  ( <. x ,  y >.  .x.  z )  =  (
<. X ,  Y >.  .x. 
Z ) )
4948, 42, 44oveq123d 6214 . . . . . . . . . . . . 13  |-  ( ( ( ( ( ( ( ( ph  /\  x  =  X )  /\  y  =  Y
)  /\  z  =  Z )  /\  f  =  F )  /\  g  =  G )  /\  w  =  W )  /\  k  =  K )  ->  (
g ( <. x ,  y >.  .x.  z
) f )  =  ( G ( <. X ,  Y >.  .x. 
Z ) F ) )
5047, 41, 49oveq123d 6214 . . . . . . . . . . . 12  |-  ( ( ( ( ( ( ( ( ph  /\  x  =  X )  /\  y  =  Y
)  /\  z  =  Z )  /\  f  =  F )  /\  g  =  G )  /\  w  =  W )  /\  k  =  K )  ->  (
k ( <. x ,  z >.  .x.  w
) ( g (
<. x ,  y >.  .x.  z ) f ) )  =  ( K ( <. X ,  Z >.  .x.  W ) ( G ( <. X ,  Y >.  .x.  Z ) F ) ) )
5145, 50eqeq12d 2473 . . . . . . . . . . 11  |-  ( ( ( ( ( ( ( ( ph  /\  x  =  X )  /\  y  =  Y
)  /\  z  =  Z )  /\  f  =  F )  /\  g  =  G )  /\  w  =  W )  /\  k  =  K )  ->  (
( ( k (
<. y ,  z >.  .x.  w ) g ) ( <. x ,  y
>.  .x.  w ) f )  =  ( k ( <. x ,  z
>.  .x.  w ) ( g ( <. x ,  y >.  .x.  z
) f ) )  <-> 
( ( K (
<. Y ,  Z >.  .x. 
W ) G ) ( <. X ,  Y >.  .x.  W ) F )  =  ( K ( <. X ,  Z >.  .x.  W ) ( G ( <. X ,  Y >.  .x.  Z ) F ) ) ) )
5232, 51rspcdv 3175 . . . . . . . . . 10  |-  ( ( ( ( ( ( ( ph  /\  x  =  X )  /\  y  =  Y )  /\  z  =  Z )  /\  f  =  F )  /\  g  =  G )  /\  w  =  W )  ->  ( A. k  e.  (
z H w ) ( ( k (
<. y ,  z >.  .x.  w ) g ) ( <. x ,  y
>.  .x.  w ) f )  =  ( k ( <. x ,  z
>.  .x.  w ) ( g ( <. x ,  y >.  .x.  z
) f ) )  ->  ( ( K ( <. Y ,  Z >.  .x.  W ) G ) ( <. X ,  Y >.  .x.  W ) F )  =  ( K ( <. X ,  Z >.  .x.  W )
( G ( <. X ,  Y >.  .x. 
Z ) F ) ) ) )
5326, 52rspcimdv 3173 . . . . . . . . 9  |-  ( ( ( ( ( (
ph  /\  x  =  X )  /\  y  =  Y )  /\  z  =  Z )  /\  f  =  F )  /\  g  =  G )  ->  ( A. w  e.  B  A. k  e.  (
z H w ) ( ( k (
<. y ,  z >.  .x.  w ) g ) ( <. x ,  y
>.  .x.  w ) f )  =  ( k ( <. x ,  z
>.  .x.  w ) ( g ( <. x ,  y >.  .x.  z
) f ) )  ->  ( ( K ( <. Y ,  Z >.  .x.  W ) G ) ( <. X ,  Y >.  .x.  W ) F )  =  ( K ( <. X ,  Z >.  .x.  W )
( G ( <. X ,  Y >.  .x. 
Z ) F ) ) ) )
5453adantld 467 . . . . . . . 8  |-  ( ( ( ( ( (
ph  /\  x  =  X )  /\  y  =  Y )  /\  z  =  Z )  /\  f  =  F )  /\  g  =  G )  ->  (
( ( g (
<. x ,  y >.  .x.  z ) f )  e.  ( x H z )  /\  A. w  e.  B  A. k  e.  ( z H w ) ( ( k ( <.
y ,  z >.  .x.  w ) g ) ( <. x ,  y
>.  .x.  w ) f )  =  ( k ( <. x ,  z
>.  .x.  w ) ( g ( <. x ,  y >.  .x.  z
) f ) ) )  ->  ( ( K ( <. Y ,  Z >.  .x.  W ) G ) ( <. X ,  Y >.  .x. 
W ) F )  =  ( K (
<. X ,  Z >.  .x. 
W ) ( G ( <. X ,  Y >.  .x.  Z ) F ) ) ) )
5524, 54rspcimdv 3173 . . . . . . 7  |-  ( ( ( ( ( ph  /\  x  =  X )  /\  y  =  Y )  /\  z  =  Z )  /\  f  =  F )  ->  ( A. g  e.  (
y H z ) ( ( g (
<. x ,  y >.  .x.  z ) f )  e.  ( x H z )  /\  A. w  e.  B  A. k  e.  ( z H w ) ( ( k ( <.
y ,  z >.  .x.  w ) g ) ( <. x ,  y
>.  .x.  w ) f )  =  ( k ( <. x ,  z
>.  .x.  w ) ( g ( <. x ,  y >.  .x.  z
) f ) ) )  ->  ( ( K ( <. Y ,  Z >.  .x.  W ) G ) ( <. X ,  Y >.  .x. 
W ) F )  =  ( K (
<. X ,  Z >.  .x. 
W ) ( G ( <. X ,  Y >.  .x.  Z ) F ) ) ) )
5618, 55rspcimdv 3173 . . . . . 6  |-  ( ( ( ( ph  /\  x  =  X )  /\  y  =  Y
)  /\  z  =  Z )  ->  ( A. f  e.  (
x H y ) A. g  e.  ( y H z ) ( ( g (
<. x ,  y >.  .x.  z ) f )  e.  ( x H z )  /\  A. w  e.  B  A. k  e.  ( z H w ) ( ( k ( <.
y ,  z >.  .x.  w ) g ) ( <. x ,  y
>.  .x.  w ) f )  =  ( k ( <. x ,  z
>.  .x.  w ) ( g ( <. x ,  y >.  .x.  z
) f ) ) )  ->  ( ( K ( <. Y ,  Z >.  .x.  W ) G ) ( <. X ,  Y >.  .x. 
W ) F )  =  ( K (
<. X ,  Z >.  .x. 
W ) ( G ( <. X ,  Y >.  .x.  Z ) F ) ) ) )
5712, 56rspcimdv 3173 . . . . 5  |-  ( ( ( ph  /\  x  =  X )  /\  y  =  Y )  ->  ( A. z  e.  B  A. f  e.  (
x H y ) A. g  e.  ( y H z ) ( ( g (
<. x ,  y >.  .x.  z ) f )  e.  ( x H z )  /\  A. w  e.  B  A. k  e.  ( z H w ) ( ( k ( <.
y ,  z >.  .x.  w ) g ) ( <. x ,  y
>.  .x.  w ) f )  =  ( k ( <. x ,  z
>.  .x.  w ) ( g ( <. x ,  y >.  .x.  z
) f ) ) )  ->  ( ( K ( <. Y ,  Z >.  .x.  W ) G ) ( <. X ,  Y >.  .x. 
W ) F )  =  ( K (
<. X ,  Z >.  .x. 
W ) ( G ( <. X ,  Y >.  .x.  Z ) F ) ) ) )
5810, 57rspcimdv 3173 . . . 4  |-  ( (
ph  /\  x  =  X )  ->  ( A. y  e.  B  A. z  e.  B  A. f  e.  (
x H y ) A. g  e.  ( y H z ) ( ( g (
<. x ,  y >.  .x.  z ) f )  e.  ( x H z )  /\  A. w  e.  B  A. k  e.  ( z H w ) ( ( k ( <.
y ,  z >.  .x.  w ) g ) ( <. x ,  y
>.  .x.  w ) f )  =  ( k ( <. x ,  z
>.  .x.  w ) ( g ( <. x ,  y >.  .x.  z
) f ) ) )  ->  ( ( K ( <. Y ,  Z >.  .x.  W ) G ) ( <. X ,  Y >.  .x. 
W ) F )  =  ( K (
<. X ,  Z >.  .x. 
W ) ( G ( <. X ,  Y >.  .x.  Z ) F ) ) ) )
5958adantld 467 . . 3  |-  ( (
ph  /\  x  =  X )  ->  (
( E. g  e.  ( x H x ) A. y  e.  B  ( A. f  e.  ( y H x ) ( g (
<. y ,  x >.  .x.  x ) f )  =  f  /\  A. f  e.  ( x H y ) ( f ( <. x ,  x >.  .x.  y ) g )  =  f )  /\  A. y  e.  B  A. z  e.  B  A. f  e.  ( x H y ) A. g  e.  ( y H z ) ( ( g ( <. x ,  y
>.  .x.  z ) f )  e.  ( x H z )  /\  A. w  e.  B  A. k  e.  ( z H w ) ( ( k ( <.
y ,  z >.  .x.  w ) g ) ( <. x ,  y
>.  .x.  w ) f )  =  ( k ( <. x ,  z
>.  .x.  w ) ( g ( <. x ,  y >.  .x.  z
) f ) ) ) )  ->  (
( K ( <. Y ,  Z >.  .x. 
W ) G ) ( <. X ,  Y >.  .x.  W ) F )  =  ( K ( <. X ,  Z >.  .x.  W ) ( G ( <. X ,  Y >.  .x.  Z ) F ) ) ) )
608, 59rspcimdv 3173 . 2  |-  ( ph  ->  ( A. x  e.  B  ( E. g  e.  ( x H x ) A. y  e.  B  ( A. f  e.  ( y H x ) ( g (
<. y ,  x >.  .x.  x ) f )  =  f  /\  A. f  e.  ( x H y ) ( f ( <. x ,  x >.  .x.  y ) g )  =  f )  /\  A. y  e.  B  A. z  e.  B  A. f  e.  ( x H y ) A. g  e.  ( y H z ) ( ( g ( <. x ,  y
>.  .x.  z ) f )  e.  ( x H z )  /\  A. w  e.  B  A. k  e.  ( z H w ) ( ( k ( <.
y ,  z >.  .x.  w ) g ) ( <. x ,  y
>.  .x.  w ) f )  =  ( k ( <. x ,  z
>.  .x.  w ) ( g ( <. x ,  y >.  .x.  z
) f ) ) ) )  ->  (
( K ( <. Y ,  Z >.  .x. 
W ) G ) ( <. X ,  Y >.  .x.  W ) F )  =  ( K ( <. X ,  Z >.  .x.  W ) ( G ( <. X ,  Y >.  .x.  Z ) F ) ) ) )
617, 60mpd 15 1  |-  ( ph  ->  ( ( K (
<. Y ,  Z >.  .x. 
W ) G ) ( <. X ,  Y >.  .x.  W ) F )  =  ( K ( <. X ,  Z >.  .x.  W ) ( G ( <. X ,  Y >.  .x.  Z ) F ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1370    e. wcel 1758   A.wral 2795   E.wrex 2796   <.cop 3984   ` cfv 5519  (class class class)co 6193   Basecbs 14285   Hom chom 14360  compcco 14361   Catccat 14713
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-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-nul 4522
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 2264  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ne 2646  df-ral 2800  df-rex 2801  df-rab 2804  df-v 3073  df-sbc 3288  df-dif 3432  df-un 3434  df-in 3436  df-ss 3443  df-nul 3739  df-if 3893  df-sn 3979  df-pr 3981  df-op 3985  df-uni 4193  df-br 4394  df-iota 5482  df-fv 5527  df-ov 6196  df-cat 14717
This theorem is referenced by:  oppccatid  14769  sectcan  14805  sectco  14806  sectmon  14827  monsect  14828  subccatid  14867  fuccocl  14985  fucass  14989  invfuc  14995  arwass  15053  xpccatid  15109  evlfcllem  15142  hofcllem  15179
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