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Theorem catass 13588
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 13574 . . . 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 232 . . 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 15 . 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 451 . . . . 5  |-  ( (
ph  /\  x  =  X )  ->  Y  e.  B )
11 catcocl.z . . . . . . 7  |-  ( ph  ->  Z  e.  B )
1211ad2antrr 706 . . . . . 6  |-  ( ( ( ph  /\  x  =  X )  /\  y  =  Y )  ->  Z  e.  B )
13 catcocl.f . . . . . . . . 9  |-  ( ph  ->  F  e.  ( X H Y ) )
1413ad3antrrr 710 . . . . . . . 8  |-  ( ( ( ( ph  /\  x  =  X )  /\  y  =  Y
)  /\  z  =  Z )  ->  F  e.  ( X H Y ) )
15 simpllr 735 . . . . . . . . 9  |-  ( ( ( ( ph  /\  x  =  X )  /\  y  =  Y
)  /\  z  =  Z )  ->  x  =  X )
16 simplr 731 . . . . . . . . 9  |-  ( ( ( ( ph  /\  x  =  X )  /\  y  =  Y
)  /\  z  =  Z )  ->  y  =  Y )
1715, 16oveq12d 5876 . . . . . . . 8  |-  ( ( ( ( ph  /\  x  =  X )  /\  y  =  Y
)  /\  z  =  Z )  ->  (
x H y )  =  ( X H Y ) )
1814, 17eleqtrrd 2360 . . . . . . 7  |-  ( ( ( ( ph  /\  x  =  X )  /\  y  =  Y
)  /\  z  =  Z )  ->  F  e.  ( x H y ) )
19 catcocl.g . . . . . . . . . 10  |-  ( ph  ->  G  e.  ( Y H Z ) )
2019ad4antr 712 . . . . . . . . 9  |-  ( ( ( ( ( ph  /\  x  =  X )  /\  y  =  Y )  /\  z  =  Z )  /\  f  =  F )  ->  G  e.  ( Y H Z ) )
21 simpllr 735 . . . . . . . . . 10  |-  ( ( ( ( ( ph  /\  x  =  X )  /\  y  =  Y )  /\  z  =  Z )  /\  f  =  F )  ->  y  =  Y )
22 simplr 731 . . . . . . . . . 10  |-  ( ( ( ( ( ph  /\  x  =  X )  /\  y  =  Y )  /\  z  =  Z )  /\  f  =  F )  ->  z  =  Z )
2321, 22oveq12d 5876 . . . . . . . . 9  |-  ( ( ( ( ( ph  /\  x  =  X )  /\  y  =  Y )  /\  z  =  Z )  /\  f  =  F )  ->  (
y H z )  =  ( Y H Z ) )
2420, 23eleqtrrd 2360 . . . . . . . 8  |-  ( ( ( ( ( ph  /\  x  =  X )  /\  y  =  Y )  /\  z  =  Z )  /\  f  =  F )  ->  G  e.  ( y H z ) )
25 catass.w . . . . . . . . . . 11  |-  ( ph  ->  W  e.  B )
2625ad5antr 714 . . . . . . . . . 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 716 . . . . . . . . . . . 12  |-  ( ( ( ( ( ( ( ph  /\  x  =  X )  /\  y  =  Y )  /\  z  =  Z )  /\  f  =  F )  /\  g  =  G )  /\  w  =  W )  ->  K  e.  ( Z H W ) )
29 simp-4r 743 . . . . . . . . . . . . 13  |-  ( ( ( ( ( ( ( ph  /\  x  =  X )  /\  y  =  Y )  /\  z  =  Z )  /\  f  =  F )  /\  g  =  G )  /\  w  =  W )  ->  z  =  Z )
30 simpr 447 . . . . . . . . . . . . 13  |-  ( ( ( ( ( ( ( ph  /\  x  =  X )  /\  y  =  Y )  /\  z  =  Z )  /\  f  =  F )  /\  g  =  G )  /\  w  =  W )  ->  w  =  W )
3129, 30oveq12d 5876 . . . . . . . . . . . 12  |-  ( ( ( ( ( ( ( ph  /\  x  =  X )  /\  y  =  Y )  /\  z  =  Z )  /\  f  =  F )  /\  g  =  G )  /\  w  =  W )  ->  (
z H w )  =  ( Z H W ) )
3228, 31eleqtrrd 2360 . . . . . . . . . . 11  |-  ( ( ( ( ( ( ( ph  /\  x  =  X )  /\  y  =  Y )  /\  z  =  Z )  /\  f  =  F )  /\  g  =  G )  /\  w  =  W )  ->  K  e.  ( z H w ) )
33 simp-7r 749 . . . . . . . . . . . . . . 15  |-  ( ( ( ( ( ( ( ( ph  /\  x  =  X )  /\  y  =  Y
)  /\  z  =  Z )  /\  f  =  F )  /\  g  =  G )  /\  w  =  W )  /\  k  =  K )  ->  x  =  X )
34 simp-6r 747 . . . . . . . . . . . . . . 15  |-  ( ( ( ( ( ( ( ( ph  /\  x  =  X )  /\  y  =  Y
)  /\  z  =  Z )  /\  f  =  F )  /\  g  =  G )  /\  w  =  W )  /\  k  =  K )  ->  y  =  Y )
3533, 34opeq12d 3804 . . . . . . . . . . . . . 14  |-  ( ( ( ( ( ( ( ( ph  /\  x  =  X )  /\  y  =  Y
)  /\  z  =  Z )  /\  f  =  F )  /\  g  =  G )  /\  w  =  W )  /\  k  =  K )  ->  <. x ,  y >.  =  <. X ,  Y >. )
36 simplr 731 . . . . . . . . . . . . . 14  |-  ( ( ( ( ( ( ( ( ph  /\  x  =  X )  /\  y  =  Y
)  /\  z  =  Z )  /\  f  =  F )  /\  g  =  G )  /\  w  =  W )  /\  k  =  K )  ->  w  =  W )
3735, 36oveq12d 5876 . . . . . . . . . . . . 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 745 . . . . . . . . . . . . . . . 16  |-  ( ( ( ( ( ( ( ( ph  /\  x  =  X )  /\  y  =  Y
)  /\  z  =  Z )  /\  f  =  F )  /\  g  =  G )  /\  w  =  W )  /\  k  =  K )  ->  z  =  Z )
3934, 38opeq12d 3804 . . . . . . . . . . . . . . 15  |-  ( ( ( ( ( ( ( ( ph  /\  x  =  X )  /\  y  =  Y
)  /\  z  =  Z )  /\  f  =  F )  /\  g  =  G )  /\  w  =  W )  /\  k  =  K )  ->  <. y ,  z >.  =  <. Y ,  Z >. )
4039, 36oveq12d 5876 . . . . . . . . . . . . . 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 447 . . . . . . . . . . . . . 14  |-  ( ( ( ( ( ( ( ( ph  /\  x  =  X )  /\  y  =  Y
)  /\  z  =  Z )  /\  f  =  F )  /\  g  =  G )  /\  w  =  W )  /\  k  =  K )  ->  k  =  K )
42 simpllr 735 . . . . . . . . . . . . . 14  |-  ( ( ( ( ( ( ( ( ph  /\  x  =  X )  /\  y  =  Y
)  /\  z  =  Z )  /\  f  =  F )  /\  g  =  G )  /\  w  =  W )  /\  k  =  K )  ->  g  =  G )
4340, 41, 42oveq123d 5879 . . . . . . . . . . . . 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 743 . . . . . . . . . . . . 13  |-  ( ( ( ( ( ( ( ( ph  /\  x  =  X )  /\  y  =  Y
)  /\  z  =  Z )  /\  f  =  F )  /\  g  =  G )  /\  w  =  W )  /\  k  =  K )  ->  f  =  F )
4537, 43, 44oveq123d 5879 . . . . . . . . . . . 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 3804 . . . . . . . . . . . . . 14  |-  ( ( ( ( ( ( ( ( ph  /\  x  =  X )  /\  y  =  Y
)  /\  z  =  Z )  /\  f  =  F )  /\  g  =  G )  /\  w  =  W )  /\  k  =  K )  ->  <. x ,  z >.  =  <. X ,  Z >. )
4746, 36oveq12d 5876 . . . . . . . . . . . . 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 5876 . . . . . . . . . . . . . 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 5879 . . . . . . . . . . . . 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 5879 . . . . . . . . . . . 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 2297 . . . . . . . . . . 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 2887 . . . . . . . . . 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 2885 . . . . . . . . 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 453 . . . . . . . 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 2885 . . . . . . 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 2885 . . . . . 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 2885 . . . . 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 2885 . . . 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 453 . . 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 2885 . 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 14 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 set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1623    e. wcel 1684   A.wral 2543   E.wrex 2544   <.cop 3643   ` cfv 5255  (class class class)co 5858   Basecbs 13148    Hom chom 13219  compcco 13220   Catccat 13566
This theorem is referenced by:  oppccatid  13622  sectcan  13658  sectco  13659  sectmon  13680  monsect  13681  subccatid  13720  fuccocl  13838  fucass  13842  invfuc  13848  arwass  13906  xpccatid  13962  evlfcllem  13995  hofcllem  14032
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-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-nul 4149
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-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-iota 5219  df-fv 5263  df-ov 5861  df-cat 13570
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